Strong-field Physics with Ultrafast Optical Resonators
by
William Putnam
MASSACHUSETTS INSTITUTE
OF TECHNOLOLGY
B.S. Physics and B.S. Electrical Science and Engineering,
Massachusetts Institute of Technology (2008)
JUL 0 7 2015
LIBRARIES
M.Eng. Electrical Engineering and Computer Science,
Massachusetts Institute of Technology (2008)
Submitted to the Department of Electrical Engineering and Computer Science
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Electrical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2015
0 Massachusetts Institute of Technology 2015. All rights reserved.
Author ...................
Signature redacted ..............
Department of tietrical Engineering and Computer Science
February 23, 2015
Certified by ....................
Signature redacted ..........
Franz X. Kartner
Adjunct Professor of Electrical Engineering
Thesis Supervisor
Certified by ...................
Signature redacted
Erich P. Ippen
Elihu Thomson Professor of Electrical Engineering
Professor of Physics
Thesis Supervisor
Accepted by....................
Signature redacted
I
(Qfeslie A. Kolodziejski
Chair of the Committee on Graduate Students
2
Strong-field Physics with Ultrafast Optical Resonators
by
William Putnam
Submitted to the Department of Electrical Engineering and Computer Science
on February 23, 2015, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Electrical Engineering
Abstract
Light fields from modern high-intensity, femtosecond laser systems can produce electrical forces that rival the binding forces in atomic and solid-state systems. In this
strong-field regime, conventional non-linear optics gives way to novel phenomena such
as the production of attosecond bursts of electrons and photons. Strong laser fields
are generally achieved with amplified, ultrafast laser pulses. In this thesis we explore
phenomena unique to strong-fields by using optical resonators to passively enhance
ultrafast laser pulses.
We pursue two major themes in the area of ultrafast resonator-enhanced strongfield physics. First, we use plasmonic nanoparticles as nano-optical resonators to
explore strong-field photoemission near nanostructures on the surface of a chip. We
demonstrate strong-field photoemission with our chip-scale devices under ambient
conditions. Additionally, we use the strong-field photoemission current to probe the
ultrafast temporal response of the plasmonic nano-optical field around the nanoparticle emitters. We also show a carrier-envelope sensitive component of the photoemission current and develop a simple model to predict this sensitivity. Second, we
investigate cavity-enhanced high-harmonic generation. In particular, we explore the
design of novel optical cavities based on Bessel-Gauss modes. Such cavities might
have the capability to allow perfect out-coupling for intra-cavity generated harmonics as well as to provide for extremely large mode areas on the cavity mirrors. We
prototype a particular Bessel-Gauss cavity design and discuss the limitations of this
approach.
Thesis Supervisor: Franz X. Kdrtner
Title: Adjunct Professor of Electrical Engineering
Thesis Supervisor: Erich P. Ippen
Title: Elihu Thomson Professor of Electrical Engineering
Professor of Physics
3
4
Acknowledgments
Having attended MIT as an undergraduate and graduate student, I have now spent
more than a third of my life working, playing, and living in the MIT community. The
people here at MIT are far and away the institute's most valuable resource, and I am
thankful to many of them for their help, support, and care over my many years here.
Firstly, I need to thank my doctoral advisor, Prof. Franz Kdrtner. Throughout my
years working with Franz, he has always provided energetic and enthusiastic support
for me and my research pursuits. I would also like to thank my thesis committee
members Prof. Erich Ippen and Prof. Karl Berggren. I have known both Erich
and Karl since my time as an undergraduate, and throughout my MIT career they
have always had open doors and a great willingness to share their time and knowledge. Additionally, I have to thank all my research collaborators over the years: in
particular, Richard Hobbs for all the fantastic electron-beam lithography work and
insightful discussions, Jim Daley for all the invaluable assistance through my various
catastrophic pursuits in fabrication, Xiaolong Hu for helping teach me the ropes of
nano-fabrication oh-so-many years ago, and the entire optics and quantum electronics
group, especially Andrew Benedick, Gilberto Abram, Jon Cox, Shu-Wei Huang, and
Donnie Keathley for all the stimulating conversations and good times in the lab.
Beyond my co-workers, I also need to thank my family and friends for all the love
and support they have given me through the ups and downs of my graduate studies.
My parents Mary and Peter have never wavered in their support and encouragement
through all my pursuits. My sister Julie, my brother James, and the rest of my family
as well as my friends Tim, Patrick, Eric, and Luke have always been there for me
through the good and bad times with a reassuring word or humorous distraction.
Lastly and most importantly, I need to thank Maria. Since our first days together
as graduate students at MIT, Maria has been my partner in this entire experience.
From difficult times in our early years of graduate school to right now as I write these
words, Maria has always been there for me with endless care and willingness to give.
5
6
Contents
1
Strong-fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
1.1.1
'Conventional' non-linear optics and strong-field physics . . . .
27
1.1.2
Multiphoton to strong-field photoemission
. . . . . . . . . . .
30
Reaching the strong-field regime . . . . . . . . . . . . . . . . . . . . .
34
1.2.1
Ultrafast pulses . . . . . . . . . . . . . . . . . . . . . . . . . .
34
1.2.2
Ultrafast laser amplifiers . . . . . . . . . . . . . . . . . . . . .
38
1.3
Strong-fields at the nanoscale and HHG . . . . . . . . . . . . . . . . .
40
1.4
Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
1.1
1.2
2
45
Theory of photoemission from solid surfaces
2.1
2.2
2.3
3
25
Introduction
. . . . . . . . . . . . . . . . . . . . .
45
2.1.1
Field emission . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
2.1.2
Photo-assisted field emission . . . . . . . . . . . . . . . . . . .
49
2.1.3
Multiphoton and strong-field photoemission
. . . . . . . . . .
51
Strong-field photoemission . . . . . . . . . . . . . . . . . . . . . . . .
58
. . . . . . . . . . . . . . . . .
59
. . . . . . . . . . . . . . . . . . . .
60
Electron emission fundamentals
2.2.1
Ultrafast optical pulse emission
2.2.2
Continuous-wave emission
2.2.3
Alternative emission rate formulations
. . . . . . . . . . . . .
66
2.2.4
Time-dependent emission . . . . . . . . . . . . . . . . . . . . .
67
Carrier-envelope phase effects
. . . . . . . . . . . . . . . . . . . . . .
70
Plasmonic nanoparticles and optical resonators
75
3.1
Surface Plasmons and nanoparticle resonators
. . . . . . . . . . . . .
75
3.2
Circuit model for nanoparticle resonators . . . . . . . . . . . . . . . .
79
3.2.1
Frequency domain -
susceptibility and extinction . . . . . . .
81
3.2.2
Time-domain ultrashort pulse broadening
. . . . . . . . . . .
82
. . . . . . . . . . . . .
84
3.3
Nanoparticle fabrication and characterization
7
CONTENTS
87
Photoemission from plasmonic nanoparticles
4.3
. . . . . . . . . . .
87
Previous results and ongoing work . . . .
. . . . . . . . . . .
89
. . . . . . . . . . . . . .
. . . . . . . . . . .
91
.
.
4.1.1
.
Strong-fields on a chip
Flattened nano-tips on a chip
. . . . . .
. . . . . . . . . . .
92
4.2.2
On-chip nanoparticle emitter arrays . . .
. . . . . . . . . . .
94
. . . . . . . . . . . . . . .
. . . . . . . . . . .
97
. .
. . . . . . . . . . .
97
.
4.2.1
.
4.2
. . . . . . . .
Strong-fields near nanostructures
Experimental details
.
4.1
Pulse measurement
. . . . . . . . . . . .
. . . . . . . . . . .
98
4.3.3
Carrier-envelope phase stabilization . . .
. . . . . . . . . . .
100
4.3.4
Device alignment . . . . . . . . . . . . .
. . . . . . . . . . .
101
. . . . . . . . . .
. . . . . . . . . . .
103
4.4.1
Collector and emitter currents . . . . . .
. . . . . . . . . . .
103
4.4.2
Photoemission current versus pulse energy
. . . . . . . . . . .
104
4.4.3
Device degradation . . . . . . . . . . . .
. . . . . . . . . . .
108
4.5
Probing the plasmonic field via IAC . . . . . . .
. . . . . . . . . . .
110
4.6
Carrier-envelope phase sensitivity . . . . . . . .
. . . . . . . . . . .
113
.
.
.
.
.
.
Photoemission measurements
.
4.4
.
4.3.2
119
119
.
Few-cycle Er:fiber based laser source
.
4.3.1
.
4
121
5 Enhancement cavities for high-harmonic generation
5.1 Enhancement cavities for strong-field physics . . . .
5.2 Enhancement cavity design for HHG . . . . . . . .
6
Bessel-Gauss beams
6.1 Bessel beams to Bessel-Gauss beams . . . . . .
6.2 Constructing Bessel-Gauss beams . . . . . . . .
6.3 Focal properties of Bessel-Gauss beams . . . . .
6.4 Bessel-Gauss beams and simple optical elements
125
155
B Evolution operator basics
159
C Volkov waves
165
.
135
141
.
133
143
.
.
A Pulse trains in the time and frequency domains
.
141
128
.
7 Bessel-Gauss beam enhancement cavities
7.1 Bessel-Gauss beam cavity design . . . . . . . . . . .
7.1.1 Confocal Bessel-Gauss cavity . . . . . . . .
7.2 Confocal Bessel-Gauss cavity demonstration . . . .
125
.
.
.
.
.
8
.
.
.
.
147
CONTENTS
D Bessel-Gauss beam spatial phase
169
9
CONTENTS
10
List of Figures
1-1
1-2
1-3
1-4
Multiphoton photoemission. Photons of different colors excite electrons from the Fermi level to an energy above the vacuum level, and
the excited electrons subsequently leave the metal. The violet, red, and
green squiggles represent violet, red, and green photons of energy hvv.
hvr, and hvg respectively. U represents energy, and x is the spatial
coordinate normal to the metal surface. Urn is a sketch of the metal's
binding potential, and WF represents the work function. . . . . . . .
31
Strong-field photoemission. At high field strengths, the strong
fields distort the metal's binding potential and results in electron tunneling emission from the metal. . . . . . . . . . . . . . . . . . . . . .
33
Progress in ultrafast laser sources and amplifiers. a. The
achievable minimum laser pulse duration as a function of year. The
arrow indicates the first demonstration of SHG [2]. b. The achievable maximum focused laser intensity as a function of year. The 107
W/cm2 mark shows the intensity used in the first demonstration of
SHG [2]. Note the tremendous growth in achievable intensity since the
development of chirped pulse amplification (CPA). (Images borrowed
from Ref. [5] without permission). . . . . . . . . . . . . . . . . . . . .
34
Ultrashort optical pulse train from a mode-locked laser. A
single optical pulse circulates in the laser cavity (sketched in the upper
left) with a circulation period TR. The pulse periodically leaks out of
the cavity forming a pulse train with temporal spacing TR = 1fR.
Each pulse contains an energy Ep and has a full-width at half maximum duration of TFWHM . . . . . . . . . . . . . . . . . . . . . . . . .35
11
LIST OF FIGURES
1-5
Ultrafast optical pulses and pulse trains in the frequency domain.
a.
The spectrum of an isolated ultrafast optical pulse with
central frequency
f,
and CEP
,CCEO.
b. The spectrum of an ultrafast
optical pulse train with repetition rate fR, central frequency f,, and
carrier-envelope offset frequency fCEO . . . . . . . . . . .
1-6
.. . . . .36
Ultrashort laser pulse amplifiers and resonators. a. Picture of a
typical commercial CPA laser system capable of generating multi-mJ
femtosecond laser pulses in a strong-field physics laboratory (photo
courtesy of P. D. Keathley).
b.
Sketch on an ultrafast optical en-
hancement cavity. An optical pulse circulates in the resonator with
amplitude ares. It is excited by an incident pulse train with amplitude
a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-7
Strong-field photoemission near nanostructures.
a.
39
Strong-
field photoemission from plasmonic nanoparticles. A femtosecond laser
pulse illuminates the nanoparticles, and they photo-emit electrons (illustration borrowed from Ref. [22] without permission).
b. Electron
energy spectra measured from strong-field photoemission experiment
with differing excitation pulse energies (picture borrowed from Ref.
[23] without permission). . . . . . . . . . . . . . . . . . . . . . . . . .
1-8
41
Characteristics of HHG and cavity-enhanced HHG. a. Example of an HHG spectrum.
For this example, the driving laser light
wavelength varies but for the pink trace A = 1.8 prm or the photon
energy ~ 0.7 eV. Note the large plateau of hundreds of harmonics; this
plateau resembles the plateau in the electron energy spectra from Figure 1-7b (this data was borrowed without permission from Ref. [3]).
b.
Basic arrangement for cavity-enhanced HHG. An ultrafast laser
pulse is enhanced in a bow-tie ring cavity where high-harmonics are
produced and out-coupled via a sapphire plate (the pictured harmonics
are borrowed from Ref. [26] without permission).
12
. . . . . . . . . . .
42
LIST OF FIGURES
2-1
2-2
Field emission and models. a. Basic model for metallic surface.
The interior of the metal is treated as a free electron gas, and the
metal surface is modeled as a simple rectangular step of height WF. As
before, the metal potential is denoted as Um(x) with x the coordinate
normal to the surface. The real part of the electron wavefunction at the
Fermi energy is drawn in red (calculated numerically). b. Potential
with an applied static bias. With an applied bias the rectangular step
is deflected, and electrons can tunnel through the barrier and into
the vacuum. The real part of the wavefunction is again drawn in red
(numerical calculation). Also, note that we assume no field penetration
into the metal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
Photo-assisted field emission. a. Photo-thermal emission. An
optical excitation locally heats the electron gas in a metal (see the
lightly shaded region labeled THot). This heating results in a stretched
Fermi-Dirac distribution. Higher energy levels are now increasingly
thermally occupied and can undergo field emission. The real part of
the electron wavefunction at 2 eV above the Fermi energy is drawn
in red (calculated numerically) with a static field of Estat = E0 =
10 V/nm. b. Photo-field emission. An optical excitation resonantly
excites electrons to higher energy levels, and these excited electron
field-emit. The real part of the electron wavefunction at 3 eV (three
photons at a wavelength of 1.2 pm) above the Fermi energy is drawn
in red (calculated numerically) with a static field of Estat = Eo = 10
V /nm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
Multiphoton photoemission. The rectangular step potential is illustrated in black. We see it wiggles only a small amount in response to
the relatively weak field E(t) (see dashed). The real part of the initial
wavefunction #(x, t) is drawn in red, and the real part of a potential
final state (that is also a eigenstate of the unperturbed step potential)
is drawn in blue. This state has an energy of 3 eV above the drawn
vacuum level (both wavefunctions are numerically calculated). .....
55
2-4 Strong-field perturbation theory. The terms in Eq. (2.25) can be
interpreted via these simple diagrams . . . . . . . . . . . . . . . . . .
56
2-3
13
LIST OF FIGURES
2-5
Strong-field photoemission. The rectangular step potential is illustrated in black. We see it wiggles dramatically with the strong field
E(t) with Eo = 10 V/nm (see dashed sketch). The real part of the
initial wavefunction O(x) is drawn in red (numerically calculated), and
the real part of a potential final state (that is a length gauge Volkov
wave) is drawn in blue. This state has an energy of 3 eV above the
drawn vacuum level and an additional pondermotive energy of 1.8 eV.
2-6
58
Multiphoton and tunneling emission. At low fields (high -y), we
expect the photoemission current to resemble the multiphoton scaling
i.e. oc En (drawn in red). Above a critical field (y a 1) and for low y,
we expect the photoemission current scaling to resemble the FowlerNordheim equation (drawn in black). The parameters used above are
for gold being illuminated by light with a wavelength of 1.2 pm. . . .
2-7
59
Strong-field photoemission with an ultrafast pulse. The photoemission current is plotted for an ultrafast pulse with a central wavelength A, = 1.2 pm and a pulse duration r=
The pulse is of a cos
2
9.5 fs (WF= 5.1 eV).
shape. Additionally, the photoemission rate for
the continuous wave case with a wavelength A = 1.2 pm is plotted for
com parison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-8
61
Strong-field photoemission with continuous-wave excitation.
The photoemission current is plotted for a continuous-wave excitation
at a central wavelength of A = 1.2 Am. As before WF= 5.1 eV. The
numerical calculation is plotted in pink and the analytical result is
draw n in blue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-9
64
Strong-field photoemission in the time domain. The pink shows
the electric field that drives photoemission from a gold surface with
WF
=
5.1 eV. The field has strength E0 = 30 V/nm and wavelength
A = 1.2 pm.
The blue shows the probability current as found via
numerical solution of the TDSE. The dashed black shows the quasistatic tunneling current. The peaks of these currents are shifted from
that of the field as they are calculated a short distance from the surface.
Additionally, the inset shows the potential model for the simulation
(note that the potential is truncated = 0.5 nm from the surface for
simplicity in caculation). . . . . . . . . . . . . . . . . . . . . . . . . .
14
68
LIST OF FIGURES
2-10 Strong-field photoemission in space and time. The wavefunction
amplitude for the simple rectangular step potential modeled in Figure
2-9 is displayed in space and time. Note the surface is located at x = 0
nm. Also, note the large current spike near t = 0 fs. . . . . . . . . . .
69
2-11 Threshold nature of CEP sensitivity. Two short pulses and two
long pulses are illustrated with carrier-envelope phases p = r and 7r/2.
The shifted CEP strongly affects the peak field for the short pulses
(the CEP dictates whether the peak field in this case is above or below
Et), while the CEP shift has a minimal effect on the peak field of the
longer pulses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
2-12 CEP model for a simple pulse. The top plots show a simple 9.5
fs cos 2 shaped laser pulse (red) and the resulting quasi-static emission
current (Fowler-Nordheim) for the one- and two-sided emission cases.
The middle plots show the emitted charge as a function of the CEP
(note the two-sided charge has been magnified by a factor of 103). The
bottom plot shows the Fourier series coefficients for the emitted charge. 73
2-13 CEP model for a real pulse. The top plots show the measured 9.5
fs, 1.2 pm wavelength laser pulse used in our experiments (red) and
the resulting quasi-static emission current (Fowler-Nordheim) for the
one- and two-sided emission cases. The middle plots show the emitted
charge as a function of the CEP (note the two-sided charge has been
magnified by a factor of 10). The bottom plot shows the Fourier series
coefficients for the emitted charge. . . . . . . . . . . . . . . . . . . . .
74
Surface plasmons on a metal surface. a. At a metal-dielectric
interface (dielectric above the dashed line and metal below the dashed
line), longitudinal surface charge oscillations can be excited. These
charge oscillations are the origin of the surface plasmon polariton modes
and have wavelength Aspp. b. The electric and magnetic fields produced from these longitudinal surface charge oscillations. Since the
fields originate from the surface charge oscillations, they are largely
confined to a region near the metal's surface (see green sketch of mode
profile). (Images in a. and b. were borrowed without permission from
[401). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
3-1
15
LIST OF FIGURES
3-2
Optical cavities and plasmonic nanoparticle resonators. a. The
fundamental operation of a familiar optical cavity. Between the cavity
mirrors an optical pulse circulates. Energy is fed into the intra-cavity
pulse by an external pulse train. b. Plasmonic nanoparticle resonators,
for example nano-rods or nano-triangles, are analogous to the familiar
optical cavity. They support surface plasmonic electromagnetic modes
that can propagate up and down the nanoparticles.
Inset shows the
transverse intensity profile of such a mode on a nano-rod resonator
with a circular cross-section and a diameter of 20 nm (image borrowed
from Ref. [41] without permission).
3-3
. . . . . . . . . . . . . . . . . . .
78
Circuit model for plasmonic nanoparticle resonators. a. Image
of nano-rod resonator and cartoon of basic optical cavity-like operation.
b. Second-order RLC circuit model for nano-rod resonator.
3-4
. . . . .
80
Example extinction spectrum and fit. The blue curve represents
the measured extinction spectrum for an array of nano-triangles with
pitch 400 nm, altitude 200 nm and base 150 nm. The dashed red curve
shows the fit via Eq. (3.3). . . . . . . . . . . . . . . . . . . . . . . . .
3-5
82
Ultrafast optical pulse broadening in a nanoparticle resonator.
The resonator used has a resonant wavelength of 1256 nm and a damping time of 7.4 fs. The black trace shows the excitation pulse envelope
(central wavelength of 1.2 pm). The blue trace shows the envelope of
the response on the nanoparticle resonator. . . . . . . . . . . . . . . .
3-6
83
Chip overview. a. Picture of the actual chip. b. Zoomed in microscope image of the arrays of devices. c. Zoom in of the purple region
outlined in part b. d., and e. electron micrographs of the blue and
red regions outlined in part c.
3-7
. . . . . . . . . . . . . . . . . . . . . .
Extinction spectra for the nano-rod devices.
84
The blue traces
are measurement and the red dashed traces are model fits according
to Eq.
(3.3).
The dark blue trace is for the nano-rod with resonant
wavelength 1041 nm . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-8
Extinction spectra for the nano-triangle devices.
85
The blue
traces are measurement and the red dashed traces are model fits according to Eq. (3.3). The dark blue trace is for the nano-triangle with
resonant wavelength 1059 nm. . . . . . . . . . . . . . . . . . . . . . .
16
85
LIST OF FIGURES
3-9
Image and sketch of chip layout.
a.
Microscope image of set
of eighteen fabricated device arrays. b. Sketch showing the resonant
wavelengths of each array. The R and T labels indicate whether the
array is a nano-rod array or a nano-triangle device array.
4-1
. . . . . . .
86
Strong-field physics with nanostructures. a. Schematic of nanotip emitter operation. b. Basic layout for nanoparticle emitter experiments.
c. Example data set showing carrier-envelope phase effects
in the emitted electron's energy spectra from nano-tip emitters.
d.
Energy spectra showing large re-scattered plateau of emitted electrons
from nanoparticles.
(illustrations and pictures in a-d borrowed from
Refs. [10, 22] without permission).
4-2
. . . . . . . . . . . . . . . . . . .
90
Nano-tip emitter experiments on a chip. a. Conventional nanotip emitter experimental arrangement (illustration borrowed from Ref.
[10] without permission). b. Basic layout of one of our flattened onchip nano-emitting tips.
4-3
. . . . . . . . . . . . . . . . . . . . . . . . .
92
Electrical currents across large gaps under ambient conditions. a. Basic experimental arrangement. Electrical emission from
gold whisker on photo-lithographically defined pad. b. Measured electrical currents: emitter current, collector current, and current difference. All currents were measured sequentially, i.e. one after the other.
4-4
94
Nanoparticle emitter device layout (optical microscope image). The substrate is composed of a sapphire chip coated in indium
tin oxide (ITO). An array of nanoparticle emitters is fabricated on
the ITO layer (enclosed by the red box in the image and an example
show in the inset). The ITO is patterned into two regions: an emitter
that connects to the nanoparticles and a collector.
The collector is
separated from the emitter by a few micron gap. . . . . . . . . . . . .
4-5
95
Nanoparticle emitter device layout (optical microscope image) and basic experimental setup. Femtosecond laser pulses are
focused by an objective (Obj.) to a small spot-size on the nanoparticle
array. The excitation laser pulses result in strong-field photoemission
from the devices, and the photo-emitted electrons jump from emitter
to collector. The three main elements of the following few chapters are
num bered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
96
LIST OF FIGURES
4-6
Femtosecond laser system overview.
Schematics and pictures of
the Er:fiber oscillator and the supercontinuum generation stages are
shown. At the output of the femtosecond laser system is the dispersive wave spectrum spanning 1-1.4 pm as shown. The dispersive wave
pulses are sent from the laser system output to the pulse measurement
setup, the carrier-envelope phase stabilization/characterization setup,
and the actual strong-field experiments. . . . . . . . . . . . . . . . . .
4-7
Dispersive wave pulse measurement.
a.
persive wave pulse used in the experiments.
98
Spectrum of the disb.
Measured optical
pulse shape from the 2DSI (blue) and the ideal transform-limited pulse
(dashed-red).
c. Interferometric autocorrelation measurements (note
the early delay data is poor due to an error in the calibration at the
start of this trace). d. Image of the 2DSI setup. . . . . . . . . . . . .
4-8
99
Carrier-envelope phase stabilization and characterization. a.
Overview of the modifications to the femtosecond laser system to allow
carrier-envelope phase locking. The physical meaning of the CEP is
also illustrated to the right. b. Results from the out-of-loop
f
- 2f
interferometer when the carrier-envelope offset frequency is locked to
o Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1
4-9
Alignment microscope setup and focused spot characterization. a. Picture of the confocal alignment microscope and the device
mount (right). b. Knife-edge measurements of the focused laser spot
in both transverse planes (measurement made by scanning a 10 Am
wide gold wire across the laser spot).
c. Example microscope image
recorded on the CCD. Old devices are shown with very high spatial
resolution.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
102
4-10 Collector and emitter currents from a nano-triangle array.
The upper plot shows the simultaneous measurement of the collector
and emitter currents and the stability of these currents. The bottom
plot shows the relative phase between the collector and emitter currents.
The right image shows a reminder of our basic experimental
arrangement. The nano-triangle array has ARes = 1059 nm and
5 .8 fs.
Trd =
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
104
LIST OF FIGURES
4-11 Photoemission current versus pulse energy for a nano-triangle
array. The array has
ARes =
1105 nm and
rsad
= 6.4 fs. The current
scaling is measured for four different collector biases.
energies the current scales as ~
scaling falls off to
-
I.
5.,
At low pulse
while at higher pulse energies this
The blue dot at the top right of the graph
labels a data point that shows 42.6 nA of current which corresponds
to approximately 37 electrons per pulse per emitter. . . . . . . . . . .
105
4-12 Photoemission current versus pulse energy for a nano-rod array. The array has
1041 nm and
ARes =
Trad
= 4.8 fs. The current
scaling is measured for four different collector biases. At low pulse energies the current scales as
-
I" , while at higher pulse energies this
scaling falls off to ~ I2. The blue dot at the top right of the graph
labels a data point that shows 34.3 nA of current which corresponds
to approximately 21 electrons per pulse per emitter. . . . . . . . . . .
106
4-13 Photoemission current versus pulse energy (nano-triangle).
The four nano-triangle arrays have
ARes
= 951, 1059, 1158, and 1256
nm. The measurements were made at 30 V bias. . . . . . . . . . . . .
107
4-14 Photoemission current versus pulse energy (nano-rod). The
four nano-rod arrays have AR,, = 968, 1041, 1177, and 1238 nm. The
measurements were made at 30 V bias.
. . . . . . . . . . . . . . . . .
4-15 Repeatable photoemission current scalings.
was made at 30 V from a nano-triangle array with
and
TRad
108
The measurement
ARes =
1105 nm
= 6.4 fs. The red trace is the same measurement as shown
from Figure 4-7. The blue trace is measured by sweeping the intensity
in the opposite direction several minutes later. . . . . . . . . . . . . .
4-16 Emitter array degradation.
a.
109
Microscope image of extensively
used device arrays (the labeled array was used in this condition to
measure the
ARes
= 1158 nm trace from Figure 4-13). The light colored
strip near the collector edge forms during device operation. b. SEM
image of the labeled device array.
The light colored strip from the
microscope image appears to be ITO de-lamination. . . . . . . . . . .
19
110
LIST OF FIGURES
4-17 Interferometric autocorrelation measurement performed with
the strong-field photoemission current. The device used in this
7.4 fs.
ARes =
1256 nm and r7ad
-
measurement is a nano-triangle array with
The top trace shows the measured autocorrelation (red), the
expected autocorrelation from our time-domain model (blue), and the
second-harmonic generation (SHG) autocorrelation measured previously. The bottom panel shows the measured pulse (black) that excites
the nano-resonator array and the expected pulse from our time-domain
m odel (blue).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
111
4-18 Interferometric autocorrelation measurement performed with
the strong-field photoemission current. The device used in this
measurement is a nano-triangle array with
ARes =
1105 nm and
'rad =
6.4 fs. The top trace shows the measured autocorrelation (red), the
expected autocorrelation from our time-domain model (blue), and the
second-harmonic generation (SHG) autocorrelation measured previously. The bottom panel shows the measured pulse (black) that excites
the nano-resonator array and the expected pulse from our time-domain
m odel (blue).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
112
4-19 Interferometric autocorrelation measurement performed with
the strong-field photoemission current. The device used in this
measurement is a nano-triangle array with
4.8 fs.
ARes =
1041 nm and rad
=
The top trace shows the measured autocorrelation (red), the
expected autocorrelation from our time-domain model (blue), and the
second-harmonic generation (SHG) autocorrelation measured previously. The bottom panel shows the measured pulse (black) that excites
the nano-resonator array and the expected pulse from our time-domain
m odel (blue).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
113
4-20 Geometry of strong-field emission. Nano-triangles and nano-rods
are illuminated by femtosecond laser pulses (here labeled F(t)). The
Nano-triangles will only emit from their apex and therefore only emit
for half of the pulse's optical cycles, while the nano-rods emit from
every cycle.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
114
LIST OF FIGURES
4-21 Initial CEP sensitivity measurement.
The top trace shows the
RF spectrum of the emitter current for a nano-triangle array and the
bottom trace for a nano-rod array with the
fCEO
locked to 2 kHz. The
noise data corresponds to when the fCEO is unlocked. The units dBpA
are equivalent to 20logio(JE/1 pA). . . . . . . . . . . . . . . . . . . .
115
4-22 Absolute phase stepping measurement. A barium fluoride wedge
is stepped through the excitation pulse train shifting the absolute CEP
of the pulse train. The response is measured via lock-in detection. . .
4-23 CEP sensitivity versus
ARes.
116
a. RF traces of the emitter currents
from five different nano-triangle arrays are displayed (note each trace
is measured from 1.94 - 2.06 kHz). b. The corresponding sensitivity is
plotted for these five devices and two additional ones. . . . . . . . . .
5-1
Operation of a femtosecond enhancement-cavity.
a.
118
In the
time domain, small portions of the incident pulse train are transmitted
into the cavity and add to the circulating intra-cavity pulse.
If the
cavity parameters are properly tuned, these transmitted pulses will
constructively add and build-up the energy of the intra-cavity pulse.
b. In the frequency domain, the cavity has a comb of resonances spaced
by the free-spectral range of the cavity. If each spectral mode of the
incident optical pulse train overlaps a cavity resonance, the pulse train
will be enhanced.
5-2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
120
Bow-tie ring cavities and popular out-coupling schemes for
intra-cavity HHG. a. A sapphire plate is placed in the cavity and
Brewster's angle to couple out the generated HHG beam. b. An EUV
grating is etched on to a highly reflecting cavity mirror to diffract out
the generated high harmonics. . . . . . . . . . . . . . . . . . . . . . .
6-1
123
k-space distribution for a Bessel and a Bessel-Gauss beam. a.
The k-space distribution for a Bessel beam at the focal plane z
=
0.
b. The k-space distribution for a Bessel-Gauss beam at the focal plane
z = 0.
6-2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
127
Decentered Gaussian beam. At the focal plane z = 0 (shaded),
the beam has a Gaussian distribution that is displaced from the origin
by rd
= (rd, -/).
Away from the focal plane the beam resembles a tilted
Gaussian beam, propagating at an angle p to the optical axis, i.e. the
z-ax is. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
129
LIST OF FIGURES
6-3
Constructing Bessel-Gauss beams. a. We superpose many decentered Gaussian beams with differing -y. This amounts in superposing
many decentered Gaussian beams along the surface of a cone
or a frustum (rd 74= 0).
b.
(rd =
0)
An overlay of the transverse intensity
profile after the superposition. Note the annular form of the generalize
Bessel-Gauss beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-4
131
Types of Bessel-Gauss beams. a.-c. Illustrations of r - z plane
cross-sections of gBG, BG, and mBG beams respectively. d.-f. Plots
of the amplitude in the r - z plane for gBG (A
p = 0.21 , r=
and mBG (A
6-5
=
1 pm, wo = 200 pm,
0.25 mm), BG (A = 1 pm, wo = 200 pm, p = 0.29'),
=
1 pm, wo = 200 pm,
rd =
1 mm) beams respectively..
132
BG beam focal properties and intensity gain. a. Plot of amplitude cross-section in the z = 0 plane of a BG beam with A = 1 pm,
wo = 200 pm, and semi-aperture angle p
=
0.29' (same parameters
from BG beam plotted in Figure 6-4e). Cross-section of the focus in the
y-direction is on the right with
2
WB
labeled. b. Plot of approximate
(orange dashed) and exact (solid green) intensity gain of BG beams
with A = 1 pm, wo = 30 pm, and semi-aperture angles p of 10, 20,
30 ,
and 40 at distance z. The intensity gain of a Gaussian beam with
A = 1 pm and wo = 30 pm (blue curve) is also included. c. Plot of the
amplitude cross-section in the z = 20 cm plane of the BG beam from
plot a. Cross-section in the y-direction is included on the right with w
and r, labeled . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-6
gBG beam transformations.
135
a. Example 1 geometry: an mBG
beam reflecting from a curved mirror.
b.
r - z plane cross-section
of numerically simulated amplitude for example 1 (note z-axis corresponds to reflecting geometry). c. r-direction cross-sections of field's
spatial amplitude and phase at the end of propagation (numerically
simulated (blue) and analytical (red-dashed)). d. Example 2 geometry: an mBG beam reflecting from a reflecting axicon. e and f are as
b and c but for example 2. g Example 3 geometry: an mBG beam
reflecting from a toroidal optic. h and i are as b and c) but for example
138
3..............................................
22
LIST OF FIGURES
7-1
Gaussian cavities and Bessel-Gauss cavities. a. Illustration of a
Gaussian beam enhancement cavity. Note that the harmonics (purple
pulse) are generated collinearly with the driving beam. b. The intracavity Gaussian mode intensity on the cavity mirrors in the x - y
plane. The dashed white circles indicate roughly where two of the
cavity mirrors lie. c. Illustration of a Bessel-Gauss enhancement cavity.
This cavity is rotationally symmetric about the z-axis (as indicated by
the red circle). Also, note that the harmonics propagate along the zaxis. d. Intra-cavity Bessel-Gauss mode intensity on the segmented
cavity mirror in the x - y plane. The dashed white circles roughly show
the boundaries between the different sections of the segmented mirror. 142
7-2
Single-mode selection in the confocal BG cavity. a. Cavity mirror with patterned annular (donut-shaped) region of high-reflectivity.
b. Cross-section of patterned cavity mirror with incident beams. . . . 144
7-3
Patterned-mirror confocal BG cavity simulation. a. r - z plane
cross-section of fundamental BG mode amplitude. b. Normalized
mode intensity at mirror surface plotted against r (as labeled in (a)).
c. Mode intensity at focus plotted against r (same normalization as
(b) and labeled in (a)). . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7-4 Patterned mirror confocal cavity scaling. a. Intensity gain, Ig,
scaling with repetition rate (i.e. cavity length and mirror radius of
curvature). b. Effective waist, weff, scaling with repetition rate (i.e.
cavity length and mirror radius of curvature). For all cavities in these
plots Ar = 3.1Wmin . .
. . . .. .. . . .146
. . . . . . .. .
. . . . . .
7-5
Confocal Bessel-Gauss cavity mirrors a. Illustration of patterned
cavity mirror with intra-cavity mode intensity overlaid (not to scale).
Inset is a microscope image of a section of a patterned mirror used in
the experiment. b. Photograph of actual patterned cavity mirror in
the experimental setup. . . . . . . . . . . . . . . . . . . . . . . . . . 148
7-6
Confocal Bessel-Gauss prototype cavity. a. Experimental arrangement (sampling pellicle and CCD not shown). The photodiode
signal is used to lock the cavity (input-coupling mirror is actuated with
a piezo). b. Simulated field (r - z plane cross-section) traversing the
coupling optics/cavity system. . . . . . . . . . . . . . . . . . . . . . . 149
23
LIST OF FIGURES
7-7
Images of the transmitted cavity mode. Images are normalized
and taken when the cavity is a. misaligned and the cavity length is
swept (the dashed ring is added for ease of illustration), b. well-aligned
and the cavity length is swept c. locked from a well-aligned state.
7-8
. .
150
Effective finesse in the presence of curvature variations. The
solid line (green) is the analytical model i.e.
Eq.
(7.4).
The dots
(black) represent simulation results. The dashed line (red) shows the
value of
ravg/w
used in the experiment. The inset illustrates the simple
model for estimating curvature variations.
A-1
. . . . . . . . . . . . . . .
151
An optical pulse train in the time and frequency domains.
In the above, we proceed step by step from the time domain to the
frequency dom ain . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
156
A-2 The basic f-2f interferometer. In the basic f-2f interferometer a
low frequency component from the spectrum is frequency doubled via
second-harmonic generation (SHG) and mixed with a high frequency
component. The mixing process yields a beat-note at the fCEO-
24
- -.
157
Chapter 1
Introduction
These days one would be hard pressed to walk down a corridor in a university's
applied physics department and not glimpse an advertisement for a talk or see a
research poster with a title involving the prefixes pico-
(10-12),
femto- (10-15), or
atto- (10-18). Since the advent of scientific investigation, researchers have sought
instruments to explore and probe the world around them with ever finer resolution,
and in the past several decades, the laser physics community has pushed the temporal
resolution of our measurement capabilities, i.e. the temporal duration of the shortest
achievable bursts of light, from the picosecond to the femtosecond scale and, recently,
towards the attosecond level.
The advantages of short pico- or femtosecond laser pulses extend beyond just the
temporal resolution they can provide. Femtosecond laser pulses from a mode-locked
laser oscillator are typically emitted as a train of pulses with each pulse carrying an
optical energy on the order of nanojoules (nJ). Consider the optical power at the peak
of such a laser pulse. This peak power is on the order of nanojoules, i.e. 10- J (1
nJ), divided by femtoseconds, i.e. 1015 s (1 fs); that is, this peak power is on the
order of megawatts, i.e. 106 W (1 MW)! Over the past several decades, developments
in short pulse laser amplifiers have made the amplification of these nanojoule laser
pulses to the millijoule level commonplace. Accordingly, remarkable peak powers
in the range of terawattsi, i.e. 1012 W (1 TW), can be achieved with commercial,
table-top femtosecond laser systems.
Just over fifty years ago, and only a year after the development of the first laser,
researchers illuminated a quartz crystal with millisecond-duration laser pulses and
produced second-harmonic generation [2]. The illuminating laser pulses contained
'To establish a sense of scale for a terawatt, note that the mean global power consumption in the
year 2012 was approximately 2.6 terawatts [1].
25
CHAPTER 1. INTRODUCTION
three joules (3 J) of energy in their one millisecond (1 ms) duration and had a
peak power of around three kilowatts (3 kW). This was the first demonstration of
second-harmonic generation (SHG) and a major milestone in the early days of nonlinear optics.
Nowadays, with near terawatt peak power femtosecond laser pulses,
researchers have pushed non-linear optics to a new frontier. With such extreme optical peak powers, harmonics of the hundredth order and higher have been generated
from non-linear optical media [3, 4, 5, 6, 7]; pulses of extreme-ultraviolet (EUV) light
under a hundred attoseconds in duration have been produced in this high-harmonic
generation (HHG) process [8, 9]; and individual optical cycles of femtosecond laser
pulses have been used to switch on-and-off photoemission currents near nanostructures and to steer the resulting sub-optical cycle duration electron bursts near such
nanostructures [10, 11, 121. This extreme regime of non-linear optics is known as the
'strong-field' regime.
In this thesis we will explore the strong-field regime of non-linear optics with
ultrafast, femtosecond laser pulses and passive optical resonators.
Passive optical
resonators offer a means to enhance the optical energy in a femtosecond laser pulse and
achieve the high peak powers necessary for strong-field physics without the complexity
and limitations of traditional femtosecond laser amplifiers.
In particular, in this
thesis we will focus on two different areas in the broad topic of strong-field physics
with ultrafast optical resonators. First, we will make use of plasmonic nanoparticle
optical resonators to explore strong-field physics near nanostructures. We will
switch on and off photoemission currents from nanoparticle resonators with individual
optical cycles of a femtosecond laser pulse. We will use this photoemission current
to characterize the femtosecond dynamics of the nanoparticle's excited plasmonic
field and demonstrate carrier-envelope phase sensitivity of the photoemission current.
Second, we will explore novel optical resonator designs for cavity-enhanced highharmonic generation. We will design and prototype optical enhancement cavities
supporting Bessel-Gauss intra-cavity modes, and we will explore the advantages and
limitations of these cavities for cavity-enhanced HHG.
We begin this thesis by reviewing the what, how, and why of strong-field physics.
First, we trace the origins of strong-field physics from traditional non-linear optics
and define the strong-field regime. We next discuss how the field strengths necessary
for strong-field phenomena can be achieved. In particular, we will review the basics
of ultrafast, femtosecond laser pulses and femtosecond pulse amplification and resonator techniques. Next, we motivate our interest in this high-intensity regime and
explain why strong-field physics has garnered such attention from the laser physics
26
CHAPTER 1. INTRODUCTION
community. Lastly, we provide an overview of the specific goals of this thesis and an
outline of the coming chapters.
1.1
Strong-fields
In this section, we develop the concept of the strong-field regime from the basic
principles of linear and non-linear optics. Additionally, we provide a brief detour to
consider just how strong strong-fields really are, and we go through a simple, relevant
first example of strong-field phenomena: strong-field photoemission.
1.1.1
'Conventional' non-linear optics and strong-field physics
Since the 'field' part of strong-field refers to the electric field of an electromagnetic
wave, and we are interested in light interacting with matter, a reasonable starting
point is the wave equation for an electromagnetic wave in some material.
V2E -
a2p
1 (92E
c2
E=
[-
(9t2
at2
(1.1)
In the above, E is the electric field of the wave, c is the speed of light in vacuum,
and P is the polarization response of the material, i.e. the dipole moment per unit
volume produced in the material in response to the electromagnetic wave.
The wave equation in Eq. (1.1) takes the form of a 'driven' or inhomogeneous wave
equation. Qualitatively, the left-hand side of the equation describes freely propagating electromagnetic waves, and the right-hand side represents a driving term that
can provide sources or sinks for these waves. In linear optics, we assume that the
polarization response is proportional to the electric field
P = CoX( E
In the above, X(
(1.2)
is the linear susceptibility. In the familiar regime of linear optics,
with the driving polarization term proportional to the electric field, the right hand
side of Eq. (1.1) can be lumped in with the time derivative of the electric field on
the left-hand side. The result is a homogeneous wave equation with a modified wave
speed: c/n = c//1
+ X), where n is the familiar index of refraction. Physically,
in the linear optical regime, the electric field of the wave produces a proportional
polarization in the material which subsequently radiates electromagnetic waves of the
same frequency.
27
CHAPTER 1. INTRODUCTION
As the strength of the electric field increases, the polarization response deviates
from the simple linear form. This is the regime of non-linear optics. The polarization
is now a non-linear function of the electric field, i.e. P = P(E). In conventional nonlinear optics, we assume that the polarization's departure from the linear response is
small. We can therefore expand P(E) in the perturbative form
P(E) = co(X( 1)E + X )E2 + X 3 )E 3 +.. .)
(1.3)
In the above, XM again corresponds to the linear susceptibility, and the higherorder X(') terms (i.e. with n > 1) correspond to the non-linear susceptibilities. From
these non-linear susceptibilities and higher-powers of the electric field come the classic
phenomena of conventional non-linear optics, e.g. second-harmonic generation derives
from the X) E 2 term. Note that in the perturbative expansion we have implicitly
assumed that X(1) >> x() >> X(). We can re-write Eq. (1.3) in a more intuitive
form as
P(E) = coX(')E
+ a2
I + ai
+ . .2.
(1.4)
Here we have factored the linear response out of the polarization and expressed the
higher-order terms as ratios of the electric field to some critical field strength Ec. The
parameters a1 and a 2 for many materials are comparable and close to unity. In this
form, the assumption of the perturbative expansion is very clear. When E << Ec
the higher-order terms are small and the series expansion is valid.
From this analysis we see that when E approaches E, the conventional, perturbative approach of non-linear optics breaks down.
This is the strong-field regime.
Here the non-linear response takes on a non-perturbative character and fundamentally new non-linear optical phenomena can be observed.
We will discuss some of
these phenomena and exciting applications in the following sections and in the course
of this thesis. However, before this discussion let us first consider the boundary of
the strong-field regime. Let us consider the value of Ec.
A first guess at the value of E, might be the characteristic binding field strength
of the material 2 . However, the characteristic binding field strength is not exactly a
2
If we use for Ec the atomic unit of electric field Eat = e/47rcoa2, i.e. the electric field experienced
by an electron in a Coulomb potential at a Bohr radius (ao), and note that XM) is of order unity
1
1
for most materials, we can approximate X
xM
/E2
2 im/V and xj) =
)/E2
3.78 x 10-24. Note that here we have assumed that a,, a 2 : 1. These values are actually comparable
to the X(2 ) and X(3 ) values for many materials (see Ref. [13]).
28
CHAPTER 1. INTRODUCTION
familiar quantity. Instead, let us compare the characteristic binding energy Eb of
the material to the characteristic energy of the oscillating electric field, i.e. to the
pondermotive potential Up.
The pondermotive potential of an oscillating electric field represents the average
kinetic energy of an electron wiggling in this field. Consider an electron (mass m and
charge -e) in an electric field E(t) = Eo cos(wt). The electron's spatial position x(t)
is given as a solution to the simple equation of motion
d2 x2 (t)
-d E(t)
x(t)
=
e E02
e
cos(wt)
(1.5)
Note that in the above we have disregarded any initial velocity of the electron. From
the electron's spatial trajectory x(t) we can calculate the electron's average kinetic
energy, i.e. Up = m(,(t)2 )/2, and we find
UP
e 2 E2
= 4mW 2
(1.6)
This pondermotive potential Up is the characteristic energy scale associated with the
electric field. We expect that when this energy exceeds the characteristic binding
energy of the material Eb, the field of the electromagnetic wave will be strong relative
to the fields in the material, and we will enter the non-perturbative regime of strongfield physics. Therefore we expect the transition to the strong-field regime to be
defined by Up ~ Eb. In fact, as we will elaborate on in Chapter 2, the boundary to
the strong-field regime will be defined by a slightly modified parameter known as the
Keldysh parameter -y. This parameter is defined as
ly
=
Eb
U--=
2Up
2mEb
(1.7)
w
eE0
When -y < 1 we expect the electric field strength of the incident electromagnetic
wave to exceed the field strengths in the material, and we expect non-perturbative,
strong-field effects. Now, to provide a more tangible physical picture of the strongfield regime let us consider a specific physical example: strong-field photoemission.
However, before moving on to this example let us actually consider some numbers
here and consider how strong strong-fields really are.
29
CHAPTER 1. INTRODUCTION
How strong are strong-fields?
Consider light with a wavelength of 1.2 pm illuminating a gold surface (gold has a
5.1 eV [14]). With these numbers 4
we see that to reach -y ~ 1, we must have Eo
12 V/nm. This field strength is on
,
binding energy or work function3 of Eb = WF
the order of volts per Angstrom. For solid-state systems like the gold surface we are
considering, typical lattice constants are at the Angstrom level and typical binding
energies are at the volt level, so this field strength is exactly in the expected range.
Let us now consider how strong a 12 V/nm electric field is compared to a force
and field we are more familiar with and experience everyday: gravity. Typical gravitational acceleration here on earth is g ~ 9.8 M/s 2 . The acceleration of an electron
in a 12 V/nm electric field is approximately a ~ 2.2 x 1021 M/s 2
This is obviously a tremendous acceleration.
near the edge of a black hole is ~ 1027 g's
2.2 x 1020 g's.
In fact, the gravitational acceleration
[5],
so we are talking about some pretty
extreme conditions!
1.1.2
Multiphoton to strong-field photoemission
Photoemission is the emission of electrons from a material surface by light.
The
conventional picture of photoemission involves energy, i.e. photon, absorption (see
Figure 1-1).
An optical field illuminates a metal surface (for the purposes of this
thesis, the photo-emitting material will be metallic, in our case gold), and the optical
field wiggles an electron in the metal. We model the metallic binding potential as
a simple, smoothed rectangular step of a height given by the work function WF.
When illuminated by light (in the case of Figure 1-1, violet, red, and green light),
the electrons in the metal slowly pick up energy. Eventually they can absorb enough
energy to hop over the binding potential barrier and out into the vacuum.
The photo-emitted electrons can absorb one or more photons from the illuminating optical field.
Conventionally, the electrons only absorb the number of photons
required to surmount the work-function barrier 5 ; see, for example, that in Figure
1-1 three red photons are required to overcome the barrier, two green photons are
required, and just one violet photon is required.
This multiphoton absorption and
3
Depending on various conditions, the work function of gold varies around 5 eV by several tenths
of an eV. For the purposes of this thesis we will use the value of 5.1 eV [14].
'These numbers are typical of our experiments described in Chapter 4.
'We are taking a very simplistic approach to photoemission here for the purpose of illustrating the
essential physical phenomena. Photo-emitted electrons can absorb more photons than are required
to surmount the work-function. This phenomena is known as above-threshold photoemission.
30
CHAPTER 1. INTRODUCTION
hv,+ hV,+ h,
vacuum
i'J
- -(x
U
1
W_
Figure 1-1: Multiphoton photoemission. Photons of different colors excite
electrons from the Fermi level to an energy above the vacuum level, and the excited electrons subsequently leave the metal. The violet, red, and green squiggles
represent violet, red, and green photons of energy hvv. hvr, and hvg respectively.
U represents energy, and x is the spatial coordinate normal to the metal surface. Un is a sketch of the metal's binding potential, and WF represents the
work function.
photoemission is a well-known non-linear optical effect [13, 15], and we can write the
total photoemission current as
J = a 1,
+ 02)I2 + C()13
(1.8)
In the above, J represents the photo-emitted electrical current, and I, Ir, and I
are the optical intensities of the violet, red, and green light respectively. The constants a(), a(2), and a() describe the relative contributions of the one-, two-, and
three-photon components to the total current, and, similar to the non-linear susceptibilities we previously discussed, a0) >> a 2 ) >> a(3 ).
Paralleling our preceding
development, we can re-write Eq. (1.8) as
J=
(1
(v)
+ 0(2)
g
+
p(
(1.9)
Where in the above, O(n) = a()I . Here we have factored out a critical intensity,
Ic, and in this form the /(n) terms for many materials are of the same order, i.e.
31
CHAPTER 1. INTRODUCTION
#3(1)
~
(2)
,~
(3).
It is clear that when the optical intensity is less than the critical
intensity, I << Ic, the higher-order multiphoton terms will provide relatively small
contributions to the total photo-emitted current.
This is exactly what is observed
in conventional photoemission experiments. However, when the optical intensity approaches the critical intensity I ~Ic, or when the electric field of the optical wave
approaches the associated critical field E ~ Ec, we see that all the multiphoton orders
will become comparable; here we enter the strong-field photoemission regime, and as
we will see, the basic physical picture of photoemission changes.
Something not considered in the previous discussion is that the optical field actually wiggles the metallic binding potential. In the vacuum half-space the electric
field of the illuminating light creates a potential Um(x) - exE(t) where Um(x) is the
metal's binding potential, x is the spatial coordinate normal to the metal surface, and
E(t) is the electric field of the illuminating light' (see Figure 1-2). As the intensity
approaches the critical intensity 1, and the field strength grows toward Ec, the barrier
can become extremely distorted as drawn in Figure 1-2.
Near the critical field strength Ec, the potential barrier is collapsed to such an
extreme degree that electrons from the metal can tunnel through. We imagine that
this tunneling current will become significant when the barrier is collapsed for a sufficiently long duration such that a sizable number of electrons can traverse the barrier.
Assume that it takes an electron approximately the time
Tt
to tunnel through the
barrier, and let us define the oscillation period of illuminating electric field, i.e. the
cycle-time, to be
Tcyc.
Then if Ftun < 7cy,, we expect a sizable tunneling current. We
can quantify this condition by calculating the tunneling time rtss. However, tunneling
time is a tricky and historically debated quantity
[16, 17]. We will elaborate on how
to estimate the tunneling time in Chapter 2; however, for now we will just use the
result
Ttn =
V2mWF/eEo where E0 is the electric field deflecting the barrier. Using
this result and writing the cycle time
Tcyc
= 1/w where w is the angular frequency of
the optical field 7 , we find that the tunneling current becomes significant when
Ttun
y = -
rcyc
w
__-
2mWF
eEo
E
=
<1
(1.10)
2Up
Note that this ratio that defines when the optically-driven tunneling current becomes
6
For simplicity in this intuitive discussion, we are assuming that the field is constant in space.
This is a reasonable assumption if we are considering dynamics occurring only very close to the
metallic surface.
'This cycle time is the optical period divided by 27r. This definition is to maintain consistency
with the standard form of the Keldysh parameter [7, 18].
32
CHAPTER 1. INTRODUCTION
hvv
vacuum
U
----
e
o- -- +- -
UmJz)
----------
cxE(t)
Figure 1-2: Strong-field photoemission. At high field strengths, the strong
fields distort the metal's binding potential and results in electron tunneling
emission from the metal.
sizable is the same Keldysh parameter we defined earlier (note WF = Eb).
We
previously saw that this parameter defined a boundary where the energy scale of
the field, Up, began to compare to the energy scale of the material, Eb.
We see
here that the boundary to the strong-field regime also defines the boundary to the
optically-driven tunneling regime. In other words, when -y ~ 1, the physics of the
photoemission process begin to depart from the simple picture of electrons wiggling
and grabbing energy from the optical field and begin to resemble an electron tunneling
process in which the binding potential is deflected by the optical field and electrons
tunnel through.
This picture of optically driven tunneling is central to strong-field photoemission.
There are several interesting characteristics to this optically-driven tunneling regime I
that largely explain the tremendous interest in this physical process. In the coming
chapters we will elaborate theoretically and experimentally on these characteristics.
'in the following photoemission related discussions we will use the terms 'strong-field regime' or
'optically-driven tunneling regime' interchangeably; they both refer to the same extreme-field region
of photoemission.
33
CHAPTER 1. INTRODUCTION
Reaching the strong-field regime
1.2
Now that we have a sense for what defines the strong-field regime, let us explore how
we actually go about reaching the field strengths necessary for strong-field physics
like strong-field photoemission. As mentioned, it was the development of the first
laser that allowed researchers to reach the required peak optical powers and field
strengths to explore some of the first non-linear optical phenomena, namely secondharmonic generation. It has largely been the development of ultrafast, femtosecond
laser and amplifier technologies that have likewise provided researchers with the tools
to achieve the peak optical powers and field strengths necessary to explore the strongfield regime of non-linear optics. In Figure 1-3, we chart the progress of ultrafast laser
sources and amplifiers over the past few decades. In this section we provide a brief
review of some fundamental concepts in the treatment of ultrafast optical pulses as
.
well as some basic information on ultrafast laser amplifiers9
b
a
r lops
CPA
1021
1020
ps--
E
.1019
100 fs . .
10
p 1017
W
0J
010
fs -
107 W/cm
2
6isfs
1960
1970
1990
1980
2000
1960
2010
1970
1980
1990
2000
2010
Year
Year
Figure 1-3: Progress in ultrafast laser sources and amplifiers. a. The achievable minimum laser pulse duration as a function of year. The arrow indicates
the first demonstration of SHG [2]. b. The achievable maximum focused laser
2
7
intensity as a function of year. The 10 W/cm mark shows the intensity used
in the first demonstration of SHG [2]. Note the tremendous growth in achievable
intensity since the development of chirped pulse amplification (CPA). (Images
borrowed from Ref. [5] without permission).
1.2.1
Ultrafast pulses
For the purposes of this thesis, we consider only the electric field of an ultrashort
optical pulse and model this field as a slowly varying pulse envelope modulating a
rapidly varying optical carrier-wave.
Our model for an ultrashort optical pulse can
then be written as
9
For a more elaborate treatment on these subjects see Refs. [4, 19]
34
CHAPTER 1. INTRODUCTION
E(t) = E0 x P(t) cos(27rfct +
(OCEO)
(1.11)
where E(t) is the electric field of the pulse, E0 is the peak electric field, P(t) is
a normalized pulse envelope shape, f, is the central frequency of the carrier-wave,
and LCEO is a phase-shift called the carrier-envelope offset phase (CEP). The CEP
describes the displacement of the carrier-wave maximum, or field maximum, from the
pulse envelope maximum (see Figure 1-4). Note that the CEP defines the shape of
the optical electric field of each pulse.
'rFWHfM
Ewas
TR
=A
IfR
Figure 1-4: Ultrashort optical pulse train from a mode-locked laser. A single
optical pulse circulates in the laser cavity (sketched in the upper left) with a
circulation period TR. The pulse periodically leaks out of the cavity forming a
pulse train with temporal spacing TR = 1/fR. Each pulse contains an energy
Ep and has a full-width at half maximum duration of TFWHM
A train of optical pulses as emitted by a mode-locked laser is illustrated in Figure
1-4. In Figure 1-4, we see that a single optical pulse circulates in the mode-locked
laser's resonator with period TR (illustrated in the upper left of Figure 1-4). The pulse
leaks out and forms the illustrated train of pulses with each pulse copy separated by
the repetition rate period TR (the repetition rate frequency is fR = 1/TR). Each pulse
has an associated pulse energy Ep and an associated full-width at half maximum
duration rFWHM1 . From Figure 1-4, we also see that from pulse to pulse ,0 CEO
shifts by some fixed amount AZo. This shift is due to group and phase velocity
mismatch in the mode-locked laser resonator.
The frequency at which this phase
"0 The pulse duration will often also be written as just r and is conventionally defined as the fullwidth at half maximum of the intensity envelope (hence the seemingly odd location of the labeling
in Figure 1-4).
35
CHAPTER 1. INTRODUCTION
shifts is known as the carrier-envelope offset frequency fCEO, and can be defined as
(A/27r) x fR. The CEP and the fCEO will be important parameters in our
experiments (further discussion on their properties is provided in Appendix A).
fCEO =
It is worthwhile to make some brief comments on the frequency domain structure
of the considered ultrafast optical pulses and pulse trains. A sketch of the spectrum
for an isolated ultrafast optical pulse and a pulse train are provided in Figure 1-5a and
b respectively. Note that the spectrum of the isolated ultrafast pulse is clearly very
broad and the carrier-envelope phase appears as a constant, absolute phase across
the entire spectrum. For the pulse train the broad spectrum is broken into harmonics
appearing at the repetition rate fR. The fCEO becomes important as it provides an
offset to each of the comb lines comprising this frequency comb spectrum (more details
on time and frequency domain structure as well as WCEO and fCEO are provided in
Appendix A).
a
:.f
b
fR
xe~&
nfA+fcEo
Figure 1-5: Ultrafast optical pulses and pulse trains in the frequency domain.
a. The spectrum of an isolated ultrafast optical pulse with central frequency
f, and CEP WCEO. b. The spectrum of an ultrafast optical pulse train with
repetition rate fR, central frequency f,, and carrier-envelope offset frequency
fCEOConsidering the above definitions, let us now look at some typical parameters for
an ultrafast optical pulse train. Consider the femtosecond laser pulse train we will
use in our experiments in Chapter 4: the repetition rate is fR = 78 MHz; the pulse
energy is Ep - 0.2 nJ; and the pulse duration is 'FWHM ~ 9 fs. The average power
of this pulse train is then Pvg = fR x Ep = 15.6 mW. This relatively low power is
comparable to the average power emitted by many compact continuous-wave lasers,
22 kW.
e.g. laser pointers"1 . However, the peak power is Pp = Ep/TFWHM
Consider tightly focusing one of these laser pulses to a beam radius of wo
2.6 pm
(typical of our experimental conditions in Chapter 4). With this beam radius and
"This is not a coincidence. The laser pulse energy in a mode-locked laser is dictated by the energy
storage capabilities of the laser gain medium, so we expect these powers to be comparable [13].
36
CHAPTER 1. INTRODUCTION
the above peak power, we can estimate the peak optical intensity Ip and the peak
electric field E0 , of one of these laser pulses. We find"
~ 2E 7T'UO2/FWHM
2.1 x 10" W/cm2
--
+
Eo
1.3 V/nm
(1.12)
Although tremendous optical fields are achievable directly at the output of femtosecond mode-locked laser oscillators, the field strengths still generally fall short of the
strong-field regime levels (for our above example, 1.3 V/nm << 12 V/nm). The second major technological development that has given researchers the ability to reach
the strong-field regime is ultrafast laser pulse amplification, in particular chirped pulse
amplification. However, before considering ultrafast laser amplifiers let us take a brief
detour to consider just how fast ultrafast optical pulses are.
How fast is ultrafast?
For our purposes, ultrafast or ultrashort laser pulses are on the femtosecond scale. Although increasingly common in research labs and industrial applications, it is difficult
to grasp just how short a femtosecond is. The otherworldly nature of the femtosecond
is not terribly surprising considering that the average firing rates of most synapses in
our brains are only on the order of 10-100 Hz i.e. with a period of 10-100 ms [20]. To
get a sense for how small a femtosecond is, let us consider the following ratio' 3
8 fs
1 minute
1 minute
Age of the universe
With this sense of scale for the femtosecond, one is left wondering how we are able
to make lasers emitting such short pulses or why we might be interested in such an
absurdly small unit to begin with. The answer is found simply through some Planck's
constant gymnastics. Considering the energy-time uncertainty relation, AtAE - h,
we find that for At ~1 fs, AE ~ 0.7 eV. Energy-level spacings in molecules and
solids are generally on the eV-level, so accordingly, the dynamics of molecules and
solids occur in the femtosecond domain.
Additionally, by no coincidence, photon
energies of visible light are in the few eV regime, and therefore, the oscillation pe2
The factor of two in the Ip expression in Eq. (1.12) emerges for Gaussian beams. Additionally,
the relation used here (and for the rest of this thesis) between the intensity and the electric field
strength is the usual expression for plane electromagnetic waves: I = E 2 /2ZO where Zo is the
impedance of free-space, Zo = /po/co ~ 377 Q.
1 3 The age of the universe approximation comes from Ref. [21].
37
CHAPTER 1. INTRODUCTION
riod of optical electromagnetic waves must be on the order of a few femtoseconds. So,
although seemingly bizarrely small, the femtosecond is a very natural unit for interactions, transitions, and dynamics in molecules and solids. It is with these interactions,
transitions, and dynamics that ferntosecond laser pulses are made and a large reason
why (aside from the tremendous peak power they offer) ultrafast, femtosecond laser
sources are of great interest.
1.2.2
Ultrafast laser amplifiers
In addition to ultrafast laser pulses, the development of ultrafast laser amplifiers
have been critical to achieving the field strengths necessary for strong-field physics.
The most predominant form of ultrafast laser amplifier is the chirped pulse amplifier
(CPA). The basic mechanism behind chirped pulse amplification is as follows: CPAs
broaden ultrafast laser pulses to limit their peak power, amplify or build-up the pulse
energy of the resulting broadened pulses, and then re-compress the amplified pulses
into extremely high peak power ultrashort optical pulses [4, 19]. Typical commercial
CPA systems can provide millijoule pulse energies in laser pulses of around 20-30
fs.
This amounts in peak optical powers of Pp a 1 mJ/20 fs e 50 GW. Imagine
increasing the tightly focused 0.2 nJ laser pulse discussed previously to 1 mJ pulse
energy. This would result in a peak intensity of Ip
1018 W/cm
2
and a peak field of
E0 - 2800 V/nm, well into the strong-field regime!
Although chirped pulse amplifiers offer access to tremendous optical peak intensities and field strengths (see Figure 1-5), they are not without drawbacks.
Their
primary limitations come in two forms: repetition rate restrictions and complexity.
Firstly, chirped pulse amplifier systems reduce the repetition rate of the amplified optical pulse train. CPAs generally take a e 100 MHz optical pulse train from a modelocked oscillator seed and dramatically amplify the pulse train but also dramatically
reduce the repetition rate of the train to just a few kHz. This drop in repetition rate
is largely associated with average power limitations: a 1 mJ pulse train at 100 MHz
repetition rate would have an average power of 100 kW! This decrease in repetition
rate however dramatically limits the potential flux of the products of strong-field interactions. For example, if we are interested in using the high-harmonics produced
via a strong-field non-linear optical process as a source of short-wavelength light, then
this source will have a flux limited by this kHz repetition rate. The second major restriction of CPA systems is complexity. Looking at Figure 1-6a, although these CPA
systems are table-top in size, they can sometimes stretch the definition of a table-top.
38
CHAPTER 1. INTRODUCTION
b
a
a,
r. t
Figure 1-6: Ultrashort laser pulse amplifiers and resonators. a. Picture of a
typical commercial CPA laser system capable of generating multi-mJ femtosecond laser pulses in a strong-field physics laboratory (photo courtesy of P. D.
Keathley). b. Sketch on an ultrafast optical enhancement cavity. An optical
pulse circulates in the resonator with amplitude ares. It is excited by an incident
pulse train with amplitude aj.
An alternative route to achieving the field strengths necessary for the strong-field
regime while bypassing repetition rate restrictions or the complexity of these large
amplifier systems is therefore very desirable. To this end researchers have pursued
using passive optical resonators to achieve these strong-fields directly from the output
of mode-locked laser oscillators. The essential idea behind passive optical resonators is
illustrated in Figure 1-6b. In Figure 1-6b, we have a two mirror arrangement confining
an optical pulse. The pulse circulates between the two mirrors with amplitude ares.
If we assume in this simplistic treatment that one of the mirrors is a perfect reflector
and the other has reflectivity, r, and transmissivity, t, then we find that the pulse
amplitude in the resonator, are,, must be related to the incident pulse train amplitude
ai by
ares
=
-
rares + itai
it
i
ai
1 - r
(1.13)
(1.14)
Therefore, tremendous intra-resonator pulse amplitudes can be built up without active
amplification! The above analysis has obviously been a very simplistic one, and there
are many restrictions to passive pulse amplification with optical resonators. We will
discuss these throughout this thesis as we pursue strong-field physics with optical
39
CHAPTER 1. INTRODUCTION
resonators. In the following section, we first will give some details on applications of
strong-field physics and motivate interest in this resonator approach.
1.3
Strong-fields at the nanoscale and HHG
Thus far we have described the what and the how of strong-field physics. We have
defined the strong-field regime and provided a relevant example, and we have reviewed
how we can reach this strong-field regime with ultrafast laser pulses and amplifiers.
Additionally, we suggested the possible advantages in using optical resonators to reach
the necessary field strengths for strong-field physics. In this section we will build on
this discussion; we highlight the two application areas of strong-field physics that will
be the focus of this thesis, and we describe the utility of optical resonators in these
applications.
The first application area of interest is strong-field photoemission near nanostructures. As we mentioned earlier, when an optical field reaches a critical strength the
fundamental physical picture of photoemission changes.
In the strong-field regime
of photoemission, sub-optical cycle bursts of electrons can be ripped from material
surfaces.
These electrical bursts can be switched on and off with individual cycles
of the excitation laser pulse, and after emission these electrical bursts wiggle in the
driving laser field. They can be driven back into the material surface where they
can re-scatter and extract more energy from the laser field. The prospect of using
these sub-optical cycle electron bursts for scientific investigations of solid-state surface dynamics as well as the prospect of steering these electron bursts with optical
waves for technological purposes have made strong-field photoemission a hot research
topic in the past decade. In particular, nanostructure surfaces offer several particular
advantages to the strong-field photoemission experiments. One such advantage is the
field-enhancement present at a nanostructure. This field-enhancement eliminates the
need for repetition rate reducing, complex laser amplifiers. This field-enhancement
near plasmonic nanoparticles also takes on an optical resonator like character. Figure
1-7 illustrates the basic arrangement for strong-field photoemission from nanostructures and a representative dataset.
The second application area of interest is high-harmonic generation. As discussed,
when the optical fields approach a critical strength, the perturbative description of
non-linear optical processes breaks down.
In this strong-field regime familiar phe-
nomena like second-harmonic generation are replaced by non-perturbative phenomena. Illuminating gas jets [3, 4, 5, 6, 7] with high peak power, ultrafast laser pulses,
40
CHAPTER 1. INTRODUCTION
b
0.2 pi
0.2fp.
fsIgr0
2
4
G
8~
10
1
Electron energy (eV)
Figure 1-7: Strong-field photoemission near nanostructures. a. Strong-field
photoemission from plasmonic nanoparticles. A femtosecond laser pulse illuminates the nanoparticles, and they photo-emit electrons (illustration borrowed
from Ref. [22] without permission). b. Electron energy spectra measured from
strong-field photoemission experiment with differing excitation pulse energies
(picture borrowed from Ref. [23] without permission).
researchers have generated hundreds of harmonics. The process that leads to this
generation is creatively named high-harmonic generation (HHG).
The essential physics of HHG is closely linked to strong-field photoemission. HHG
can be understood as a simple three-step process [24, 25]. First, an intense laser
field results in strong-field ionization of an electron from a host atom. The emitted
electron, like those in strong-field photoemission, is sub-optical cycle in duration and
subsequently wiggles in the oscillating laser field acquiring energy. This electron can
be sent by the laser field back towards its parent atom where it can recombine and
emit a high-energy photon. From this simple physical picture of HHG, we predict
a broad spectrum of harmonics with a plateau-like structure (see example in Figure
1-8a).
The generated high harmonics have excited researchers as they might furnish
routes towards table-top, coherent sources of light in the hard-to-reach spectral region
of the extreme ultraviolet (EUV) or soft x-ray range. However, as we have mentioned,
the amplification necessary to achieve the required field strengths for HHG limits
the achievable flux of HHG based short-wavelength sources. In order to push the
repetition rate of HHG sources from the kHz regime of CPA systems to the 100 MHz
regime of mode-locked laser oscillators, passive optical resonators, i.e. enhancement
cavities have been employed (see Figure 1-8b). We will discuss the benefits and
drawbacks of these cavities and the general approach in subsequent chapters.
41
CHAPTER 1. INTRODUCTION
a
b
20
40i
60'
so
100
120
140
Out-coupWd high harmonics
160
Pluom~ ensrgy (sy)
Figure 1-8: Characteristics of HHG and cavity-enhanced HHG. a. Example of
an HHG spectrum. For this example, the driving laser light wavelength varies
but for the pink trace A = 1.8 Mm or the photon energy ~ 0.7 eV. Note the
large plateau of hundreds of harmonics; this plateau resembles the plateau in
the electron energy spectra from Figure 1-7b (this data was borrowed without
permission from Ref. [3]). b. Basic arrangement for cavity-enhanced HHG. An
ultrafast laser pulse is enhanced in a bow-tie ring cavity where high-harmonics
are produced and out-coupled via a sapphire plate (the pictured harmonics are
borrowed from Ref. [26] without permission).
1.4
Outline of this thesis
As a final section to this introductory chapter, we reiterate the purpose of this thesis
and outline the following chapters. The course of this thesis research has spanned
many projects and a wide range of topics. We have unified two of the main projects
in the broad category of strong-field physics with ultrafast optical resonators.
We
present work on the two main application areas outlined above. First, we use plasmonic nanoparticle optical resonators to explore strong-field physics near nanostructures. We will switch on and off photoemission currents from nanoparticle resonators with individual optical cycles of a femtosecond laser pulse. We will use this
photoemission current to characterize the femtosecond dynamics of the nanoparti-
cle's excited nanoplasmonic field and demonstrate carrier-envelope phase sensitivity
of the photoemission current. Second, we will explore novel optical resonator designs
for cavity-enhanced high-harmonic generation. We will design and prototype
optical enhancement cavities supporting Bessel-Gauss intra-cavity modes, and we
will demonstrate that such cavity modes might allow the major obstacles to efficient
cavity-enhanced HHG to be overcome.
The outline of the following chapters is as
follows:
42
CHAPTER 1. INTRODUCTION
Chapter 2 -
Theory of photoemission from solid surfaces
The main goal of this chapter is to develop a qualitative physical picture and
quantitative model for strong-field photoemission. We first consider some fundamental elements of electron emission; in particular, we discuss field emission
and photo-assisted field emission.
We develop a basic theoretical framework
to study photoemission, and we analyze photoemission rates in the multiphoton and strong-field regimes. We additionally consider the time-dependence of
strong-field photoemission currents and the carrier-envelope phase sensitivity of
the strong-field photoemission process.
Chapter 3 -
Plasmonic nanoparticles and optical resonators
Here we discuss the fundamentals of plasmonic nanoparticle resonators. We begin with an intuitive review of surface plasmons and build a conceptual physical
picture of the operation of nanoparticle resonators, and we construct a simple
circuit model that can accurately describe and predict the response of plasmonic nanoparticles when excited by femtosecond optical pulses.
Lastly, we
mention some details of our nanoparticle fabrication procedure, describe the
overall layout of our nanoparticle emitter chips, and present fits of our nanoparticle properties to the simple, second-order circuit model.
Chapter 4 -
Photoemission from plasmonic nanoparticles
In this chapter we present the experimental results from our investigations into
on-chip strong-field photoemission from plasmonic nanoparticle emitters. We
motivate our results by providing some background on strong-field physics with
nanostructures.
We then move to provide details regarding the experimental
system: we discuss the femtosecond laser source, the laser pulse measurement
system, the carrier-envelope phase stabilization technique, and the device alignment microscope. We then discuss basic scaling properties of the photoemission
from the nanoparticles and demonstrate signatures of the strong-field regime.
We use the strong-field photoemission current to perform interferometric autocorrelations and characterize the nano-plasmonic, ultrafast optical field on
the nanoparticles.
Lastly, we demonstrate the carrier-envelope phase sensitiv-
ity of the photoemission signal and develop a simple model for predicting this
carrier-envelope phase response.
43
CHAPTER 1. INTRODUCTION
Chapter 5 - Enhancement cavities for high-harmonic generation
In this chapter we outline the basics of cavity-enhanced high-harmonic generation. The essential motivation for enhancement cavities for strong-field physics
is discussed, and the concept of passive amplification inside an ultrafast, femtosecond enhancement cavity is reviewed. We discuss the requirements for an
effective enhancement cavity for HHG and the challenges associated with the
design of such a cavity.
Chapter 6 -
Bessel-Gauss beams
Here we introduce Bessel-Gauss beams. We construct Bessel-Gauss beam solutions by superposing familiar Gaussian-like beams. We describe the focal and
far-field properties of Bessel-Gauss beams and illustrate their advantages for
strong-field enhancement cavities. Lastly, in preparation for design of such cavities with Bessel-Gauss modes, we analyze how Bessel-Gauss beams transform
as they traverse simple optical elements.
Chapter 7 - Bessel-Gauss beam enhancement cavities
In this chapter we build upon the properties of Bessel-Gauss beams to analyze Bessel-Gauss beam enhancement cavities. We consider the overall design principles of Bessel-Gauss beam cavities and how to build such cavities
from fundamental Gaussian designs. We discuss a specific design: the confocal Bessel-Gauss cavity and possible arrangements for high-harmonic generation applications. We discuss a continuous-wave experimental demonstration of
the confocal Bessel-Gauss cavity and highlight the limitations of Bessel-Gauss
modes illuminated by this demonstration.
44
Chapter 2
Theory of photoemission from solid
surfaces
In the next three chapters, we will discuss strong-field photoemission from nanostructures, in particular plasmonic nanoparticle resonators. We begin the discussion
in this chapter with an introduction to the theory underlying photoemission from
solid surfaces. Our main goal will be to develop a qualitative physical picture and
quantitative model for photoemission in the strong-field regime. Towards this end,
we first look at some fundamental elements of electron emission. We discuss the
basics of field emission i.e. electron emission from a material in the presence of a
static bias field; we briefly discuss photo-assisted field emission mechanisms; and we
consider the basic theoretical formalism behind photoemission. We build on this discussion to analyze the basics of strong-field perturbation theory, and we calculate
average photoemission currents in the strong-field regime. We carry out these calculations for illumination with an ultrashort optical pulse and with a continuous-wave,
monochromatic excitation. Next, we discuss the time-dependence of photoemission in
the strong-field regime and develop a simple quasi-static model. Lastly, we consider
carrier-envelope phase effects in strong-field photoemission. We briefly discuss their
origins and present a simple model to predict their strength.
2.1
Electron emission fundamentals
Electrons can be coerced to leave solid surfaces by a variety of means. As mentioned
in the previous chapter for the purposes of this thesis we will be concerned with metallic surfaces, in particular gold surfaces. In the previous chapter, we briefly discussed
45
CHAPTER 2. THEORY OF PHOTOEMISSION FROM SOLID SURFACES
a particular electron emission mechanism, multiphoton photoemission, in which electrons in a metal can absorb optical energy and hop over their binding potential and
out of the metal and into vacuum. In addition to this optically-driven photoemission
mechanism, many other physical mechanisms for electron emission exist. We divide
these emission mechanisms into three logical categories: field emission, photo-assisted
field emission, and photoemission.
These three categories of electron emission are divided according to whether they
rely on a static bias field, a static field and an optical field, or just an optical field.
The field emission mechanism relies on a static bias applied to the metal surface
that allows electrons to tunnel from the metal and into vacuum. The photo-assisted
field emission mechanism involves a mixture of optical and static field effects. For
example, in photothermal emission an optical source heats the electrons in a metal,
and these hot electrons then field-emit from the metal surface in the presence of a
static field. Lastly, the photoemission mechanism, as has been discussed, involves an
optically driven form of emission. In the following, we discuss each of these electron
emission mechanisms, and we build towards an understanding of photoemission in
the strong-field regime.
2.1.1
Field emission
In the field emission regime, static fields lead to electron emission from a metal
surface.
The static electric field deflects the electron's binding potential such that
electron's can tunnel from their host material. We can make an extremely simple, yet
insightful, model for this phenomenon. We can model the metal's binding potential at
the surface as a simple rectangular step (as we did for our discussion of multiphoton
photoemission in the previous chapter).
We can then write the time-independent
Schr6dinger equation
h2
d2
2
2m dx + (WF + EF - eXEstat)U(X)
In the above,
#(x)
O(x)
= EF(x)
(2.1)
is the wavefunction for a single electron in the metal at the Fermi
energy EF, WF is the work function, and u(x) is a step-function. Additionally, as
before -e
and m are the electron charge and mass 2 respectively. The coordinate x
denotes distance in the direction normal to the metal's surface, and Estat denotes the
'We define the step-function as u(x) = 1 if x > 0 and u(x) = 0 if x < 0.
2
Note that we use the bare electron mass, m, in Eq. (2.1) instead of the effective mass, m*U, in
gold. The reason for this will be elaborated upon in a later section.
46
CHAPTER 2. THEORY OF PHOTOEMISSION FROM SOLID SURFACES
static applied bias field'. The basic model and wavefunction for the Etat = 0 and
Estat = Eo = 10 V/nm cases are illustrated in Figure 2-1. Note that from the form
of Eq. (2.1), we are assuming there is no field penetration in the metal. Here and in
the following, we use the normal parameters for a gold surface [14, 27], EF = 5.5 eV
and WF
=
5.1 eV.
b
a
vacuum
vacuum
5
-5(x)
S
-
E= 10 V/-
W?
--
-
EF
----
-51
-5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
X (1m1)
:r(1111)
Figure 2-1: Field emission and models. a. Basic model for metallic surface.
The interior of the metal is treated as a free electron gas, and the metal surface
is modeled as a simple rectangular step of height WF. As before, the metal
potential is denoted as Urnm(x) with x the coordinate normal to the surface.
The real part of the electron wavefunction at the Fermi energy is drawn in red
(calculated numerically). b. Potential with an applied static bias. With an
applied bias the rectangular step is deflected, and electrons can tunnel through
the barrier and into the vacuum. The real part of the wavefunction is again
drawn in red (numerical calculation).
With Estat = 0, Eq. (2.1) can easily be solved, and the wavefunction is found
to be oscillatory in the metal; it is composed of forward and backward plane wave
components reflecting off the potential step.
Under the barrier the wavefunction
takes on an evanescent form. Under the barrier (for x > 0) the wavefunction takes
the general form
#(x) oc exp(-ax)
with
3 We
a=
2mWF/h
(2.2)
should make a short note concerning notation. Tragically, the poor choice was made to use
the letter E for both energies and for electric field. Hopefully, context will indicate which quantity
we are discussing.
47
CHAPTER 2. THEORY OF PHOTOEMISSION FROM SOLID SURFACES
This general behavior is illustrated in Figure 2-la.
With an applied static field,
(2.1) becomes more challenging to solve analytically. However, we can use
Eq.
the Wentzel-Kramers-Brillouin (WKB) approximation to estimate the wavefunction
under the barrier as
#(x) oc exp
-
dx'P(x'))
with
p(x) =
2m,(WF - exEstat)/h
(2.3)
The above form of the wavefunction only applies to the under-barrier region i.e.
0 < x < d where d is the barrier exit (or tunnel exit).
The essence of the WKB approximation used above is to separate the problem into
two length scales. The relatively large length scale over which variations in V(x) =
WF + EF - exEstat occur, and the far shorter length scale defined by A = h/p(x).
Since this wavelength A is generally very small compared to the scale of variations in
V(x), the potential can be approximated as quasi-static, and the above form of the
wavefunction is found. The general intuition behind this quasi-static concept will be
applied in future sections to analyze the time-dependent behavior of photoemission.
Using the WKB form of the wavefunction, we can estimate the energy-dependent
barrier transmission, T(E), by considering the wavefunction at the barrier or tunnel
exit
T(EF) c
(d) I2 cx exp
-
dx N2m(WF
-
exEstat)
(2.4)
In the above, d again refers to the tunnel exit, and T(EF) is the barrier transmission
at the Fermi energy. The above integration can easily be performed, and we find
T(EF) oc exp
where Et,, = 4/2mWF/3he
-tun
(_Estat)
(2.5)
(~ 78.7 V/nm for our parameters) is a characeristic
tunneling field strength. The form of T(EF) gives a sense for the extreme exponential
nature and sensitivity of the tunneling process. A complete field emission tunneling
current can be calculated from the above transmission function by integrating the
transmission of all the electron states in the metal weighted by their incident electron
velocity and probability of occupation. Carrying out this operation [28], we find the
well-known Fowler-Nordheim equation for the tunneling current as a function of the
applied static field
48
CHAPTER 2. THEORY OF PHOTOEMISSION FROM SOLID SURFACES
Jitn(Estat) =
2
( Estat
(Etun
exp
)
Eun
(2.6)
(Estat
The above simple model and the Fowler-Nordheim equation have been used extensively to model field emission devices in their many applications. In later sections we
will make use of this simple model to trace the time dynamics of strong-field photoemission. However, before proceeding towards this goal and our discussion of other
emission mechanisms, we look at one last quantity of interest that we can extract
from this simple static model: the tunneling time.
Tunneling time
Loosely defined, the tunneling time here refers to the duration the electron takes to
traverse the barrier. Tunneling time is a tricky and debated quantity however with this
simple definitition in mind, we can formulate an esimate for this time via our WKB
expression for the wavefunction [16, 17]. Previously, we found that under the barrier
the electron momentum in out model takes the form p(x) = -/2m(WF
-
exEstat).
We can estimate the velocity under the barrier as v(x) = p(x)/n, and accordingly
we can crudely estimate the time it takes for an electron to traverse the barrier as
fdx
n
To
v(x)
0
d
n(2.7)
2(WF - ex Estat)
e Estat
In the above, d again refers to the tunnel exit position, and we see that we obtain the
.
tunneling time used in the previous chapter4
2.1.2
Photo-assisted field emission
The next category of electron emission processes involves a mixture of static and
optical field effects. These emission mechanisms generally involve two basic steps.
In the first step, light excites the electrons in a metal in some way. At their higher,
excited energy levels, the electrons see an effectively lowered work function, and, when
a static field is applied, they field-emit out of the metal. In Figure 2-2 we illustrate
two common photo-assisted field emission mechanisms: photo-thermal emission and
photo-field emission. In the following, we briefly discuss these mechanisms.
4We should reiterate that this tunneling time derivation is not
intended to be rigorous, but
intuitive. For more thorough treatments and discussion of tunneling time see Refs. [7, 16, 17].
49
CHAPTER 2. THEORY OF PHOTOEMISSION FROM SOLID SURFACES
hv,
hv,
a
b
vacuum
vacuum
5
F4)= 10
E0
10 V/11111'
----------VV VV V ---
----------------- ---- -- ------
3hv
0
Ep
EF
---------------------------------- --------------- -------------------
-5
-2
ft
-------------------------- n TH.t.
-1.5
-1
-0.5
-5
0
0.5
1
1.5
-2
x (niii)
J
-1.5
-1
-0.5
0
0.5
1
1.5
x (nimi)
Figure 2-2: Photo-assisted field emission. a. Photo-thermal emission. An
optical excitation locally heats the electron gas in a metal (see the lightly shaded
region labeled THot). This heating results in a stretched Fermi-Dirac distribution. Higher energy levels are now increasingly thermally occupied and can
undergo field emission. The real part of the electron wavefunction at 2 eV above
the Fermi energy is drawn in red (calculated numerically) with a static field
of Estat = E0 = 10 V/nm. b. Photo-field emission. An optical excitation
resonantly excites electrons to higher energy levels, and these excited electron
field-emit. The real part of the electron wavefunction at 3 eV (three photons
at a wavelength of 1.2 pm) above the Fermi energy is drawn in red (calculated
numerically) with a static field of Estat = E0 = 10 V/nm.
Photo-thermal emission
As mentioned, in photo-thermal emission an optical excitation heats the electron gas.
In our simple model we can think of the Fermi-Dirac distribution broadening dramatically in response to the heating (see Figure 2-2a). At the high optical field strengths
achievable with an ultrafast laser pulse, this heating can be dramatic. The electron
gas can be heated to thousands of degrees K before thermalizing with the lattice over
hundreds of femtoseconds (note the electron temperature is distinct from the bulk
temperature of the gold i.e. the lattice temperature, so the electron temperature can
be thousands of K with no apparent damage or melting occurring). When a static
field is applied this hot electron gas can result in large field emission currents. These
currents should occur over the duration of the electron gas thermalization and should
be extremely sensitive to the applied static bias due to the field emission nature of
the process [29]. Although important to mention, photothermal emission will not
50
CHAPTER 2. THEORY OF PHOTOEMISSION FROM SOLID SURFACES
play a major role in our experiments as will be evidenced by the lack of static bias
sensitivity mentioned later in Chapter 4.
Photo-field emission
Photo-field emission is similar to photothermal emission. In photo-field emission, incident photons excite electrons in the metal to higher energy levels, and these electrons
subsequently field-emit in the presence of a static bias field. This emission mechanism
is distinct from photo-thermal emission in that it involves the generation of a highly
non-equilibrium electron distribution. In other words, electrons are optically excited
and field-emit before the electron gas can thermalize into a hot Fermi-Dirac distribution. This emission mechanism can also involve the resonant excitation of electrons
between bands. In Figure 2-2b, the photo-field emission process is illustrated with a
three-photon, 3 eV excitation. This gap actually closely corresponds to an available
transition in gold between the d-band and the sp-band [30]. As for photo-thermal
emission, we expect that photo-field emission will be very sensitive to the applied
static bias due to its field emission nature.
As we mentioned, we will largely rule
out photo-field emission due to the relatively weak static field effects we will observe
in our experiments in Chapter 4; however, future studies of photo-field emission are
warranted.
2.1.3
Multiphoton and strong-field photoemission
Our final category of electron emission is photoemission. As we have mentioned, photoemission involves the optical exciation of electrons from the metal, and in photoemission, static bias field effects are relatively weak. In the following, we first formulate
the basic framework for the calculation of photoemission currents. Next, we discuss
photoemission currents and calculation techniques in the multiphoton regime and in
the strong-field regime.
Photoemission involves a time-dependent, optical excitation.
Therefore, all the
interesting behavior will emerge from the time-dependent Schrddinger equation
i
)= H(t) 10)
(2.8)
In the above our state ket is 10). We should mention that in the following we will be
rather loose with our state definitions and bounce between the spatial or momentum
domain wavefunction representations. Additionally, our problem will be entirely one-
51
CHAPTER 2. THEORY OF PHOTOEMISSION FROM SOLID SURFACES
dimensional, and we will denote operators with a hat (e.g.
k
is the momentum
operator). Our time-dependent Hamiltonian is written as H(t). We consider again
the simple rectangular step potential from the preceding discussions. In this case our
Hamiltonian can be written as the sum of several parts
H
K+V +-F
p2
-
2m
(2.9)
(WF + EF)u(-x)
-
exE(t)
(2.10)
In the above, K corresponds to the kinetic portion of the Hamiltonian, V corresponds
to the potential component (note that to simplify later calculations, we have shifted
V from our earlier Hamiltonian such that the vacuum level is set to an energy of
zero), and F corresponds to the field component. Note that the field component is
written in the length gauge with the dipole approximation 5 . Now denoting the time
evolution operator associated with H(t) as U(t, to), we can write a simple expression
for the time evolution of an initial state 1#) (see Appendix B)
|V)(0)) = 1#( )) -
(2.11)
dt' U(t, t') F(t') #(t'))
-
In the above, the state J(t)) starts in the initial state 1#) at time t = to. Additionally,
1#(t)) denotes the time evolution of the state 1#(to)) in the absence of F (see Appendix
B). For the photoemission problem we are considering, the wavefunction of this initial
state is given by the O(x) expression we found in Eq.
(2.2).
We can then also
easily find q(t)) by incorporating the time-dependence into O(x) in the absense of F.
Incorporating this time-dependence into the wavefunction, we find
0(x, t) oc exp(-ax + iWFlh)
(2.12)
We are interested in excitations from this initial state to higher energy states that are
not bound to the metal; we are interested in the emission of electrons. This excitation
is represented by the evolution of 1#) in the presence of F to 10).
To analyze the
excitation of 1#) into these higher energy unbound states, we take the overlap of
the time evolution of 1') with a family of such unbound final states that we denote
'Also, note that unlike our previous Hamiltonian, in Eq. (2.10), our time-dependent field now
penetrates the metal (there is not a step-function multiplying the field component). This follows
because, as we will see, we are only concerned with the x > 0 region, so omitting the step-function
is unimportant.
52
CHAPTER 2. THEORY OF PHOTOEMISSION FROM SOLID SURFACES
as {IXa)} where a is some continuous or discrete label.
This overlap gives martix
elements describing the excitation
AMa(t)
(2.13)
(Xa(t)#(t))
=
-t
dt' (Xa|I U (1, i') F(t') #(t'))
(2.14)
In the above, we have shifted the initial time such that to = -oc.
Additionally,
we have assumed the initial state is orthogonal to the family of excited states, i.e.
(Xa|1) = 0. The matrix elements Ala describe the amplitude of the transition from
the intial state 1#) to the unbound, excited state IXa); however, we are interested in
transition rates or currents. We can define a transition or exctiation rate or current
in the usual way
1
t-+oo t
Pa = lim - Ma(t)
2
(2.15)
In the above, Fa is the photoemission rate from the initial state 10) to the final,
excited state IXa).
We have now established a basic framework for considering photoemission, and
thus far, all our developements have been exact. We now turn to approximate methods, in particular time-dependent perturbation theory, to provide us with an estimate
for the time evolution operator U(t, t') in the above equations and point us towards an
appropriate set of final states,
{ IXa) }. In the following, we look at this perturbation
theory for two particular examples of photoemission under different approximations.
In particular we consider multiphoton and strong-field photoemission.
Multiphoton photoemission
In the regime of multiphoton photoemission the essential approximation we make is
that the field component of the Hamiltonian is small. We treat the field portion of the
Hamiltonian as a perturbation. We can then rewrite our photoemission Hamiltonian
in Eq. (2.10), as
53
CHAPTER 2. THEORY OF PHOTOEMISSION FROM SOLID SURFACES
H
K-+V+F
=
(2.16)
(WF + EF)U(-X)
_
2m
Ho
-
+
exE(t)
(2.17)
F
(2.18)
+ V. Now treating F as a small quanity, we can
Where we have defined Ho as K
write a perturbative expansion for the evolution operator (see Appendix B)
U(t, to)
Uo(t, to) - i
dt'Uo(t, t')F(t')
(2.19)
to
dt'
dt"Uo(t, t')F(t')Uo(t', t")F(t")
+
-
In the above expression, we make use of the unperturbed time evolution operator
Uo(t, to). This is the time evolution operator for the unperturbed Hamiltonian Ho.
Now before proceeding we must define the form of the excitation field.
Using a
continuous-wave, monochromatic field for simplicity we can write
F(t) = -exEo cos wt
(2.20)
With this field Hamiltonian, we now have a complete problem we can solve. However,
we still have two decisions to make before we proceed. First, we must decide on a
suitable set of final states to consider emission into, and second, we must decide how
to truncate our expansion for the evolution operator in Eq. (2.19).
The first decision is rather straightforward.
Since we are considering the field
portion of the Hamiltonian to be weak, we will just consider emission into eigenstates
of the unperturbed Hamiltonian, Ho. We can call this family of eigenstates { #q)}
where q is a continuous parameter denoting the momentum of the states. The initial
wavefunction and a candidate final wavefunction are illustrated in Figure 2-3.
The second decision actually makes itself.
If we consider the actual expansion
in Eq. (2.19) under the rotating wave approximation6 , we find that a single term
dominates. This term is the n - 1 term where n is defined as the lowest term such
that nhw - Eq where Eq is the energy of 10q). This term involves n - 1 integrations
6 For a more elaborate description
of the various approximations that go into multiphoton pho-
toemission calculations see Refs. [13, 15].
54
CHAPTER 2. THEORY OF PHOTOEMISSION FROM SOLID SURFACES
1210-
2- -EF
-2
-4
-0.5
0
0.5
1
1.5
2
xi (ii)
Figure 2-3: Multiphoton photoemission. The rectangular step potential is
illustrated in black. We see it wiggles only a small amount in response to the
relatively weak field E(t) (see dashed). The real part of the initial wavefunction
#(x, t) is drawn in red, and the real part of a potential final state (that is also
a eigenstate of the unperturbed step potential) is drawn in blue. This state
has an energy of 3 eV above the drawn vacuum level (both wavefunctions are
numerically calculated).
and accordingly the resulting matrix element is proportional to E01. Making use of
this term to then calculate the total photoemission rate, we find that
fMP
oc0
(2.21)
This general form the multiphoton photoemission rate, rMP, is exactly of the form
we described in the previous chapter. Note that in deriving this form, we relied on
the assumption that the field component of the Hamiltonian is small relative to the
potential and the kinetic components. When this assumption breaks down our above
expression will fail. We will explore this regime in the following.
Strong-field photoemission
As we discussed in the previous chapter, in the strong-field regime the field strength
becomes such that the characteristic energy associated with the field is stronger than
the binding potential itself. In this regime, we must shift the approximations made in
the multiphoton case. In the multiphoton case we treated the field as weak relative
to the potential. Here we treat the potential as weak relative to the field. We can
then split our Hamiltonian as
55
CHAPTER 2. THEORY OF PHOTOEMISSION FROM SOLID SURFACES
H =
=
K+F+V
(ex
2m
=
'4
(2.22)
E(t))
(2.23)
- (WF + EF)U(-X)
V
HF
(2.24)
Where we have defined HF as the 'free-field' Hamiltonian K + F. Now treating V as
a small quanity, we can write a perturbative expansion for the evolution operator
U(t, to)
=
UF(t, 0o) -
I
hto
dt'
-
(2.25)
dt'UF(t, t)V
dt"UF (t, t')VUF(t
+
..-
In the above, we use the evolution operator UF(t, to). This is the evolution operator
associated with the free-field Hamiltonian, HF, only. The above perturbative expansion is very interesting. We can diagrammatically interpret each term as shown in
Figure 2-4.
1
2
+
+
0
Figure 2-4: Strong-field perturbation theory. The terms in Eq. (2.25) can be
interpreted via these simple diagrams.
In Figure 2-4 we see each term in the expansion corresponds to an interaction, or
scattering event, with the binding potential. The zeroth order term corresponds to
no interaction with the potential, V; the first order term corresponds to a single
interaction; and the second order term corresponds to two interactions. Here we are
interested in only the total current, so we make an approximation and only consider
the direct electrons i.e. we only consider the zeroth order term (more detailed analysis
shows that the higher order terms are primarily of interest in characterizing the energy
spectrum of the emitted electrons [7]). In other words, we make the approximation
56
CHAPTER 2. THEORY OF PHOTOEMISSION FROM SOLID SURFACES
U (t, tO) ~_ UF (t, tO)
(2.26)
We totally ignore the binding potential, and only treat the field component of the
Hamiltonian in the evolution operator. The binding potential only plays a role in
setting the initial state. Following Eq. (2.11), we can then write an expression for
our final state as
=|(t))
q5t)-
t
]dt'UF-(t, t')F(t')
|5(t'))
(2.27)
Additionally, like in the multiphoton case we must choose a final set of states to
consider emission into. Since we are assuming the field is very strong compared to
the potential, we want to choose our family of final states to be the exact eigenstates
of the HF Hamiltonian. These eigenstates of the free-field Hamiltonian are known as
Volkov waves (see Appendix C), and in the length gauge, we can write the Volkov
wave as
q =
In the above, the ket Iq
q + eA(t)) exp(-iS(t)/h)
(2.28)
+ eA(t)) represents the free electron state with momentum
q + eA(t), i.e. the wavefunction of this state is exp(i(q + eA(t))x/h). Additionally,
S(t) denotes the is defined as ft dt'(q + eA(t')) 2 /2m. In Figure 2-5, a length gauge
Volkov wave is illustrated showing the basic mechanism of our strong-field perturbation theory calculation.
Using these length gauge Volkov waves as our set of final states and inserting them
into the expression for our matrix element, we find
(t)
/
=
-
f
dt' ("
(t) UF(t, t')F(t') |#(t'))
dt' K4' (t')| F(t') |#(t'))
dt' (q + eA (t')I exF(t') |#(t')) exp(iS(t')/h)
(2.29)
(2.30)
(2.31)
In the above we have noted that UF is the evolution operator for the Volkov waves,
and we have re-labeled our matrix element as Mq denoting emission into the Volkov
state with momentum q.
Unlike the photoemission case, it is not directly visible what the general form of
57
CHAPTER 2. THEORY OF PHOTOEMISSION FROM SOLID SURFACES
12-
--- -- --- -- --- --
10
4E
0
-0.5
= 10 V/1111
-EF
0
0.5
1
1.5
2
.r (iiiii)
Figure 2-5: Strong-field photoemission. The rectangular step potential is illustrated in black. We see it wiggles dramatically with the strong field E(t) with
Eo = 10 V/nm (see dashed sketch). The real part of the initial wavefunction
O(x) is drawn in red (numerically calculated), and the real part of a potential
final state (that is a length gauge Volkov wave) is drawn in blue. This state has
an energy of 3 eV above the drawn vacuum level and an additional pondermotive
energy of 1.8 eV.
this matrix element or the resulting emission rate I'SF will be. In the following we
will calculate numerically the above matrix element and develop an analytic approximation to formulate a closed form expression for photoemission in the strong-field
regime.
2.2
Strong-field photoemission
Having established a basic framework and intuition for photoemission calculations,
we now proceed to calculate photoemission currents in the strong-field regime. In
the following, we proceed along the lines outlined above and calculate the matrix
element and emission rate for strong-field photoemission assuming excitation via an
ultrafast optical pulse and a continuous-wave, monochromatic excitation. However,
before describing these calculations, let us first consider what we expect.
We will be looking at emission current versus optical excitation field. As we have
described, we expect that at some critical field strength the photoemission current
will depart from the multiphoton regime and eneter the strong-field regime. We
expect this critical field to be associated with a Keldysh parameter of around one. In
58
CHAPTER 2. THEORY OF PHOTOEMISSION FROM SOLID SURFACES
other words, when the tunneling time shrinks below the cycle time, that is when the
optically driven tunneling current begins to dominate, we expect a departure from the
multiphoton regime. Therefore, above the multiphoton regime we expect a current
versus field scaling comparable to that for the field emission case. Therefore we expect
the photoemission current to resemble the behavior of the Fowler-Nordheim equation.
This is illustrated in Figure 2-6.
104
102
-Ivacuum
100P
10-21
4-J
10
U
10
C:
iacuum
0
108
U
10~
7F.
1012
101
100
101
Figure 2-6: Multiphoton and tunneling emission. At low fields (high -y),
we expect the photoemission current to resemble the multiphoton scaling i.e.
oc E02 (drawn in red). Above a critical field (-y
1) and for low -y, we expect
the photoemission current scaling to resemble the Fowler-Nordheim equation
(drawn in black). The parameters used above are for gold being illuminated by
light with a wavelength of 1.2 4m.
2.2.1
Ultrafast optical pulse emission
Let us first consider the strong-field photoemission from an ultrafast laser pulse. The
laser pulse can have an arbitrary shape, E(t), as our methods will rely on numerical
calculation. Let us consider the matrix element of interest
59
CHAPTER 2. THEORY OF PHOTOEMISSION FROM SOLID SURFACES
-
Mq(t)
ft
dt'(q+eA(t')|exF(t')|#(t'))exp(iS(t')/h)
=
h
-
dx
dt' exp(i(q + eA(t'))x/h)exF(t')#(x) exp(iS(t')) (2.33)
dx
dt'+a (exp(i(q + e A(t'))x/h)) q(x) exp(iS(t'))
Where in the above we have used
dence of 10(t)).
(2.32)
5(I)
(2.34)
= (S(t) + WFi)/h to remove the time depen-
Reordering the above integration and inserting the expression for
Oo(x), we find
Mq (t)
dt' exp(i(t'))
-
-
j
t dt'exp(iS(t'))( eFt)
_Q
dx exp(-zx - i(q + eA(t'))x/h)
(2.35)
(q +eA (t')) +a)
This matrix element can then be numerically calculated.
Figure 2-7.
2
The results are shown in
From Figure 2-7, we see that the strong-field photoemission current'
closely resembles our expectations. At low fields it follows the multiphoton scaling,
and at high fields it traces the tunneling emission curve. Note that additionally in
Figure 2-7 we include an analytical emission rate that we will calculate in the next
section for a continuous-wave excitation.
The current for the ultrafast pulse very
closely resembles this monochromatic calculation.
2.2.2
Continuous-wave emission
For the case of continuous wave emission, an analytical emission rate can be derived
as we us a far less general form of the electric field. Here we assume
E(t) = Eo cos(wt)
A(t) =-
sin(wt)
(2.36)
With this simple field we can write an explicit form of the Volkov wavefunction by
carrying out the integration in the S(t) term. We find
'It should be noted that in the short pulse scenario the matrix element leads directly to the
actual emission rate as we assume each pulse arrives at some fixed interval, and therefore, the total
integration over all time of this matrix element corresponds to the emission probability for that
interval and is accordingly proportional to the actual emission current.
60
CHAPTER 2. THEORY OF PHOTOEMISSION FROM SOLID SURFACES
10
cw (analaytical)
Ac= 1.2 gm, r =9.5fs
102
10 0,
100
10
U
10-
I
10
1010.1
10-12I
I
10o1
10-1
100
Figure 2-7: Strong-field photoemission with an ultrafast pulse. The photoemission current is plotted for an ultrafast pulse with a central wavelength
A, = 1.2 pm and a pulse duration T = 9.5 fs (WF = 5.1 eV). The pulse is of a
cos 2 shape. Additionally, the photoemission rate for the continuous wave case
with a wavelength A = 1.2 pm is plotted for comparison.
S(t)
=
=
dt'
q2
\2m
-
2m
+
e 2 E2
)t
4mw 2 J
- sinwt')
W
+
qeE0
mw 2
(2.37)
e 2 E2
cos wt -
8mw 3
sin 2wt
(2.38)
We can then write the Volkov wavefunction with momentum q in the length gauge as
0"(x,t) =
q
exp
+
(i
h
mW 2
eEo
-sinwt
Cos Wt -
q2
)W(772m
x-
8mw 3
+
e2 ES
4mw 2
t
sin 2wt
With this expression for the Volkov waves, we can write our matrix element as
61
(2.39)
CHAPTER 2. THEORY OF PHOTOEMISSION FROM SOLID SURFACES
Mq (t)
dt' exp(iS(t')) (q + eA(t') I eEo cos wt 1#)
=
j
=
dt'exp i(t')+
_=
+UP+WF
h
(2.40)
) x
2m
L,(q~-io"'(2.41)
In the above we have again used S(t) = (S(t)+WFt)/h to remove the time dependence
of 1#(t)), and in the second line we have expanded the matrix element into a Fourier
series with Fourier coefficients L,(q). These coefficients are defined as
L,(q)
9
=
27
2ir/w
dt' (q + eA(t')I exEo cos wt'|#) x
0
exp
iwnt' + h-
,
e 2 E02
w')
cos
wt
mrjw2
'
8mIW3 sin 2L t
, qe Eo
(2.42)
From the simple form of Aq(t), we can massage the total photoemission rate into the
familiar form
,q
|Ln(q)126
X
+ Up + WF - nhW
(2.43)
n(2
In the above, we see that to calculate the total photoemission rate, we must find
Ln(q). Before proceeding to this calculation, we should briefly review the physics
contained in Eq. (2.43). Firstly, the general form of this equation looks familiar.
It resembles Fermi's Golden Rule. The L,(q) function resembles the perturbation
matrix element in Fermi's Golden Rule, and it is followed by a delta-function enforcing
energy conservation. To obtain the total emission rate, we sum over all the possible
multiphoton transitions and final state momenta that conserve energy. An interesting
point to note is that using the free-field, Volkov states as our final states has resulted in
the ponderomotive potential appearing in the delta-function. So, to make a transition
from our initial state to a final Volkov state, we must not only provide the energy to
surmount the binding potential and reach the kinetic energy of the free-field state, but
we also must provide the pondermotive energy of the Volkov state in the oscillating
field (recall our previous discussion and Figure 2-5).
We now proceed with our calculation of Ln(q) in two alternative ways. We can
62
CHAPTER 2. THEORY OF PHOTOEMISSION FROM SOLID SURFACES
take a numerical approach or we can analytically estimate these Fouriere coefficients
via a saddle-point analysis.
Before proceeding with both of these approaches, let
us manipulate L,(q) to a slightly simpler form. First, we can use a similar trick as
we used in Eq. (2.34), and we can simplify the spatial matrix element in L,(q) by
re-writing it as a derivative
(q + eA(t')I ex E0 coswt' \#)
dx exp
=
J
(
- i(q + eA(t'))x/h) exE0 cos(wt')#(x)
dx(-ih) 0 (exp
(-
i(q + eA(t'))x/h))#(x)
(2.44)
(2.45)
Inserting this new form of the spatial matrix element into our expression for Ls(q) in
Eq. (2.42), we can then integrate by parts and find a simplified form for the Fourier
coefficient as given by
L,(q)
oc
dO(iq- i
x
exp i (n
sin0 - a)
+qe
q+ osE0
0mhw 2
sin 2))
e2 E 2
(2.46)
8hmw 3
In the above, we have made the substitution 0 = wt', and a = V2mWF/h as before.
With this simplified form, we can now easily numerically calculate L,(q) (note a
similar form for L,(q) appears in Refs. [31, 32]). Summing the different L"(q) Fourier
components and enforcing the energy conservation condition in the 6-function of Eq.
(2.43), we can then find an expression for the total photoemission rate. This rate is
plotted in Figure 2-8.
In Figure 2-8, we see some interesting behavior in the total photoemission rate.
Step like features emerge in the emission rate scaling.
These features are 'channel
closings', and they arise as a particular multiphoton channel closes. From the energy
conservation condition, we know that nhw = q2 /2m+ Up + WF. As the field increases
so Up increases. When n < (Up + WF)/hw an n-photon transistion can no longer
provide the minimum energy to excite an electron, and this n-photon channel closes.
This is the origin of these step-like features. This interplay of different n-photon
channels also explains the roll-off in the emission rate current as the field increases
and we move into the strong-field, tunneling regime.
63
CHAPTER 2. THEORY OF PHOTOEMISSION FROM SOLID SURFACES
10
cw (analaytical)
102
--
cw (numerical)
100
10-2
4-J
104
U
0-6
10
channel closings
10-0
10-12
101
100
10-
y
Figure 2-8: Strong-field photoemission with continuous-wave excitation. The
photoemission current is plotted for a continuous-wave excitation at a central
wavelength of A = 1.2 pm. As before WF = 5.1 eV. The numerical calculation
is plotted in pink and the analytical result is drawn in blue.
Analytical emission rate
We can calculate an analytical form of the total photoemission rate by approximating
L,(q) via a saddle-point analysis. The general form of L,(q) can be estimated by a
saddle-point analysis as follows
Ln(q) =
/2
g(9)eS(6 )
x g(Oi)eS)
(2.47)
i5"(Oi)
Where in the above the Oi's are the saddle-points, and we have encapsulated the major
components of Ln(q) in the functions g(9) and S(O). For reference, S(9) is
S(o) = no +
eq E0
2Cos 0
-
2E
e2 EW sin 20
(2.48)
When this term is sufficiently rapidly varying, a saddle-point analysis is reasonable.
Therefore, for large fields, i.e. large E0 , we can expect our saddle-point approximation
64
CHAPTER 2. THEORY OF PHOTOEMISSION FROM SOLID SURFACES
to be valid. From the above we can calculate the saddle-points, and we find they can
approximately be written as
~i
sinh-'-y +
VI
0-
~F
+ i sinh-1
+ y2 (2
-
1
1/+,
q
+
mz)
+ 2(1+
-i
+
(
2
y2)3/2
(-y
2(1+
(2Vmowz
q(2.49)
nz)
(2
(2.49
(
_2)3/2 (2
( 2 .5 0
)
0+
/m__z
Note that in the calculation of the above saddle-points the Keldysh parameter, -1
w V2mWF/eEo emerges naturally. With these saddle-points we can then approximate
the magnitude of our Fourier coefficients
)-
'
(W
)WE
Ln(q)
(2.51) 12
)
2
hw
#++
/I+2 -1+2
Finally, using these coefficients we can estimate the sum over different multiphoton
orders as an integral and calculate a complete emission rate. In other words, we can
estimate
(2.52)
F
pSF
JNmi n
In the above FSF is the photoemission rate associated with the n-photon transition,
and Nmin gives the minimum number of photons necessary at a speficied field. Carrying out the above integration, we find
-y
pSF
2
1 +-y sinh-' y - 1
x exp
2 F
(_hw
((I +
2
) simh-
2-y
V
2,
(2.53)
(
7 exp 2n(sinh-11 -
The above total photoemission rate corresponds to the curve drawn in Figures 2-7
and 2-8 (note that similar rates have been derived previously; see Refs.
[18, 33]).
It is interesting to note that the Keldysh parameter appears naturally in this rate
as it does in the saddle-points. Additionally, from the plots in Figures 2-7 and 2-8,
we see see that the above photoemission rate agrees well with the expected emission
rates in the multiphoton and strong-field regimes.
65
At first glance, it might seem
CHAPTER 2. THEORY OF PHOTOEMISSION FROM SOLID SURFACES
surprising that this rate agrees with the multiphoton one as we have neglected the
potential. However, when we calculate the multiphoton emission rate the excited
states very closely resemble entirely free-states. In other words, the potential plays
virtually no role in the multiphoton case aside from determining the initial state.
Additionally, it might seem somewhat surprising that in the tunneling regime our
total photoemission rate agrees asymptotically with the Fowler-Nordheim equation
since here we are assuming emission from just a single level, the Fermi level, while for
the Fowler-Nordheim equation we integrated over all initial energies. It is interesting
that these emission rates still agree; however, it is worthwhile to note that although
we only consider a single initial state in the strong-field photoemission case above, we
integrate over all final momenta. This parallel between the strong-field photoemission
rate and the Fowler-Nordheim equation merits further investigation.
We should additionally make some mention of the characteristic roll-off we observe
in the photoemission current in Figures 2-7 and 2-8. As we previously discussed, there
is a channel closing phenomena occurring as we increase the field strength. As we
move to higher and higher fields, higher and higher photon orders contribute and
close. The contributions from each of these orders grows smaller and only the top of
the transition contributes. This can be seen by looking at each of the L,(q) terms.
This behavior is responsible for the characteristic current roll-off.
2.2.3
Alternative emission rate formulations
In the preceding, we have discussed numerical and analytical calculations of the
strong-field photoemission rate for utlrafast optical pulse excitation and continuouswave, monochromatic illumination. We have found that under both of these conditions the emission rate roughly follows a simple analytical form derived in the above
and connects the multiphoton and tunneling regimes of emission. Before moving forward to discuss the time-dependence of the photoemission current in the strong-field
regime. It is worth mentioning two other scenarios of relevance for strong-field photoemission calculations. The first scenario is when field penetration into the emitting
metal is substantial. In this case the general behavior of the emission rate is altered.
The photoemission tends to roll-off more rapidly than our derived rates. This model
has been pursued previously and can be calculated numerically [31]. The second
scenario worth mentioning is when there is a fixed limit to the minimum acceptable
kinetic energy of the emitted electrons. This scenario might arise when image charge
or space charge effects restrict the emission. To calculate the strong-field photoe-
66
CHAPTER 2. THEORY OF PHOTOEMISSION FROM SOLID SURFACES
mission rate in this case, we can use our expressions for L,(q), but only consider
contributions with q > q . It can then be seen that at low fields the emission rate
scales at a steeper rate and rolls-off at higher fields at a more rapid rate. The overall
photoemission rate more closely resembles the tunneling emission rate. As it turns
out this model seems to best agree with our experimentally measured photoemission traces; however, our development of this model is still in its early stages, so we
postpone further discussion.
2.2.4
Time-dependent emission
Thus far we have only considered average photoemission rates in the strong-field
regime; here we consider the actual time-dependence of strong-field photoemission.
We have repeatedly stressed that the strong-field photoemission regime begins when
7
- 1, and when the optically-driven tunneling current becomes substantial.
reiterate, -y
To
1 means that the metal surface's potential barrier is collapsed for a
sufficiently long duration for a sizable amount of electrical current to be emitted via
tunneling, i.e. -s
< rcyc.
The key point to note is that this electrical tunneling
current is emitted only when the barrier is dramatically distorted, and considering
that this distortion is most dramatic near the peak of the optical field, we must have
electrical bursts being emitted that are sub-optical cycle in duration. In other words,
in this tunneling regime, we can produce electrical bursts that are significantly shorter
in duration than the optical pulse or even than an optical cycle. Considering typical
optical cycle times are on the order of femtoseconds, we likely can produce attosecond
bursts of electrons via strong-field photoemission.
The time-dependence of strong-field photoemission and the emission of sub-optical
cycle electron bursts are illustrated in Figure 2-9.
Here, we show the solution to
the time-dependent Schr6dinger equation' (TDSE) for our standard rectangular step
potential illuminated by an oscillating light field with a wavelength of A = 1.2 /Im.
The height of the step is the usual value for gold, WF= 5.1 eV. Additionally, the
strength of the optical field is E0 = 30 V/nm (this field strength is typical of our
experiments in Chapter 4 and leads to a Keldysh parameter value of 'y = 0.4). The
TDSE solution in Figure 2-9 confirms our intuition. We see the electrical current is
emitted in a sub-optical cycle burst with a full-width-at-half-maximum of ~ 860 as
(optical cycle time
Tcyc
= 4 fs at A = 1.2 Aim). This same solution to the TDSE is
8
The TDSE was solved for here using a custom TDSE solver built to handle unbounded domains
by way of discrete transparent boundary conditions (DTBCs) [34, 35].
67
CHAPTER 2. THEORY OF PHOTOEMISSION FROM SOLID SURFACES
W--%
Optical field (A = 1.2 gm)
-20
0.4
-
TDSE solution
--
Quasi-static solution
-
15
10
5
0.3
I
-
0
-5
0.2-
-0.5
4a,
0
x (nmii)
0.5
1
0.1
0
(13
-0.11.
-4
-3
-2
-1
1
0
2
3
4
time (fs)
Figure 2-9: Strong-field photoemission in the time domain. The pink shows
the electric field that drives photoemission from a gold surface with WF = 5.1
eV. The field has strength Eo = 30 V/nm and wavelength A = 1.2 ym. The
blue shows the probability current as found via numerical solution of the TDSE.
The dashed black shows the quasi-static tunneling current. The peaks of these
currents are shifted from that of the field as they are calculated a short distance
from the surface. Additionally, the inset shows the potential model for the
simulation (note that the potential is truncated ~ 0.5 nm from the surface for
simplicity in caculation).
plotted again in Figure 2-10. In Figure 2-10, however, the wavefunction amplitude is
displayed against space and time.
Quasi-static emission rates
Another intriguing characteristic of strong-field photoemission is readily visible in
Figure 2-9. In addition to the TDSE solution, we also plot the quasi-static opticallydriven tunneling current. This quasi-static current is the current given by the static
field emission tunneling formula when the optical field is inserted for the static field.
For example, the quasi-static optically-driven tunneling current with the Fowler-
Nordheim emission current is given by
J (t) = J2
exp
Etun
68
-
tun
E(t)
(2.54)
CHAPTER 2. THEORY OF PHOTOEMISSION FROM SOLID SURFACES
-3
-2-
(U
E
0
1
3
4
-0.5
0
0.5
1
Figure 2-10: Strong-field photoemission in space and time. The wavefunction
amplitude for the simple rectangular step potential modeled in Figure 2-9 is
displayed in space and time. Note the surface is located at x = 0 nm. Also,
note the large current spike near t = 0 fs.
Where in the above, E(t) is the field of the optical pulse, and Et, is the usual critical tunneling field'. To reiterate, the quasi-static optically-driven tunneling current
follows by simply inserting the optical field into the static field emission current form
and then computing the current at each time step.
The quasi-static optically-driven current model assumes that the tunneling dynamics through the barrier are so fast that the oscillating optical field looks effectively static throughout. This should correspond to the regime where y << 1. Also,
note the similarity between the separation of time scales here and the separation of
length scales in the WKB approximation used previously. In Figure 2-9 the quasistatic current comes from calculating numerically the solution to the time-independent
Schr6dinger equation when the barrier is deflected by the optical field at each time
step. Inspecting the quasi-static optically-driven current, we see that it very closely
resembles the TDSE solution. This implies that for this tunneling emission process,
the optical field can almost be treated as a static field. Additionally, it suggests that
this tunneling emission process is sensitive to the actual optical field itself. After the
9
In the calculation of this critical field the electron mass comes into play. We use here and
throughout the thesis the bare electron mass because we are interested in dynamics occurring at the
femtosecond level. The effective mass is an intraband effect and does not respond at this speed [36].
69
CHAPTER 2. THEORY OF PHOTOEMISSION FROM SOLID SURFACES
preceding build-up this may seem like an obvious point, however it is an important
one.
Recall that in the multiphoton picture, the physical mechanism for electron
emission was based on energy absorption. Therefore, the number of emitted electrons
only depended on the total number of photons or the total energy of the illuminating
laser or laser pulse. However, in the strong-field regime the emitted electrical current
depends sensitively on the actual electric field of the light pulse.
2.3
Carrier-envelope phase effects
In the final section of this chapter we discuss carrier-envelope phase effects in strongfield photoemission.
In our preceding discussion, we saw that strong-field photoe-
mission is a field sensitive effect.
We argued that strong-field photoemission cur-
rents can be accurately approximated by quasi-static tunneling models. We saw that
J(t) ~ Jtia(E(t)) where J(t) is the time-dependent strong-field photoemission current, Jt,,(-) is a field emission tunneling formula that typically takes a static electric
field as its argument, and E(t) is the electric field of the exciting optical pulse (see Eq.
(2.53)).
Now let us consider the total charge, Q, photo-emitted during illumination
by this optical pulse. We can write a simple expression for Q
Q( ) =
=
fpledt Jtun (E (V, t))
dtJo
jf
pulse
(
-
EYJ)
E ( , t) )
exp (-E
(E
t
( , t)
(2.55)
In the above, we have written the total emitted charge, Q(p) and the electric field of
the pulse, E(o, t), as explicit functions of the carrier-envelope phase, 0. Additionally,
on the right we have replaced Jtn (') with the quasi-static Fowler-Nordheim emission
formula from Eq. (2.53). Lastly, the integration label is to indicate the integration is
taken over the duration of the optical pulse.
Inspection of Eq. (2.54) suggests several interesting properties of Q( O). Firstly,
the exponential term in the tunneling current suggests this current will be very sensitive to the height and shapes of the peaks of the electric field' . We can think of
this expoentially sensitive tunneling as a threshold effect. Only when a certain field
value is reached will substantial current be produced, and therefore, as illustrated
0 A clear example of this
sensitivity was provided in Figure 2-9.
70
CHAPTER 2. THEORY OF PHOTOEMISSION FROM SOLID SURFACES
in Figure 2-11, for short pulses variation in the carrier-envelope phase can lead to
dramatic differences in the total emitted charge. As written,
Et
IK--
u
Q
is a function of W.
-- -t
N y
q =zx/2
Vr
t
Figure 2-11: Threshold nature of CEP sensitivity. Two short pulses and
two long pulses are illustrated with carrier-envelope phases W =r and -r/2. The
shifted CEP strongly affects the peak field for the short pulses (the CEP dictates
whether the peak field in this case is above or below Et.,), while the CEP shift
has a minimal effect on the peak field of the longer pulses.
An interesting property of the strong-field emission current is that it depends on
the direction of the field. Since it is a tunneling-like current the field must deflect the
barrier downwards for emission, so only one field direction will emit from a surface
(see for example Figure 2-9). In our experiments however, we will emit from different
structures that allow for not only this one-sided emission but also two-sided emission,
in which both field directions emit. We denote the charge emitted in the one-sided
emission case as QT and that for the two-sided emission case as QR (the reason for
this nomenclature will be clarified in Chapter 4). More specifically we define these
two charges as
QT&O) =
Ipulse dt Jun (E - ( , t))
and
71
QR(p)
=
dtJtun(E(p, t)) (2.56)
CHAPTER 2. THEORY OF PHOTOEMISSION FROM SOLID SURFACES
Where in the above, E-(o, t) refers to the optical field that consists of only the
negative valued part of the total optical pulse E(o, t). Similarly let us define E+(p, t)
as only the positive valued part of the total optical pulse.
Since E(p, t) = E(p + 27r, t), we expect QT(p) and QR(O) to be periodic with
period 2-F. However, also note that
QRP)
=
JusedtJtun(E(ot))
ju=se dt J.(E-
J=use dt (Jun(E-(sp, t))
=
(2.57)
(p, t) + E+(so, t))
(2.58)
+ Jun (E+(W, t)))
ulse dt (Jun(E-(o, t)) + Jtun(E- (p + 7, t)))
QT((P)+QT(O+7)
(2.59)
(2.60)
The transition from Eq. (2.56) to Eq. (2.57) follows from the behavior of the tunneling
current in the general exponential form (the dominant current contributions come
from field peaks and zero field produces zero current). From the above development
we see then that QR(W) is in fact periodic with period r. The Fourier series expansion
of QT (periodic with period 27) contains terms of the form exp(ina); however, the
Fourier series expansion of QR (periodic with period r) contains terms of the form
exp(i2no). In fact, we see that the expansion coefficients for QR are exactly twice
those for the even harmonic of QT. This behavior and the basic model for the CEP
sensitivity are demonstrated in Figure 2-12 for a 9.5 fs cos2 shaped optical pulse with
a center wavelength of 1.2pm and in Figure 2-13 for the actual measured pulse used
in our experiments in Chapter 4.
From Figure 2-12 and 2-13, we see that our simple model predicts a carrierenvelope phase modulation depth of a few parts in a thousand.
Additionally, our
basic considerations of the shape and behaviors of the total emitted charge are wellfounded. We will elaborate on our model and compare it to our measurements in
Chapter 4.
72
CHAPTER 2. THEORY OF PHOTOEMISSION FROM SOLID SURFACES
1-sided
30,
30
20-
20
10,
10
'k./I A
v
0
-10
A A11C.",
v
0
-10
-20
-20
-30
-30
-1 5
-10
-5
A
2-sided
0
5
10
5
-1
15
-10
-5
t(fs)
0
t(fi.)
5
10
15
1
0.999
0.999
0.999
0.9
0.9985
0.9985
0.998
0.998
0.9975
0.9975
x 1000
Z-
0
0.5
1
V.Vv
1.5
0
0.5
1
,P/7[
1.5
1P/ 7
10
0
10*
1-sided
0 2sided
ti
10
10
0
-0-
-
10
2
3
4
5
fczo harmonic
Figure 2-12: CEP model for a simple pulse. The top plots show a simple 9.5
fs cos 2 shaped laser pulse (red) and the resulting quasi-static emission current
(Fowler-Nordheim) for the one- and two-sided emission cases. The middle plots
show the emitted charge as a function of the CEP (note the two-sided charge
has been magnified by a factor of 10'). The bottom plot shows the Fourier series
coefficients for the emitted charge.
73
CHAPTER 2. THEORY OF PHOTOEMISSION FROM SOLID SURFACES
1-sided i
30
2-sided
30
20
20
101
10
0
S0
0
'VI
-10
A
I
-10
-20
-20
0
-0
50
100
-5
t(fs)
-
1
I
0.9995.
0.9995
0.9990
0.999
0.9985
0.9985
0.996
0.998
Z
-N.
0.9975
0.997
L
)
100
50
t (fs)
x 10
0.9975
0.997
.
0.9985
0
0.5
1
1.5
0
0.5
1
,P/7
1.5
IP/7
105
* 1-sided
0 2-sided
N
10
*-.--..--....- I ...........
....
..........
10
10
10
212
0
1
2
fcEO
3
4
5
harmonic
Figure 2-13: CEP model for a real pulse. The top plots show the measured
9.5 fs, 1.2 pm wavelength laser pulse used in our experiments (red) and the
resulting quasi-static emission current (Fowler-Nordheim) for the one- and twosided emission cases. The middle plots show the emitted charge as a function of
the CEP (note the two-sided charge has been magnified by a factor of 10). The
bottom plot shows the Fourier series coefficients for the emitted charge.
74
Chapter 3
Plasmonic nanoparticles and optical
resonators
In this chapter we discuss the fundamentals of plasmonic nanoparticle resonators. As
mentioned in the previous chapter, we will be building strong-field photo-emitting
devices with such plasmonic nanoparticles; so, it will be critical to have a conceptual
understanding and fundamental model for the optical behaviors of these tiny nanoresonators'. Our focus in this chapter will center on constructing a simple circuit
model that can accurately describe and predict the response of plasmonic nanoparticles when excited by femtosecond optical pulses. We begin with a brief, intuitive
review of surface plasmons and build a conceptual physical picture of the operation
of nanoparticle resonators. Next, we sketch a simple, second-order circuit model that
captures the essential behaviors and optical properties of the nanoparticles. The
circuit model is analyzed in the frequency and time domains, and the spectral and
temporal response of our nano-resonators to femtosecond optical pulses is discussed.
Lastly, we mention some details on our nanoparticle fabrication procedure, describe
the overall layout of our nanoparticle emitter chips, and present fits of our nanoparticle properties to the simple, second-order circuit model.
3.1
Surface Plasmons and nanoparticle resonators
Certain material surfaces can support confined electromagnetic modes; these modes
are known as surface plasmon polaritons or simply surface plasmons. Over the past
'We should mention that in this chapter and the following we will interchangeably use the terms
nanoparticle, nanoparticle resonator, and nano-resonator. Additionally, these terms will often be
preceded by the descriptor 'plasmonic'.
75
CHAPTER 3. PLASMONIC NANOPARTICLES AND OPTICAL RESONATORS
decades numerous exciting applications involving these confined surface electromagnetic modes have emerged, and there are many excellent and complete references
outlining their origins and behaviors [37, 38, 39]. Here, we provide just a brief, intuitive review of surface plasmons, highlighting several key properties of interest for
our purposes, and we describe how nanoparticles can act as tiny optical resonators
for these modes.
Surface plasmons result from longitudinal oscillations in a material surface's freeelectrons. Such oscillations are illustrated in Figure 3-la. Under particular conditions,
an optical wave can excite these longitudinal oscillations, and the wavelength of the
oscillations, i.e. the surface plasmon polariton wavelength, Aspp, is generally in the
optical domain, i.e. hundreds of nanometers. These longitudinal charge oscillations
produce electric and magnetic fields as sketched in Figure 3-2b.
Since the charge
oscillations occur largely near the material surface, these fields are confined in near
proximity to the surface. A material surface can support these charge oscillations
and resulting surface plasmon modes when the real part of the dielectric function
changes sign across the surface boundary. Therefore, surface plasmons commonly exist at metal-dielectric interfaces: the dielectric has a positive real part to its dielectric
function, i.e. Ed > 0, and the metal has a negative real part to its dielectric function,
i.e. Em < 0 (illustrated in Figure 3-la).
Let us now consider a metallic, rod-shaped nanoparticle on a dielectric substrate.
Since such a nanoparticle is composed of metal-dielectric interfaces on all sides, we
expect the particle to support surface plasmonic, confined electromagnetic modes.
These modes can propagate up and down the body of the rod. Additionally, at the
truncated ends of the rod, we expect these modes will reflect back and forth.
This basic picture of the optical behavior of the metallic nano-rod is analogous
to that of the familiar optical cavity (see Figure 3-2).
The fundamental operation
of an optical cavity is reiterated in Figure 3-2a. The cavity consists of two mirrors
that confine free-space electromagnetic modes to propagate in the intermediate space.
These modes reflect back and forth off these mirrors and circulate in the cavity. When
properly tuned, an external optical source can feed energy into these modes and buildup a tremendous amount of optical energy in the cavity (as mentioned in Chapter 1).
The rod-shaped nanoparticle behaves in much the same way. An optical source can
excite plasmonic modes that are confined to the nano-rod's surface (the transverse
optical intensity profile for such a mode is shown in the inset of Figure 3-2b). These
modes propagate up and down the body of the nano-rod, reflect from the nano-rod's
end-caps, and, when fed by a properly tuned optical source, build-up a large amount
76
CHAPTER 3. PLASMONIC NANOPARTICLES AND OPTICAL RESONATORS
_-__-_-_-_-_-
------
AS
C, < 0
-
- - +
- - - -+
+
dielectric
P
metal
-X
b
Aspp1
Figure 3-1: Surface plasmons on a metal surface. a. At a metal-dielectric
interface (dielectric above the dashed line and metal below the dashed line),
longitudinal surface charge oscillations can be excited. These charge oscillations
are the origin of the surface plasmon polariton modes and have wavelength
Aspp. b. The electric and magnetic fields produced from these longitudinal
surface charge oscillations. Since the fields originate from the surface charge
oscillations, they are largely confined to a region near the metal's surface (see
green sketch of mode profile). (Images in a. and b. were borrowed without
permission from [40]).
of optical energy in such a plasmonic nanoparticle optical resonator.
This analogy between the familiar optical cavity and the nanoparticle resonators is
a powerful one. The optical cavity is characterized by two basic parameters: the cavity's resonant wavelength or frequency and the cavity's confinement time or quality.
The resonant wavelength defines what particular wavelengths will excite the resonant
modes of the cavity. The confinement time or quality define how long these modes will
resonate before decaying and gives a measure of what level of optical energy can be
built-up in the cavity. Similarly, we expect the nanoparticle resonators to be defined
by a resonant wavelength, i.e. the wavelength at which the resonant nano-plasmonic
modes will be excited, and a damping time, i.e. a timescale over which these modes
will decay.
77
CHAPTER 3. PLASMONIC NANOPARTICLES AND OPTICAL RESONATORS
a
Optical cavity
b
Plasmonic nanoparticle
resonator
Nano rods
Nano-triangles
Figure 3-2: Optical cavities and plasmonic nanoparticle resonators. a. The
fundamental operation of a familiar optical cavity. Between the cavity mirrors
an optical pulse circulates. Energy is fed into the intra-cavity pulse by an
external pulse train. b. Plasmonic nanoparticle resonators, for example nanorods or nano-triangles, are analogous to the familiar optical cavity. They support
surface plasmonic electromagnetic modes that can propagate up and down the
nanoparticles. Inset shows the transverse intensity profile of such a mode on a
nano-rod resonator with a circular cross-section and a diameter of 20 nm (image
borrowed from Ref. [41] without permission).
Although powerful, the nanoparticle resonator and optical cavity analogy is not
exact. If we consider the confinement time of a reasonable quality enhancement cavity,
we find it can take on values in the ps range. Feeding such a cavity with an optical
pulse train with pulse spacing on the order of 10 ns (recall that typical repetition rates
from mode-locked laser oscillators are on the order of 100 MHz), we expect hundreds
of consecutive pulses to constructively interfere and add to each other inside the
cavity. However, considering a nanoparticle optical resonator, as we will see, typical
damping or confinement times are only on the order of 1-10 fs. Therefore the excited
plasmonic mode on the nanoparticle produced by an incident laser pulse will long
have damped out before the next excitation pulse arrives ~ 10 ns later. The buildup of optical energy in the nanoparticle resonator follows from a different physical
picture. Energy is not built up in the plasmonic nanoparticle modes from consecutive
excitation laser pulses, but from consecutive laser pulse cycles. When excited by
a laser pulse, each cycle of the pulse launches a surface plasmonic electromagnetic
78
CHAPTER 3. PLASMONIC NANOPARTICLES AND OPTICAL RESONATORS
mode into the nanoparticle. The nanoparticle resonators are so short (lengths ~ 100
nm) that these excited modes can reflect and interfere with the modes excited by
consecutive cycles of the pulse. Therefore, the nanoparticle optical resonators buildup optical energy in their surface plasmonic modes within a single optical excitation
pulse2 . We expect from this process that the temporal shape of the excited field on
the nanoparticle resonator will look substantially different from that of the exciting
laser pulse. In the following section, we will construct a simple circuit model for these
nanoparticle resonators that can accurately predict this temporal reshaping.
3.2
Circuit model for nanoparticle resonators
Building on the intuitive understanding developed in the preceding section, here we
seek to develop a quantitative model that can accurately predict behavior of plasmonic
nanoparticles when excited by an optical pulse. Considering our preceding description
of plasmonic nano-resonators, a reasonable route towards modeling these devices is to
calculate the plasmonic modes supported by the nanoparticles, determine how these
modes interact with the end-caps of the nano-resonators, and put these properties
together to establish a model for the nano-resonator. This approach has been carried
out by numerous researchers; however, it generally requires substantial numerical
calculation.
Additionally, numerous other approaches towards modeling plasmonic
nano-resonators have been developed ranging from simple and intuitive spring-mass
models [42] to sophisticated, entirely numerical approaches [41, 43]. In this section we
will follow a more recent approach; we will concisely describe a simple circuit model
that can accurately account for the nano-resonators spectral and temporal response
to an optical pulse [40, 44, 45, 46].
We can model the plasmonic nano-resonators with a simple distributed element
circuit model. For simplicity, let us consider the metallic nano-rods.
Looking at a
small section of the body of a nano-rod, the metallic rod and the optical excitation
can be treated as an inductance in series with a resistance and an oscillating voltage
source. The inductance corresponds to the Faraday and kinetic inductance of this
small section; the resistance corresponds to the material and the radiation resistance
of the small section; and the oscillating voltage source corresponds to the field of the
2
Note that the cartoon depiction of the circulating pulse along the nano-resonators in Figure 3-2b
is actually inaccurate as it does not show this cycle to cycle interference. The cartoon in 3-2b was
only to build the analogy between the familiar optical cavity and the nano-resonators; however, it
fails to capture this subtlety.
79
CHAPTER 3. PLASMONIC NANOPARTICLES AND OPTICAL RESONATORS
optical excitation pulse on this small section of the nano-rod (here we disregard a
distributive capacitance). The nano-rod is composed of many such small sections all
in series. Looking at the Thevenin equivalent of this distributed element circuit, we
find that we can replace this distributed model with a simple lumped element model
consisting solely of an inductor in series with a resistor and an oscillating voltage
source. The total inductance now corresponds to the total Faraday inductance and
kinetic inductance of the metal nano-rod, the resistance to the total material and
radiation resistance, and the voltage source to the total field applied across the rod.
Additionally, at the end-caps of the nano-rod we place a capacitor to model charge
accumulation at the nano-rod ends.
The complete circuit model for the nano-rod
resonator' is illustrated in Figure 3-3.
a
b
Nano-rods
Circuitmodel
Vs
Rrad
+
LK
Cend
Figure 3-3: Circuit model for plasmonic nanoparticle resonators. a. Image
of nano-rod resonator and cartoon of basic optical cavity-like operation. b.
Second-order RLC circuit model for nano-rod resonator.
To extract the optical properties of the plasmonic nanoparticle from the circuit
model, consider the voltage on the model's capacitor. The voltage on the capacitor is
proportional to the charge on the capacitor, i.e. VC = Qend/Ced. Since the capacitor
models charge accumulation at the nanoparticle's end-caps, the capacitor voltage is
proportional to this end-cap charge Qend. This end-cap charge is relevant for two main
reasons: firstly, the end-cap charge will dictate the dipole moment of the nanoparticle;
secondly, the end-cap charge will determine the strength of the local electric field near
the nano-resonator's end-caps. The importance of these two values will be illuminated
as we consider our circuit model in the frequency and time domains.
3 Although we have motivated this circuit model by looking at a nano-rod resonator,
we will use
the same basic model for our nano-triangle resonators.
80
CHAPTER 3. PLASMONIC NANOPARTICLES AND OPTICAL RESONATORS
3.2.1
Frequency domain -
susceptibility and extinction
Here we focus on the capacitor voltage in the frequency domain, Vc(w). This voltage
is proportional to the end-cap charge, i.e. Vc(w) OC Qend. Additionally, since the nanorod is of a fixed length, 1,od, the dipole moment of the rod, prod(w), is also proportional
to this end-cap charge, and accordingly, to the capacitor voltage, prod(w) Oc Vc(w).
As we have mentioned, we will be fabricating arrays of nanoparticle emitters. If we
consider an array of nano-rods, the polarization of the array, P(W), i.e. the dipole
moment per unit volume of the array, must be proportional to prod(w). Combining
all these proportionalities, we find
P(w) = coX(w)E(w) cX Prod(W) Oc Vc(w)
(3.1)
Where in the above, X(w) is the susceptibility, and E(w) is the electric field of the
optical excitation. Noting that from our model the laser voltage should be proportional to the optical excitation field, we can then note that x(w) c VC(W)/Vas(W).
Solving for this transfer function from our simple second-order circuit we then find
an expression for the susceptibility of the nanoparticle array
x(W) C Vc(w)
ZC
=
Vas(W)
ZC+ZL +ZR
1
2
1 - W LCend + ZiwRCend
2
2
In the above, Zc, ZL, and
ZR
resistor respectively, and wo
=
w
/
(3.2)
are the impedances for the capacitor, inductor, and
1//LCend and
we have used L and R instead of the
LK
and
T
= L/R.
Rrad.
Note that in the above,
The latter terms were used in
the circuit model schematic to emphasize the dominant contributions to each circuit
element.
The inductance is dominantly composed of a kinetic inductance, and the
resistance is primarily a radiation resistance. L and R in the above refer to the total
inductance and resistance respectively; however, it is worth mentioning that since
R ~ Rrad, the damping in the nano-resonators is primarily radiative. We will thus
often refer to T as Trad.
With a model for the susceptibility, we now will measure and fit the nanoparticle
resonators extinction spectrum.
The extinction spectrum is a commonly measured
optical characteristic for plasmonic nanoparticle arrays and will allow us to fit the
81
CHAPTER 3. PLASMONIC NANOPARTICLES AND OPTICAL RESONATORS
parameters of our model. We define the optical extinction a(w) as exp(-a(w)) =
T(w)/Tref(w) where T(w)/Tref(w) is the optical spectrum transmitted through the
device array divided by the optical spectrum transmitted through the substrate, i.e.
the reference transmission. The extinction a(w) can be related to x(w) through the
imaginary part of the refractive index. From this emerges the well-known expression
a(w) = CIm(X(w)). From this expression and our expression for x(w) in Eq. (3.2),
we find
a (W) oc w)
-
(W/)(3.3)
w2
+ (w 2 /r)
The measurement of a nano-triangle array's extinction spectra along with a fit to the
lineshape in Eq. (3.3) is presented in Figure 3-4. The agreement between measurement and fit is relatively good. We find for this particular nano-triangle array that
the resonant wavelength is Are
1000 nm, and the damping time is rad= 5.2 fs.
1.2
Extinction
--
-
log T/T, f
-
measured
model
1
0.8
ic
0.6
Ig 0.4
0.2
0
700
800
900
1000
1100
1200
1300
1400
wavelength (nm)
Figure 3-4: Example extinction spectrum and fit. The blue curve represents
the measured extinction spectrum for an array of nano-triangles with pitch 400
nm, altitude 200 nm and base 150 nm. The dashed red curve shows the fit via
Eq. (3.3).
3.2.2
Time-domain ultrashort pulse broadening
Here we will concentrate on the capacitor voltage in the time domain, Vc(t). This
voltage is proportional to the end-cap charge, i.e. Vc(t) OC Qend(t). From our preceding discussions, we have seen that these nano-resonators resemble tiny optical
82
CHAPTER 3. PLASMONIC NANOPARTICLES AND OPTICAL RESONATORS
dipoles. We therefore expect the peak fields near our nano-resonators to occur right
near the end-caps and for this peak field to be directly related to the end cap charge,
i.e. Epl(t) OC Qend(t) oc Vc(t), where Ep,(t) refers to the peak nano-plasmonic optical
field. Therefore, we expect that the time-domain behavior of the nano-plasmonic field
will follow that of the capacitor voltage. Returning to our circuit model, the capacitor
voltage obeys the familiar, second-order differential equation
d2 Vc(t)
2
dt
1 dVc(t)
+ r dt +w!Vc(t) = W0Vias(t) OC E(t)
(3.4)
where E(t) in the above is the field of optical excitation pulse. Therefore, by solving
the above equation with a specified optical driving pulse, we can estimate the temporal
behavior of the localized plasmonic field around our nanoparticle resonators. An
example of such a solution is provided in Figure 3-5. In this example an excitation
laser pulse with the black envelope 4 and a central wavelength of 1.2 pm is input for
Vlas. The blue pulse is the resulting envelope of the response Epi(t) Oc Vc(t). The
.
nanoparticle used in this example has a resonant wavelength of Ares = 1256 nm and
a damping time of Trad = 7.4 fs 5
1
I
AR,, = 125 6 nm
0.8
0.6
Trad
I
14
.
-
0.41
Excita tion pulse
- Nano -res. pulse
0.2
I
a
-40
-30
-20
0
-10
10
20
30
40
time (fs)
Figure 3-5: Ultrafast optical pulse broadening in a nanoparticle resonator.
The resonator used has a resonant wavelength of 1256 nm and a damping time of
7.4 fs. The black trace shows the excitation pulse envelope (central wavelength of
1.2 pm). The blue trace shows the envelope of the response on the nanoparticle
resonator.
'The black excitation pulse used here is an actually measured laser pulse that will be described
in greater detail in Chapter 4. It is the pulse displayed in Figure 2-13.
'These numbers are from an actual fabricated nano-triangle array.
83
CHAPTER 3. PLASMONIC NANOPARTICLES AND OPTICAL RESONATORS
3.3
Nanoparticle fabrication and characterization
For the experiments that will be described in the following chapter, we fabricate a
collection of different nanoparticle resonators. As described, the nanoparticles take
on two main shapes: nano-rods and nano-triangles. The nanoparticles are fabricated
on a 1 cm2 sapphire chip. The sapphire is coated in a 50 nm layer of indium tin
oxide (ITO) that is patterned to separate the chip into emitters and collectors (to be
discussed in Chapter 4). An overview of the chip architecture is shown in Figure 3-6.
A total of eighteen different device arrays were fabricated. Of these arrays there were
seven different nano-rod size arrays and nine different nano-triangle arrays. Extinction
spectra for all the nano-rods are shown in Figure 3-7. Extinction spectra for all the
nano-triangles are shown in Figure 3-8. Finally, a microscope image of the chip layout
as well as a sketch of all the different resonances for the arrays is included in Figure
3-9.
500 nm
Figure 3-6: Chip overview. a. Picture of the actual chip. b. Zoomed in
microscope image of the arrays of devices. c. Zoom in of the purple region
outlined in part b. d., and e. electron micrographs of the blue and red regions
outlined in part c.
84
CHAPTER 3. PLASMONIC NANOPARTICLES AND OPTICAL RESONATORS
model - -
-
-
measured
1
0.8
C
S0.6
o.4
0.2
0
700
800
900
1000
1100
1200
1300
1400
wavelength (nm)
Figure 3-7: Extinction spectra for the nano-rod devices. The blue traces are
measurement and the red dashed traces are model fits according to Eq. (3.3).
The dark blue trace is for the nano-rod with resonant wavelength 1041 nm.
-
model - -
-
measured
1
0.8
S0.6
0.4
0.2
700
800
900
1000
1100
1200
1300
wavelength (nm)
Figure 3-8: Extinction spectra for the nano-triangle devices. The blue traces
are measurement and the red dashed traces are model fits according to Eq. (3.3).
The dark blue trace is for the nano-triangle with resonant wavelength 1059 nm.
85
1400
CHAPTER 3. PLASMONIC NANOPARTICLES AND OPTICAL RESONATORS
Figure 3-9: Image and sketch of chip layout. a. Microscope image of set of
eighteen fabricated device arrays. b. Sketch showing the resonant wavelengths
of each array. The R and T labels indicate whether the array is a nano-rod array
or a nano-triangle device array.
86
Chapter 4
Photoemission from plasmonic
nanoparticles
In the previous two chapters we discussed the theory behind photoemission in the
multiphoton and strong-field regimes, and we described the basic properties and behaviors of plasmonic nanoparticles. In this chapter we put these ideas together and
describe photoemission from plasmonic nanoparticle resonators. We begin by discussing the motivation for investigations into photoemission near nanostructures and
by providing an overview of past as well as on-going work in this area. We next briefly
describe our early experiments and sketch the experimental approach we follow in this
work. Next, we provide some details regarding the various components of our experimental system: we discuss the femtosecond laser source, the laser pulse measurement
system, the carrier-envelope phase stabilization technique, and the device alignment
microscope. We then proceed to present our experimental results. First, we investigate basic scaling properties of the photoemission signal and demonstrate signatures
of the strong-field regime. Next, we use the strong-field photoemission current to perform interferometric autocorrelations and characterize the nano-plasmonic, ultrafast
optical field excited by the femtosecond laser pulse on the nanoparticles. Lastly, we
demonstrate the carrier-envelope phase sensitivity of the photoemission signal and
develop a simple model for predicting this carrier-envelope phase response.
4.1
Strong-fields near nanostructures
As discussed in Chapter 1, over the past decade, photoemission, in particular strongfield photoemission, from nanostructures has garnered tremendous interest for its po87
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
tential scientific and technological impact. From a scientific perspective, the prospect
of generating sub-optical cycle electron bursts or even attosecond electron packets
(as we discussed in Chapter 2) suggests exciting possibilities in probing ultrafast phenomena near solid-surfaces at unprecedented temporal resolutions. For example, such
ultrashort electron bursts might allow the probing of electron motion in solid-state
surfaces and nanostructures as well as the probing of the ultrafast fields or plasmons
excited near such surfaces [47].
From a technological standpoint, the prospect of
switching electrical currents with the optical field of light might offer opportunities
for novel opto-electronics or detectors such as on-chip carrier-envelope phase detectors
[10, 12, 48, 49, 50]. Additionally, as we will discuss, there are exciting applications
for strong-field photo-emitting nanostructures in electron-beam technologies as novel
photo-cathodes.
Central to the applications for strong-field photoemission outlined above are two
fundamental optical properties of nanostructures: field-enhancement and localization.
Nanostructures can offer field enhancement to an incident optical wave. This field enhancement generally arises from two phenomena: geometric focusing and resonance.
The geometric focusing or geometric enhancement emerges from the nanostructure's
lightning-rod-like behavior when illuminated with optical fields. Intuitively, at sharp
corners and tips with nanometer-scale radii of curvature electric field lines will become very crowded as the field attempts to bend around the corner all the while
remaining normal to the surface. This 'field-line crowding' results in a concentration
of the optical energy and a resulting electric-field enhancement [51]. The physics is
conceptually the same to the behavior of a lightning-rod. Additionally, this geometric
enhancement is the same process that leads to localization. As described, the electric
field-lines crowd at sharp corners and tips, and the optical energy becomes concentrated or localized. This localization or this concentrated spot of high electric-field
can result in highly-localized photoemission. This highly-localized electron emission
from the geometrically enhanced field near a sharp tip or corner of a nanostructure is
of great technological interest for future photocathode technologies as it might serve
as a very low-emittance virtual point-source of electrons [52, 53].
The second source of field-enhancement is resonance. As discussed in the previous
chapter certain material surfaces can support confined electromagnetic modes that
result from oscillations of the material's electrons and are seemingly bound to the
material surface. As we discussed, metallic nanoparticles can act as resonators for
these surface-plasmonic modes, and when fed by a properly tuned optical source, they
can build up a tremendous optical energy and field.
88
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
In the following, we leverage the field-enhancement properties of nanostructrues, in
particular plasmonic nanoparticle resonators, to achieve the necessary field strengths
to explore some of the interesting strong-field photoemission phenomena described
above. Before discussing our particular approach and results however, let us review
some of the major results to date and some of the on-going work in the study of
strong-field photoemission near nanostructures.
4.1.1
Previous results and ongoing work
In studies of strong-field photoemission near nanostructures, there have been two major classes of nanostructures that have been used: nano-tips and plasmonic nanoparticles. These two basic geometries nicely follow the division laid out previously in our
field-enhancement discussion, i.e.
nano-tips are devices largely dominated by geo-
metric enhancement effects and plasmonic nanoparticles are largely resonant devices.
The big-picture approaches to strong-field photoemission experiments with these two
device types are illustrated in Figure 4-la and b. The essential experimental concept
is similar for both device types: a nano-tip or an array of plasmonic nanoparticles
is mounted in a vacuum setting. The nano-tip or array is then illuminated with a
femtosecond laser pulse, and the photo-emitted electrons are collected and analyzed.
In the following, we will briefly describe some of the prior results and on-going work
with each of these device types.
Nano-tip emitters comprise sharp metallic nano-tapers with radii of curvature at
the very tip of only a few nanometers. Coated atomic-force microscopy tips or other
similarly sharp tips, commonly composed of tungsten or gold, are used in these experiments. As mentioned, these nano-tips show primarily a geometric, lightning-rod-like
field-enhancement effect at the tip, and when illuminated by femtosecond laser pulses,
photo-emit large, short bursts of electrons. Since the first experiments demonstrating
femtosecond-laser triggered photoemission from nano-tips [54, 55, 56], there has been
an explosion of experimental works in this area, with most investigations targeting
the strong-field regime. Thus far there have been demonstrations of the roll-off in the
current yield as the strong-field regime is approached [31, 12, 57] (discussed previously in Chapter 2); there have been measurements of the electron energy spectra of
the emitted electrons showing re-scattering behavior[10, 11, 12, 23, 57, 58, 59, 60]
1;
'This electron re-scattering phenomena is a classic signature of the strong-field regime [7]. The
essential concept is sub-optical cycle electrons are emitted from the metal surface, and after emission,
wiggle in the nearby laser field. Some of these electrons will be redirected back towards the metal
surface and scatter off the surface. These re-scatter electrons can pick up large amounts of energy
89
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
Single nano-tip emitters
a
Nanoparticle emitter arrays
b
fslaser pulse
eectrons
electron pulse
d
c
10
.
-rod
red
20
25
10
.2
1
-
0
5
Energ
(W)
10
0
Cu-ff posat~on
5
10
15
Enwxy(ov)
4
10
16 0
10
Noemaked
Nomtnakod
co t rate (au.) cont rate (a.u.)
Figure 4-1: Strong-field physics with nanostructures. a. Schematic of nanotip emitter operation. b. Basic layout for nanoparticle emitter experiments. c.
Example data set showing carrier-envelope phase effects in the emitted electron's
energy spectra from nano-tip emitters. d. Energy spectra showing large rescattered plateau of emitted electrons from nanoparticles. (illustrations and
pictures in a-d borrowed from Refs. [10, 22] without permission).
and there have been a number of works exploring carrier-envelope phase sensitivity
in the emission process [10, 121. As discussed in Chapter 2, the field-sensitive nature of the strong-field emission process has excited researchers about the prospect of
being able to detect carrier-envelope phase effects in strong-field photoemission currents. Detection of the carrier-envelope phase of ultrafast laser pulses (see discussion
in Appendix A) is a classic problem in ultrafast optics, and a compact, chip-scale
detector could be revolutionary. Additionally, carrier-envelope phase effects are of
interest from a purely scientific perspective as such effects are a hallmark of true
optical field-sensitivity. So far, carrier-envelope phase signatures have appeared in
the energy spectra of electrons emitted from nano-tips [10] and more recently have
been shown in the total emission current as well [12]; however although scientifically
intriguing, thus far the overall strength of these effects has been relatively weak and
not suitable for an effective carrier-envelope phase detector (see Figure 4-1c).
from the laser field and have particular signatures in the emitted electron energy spectra.
90
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
Although also of great interest for strong-field photoemission, plasmonic nanoparticle emitters have received much less attention to date. These emitters frequently are
composed of arrays of gold nanoparticles of spatial dimensions on the order of ten to
a hundred nanometers patterned on a transparent substrate. These nanoparticles experience both resonant and geometric enhancement effects and, like nano-tips, photoemit large electrical currents when illuminated by femtosecond laser pulses. Although
there have been fewer experimental investigations into these emitters, the emitted
electron energy spectra has been studied and shown to demonstrate re-scattering effects characteristic of the strong-field regime [22] (see Figure 4-1d). Additionally, these
emitters have attracted particular attention for futuristic photocathode applications
as discussed previously [52, 53].
4.2
Strong-fields on a chip
Our approach to strong-field photoemission involves moving the experiments to a chip.
As discussed in Chapter 1, strong-field physics generally involves complex laser amplifiers, large vacuum setups, and generally complex experimental machinery. Strongfield experiments with nanostructures are frequently simpler than other strong-field
science investigations; the field-enhancement provided by the nanostructures often
means complex laser amplifiers are unnecessary. The field from tightly focused laser
pulses directly from an ultrafast, mode-locked oscillator are often sufficient to reach
the strong-field regime. However, although simpler in this regard, strong-field experiments with nanostructures do require much of the other large, expensive experimental machinery of traditional strong-field physics, in particular vacuum setups.
Our general concept at the start of this work was to remove the cost and complexity
of these experimental setups by putting our nanostructure, our detectors, and our
entire experiments on the surface of a micro-scale chip.
With collecting electrodes
and detectors only nanometers to tens of microns from our emitting nanostructures
vacuum might not be required, and strong-field science experiments can be carried
out under ambient conditions with tightly focused laser pulses directly from a modelocked, ultrafast laser oscillator. Such on-chip strong-field experiments are not only
interesting from the perspective of a simplified experimental platform but also could
provide the opportunity to make compact, novel opto-electronic devices and detectors
that operate in the strong-field regime.
In the following, we discuss some of our early attempts towards on-chip strongfield physics. Next, we extract valuable lessons from this early work and outline the
91
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
essential approach and device design we use in our on-chip strong-field experiments
presented in the remainder of this chapter.
Flattened nano-tips on a chip
4.2.1
Our initial work towards on-chip strong-field physics involved moving nano-tip emitters to a chip. The basic approach to these on-chip nano-tip emitters was to flatten
to make a two-dimensional nano-tip emitter, on a chip with a
of an
nearby collector electrode. An illustration of the basic concept and an image
the nano-tip, i.e.
early device geometry are provided in Figure 4-2. Our flattened, two-dimensional tips
were composed of a 30 nm thick layer of gold and showed a small ~ 10 nm radius of
curvature near the tip apex. They were estimated to show similar geometric optical
field-enhancements to the free-standing, three-dimensional nano-emitter tips used in
many previous experiments [61]. Additionally, the collector electrodes were placed
only around a hundred nanometers from the flattened nano-tips and small bias voltages were applied to them (on the order of a few volts). Although small, these bias
voltages led to tremendous bias fields due to the small emitter-collector gap distance
so close
~ 1 V/(100 nm) = 10 MV/m). The collector electrodes were placed
to the emitters, only on the order of a hundred nanometers, because the devices were
operated under ambient conditions. The mean-free path of a low-energy, few eV,
electron in air is typically mf ~ 200-300 nm [62]. In order to ensure the ambient
conditions would not affect our ability to collect the emitted electrons, the collector
(EBIAs
was placed within this mean-free path.
a
b
v's
TiS
e
Figure 4-2: Nano-tip emitter experiments on a chip. a. Conventional nano-tip
emitter experimental arrangement (illustration borrowed from Ref. [10] without
permission). b. Basic layout of one of our flattened on-chip nano-emitting tips.
92
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
The preliminary photoemission results from these flattened nano-tips showed only
miniscule currents. The tips were illuminated with tightly-focused a 6 fs laser pulses
with a central wavelength of A ~ 800 nm emerging from a home-built Ti:sapphire
mode-locked oscillator 2 . The laser pulse were scanned in energy up to 5 nJ; however,
only near negligible currents were observed, and these currents fluctuated and were
unreliable. Additionally, at the higher pulse energies device damage was consistently
an issue. The exact issues and reasons for this poor performance are not entirely
understood; possible explanations include fabrication and electrical contact issues
with the flattened nano-tips, backwards emission from the edges on the collector,
among other effects.
Investigating a set of these flattened nano-tip emitters, an interesting observation
was made. The fabrication of the flattened nano-tips involved an initial electron beam
lithography process to pattern the gold nano-tips which was subsequently followed by
an optical lithography process to create large gold pads to electrically connect to the
flattened nano-tips (see Figure 4-3a). When the edge of one of the large connecting
gold pads was illuminated, we measured a relatively large photoemission current at
the collector (see Figure 4-3a). Looking at the SEM image in Figure 4-3a, we see that
the photolithographically defined pads have very rough edges. The photolithography
was done hastily, and imperfect resist sidewalls coupled with the lift-off process led to
the formation of the apparent roughness. This roughness takes the form of thin gold
whiskers at the edges of these large gold pads. When illuminated with the femtosecond
Ti:sapphire laser pulse, these whiskers show the same geometric field-enhancement
that the nano-tips demonstrate, and accordingly they photo-emit electrons.
The
remarkable fact here is that the emitting whisker was tens of microns away from the
collector electrode, yet despite this gap, many mean-free-paths in length, the electrons
still reliably traversed the ambient conditions from the emitter to the collector (see
Figure 4-3b).
This observation fundamentally changed a crucial element of our on-chip flattened
nano-tip emitter design. The collector electrodes had all been placed only around a
hundred nanometers from the nano-tip emitters and mostly only single emitters had
been used because the assumption was the emitted electrons would not effectively
traverse distances much greater than a mean-free-path,
mf
~ 200-300 nm, under
ambient conditions. However, the above experiment revealed that this was false. The
length scale of interest is not the electron's mean-free-path, but the 'capture-free'
2
This home-built laser operates at 84 MHz and makes use of a similar layout and design as Refs.
[63, 64].
93
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
path, i.e. the mean distance the electron can travel before being captured. For our
initial experiments, we are only concerned with the total emitted electrical current.
Therefore, for our purposes it is irrelevant whether the emitted electrons scatter in
air; it only matters that the electrons can travel from emitter to collector before
being captured by a volatile organic, electronegative species, or something else in
the intermittent air. Additionally, this capture-free path is far greater for low-energy
electrons in ambient conditions than the mean-free path. Low-energy electrons have
been demonstrated to have nanosecond lifetimes in ambient conditions which for a 1
eV electron could translate into a path length approaching the mm-scale [65, 661.
This revelation of the importance of the capture-free-path versus the mean-free
path lifted a critical restriction in our on-chip strong-field emitter design and, as we
will now discuss, pushed us towards plasmonic nanoparticle emitters.
b
TiS
o emitter current (4)
2
-
@
collector current (Jd
0
s-Jc
'Eand Ic measured consecutively
-os
1
2
3
4
S
a
7
8
-
a
0
10
Time (s)
Figure 4-3: Electrical currents across large gaps under ambient conditions.
a. Basic experimental arrangement. Electrical emission from gold whisker on
photo-lithographically defined pad. b. Measured electrical currents: emitter
current, collector current, and current difference. All currents were measured
sequentially, i.e. one after the other.
4.2.2
On-chip nanoparticle emitter arrays
Without the mean-free path length scale restriction, we shifted our device design
towards nanoparticle emitters. With the nanoparticle emitters, we could place many
94
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
emitters in the focused laser spot, and without the mean-free path restriction, we
could place the collector electrode microns away from the emitters and the laser
illumination.
The basic layout of our modified on-chip strong-field emitter devices is illustrated
in Figure 4-4. The chips are composed of a sapphire substrate coated in a 50 nm thick
layer of the transparent conductor indium tin oxide (ITO). The nanoparticle emitters
are patterned on the ITO layer via electron-beam lithography. The nanoparticles are
composed of 20 nm thick layer of gold with no adhesion layer to the ITO. As we
described in the previous chapter, several different nanoparticle shapes and sizes are
fabricated on the chips.
Nanoparticle emitter arrays
substrate
(sapphire)
Emitter
(ITO)
Collector
(ITO)
electrode gap
(3 pm)
Figure 4-4: Nanoparticle emitter device layout (optical microscope image).
The substrate is composed of a sapphire chip coated in indium tin oxide (ITO).
An array of nanoparticle emitters is fabricated on the ITO layer (enclosed by the
red box in the image and an example show in the inset). The ITO is patterned
into two regions: an emitter that connects to the nanoparticles and a collector.
The collector is separated from the emitter by a few micron gap.
The essential surrounding experimental components along with the basic layout of
our strong-field emitter chips are illustrated in Figure 4-5. Figure 4-5 shows the same
optical microscope image Figure 4-4 from a section of one of our chips (for a largerscale view of the chip see Figures 3-6 and 3-9). Femtosecond excitation laser pulses are
tightly focused through an objective and illuminate these nanoparticle arrays. Under
95
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
2. Nanoparticle emitter arrays
Emitter
SF phenomena on a chip
under ambient conditions I
Collector
Figure 4-5: Nanoparticle emitter device layout (optical microscope image)
and basic experimental setup. Femtosecond laser pulses are focused by an objective (Obj.) to a small spot-size on the nanoparticle array. The excitation
laser pulses result in strong-field photoemission from the devices, and the photoemitted electrons jump from emitter to collector. The three main elements of
the following few chapters are numbered.
this intense optical illumination, the nanoparticles emit electrons in the strong-field
regime. These emitted electrons jump across the small (~z3 pm) gap from the emitter
to the collector. In Chapter 3 we analyzed the plasmonic nanoparticle emitter arrays
and characterized our fabricated arrays (i.e. we discussed item 2 from Figure 4-5). In
the following sections we will provide details on the femtosecond laser source (item 1 in
Figure 4-5) and will present our experimental observations of strong-field phenomena
on our chips under ambient conditions (item 3 in Figure 4-5).
Our main objectives in these experiments will be to observe novel strong-field
photoemission phenomena from our plasmonic nanoparticle emitters under ambient
conditions. As we discussed in Chapter 2, there are several intriguing effects that
emerge from photoemission in the strong-field regime. Most notably, in the strongfield regime, sub-optical cycle electron bursts can be produced and these brief electrical currents can be switched on and off with each optical cycle of a laser pulse.
This electric field, or optical waveform, production and control of electrical currents
near nanostructures has been intensely pursued by researchers. As described, our
experimental arrangement, or on-chip strong-field laboratory, is only capable of mea96
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
suring total currents; however, we can look for signatures of this waveform control by
looking for carrier-envelope phase sensitivity of the total photoemitted current. This
will be the main thrust of our experiments: the exploration of optical waveform controlled electrical currents from plasmonic nanoparticles in the form of carrier-envelope
phase (CEP) sensitivity. Along the way we will also demonstrate other signatures of
strong-field photoemission and use the strong-field photoemission signal to precisely
characterize the temporal behavior of the plasmonic field near the nanostructures.
4.3
Experimental details
In this section we present some details on the various components of the experimental
setup. We first mention some details regarding the femotsecond laser source. Next,
we discuss the measurement and characterization of the ultrashort, femtosecond laser
pulses emerging from this source and used in the experiments. We move on to describe
the carrier-envelope phase stabilization procedure. Finally, we mention some details
regarding the device alignment microscope.
This section is intended to provide a
large-scale overview of the different elements that go into the experiment.
4.3.1
Few-cycle Er:fiber based laser source
The femtosecond laser source consists of two main parts: an Er:fiber oscillator and
a supercontinuum generation stage. An overview of the basic system is sketched in
Figure 4-6 (similar systems have been described in Refs. [67, 68, 69, 70, 711). The
Er:fiber laser produces optical pulses at
fR
= 78.4 MHz with approximately Ep
=
0.375 pJ. The emitted pulse train then passes through an acousto-optic frequency
shifter (AOFS), and the first diffracted order is sent into an Er:doped fiber amplifier
(EDFA). As we will describe in greater detail in the following, the AOFS tunes the
carrier-envelope offset frequency of pulse-train.
a pulse energy Ep 1
The pulses out of the EDFA have
4.5 nJ and are compressed in a silicon prism compressor to a
duration of - 90 fs. These pulses are then focused into a specially designed highly nonlinear fiber 3 (HNF). In the HNF the interplay of various non-linear optical processes
lead to supercontinuum generation. In particular a two-part spectrum is generated; a
Raman-shifted soliton is produced near a wavelength of 1.9 pm, and a dispersive wave
spectrum is also generated with a center wavelength near 1.2 Am. The dispersive wave
3
This fiber consists of a germanium-doped highly non-linear segment and several other different
fiber stretches that have been spliced together. The fibers are designed by the group of Prof. A.
Leitenstorfer [68, 70].
97
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
spectrum is relatively flat from 1-1.4 pm and illustrated in Figure 4-6. The dispersive
wave will be used for our experiments and is compressed post-HNF in an SF10 prism
compressor. The SF10 prism compressor also contains a spatial filter that picks off
the Raman-shifted soliton so that at the output of the entire laser system we are left
with just the dispersive wave spectrum with a pulse energy of around Ep = 0.3 nJ,
a repetion rate of the Er:fiber oscillator seed at fR = 78.4 MHz, and a relatively flat
spectrum spanning the wavelength range of 1-1.4 pm.
Erfiber oscillator
Supercontinuum Gen.
si pc
HNF
Optical Spectrum
0.
40.1
nEp= 0.3 nJ
f.=75. MHz
02
AOSISpIO9PC
01
9
AV
3
1000
1100
1200
1300
1400
1500
wavelength (nm)
Pulse measurement
CEP stabilization
Experiments
Figure 4-6: Femtosecond laser system overview. Schematics and pictures
of the Er:fiber oscillator and the supercontinuum generation stages are shown.
At the output of the femtosecond laser system is the dispersive wave spectrum
spanning 1-1.4 pm as shown. The dispersive wave pulses are sent from the
laser system output to the pulse measurement setup, the carrier-envelope phase
stabilization/characterization setup, and the actual strong-field experiments.
4.3.2
Pulse measurement
The dispersive wave pulses out of the femtosecond laser system are measured with
a two-dimensional spectral shearing interferometer (2DSI)'. The results of the pulse
measurement are shown in Figure 4-7. Figure 4-7a shows a repeat of the dispersive
4Two-dimesional spectral shearing interferometry is a spectral-shearing pulse measurement technique similar to SPIDER but with simpler and less-sensitive calibration requirements. For further
information see Ref. [72].
98
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
wave spectrum, and Figure 4-7b shows the measured pulse shape (blue) and the
transform-limited pulse (red-dashed). The central spike of the measured pulse has a
full-width-at-half-maximum of r ~ 9.6 fs, close to that of the ideal transform-limited
pulse.
a
b
Optical Spectrum
Retrieved Pulse
I
0.8
0.8
0.6
0.6
-Measured
=9.6fs
.
1
C
0.4
0.4
0.2
, = 0.3 M
fR =78.4MHz
0.2
a
900
1000
C
0
1400
1100
1200
1300
wavelength (nm)
1500
-40
-20
0
time (fS)
20
40
d
InterferometricAutocorrelation
Predicted
-FnL
*Measured
.4
0
-40
-30
-20
-10
0
delay (f6)
10
20
30
40
Figure 4-7: Dispersive wave pulse measurement. a. Spectrum of the dispersive wave pulse used in the experiments. b. Measured optical pulse shape from
the 2DSI (blue) and the ideal transform-limited pulse (dashed-red). c. Interferometric autocorrelation measurements (note the early delay data is poor due
to an error in the calibration at the start of this trace). d. Image of the 2DSI
setup.
To confirm the measured pulse shape, interferometric autocorrelation (IAC) measurements were additionally performed. The IACs were taken by focusing two pulses,temporally
spaced by a carefully calibrated delay, into a 40 pm thick Beta barium borate (BBO)
crystal and recording the produced second-harmonic generation signal 5 . The IAC
measurement for the optical pulse displayed in Figure 4-7b is given in Figure 4-7c.
'We should additionally mention that in the interferometer that composes the IAC specially
designed group-delay dispersion matched beamsplitters are used. These beamsplitters show a group
delay equivalent to 0.75 mm of fused silica (up to an arbitrary constant) on both reflection and
transmission. The effect of these beamsplitters is incorporated in the calculations of the expected
IAC traces provided in Figure 4-7c.
99
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
In Figure 4-7c the measured IAC data is displayed (red dots); the expected IAC for
an ideal transform limited pulse is shown (light blue); and the expected IAC for the
measured pulse given in Figure 4-7 is displayed (dark blue). The expected IAC for
our measured pulse fits the measurement reasonably well, so we have some confidence
in our measured pulse shape and characteristics.
Carrier-envelope phase stabilization
4.3.3
As we have mentioned one of our primary goals will be to detect carrier-envelope
phase sensitivity in our photoemission experiments. Therefore, it is critical to stabilize the carrier-envelope phase of our femtosecond laser pulses.
The CEP of the
laser pulse train emitted by our femtosecond laser system is illustrated in Figure
4-8a. Ideally, from pulse to pulse the CEP shifts by some specific amount A&p (as
mentioned in Chapter 1, this shift is due to a mismatch the between group velocity
and phase velocity in the laser cavity). However, with noise and other perturbations
this progression of the CEP can be irregular; therefore, we need to actively control
the CEP to ensure we have reproducible and predictable electric field shapes from
pulse to pulse.
To stabilize the CEP of our laser pulse train we must first detect the CEP of the
pulse train and then feedback the detected signal to an actuator that can tune the
CEP to the desired value. In our setup, the CEP is detected in the conventional way
with an
f
- 2f interferometer. The basic layout of our f - 2f interferometer is shown
in Figure 4-8a. The long-wavelength Raman-shifted soliton and a very small portion
of the shortest wavelength light from the dispersive wave are picked off in the SF10
prism compressor. These two optical signals are sent to the
f
- 2f. The output of
the interferometer is then sent to the AOFS which tunes the CEP of the laser pulse
train. The AOFS uses acoustic waves to Doppler shift the pulse train and accordingly
tune the CEP of the laser pulse train [67, 73].
The results of the CEP lock are presented in Figure 4-8b. These results come from
an out-of-loop
f
- 2f interferometer and show good fringe stability and a relatively
low rms drift in the CEP of only 167 mrad.
f
We should mention that our in-loop
- 2f interferometer contains an additional acousto-optic modulator in one arm to
allow for locking of the carrier-envelope offset phase down to 0 Hz.
100
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
a
VCZ- A
)
V
quLo
A
9CEO
b
Out-of4aopyf-2
C
mrad
V
6rm,=167
0
- 1.5L
0
10
20
30
40
50
60
time (s)
Figure 4-8: Carrier-envelope phase stabilization and characterization. a.
Overview of the modifications to the femtosecond laser system to allow carrierenvelope phase locking. The physical meaning of the CEP is also illustrated
to the right. b. Results from the out-of-loop f - 2f interferometer when the
carrier-envelope offset frequency is locked to 0 Hz.
4.3.4
Device alignment
The final component of the experimental setup is the device alignment microscope.
In our experiments, we illuminate a 20 pm x 20 pm array of plasmonic nanoparticles
with focused femtosecond laser pulses that have a beam diameter of only around 4
pm. To reliably and consistently place this tiny laser spot on our emitter array, we
build a simple confocal microscope.
The microscope is illustrated in Figure 4-9a. A reflecting objective (Obj. 1 at
the top of the picture in Figure 4-9a) focuses the laser light onto our devices (device
mount image included to the right). The laser light is then collected by a second
objective (Obj. 2 at the bottom of the picture in Figure 4-9a) and imaged onto
101
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
a
b
fs-source
Fit (Gaussi.n)
0 Meuremert
-
2.
0.53
W,=1.9
-10
White
Pm
-5
foci
0
5
10
podtrn (-)
light
Figure 4-9: Alignment microscope setup and focused spot characterization.
a. Picture of the confocal alignment microscope and the device mount (right).
b. Knife-edge measurements of the focused laser spot in both transverse planes
(measurement made by scanning a 10 pm wide gold wire across the laser spot).
c. Example microscope image recorded on the CCD. Old devices are shown
with very high spatial resolution.
a CCD. Additionally, a white light source illuminates our mounted device through
the collecting objective (Obj. 2). This light reflects from our devices and is also
imaged on the CCD. This simple confocal microscope allows us to simultaneously
image our devices and the position of our focused laser spot. An example image of
an early set of our devices is included in Figure 4-9c. From the scale bar, we see that
our confocal microscope can accurately image fine features of our devices and enable
precise alignment of our focused laser spot. The confocal microscope setup offers the
additional opportunity to characterize the focused laser spot. Gold alignment pads
or wires can be swept across the focused laser spot to record knife-edge traces of
the focused spot. Such knife-edge measurements for both transverse directions are
included in Figure 4-9b. In these measurements a 10 prm wide gold wire was scanned
across the focused laser spot, and the laser spot is found to have beam waists in the
two transverse directions of w, = 3.5 tm and wy = 1.9 Pm.
102
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
4.4
Photoemission measurements
In this section we present measurements of photoemission currents from our plasmonic nanoparticle devices. First, to confirm our fundamental picture of the device
operation, we make simultaneous measurements of the current leaving the emitter
electrode and the current arriving at the collector electrode and compare the relative
phases of these currents. Next, we will analyze how the photoemission current varies
as the excitation pulse intensity is increased. Finally, we will discuss degradation and
changes to our devices through use.
Collector and emitter currents
4.4.1
First and foremost, we want to confirm the basic picture of our device operation.
We want to be sure that our devices are emitting electrons and that these electrons
are reliably crossing the few micron gap from emitter to collector.
To this end,
we illuminate our nanoparticle emitters and simultaneously measure the current at
the collector electrode, JC (the collector current), and the current at the emitter
electrode,
JE
(the emitter current). The results of this first measurement are shown
in Figure 4-10.
In the top panel of Figure 4-10, we see that Jc and
JE
match
extremely well, and both currents are relatively stable over the approximately three
and a half minute measurement period. These measurements are made via lock-in
detection6 , so we can additionally compare the relative phase of the emitter and
collector currents.
This relative phase is shown in the lower panel of Figure 4-10.
We see that the relative phase of the currents is almost exactly 180'. Therefore, Jc
and
JE
must have opposite signs.
Additionally, we observe that a current is only
produced when the laser illuminates the device array and when there is a positive
bias on the collector electrode relative to the emitter (for the experiment shown in
Figure 4-10 this bias was
VBIAS
=
10 V). When the laser illuminates other regions
of the chip or when the laser is turned off there is no current.
Additionally, when
the bias between on the collector is increased relative to the emitter the current is
increased, while when this bias is decreased or made negative, the current can be
decreased dramatically. From these results, we can conclude that electron are being
emitted from our plasmonic nanoparticle devices, traveling across the few micron gap
between the emitter and the collector, and reliably arriving at the collector. Lastly,
we should mention that the data presented in Figure 4-10 was recorded from a nano'The illuminating laser beam is chopped at fchp = 200 Hz, and the collector and emitter currents
are independently measured on two different lock-in amplifiers.
103
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
triangle array with ARes = 1059 nm and TRad = 5.8 fs. Similar behavior was observed
for all the emitter arrays.
Collector current (d
6
J
Emitter current (Ji)
.2c
12n
z
C
Emitter
VBWs =
0
10 V
50
100
150
20
Time (s)
V
a
181.
Am=1059 nm
X = 5.8 fs
O= 179.9' 0.2IJC'- 5I 4.2 8.5 pA
1C -
18(1
1 79
0
50
100
150
200
Time(s)
Figure 4-10: Collector and emitter currents from a nano-triangle array. The
upper plot shows the simultaneous measurement of the collector and emitter
currents and the stability of these currents. The bottom plot shows the relative
phase between the collector and emitter currents. The right image shows a
reminder of our basic experimental arrangement. The nano-triangle array has
ARes = 1059 nm and rad = 5.8 fs.
4.4.2
Photoemission current versus pulse energy
Here we sweep the excitation laser pulse intensity and observe the photoemission
current yield for different biases and different device types. Starting at the laser's full
pulse energy and slowly decreasing it with a neutral density filter, we measure the
collector current. We find that at low pulse energies the photoemission current follows
a power-law scaling of
-
I
where I is the excitation pulse intensity (I oc Ep where
Ep is the pulse energy). At higher pulse energies, the photoemission current rollsoff from this scaling and, for a range of pulse energies, scales like J2. An example
of this behavior is plotted in Figure 4-11 and Figure 4-12. Figure 4-11 shows the
photoemission current at the collector versus pulse energy for a nano-triangle array
with ARes = 1105 nm and rad = 6.4 fs. Data for four different collector bias voltages:
5 V, 10 V, 20 V, and 30 V are displayed. Figure 4-8 shows a similar presentation of
data but for nano-rod devices with ARes = 1041 nm and rad = 4.8 fs.
The data displayed in Figure 4-11 and Figure 4-12 is representative of similar
104
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
5. 5I
A,=1105nm
-cd =6.4fs
101
10
C
CP
-
40
VBus= 30 V
= 20V
V
0
2=
1042.6 nA
0.0125
0.025
37 e-
pulse /emitter
0.05
0.1
0.2
Pulse energy (ni)
Figure 4-11: Photoemission current versus pulse energy for a nano-triangle
array. The array has ARes = 1105 nm and Tra = 6.4 fs. The current scaling is
measured for four different collector biases. At low pulse energies the current
scales as ~ 1 5, while at higher pulse energies this scaling falls off to - I2. The
blue dot at the top right of the graph labels a data point that shows 42.6 nA of
current which corresponds to approximately 37 electrons per pulse per emitter.
photoemission current scaling measurements performed on our other devices. There
are several important trends to mention. Firstly, the I5 5 scaling at low pulse energies
agrees with our expectation for multiphoton photoemission. The central wavelength
of our excitation laser is 1.2 pm, and, as mentioned, the work function of gold is
around 5.1 eV, so we expect to need 5 - 6 photons to surmount this barrier. We
therefore expect the multiphoton photoemission current to follow an P scaling with
n = 5 - 6. This is exactly what we observe at low pulse energies. At higher pulse
energies and intensities, we expect that the photoemission mechanism will shift from
multiphoton to strong-field photoemission. As discussed in Chapter 2, we expect a
resulting roll-off of the current scaling. This is again what we observe. The turning
point for this current roll-off occurs near Ep ~~25 pJ, which implies at Ep ~ 25 pJ
the Keldysh parameter is close to unity. With our focal conditions and wavelength,
105
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
this implies a field enhancement of ~ 20 for our devices, which is typical of what has
been measured for similar plasmonic nanoparticles [22].
Another observation is that all of the current scaling curves at the different biases
share very similar shapes. They all have similar turning points and scaling behavior.
An initial thought may have been that the current roll-off at higher pulse energies is
due to a space-charge effect which, at high photoemission current yields, begins to
limit the scaling; however, the fact that these curves are independent of bias indicates
otherwise. This suggests that strong-field effects are indeed responsible for the current
roll-off behavior.
15.5
Aae, =1041 nm
t,
= 4.8 fs
,'
,
101
I
C
100
-
110
C
101
.J2 -0oo
0040
0
VBs=
10-2
I
= 20 V
= 10 V
= 5V
U
10~
30 V
if
= 34.3 nA = 21 e- / pulse / e mitter
0.0125
0.025
0.05
0.1
0.2
Pulse energy (ni)
Figure 4-12: Photoemission current versus pulse energy for a nano-rod array.
The array has ARe, = 1041 nm and rad = 4.8 fs. The current scaling is measured
for four different collector biases. At low pulse energies the current scales as
~ I 5, while at higher pulse energies this scaling falls off to ~ 12. The blue dot
at the top right of the graph labels a data point that shows 34.3 nA of current
which corresponds to approximately 21 electrons per pulse per emitter.
In the preceding measurements we looked at photoemission current scaling behavior from nano-rod and the nano-triangle emitters under different bias conditions.
Now, let us consider the photoemission current scaling for emitters with different sizes
106
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
and resonant behaviors. In Figure 4-13 and Figure 4-14, we show currents scaling
data for four different nano-triangle arrays and four different nano-rod arrays (all
measured at a 30 V bias). In Figure 4-13 we see that for three of the four different
nano-rod arrays, the current scaling curves take on a very similar shape. This again
implies that the field-enhancement for these arrays is comparable. This is reasonable
as these three arrays all have their resonant wavelengths in the bandwidth of the
excitation laser pulse (AR, = 1256 nm, 1158 nm, and 1059 nm). The off-resonant
array takes on a very different current scaling shape and emits far fewer electrons.
In Figure 4-14 we see similar behavior for nano-rods. The on-resonant arrays show
similar current scaling behaviors, while the off-resonant array shows slightly different
behavior.
101
ARes = 1256 nm
100
= 951 nm
J5.5
= 1158 nm
= 1059 nm
4..
C
I.-1
a
0
10-1
0
0
= 10'
0.0125
0.025
0.05
0.1
0.2
Pulse energy (ni)
Figure 4-13: Photoemission current versus pulse energy for four different
nano-triangle arrays. The four arrays have ARe, = 951, 1059, 1185, and 1256
nm. The measurements were made at 30 V bias.
107
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
10
AR= 1238 nm
= 1177 nm
15.51'
= 1041 nm
=
<
968nm
4
10
5 10
%=0
10
0.0125
0.025
0.05
0.1
0.2
Pulse energy (ni)
Figure 4-14: Photoemission current versus pulse energy for four different
nano-rod arrays. The four arrays have Ages = 968, 1041, 1177, and 1238 nm.
The measurements were made at 30 V bias.
4.4.3
Device degradation
.
The final photoemission measurements we present in this section concern stability and
degradation. One possible issue with our on-chip emitters operating under ambient
condition could be stability. Considering the extremely dirty, out-of-vacuum operating environment, we first look at how repeatable our photoemission measurements
are.In Figure 4-15 two measurements are shown of the current scaling behavior for
a nano-triangle array. The first measurement (red) shows the same data displayed
in Figure 4-11 (30 V collector bias). The second measurement (blue) was made by
increasing the optical intensity. The sweep direction of the neutral density filter was
reversed and the photoemission current was measured. This measurement was taken
several minutes after the first. Over this time window the current scaling measurements are clearly very stable and repeatable 7
7 We
should also note that our emitter arrays are used many times over month long periods and
continue to show good, consistent performance.
108
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
As=1105nm
101
xm
=6.4fs
100
0
am
a 10-2
10'
9
Vus= 30 V
10
42.6 nA = 37 e- / pulse
0.0125
0.025
0.05
I emitter
0.1
0.2
Pulse energy (ni)
Figure 4-15: Repeatable photoemission current scalings. The measurement
was made at 30 V from a nano-triangle array with Ages = 1105 nm and TRad =
6.4 fs. The red trace is the same measurement as shown from Figure 4-7. The
blue trace is measured by sweeping the intensity in the opposite direction several
minutes later.
The second concern when it comes to device repeatability is degradation. Under
the ambient conditions the devices might degrade due to the dirty environment, ion
bombardment, etc. Figure 4-16 shows a microscope image and an SEM image of
the devices after fairly extensive use8 . From the microscope image the devices do
not seem to change their optical properties. They appear identical under the optical
microscope; however, near the edge a light colored layer is appearing on the collector
side. Looking at the SEM, this light colored layer seems to be due to a de-lamination
of the ITO layer from the collector. Additionally, looking at the SEM, the device
quality on the emitter side does not seem to have degraded at all. This degradation
8For our purposes, extensive use means many experimental runs, but likely only on the order of
tens of minutes of continuous exposure to the laser at the highest pulse energy.
109
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
of the ITO does not have a tremendous effect on the emission (the ARe, = 1158 nm
trace from Figure 4-13 was measured with the labeled array). The only effect of this
degradation seems to be to lower the effective bias field; as the ITO de-laminates,
the high-quality ITO pushes further from the emitter or, effectively, the collector is
pushed further from the emitter. The origin of this degradation is unclear, but likely
relates to technical details associated with the ITO deposition process or possibly the
ITO etch process.
a
b
Figure 4-16: Emitter array degradation. a. Microscope image of extensively
used device arrays (the labeled array was used in this condition to measure
the ARe, = 1158 nm trace from Figure 4-13). The light colored strip near the
collector edge forms during device operation. b. SEM image of the labeled
device array. The light colored strip from the microscope image appears to be
ITO de-lamination.
4.5
Probing the plasmonic field via IAC
In this section we describe interferometric autocorrelation (IAC) experiments performed with our devices. In our preceding discussion we saw that for a range of pulse
energies, as the current scaling begins to roll-off into the strong-field regime, the photoemission current scales as 1'. We can use this second-order non-linearity to carry
out second-order IACs with our plasmonic nanoparticle devices'. These autocorrelation measurements allow us to characterize the temporal shape of the plasmonic field
around the or nanoparticles and may validate the time-domain predictions from our
'Similar measurements have been performed using the non-linear light produced when plasmonic
nanoparticles are illuminated by femtosecond excitation pulses. However, these studies have been
restricted to third-order IACs [30, 74, 751.
110
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
simple nano-resonator circuit model. The results from three such IAC measurements
are given in Figure 4-17, Figure 4-18, and Figure 4-19.
AR, = 1256 nm
.
T,.d= 7A fs
-
-40
-30
-20
-10
0
Measured IAC
expected IAC
SHG IAC
10
20
30
40
10
20
30
40
delay (fs)
0. 8
-
Excitation pulse
Nano-res. pulse
0.6
EU
0.4
0.2
-40
-30
-20
-10
0
time (fs)
Figure 4-17: Interferometric autocorrelation measurement performed with
the strong-field photoemission current. The device used in this measurement
is a nano-triangle array with ARe, = 1256 nm and Trad = 7.4 fs. The top
trace shows the measured autocorrelation (red), the expected autocorrelation
from our time-domain model (blue), and the second-harmonic generation (SHG)
autocorrelation measured previously. The bottom panel shows the measured
pulse (black) that excites the nano-resonator array and the expected pulse from
our time-domain model (blue).
The results in Figure 4-17 agree extremely well with the time-domain predictions
of our simple circuit model. Note the data shows a dramatically broadened autocorrelation compared to what was measured in section 4.3.2 from the excitation laser
pulse directly out of the femtosecond laser system. When the pulse excites the plasmonic mode on the nanoparticle resonator, it broadens as the resonator filters the
pulse. The excellent agreement between the predicted autocorrelation, derived from
the time-domain response of our simple circuit model, and the measured result give
confidence in the validity of our simple frequency and time-domain model for the
111
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
nano-resonators as well as our measured pulse shape.
8
AR. =1105 nm
= 6.4 fs
T
7
6
_____
*
expected IAC
-
______
Measured IAC
-
SHG IAC
5
4
3
1
-40
-30
-20
-10
0
10
20
30
40
10
20
30
40
delay (fs)
-
0.8
-
Excitation pulse
Nano-res. pulse
0.6
0.4
0.2
0
-40
-30
-20
0
-10
time (fs)
Figure 4-18: Interferometric autocorrelation measurement performed with
the strong-field photoemission current. The device used in this measurement
is a nano-triangle array with ARes = 1105 nm and Trad = 6.4 fs. The top
trace shows the measured autocorrelation (red), the expected autocorrelation
from our time-domain model (blue), and the second-harmonic generation (SHG)
autocorrelation measured previously. The bottom panel shows the measured
pulse (black) that excites the nano-resonator array and the expected pulse from
our time-domain model (blue).
The results in Figure 4-18 show the data for a second autocorrelation on a nanoresonator array shifted slightly off-resonance.
Again, there is excellent agreement
between our predicted autocorrelation and the measurement. Note, that for this array the pulse broadening is less significant as the resonance is slightly off the central
wavelength of the excitation pulse. Finally, in Figure 4-19, we present one last autocorrelation measurement that further corroborates the accuracy of our circuit model.
This trace shows a pulse that is barely broadened by a nearly off-resonant nanoparticle resonator. Lastly, we should mention that similar results have been found for
autocorrelations from all of our device arrays, and such that overall, we have high112
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
confidence in the predictive abilities of our simple circuit model for our plasmonic
nano-resonators and in our measured pulse shape.
8
7
6
AR,, = 1041 n
,,d = 4.8 fs
I
Measured HAC
-
-SHG
5
EU
expected IAC
IAC
4
11
-40
-30
-20
-10
0
10
20
30
40
10
20
30
40
delay (fs)
1
0.8
-
Excitation pulse
Nano-res. pulse
0.6
Ed
0.4
0.2
0
-40
-30
-20
-10
0
time (fs)
Figure 4-19: Interferometric autocorrelation measurement performed with
the strong-field photoemission current. The device used in this measurement
is a nano-triangle array with ARes = 1041 nm and Trad = 4.8 fs. The top
trace shows the measured autocorrelation (red), the expected autocorrelation
from our time-domain model (blue), and the second-harmonic generation (SHG)
autocorrelation measured previously. The bottom panel shows the measured
pulse (black) that excites the nano-resonator array and the expected pulse from
our time-domain model (blue).
4.6
Carrier-envelope phase sensitivity
The final set of experiments we will carry out with our nano-emitter devices will be
targeted at looking for carrier-envelope phase effects in the photoemission current. As
discussed, carrier-envelope phase effects are a signature of optical waveform control,
and this area has been one of the more hotly targeted measurements researchers have
sought to perform with strong-field photo-emitting nanostructures. In the following
113
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
section, we will describe preliminary carrier-envelope phase sensitivity measurements,
and we will discuss modeling the CEP effect.
First, recall our simple model for CEP sensitivity from Chapter 2, and let us
consider the more specific form of Q(p) for our nano-triangle and nano-rod devices.
The electric field sensitive strong-field photoemission from our two device types is
illustrated in Figure 4-20. An interesting property of the strong-field emission current
that we noted in Chapter 2, is that it depends on the direction of the field. Since it is
a tunneling-like current the field must deflect the barrier downwards for emission, so
only one field direction will emit from a surface. With an optical pulse as illustrated
in Figure 4-20, our nano-rods will emit during every half-cycle of the optical pulse
and will emit from both ends (two-sided emission), while our nano-triangles, which
experience significant field enhancement only at their sharp apex1 0 , will only emit
from their apex and, therefore, will only emit for half of the optical cycles (onesided emission).
We can then use the results from Chapter 2 regarding one- and
two-sided emitters to model the charge emitted by the nano-rods, QR(p), and the
nano-triangles, QT (W).
F+()
F(t)
F-()
Figure 4-20: Geometry of strong-field emission. Nano-triangles and nanorods are illuminated by femtosecond laser pulses (here labeled F(t)). The Nanotriangles will only emit from their apex and therefore only emit for half of the
pulse's optical cycles, while the nano-rods emit from every cycle.
As described, we expect for our pulses the carrier-envelope phase sensitive current
to be relatively small. For the measurement, we therefore stabilize the carrier-envelope
offset frequency, i.e. fCEO = 2 kHz, and look at the radiofrequency (RF) spectrum
of the emitted current with a narrow resolution bandwidth near this frequency. The
results from an initial measurement are shown in Figure 4-21. In the top panel of
Figure 4-21, we see the RF spectrum at 1 Hz resolution bandwidth of the emitter
'0 This is the sharp apex that is aligned with the polarization direction of the field.
114
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
current from the nano-triangles (blue). We see a strong ~ 23 dB component at the
fCEO. This component disappears when the fCEO is unlocked (red). In the bottom
panel of Figure 4-21, we see the same plot for the nano-rods. Since we expect no
odd-harmonics of
fCEo
in the current from the nano-rods, it is no surprise that no
peaks are visible.
10
A for triangles
0
J.ne
10RBW
CL.
~23 dB
= 1 Hz
1.9
_1
-A
20
1.95
2
Frequency (kHz)
2.05
2.1
2
2.05
2.1
Ifia
0
to
go -10
JE for rods
JEfnoise
RBW = 1 Hz
-20
1.9
1.95
Frequency (kHz)
Figure 4-21: Initial CEP sensitivity measurement. The top trace shows the
RF spectrum of the emitter current for a nano-triangle array and the bottom
trace for a nano-rod array with the fCEO locked to 2 kHz. The noise data
corresponds to when the fCEO is unlocked. The units dBpA are equivalent to
20loglo(JE/1 pA).
We next confirm the CEP sensitivity of the current by carrying out a simple phasestepping experiment. In this experiment a barium fluoride wedge is progressively slid
through the exciting laser pulse. The mismatch between the group velocity and phase
velocity in the wedge will lead to a shift in the absolute CEP of the laser pulse train
as it passes through the wedge. In this experiment we again lock the fCEo to a 2 kHz
local oscillator. We then use the signal from this local oscillator as the reference to a
lock-in amplifier. The lock-in amplifier measures the carrier-envelope phase sensitive
115
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
component of the emitter current from a nano-triangle device (same device used in
the data from Figure 4-21). The relative phase measurement on the lock-in should
give us the phase difference between the local oscillator and the CEP of the pulse
train. Therefore, we expect that if the response on the lock-in is actually due to a
CEP sensitive effect, then sliding the wedge through the pulse train will step this
relative phase. In our experiment we shift the wedge 2.5 mm with a translation stage
every 10 s.
From the expected group and phase velocity mismatch at the central
wavelength of our pulses i.e. 1.2 ptm, we expect this phase to step by ~ 560. This is
almost exactly what we observe. The deviations from this behavior are explained by
the warm-up or settling time of the lock-in and the 167 mrad rms deviations expected
in the stabilized carrier-envelope phase (see Figure 4-8).
7
450
400
BaF wedge
"0 350
.
300
250
200,
Insert .75* B F 2 wedge
2.5 mm every 10 s
100
50
56 qCEP shift
10
20
30
40
time (s)
50
60
70
80
Figure 4-22: Absolute phase stepping measurement. A barium fluoride wedge
is stepped through the excitation pulse train shifting the absolute CEP of the
pulse train. The response is measured via lock-in detection.
Thus far we have seen convincing evidence that our strong-field nano-devices show
carrier-envelope phase sensitive properties in their photoemission currents. Here we
would like to see if our simple model can accurately predict this sensitivity. To this end
let us define the carrier-envelope phase sensitivity. This sensitivity, S, will be defined
as the CEP sensitive current divided by the average current.
116
The CEP sensitive
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
current
JCEP
is the magnitude of the first harmonic of fCEO in the photoemission
current spectrum. The average current is just the mean photoemission current, i.e.
JE.
If we calculate the sensitivity for our measured pulse shape (see Figure 4-7b or
the field in Figure 2-13 and the model in Chapter 2), we find that S ~1-4. Now
if we input our measured pulse shape into our time domain circuit model for our
plasmonic nanoparticles and use the resulting pulse to calculate the sensitivity, then
we find a different sensitivity for each nanoparticle type. These different sensitivities
are plotted in Figure 4-23b. We refer to this model as the 'super-simple model' for
the CEP sensitivity or the SSM model.
In a next experiment we measure the carrier-envelope phase sensitive current
from seven different nano-triangle devices. Five of these traces are shown in Figure
7-23a. From Figure 7-23, note that the magnitude of the CEP sensitive current varies
as we shift the resonant wavelength of the device.
The off-resonant devices show
relatively little CEP sensitive current, the CEP sensitive current is maximized just as
the resonant wavelength of the devices begins to approach the excitation spectrum,
and then the CEP sensitive current falls off as the devices move on resonance. We
then translate these measurements of
is then plotted in Figure 7-23b.
JCEP
to CEP sensitivity numbers. This data
From Figure 7-23b, we see that remarkably our
extremely simple model for the CEP sensitivity very accurately predicts the measured
sensitivity. This is a very initial result, and it still needs to be seen how reproducible
this agreement is as the pulse shape shifts or more nanoparticle arrays are used.
117
CHAPTER 4. PHOTOEMISSION FROM PLASMONIC NANOPARTICLES
a
A,= 951 nm A4= 1000 nm Af = 1059 nm AR.= 1105 nm A=
I
I
*
Salft
*
CL
-
...
*
1.95
2
2.05
I
*
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
2
I
I
I
I
I
*
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
2.05
2.05
2
Frequency (kHz)
2
2.05
2
2.05
b
-----------------------------
----------------o
0
08
-.
8
*
0
600
SSM
0
8
model
850
900
950
1000
1050
1100
1150
1200
1250
1300
AR. (nm)
Figure 4-23: CEP sensitivity versus ARes. a. RF traces of the emitter
currents from five different nano-triangle arrays are displayed (note each trace
is measured from 1.94 - 2.06 kHz). b. The corresponding sensitivity is plotted
for these five devices and two additional ones.
118
Chapter 5
Enhancement cavities for
high-harmonic generation
In this chapter and the following ones, we depart from strong-field physics at the
nanoscale and move to discuss cavity-enhanced high-harmonic generation (HHG)1.
We will outline the basics of cavity-enhanced HHG. The essential motivation for
enhancement cavities for strong-field physics will be described, and the concept of
passive amplification inside an ultrafast, femtosecond enhancement cavity will be
reviewed. Next, we will discuss the requirements for an effective enhancement cavity
for HHG and the challenges associated with designing such a cavity. Finally, previous
and ongoing work in this area will be discussed, and we will outline our novel approach
to designing enhancement cavities for strong-field physics.
5.1
Enhancement cavities for strong-field physics
Interest in strong-field physics has exploded over the past several decades.
HHG
has provided a route to compact, coherent sources of short-wavelength light in the
extreme-ultraviolet (EUV) and soft x-ray regime, attosecond science is pushing temporal resolution to the atomic scale, and new applications are constantly appearing.
However, as mentioned in Chapter 1, to reach the necessary intensities for strong-field
physics, complex amplifier systems are generally required. As we have discussed, such
amplifiers can readily produce millijoule pulses of tens of femtoseconds in duration,
however only after reducing the seed oscillator's repetition rate from near 100 MHz
down to the kHz regime.
In recent years femtosecond enhancement cavities have
'We should note that much of the following chapters parallels previous published work by the
author. See Refs. [76, 77].
119
CHAPTER 5. ENHANCEMENT CAVITIES FOR HIGH-HARMONIC GENERATION
emerged as an alternative route to achieving the high-intensities necessary for strongfield physics with the additional advantage of maintaining the driving oscillator's high
repetition rate [26, 78, 79, 80, 81, 821.
The coupling of an ultra-short pulse train from a mode-locked laser to a femtosecond enhancement cavity is sketched in Figure 5-1. Inside the femtosecond enhancement cavity, pulses are enhanced through constructive interference.
After a single
round-trip through the cavity, a pulse constructively interferes, adds to, the next
pulse in the optical pulse train (see Figure 5-1a). When the cavity/mode-locked laser
system is appropriately stabilized such that each mode in the frequency comb overlaps
with a cavity resonance, then each comb mode experiences the natural enhancement
inside the cavity and, accordingly, the entire pulse is amplified (see Figure 5-1b).
a.
Time Domain
b.
Frequency Domain
ci
ii
*1
Resomwe
Frquency
*~ComibMode
Frequency
Figure 5-1: Operation of a femtosecond enhancement-cavity. a. In the time
domain, small portions of the incident pulse train are transmitted into the cavity and add to the circulating intra-cavity pulse. If the cavity parameters are
properly tuned, these transmitted pulses will constructively add and build-up
the energy of the intra-cavity pulse. b. In the frequency domain, the cavity
has a comb of resonances spaced by the free-spectral range of the cavity. If each
spectral mode of the incident optical pulse train overlaps a cavity resonance, the
pulse train will be enhanced.
The utility of this intra-cavity pulse enhancement for HHG can be understood in
two-ways. First, the cavity can be interpreted as a passive amplifier that increases
the power of the incident pulse train to levels suitable for HHG. Since the intensity
enhancement in a cavity is ~ 1/loss, the intra-cavity loss must be low for significant
amplification.
Accordingly, the loss associated with the HHG process or the HHG
conversion efficiency must be low; however, since HHG energy conversion efficiencies
2
2
Although conversion efficiencies for HHG are typically < 10- 5 losses may be much higher due
120
CHAPTER 5. ENHANCEMENT CAVITIES FOR HIGH-HARMONIC GENERATION
are typically < 10-5 one expects suitable amplification [83]. This interpretation of
the cavity operation is generally adopted by the cavity-enhanced HHG community;
however, an alternative interpretation is to consider HHG as a general non-linear
process with some conversion efficiency
as L.
i]
and some total loss which we will now define
Considering the cavity enhancement of a 1/L, the effective HHG efficiency
becomes ~ 7/L (>> 7). In this interpretation the cavity's function is to increase the
efficiency of HHG production from the high-repetition rate, un-amplified pulse-train
coupled into the cavity by recycling the un-converted portion of each pulse for further
HHG generation.
Given the preceding discussion, high-intensity femtosecond enhancement cavities
are of great interest for strong-field physics.
Recently, femtosecond enhancement
cavities have been demonstrated with intra-cavity average powers of kilowatts, and
intra-cavity peak intensities > 10" W/cm2 have been achieved in such cavities operating at typical megahertz oscillator repetition rates [26, 80, 84]. Intra-cavity HHG
(i.e. cavity-enhanced HHG) has been demonstrated with tens of microwatts of average power per harmonic in the extreme-ultraviolet [26, 80].
These sources have
allowed for novel EUV spectroscopic studies [80] and may make way for higher flux
sources of EUV and soft x-ray light in years to come.
5.2
Enhancement cavity design for HHG
Although tremendous progress has been made with high-intensity femtosecond enhancement cavity technology, many strong-field physics applications demand higher
intra-cavity peak intensities than have been achieved. Intra-cavity peak intensity is in
part limited by the damage threshold of the cavity mirrors and the achievable mode
waist on the cavity mirrors (~ 1 mm for high-intensity bow-tie cavities) [78, 79, 26, 80].
Additionally, the next generation of high-intensity femtosecond enhancement cavities
for HHG require higher harmonic yields. Harmonic yields in cavity-enhanced HHG
are largely limited by the out-coupling optics. Sapphire plates [78, 79, 26], EUV gratings [81, 80], and small apertures in the cavity mirrors [82, 85, 86] have been used
for coupling high-harmonics out of enhancement cavities, but the best demonstrated
out-coupling efficiencies remain relatively low (a 10%).
Considering the above requirements, the ideal enhancement cavity for HHG must
possess three main characteristics:
to ionization that does not contribute to the HHG output. This is an ongoing area of study.
121
CHAPTER 5. ENHANCEMENT CAVITIES FOR HIGH-HARMONIC GENERATION
1. Significant intensity gain from the mirror surfaces to the focus.
In
the following, we will use the term intensity gain to refer to the ratio of the
optical intensity at the intra-cavity focus to the intensity at the cavity mirrors.
Dielectric mirrors can only withstand intensities of around 1010 - 1011 W/cm 2
without damage.
Additionally, since phase-matching considerations limit the
minimum spot-size of the driving laser pulse to be = 30 pm, significant intensity
gain means a large spot-size on the cavity mirrors.
2. A method to couple the HHG beam out of the enhancement cavity.
Due to momentum conservation, the HHG beam is generated collinearly to the
driving beam. Since EUV/soft x-ray light is strongly absorbed by virtually all
materials, out-coupling the HHG is a non-trivial task.
3. A desirable intra-cavity beam for phase-matching the HHG process.
For brevity, this last requirement will not be discussed here; it suffices to say
that phase-matching HHG is a complicated, still-debated subject, and for our
purposes this requirement translates simply to a beam with a spot size > 30
pm.
The bow-tie ring cavity has been the workhorse of previous studies on cavityenhanced HHG. A sketch of two bow-tie ring cavities is presented in Figure 5-2.
The modes of these resonators are the conventional Hermite-Gaussian beams. These
resonators allow spot sizes of several mm 2 on the cavity mirrors while having focused
waists of e
30 pm. Such intensity gain is suitable for HHG at the lower end of the
intensity range = 10
W/cm 2 , but scaling intensities to > 1014 W/cm
2
pushes the
required intensity on the cavity mirrors beyond the damage threshold.
An additional challenge with the bow-tie cavity geometry has been out-coupling
the harmonics.
As mentioned, a popular out-coupling method has been to use an
intra-cavity sapphire plate (see Figure 5-2a) oriented at Brewster's angle for the
driving laser light. The plate then presents virtually no loss for the pump light while
giving a small Fresnel reflection to the EUV. The disadvantages of this technique are:
the Fresnel reflection is relatively small for the EUV giving an out-coupling efficiency
of < 10%; and, for high-intensities, non-linearities in the plate [87] are unavoidable,
and these non-linearities limit the in-coupling to the cavity.
An alternative out-
coupling method uses an EUV grating etched onto a highly-reflecting cavity mirror
(see Figure 5-2b) [81, 80]. The small pitch of the grating does not affect the driving
122
CHAPTER 5. ENHANCEMENT CAVITIES FOR HIGH-HARMONIC GENERATION
Out-coupled
High Harmonics
a.
Brewster Plate
arculating
Driving Ught
b.
Diffracted High Harmonics
Circulating
Driving Ught
EUVGrating/High
Reflector
Figure 5-2: Bow-tie ring cavities and popular out-coupling schemes for intracavity HHG. a. A sapphire plate is placed in the cavity and Brewster's angle
to couple out the generated HHG beam. b. An EUV grating is etched on to a
highly reflecting cavity mirror to diffract out the generated high harmonics.
light, but diffracts the generated harmonics out of the cavity. This technique avoids
the non-linearities of the sapphire plate, but still presents an out-coupling efficiency
of only ~ 10%. Finally, some additional work has been done recently using higherorder Hermite-Gaussian modes in bow-tie cavities to allow for small apertures in the
cavity mirrors [82, 85, 861. These studies have been restricted to using only several
Hermite-Gaussian modes however and have had limited success primarily due to the
very small apertures < 100 tm they allow.
In the coming chapters, we will describe high-intensity cavities based on BesselGauss beams. We will design and prototype such cavities, and as we will see, Bessel-
Gauss intra-cavity modes might allow us to bypass the major hurdles and challenges
of cavity-enhanced strong-field physics [76, 77]. These cavities allow for large (> 1
123
CHAPTER 5. ENHANCEMENT CAVITIES FOR HIGH-HARMONIC GENERATION
mm) diameter holes in the cavity mirrors as well as centimeter size effective mode
diameters on the cavity mirrors. These Bessel-Gauss type cavities might allow for
efficient out-coupling as well as increased intensity gain for future cavity-enhanced,
high-intensity physics.
In the following two chapters, we will first derive the basic
properties of Bessel-Gauss beams and develop the tools to work with these beams
(Chapter 6). We next will put these developments to use and design Bessel-Gauss
cavities for high-intensity applications (Chapter 7).
Additionally, we will describe
preliminary demonstrations of these cavities as well as highlight the new challenges
these Bessel-Gauss designs present (Chapter 7).
124
Chapter 6
Bessel-Gauss beams
In this chapter we introduce Bessel-Gauss beams. We begin by deriving the somewhat more familiar Bessel beam and show how the Bessel-Gauss beam-type emerges
as a paraxial and physically-realizable approximation to the Bessel beam. Next, we
construct Bessel-Gauss beam solutions by superposing familiar Gaussian-like beams.
We then describe the focal and far-field properties of Bessel-Gauss beams and illustrate their advantages for strong-field enhancement cavities. Lastly, in preparation
for design of such cavities with Bessel-Gauss modes, we analyze how Bessel-Gauss
beams transform as they traverse simple optical elements.
6.1
Bessel beams to Bessel-Gauss beams
In deriving the Bessel beam, or any optical beam-type, the starting point is the
Helmholtz equation
(6.1)
(V 2 +k 2 ) U(, z) =zO
where U(', z) is the usual complex scalar amplitude of the optical beam, and the
equation is written in cylindrical coordinates (j', z) = (r, 0, z). Let us look for beamlike solutions to Eq. (6.1) with planar wavefronts, i.e. let us look for solutions of the
form
UB(-, z) = A (r)eikzz
(6.2)
Inserting this ansatz into Eq. (6.1), we find that A(f) must satisfy the two-dimensional
Helmholtz equation
(6.3)
(V2 + 2 ) A(-) = 0
where VT = r
+
+
12
is the Laplacian in polar coordinates, and
125
=
k2 - kz.
CHAPTER 6. BESSEL-GAUSS BEAMS
Eq. (6.3) is of a very familiar form and can readily be solved by separation of variables.
The solution takes on the well-known form
A
= AmJm(/3r)eimO,
m = 0,
2, ...
1,
(6.4)
where Jm(-) is the Bessel function of the first kind and of mth order, and Am is a
constant. Considering m = 0, our complex scalar wave amplitude is then
UB(r-,
Our solution
UB(', z)
z)
(6.5)
A 0 J 0 (,8r)eikzz
is the well-known Bessel beam [88, 89]. Note that the trans-
verse amplitude of the beam takes the form of a familiar Bessel function while the
wavefronts of the beam are planar. In other words, the beam does not diffract. This
seemingly remarkable result has contributed to the tremendous interest in Besseltype beams.
r(,
)
However, considering the behavior of IUB
AJ2(r)
2
~ A0/r x cos (8r - 7r/4). So,
(, z)
2
for large r, we find
UB(rz)I 2 decays for large r
like l/r. Therefore, the energy in the Bessel beam, i.e. E
-
f0
rdrIUB (,
z)1 2 , must
be infinite.
This divergent behavior is readily visible in the k-space distribution of the Besselbeam.
In Figure 6-la we have plotted the two-dimensional Fourier transform of
UB(', z) = AoJo(Or). In k-space, the Bessel beam corresponds to a singular ring
This singular ring implies that the Bessel beam is composed of a superposition of
plane-waves with each plane-wave component having its central wavevector directed
along the surface of a cone.
This plane-wave picture not only suggests the infinite energy content of the Bessel
beam, but it also hints at a construction for a physically-realizable Bessel-like beam.
Since it is the singular nature of the k-space that leads to the infinite energy demands,
let us add some width to the Bessel beam's singular k-space ring. Mathematically,
we can convolve a smooth function with the singular ring to provide this width.
Convolving with a Gaussian function, we obtain a k-space distribution as illustrated
in Figure 6-1b.
Convolving with a Gaussian in k-space translates to multiplying by a Gaussian
in real space. Therefore the real-space beam corresponding to the 'physical' k-space
'This Fourier transform result follows directly from the integral representation of Jo(r),
Jo(r) ~ J0 exp(-irsinO)dO (see Ref. [901 pp. 140).
126
i.e.
CHAPTER 6. BESSEL-GAUSS BEAMS
distribution illustrated in Figure 6-1b, takes on the following form at the focal plane
UBG(iz=O)
=
UB(F,z = 0) x exp(-r2 /w2)
= AoJo(#r) x exp(-r2 /w )
(6.6)
where in the above we have used a real-space Gaussian with width wo. This beam
type is known as the Bessel-Gauss beam, and it turns out that it is an exact solution
to the paraxial Helmholtz equation [91]. The Gaussian windowing function provides
for a beam with finite energy that is a physically-realizable approximation to the
Bessel beam.
Figure 6-1: k-space distribution for a Bessel and a Bessel-Gauss beam. a.
The k-space distribution for a Bessel beam at the focal plane z = 0. b. The
k-space distribution for a Bessel-Gauss beam at the focal plane z = 0.
Unlike the Bessel beam, we expect the Bessel-Gauss beam to diffract. From Eq.
(6.6), we see that near the focal plane at z = 0, the optical wave amplitude is peaked
near r = 0, i.e. the z-axis or the optical axis. As the beam diffracts, we expect it
to spread, and somewhere in the far-field we expect the amplitude will resemble the
spatial Fourier transform of the amplitude at the focus. In other words, we expect
somewhere in the far-field, for the beam to resemble the k-space distribution near
the focus. Since the k-space distribution near the focus takes on the annular shape
illustrated in Figure 6-1b, we expect the Bessel-Gauss beam will go from a tightly
focused spot to a large annular beam.
From this simple analysis we already can see the advantages of Bessel-Gauss beams
127
CHAPTER 6. BESSEL-GAUSS BEAMS
for cavity-enhanced high-harmonic generation (HHG). Recall from Chapter 5 that the
main requirements for enhancement cavities for HHG were 1. Large mode areas
on the cavity mirrors (hence high intensity gains or intensity ratios from cavity
focus to mirror surface) and 2. A means to out-couple the collinearly generated
high-harmonics. The Bessel-Gauss beam diffracts from a tightly focused spot to
a large annulus.
As we will see in the coming sections, the area of the annulus
is large enough to allow for significant improvement in intensity gain compared to
the conventional Gaussian modes of bow-tie enhancement cavities. Additionally, the
out-coupling advantages of the Bessel-Gauss beam are obvious; in the far-field the
Bessel-Gauss beam will have very little on-axis amplitude, and a large hole could
be placed in the cavity mirrors for out-coupling the generated harmonics while not
perturbing the intra-cavity Bessel-Gauss mode.
6.2
Constructing Bessel-Gauss beams
In the following section we will provide a more rigorous derivation of Bessel-Gauss
beams. We will take a constructive approach and build up Bessel-Gauss beams by
superposing Gaussian-like beams 2 . The Gaussian-like beams we will superpose are
called 'decentered' Gaussian beams, and as we will see, they share many familiar
attributes with the Gaussian beam and will provide valuable intuition for BesselGauss-type beams
[93]. In the following, we first introduce these decentered Gaussian
beams and then use them to construct Bessel-Gauss beams.
Decentered Gaussian beams
The decentered Gaussian beam is an intuitive generalization of the Gaussian beam.
The basic decentered Gaussian beam is illustrated in Figure 6-2. At the focal plane,
z = 0, the decentered Gaussian beam in a cylindrical coordinate system (r, z) =
(r, 0, z), takes on the form,
UdG(', z =
0)
a0 exp(-( i?- id1/wo) 2 ) x exp(i/r cos(O
-
-y))
(6.7)
where a 0 is a constant, ia describes the off-center position of the focus, and the
modulating plane wave with wavevector of magnitude /3 and direction parallel to Fd
provides some tilt to the beam. Note, referencing Figure 6-2, that Td = (rd, Y). In
2
The approach in this section is similar to that followed in prior work. In particular see Refs.
[76, 92].
128
CHAPTER 6. BESSEL-GAUSS BEAMS
other words the angle -y in Eq. (6.7) defines the inclination angle of the focal center
position with respect to the x-axis (this angle is not labeled in Figure 6-2).
0
Op
z
I
Figure 6-2: Decentered Gaussian beam. At the focal plane z = 0 (shaded),
the beam has a Gaussian distribution that is displaced from the origin by r'd =
(rd, -y). Away from the focal plane the beam resembles a tilted Gaussian beam,
propagating at an angle p to the optical axis, i.e. the z-axis.
As illustrated in Figure 6-2, at the focal plane the decentered Gaussian beam,
UdG(T, 0, z = 0), resembles a Gaussian beam of waist wo whose central wavevector has
a component of magnitude
in the transverse plane. Intuitively, it seems that this
beam might resemble a Gaussian beam that simply has a displaced focal spot and
a tilted direction of propagation. In fact this is exactly correct! If the decentered
Gaussian beam of the form above is propagated by means of a Fresnel diffraction
integral, we find the following Gaussian-like beam
129
CHAPTER 6. BESSEL-GAUSS BEAMS
2
O exp (_j k(r
q(z)
2q(z)
udG(r, 0, z) =
+
2rrc(z) cos(O
rc(z)2 -
i
-
-))
\(6.8)
x exp(i3r cos(O -
q(z) = qo
+ z
y))
rc(z) = rd + z sin p
;
(6.9)
In the above, the qo and q(z) terms correspond to the q parameter of the beam decentered Gaussian beam. This parameter transforms just like that of the conventional
Gaussian beam. Also, note the parameter rc(z) describes the 'center of mass' of the
beam or, essentially, the beam center position as a function of z. Although, Eq. (6.8)
can see opaque at first glance, closer inspection reveals that the decentered Gaussian
beam very closely resembles a conventional Gaussian beam that propagates with at
an angle r to the optical axis. Finally, note that in the above we neglect exclusively z-dependent phase terms; in the following, we will adhere to this convention
for simplicity.
Bessel-Gauss beams
Let us now consider the different decentered Gaussian beams produced as we let -,
the inclination angle of r' and # with respect to the x-axis, vary (see Figure 6-3a).
In particular, let us first consider these beams to have rd = 0. In this case, we
see the central wavevectors of the different decentered Gaussian beams trace out the
surface of a cone with semi-aperture angle o. Superposing these different decentered
Gaussian beams with variable -y, we obtain
dyudG(r,
= Ao qoexp
q(z)
0, z)
r2 ~_kr (z)( 6.10_ i k
k(r 2 + r2 (Z))
_
_
_
)
UBG(r, z) = jr
CZq
2q(z)
q(z) = qo
+ z
;
x
JO
8r -
rc(z) = z sin o
(z)
r
(6.11)
In the above we have used the integral representation of Jo, the zeroth order Bessel
function of the first kind (see pp. 140 of Ref. [90]), Ao is a constant, and q(z) and
rc(z) are given by the same expressions as before with rd = 0 (Eq. (6.9) and Eq.
(6.11) are identical with rd = 0). Looking closely at the form of Eq. (6.10), we see
130
CHAPTER 6. BESSEL-GAUSS BEAMS
that we have a beam that at the focus (z = 0) resembles a Gaussian modulating a
Bessel function. This is the Bessel-Gauss beam (abbreviated as BG beam from here
on).
b
TY
)
)
a
V
Z
Z
* I!
~*1~
Figure 6-3: Constructing Bessel-Gauss beams. a. We superpose many decentered Gaussian beams with differing -y. This amounts in superposing many
decentered Gaussian beams along the surface of a cone (rd = 0) or a frustum
(rd = 0). b. An overlay of the transverse intensity profile after the superposition. Note the annular form of the generalize Bessel-Gauss beam.
We can generalize the BG beam by letting rd take on a non-zero value. We now
superpose many decentered Gaussian beams with centers lying on a circle of radius
rd.
This superposition is again schematically illustrated in Figure 6-3a, and a sketch
of the resulting transverse intensity profile is overlaid in Figure 6-3b. The central
wavevectors of these decentered Gaussians make up the surface of a frustum (i.e. a
truncated cone) with semi-aperture angle p.
This form of beam is known as the
generalized Bessel-Gauss beam (called the gBG beam from here on):
(2
+
rT2z)
X
Jo (3
C
q(z) = qo + z
r (z)
Note that Eq. (6.12) is of the form of Eq. (6.10).
131
= Td
+ z sin p
k3r
q (z)
)
= Ao q) exp (2
q(z)
(2q(z)
(6.12)
)
UgBG(T, z)
(6.13)
CHAPTER 6. BESSEL-GAUSS BEAMS
Although the expressions for UBG and UgBG(r, z), with a Bessel functions of a
complex argument, may not easily reveal their essential properties and behaviors,
theses beam can intuitively be understood by recalling that they are superpositions
of physically-intuitive decentered Gaussian beams. The r - z plane cross-section of
a gBG beam, consisting of intersecting decentered Gaussian beams, is illustrated in
Figure 6-4a, and the amplitude is plotted for a specific gBG beam in Figure 6-4d.
Similarily, the r - z plane cross-section of a BG beam, consisting of intersecting decentered Gaussian beams with rd = 0, is illustrated in Figure 6-4b, and the amplitude
is plotted for a specific gBG beam in Figure 6-4e.
Note that the BG beam is a special case of the gBG beam. The BG beams is the
gBG beam with rd = 0. There is another special case of the gBG beam that is of
interest. Consider a gBG beam built with decentered Gaussian components that have
$
0 and o = 0. This is the modified Bessel-Gauss beam (mBG beam from here
on) and is a superposition of decentered Gaussian beams lying along the surface of a
cylinder with radius rd. The r - Z plane cross-section of a mBG beam is illustrated
rd
in Figure 6-4c, and the amplitude is plotted for a specific mBG beam in Figure 6-4f.
a
b
Generaized Besse-auis
a
d
Modified Bessel-Gauss
C
Bessel-Giuss
f
2
2
2
1
1
1
2=0
E20
E20
-20
-1--m.
-1
-1
5
10
15
20 25 30
z (CM)
35
40
05
10
15
20
z (cm)
25 30
35
40
0
.
5
10
15 20
.-
25
30 35 40
z(cM)
Figure 6-4: Types of Bessel-Gauss beams. a.-c. Illustrations of r - z plane
cross-sections of gBG, BG, and mBG beams respectively. d.-f. Plots of the
amplitude in the r - z plane for gBG (A = 1 pm, wo = 200 pm, W = 0.210, rd
= 0.25 mm), BG (A = 1 pm, wo = 200 pm, W = 0.290), and mBG (A = 1 pm,
wo = 200 pm, rd = 1 mm) beams respectively.
132
CHAPTER 6. BESSEL-GAUSS BEAMS
6.3
Focal properties of Bessel-Gauss beams
Having established some basic intuition for Bessel-Gauss type beams, we now quickly
summarize the focal properties of BG beams essential for our purposes.
Here we
discuss only BG beams as our initial cavity designs in Chapter 7 will consist of BG
beams at the foci, and BG beams are sufficient to illustrate our main results.
As already described, at the focal plane the BG beam takes the form of a Gaussian
component modulating a Bessel function. Looking at the focal plane, from Eq. (6.10),
UBG(r,
z
=
0) = AO exp(-r/w')Jo(3r). An r -0
plane cross-section of the amplitude
of a BG beam at its focus is plotted in Figure 6-5a. The peak intensity of a BG beam
at its focus, I'
can then readily be found3 from Eq. (6.10) with z = 0
2P
Ifoc
(6.14)
In this form the peak intensity of a BG beam resembles the general form of that for
a Gaussian beam with effective waist defined as
Wfoc
=
wo exp
In the above P is the beam power,
of the first kind, and
WB
I1
1
(6.15)
OWB
o is the zeroth order modified Bessel function
= 2.4/3 is the approximate waist of the Bessel component
(i.e. the first zero of Jo(#3r), illustrated in Figure 6-5a). The approximate form of
wffl given in Eq. (6.15) follows from inspecting the argument of the Bessel function
in Eq.
(6.15).
Looking at this argument, we see that /32 W2 /4
=
/svG)
2
where
o
//k is the semi-aperture angle of the BG beam (as already described) and
OG
27kwo is the divergence angle of the component decentered Gaussian beams.
Since we are interested in BG beams that result in an annular (i.e. donut) shape far
from the focus, we must have
LPG <<
o, i.e the Gaussian components must diverge
slower than their peak intensity axes spread apart.
So, for the regime of interest
3 2 W2/4 = ( O/G) 2 >> 1, and the asymptotic expansion of Io (pp. 116 of Ref. [901)
yields the approximate form of.
As mentioned, far from the focus, the BG beams of interest resemble an annular
shape. The amplitude of a BG beam far from the focus is plotted in Figure 6c. The
amplitude of the BG beam far from the focus can be approximated as an annulus
with Gaussian cross-section, i.e.
UBG(r,
z)
(Bo/Vr~) exp(-(r -- r(z)) 2 /w(z) 2 ) where
3In the following we make use of the
integral in Eq. (2.3) of Ref. [91].
133
CHAPTER 6. BESSEL-GAUSS BEAMS
w(z) =
N1
+ (z/zo) 2 , z >> zo, and Bo is a constant [94]. Using this expression, the
peak-intensity of the BG beam far from the focus at position z can be approximated
2P
P
Z r=7(W 11(Z)) 2
Again, in this form the peak intensity of a BG beam resembles that of a Gaussian
beam with effective waist defined as
)(z)
2 2- x w(z) x rc(z)
(6.17)
where rc(z) is as in Eq. (6.11), i.e. the peak-intensity axes of the component decentered Gaussian beams (illustrated in Figure 6-5c), and w(z) is as defined above, i.e.
the waist of the component decentered Gaussian beams at z (illustrated in Figure
6-5c).
We can now put together a simple expression for the intensity gain of a BG
beam.
In line with our earlier definitions, we here define the intensity gain of a
beam at a position z as the ratio of the peak intensity at the focus to the peak
intensity at the position z, so Ig(z) = IOC/IFF(z) where Ig(z) is the intensity gain.
As we have emphasized, intensity gain is a parameter of great relevance for highintensity enhancement cavities.
For a Gaussian beam, we easily see that
IG(z)
(z/zo) 2 when z >> zo. Combining Eq. (6.14) and Eq. (6.16) we find
'7rw(z)2 /7rw
the intensity gain for a BG beam
2
(G
where C = 12
2
(6.18)
zo
2w/2.4 is a constant. For comparison, we repeat the intensity gain of
a Gaussian beam
Ig()
Recall that for the beams of interest
zo
>>
OG
(6.19)
since we want annular shapes in the
far-field. Comparing the intensity gain expressions in Eq. (6.18) and Eq. (6.19), we
see that the BG beam's intensity gain can exceed that of the Gaussian beam by orders
of magnitude. In Figure 6-5b, the intensity gain of a Gaussian beam with wo = 30 pm
is compared to that of BG beams with Gaussian component wo = 30 pm and semiaperture angles o of 1', 2',
30,
and 4'. In Figure 6-5b, the green curves represent
exact numerical calculations and the orange curves are based on the approximate
134
CHAPTER 6. BESSEL-GAUSS BEAMS
b
a 0.5 BG: wo 200 pn, p =.29*
=40
10
0.25106
1'1,2wI
E 0
--
30
Exact (Numerical)
Approximate (Analytical)
V =20
1*
10
-0.25
SC
-0.5
0.5
103
(U
-A
2
10
-
C
-0.5
0
0.25
-0.25
x(mm)
10
10
10-1
-21
-2
-1
0
x(mm)
1
10
2
1
0
10
10
2
Z/z7
--
Figure 6-5: BG beam focal properties and intensity gain. a. Plot of amplitude
cross-section in the z = 0 plane of a BG beam with A = 1 pim, wo = 200 pm,
and semi-aperture angle o = 0.29' (same parameters from BG beam plotted in
Figure 6-4e). Cross-section of the focus in the y-direction is on the right with
2
WB labeled. b. Plot of approximate (orange dashed) and exact (solid green)
intensity gain of BG beams with A = 1 pm, wo = 30 pm, and semi-aperture
angles o of 1', 20, 30, and 40 at distance z. The intensity gain of a Gaussian
beam with A = 1 ym and wo = 30 pm (blue curve) is also included. c. Plot of
the amplitude cross-section in the z = 20 cm plane of the BG beam from plot a.
Cross-section in the y-direction is included on the right with w and r, labeled.
form in Eq. (6.17). From Figure 6-5b we see that our approximate expression is very
accurate far from the focus (z >> zo). Additionally, we see that for the reasonable
parameters plotted, the intensity gain of a BG beam may far exceed that of a normal
Gaussian, and therefore, the BG beam may allow cavity geometries with intensity
gains far exceeding those of bow-tie Gaussian cavities.
6.4
Bessel-Gauss beams and simple optical elements
With some intuition with regards to Bessel-Gauss beams and with a feel for the
basic focal properties of these beams, we now move on to consider the transformation
of Bessel-Gauss beams by spherical and conical optical elements. Spherical optical
135
CHAPTER 6. BESSEL-GAUSS BEAMS
elements are those that impart a quadratic spatial phase to a wavefront e.g. a thin
lens or a spherical mirror. Conical optical elements give a linear (i.e.
~ exp(icakr))
spatial phase to wavefronts e.g. transmitting or reflecting axicons. In the following,
the importance of these elements in manipulating gBG beams will be discussed.
Consider the spatial phase, i.e. the r-dependent phase, of a gBG beam at plane
z = L. Denoting this phase by
$gBG(r)
=
#gBG(r)
we find (from Eq. (6.12))
ik r 2 + i arg (JO (3r
2R(L)
-
k
(L)
rc Y)
q(L)
(6.20)
where we have expanded the Gaussian term in the conventional way so that R(L)
L + z2/L. The Gaussian part of the gBG beam gives a quadratic phase while the
Bessel part contributes the last-term in Eq. (6.20). For a large class of gBG beams,
we can accurately approximate (as shown and discussed in Appendix D and in Ref.
[95]) the last term in Eq. (6.20) as
OgBG
ik
2R(L)
T2
+ (Lro
2
z
#
k
ik
Z=I
1 + (L/zo) 2
r
(6.21)
Therefore, the spatial phase of the gBG beam is well-approximated as the sum of
a quadratic part and a linear part.
#con(r)
A conical optical element, with spatial phase
= -iozkr at z = L, changes the linear part of an incident gBG beam's spatial
phase. The overall functional form of this phase remains unchanged however, and
to account for the new linear part of the spatial phase, the gBG beam transforms
to a new gBG beam with altered parameters (q6, r, /').
From a straightforward
calculation we determine these altered parameters; they are included in Eq. (6.22).
A spherical element, with spatial phase
q5ph(r)
= -ikr 2 /2f at z = L, changes the
quadratic part of the gBG beam's spatial phase while leaving the overall functional
form unchanged.
Similarly, a gBG beam transforms after a spherical element into
another gBG beam with new parameters (q', r', 0'). This transformation has been
previously described in detail [96]. Collecting our results, we find that after conical
or spherical optical elements the gBG beam parameters transform as:
Conical (Ocon(r)
qO = q(L)
=
-iekr)
;
ro = rc(L)
136
;
-k& a=
(6.22)
CHAPTER 6. BESSEL-GAUSS BEAMS
-ikr 2 /2f)
Spherical (Obph(r)
qO =
_
q(L)
kreL_
;
-q(L)/f + 1
)
(Lro
=rc
(L)
; #= L (6.23)
The gBG beam, after a conical or spherical element, can then be written (up to a
constant phase factor [96]) in the standard form of Eq. (6.12) and Eq. (6.13) with
the substitutions qo
-*
q, ro
--
r',
#
-
13', and where z' = z - L, i.e. z' is the
distance to the optical element at z = L.
Considering the gBG transformation properties in Eq.
(6.22) and Eq.
(6.23),
we can formulate an intuitive picture of gBG beam propagation through conical and
spherical optical elements.
Propagation through such elements can be compactly
summarized as follows: 1.
Through conical optical elements, (a) the Gaussian
component i.e.
the q parameter of a gBG beam is unaffected; and (b) the peak-
intensity axes of the decentered component beams follow the trajectories of meridional rays through the element. 2. Through spherical optical elements, (a) the
Gaussian component i.e. the q parameter transforms like that of an on-axis Gaussian
beam; and (b) the peak-intensity axes of the decentered Gaussian component beams
follow the trajectories of meridional rays.
These basic propagation rules are demonstrated and tested through three examples
illustrated in Figure 6-6. In the first example (illustrated in Figure 6-6a, b, and c),
consider a spherical mirror with radius of curvature R = 20 cm at position z = R = 20
cm and an incident mBG beam of wavelength A = 1 pm, Gaussian component waist
wo = 300 pm, and ro = 1 mm (the focal plane is z = R/2 = 10 cm as shown in
Figure 6-6b). The mBG beam propagates, reflects from the mirror, and transforms
to a new gBG type beam. From Eq. (6.23) and the above discussion, we expect (a)
the Gaussian component waist of the new gBG beam to be w' = Af/ rwo = 106 pm
(as for an on-axis Gaussian) and (b) the mBG beam to transform into a BG beam
with its focus at z = R/2 (meridional rays parallel to the optical axis transform to
meridional rays intersecting the optical axis at the focus). In Figure 6-6b we plot an
r - z plane cross-section of the numerically simulated amplitude in this scenario and
observe the expected behavior. In Figure 6-6c we plot cross-sections in the r direction
of the spatial amplitude and phase of the field at the end of propagation and see our
numerical (blue) simulation agrees to a high degree of accuracy with our analytical
prediction from Eq.
(6.12) and Eq.
(6.23) (red-dashed).
137
The wave-propagation
CHAPTER 6. BESSEL-GAUSS BEAMS
software used for numerical simulation will be discussed in the next section.
b
a
C
1
15
0 .8
10
~0.6
V 5
.4
L0
0 .2
-5
-
E
C
0
z (cm)
d
1
r(mm)
r(mm)
30
-
.8
0
.6
.-g
2
Numerical
Analytilca
20
810
E
.4
0
0
.2
0
g
Numerical
Analtca
1
2-10
1
f
--
h
5
10 15
20
z(cm)
15
10
5
0
0 04-10
0
.A0
2
4
01
r (mm)
0
2
r(mm)
4
1
0
1
25
.8
20
-
-
Numerical
Analytical
C0 .6
Er-=
810
.4
.2
0
0
zICM)
0
1
r(mm)
2
-5
0
1
r(mm)
2
Figure 6-6: gBG beam transformations. a. Example 1 geometry: an mBG
beam reflecting from a curved mirror. b. r - z plane cross-section of numerically
simulated amplitude for example 1 (note z-axis corresponds to reflecting geometry). c. r-direction cross-sections of field's spatial amplitude and phase at the
end of propagation (numerically simulated (blue) and analytical (red-dashed)).
d. Example 2 geometry: an mBG beam reflecting from a reflecting axicon. e
and f are as b and c but for example 2. g Example 3 geometry: an mBG beam
reflecting from a toroidal optic. h and i are as b and c) but for example 3.
For the second example (illustrated in Figure 6-6d, e, and f), the same mBG
beam from above propagates through the same geometry and reflects from a reflecting
axicon of apex angle a = 0.570. In this example we expect the mBG beam to become
a gBG beam (we do not expect the Gaussian waist of the component decentered
beams to occur at their intersection point).
In Figure 6-6e an r - z plane cross-
section of a numerical simulation of the amplitude is plotted, and we observe the
138
CHAPTER 6. BESSEL-GAUSS BEAMS
expected behavior.
In Figure 6-6f an r direction cross-section of the field's spatial
amplitude and phase at the end of propagation are plotted, and our numerical (blue)
simulation agrees well with our analytical prediction from Eq. (6.12) and Eq. (6.22)
(red-dashed).
Finally, the third example (illustrated in Figure 6-6g, h, i) contains a hybrid
conical-spherical optic. The optic is a general toroidal optical element i.e. a spherical
and a conical element separated by zero distance. The same mBG beam from the prior
two examples propagates through the same geometry and reflects from this toroidal
element. Recall that a conical optical element adjusts the tilt parameters of a gBG
beam while leaving the Gaussian parameters alone (i.e. the conical element affects
only the peak-intensity axes of the decentered component beams), and a spherical
optical element adjusts all the parameters of a gBG beam. Therefore, by combining
a conical and spherical element into a general toroidal optic, the tilt parameters
(i.e. ro and /)
and Gaussian parameter (i.e. q parameter) of a gBG beam can be
independently adjusted by one optical element. Our final example illustrates this as
the toroidal element consists of a spherical part of radius of curvature R = 20 cm
and a conical part with tilt such that the focus (i.e. the point of intersection for the
decentered component beams) will lie at exactly z = 2R/3. In Figure 6-6h an r - z
plane cross-section of a numerical simulation of the amplitude is plotted, and again,
we observe the expected behavior. Figure 6-6i shows an r direction cross-section of
the field's spatial amplitude and phase at the end of propagation, and our numerical
(blue) simulation agrees well with our analytical prediction from Eq.
(6.22), and Eq. (6.23) (red-dashed).
139
(6.12), Eq.
CHAPTER 6. BESSEL-GAUSS BEAMS
140
Chapter 7
Bessel-Gauss beam enhancement
cavities
In this chapter we build upon the properties of Bessel-Gauss beams derived and discussed in the previous section to analyze Bessel-Gauss beam enhancement cavities.
We begin our discussion by considering the overall design principles of Bessel-Gauss
beam cavities and how to build such cavities from fundamental Gaussian designs.
We then specifically discuss the design of the confocal Bessel-Gauss cavity and possible arrangements for high-harmonic generation applications. Next, we discuss a
continuous-wave experimental demonstration of the confocal Bessel-Gauss cavity and
highlight the limitations of Bessel-Gauss modes illuminated by this demonstration.
Finally, we conclude this chapter with a summary of the work pursued on BesselGauss beams and the outlook for this nascent research field.
7.1
Bessel-Gauss beam cavity design
Prior work has explored Bessel-Gauss beam cavities with axicons and flat mirrors [98],
axicons and curved mirrors [95, 99], and general phase-conjugating optics [100]; however, this past work has focused on Bessel-Gauss cavities for use as laser resonators.
In the following section, we extend this body of work on Bessel-Gauss cavities to enhancement cavities. We outline a different general approach to designing gBG beam
cavities, discuss in detail a particular novel gBG cavity i.e. the confocal BG cavity,
and comment on future challenges in realizing high-intensity gBG cavities.
Consider an enhancement cavity composed of two spherical and two flat mirrors
supporting a Gaussian beam solution as illustrated in Figure 7-la. The Gaussian
141
CHAPTER 7. BESSEL-GAUSS BEAM ENHANCEMENT CAVITIES
beam is re-imaged as it traverses the cavity i.e. q(z + 2L) = q(z) where 2L is the
round-trip cavity length. Additionally, for small-angles, the Gaussian beam's peak
intensity axis follows that of a ray through the system.
a
Gaussian Cavity
C
Bessel-Gauss Cavity
Figure 7-1: Gaussian cavities and Bessel-Gauss cavities. a. Illustration of
a Gaussian beam enhancement cavity. Note that the harmonics (purple pulse)
are generated collinearly with the driving beam. b. The intra-cavity Gaussian
mode intensity on the cavity mirrors in the x - y plane. The dashed white circles
indicate roughly where two of the cavity mirrors lie. c. Illustration of a BesselGauss enhancement cavity. This cavity is rotationally symmetric about the
z-axis (as indicated by the red circle). Also, note that the harmonics propagate
along the z-axis. d. Intra-cavity Bessel-Gauss mode intensity on the segmented
cavity mirror in the x - y plane. The dashed white circles roughly show the
boundaries between the different sections of the segmented mirror.
For an enhancement cavity to support a gBG mode, the intra-cavity gBG beam's
Gaussian parameter (i.e. q parameter) and tilt parameters (i.e. ro and 3) must repeat after every round-trip. (Recall that the gBG beam's q parameter is associated
with the Gaussian properties (e.g. waist) of the component decentered Gaussian
beams, and the tilt parameters are associated with the peak-intensity axes of the
component decentered beams.) Consider the r - z plane cross-section of our conventional Gaussian cavity. If we revolve this cross-section about its central axis (as
illustrated in Figure 7-1b), the tilted flat mirrors become conical optical elements,
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CHAPTER 7. BESSEL-GAUSS BEAM ENHANCEMENT CAVITIES
and the spherical mirrors become toroidal optical elements (these elements can be
imagined as different sections of one complex, segmented mirror structure as illustrated in Figure 7-1b).
Recalling the transformation properties of gBG beams, we
see this cylindrically symmetric cavity structure supports a gBG mode that is composed of decentered component beams that closely resemble the Gaussian mode of
the conventional Gaussian cavity. This link between conventional Gaussian cavities
and Bessel-Gauss cavities is a powerful one. It allows us to directly generate gBG
cavity designs from well-known Gaussian ones (albeit the gBG cavities may demand
sophisticated mirror structures that are non-trivial to fabricate). In this initial discussion, we restrict our focus to gBG cavities that require only spherical mirrors (in
particular, the confocal BG cavity). Before embarking on this discussion, we should
mention that a brief description of the cavity mode-solver we will use in the following
is included at the end of this chapter.
7.1.1
Confocal Bessel-Gauss cavity
In the following we will discuss the confocal cavity and show it supports BG type
modes. The confocal cavity is degenerate i.e. every other Hermite-Gaussian mode
of the confocal cavity shares the same resonance frequency. These modes can then
simultaneously resonate in the cavity and superpose to form different field profiles. To
restrict the cavity to operate only in a single BG type mode, we consider patterning
the cavity mirrors in an annular (i.e. donut) shape. The annulus is highly reflective
(reflectivity RH) and has average radius ravg and thickness Ar (illustrated in Figure
7-2a); the rest of the mirror surface has a low reflectivity (RL). The highly reflective
annular pattern yields low-loss to only a single BG mode: the BG mode composed of
minimally-divergent decentered Gaussian beams (illustrated in Figure 7-2b).
From the cavity center to the mirror surface (a distance of R/2) there exist minimally divergent decentered Gaussian beams.
These beams have a waist wo,min
=
AR/2ir at the cavity center, and Wmin = V/2wo,min at the mirror surfaces. All other
decentered Gaussians and higher-order decentered Hermite-Gaussians have a larger
waist at the mirror surface. Therefore, if the width of the patterned annulus is chosen
to be small enough (i.e. Ar ~ 3w), then only the BG mode composed of decentered
Gaussians with waist wo,min will have low-loss. This method of single-mode selection
is analogous to inserting an iris in a laser resonator to restrict the output to the
fundamental Gaussian mode. The average radius of the annulus determines the tilt
angle of each decentered component beam, so for the BG mode, p = tan'(2ravg/R).
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CHAPTER 7. BESSEL-GAUSS BEAM ENHANCEMENT CAVITIES
b
a
Ar
least divergent mode
RL, low-reflectivity
RH, high-reflectivity
Figure 7-2: Single-mode selection in the confocal BG cavity. a. Cavity
mirror with patterned annular (donut-shaped) region of high-reflectivity. b.
Cross-section of patterned cavity mirror with incident beams.
Using our cavity mode-solver, we simulate an example patterned-mirror confocal
cavity (shown in Figure 7-3). This cavity's patterned mirrors have parameters: ravg =
8 mm, Ar = 3 .1wmin = 1.2 mm, RH = 1, and RL = 0.1. The mirror radius of
curvature is R = 50 cm and spacing is L = 49.97 cm. The cavity is simulated at
wavelength A = 1 Am. From the r - z plane cross-section plot of the mode amplitude
in Figure 4-3a, we see, as expected, the cavity mode resembles a BG beam through
one pass of the cavity (through the focus) and transforms at the cavity mirror to a
mBG beam for the return trip. In Figure 7-3b and Figure 7-3c, we plot the intensity
in the radial direction at the cavity mirror and at the focus, respectively (labeled
in Figure 7-3a). From these plots we see our numerical simulation (blue) agrees well
with the analytically expected mode (red-dashed). Additionally, normalizing the peak
intensity at the cavity mirror, we see the peak intensity at the focus is I = 1.5 x 104
(this is the intensity gain). We also see the effective waist at the focus is we! f = 33
lim. From our mode-solver, we find the loss of the fundamental mode plotted in
Figure 7-3 is < 0.0011%, and the loss of the next higher-order mode is > 2.5% (note
this is exclusively diffraction-loss as RH = 1). These losses can be fine tuned by
adjusting Ar. Additionally, we note that although we simulate a continuous-wave
cavity, the patterned mirror confocal cavity supports a wide bandwidth. Simulating
the example cavity above at A = 950 nm and A = 1050 nm, we find the fundamental
mode has < 0.0015% loss and the next higher-order mode has > 1.7% loss. Finally, we
should note for our example cavity L = R; this is due to a non-paraxial propagation
effect. For even modest tilt angles (for this cavity, o = tan- 1 (2rag/R)), non-paraxial
propagation leads to small spatial phase shifts, and to maintain low-loss modes we
must have L = R cos V.
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CHAPTER 7. BESSEL-GAUSS BEAM ENHANCEMENT CAVITIES
aM
a
-10
-
Nurnerical
-
C
Analytical
Numerical
-Analytical
16000
1
14000
12000
~0.6
E-2
8000
0~
W 0.4
10
20004
010
0.2!
2
100~~
4C
6
8
0
w6000
4000
.
E
2000
10100
20
z (cm)
30
40
6
C
r (mm)
CLM
-0.2
0
0.2
r (mm)
Figure 7-3: Patterned-mirror confocal BG cavity simulation. a. r - z plane
cross-section of fundamental BG mode amplitude. b. Normalized mode intensity at mirror surface plotted against r (as labeled in a). c. Mode intensity at
focus plotted against r (same normalization as b and labeled in a).
The example cavity above shows virtually no intensity on the optical axis at the
cavity mirrors. With millimeter-sized holes at the centers of the cavity mirrors, the
modes are unaffected. The above cavity, which corresponds to a repetition rate of
= 300 MHz, provides near-perfect out-coupling for intra-cavity HHG. Additionally, with its high-intensity gain, this cavity may support peak intensities at the focus
fR
approaching 10" W/cm2 without damage to the cavity mirrors. We can use our ana-
lytical understanding of the example cavity above and our mode-solver to see how the
properties of the patterned mirror confocal cavity scale as we shift the cavity's geometry. In particular, we are interested in how the intensity gain, Ig, and effective waist,
weff, scale with varying repetition rate and ravg. The results of an analytical and
numerical scaling are given in Figure 7-4 where we plot I and weff of the simulated
example cavity above (red dot) and other numerically simulated cavity geometries
(black dots) and the analytical scaling results for I and weff using numerical integration (green) and the approximate expressions from section 2.2 (orange-dashed).
From the scaling results, we see that the approximate and exact analytical expressions agree well with each other and with the numerically simulated cavities. All
numerically simulated modes have fundamental mode loss < 0.0016% and higherorder mode loss > 2.5%. A limitation of the patterned-mirror confocal BG cavity is
also apparent. As the repetition rate grows so does the intensity gain, and so shrinks
the effective waist. This is due to the connection between the Gaussian component
AR/2ir). For
of the BG mode and the repetition rate (connected through wo,min =
lower repetition rates, the Gaussian component is large. The intensity gain can still
145
CHAPTER 7. BESSEL-GAUSS BEAM ENHANCEMENT CAVITIES
a
4b
_
100
o Simulation points
5
-
-
_
_
__
120
x 10
o 300 MHz, I = 1.5x10 4
Analytical (Exact)
- Analytical (Approx.)
o 300 MHz, wr = 33 Am
o Simulation points
-
80
-
Analytical (Exact)
Analytical (Approx.)
60
f3
C4
40
2
20
1~
_
400
200
300
Repetition Rate (MHz)
YO
4mm20
00
500
r.,
2rm
400
200
300
Repetition Rate (MHz)
500
Figure 7-4: Patterned mirror confocal cavity scaling. a. Intensity gain, Ig,
scaling with repetition rate (i.e. cavity length and mirror radius of curvature).
b. Effective waist, weff, scaling with repetition rate (i.e. cavity length and
mirror radius of curvature). For all cavities in these plots Ar = 3.1Wmin.
be made high and the effective waist small by making a very tight Bessel focus (i.e.
small
WB)
by increasing ra,; however non-paraxial effects ultimately limit rayg, and
the patterned-mirror confocal BG cavity is likely best suited for higher repetition
rates.
There are two clear possible future challenges of gBG cavities: stability and mirror surface variations. When considering cavities with only spherical mirrors, the
requirements to support a gBG type mode lead directly to the confocal and concentric cavity (both of which, as conventional Gaussian cavities, lie on the stability
boundary). Performing a stability analysis with our cavity mode-solver on the example cavity discussed previously, we find that for a range of AL = 50 Am about the
cavity length L = 49.97 cm, the fundamental mode loss can be kept < 0.002% while
the next higher-order mode loss > 2.4%. This relatively narrow stability regime may
make realization of the patterned-mirror confocal cavity challenging.
However, we
note that gBG type cavities with more sophisticated mirror structures (not restricted
to only spherical mirrors) can easily avoid these stability regime boundary issues.
The challenges associated with mirror surface variations may prove more difficult
to remedy. Consider a cavity geometry supporting a gBG type mode; the mode is
composed of decentered Gaussian component beams. The cavity mirrors have some
surface variations associated with the manufacturing process. If one localized region
of the mirror surface varies e.g. the local radius of curvature, designed to be R, is
actually R + AR, then the decentered Gaussian component beam situated in this
146
CHAPTER 7. BESSEL-GAUSS BEAM ENHANCEMENT CAVITIES
region may not be resonant in the cavity.
Across the entire mirror, depending on
surface variations, only a subset of the entire family of decentered component beams
may resonate and, accordingly, the entire gBG beam may not resonate. This problem
is associated with azimuthal degeneracy. Returning to our derivation of gBG beams
in Chapter 6, if we vary the amplitudes of the component decentered Gaussian beams
as we superpose them, we can produce azimuthal modulation in the final gBG beam
and a higher-order (azimuthal) gBG beam [94].
Returning to our general Bessel-
Gauss cavity design strategy, we see that such higher-order azimuthal gBG beams
are also modes of gBG cavities. Therefore, mirror surface variations in a gBG cavity
may prefer a particular higher-order azimuthalgBG beam (or superposition of such
beams) over the fundamental mode. Issues and restrictions associated with mirror
surface variations will be exposed in the subsequent section.
7.2
Confocal Bessel-Gauss cavity demonstration
As described in the previous section, the confocal Bessel-Gauss cavity consists of two
mirrors with radii of curvature R, separated by a distance L = R. We outlined a
procedure to isolate a single radial cavity mode in the previous section by patterning
the cavity mirrors to consist of a highly reflective ring-shaped pattern (reflectance
RH) and a low reflectivity (reflectance RL) background region (see Figure 7-2a and
7-5a). To reiterate, the ring-shaped patterns act as effective apertures and provide
low-loss to only a single, fundamental cavity mode. This fundamental mode of the
cavity resembles a BG beam as it traverses the focus and a mBG beam on its return
trip (see Figure 7-3a).
Our confocal Bessel-Gauss demonstration cavity operates at A = 633 nm, and
the cavity mirrors have a radius of curvature R = 15 cm. The patterned rings have
average radius
ravg
1.3 mm and thickness Aric = 534 um (input coupler mirror)
and Arc = 440 pm (output coupler mirror). The cavity mirrors were fabricated via
a photolithography/lift-off process in which a 15 nm thick Cr layer was deposited on
the surface of a dielectric mirror with reflectance RH = 99.1%. The Cr layer coats
the entire mirror surface except for the desired ring-shaped pattern; the pattern edge
roughness is < 1 pm. The thin Cr layer interrupts the operation of the dielectric
mirror such that the reflectance outside the ring patterns is RL ~ 30%. Simulations
of the cavity geometry predict a finesse of ~ 288 for the fundamental BG/mBG mode.
Coupling to the cavity is performed via an axicon-based imaging system (see
Figure 7-6).
The output of a HeNe laser is transformed through an axicon into a
147
CHAPTER 7. BESSEL-GAUSS BEAM ENHANCEMENT CAVITIES
a
RL, low-reflectivity
RH, high-reflectivity
Figure 7-5: Confocal Bessel-Gauss cavity mirrors a. Illustration of patterned
cavity mirror with intra-cavity mode intensity overlaid (not to scale). Inset is a
microscope image of a section of a patterned mirror used in the experiment. b.
Photograph of actual patterned cavity mirror in the experimental setup.
BG-like beam. This beam is imaged through a 4f imaging system and sent into
the confocal Bessel-Gauss cavity. Behind the cavity back mirror, the cavity outputcoupling mirror, a pellicle is placed to sample the transmitted intra-cavity mode. The
transmitted mode sampled by the pellicle is imaged on a CCD camera. The main
results of this imaging are included in Figure 7-7. The fraction of the mode that is
not sampled by the pellicle is focused onto a photodiode and used for observing the
cavity transmission peaks and locking the cavity via a dither lock.
In Figure 7-7a, we see that when the cavity is slightly misaligned and the cavity
length is swept back and forth over the main resonance, i.e. the main transmission
peak, the cavity mode takes on an odd shape and the measured finesse is ~ 300, near
the expected finesse of the cavity. Recalling that a BG/mBG beam is a superposition
of decentered Gaussian (dG) beams, the odd mode shape corresponds to only a subset
of the dG component beams being well-aligned and resonant. We should also note
that we only coarsely measure the finesse as the ratio of the free-spectral range to
the full-width at half maximum of the transmission peak recorded on the photodiode.
As the cavity is tuned to an aligned state (Figure 7-7b), more dG component beams
become aligned, and the fundamental mode appears; however, the transmission peak
is broadened (and takes on a multi-peaked structure), and the finesse drops to
40. When the cavity is then locked (Figure 7-7c), the mode profile changes from the
fundamental BG/mBG mode to a ring-shaped mode with some azimuthal variation.
These experimental results can be explained by small surface variations in the
cavity mirrors. Consider a region of the mirror surface in which the local radius of
curvature, Riocai, is perturbed, Riocal = R + JR. The dG component occupying this
148
CHAPTER 7. BESSEL-GAUSS BEAM ENHANCEMENT CAVITIES
a
Locking Electronics
HeNe source
4f imaging
0
b
i- 21A cm - 20.7 cm----- 40 cm-
13.6
i
Patterned mirror
confocal cavity
E
%E
2
4
0
80
60
40
20
100
z(cm)
Figure 7-6: Confocal Bessel-Gauss prototype cavity. a. Experimental arrangement (sampling pellicle and CCD not shown). The photodiode signal is
used to lock the cavity (input-coupling mirror is actuated with a piezo). b.
Simulated field (r - z plane cross-section) traversing the coupling optics/cavity
system.
region will have a slightly shifted resonant frequency. With many perturbed regions,
we expect the transmission peak to broaden and take on a multi-peaked structure, as
observed. Additionally, when locked, we expect the cavity mode to no longer resemble
the fundamental BG/mBG mode, but to include some azimuthal variation as only a
subset of the dG component beams will be resonant at the lock point.
Quantitatively, we can estimate the impact of mirror surface variations by a simple
analysis. The resonant frequency, v, of the TEMmn mode, i.e. the m, n HermiteGaussian mode, of a two mirror symmetric cavity with mirror curvature R and spacing
L can be written (pp. 762 of Ref. [971)
v=ovF
q
M
COs'(1 - LIR)
(7.1)
where vF is the free-spectral range of the cavity and q is the longitudinal mode
number. Evaluating the derivative of Eq. (7.1) with respect to R at the confocal
spacing L
=
R, we find
_-L/F
(mrn r1)
7r
149
(
__R
R
)=x-- (7.2)
CHAPTER 7. BESSEL-GAUSS BEAM ENHANCEMENT CAVITIES
Figure 7-7: Images of the transmitted cavity mode. Images are normalized
and taken when the cavity is a. misaligned and the cavity length is swept (the
dashed ring is added for ease of illustration), b. well-aligned and the cavity
length is swept c. locked from a well-aligned state.
where we have used the notation 6v (6R) to refer to the infinitesimal variation in v
(R). This expression gives an estimate for the shift in the resonant frequency of the
TEMmn mode when there is a small shift in the cavity mirror curvature.
Now consider the intra-cavity BG/mBG mode of the confocal Bessel-Gauss cavity
as a superposition of TEM,,. modes. At the mirror surfaces, the radial extent of
the intra-cavity BG/mBG mode is ~~ravg; the radial extent of a TEMmn mode is
w V/m + n where w is the Gaussian spot-size of the mode (pp. 691 of Ref. [97]).
We therefore expect the largest value of m + n in the superposition to be ~ (rav,/w)2
(note that here w is also the spot-size of the intra-cavity BG/mBG mode's component
dG beams and is directly related to Ar). In the presence of a small shift in the
mode composing the superposed intra-
cavity mirror curvature, 6r, each TEM.U
cavity BG/mBG mode suffers a different resonant frequency shift with the maximum
shift given by
S6lV m
ax I=
$
Wr
W
x
(-)
R
(7.3)
We thus expect the resonance peak of the intra-cavity BG/mBG mode to broaden,
and we can identify the quantity Feff = vF/2IvimaxI as the effective finesse of this
new resonance, broadened due to curvature variations throughout the mirror surface.
We can then write a simple expression for the effective finesse
Feff ~
2
((raVg/w) 2 x (6R/R))
150
(7.4)
CHAPTER 7. BESSEL-GAUSS BEAM ENHANCEMENT CAVITIES
The validity of Eq. (7.4) is demonstrated numerically in Figure 7-8. The simulated
points in Figure 7-8 follow from numerically solving for the cavity mode of our cavity
geometry with mirrors of curvature R + SR and differing values of rav,/w.
The
simulations agree very well with our simple analysis.
We can also compare this simple model to our experimental results. The mirrors
used in the experiment were specified to have A/10 surface quality, so the height
difference from the center of the mirror to the mirror edge, h, should not vary around
the mirror's edge by more than ~ A/10 ~ 63 nm. Letting h' = h + Jh with Jh = 63
nm (Figure 7-8), and finding the radius of curvature R' = R+JR, associated with this
height difference from the mirror center to the mirror edge, we see that for our mirrors
we can roughly estimate 6R/R ~ 0.05%. These curvature variations predict a finesse
in our cavity of ~ 55.7, relatively close to the value of ~ 40 that we experimentally
observe.
4
10
-
LA
Analytical Result
0 Simulation Points
rIW-R=7.47(exp.)
......
S10
h+6h
>
10
o
0.0's
Finesse=.5S.7
.2
10
1
101
w
lr
Figure 7-8: Effective finesse in the presence of curvature variations. The solid
line (green) is the analytical model i.e. Eq. (7.4). The dots (black) represent
simulation results. The dashed line (red) shows the value of r,,,g/w used in
the experiment. The inset illustrates the simple model for estimating curvature
variations.
In the preceding analysis and discussion, we have seen that cavity mirror surface variations can dramatically affect the finesse of the confocal Bessel-Gauss cavity.
Returning to the decentered Gaussian beam picture, in order to maintain the fundamental BG/mBG mode, we must ensure that the resonance frequency shifts of
151
CHAPTER 7. BESSEL-GAUSS BEAM ENHANCEMENT CAVITIES
the differing dG component beams are small compared to the width of the cavity's
natural resonance. In other words, the broadening of the cavity resonance associated
with losses due to the imperfect mirror reflectivity should far exceed any broadening
associated with mirror surface variations i.e. the reflectivity-limited finesse must be
much less than the surface-variation-limited effective finesse. This criterion was not
met for our experiment: the reflectivity-limited finesse is
- 300 while the surface-
variation-limited effective finesse is ~ 40; and clearly, the fundamental BG/mBG
mode was not maintained. Referring to Figure 7-8, we see this criterion may pose
a strict requirement on high-intensity Bessel-Gauss enhancement cavities as small
fractional curvature variations and relatively small values of ravg/w yield low effective finesses i.e. significantly broadened resonances. For future applications involving
high-intensity Bessel-Gauss cavities, extremely precise mirror surfaces may be necessary. Additionally, cavity designs beyond the confocal Bessel-Gauss cavity that may
be less sensitive to curvature variation issues are a possibility.
As a final note, we mention that although we have primarily used the dG beam
picture to understand the confocal Bessel-Gauss cavity, an equivalent picture can be
formulated in terms of higher-order azimuthal Bessel-Gauss beams [92]. The challenges associated with mirror surface variations can then be connected to azimuthal
mode degeneracy in the confocal Bessel-Gauss cavity.
In conclusion, we have reported an experimental demonstration of a continuouswave confocal Bessel-Gauss cavity. We have highlighted mirror surface variations as
the major challenge associated with scaling the confocal Bessel-Gauss cavity to highintensity applications and provided a simple analytical model for understanding this
effect.
Cavity mode-solver
Our cavity mode solver is based on the scattering matrix method for optical systems
and is designed for cylindrically symmetric cavity geometries [104].
Cylindrically
symmetric cavity modes are represented as N-dimensional column vectors (the radial
coordinate is discretized into N points). Each optical element composing the cavity,
including lengths of dielectric or vacuum, is described by a 2N x 2N scattering matrix.
(Scattering matrices for optical systems are generally 2 x 2 matrices relating incoming waves to outgoing ones [104, 103]; here, each radial point has its own scattering
matrix and lumping all the points together, we represent each element as a 2N x 2N
scattering matrix). Lengths of dielectric or vacuum have block diagonal scattering
152
CHAPTER 7. BESSEL-GAUSS BEAM ENHANCEMENT CAVITIES
matrices where each block is a matrix describing propagation. Using the exact matrix
representation of the quasi-discrete Hankel transform (denoted here as F) [102, 101],
each propagation block can be written as where A is the wavelength, k is the wavevector, v is the spatial frequency, and z is the propagation length. Propagation amounts
to transforming the wavefront to the spatial frequency domain, weighting each spatial
frequency by the correct phase factor for propagation, and transforming back to the
spatial domain (the matrices PA, were used for propagation in the simulations). After
forming scattering matrices for each individual cavity component, these matrices can
be composed to form a scattering matrix of the complete cavity system [1041. The
entire cavity can then be represented by a single 2N x 2N dimensional matrix. The
cavity modes correspond to the eigenvectors of this matrix and can be found by any
standard numerical eigenvalue solver. An obvious advantage of our mode solver is the
ability to immediately solve for all the higher order modes of a cavity system. This
does come with the disadvantage of having to store and manipulate a possibly large
2N x 2N matrix; for all simulations in this thesis however, the modes were solved for
on a desktop computer with a radial step-size of < 1 pm in a matter of minutes.
153
CHAPTER 7. BESSEL-GAUSS BEAM ENHANCEMENT CAVITIES
154
Appendix A
Pulse trains in the time and
frequency domains
In this appendix we review some of the basic properties of optical pulse trains in
In particular, we focus on the properties of the
the time and frequency domains.
carrier-envelope offset phase (CEP) and the carrier-envelope offset phase frequency
fCEO-
In Chapter 1, we defined the CEP of an optical pulse as the phase offset between
the carrier wave maximum and the optical pulse envelope.
We described how the
CEP of consecutive pulses in a pulse train shift by some fixed amount Ap (in the
absence of noise). We defined the carrier-envelope phase offset frequency,
fCEO,
as
the rate at which the CEP changed. More specifically
JCEO =
As in Chapter 1, in the above,
fR
27
(A.1)
X fR
is the pulse repetition rate. Recalling that our
optical pulse train can be described as a train of pulse envelopes multiplying a carrier
wave, we can think of the CEP as arising from the mismatch between the carrier
wave frequency,
f,
fe, and the repetition rate frequency, fR (think of the scenario when
= nfR; in this case
o = 0). In this picture we can then write
Ap = (TR
In the above,
TR
TR =
mod Tc) x
1/fR and T, = 1/f,.
Note that we can write
(A.2)
TR
- nTc where n is the integer value such that nT is closest to TR.
consider
f,
mod fR. We can write
f,
mod
155
fR
mod T=
Now let us
= (TR - nTc)(TRTc) with n defined
APPENDIX A. PULSE TRAINS IN THE TIME AND FREQUENCY DOMAINS
as above. Therefore, we must have
f,
mod fR = (TR mod Tc) x fR/T. Plugging
this into Eq. (A.2) and subsequently plugging into Eq. (A.1), we find
fCEO =
f,
mod
(A.3)
fR
With this simple development, we see a direct relationship between the fCEO and
the frequency domain picture of the pulse train. This relationship is emphasized in
Figure A-1 in which we sketch the connection between the time and frequency domain
pictures of an optical pulse train.
F
I
Optical pulse train - Time domain
TA
1.
In the time domain, the optical pulse train
resembles an array of regularly spaced
pulses where each pulse can be described
as a carrier-wave modulated by a pulse
envelope.
2.
We can decompose the pulse train into a
train of pulse envelopes (black)
x
multiplied
by a carrier wave (orange).
II
3.
P(t)
decomposed into a pulse envelope function
P(t) convolved with a train of delta
functions (green).
X
A* 1
The pulse envelope train can be further
t
Time domain
............ .... ------------..
.... ------------------------------...
Frequency domain
f -0
|ll
&
IA
~
spacing 1/T = fR). The carrier wave
becomes a delta function at carrier
frequency f, and the multiplications and
convolutions exchange places.
f
)
XI
5.
Optical pulse train - Frequency domain
.
nf+fcro
..
fi.t
Moving to the frequency domain, P(t)
transforms to a broad spectrum P(q)
(blue). The delta-function train transforms
into another delta function train (with
*
(P
4.
--------. -------------
f
Carrying out the above operations, we find
that the frequency domain representation of
the optical pulse train is a broad spectrum
modulating a delta function comb. The
center of the spectrum is f, the comb
spacing isfa, and each line in the frequency
comb can be represented as nfR + fao
where fco= f, modfR.
Figure A-1: An optical pulse train in the time and frequency domains. In the
above, we proceed step by step from the time domain to the frequency domain.
As we mentioned, in the absence of noise, A& is constant, and so fCEO is fixed.
156
APPENDIX A. PULSE TRAINS IN THE TIME AND FREQUENCY DOMAINS
However, for any real ultrafast laser system perturbations will lead to drifts in Ap
and
fCEO.
For many applications, our experiments included, it is important to have a
fixed, measurable fCEO. The most common method to measure the fCEO for feedback
stabilization is via f-2f interferometery.
The basic operation of an f-2f interferometer is illustrated in Figure A-2. The
basic concept is to frequency double a low frequency component of the optical pulse
train and mix this with a high frequency component. This frequency doubling and
mixing ultimately produces a beat note at the
fCEO.
Note that a critical requirement
of an f-2f interferometer is for at least an octave of optical bandwidth (that is an f
and 2f component must be present in the spectrum).
fR
nflR+fCEO
SHG
2nfA+ 2 fCEo
--
+fCEO
Figure A-2: The basic f-2f interferometer. In the basic f-2f interferometer
a low frequency component from the spectrum is frequency doubled via secondharmonic generation (SHG) and mixed with a high frequency component. The
mixing process yields a beat-note at the fCEo-
157
APPENDIX A. PULSE TRAINS IN THE TIME AND FREQUENCY DOMAINS
158
Appendix B
Evolution operator basics
In this appendix we review the basics of the evolution operator in non-relativistic
quantum mechanics.
We formulate a perturbative expansion for the evolution op-
erator and discuss this operator in the interaction picture. We define the evolution
operator U(t, to) as an operator that evolves an initial state at time
t
o to its form at
time t.
) = U(t, to) 0(to))
(B.1)
Considering the above definition, the evolution operator has several immediately obvious properties
U(t, to) =
1
(for t = to)
(B.2)
=
U(tt1)U(ti, to)
(B.3)
=
U-I(to, t) = Ut (to, t)
(B.4)
The last property (unitarity) follows from conservation of probability. Inserting Eq.
(B.1) into the time-dependent Schr6dinger equation with Hamiltonian H(t), we find
ih
U(t, to) = H(t)U(t, to)
(B.5)
Integrating this equation, we can move to an integral equation for the evolution
operator
U(t, to) = 1 -
dt'H(t')U(t', to)
159
(B.6)
APPENDIX B. EVOLUTION OPERATOR BASICS
U(t, o) = 1 -
-
dt'U(t, t')H(t')
(B.7)
The second equation above follows from taking the Hermitian conjugate of the first,
applying the properties in Eq. (B.4), and swapping t and to. Repeatedly substituting
U(t', to) on the right-hand side of Eq. (B.6) with the full form of Eq. (B.6), we can
find an iterative expression for the evolution operator
U(t, to) = 1 - i
]
t
2
(
dt'H(t') +
)dt'
t
dt"H(t')H(t") +
(B.8)
The above perturbative expansion for the evolution operator U(t, to) is the well-known
Dyson series.
In this thesis we work with Hamiltonians that can be separated into well-understood
constant portions and time-varying parts. We can write the general form of these
Hamiltonians as
H = HO + Hint(t)
We work with these Hamiltonians in the 'interaction picture'.
(B.9)
In the interaction
picture, we separate the influence of Ho and Hint on the initial state. We define the
interaction picture state as
1|1,(0)) = Uo (tI to) 10(M))(.)
=
exp (iHo(t - to)/h) 4'(t))
(B.11)
In the above, we make use of Uo, the evolution operator for the time-independent
-
portion of the Hamiltonian, Ho. It can easily be seen that Uo(t, to) = exp (iHo(t
to)/h). Now plugging Eq. (B.11) into the time-dependent Schr6dinger equation, we
find
(t)
(t) wh
iIwuse
(B.
12)
In the above, we have made use of Hi'nt(t) which we define as
160
APPENDIX B. EVOLUTION OPERATOR BASICS
Hit(t)
=
exp (iHo(t
=
Uot(t, to)Hint(t)Uo(t, to)
-
to)/h)Hint(t) exp
(-
iHo(t
-
to)/h)
(B.13)
(B.14)
In a parallel with our earlier developments, we can also define a time evolution operator in the interaction picture. We define this operator as
(B.15)
) =U
1J
1(t, to) [|41 (to))
Similar to the normal evolution operator, this operator evolves interaction picture
states from time
t
Inserting the definition in Eq.
o to time t.
(B.15) into the
Schr6dinger equation, we find
a
at
ih U(t, to) = Hit(t)U1 (t, to)
(B.16)
As before, we can integrate this differential equation and convert it into integral form
-t
U,(t, to)
1 -
U1 (t, to)
1- i
-
I
dt'Hit(t')U1 (t' to)
(B.17)
j
dt'U1 (t, t')Hit (t')
(B.18)
hto
hto
Again, the second equation above follows from taking the Hermitian conjugate of the
first, applying the properties in Eq. (B.4), and swapping the t and to variables.
Let us now express the total evolution operator, U(t, to), in terms of the interaction picture evolution operator, U1 (t, to), and the evolution operator for the timeindependent part of the Hamiltonian, Uo(t, to). First, note that
110,(W)
=
Uil (t0t) |10A(0))
=
U(t, to) '?7(to))
(B. 19)
(B.20)
The second equation above follows since |@b(to)) = JV(to)) from Eq. (B.10). Additionally, note that from the definition of the interaction picture state, we also have
161
APPENDIX B. EVOLUTION OPERATOR BASICS
I
() =U0t
(t, to) 0(t))
(B.21)
-
=
U1(t, to)U(t, to) 4(to))
(B.22)
Combining Eq. (B.20) and Eq. (B.22), we then find that
U(t, to) = Uo(t, to)U, (t, to)
(B.23)
With this relationship between the different evolution operators and with Eq. (B. 17)
and Eq. (B.18), we can obtain two integral expressions for the total time-evolution
operator
U(tto) = Uo (t, to) -
dt'U(tt')Hint(t')U(t' to)
(B.24)
U(t,to) = Uo(t, to) - i
dt'U(t, t')IHint(t')Uo(t', to)
(B.25)
Now let us consider a situation in which a system starts in state 16).
Under the
action of the time-independent part of the Hamiltonian, Ho, this initial state evolves
to 10(t). However, under the action of the complete Hamiltonian (including Ho and
Hint), this state evolves to 1,0(t)). For this general system, we can write an expression
for |/(t)) from Eq. (B.25). We find
-
t
We make use of this expression in Chapter 2 (also note an complimentary expression
can be found from Eq. (B.24)).
Lastly, we should note that if we iteratively solve Eq. (B.17), we can generate
a perturbative expansion for the evolution operator in the interaction picture (this
directly parallels our development of Eq. (B.8)). If we employ the relations (B.14)
and (B.23), we then find
0 U(t, t1) - ftdt'U0 (t, t')Hint W)
ht
+
=
"
U (t, t')
tdtl
2 I t'dt" Uo (t, t')H in (t') UO (t', t") H in(t")
-+
162
.
(B. 27)
APPENDIX B. EVOLUTION OPERATOR BASICS
This perturbative expansion for the total evolution operator is used extensively in
our calculations in Chapter 2.
163
APPENDIX B. EVOLUTION OPERATOR BASICS
164
Appendix C
Volkov waves
a
at
i'h/,
=
F
)
In the following we derive the Volkov wave solutions to the Schr6dinger equation with
the 'free-field' Hamiltonian, HF. Recall that HF consists only of the kinetic part of
the Hamiltonian and the potential associated with an electromagnetic field. We look
for solutions to this equation then with the Hamiltonian
(f + eA (t)) 21)
=
(C.1)
In the above, as usual, A(t) is the vector potential. Not that we have written HF in
the velocity gauge i.e. with the electrostatic potential equal to zero, (p = 0. In this
gauge we can solve Eq. (C.1) with a wavefunction of the form
79/L(x, t) = exp(iqx/h)#(t)
(C.2)
Inserting Ov"(x, t) into Eq. (C.1), we find
do
S
=
1
2m
diet
(q + eA(t))2 #(t)
(C.3)
This equation can then be directly integrated, and we find
qvi(x,
t) = exp ((iqx - iS(t)) /h)
(C.4)
In the above, we have used the quantity S(t) which we define as
S(t)
=
J
(q + eA(t')) 2 dt'
165
(C.5)
APPENDIX C. VOLKOV WAVES
Note that in the above S(t) closely resembles the action of the electron wiggling in
the laser field. Additionally, in the above we have used the superscript V to denote
that this is a 'Volkov' wave solution, and we have used the superscript V to denote
that we are working in the velocity gauge. Additionally, the subscript q denotes the
momentum of the wave.
In our calculations we make use of, not the velocity gauge Volkov wave, but
the length gauge version. We now convert ov(x, t) to its length gauge counterpart
2bv(x, t). Recall the gauge transformation rules for electromagnetic potentials
=
'=
A' - VA(x, t)
(C.6)
+ -tA(x, t)
(C.7)
In the above p denotes the electrostatic potential.
To transform from the velocity
gauge (o = 0) to the length gauge (A = 0), the function A =
- A is required. For
our simple one-dimensional case, this function takes the form
A(x, t) = xA(t)
(C.8)
Now let us consider the transformation of a wavefunction under a gauge transformation. Considering the Hamiltonian for an electron in an electromagnetic field given
with potentials (p, A), we find that when the potentials transform to (p', A') as given
by Eq. (C.6) and Eq. (C.7), the wavefunction transforms as
<'(x, t) = exp (ieA(x, t)/h) >'(x, t)
(C.9)
This can easily be confirmed via direct substitution. Using this result, we can transform 04v(x, [) to the length gauge. We find that the length gauge form of the Volkov
wave of momentum q is given by
04(x, t)
=
exp (ieA(x, t)/h)/'(x, t)
(C.10)
=
exp (iexA(t))44'"(x, t)
(C.11)
-
exp (i((q+eA(t))x
-
S(t))/h)
(C.12)
Lastly we should note that we can rewrite the length gauge Volkov wave in ket
166
APPENDIX C. VOLKOV WAVES
notation as
q
q + eA(t)) exp
(-
iS(t)/h)
(C.13)
Where in the above Iq + eA(t)) is the momentum eigenstate with eigenvalue q +-eA(t).
In other words, (xIq + eA(t)) = exp (i(q + eA(t))x/h).
167
APPENDIX C. VOLKOV WAVES
168
Appendix D
Bessel-Gauss beam spatial phase
In this short appendix, we show that the spatial phase of a large class of generalized
Bessel-Gauss (gBG) beams can accurately be represented as the sum of a quadratic
component and a linear component. More specifically, we show that the approximate
form of Eq. (6.20) provided in Eq. (6.21) is reasonable. We should note that a similar
approximation has previously been used for Bessel-Gauss (BG) beams [95]. First, we
write the second term in Eq. (6.20) as
arg (JO
(
arg(Jo(u + iv))
-_k-rc( )
(D.1)
where u and v are given by
kr
3r
=
1 +
(L/zo) 2
1 + (L/zo)
or
1 + (L /zo)
(L
2
2
L
zo
)
zo
+
(D.2)
ZO
kr
I
(rd)
(rd'\
(L/zo) 2
zo
(D.3)
The relation rc(L) = rd+ L sin p = rd + (t/k)L has been used in the above (from
Eq. (6.13)). Returning to Eq. (6.23), with Iu + ivl >> 1, we can use the asymptotic
form of the Bessel function (pp. 114 of Ref. [90]), and we find
arg(Jo(u + iv))
arg(cos(u+ iv -7r/4)
- tan-- (tan(u - 7r/4) tanh v)
-u+7r/4
1/k
zo
169
1 + (L/zo) 2 +4
(D.4)
APPENDIX D. BESSEL-GAUSS BEAM SPATIAL PHASE
The last approximation is very accurate when jvj > 3 (tanh(3) ~ .995). Therefore,
when jvt >> 1, Eq. (6.27) is an accurate approximation. Inspecting Eq. (6.26),
we see a variety of different conditions can lead to jvj >> 1, and a large class of
gBG type beams have a phase term accurately approximated by Eq.
(6.27).
Two
particular cases of this class are BG-like beams, i.e. rd is very small, and mBG-like
beams, i.e. / is very small. To see this, note that we are primarily interested in the
region where there is significant intensity, i.e. r ~ rc(L). Plugging r ~ rc(L) into Eq.
(6.26) for these two beam types we find
BG-like beams:
U
2(L/zo) 2
4 0 (1 + (L/zo)2
2W 2
(D.5)
mBG-like beams:
r(
2
2
0
2
L=
1 + (L/zo)2
From the above, we see that for BG-like beams with /3 2 W2 /4
(D.6)
(p/pG 2
L/zo not too small, jvi >> 1, and for mBG-like beams with r2/w
not too large,
lvj
and
>> 1 and L/zo
>> 1. Plugging Eq. (6.27) into Eq. (6.20) produces Eq. (6.21) (up
to a constant phase offset), so we have validated our approximation.
170
Bibliography
[1] The International Energy Agency. Key World Energy Statistics -
2014.
[2] P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, "Generation of optical
harmonics," Phys. Rev. Lett. 7, 118 (1961).
[3] B. E. Schmidt, A. D. Shiner, M. Giguerel, P. Lassonde, C. A. Trallero-Herrero, J.C. Kieffer, P. B. Corkum, D. M. Villeneuve, and Frangois Legar6, "High harmonic
generation with long-wavelength few-cycle laser pulses,"
J. Phys. B 45, 074088
(2012).
[4] F. X. Kirtner. Ultrafast Optics -
MIT course 6.638 notes.
[5] M. Wegener, Extreme Nonlinear Optics, Springer-Verlag, 2005.
[6] T. Brabec, Strong Field Laser Physics, Springer-Verlag, 2008.
[7] C. J. Joachain, N. J. Kylstra, and R. M. Potvliege, Atoms in Intense Laser Fields,
Cambridge University Press, 2012.
[8] P. B. Corkum and F. Krausz, "Attosecond science," Nat. Phys. 3, 381 (2007).
[9] F. Krausz and M. Ivanov, "Attosecond physics," Rev. Mod. Phys. 81, 163 (2009).
[101 M. Krnger, M. Schenk, and P. Hommelhoff, "Attosecond control of electrons
emitted from a nanoscale metal tip," Nature 475, 78 (2011).
[11] G. Herink, D. R. Solli, M. Gulde, and C. Ropers, "Field-driven photoemission
from nanostructures quenches the quiver motion," Nature 483, 190 (2012).
[12] B. Piglosiewicz, S. Schmidt, D. J. Park, J. Vogelsang, P. Grog, C. Manzoni, P.
Farinello, G. Cerullo, C. Lienau, "Carrier-envelope phase effects on the strongfield photoemission of electrons from metallic nanostructures," Nat. Photonics 8,
37 (2014).
[13] R. W. Boyd, Nonlinear Optics, 3rd Ed., Elsevier, 2008.
[14] P. A. Tipler and R. A. Llewellyn,
Freeman, 1999.
Classical Electrodynamics, 3rd Ed., W. H.
171
BIBLIOGRAPHY
[15] V. Nathan, A. H. Guenther, and S. S. Mitra, "Reviw of multiphoton absorption
in crystalline solids," J. Opt. Soc. Am. B 2, 294 (1985).
[16] M. Biittiker and R. Landauer, "Traversal Time for Tunneling," Phys. Rev. Lett.
49, 1739 (1982).
[17] R.
Landauer
and Th.
Martin,
"Barrier interaction
time in tunneling,"
Rev. Mod. Phys. 66, 217 (1994).
[18] L. V. Keldysh, "Ionization in the field of a strong electromagnetic wave," JETP
20, 1307 (1965).
[19] J. C. Diels and W. Rudolph,
Press, 1996.
Ultrashort Laser Pulse Phenomena, Academic
[20] A. Roxin, N. Brunel, D. Hansel, G. Mongillo, and C. van Vreeswijk, "On the
Distribution of Firing Rates in Networks of Cortical Neurons," J. of Neurosci.
31, 16217 (2011).
[21] Plank Collaboration, "Plank 2013 results: I. Overview of products and scientific
results," arXiv:1303.5062 (2013).
[22] P. Dombi, A. Hbrl, P. Racz, I. MArton, A. Trigler, J. R. Krenn, U. Hohenester,
"Ultrafast Strong-Field Photoemission from Plasmonic Nanoparticles," Nano Lett.
13, 674 (2013).
[23] P. D. Keathley, A. Sell, W. P. Putnam, S. Guerrera, L. Velasquez-Garcia, and
F. X. Kirtner,
"Strong-field photoemission from silicon field emitter arrays,"
Ann. Phys. 525, 144 (2013).
[24] P. B. Corkum,
"Plasma perspective on strong field multiphoton ionization,"
Phys. Rev. Lett. 71, 1994 (1993).
[25] M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L'Huillier, and P. B. Corkum, "The-
ory of high-harmonic generation," Phys. Rev. A 49, 2117 (1994).
[26] J. Lee, D. R. Carlson, and R. J. Jones, "Optimizing intracavity high harmonic
generation for XUV fs frequency combs," Opt. Exp. 19, 23315 (2011).
[27] N. W. Ashcroft and N. D. Mermin, Solid State Physics, Saunders Colleg Publishing, 1976.
[28] R. H. Fowler and L. Nordheim, "Electron emission in intense electric fields,"
Royal Society of London Proceedings Series A 119, 173 (1928).
[29] C. Kealhofer, S. M. Foreman, S. Gerlich and M. A. Kasevich, "Ultrafast lasertriggered emission from hafnium carbide tips," Phys. Rev. B 86, 035405 (2012).
172
BIBLIOGRAPHY
130] T. Hanke, G. Krauss, D. Triutlein, B. Wild, R. Bratschitsch, and A. Leitenstorfer, "Efficient Nonlinear Light Emission of Single Gold Optical Antennas Driven
by Few-Cycle Near-Infrared Pulses," Phys. Rev. Lett. 103, 257404 (2009).
[31] R. Bormann, M. Gulde, A. Weismann, S. V. Yalunin, and C. Ropers,
"Tip-
Enhanced Strong-Field Photoemission," Phys. Rev. Lett. 105, 147601 (2010).
[32] S. V. Yalunin, M. Gulde, and C. Ropers, "Strong-field photoemission from surfaces: Theoretical approaches," Phys. Rev. B 84, 195426 (2011).
[33] F. V. Bunkin and M. V. Fedorov, "Cold emission of electrons from the surface
of a metal in a strong radiation field," JETP 21, 896 (1965).
[34] M. Ehrhardt, "Discrete transparent boundary conditions for Schr6dinger-type
equations for non-compactly supported initial data," Appl. Num. Math. 58, 660
(2008).
[35] A. Arnold, M. Ehrhardt, and I. Sofronov, "Discrete transparent boundary conditions for Schr6dinger-type equations: fast calculation, approximation, and sta-
bility," Comm. Math. Sci. 1, 501 (2003).
[36] R. Chang, S. Potnis, R. Ramos, C. Zhuang, M. Hallaji, A. Hayat, F. DuqueGomez, J. E. Sipe, and A. M. Steinberg, "Observing the onset of effective mass,"
Phys. Rev. Lett. 112, 170404 (2014).
[37] E. N. Economou, "Surface Plasmons in Thin Films," Phys. Rev. 539, 992 (1969).
[38] H. Raether, Surface Plasmons, Springer-Verlag, 1988.
[39] S. A. Maier, Plasmonics: Fundamentals and Applications, Springer-Verlag, 2007.
[40] M. Staffaroni, J. Conway, S. Vedantam, J. Tang, and E. Yablonovitch, "Circuit
Analysis in metal-optics," Photonics and Nanostructures - Fund. and Apps. 10,
166 (2012).
[41] P. Biagioni, J.-S. Huang, and B. Hecht, "Nanoantennas for visible and infrared
radiation," Rep. Prog. Phys. 75, 024402 (2012).
[42] M. Hentschel, T. Utikal, H. Giessen, and M. Lippitz, "Quantitative Modeling of
the Third Harmonic Emission Spectrum of Plasmonic Nanoantennas," Nano. Lett.
12, 3778 (2012).
[43] L. Novotny and N. van Hulst, "Antennas for light," Nat. Photonics 5, 83 (2011).
[44] C.-P. Huang, X.-G. Yin, H. Huang, and Y.-Y. Zhu, "Study of plasmon resonance
in a gold nanorod with an LC circuit model," Opt. Exp. 17, 6407 (2009).
[45]
D. Zhu, M. Bosman, and J. K. W. Yang, "A circuit model for plasmonic res-
onators," Opt. Exp. 22, 9809 (2014).
173
BIBLIOGRAPHY
[46] M. Staffaroni, CircuitAnalysis in Metal-Optics, Theory and Applications, Ph.D.
Thesis, University of California Berkeley 2011.
[47] S. Thomas, M. Kruger, M. F6rster, M. Schenk, and P. Hommelhoff, "Probing
of optical near-fields by electron rescattering on the 1 nm scale," Nano Lett. 13,
4790 (2013).
[48] A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C.
Lemell, K. Torizuka, J. Burgd6rfer, T. W. Hdnsch, and F. Krausz, "Observation
of light-phase-sensitive photoemission from a metal," Phys. Rev. Lett. 92, 073902
(2004).
[491
C. Lemell, X.-M. Tong, F. Krausz, and J. Burgd6rfer, "Electron emission from
metal surfaces by ultrashort pulses: Determination of the carrier-envelope phase,"
Phys. Rev. Lett. 90, 076403 (2003).
[50] M. I. Stockman and P. Hewageegana, "Absolute phase effect in ultrafast optical
responses of metal nanostructures," Appl. Phys. A 89, 247 (2007).
[51] Y. C. Martin, H. F. Hamann, and H. K. Wickramasinghe, "Strength of the
electric field in apertureless near-field optical microscopy," J. Appl. Phys. 89,
5774 (2001).
[52] A. Polyakov, C. Senft, K. F. Thompson, J. Feng, S. Cabrini, P. J. Schuck, H. A.
Padmore, S. J. Peppernick, and W. P. Hess, "Plasmon-Enhanced Photocathode
for High Brightness and High Repetition Rate X-Ray Sources," Phys. Rev. Lett.
110, 076802 (2013).
[53] R. K. Li, H. To, G. Andonian, J. Feng, A. Polyakov, C. M. Scoby, K. Thompson, W. Wan, H. A. Padmore, and P. Musumeci, "Surface-Plasmon ResonanceEnhanced Multiphoton Emission of High-Brightness Electron Beams from a
Nanostructured Copper Cathode," Phys. Rev. Lett. 110, 074801 (2013).
[54] P. Hommelhoff, Y. Sortais, A. Aghajani-Talesh, and M. A. Kasevich,
emission tip as a nanometer source
of free electron femtosecond
"Field
pulses,"
Phys. Rev. Lett. 96, 077401 (2006).
[55] P. Hommelhoff, C. Kealhofer, and M. A. Kasevich, "Ultrafast electron pulses from
a tungsten tip triggered by low-power femtosecond laser pulses," Phys. Rev. Lett.
97, 247402 (2006).
[56] C. Ropers, D. R. Solli, C. P. Schulz, C. Lienau, and T. Elsaesser,
"Local-
ized multiphoton emission of femtosecond electron pulses from metal nanotips,"
Phys. Rev. Lett. 98, 043907 (2007).
[57] M. E. Swanwick, P. D. Keathley, A. Fallahi, P. R. Krogen, G. Laurent, J. Moses,
F. X. Kdrtner, and L. F. Velasquez-Garcia, "Nanostructured Ultrafast Silicon-Tip
Optical Field-Emitter Arrays," Nano. Lett. 14, 5035 (2014).
174
BIBLIOGRAPHY
[58] M. Kruger, M. Schenk, M. Fbrster, and P. Hommelhoff, "Attosecond physics in
photoemission from a metal nanotip," J. Phys. B: At. Mol. Opt. Phys. 45, 074006
(2012).
[591 M. Kruger, M. Schenk, P. Hommelhoff, G. Wachter, C. Lemell, and J. Burgd6rfer,
"Interaction of ultrashort laser pulses with metal nanotips: a model system for
strong-field phenomena," New J. Phys. 14, 085019 (2012).
[60] M. Schenk, M. Kruger, and P. Hommelhoff, "Strong-Field Above-Threshold
Photoemission from Sharp Metal Tips," Phys. Rev. Lett. 105, 257601 (2010).
[61] E. Verhagen, A. Polman, and L. Kuipers, "Nanofocusing in laterally tapered
plasmonic waveguides," Opt. Exp. 16, 45 (2008).
[62] T. K. S. Wong and S. G. Ingram, "Observation of Fowler-Nordheim tunnelling at
atmospheric pressure using Au/Ti lateral tunnel diodes," J. Phys. D: Appl. Phys.
26, 979 (1993).
[63] L. Matos, D. Kleppner, 0. Kuzucu, T. R. Schibli, J. Kim, E. P. Ippen, and F. X.
Kdrtner, "Direct frequency comb generation from an octave-spanning prismless
Ti:sapphire laser," J. Phys. D: Appl. Phys., 26, 979 (1993).
[64] L. J. Chen, Design, optimization, and applications of few-cycle Ti:sapphire
lasers, Ph.D. Thesis, MIT 2012.
[65] A. Dogariu, M. N. Schneider, and R. B. Miles, "Direct measurement of electron
loss rate in air," CLEO 2010, JTuD2.
[66] A. Dogariu, M. N. Schneider, and R. B. Miles, "Versatile radar measurement of
the electron loss rate in air," Appl. Phys. Lett. 103, 224102 (2013).
[67] J. A. Cox, W. P. Putnam, A. Sell, A. Leitenstorfer, and F. X. Kdrtner, "Pulse
Synthesis in the single-cycle regime from independent mode-locked lasers using
attosecond-precision feedback," Opt. Lett. 37, 3579 (2012).
[68] A. Sell, G. Krauss, R. Scheu, R. Huber, and A. Leitenstorfer, "8-fs pulses from
a compact Er:fiber system: quantitative modeling and experimental implementation," Opt. Exp. 17, 1070 (2009).
[69] G. Krauss, D. Fehrenbacher, D. Brida, C. Riek, A. Sell, R. Huber, and A. Leitenstorfer, "All-passive phase locking of a compact Er:fiber laser system," Opt. Lett.
36, 540 (2011).
[70] D. Brida, G. Krauss, A. Sell, A. Leitenstorfer, "Ultrabroadband Er:fiber lasers,"
Laser Photonics Rev. 8, 1 (2014).
[71] R. Selm, G. Krauss, A. Leitenstorfer, and A Zumbusch,
"Simultaneous
second-harmonic generation, third-harmonic generation, and four-wave mixing
microscopy with single sub-8 fs laser pulses," Appl. Phys. Lett. 99, 181124 (2011).
175
BIBLIOGRAPHY
[721 J. R. Birge, R. Ell, and F. X. Kdrtner,
"Two-dimensional spectral shearing
Opt. Lett. 31, 2063 (2006).
pulse
characterization,"
for
few-cycle
interferometry
[73] J. Lim, H.-W. Chen, G. Chang, and F. X. Kdrtner, "Frequency comb based on a
narrowband Yb-fiber oscillator: pre-chirp management for self-referenced carrier
envelope phase offset frequency stabilization," Opt. Exp. 21, 4531 (2013).
[74] B. Lamprecht, J. R. Krenn, A. Leitner, and F. R. Aussenegg, "Resonant and OffResonant Light-Driven Plasmons in Metal Nanoparticles Studied by Femtosecond-
Resolution Third-Harmonic Generation," Phys. Rev. Lett. 83, 4421 (1999).
[75] T. Hanke, J. Cesar, V. Knittel, A. Triigler, U. Hohenester, A. Leitenstorfer, and
R. Bratschitsch, "Tailoring Spatiotemporal Light Confinement in Single Plasmonic
Nanoantennas," Nano. Lett. 12, 992 (2012).
[76] W. P. Putnam, D. N. Schimpf, G. Abram, and F. X. Kdrtner, "Bessel-Gauss
beam enhancement cavities for high-intensity applications,"
Opt. Exp. 20, 24429
(2012).
[77] W. P. Putnam, D. N. Schimpf, and F. X. Kdrtner, "Extending cavity-enhanced
high-harmonic generation with Bessel-Gauss beams," SPIE Newsroom, 2013.
[781 R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, "Phase-coherent frequency
combs in the vacuum ultraviolet via high-harmonic generation inside a femtosec-
ond enhancement cavity," Phys. Rev. Lett. 94, 193201 (2005).
[791 C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A.
Schuessler, F. Krausz, and T. W. Hinsch, "A frequency comb in the extreme
ultraviolet," Nature 436, 234 (2005).
[801 A. Cingdz, D. C. Yost, T. K. Allison, A. Ruehl, M. E. Fermann, I. Hartl, and
J. Ye, "Direct frequency comb spectroscopy in the extreme ultraviolet,"
Nature
482, 68 (2012).
[81] D. C. Yost, T. R. Schibli, and J. Ye, "Efficient output coupling of intracavity
high-harmonic generation," Opt. Lett. 33, 1009 (2008).
[82] S. Holzberger, I. Pupeza, D. Esser, J. Weitenberg, H. Carstens, T. Eidam, P.
Russbiildt, J. Limpert, T. Udem, A. Tiinnermann, T. Hdnsch, F. Krausz, and
E. Fill, "Sub-25 nm high-harmonic generation with a 78-MHz repetition rate
enhancement cavity," QELS 2012, Postdeadline Paper QTh5B.7.
[831 E. Constant, D. Garzella, P. Breger, E. Mevel, Ch. Dorrer, C. Le Blanc, F.
Salin, and P. Agostini, "Optimizing High Harmonic Generation in Absorbing
Gases: Model and Experiment," Phys. Rev. Lett. 82, 1668 (1999).
[84] I. Pupeza, T. Eidam, J. Rauschenberger, B. Bernhardt, A. Ozawa, E. Fill, A.
Apolonski, T. Udem, J. Limpert, Z. A. Alahmed, A. M. Azzeer, A. Tiinnermann,
T. W. Hdnsch, and F. Krausz, "Power scaling of a high-repetition-rate enhance-
ment cavity," Opt. Lett. 35, 2052 (2010).
176
BIBLIOGRAPHY
[85] K. D. Moll, R. J. Jones, and J. Ye, "Output coupling methods for cavity-based
high-harmonic generation," Opt. Exp. 14, 8189 (2006).
[86] J. Weitenberg, P. RuEbiildt, T. Eidam, and I. Pupeza, "Transverse mode tailoring
in a quasi-imaging high finesse femtosecond enhancement cavity," Opt. Exp. 19,
9551 (2011).
[871 K. D. Moll, R. J. Jones and J. Ye,
"Nonlinear dynamics inside femtosecond
enhancement cavities," Opt. Exp. 13, 1672 (2005).
[88] J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free bams," Phys. Rev. Lett.
58, 1499 (1987).
Fundamentals of Photonics, John Wiley
&
[89] B. E. A. Saleh and M. C. Teich,
Sons, 2007.
[90] J. D. Jackson, Classical Electrodynamics, John Wiley & Sons, 1999.
[911 F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Comm. 64,
491 (1987).
[92] V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo,
"Generalized Bessel-Gauss beams," J. Mod. Opt. 43, 1155 (1996).
[93] A. R. Al-Rashed and B. E. A. Saleh, "Decentered Gaussian beams," Appl. Opt.
34, 6819 (1995).
[94] C. J. R. Sheppard and T. Wilson, "Gaussian-beam theory of lenses with annular
aperture," Microwaves, Optics and Acoustics 2, 105 (1978).
[95] A. N. Khilo, E. G. Katranji, and A. A. Ryzhevich, "Axicon-based Bessel resonator:
analytical description and experiment,"
J. Opt. Soc. Am. A 18, 1986
(2001).
[96] M. Santarsiero, "Propagation of generalized Bessel-Gauss beams through ABCD
optical systems," Opt. Comm. 132, 1 (1996).
[97] A. E. Siegman, Lasers, University Science Books, 1986.
[98] J. Rogel-Salazar, G. H. C. New, and S. Chavez-Cerda,
optical resonator," Opt. Comm. 190, 117 (2001).
"Bessel-Gauss beam
[99] J. C. Gutierrez-Vega, R. Rodriguez-Masegosa, and S. Chavez-Cerda,
"Bessel-
Gauss resonator with spherical output mirror: geometrical- and wave-optics anal-
ysis," J. Opt. Soc. Am. A 20, 2113 (2003).
[100] P. Pdskk6nen and J. Turunen,
"Resonators with Bessel-Gauss modes,"
Opt. Comm. 156, 359 (1998).
177
BIBLIOGRAPHY
[101] L. Yu, M. Huang, M. Chen, W. Chen, W. Huang, and Z. Zhu, "Quasi-discrete
Hankel transform," Opt. Lett. 23, 409 (1998).
"Computation of quasi[102] M. Guizar-Sicairos and J. C. Guti6rrez-Vega,
discrete Hankel transforms of integer order for propagating optical wave fields,"
J. Opt. Soc. Am. A 21, 53 (2004).
[103] H. A. Haus, Waves and Fields in Optoelectronics, CBLS, 2004.
[104] G. Abram, High intensity femtosecond enhancement cavities, M. Eng. Thesis,
MIT 2009.
178