Ultrafast Optical Physics II (SoSe 2015)
Noah Chang & Franz X. Kärtner, Bldg. 99, Room O3.111 & O3.097
Email & phone: guoqing.chang@cfel.de, 040 8998 6365
franz.kaertner@cfel.de, 040 8998 6350
Office hour: Tuesday, 9-10 am
Lectures: Fr 08.30-10.00 SemRm 1
Recitations: Fr 10.30-12.00 SemRm 1
Start: 10.04.2015
Teaching Assistant:
Kemal Shafak, Office O3.105, phone 040 8998 -6370,
E-mail: kemal.shafak@cfel.de
Qian Cao, Office O3.125, phone 040 8998 -6325,
E-mail: qian.cao@cfel.de
Office hour: Thursday 9:30-11am
Course Secretary: Christine Berber
O3.095, phone x-6351, E-mail: christine.berber@cfel.de.
Class website:http://http://desy.cfel.de/ultrafast_optics_and__x_rays_
division/lecture_notes/summer_semester_2013/
1
Prerequisites: Ultrafast Optical Physics I or basic course in Electrodynamics
Required Text: Class notes will be distributed in class.
Requirements: 10 Problem Sets, Term Paper, and Term paper presentation
Collaboration on problem sets is encouraged.
Grade breakdown: Problem set (30%), Participation (30%), Term paper (40%)
Recommended Text: Ultrafast Optics, A. M. Weiner, Hoboken, NJ, Wiley 2009.
Additional References:
•  Waves and Fields in Optoelectronics, H. A. Haus, Prentice Hall, NJ, 1984
•  Ultrashort laser pulse phenomena: fundamentals, techniques, and applications
on a femtosecond time scale, J.-C. Diels and W. Rudolph, Academic Press,
2006.
•  Few-Cycle Laser Pulse Generation and Its Applications, Ed. F. X. Kärtner,
Topics in Applied Physics Vo. 95, Springer Verlag, 2004.
•  Principles of Lasers, O. Svelto, Plenum Press, NY, 1998
•  Optical Resonance and Two-Level Atoms, L. Allen & J. H. Eberly, J. Wiley, NY,
1975.
•  Prof. Rick Trebino’s course slides on ultrafast optics:
http://frog.gatech.edu/lectures.html
2
The metric system
We will meet with both small and large numbers in this course. For
example, the optical pulses discussed later are incredibly short and the
powers and intensities can be incredibly high.
Prefixes:
Small
Milli (m)
Micro (µ)
Nano (n)
Pico (p)
Femto (f)
Atto (a)
Big
10-3
10-6
10-9
10-12
10-15
10-18
Kilo (k)
Mega (M)
Giga (G)
Tera (T)
Peta (P)
Exa (E)
Adapted from Rick Trebino’s course slides
10+3
10+6
10+9
10+12
10+15
10+18
Time scale of physical events
§  Ionization of K electrons 5x10-18 - 10-17s
§  Shortest event ever created: 67 attosecond (10-18s) x-ray pulse
(2012)
§  Bohr orbit period in hydrogen atom: 150 attoseconds
§  Single oscillation of 600nm light: 2 fs (10-15s)
§  Vibrational modes of a molecule: ps (10-12s) timescale
§  Electron transfer in photosynthesis: ps timescale
§  Period of phonon vibrations in a solid: ps timescale
§  Mean time between atomic collisions in ambient air: 0.1 ns (10-9s)
§  Period of mid-range sound vibrations: ms
How to measure a fast event?
Birth of ultrafast technology
$25,000 bet: Do all four
hooves of a running horse
ever simultaneously leave
the ground? (1872)
Leland Stanford Eadweard Muybridge
Muybridge's first picture sequence
Adapted from Rick Trebino’s course slides
What do we need to probe a fast event?
Optical power
§  The light signal received by the camera film is a train of optical
pulse.
Shutter
open
Shutter Shutter
open
close
Shutter Shutter
open
close
Shutter Shutter
open
close
t
§  We need a FASTER event to freeze the motion. Here the FASTER
event is shutter opening and closing.
§  If we have an optical pulse source, we can record images of a
running horse in a dark room.
Early history of lasers
§  1917: on the quantum theory of radiation – Einstein’s paper
§  1954: MASER by Charles Townes (1915—) et al.
If you’re
a nobel prize
winner,Prize
and 100in
years
old, youand
can Schawlow
§  1958: Charles
Townes
(Nobel
1964)
other winners using harsh words:
(Nobel Prizecomment
in 1981)
conceive basic ideas for a laser.
§  1960: LASER coined by Gordon Gould (1920-2005).
§  1960: First laser (Ruby) by Maiman
§  1961: First HeNe laser, followed by invention of most lasers.
§  1977: Gordon Gould awarded the patent for the laser.
Optics & Photonics News, 2014
MASER: Microwave Amplification by Stimulated Emission of Radiation
(Means of Acquiring Support for Expensive Research)
Laser basics: three key elements
Back
mirror
Ip
Output
mirror
gain medium
Continuous wave (CW) laser
§  Gain medium
- Enable stimulated emission to produce identical copies of photons
- Determine the light wavelength
§  Pump
- Inject power into the gain medium
- Achieve population inversion
§  Resonator cavity
- make light wave oscillating to efficiently extract energy stored in the gain
medium
- Improve directionality and color purity of the light
Laser basics: three key elements
Back
mirror
Output
mirror
Ip
gain medium
Continuous
wave output
time
Spectrum
power
power
Continuous wave (CW) laser
frequency/wavelength
Ultrafast laser: the 4th element—mode locker
Back
mirror
Output
mirror
Ip
gain medium
Ultrafast laser
Ultrashort
pulse output
Mode
locker
t
Ultrafast laser: the 4th element—mode locker
Back
mirror
Ultrashort
pulse output
Output
mirror
Ip
Mode
locker
gain medium
Ultrafast laser
τ
t
World shortest pulse: 67 attoseconds. The
center wavelength is 20 nm. It is generated
by high harmonic generation.
SHORTEST PULSE DURATION
t
10ps
Nd:glass
Dye
S-P Dye
Nd:YLF
1ps
CW Dye
Color
Center
100fs
Cr:LiS(C)AF
Er:fiber
CPM Dye
Nd:fiber Cr:YAG
10fs
K. Zhao et al., “Tailoring a 67 attosecond pulse
through advantageous phase-mismatch,” Opt. Lett.
37, 3891 (2012)
Nd:YAG
Diode
Cr:forsterite
w/Compression
Ti:sapphire
1965 1970
1975 1980 1985 1990
YEAR
1995 2000 2005
Long vs. short pulses of light
Longer pulse
corresponds to
narrower
spectrum.
power vs. time
Spectrum
time
frequency
time
frequency
Shorter pulse
corresponds to
broader spectrum.
But a light bulb is also broadband.
What exactly is required to make
an ultrashort pulse?
Answer:
A Mode-locked Laser
Adapted from Rick Trebino’s course slides
Ultrafast laser: the 4th element—mode locker
Back
mirror
Output
mirror
Ip
Ultrashort
pulse output
Mode
locker
gain medium
Ultrafast laser
t
τ
t
Ultrafast laser: the 4th element—mode locker
Back
mirror
Output
mirror
Ip
Ultrashort
pulse output
Mode
locker
gain medium
Ultrafast laser
t
τ
t
Examples of ultrafast solid-state laser media
Solid-state laser media have broad bandwidths and are convenient.
Two-photon absorption properties of
fluorescent proteins
M. Drobizhev et al., “Two-photon absorption properties of fluorescent proteins” Nat. Methods 8, 393 (2011).
Main workhorse: Ti:sapphire oscillator
Typical parameters of a commercial product
- Pulse duration: ~100 fs
- Pulse energy: 1-10 nJ
- Pulse rep-rate: 50-100 MHz
- Average power: 300-1000 mW
- Center wavelength: tunable in 700-1000 nm.
Ultrafast: pump-probe spectroscopy
18
Applications of ultrafast lasers: femtosecond
chemistry
Prof. Ahmed Zewail from Cal Tech used ultrafast-laser
techniques to study how atoms in a molecule move during
chemical reactions (1999 Nobel Prize in Chemistry).
Ultra- accurate: timing distribution
Jungwon Kim et al., Nature photonics 2, 733 (2008)
20
Ultra- accurate: fs laser frequency comb
Time domain
f CEO
⎛ Δ φ ⎞
= f R m o d ⎜ C E ⎟
⎝ 2 π ⎠
fR =
1
TR
0
Frequency comb has two degrees of freedom:
Frequency domain
f
f = nf R + f CEO
Ultra- accurate: fs laser frequency comb
C.-H. Li et al., Nature 452, 610 (2008).
22
Ultra- intense: femtosecond laser machining
§  Sub-micron material processing: Material
milling, hole drilling, grid cutting
§  Surface structuring: Photolithographic
mask repair, surface removal or smoothing
without imparting any thermal influence into
the underneath sub-layers or the substrate
§  Photonics devices: Machining of optical
waveguides in bulk glasses or silica, and
inscription of grating structure in fibers
§  Biomedical devices: Use of femtosecond
lasers for stent manufacture or eye surgery
§  Microfluidics: Microfluidic channels and
devices
§  Displays and solar: Thin-film ablation, solar
cell edge isolation
Laser processing examples on
glass with a 266 nm (UV) ns-laser
(left side) and with a 780 nm 100fs laser (right side).
L. Lucas and J. Zhang, “Femtosecond laser micromachining: A back-to-basics primer,”
Industrial Laser Solutions (2012)
Ultra-intense: protection from lightning
The pulse induces a conducting path, discharging the cloud charge
before lightning can occur.
“Lightning control by lasers,” Nat. Photonics 3, 120 (2009)
Ultra-intense: extreme light infrastructure (ELI)
ELI’s goal is to explore
new frontiers in
physics, including
trying to trigger the
breakdown of the
vacuum—the fabric of
Designs of the three pillars are shown. Top: Hungarian attosecond
space-time itself.
pillar. Middle: Czech beam-line pillar. Bottom: Romanian
photonuclear pillar.
J. Hecht,“Photonic frontiers: the extreme light infrastructure: the ELI aims to break down the vacuum,”
Laser Focus World (2011)
Nonlinear optical microscopy
§  Intrinsic sectioning ability, making 3D imaging possible.
§  Longer excitation wavelength, which reduces tissue
scattering and allows larger penetration depth.
§  New contrast mechanisms: N-photon excitation
fluorescence, Harmonic generation, Coherent Raman
scattering, etc.
§  Ultrashort pulses as the excitation source – many ultrafast
optics technologies can be employed.
W. R. Zipfel et al., Nat. Biotechnol. 21,1369(2003). N. G. Norton et al, Nat. Photonics 7, 205 (2013).
Three channel OPA (optical parametric
amplification) system: what optics lab looks like
20W green light in our lab
Topics to cover
§  Linear and nonlinear pulse propagation: Optical solitons and pulse
compression.
§  Laser dynamics: CW operation, Q-switching, mode locking.
§  Pulse characterization: Autocorrelation, FROG, SPIDER and 2DSI.
§  Noise in mode-locked lasers and frequency combs.
§  Laser amplifiers and optical parametric amplifiers/oscillators.
§  Mid-infrared sources and terahertz sources
§  Soft and hard X-ray sources including attosecond pulse generation.
29
Ultrafast lasers emit pulse train
5fs
t
TR : pulse repetition rate
2.7fs
W : pulse energy
Pave = W/TR : average power
τFWHM : Full Width Half Maximum pulse width
Pp : peak power
30
Some physical quantities
Average power:
Repetition rate:
Pulse energy:
(Average power = Rep-rate X Pulse energy)
Pulse width (duration):
Peak power:
(Peak power = Pulse energy / duration)
(peak) Intensity:
If an optical beam with 1 PW peak power is focused to 1um2 area,
the peak intensity is 1023 W/cm2.
31
Neglecting the spatial dependence for
now, the pulse electric field is given by:
Electric field E (t)
An ultrashort laser pulse
has an intensity and phase
vs. time.
I (t )
Time [fs]
E (t ) ∝
1
2
Intensity
I (t ) exp{i [ω0t − φ (t )]} + c.c.
Carrier
frequency
Phase
A sharply peaked function for the intensity yields an ultrashort pulse.
The phase tells us the color evolution of the pulse in time.
Adapted from Prof. Rick Trebino’s course slides
Removing the 1/2, the c.c., and the
exponential factor with the carrier
frequency yields the complex
amplitude, E(t), of the pulse:
Electric field E (t)
The real and complex pulse
amplitudes
I (t )
E(t ) ∝ I (t ) exp{−iφ (t )}
Time [fs]
This removes the rapidly varying part of the pulse electric field and
yields a complex quantity, which is actually easier to calculate with.
I (t ) is often called the real amplitude, A(t), of the pulse.
Adapted from Prof. Rick Trebino’s course slides
The Gaussian pulse
For almost all calculations, a good first approximation for any
ultrashort pulse is the Gaussian pulse (with zero phase).
E (t ) = E0 exp ⎡⎣ −(t / τ HW 1/ e ) 2 ⎤⎦
= E0 exp ⎡⎣ −2 ln 2 (t / τ FWHM ) 2 ⎤⎦
= E0 exp ⎡⎣ −1.38(t / τ FWHM ) 2 ⎤⎦
§  where τHW1/e is the field half-width-half-maximum, and
τFWHM is the intensity full-width-half-maximum.
§  The intensity is:
2
I (t ) ∝ E0 exp ⎡⎣ −4ln 2(t / τ FWHM ) 2 ⎤⎦
2
∝ E0 exp ⎡⎣ −2.76(t / τ FWHM )2 ⎤⎦
Adapted from Prof. Rick Trebino’s course slides
The Fourier Transform
To think about ultrashort laser pulses, the Fourier
Transform is essential.
We always perform Fourier transforms on the real or complex
pulse electric field, and not the intensity, unless otherwise
specified.
Adapted from Prof. Rick Trebino’s course slides
The frequency-domain electric field
§  The frequency-domain equivalents of the intensity
and phase are the spectrum and spectral phase.
§  Fourier-transforming the pulse electric field:
E (t ) =
yields:
1
2
I (t ) exp{i [ω0t − φ (t )]} + c.c.
Note that φ and ϕ are different!
The frequency-domain electric field has
positive- and negative-frequency components.
Adapted from Prof. Rick Trebino’s course slides
Note that these two terms
are not complex
conjugates of each other
because the FT integral is
the same for each!
The complex frequency-domain pulse field
Since the negative-frequency
component contains the same
information as the positivefrequency component, we usually
neglect it.
We also center the pulse on its
actual frequency.
Carrier Frequency
So the most commonly used
complex frequency-domain pulse
field is:
Thus, the frequency-domain electric field also has an intensity and
phase. S is the spectrum, and ϕ is the spectral phase.
Adapted from Prof. Rick Trebino’s course slides
Often Used Pulses
Pulse width and spectral width: FWHM
38
Operators used in Maxwell’s Equations
39
Operators used in Maxwell’s Equations
40
Operators used in Maxwell’s Equations
41
Maxwell’s Equations of Isotropic and homogeneous Media
Maxwell’s Equations: Differential Form
Current due to free charges
Ampere’s Law
Faraday’s Law
Gauss’s Law
Free charge density
No magnetic charge
Material Equations:
Bring Life into Maxwell’s Equations
Polarization
Magnetization
42
Derivation of wave equation
Vector Identity:
Vacuum speed of light:
43
Derivation of wave equation
No free charges, No currents from free charges, Non magnetization
Simplified wave equation:
Wave in vacuum
Source term
Laplace operator:
∂2
∂2
∂2
Δ=∇ = 2 + 2 + 2
∂x ∂y ∂z
2
44
Interaction between EM waves and materials
0.5 nm
Wavelength of green light is about 500 nm. So the optical wave experiences
an effective homogeneous medium, which is characterized by
Electric permittivity
ε
and
Magnetic permeability
The velocity of light is different from the vacuum speed by a factor called the
refractive index
Dielectric susceptibility and Helmholtz Equation
In a linear medium, dielectric susceptibility is independent of optical field
2
1 + χ (ω ) = n (ω )
Can be complex
Refractive Index
Medium speed of light (dependent on frequency):
46
Lorentz model of light-atom interaction
Important assumptions
The atomic core is
-- positively charged
-- static (heavy, fixed within the crystal
lattice)
-- with the center of charge at x = 0.
The electrons are
-- light weight
-- elastically bound by a massless spring
with spring constant with equilibrium position
at x = 0
-- carrying out a damped movement; that is,
after removing the force, the movement
decreases and finally ends.
-- the electron and atomic core form an
oscillator with a resonant frequency w0.
Juergen Popp et al., Handbook of biophotonics (2012).
H. A. Lorentz
(1853-1928)
Lorentz model: forced electron harmonic oscillator
Dipole moment is defined as the product
of magnitude of charges and the distance
of separation between the charges.
Without an applied field, the centers of the
negative and the positive charges
coincide. The dipole moment is zero.
If a field constant in time is applied,
the electrons are displaced relative to their
position in the absence of an external field.
The centers of the positive and negative
charges no longer coincide and a static
dipole moment is induced.
If a time-dependent electric field interacts with the atom, then the electron
starts to oscillate around its equilibrium position with the same frequency of the
electric field. Such an oscillating dipole moment will emit a new electromagnetic
wave.
Juergen Popp et al., Handbook of biophotonics (2012).
Lorentz model of light-atom interaction
When light of frequency w excites an atom with resonant frequency w0:
Electric field
at atom
r
E (t )
Electron
r
xe (t )
Emitted
field
r
E ʹ′(t )
Incident light
+
=
On resonance (ω = ω0)
Emitted light
Transmitted light
Incident Light excites electron oscillation à electron oscillation
emits new light at the same frequency à incident light interferes
with the new light leading to the transmitted light.
The crucial issue is the relative phase of the incident light and this
emitted light. For example, if these two waves are ~180° out of
phase, the beam will be attenuated. We call this absorption.
Adapted from Rick Trebino’s course slides
Forced oscillator and resonance
When we apply a periodic force to a natural oscillator (such as a
pendulum, spring, swing, or atom), the result is a forced oscillator.
Examples:
Child on a swing being pushed
Periodically pushed pendulum
Bridge in wind or an earthquake
Electron in a light wave
Nucleus in a light wave
Tacoma Narrows Bridge oscillating and
collapsing because oscillatory winds
blew at its resonance frequency.
(collapsed under 64 km/h wind
conditions the morning of November 7,
1940)
The forced oscillator is one of the most important problems in
physics. It is the concept of resonance.
Adapted from Rick Trebino’s course slides
One more example: child on a swing
If you give the swing a push it will swing back and forward. If you
just give it one push it will swing back and forth a few times and
then come to rest. (That’s because of friction and damping.) To
keep the swing moving you have to push again each time the
swing reaches the closest point to you. You have to match the
frequency of the swing to make it swing high.
Course note, MIT 6.007 & http://www.lifeinresonance.com/?page_id=47
The forced oscillator
The amplitude and relative
phase of the oscillator
motion with respect to the
input force depend on the
frequencies.
Let the oscillator’s resonant frequency be ω0, and
the forcing frequency be ω.
Let the forcing function be
a light electric field and the
oscillator a (positively
charged) nucleus in a
molecule.
Electric field
at nucleus Nucleus
Below
resonance
ω << ω0
Weak
vibration.
In phase.
On
resonance
ω = ω0
Strong
vibration.
90° out
of phase.
Above
resonance
ω >> ω0
Weak
vibration.
180° out
of phase.
Adapted from Rick Trebino’s course slides
The forced oscillator
The amplitude and relative
phase of the oscillator
motion with respect to the
input force depend on the
frequencies.
The electron charge is
negative, so there’s a 180°
phase shift in all cases
(compared to the previous
slide’s plots).
Electric field
at electron Electron
Below
resonance
ω << ω0
Weak
vibration.
180° out
of phase.
On
resonance
ω = ω0
Strong
vibration.
-90° out
of phase.
Above
resonance
ω >> ω0
Weak
vibration.
In phase.
Adapted from Rick Trebino’s course slides
The forced oscillator
Electric field
at atom
Electron
Below
resonance
The amplitude and relative ω << ω0
phase of the oscillator
motion with respect to the
input force depend on the
On
frequencies.
resonance
ω = ω0
Maxwell’s Equations
will yield emitted light
that’s 90° phaseshifted with respect
to the atom’s motion.
Above
resonance
ω >> ω0
Adapted from Rick Trebino’s course slides
Emitted
field
Weak
emission.
90° out
of phase.
Strong
emission.
180° out
of phase.
Weak
emission.
-90° out
of phase.
Interference depends on relative phase
When two waves add together with the same complex exponentials,
we add the complex amplitudes, E0 + E0'.
~
Constructive
interference:
1.0
Destructive
interference:
1.0
=
1.2
1.0
-0.2
Laser
+
-0.2i
=
=
0.8
time
Quadrature phase: ±90°
interference:
+
+
0.2
~
1-0.2i
time
time
Slower phase velocity
(when accumulated over
Adapted from Rick Trebino’s course slides
distance)
Absorption
Dielectric Permittivity: Lorentz model
Density (# of atoms per unit volume)
Elementary Dipole
Lorentz Model:
x(t)
p (t ) = −e x(t )
x(t ) is much smaller than the wavelength of electric field. Therefore we can
neglect the spatial variation of the E field during the motion of the charge.
Response to a monochromatic field
force
mass
quality factor
frequency of undamped oscillator
57
Real and Imaginary Part of the Susceptibility
Ne 2 1 / 2
ωp = (
)
ε 0 m0
Plasma frequency
58
Real and Imaginary Part of the Susceptibility
Q=10
Figure 2.3: Real part (dashed line) and imaginary part (solid
line) of the susceptibility of the classical oscillator model for
the dielectric polarizability
59
In a metal, free electrons between background ions
In general:
ω < ω p : Metal reflects and for ω > ω p :"transparent"
60
Real and Imaginary Part of the Susceptibility
Example: EM-Wave polarized along x-axis and propagation along z-direction:
In general:
Dispersion relation:
kr (ω ) =
ω
c0
nr (ω )
61
Sellmeier equations to model refractive index
If the frequency is far away from the absorption resonance
Ω02 − ω 2 >> 2ω
Ω0
Q
2
ω
χ~r (ω ) = 2 p 2
Ω0 − ω
Normally there are multiple resonant frequencies for the electronic oscillators.
It means in general the refractive index will have the form
2
2
ω
λ
n 2 (ω ) = 1 + ∑ Ai 2 i 2 = 1 + ∑ ai 2
2
Ω
−
ω
λ
−
λ
i
i
0
i
62
Absorption
and refractive
index Vs.
wavelength
63
Kramers-Kroenig (K-K) Relation
Causality of medium impulse response: χ (t ) = 0, for t
<0
K-K relationship between real and imaginary part of susceptibility