AN ABSTRACT OF THE DISSERTATION OF
Andrew D. Jameson for the degree of Doctor of Philosophy in Physics presented on
January 20, 2012.
Title: Generating and Using Terahertz Radiation to Explore Carrier Dynamics of
Semiconductor and Metal Nanostructures
Abstract Approved: ________________________________________________
Yun-Shik Lee
In this thesis, I present studies in the field of terahertz (THz) spectroscopy. These
studies are divided into three areas: Development of a narrowband THz source, the study
of carrier transport in metal thin films, and the exploration of coherent dynamics of quasiparticles in semiconductor nanostructures with both broadband and narrowband THz
sources. The narrowband THz source makes use of type II difference frequency
generation (DFG) in a nonlinear crystal to generate THz waves. By using two linearly
chirped, orthogonally polarized optical pulses to drive the DFG, we were able to produce
a tunable source of strong, narrowband THz radiation. The broadband source makes use
of optical rectification of an ultra-short optical pulse in a nonlinear crystal to generate a
single-cycle THz pulse.
Linear spectroscopic measurements were taken on NiTi-alloy thin films of various
thicknesses and titanium concentrations with broadband THz pulses as well as THz
power transmission measurements. By applying a combination of the Drude model and
Fresnel thin-film coefficients, we were able to extract the DC resistivity of the NiTi-alloy
thin films.
Using the narrowband source of THz radiation, we explored the exciton dynamics
of semiconductor quantum wells. These dynamics were made sense of by observing
time-resolved transmission measurements and comparing them to theoretical
calculations. By tuning the THz photon energy near exciton transition energies, we were
able to observe extreme nonlinear optical transients including the onset of Rabi
oscillations. Furthermore, we applied the broadband THz waves to quantum wells
embedded in a microcavity, and time-resolved reflectivity measurements were taken.
Many interesting nonlinear optical transients were observed, including interference
effects between the modulated polariton states in the sample.
©Copyright by Andrew D. Jameson
January 20, 2012
All Rights Reserved
Generating and Using Terahertz Radiation to Explore Carrier Dynamics of
Semiconductor and Metal Nanostructures
by
Andrew D. Jameson
A DISSERTATION
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Doctor of Philosophy
Presented January 20, 2012
Commencement June 2012
Doctor of Philosophy dissertation of Andrew D. Jameson presented on January 20, 2012.
APPROVED:
_______________________________________________________
Major Professor, representing Physics
_______________________________________________________
Chair of the Department of Physics
_______________________________________________________
Dean of the Graduate School
I understand that my dissertation will become part of the permanent collection of Oregon
State University libraries. My signature below authorizes release of my dissertation to
any reader upon request.
_______________________________________________________
Andrew D. Jameson, Author
ACKNOWLEDGEMENTS
I would like to express thanks to several people who made this work possible.
First, I would like to thank my advisor, Dr. Yun-Shik Lee. His endless patience,
guidance, and curiosity made the laboratory a place where I could truly flourish. To my
predecessor in the lab, Jeremy Danielson, I give thanks for teaching me much and passing
the torch. I also would like to acknowledge my lab partner Joe Tomaino for keeping the
lab fun and whose virtuosic MatLab skills were a key component to the success of our lab
work.
I would also like to thank my parents, Marj and Darrol Jameson. They may have
had concerns at one point about how long this work was taking, but they never flinched
in their support of me, both morally and financially.
Lastly, I thank the Brewstation, the best coffee house and pub in town, for
providing sanctuary and family atmosphere needed to get through the stressful times.
TABLE OF CONTENTS
Page
1 Overview of Terahertz Radiation, Sources, and Detectors ..................................
1
1.1 Terahertz Radiation .....................................................................................
1
1.2 THz Sources ................................................................................................
2
1.2.1 Frequency Converting Systems .........................................................
1.2.2 Photocurrent Radiation ......................................................................
1.2.3 Free Electron Sources ........................................................................
1.2.4 THz Lasers .........................................................................................
3
4
5
5
1.3 THz Detectors .............................................................................................
6
1.3.1 Coherent Detectors.............................................................................
1.3.2 Incoherent Detectors ..........................................................................
7
7
2 Electromagnetic Waves in Nonlinear Media .......................................................
9
2.1 Maxwell’s Equations in Nonlinear Media ..................................................
9
2.2 Nonlinear Susceptibility..............................................................................
11
2.3 Second-Order Nonlinear Susceptibility ......................................................
12
2.4 Nonlinear Susceptibility as a Tensor ..........................................................
15
2.5 Phase Matching ...........................................................................................
16
3 THz Generation and Detection Using a Femtosecond Laser ...............................
19
3.1 Optical Rectification of an Ultra-short Optical Pulse .................................
19
3.2 Generation of Broadband THz Using Optical Rectification in ZnTe .........
21
3.3 Generation of Narrowband THz Pulses Using Type II Difference
Frequency Generation (DFG) .....................................................................
26
3.3.1 DFG with Chirped Pulses ..................................................................
3.3.2 Experimental Arrangement ................................................................
3.3.3 Angular Dependence of THz Output .................................................
26
28
29
TABLE OF CONTENTS (Continued)
Page
3.4 Terahertz Time Domain Spectroscopy With Electro-Optic Sampling .......
31
3.4.1 Experimental Arrangement ................................................................
3.4.2 Quantitative Description of EO Sampling .........................................
3.4.3 Distortion of the Electro-Optic Signal ...............................................
32
33
39
4 Intense Narrowband THz Generation via Type-II DFG ......................................
42
4.1 Measurements and Results ..........................................................................
43
5 THz Spectroscopy of Nickel-Titanium (NiTi-alloy) Thin Films .........................
53
5.1 NiTi Alloy Thin Film Preparation and Composition ..................................
54
5.2 Experimental Setup .....................................................................................
56
5.3 Transmission Measurements .......................................................................
58
5.4 Analysis and Results ...................................................................................
61
5.4.1 Justification for the Theoretical Model ..............................................
5.4.2 The Drude Model ...............................................................................
5.4.3 Thin Film Fresnel Coefficients ..........................................................
5.4.4 Results ................................................................................................
61
64
66
72
6 Transient Optical Response of Quantum Well Excitons to Intense
Narrowband THz Pulses .........................................................................................
76
6.1 Quantum Wells ...........................................................................................
77
6.2 Excitons.......................................................................................................
81
6.3 Experimental Arrangement .........................................................................
89
6.4 Results .........................................................................................................
94
6.4.1 Experimental Results .........................................................................
6.4.2 Theoretical Results and Comparison .................................................
94
99
7 Extreme Nonlinear THz Transients in Quantum Well Microcavities .................
104
TABLE OF CONTENTS (Continued)
Page
7.1 Microcavity General Characteristics...........................................................
104
7.2 Polaritons ....................................................................................................
107
7.3 Our Sample .................................................................................................
109
7.4 Experiment ..................................................................................................
110
7.4.1 Experimental Setup ............................................................................
7.4.2 Experimental Considerations .............................................................
110
114
7.5 Results .........................................................................................................
116
8 Conclusions ..........................................................................................................
124
9 Bibliography ........................................................................................................
129
LIST OF FIGURES
Figure
Page
1.1 Electromagnetic Spectrum .............................................................................................1
1.2 Second order nonlinear effects.......................................................................................4
1.3 Schematic of THz generation by a photoconductive antenna ........................................4
1.4 Generic four-level system for a THz laser .....................................................................6
2.1 Oscillator potentials .....................................................................................................13
2.2 Graphical representation of phase matching of THz radiation generated by optical
rectification with optical pump ..........................................................................................17
3.1 Schematic of the laser system used to generate short optical pulses ...........................19
3.2 Optical rectification .....................................................................................................21
3.3 Angle-dependent THz output from optical rectification ..............................................24
3.4 Simple schematic for generating single-cycle THz .....................................................25
3.5 Demonstration of linear chirp ......................................................................................27
3.6 Experimental setup for generating narrowband THz with Type II DFG .....................29
3.7 Two orthogonal fields coincident on a [110] ZnTe crystal..........................................30
3.8 Experimental schematic for Electro-Optic sampling ...................................................33
3.9 Arrangement of ZnTe crystallographic axes................................................................36
3.10 Index ellipse for EO sampling ...................................................................................38
3.11 Filter functions for various ZnTe crystal thickness ...................................................40
4.1 Michelson interferometer setup used to determine THz power spectra ......................44
4.2 Field autocorrelation of THz pulse ..............................................................................44
4.3 Comparison of Single-Cycle TDS and DFG TDS .......................................................45
LIST OF FIGURES (Continued)
Figure
Page
4.4 Demonstration of DFG tunability ................................................................................46
4.5 Emitted THz beam power as a function of central frequency .....................................47
4.6 Emitted THz power for both negative and positive time delays ..................................49
4.7 THz power vs. optical pump and Optical-to-THz conversion efficiency ....................51
5.1 Sample EDX spectrum used to determine Ni and Ti concentration ............................55
5.2 AFM measurements of the thickness of several NiTi alloy thin films ........................56
5.3 Depiction of NiTi sample and detection schemes........................................................57
5.4 Raster-scan of two NiTi alloy thin films of different thickness on Si substrate ..........58
5.5 NiTi alloy Relative power transmission measurement using a Si bolometer ..............59
5.6 NiTi alloy time-domain data ........................................................................................60
5.7 Illustration of the interface of NiTi alloy and silicon ..................................................66
5.8 Illustration of the THz transmitted out of NiTi sample ...............................................69
5.9 Long TDS measurements taken on NiTi alloy ............................................................71
5.10 Summary of resistivity measurements made on NiTi alloy thin-film samples ..........73
5.11 Phase diagram for NiTi alloys ...................................................................................74
6.1 Simple depiction of a 1-dimensional quantum well ....................................................78
6.2 Bulk and QW GaAs band structure .............................................................................79
6.3 Absorption spectrum of GaAs/Al0.3Ga0.7As QW used in our study .........................80
6.4 Graphical depiction of the first two energy levels for both the infinite and finite
barrier quantum well systems ............................................................................................88
6.5 Experimental Arrangement for the DFG QW experiment ...........................................89
LIST OF FIGURES (Continued)
Figure
Page
6.6 Schematic of pulse compressor ....................................................................................90
6.7 Internal structure of confined excitons ........................................................................91
6.8 Depiction of exciton polarization dynamics in our QW system ..................................93
6.9 Modulated 1-T(ω) optical transmission spectrum .......................................................95
6.10 Several 1-T(ω) spectra ...............................................................................................97
6.11 Full time evolution of differential transmission.........................................................98
6.12 Theoretical calculations of 1-T(ω) ...........................................................................102
7.1 A generic resonant cavity...........................................................................................105
7.2 Gaussian gain profile .................................................................................................106
7.3 Polariton dispersion relation ......................................................................................108
7.4 Depiction of QW micro-cavity sample ......................................................................109
7.5 Reflectivity spectrum of QW micro-cavity sample ...................................................110
7.6 Experimental setup used with single-cycle micro-cavity QW ...................................111
7.7 QW micro-cavity sample mount degrees of freedom ................................................112
7.8 Simple white light continuum generation setup.........................................................112
7.9 Possible quasi-particle dynamics in micro-cavity QW ..............................................115
7.10 Microcavity QW detuning measurement .................................................................117
7.11 Peak positions of the LEP and HEP modes vs. cavity detuning ..............................118
7.12 Single-Cycle THz pulse used in time-resolved study of QW micro-cavity.............119
7.13 Time-resolved differential reflectivity .....................................................................120
7.14 Time resolved reflectivity of the microcavity sample .............................................122
For my wife Jennifer whose intelligence, grace, patience, and beauty
are both the impetus for all I do and the standard
by which I judge all I’ve done.
1. Overview of Terahertz Radiation, Sources, and Detectors
1.1 Terahertz Radiation
Terahertz (THz) radiation, also known as sub-millimeter waves or T-rays, is the
portion of the electromagnetic (E-M) spectrum that falls between infrared and microwave
bands (See Figure 1.1). Although there is some disagreement on the exact portion of the
E-M spectrum defined as THz radiation, it is generally accepted that THz frequencies
range from ~0.3-30 THz. 0.3 THz corresponds to a photon energy of about 1.24 meV,
which in turn, through the Boltzmann constant, corresponds to thermal energy at a
temperature of around 14 K. As a result, black body radiation in the THz can be present
in all but extremely cold environments such as cryogenically controlled experiments.
Although there has been much scientific interest in THz radiation since the 1920’s [1], it
has taken until the last few decades for the technology to emerge making production of
coherent THz sources possible. During this period, THz science and technology has
exploded to include a wide variety of sources, detectors, and fields of study.
Figure 1.1 Electromagnetic Spectrum
Because of the low energy of THz photons, they do not excite atomic transitions. This
makes many dielectric materials, which are opaque to visible light, transparent to THz
waves. Additionally, THz radiation is non-ionizing. Its wavelength is sufficiently short
to produce detailed images of macroscopic objects. These properties have demonstrated
2
potential for the use of THz waves in security applications such as scanning for
explosives and weapons [2, 3], and medical applications in the non-invasive imaging of
tissues [4-7]. THz radiation is also very powerful as a spectroscopic tool. THz Time
Domain Spectroscopy (THz-TDS) simultaneously gathers both amplitude and phase
information of the THz field. This allows direct access to fundamental material
properties such as permittivity without use of complicated Kramers-Kronig calculations.
THz-TDS has been applied in a variety of studies including the determination of a
material’s properties such as the dielectric properties of graphene [8], rotational and
torsion dynamics of molecules [9,10], and identification of chemical agents such as
illegal drugs [11, 12].
Another very important application of THz radiation is its use in studying carrier
dynamics. Of particular interest to many research groups including our own is the use of
THz spectroscopy to explore the dynamics of quasi-particles in semiconductor
nanostructures. Examples of these studies include the dynamics of excitons [13-15],
plasmons [16-18], and more exotic polaritons [19-21]. The relaxation times of these
excitations are very fast, typically on the order of picoseconds. Even so, THz oscillations
are rapid enough to explore these dynamics.
1.2 THz Sources
Development of THz sources was driven mainly by two scientific groups:
Ultrafast, time-domain spectroscopists in search of longer wavelengths, and radioastronomers who desired to work with shorter wavelengths [22]. From here, the
development of THz sources quickly spread to many fields. Sources in use today fall into
3
four general categories: (i) Frequency converting systems, (ii) Photocurrent radiation, (iii)
Free electron sources, and (iv) THz lasers [23].
1.2.1 Frequency Converting Systems
Frequency converting systems are one of the most commonly used table-top
sources of THz radiation used for spectroscopic purposes. They use a strongly non-linear
medium to convert incoming pump radiation to THz waves. Frequency conversion can
occur from high optical frequencies to relatively low THz frequencies (down-conversion)
or from low microwave frequencies to relatively high THz frequencies (up-conversion).
Optical rectification and difference frequency generation (DFG) in a non-linear
crystal are both second-order non-linear optical effects which result in frequency downconversion. Optical rectification involves a broadband, femtosecond optical pulse which
creates a time-dependent polarization in a non-linear crystal. With a correctly chosen
optical pump, this polarization radiates at THz frequencies, and the pulse shape is related
to the incoming optical envelope. DFG involves two continuous wave (CW) optical
sources, ω1 and ω2, whose frequency difference, Δω, lies in the THz range. Figure 1.2
depicts these processes in an oversimplified manner. Both of these methods will be
discussed more rigorously in the next chapter as they were heavily relied upon in our
experimentation. Frequency up conversion is achieved by creating high harmonics of
microwave sources in a strongly non-linear diode.
4
Figure 1.2 Second order nonlinear effects. Simple depictions of a) Optical
Rectification, and b) Difference Frequency Generation
1.2.2 Photocurrent Radiation
Another common method of producing THz radiation is through the use of a
photoconductive (PC) antenna [24-26]. A schematic of a simple antenna is shown below
in Figure 1.3.
Figure 1.3 Schematic of THz generation by a photoconductive antenna.
A bias voltage is placed across the antenna, and the gap is illuminated with a pump laser.
The combination of optical excitation and the bias generates carrier motion in the gap
giving rise to a time-dependent current, I(t), which emits THz radiation. As with nonlinear crystals, PC antennae can produce single-cycle THz pulses with a short optical
pulse as well as CW THz radiation with two CW optical inputs.
5
1.2.3 Free Electron Sources
Free electron sources are excellent for generating high power THz radiation [2729]. These sources involve creation of electrons in vacuum which are manipulated by
magnetic fields. The subsequent acceleration of the electrons results in the desired
radiation. A Free Electron Laser (FEL) is an example of this. For broadband, short THz
pulses, a population of electrons is excited with an ultra-short optical pulse. These
electrons are then accelerated to relativistic speeds causing emission of radiation. CW
THz can be achieved by accelerating electrons to relativistic speeds, then passing them
through a magnet array causing them to oscillate in a sinusoidal pattern, which in turn
radiates narrowband THz waves. These systems are widely tunable from the microwave,
THz, visible, IR, and more recently to X-ray [30], and produce very high power. Their
only drawback is the need for linear accelerators which requires large, expensive
structures to house and run.
Another free electron source is the Backward Wave Oscillator (BWO) [31]. The
BWO operates on very similar principles as the FEL, but in a much more compact
device. The basic principle of these devices involves an electron beam that travels in the
presence of a slowly oscillating, oppositely traveling, electro-magnetic field. The EM
field causes bunching and oscillation of the electrons which, in turn, emit and amplify
THz radiation. Recent advances in the fields of micro-scale manufacturing have
prompted predictions of miniaturization of BWO THz emitters down to the scale of handheld devices [32].
1.2.4 THz Lasers
6
Directly lasing systems involve a population-inverted system whose carrier
recombination directly radiates THz frequencies. Figure 1.4 below depicts this process.
There are some gas and solid state systems which have carrier recombination
characteristics allowing the construction of THz lasers. Recently, the most studied solid
state THz source is the Quantum Cascade Laser (QCL) [33-35]. The lasing action of
these devices develops from the repeated tunneling of carriers through a periodic stack of
semiconductor layers. Each tunneling is associated with a THz photon, and the repetitive
nature builds up a coherent THz output. QCL’s provide a promising technique for
miniaturizing THz sources. The only impracticality is that they are operated at cryogenic
temperatures.
Figure 1.4 Generic four-level system for a THz laser.
1.3 THz Detectors
As mentioned above, the energy associated with THz photons is very small when
compared to visible light. Because of this small photon energy, many traditional
detectors are not sensitive enough for use with THz sources. However, there are several
schemes which allow the detection of THz photons. Measurement methods which
provide both amplitude and phase information of the THz field are referred to as
7
“coherent” detection. Systems which provide intensity information on the THz field are
referred to as “incoherent” detectors [36].
1.3.1 Coherent Detectors
Coherent detection is a very powerful laboratory tool because, as mentioned
above, it simultaneously provides amplitude and phase information of THz waves. This
is important, because once the amplitude and phase information are known, the
dispersion and absorption of a sample can be uniquely determined. Another key feature
of coherent detection schemes is that they usually share similar aspects of the means by
which the THz wave is generated. In particular, they often involve similar non-linear
processes to the generation method. In addition to this, they also use a portion of the
generation pump as a probe during detection. Important examples of this are electrooptic (EO) sampling in a nonlinear crystal and photoconductive switching with a PC
antenna. In brief, EO sampling involves the THz field altering the index of refraction in a
nonlinear detection crystal, which alters the ellipticity of a probe beam. The change in
ellipticity is directly proportional to the THz field. EO sampling will be described in
much more detail in chapter 2 since it was the method used in the lab for all our time
domain measurements. PC switching is similar in that the THz pump induces a current
across a PC antenna which is proportional to the THz field.
1.3.2 Incoherent Detectors
Because of the small photon energy, and often weak intensity of THz beams,
incoherent detectors of the type used for higher energy photons have been more difficult
to produce. However a group of incoherent detectors, all of which are based on thermal
8
effects, has emerged. The most common of this class includes bolometers, pyroelectric
detectors, and Golay cells. Bolometers measure a temperature change due to a THz
photon by measuring the change in resistance in a semiconductor held at cryogenic
temperatures. Pyroelectric materials undergo a change in the polarization of the material
due to temperature changes. Pyroelectric detectors exploit that property by measuring the
small voltage change due to THz photon heating. Golay cells have a small gas chamber
with a flexible diaphragm on one side. THz photons heat the gas causing a change in
pressure which distorts the diaphragm. A separate reflectivity measurement off the
diaphragm is taken with the change in reflectivity being proportional to the distortion.
These detection methods are all sensitive to THz energy, and can be very useful down to
very small THz intensities. However, since they all rely on slow thermal effects, their
response times are correspondingly slow.
9
2. Electromagnetic Waves in Nonlinear Media
Many methods of generating and detecting THz radiation were reviewed briefly
in the previous section. Some of these methods played crucial roles in our
experimentation, and will be covered in much more detail in the following chapters.
These methods include generation of THz waves with optical rectification, generation of
THz waves with difference frequency generation, and detection of THz waves with
electro-optic sampling in a nonlinear crystal. All of these methods involve zinc telluride
(ZnTe) crystals, so a later section will include calculations with the specific material
properties of ZnTe. Before these calculations, it is necessary to cover some formalism of
wave propagation in nonlinear optical media. The next section will present a classical
wave equation/oscillator model which is sufficient to describe the light-matter
interaction.
2.1 Maxwell’s Equations in Nonlinear Media
The formulation for the behavior of electromagnetic waves in nonlinear media is
slightly different than that for linear media, but the approach is the same. We begin with
Maxwell’s equations [37]:
E
B 0 0
B
t
E
0 J
t
(2.1.1)
(2.1.2)
0
(2.1.3)
B 0
(2.1.4)
E
10
For our purposes, we can consider cases in which there are no free sources of
charge or current, and the materials are not magnetic, such that J = 0, ρ = 0, and B = H.
In addition, the relationship
D 0E P
(2.1.5)
will be useful, but we note that now P depends nonlinearly on the electric field, E .
Specifically, P now has a nonlinear component such that D can be written as
D 0 E P (1) P NL D (1) P NL
(2.1.6)
(1)
(1)
where D 0 E P is the linear part of D . Using equation 2.1.6 and Maxwell’s
equations we can derive the wave equation in the usual way yielding
2 D (1)
2 P NL
E 0
0
t 2
t 2
(2.1.7)
Introducing the relationship
Dn(1) (1) (n ) En
(2.1.8)
we can write the generalized wave equation as
NL
2 En
2 Pn
(1)
E n 0 ( n ) 2 0
t
t 2
(2.1.9)
Note that the fields now bear the subscript n indicating the summation over individual
frequencies. This is a necessary generalization when considering dispersive media. Now
the above relationship takes the form of a driven wave equation where the nonlinear
polarization is the driving term. We can further simplify equation 2.1.9 by making use of
the following identity:
11
E ( E) 2 E
(2.1.10)
In the case of a transverse plane wave, the Laplacian of E is all that is left yielding the
more familiar wave equation:
2 (1)
2 NL
E
P
n
n
2 E n 0 (1) ( n )
0
t 2
t 2
(2.1.11)
As will be seen in later sections, equation 2.1.11 can be used to qualitatively
describe various nonlinear interactions. Details of this derivation can be found in [38].
2.2 Nonlinear Susceptibility
In general, and as demonstrated by the simple derivation of the nonlinear wave
equation above, the optical response of a material can be characterized by its polarization.
We can express the polarization of a material as a power series expansion in terms of the
electric field strength. For simplicity, we consider an isotropic material that is lossless
and dispersionless. In this simple case, the polarization becomes:
P(t ) (1) E(t ) ( 2) E 2 (t ) (3) E 3 (t )...
(2.2.1)
in which the susceptibilities are scalars. The first term in equation 2.2.1 is the expression
for polarization for linear media. Higher order terms represent nonlinear contributions.
For even the simplest expressions of electric field, it is easy to see how various non-linear
terms arise. Consider the simple example of the second order effects given by the electric
field
E(t ) E0 (e it eit ) .
In this case, the second order polarization is
(2.2.2)
12
P ( 2) (t ) ( 2) E E ( 2) E02 (e it e it )(e it e it )
(2.2.3)
which yields
P( 2) (t ) 2 ( 2) E02 1 cos(2 )
(2.3.4)
Even with this simple electric field, there are two nonlinear components that are
not a function of the original frequency, ω. One is independent of frequency while the
other is dependent on 2ω. The term independent of frequency is known as the
rectification signal. Since it is independent of frequency, the rectification signal is
proportional to constant field intensity, |E0|2. Optical rectification is the basis of one of
our most important THz generation systems and will be discussed in more detail later.
The process that produces the term proportional to 2ω is known as frequency doubling.
The above derivation demonstrates how nonlinear processes can efficiently produce new
frequencies of radiation. Another of our most important THz generation schemes
depends on the related process of difference frequency generation (DFG) which will also
be discussed in more detail later. Although this example is not particularly useful for real
materials, it does introduce the concept of nonlinear optical effects and their
proportionality to the nonlinear optical susceptibility. In reality, materials and fields are
often far more complicated, and their properties must be represented by tensors. Since
second-order effects are primarily involved in the THz generation and detection schemes
mentioned, it is important to look more closely at the second order susceptibility.
2.3 Second-Order Nonlinear Susceptibility
A good place to begin exploring non-linear susceptibility is the Lorentz model in
which the material is treated as a system of oscillators. In this case, the electrons bound
13
to the ion are treated as simple harmonic oscillators (SHO) driven by an electromagnetic
field. However, 2nd order effects can only occur in noncentrosymmetric materials whose
potential is not symmetric along the oscillation direction. For sufficiently weak fields
(small oscillations), the SHO model is still a good approximation. As the field strength
increases, the electrons are driven to larger oscillations and the SHO is no longer a
sufficient approximation. One way to account for the lack of symmetry is to expand the
potential in a Taylor series and include an additional term. The potential is now
transformed from the normal SHO potential to the following:
U ( x)
1
1
1
m0 x 2 U ( x) m0 x 2 mx3
2
2
3
.
(2.3.1)
The figure below illustrates the new potential as compared with the SHO potential.
Figure 2.1 Oscillator potentials. The symmetric SHO potential compared with an
oscillator potential including an additional term. The parabolic SHO potential is a good
approximation of the new potential for small values of x, but diverges as the amplitude of
x becomes larger.
Including this new term, the equation of motion for the system becomes:
2 x(t )
x(t )
e
02 x(t ) x 2 (t ) E (t )
2
t
t
m
(2.3.2)
14
Now consider the more generalized driving field E(t ) E0 (ei1t e i2t ) c.c.
Notice now that equation 2.3.2 cannot be solved exactly due to the new potential which
has introduced a quadratic term in the displacement. However, if we assume αx2 <<
ω20x, we can find solutions to equation (2.3.2) with a perturbative approach. The
perturbative approach involves expanding the displacement in a power series such that
x(t) = λx(1)(t) + λ2x(2)(t) + λ3x(3)(t)… As long as the series is convergent, this approach is
valid. This means that each higher order component of x(t) must be smaller than the last
(x(1)(t)>> x(2)(t)>> x(3)(t)…). In other words, the nonlinearity cannot be too strong, or
these inequalities may not be valid. More details of this calculation can be found in [39].
In the second order of x(t), which we are primarily interested in, the solution has several
separable parts that are functions of different combinations of ω1 and ω2. Of particular
relevance is the portion
2e 2
E 0 E 0*
2
m
1
2 ( 2 2 2i )( 2 2 2i )
1
1
0
1
1
0 0
1
2 ( 2 2 2i )( 2 2 2i )
2
2
0
2
2
0 0
x ( 2 ) ( 0)
(2.3.3)
which corresponds to optical rectification. Using the relationships
P ( 2) (0, 1, 2 ,1, 2 ) ( 2) (0, 1, 2 ,1, 2 ) E0
2
(2.3.4)
and
P( 2) (0, 1, 2 ,1, 2 ) Nex( 2) (0)
(2.3.5)
we can show that the second order susceptibility in the case of optical rectification is:
15
( 2) (0, 1, 2 ,1, 2 )
Ne3
1
m202 02 (02 12, 2 )2 41, 2 2
(2.3.6)
All the other portions of the second order susceptibility mentioned above can be found in
a similar manner, as well as higher order susceptibilities if desired.
2.4 Nonlinear Susceptibility as a Tensor
Before progressing to the specific detection and generation methods used in my
research, we need a more rigorous definition of the susceptibility as it applies to materials
which are not isotropic. In general, the second order nonlinear polarization can be
written as:
Pi (n m ) D ijk( 2) (n m , n , m ) E j (n )Ek (m )
jk
(2.4.1)
mn
where D is a degeneracy factor that comes from summing over the frequency
components. Equation 2.4.1 can be extended to arbitrary order by simply adding more
frequency components and more field components. Notice that in this case the tensor ijk( 2)
can potentially have 27 independent components. We can simplify this using what is
known as contracted notation. First we introduce the definition:
dijk
1 ( 2)
ijk
2
(2.4.2)
We now apply symmetry conditions, known as Kleinman’s symmetry conditions,
which greatly reduce the complexity of the tensor [40]. These conditions require dijk to
be symmetric in its last two indices such that dijk = dil. The simplified notation is as
follows:
16
jk : 11 22 33 23,32 13,31 12,21
l:
1
2
3
4
5
6
(2.4.3)
Applying 2.4.3 reduces the number of independent components to 18. Applying
Kleinman permutation symmetry [40], it can be shown that the total remaining
independent components in contracted notation is only 10 such that:
d11
d il d16
d
15
d12
d13
d14
d15
d 22
d 23
d 24
d14
d 24
d 33
d 23
d13
d16
d12
d14
(2.4.4)
Equation 2.4.4 can be used in matrix calculation of values of the polarization. As will be
shown, many crystals have additional symmetry that further reduces the number of
independent components of the susceptibility tensor.
2.5 Phase Matching
As we have discussed the background of electromagnetic radiation propagating in
nonlinear media and some of the effects that arise as a result, it is important to discuss
phase matching. Consider two EM fields
~
~
E1 (r ) A1 (1 ) exp( ik1 r ), E2 (r ) A2 (2 ) exp( ik 2 r )
(2.5.1)
where Ai are complex amplitudes which are functions of different frequencies ω1 and ω2.
As the two waves propagate through a material, perfect phase matching is defined by the
simple relationship
k k1 k 2 0 .
(2.5.2)
Depending on the situation, phase matching may be referring to different processes, but
this relationship is always valid. For example, the situation may be a sum frequency
generation (SFG). Using the fields above, the output of such a process would be
17
~
E3 (r ) A3 (1 2 ) exp( ik r )
(2.5.3)
where the efficiency of the output depends on the value of k . Another situation, which
will arise later when discussing optical rectification in ZnTe, is the phase matching of a
pump field and the field it generates. In this case, the optical pulse and the THz pulse
propagate together. If the phase velocity of the THz pulse matches the group velocity of
the optical pulse, coherent amplification of the THz radiation will occur. If these
velocities are not matched, then destructive interference will eventually dominate the
process, and the output will be weak or nonexistent. Phase matching and phase mismatch
is graphically depicted below in Fig. 2.2.
Fig. 2.2 Graphical representation of phase matching of THz radiation generated by
optical rectification with optical pump. (a) Optical group velocity is equal to THz wave
phase velocity resulting in amplification of the THz radiation. (b) Velocity mismatch
between THz wave and optical pump results in interference and weak signal.
18
In principle, it is difficult to perfectly phase match multiple traveling waves inside a
material. Phase matching is especially difficult for short pulses since they contain a band
of frequency components which will travel at different velocities due to dispersion.
Additionally, for nonlinear processes, the output is often of a much different frequency,
and most materials do not have a flat frequency response over many orders of magnitude.
If the nonlinear medium is too thick, these destructive processes will dominate. If the
medium is too thin, coherent amplification will not build up, and the output will be weak.
In practice the thickness of the material must be balanced to maximize the output while
minimizing parasitic effects. The parameter by which this is characterized is called the
“walk-off length”. In general, the walk-off length can be expressed as [41]:
lw
c p
nTHz nopt
(2.5.4)
However, we must keep in mind that the optical index of refraction that we refer to is the
group index. So, in general, the task of finding a good nonlinear material comes to
finding one with optical properties that give a large walk-off length.
19
3. THz Generation and Detection Using a Femtosecond Laser
With the background of chapter 2, we can now explore in more detail a few
specific nonlinear optical processes that are central to this research. These include the
generation of THz waves through optical rectification of an ultra-short optical pulse, the
generation of THz waves through type II difference frequency generation (DFG) of two
linearly chirped, orthogonally polarized pulses, and the detection of THz waves using
electro-optic sampling.
3.1 Optical Rectification of an Ultra-short Optical Pulse
Optical rectification of an ultra-short optical pulse to generate THz radiation is the
“work horse” of the lab. This method generates a relatively broadband, strong, singlecycle THz pulse that was useful for everything from characterizing materials with linear
transmission measurements to exciting nonlinear carrier effects. To demonstrate how
THz waves are generated, it is important to know the properties of our laser system. The
simple schematic below shows the important parts of the system:
Figure 3.1 Schematic of the laser system used to generate short optical pulses.
20
As Fig. 3.1 shows, the output of our laser system is a high-power, ultra-short, optical
pulse. Previously, we only considered monochromatic plane waves as the incident
radiation. Now we treat the incident field as a transform-limited, Gaussian pulse which
approximates the output of our laser system very well. In the time domain, this pulse has
the following form:
2
E (t ) E0 cos(t ) exp t 2
(3.1.1)
In Chapter 2, by deriving the wave equation (equation 2.1.9) for nonlinear media,
2 PnNL
we saw that the nonlinear polarization, 0
acts as a source term for the
t 2
propagating wave. Using equation 2.3.4 from the previous chapter, we can rewrite the
wave equation in the following form:
2
2 (1)
E
opt
En
2 En (1) ( n )
( 2)
2
t
t 2
2
(3.1.2)
Here, we have assumed μ0 = 1 since the materials we will discuss have negligible
magnetic properties. Qualitatively, we can see an immediate relationship between the
polarization and radiation it produces. The radiated field (En = ETHz in our case) is
proportional to the second time derivative of the intensity of the driving optical field, Eopt.
Taking the second derivative of equation 3.1.1, we have:
ETHz
2 Eopt
t 2
2
2t 2
2 2
2 E0 exp 2
t
16t 2 4t
2t 2
4 2 exp 2
(3.1.3)
21
Based on equation 3.1.3, the figure below illustrates the shape of the THz waveform we
can expect as a result.
Figure 3.2 Optical rectification. Optical rectification produces a THz pulse whose
waveform is proportional to the second time derivative of the optical pulse envelope.
3.2 Generation of Broadband THz Using Optical Rectification in ZnTe
In general, the THz output due to optical rectification depends on the polarization
of the driving optical pulse with respect to the crystallographic axes. It is therefore
necessary to use the tensor formalism for susceptibility introduced in Chapter 2 to
characterize the THz radiation. In general, the 2nd order tensor polarization for optical
rectification can be expressed as [41]:
d11 d12
Px
Py 2 0 d 21 d 22
d
P
z
31 d 32
d13
d14
d15
d 23
d 24
d 25
d 33
d 34
d 35
E x2
E y2
d16
E z2
d 26
2
E
E
y z
d 36
2Ex Ez
2E E
x y
(3.2.1)
22
As ZnTe used as a source in nearly all our experiments, it is worth discussing in
more detail. ZnTe is cubic crystal of the class 4 3m . This crystal class has only three
non-vanishing matrix elements in its contracted susceptibility tensor which are all equal
such that d14 = d25 = d36 [41]:
0 0 0 d14
d il 0 0 0 0
0 0 0 0
0
d14
0
0
0
d14
(3.2.2)
In spherical coordinates, an arbitrarily polarized field is given by:
sin cos
E E0ˆ E0 sin sin
cos
(3.2.3)
Using 3.2.1, 3.2.2, and 3.2.3 the polarization becomes:
Px
0 0 0 d14
2
Py 2 0 E0 0 0 0 0
P
0 0 0 0
z
0
d14
0
sin 2 cos2
sin 2 sin 2
0
cos2
0
2
sin
sin
cos
d14
2 sin cos cos
2 sin 2 sin cos
(3.2.4)
sin cos
4 0 d14 E sin cos cos
sin sin cos
2
0
To observe 2nd order nonlinear effects we wish to maximize the THz intensity. As
the THz radiation is parallel to the polarization, we can use the expression for
23
polarization intensity to find the optimal alignment of the optical pump. The intensity of
the polarization is:
P 02 d142 E04 sin 2 4 cos2 sin 2 sin 2 2
2
(3.2.5)
Equation 3.2.5 is maximized for sin2(2φ) = 1, or φ = (2n+1)π/4 for n = 0,1,2… which
yields:
ITHz ( , ) P 02 d142 E04 sin 2 4 cos2 sin 2
2
(3.2.6)
which can be written as
ITHz ( , ) P
2
3 MAX 2
ITHz sin 4 3 sin 2
4
(3.2.7)
Equation 3.2.7 has a maximum at sin 1 2 . In terms of a ZnTe crystal cut in the
3
common fashion shown in the Figure 3.3 below, this tells us we can maximize the output
of optical rectification by aligning the optical pump polarization along the [ 1 11] axis, or
at about θ = 55.74° from the [001] axis.
24
Fig. 3.3 Angle-dependent THz output from optical rectification. (a) A [110] cut ZnTe
crystal. The white arrow represents the propagation direction of the optical pump, and θ
is the polarization angle with respect to the [001] axis. (b) Output THz intensity as
function of θ.
The figure below illustrates the simple experimental setup required to generate a singlecycle THz pulse via optical rectification. Also shown are a comparison of the theoretical
angle-dependent intensity with experimental data, and an example of typical single cycle
output obtained by electro-optic sampling which will be discussed later in this chapter.
25
Fig. 3.4 (a) Simple schematic for generating single-cycle THz. (b) Comparison of
theoretical and experimental angle-dependent THz intensity. (c) Sample THz pulse
generated using optical rectification.
Fig 3.4(b) shows that the measured THz intensity deviates slightly from the theoretical
value. This deviation from theory is due to unaccounted for higher-order effects and
inhomogeneity in the generation crystal. As can be seen in Fig 3.4 (c), the shape of the
THz pulse generated via optical rectification matches fairly well with the theoretical
shape in Fig. 3.2 with some deviations. These include a slightly narrower and deeper
oscillation after t = 0, and some subsequent “ringing” oscillations that follow. The
inhomogeneity of the pulse just after t = 0 is due to phase mismatch discussed in section
2.5, while the subsequent high frequency ringing is caused by the increase in index of
refraction as the THz frequency increases [42]. Further oscillations later in time are often
due to absorption and reemission of the THz after it leaves the crystal. These oscillations
are especially noticeable when water vapor is present as it strongly absorbs THz
radiation. The ringing is commonly countered by purging the experimental area with dry
nitrogen or performing the experiment in an evacuated chamber.
26
3.3 Generation of Narrowband THz Pulses Using Type II Difference Frequency
Generation (DFG)
As mentioned in section 2.3, when a sufficiently intense field of the form
E(t ) E0 (exp( i1t ) exp( i2t ) c.c) impinges on a nonlinear crystal, several secondorder nonlinear processes can occur including difference frequency generation and
optical rectification. As an example, the frequency independent case of optical
rectification was discussed in more detail. Now we look at the process of DFG which
involves frequency dependence proportional to exp i1 2 t . Simply put, DFG
involves the input of two sources, ω1 and ω2, which generate an output of frequency ω3 =
ω1 - ω2. Producing THz waves with this method has been around for over fifty years.
Zernike et al used a Nd:Glass laser and a quartz crystal to produce an output at 100 cm-1
(100 μm or 3 THz) in 1965 [43]. More recently, similar tunable THz sources have been
created which use DFG with a variety of sources and materials [44-47].
3.3.1 DFG with Chirped Pulses
In order to generate narrowband THz pulses, our group developed a novel,
tunable, table-top system. As will be shown, the development of our technique yielded
excellent results [48]. This method involves using two linearly chirped, ultra-short
optical pulses. To create a chirped pulse, one simply needs to introduce some group
velocity dispersion (GVD) into the pulse. Adding or compensating for GVD is most
commonly achieved by the use of prisms or diffraction gratings. A set of either of these
devices can introduce a linear chirp by causing a delay between the high frequency
27
components of the pulse and the low frequency components. Equation 3.1.1 gives us the
field for the transform limited case. To express the chirped pulse, we need to add a linear
phase term such that
t2
E (t ) E0 exp 2 exp i0 bt t
(3.3.1)
The figure below demonstrates the results of a linear chirp on a transform-limited pulse.
Fig. 3.5 Demonstration of linear chirp. (a) Transform limited optical pulse. (b) The same
pulse with a linear chirp added in which high frequency components are moved to the
right, and low frequency to the left.
By looking at figure 3.5(b), we can see that the frequency components have been spread
out in time. The instantaneous angular frequency of this pulse is defined as
ins (t ) ins
d d
0 t bt 2 0 2bt
dt dt
(3.3.2)
demonstrating that the frequency of the pulse increases with time if b is positive. By
combining two such chirped pulses with a time delay, τ, between them, it is possible to
28
overlap different frequency components in time, enabling DFG. With this configuration,
the instantaneous frequency difference between the two pulses is:
f ins
1
0 2bt 0 2bt b
2
(3.3.3)
Equation 3.3.3 suggests that with a specific chirp parameter, b, and time delay, τ, Δfins
will be constant in time, and thus the resulting DFG polarization will oscillate at this
same frequency throughout the duration of the overlap.
3.3.2 Experimental Arrangement
Our experimental arrangement is as follows. By not fully recompressing the
output of our regenerative amplifier, we generate a chirped pulse. This pulse is split with
a 50-50 beam splitter (BS), and the polarization of one pulse is rotated 90° by a double
pass through a λ/4 wave plate. A delay is introduced between the pulses, and they copropagate through a 1 mm ZnTe crystal where Type II DFG generates a narrowband THz
pulse. A depiction of the experimental setup is shown below in Figure 3.6. There are
noteworthy advantages to using this configuration. The first is that if a typical Michelson
interferometric setup were used, the maximum beam intensity that can be output is 50%
of the input due to reflective losses. By using orthogonally polarized light and a thin film
polarizer (TFP), transmitted power was maximized. The other advantage to this set up is
that it minimizes parasitic effects, such as two-photon absorption, that decrease the THz
conversion efficiency [48].
29
Fig. 3.6 Experimental setup for generating narrowband THz with Type II DFG
3.3.3 Angular Dependence of THz Output
As in the previous section, we can use the tensor formalism to characterize the
angular dependence of the DFG output. The general form of the fields can be derived
from Fig. 3.7 below. For this general arrangement, the polarization vectors for the fields
E1 and E2 can be written in terms of Cartesian coordinates as:
E1
E2
1
2
1
2
x y cos z sin
x y sin z cos
(3.3.4)
Here we have neglected to include the time dependence and amplitude for simplicity.
Making use of equation 3.3.4 and equation 2.4.1 in matrix form, we can again write an
30
expression for the polarization which is somewhat more complicated than that for optical
rectification since we are now pumping with two independent fields in the configuration
shown below.
001
E1
1 1 0
E2
1 10
Fig. 3.7 Two orthogonal fields coincident on a [110] ZnTe crystal.
Again, making use of the d-matrix for ZnTe, the expression for the polarization is:
Px
0 0 0 d14
Py 4 0 0 0 0
P
0 0 0 0
z
0
d14
0
E x (1 ) E x ( 2 )
E y (1 ) E y ( 2 )
0
E z (1 ) E z ( 2 )
0
E
(
)
E
(
)
E
(
)
E
(
)
y
1
z
2
z
1
y
2
d14
E x (1 ) E z ( 2 ) E z (1 ) E x ( 2 )
E ( ) E ( ) E ( ) E ( )
y
1
x
2
x 1 y 2
(3.3.5)
Using equations 3.3.4 and performing a bit of algebra we can write equation 3.3.5 as
31
Px
0 0 0 d14
*
Py 2 0 E1 E 2 exp( i 3 t ) 0 0 0 0
P
0 0 0 0
z
0
d14
0
cos sin
cos sin
0
2 cos sin
0
2
cos
2
d14
2 cos 2
sin 2
(3.3.6)
where we have included the time dependence of the field in which ω3 = ω1 – ω2.
Equation 3.3.6 can be simplified to
2 cos 2
Px
*
Py 2 0d14 E1E2 exp( i3t ) 2 cos 2
P
z
sin 2
(3.3.7)
As before with optical rectification, the THz output is proportional to P2, which gives us
an angular dependence for the DFG THz wave defined by
ETHz P 3 cos2 2 1
2
(3.3.8)
Equation 3.3.8 tells us two things. First, we notice that the amplitude of ETHz is
maximized when = 0. In other words, E1 and E2 must be aligned along the 1 1 0 and
001 axes respectively to maximize the THz output.
Second, we see that we should
expect a symmetric sinusoidal variation of the field amplitude with unlike optical
rectification. As the experimental output of this setup constituted my first published
work, the results of this method are discussed in detail in Chapter 4.
3.4 Terahertz Time Domain Spectroscopy With Electro-Optic Sampling
32
One very powerful measurement tool used daily in our studies was terahertz timedomain spectroscopy (THz-TDS). As we will see, THz-TDS allows us to map out the
THz field amplitude in time. The reason this tool is so potent is that it simultaneously
delivers both the amplitude and phase information of the THz field. The availability of
the phase and amplitude information grants direct access to electronic properties of
materials, such as permittivity, without the cumbersome Kramers-Kronig calculations
necessary from traditional spectroscopy [49-52].
The phase and amplitude of the electric field is mapped out by use of the linear
electro-optic (EO) effect, also called the Pockels Effect. The EO effect will be
quantitatively described in the next section. The EO effect, in essence, is a change in
index of refraction of a material proportional to a constant or low-frequency electric field.
In our case, the THz field induces a birefringence in an EO crystal (in our case ZnTe). A
low power, linearly polarized optical probe is sent through the EO crystal simultaneously.
Due to the birefringence, the probe polarization will be rotated by some amount directly
proportional to the THz field. By changing the delay between the optical probe and THz
pump, one can sweep the probe across the entire THz pulse essentially mapping out the
field amplitude.
3.4.1 Experimental Arrangement
Figure 3.8 below demonstrates a typical setup to perform EO sampling.
33
Fig. 3.8 Experimental schematic for Electro-Optic sampling. This figure demonstrates
the polarization of the optical pulse after each component (The dotted circle demonstrates
the slight deviation from circular polarization).
In the absence of the THz field the linearly polarized probe pulse is not rotated. A λ/4
plate transforms the pulse to circular polarization which is split into orthogonal
components by means of a Wollaston prism. These are measured by a balanced
photodiode which will register zero current in this case since circular polarization has
equal orthogonal components. In the presence of the THz field, the output of the EO
crystal will be slightly elliptical. This slight ellipticity translates to a slightly elliptical
input into the Wollaston prism, which will register as a small current in the photodiode
since one signal is now slightly larger than the other. The difference is proportional to
the THz field amplitude at that particular delay position. As mentioned above, one can
change the placement of the probe pulse with respect to the THz pulse logging these
differences at each delay position effectively mapping the THz waveform as a function of
time.
3.4.2 Quantitative Description of EO Sampling
34
Developing a more quantitative approach to EO sampling not only helps to understand
the underlying physical mechanisms, but allows us to optimize the experimental
arrangement. Unlike the previous sections, the linear EO effect has traditionally been
described in a different manner that will be followed here [53]. In general, the
relationship between E and D for an anisotropic material can be represented by the
following equation:
D x xx
D
y yx
D z zx
xy xz E x
yy yz E y
zy zz E z
(3.4.1)
For a lossless material, the dielectric tensor is real and symmetric, which means that it
can be expressed in a diagonal form by means of an orthogonal transformation [53]. In
other words, there is a rotation of the current axes possible that will diagonalize the
permeability tensor. The new coordinate system is known as the principle-axis system,
and yields the following expression in terms of the new axes:
D X XX
D 0
Y
DZ 0
0
YY
0
0 EX x X
0 EY , y Y
ZZ E Z z Z
(3.4.2)
Using the above expression and considering the energy density U 1 2 D E , we can
derive an expression for the surfaces of constant energy in terms of the new directions X,
Y, and Z:
35
X2 Y2
Z2
1
2
2
2
n XX
nYY
n ZZ
(3.4.3)
where we have used
1
1
1
1 2
1 2
1 2
X D X , Y DY , Z DZ , and ij nij
2
2
2
(3.4.4)
Equation 3.4.3 is known as the index ellipsoid since it describes the shape of an ellipse.
It can be used to relate the specific axes of the material to their respective indices of
refraction for a given propagation direction. Note that for a set of coordinates which are
not the principal axes, this expression will be complicated by cross terms since the
permeability tensor is not diagonal. The most general form of the index ellipsoid is
xx x 2 yy y 2 zz z 2 2 xy xy 2 yz yz 2 xz xz 1
(3.4.5)
where we have defined the impermeability tensor as ij ij1 .
The next step is to examine what happens to the index ellipsoid when an electric
field is applied to the material. The arrangement of the field and crystal are as shown
below:
36
Y 001
ETHz
φ
X 1 10
Z 1 10
Fig. 3.9 Arrangement of ZnTe crystallographic axes. The ZnTe crystal is cut along the
[110] axis to maximize nonlinear effects. The THz is aligned as shown and propagating
along the [1 1 0] axis.
The detailed procedure by Casalbuoni, et al in reference [42], shows how this formalism
applies specifically to our case of a THz field in a ZnTe detection crystal. Following
their process, we expand the impermeability tensor in the applied field. We assume that
this series converges and retain the first two terms
ij ij ij(0) rijk Ek ...
(3.4.6)
k
which will be greatly simplified by the fact that ZnTe has only three non-zero terms.
Expanding equation 3.4.6 in matrix form, we find that the permeability is no longer
diagonal. As a result, we must find the normalized eigenvectors which point along the
direction of the principal axes. These are given by Casalbuoni et al as
37
1
1
sin
U1
1
1
2
2
1 3 cos
2 2 cos
1 3 cos 2 sin
1
1
sin
U2
1
1
2
2
1 3 cos
2 2 cos
1 3 cos 2 sin
1
1
U3
1
2
0
(3.4.7)
derived from the impermeability eigenvalues:
1, 2
1 r41 ETHz
sin 1 3 cos2
2
2
n0
1
3 2 r41 ETHz sin
n0
(3.4.8)
Note that the first order parts of the eigenvalues are all equal to 1/n02 since ZnTe is
isotropic in the linear regime. From the principal axes, we immediately note the U3
points normal to the [110] plane (parallel to the propagation direction of the ETHz) and
thus U1 and U2 are the axes that define the field-induced birefringence. U1 lies in the
[110] plane, but at some angle, ψ, to the [1 10] axis defined by
cos 2
sin
1 3 cos 2
A schematic of the resulting axes along with the THz field is shown below.
(3.4.9)
38
Y [001]
U1
U2
n2
n1
ETHz
X [1 10]
Fig. 3.10 Index ellipse for EO sampling. The index ellipse for a [110] cut ZnTe crystal
pumped by a THz field propagating along the [1 1 0] axis. The resulting birefringence is
shown exaggerated along the new principal axes U1 and U2.
2
If we return to equations 3.4.8 and recall that ij 1 nij , we can obtain expressions for the index
of refraction along U1 and U2. Since ηij converges, we can use r41 ETHz 1 / n02 to obtain the
following expressions for n1 and n2
n1, 2 n0
n03 r41 ETHz
sin 1 3 cos 2
4
(3.4.10)
For a crystal of thickness d, an optical probe pulse will accumulate a phase difference
between its two orthogonal polarization vectors. The phase difference, designated Γ, due
to equation 3.4.10 is
0 d
c
n1 n2
0 dn03 r41 ETHz
2c
1 3 cos 2
(3.4.11)
39
Equation 3.4.11 shows that the phase shift is indeed directly proportional to ETHz as stated
earlier. Clearly Γ is maximized at φ = 0, which corresponds to a maximally observable
effect if ETHz is polarized along the [1 10] axis. In this maximized case, U1 would be at an
angle ψ = 45°.
3.4.3 Distortion of the Electro-Optic Signal
In practice, the EO signal mapped out by the balanced photodiode can be quite distorted
with respect to the actual THz waveform entering the EO crystal. The degree of distortion
depends on several factors including (i) the bandwidth of the optical probe in the form of the
autocorrelation of the probe spectrum, (ii) the dispersion of the 2nd order susceptibility,
( 2) () ,
and (iii) the phase mismatch between the THz and optical probe [40]. The EO signal can be
expressed as:
S ( ) ATHz () f () exp( i )d
(3.4.12)
exp( ik (0 , )d 1
f () COpt () ( 2) ( 0 ; , 0 )
ik (0 , )
(3.4.13)
where f () is given by
In this formulation [54, 55], S(τ), is the signal after the EO crystal, ATHz, is the THz
spectrum entering the crystal, and f(Ω) is the filter function of the EO crystal. The filter
function consists of the three components listed above (Copt is the spectrum of the
autocorrelation of the optical probe, χ(2) is the dispersive 2nd order susceptibility, and the
final expression in brackets is the phase mismatch characterized by Δk+. d is crystal
40
thickness). Taking into account the THz absorption in the EO crystal, the following
figure shows the filter function for several crystal thicknesses.
1.0
0.1 mm
0.5
0.8
0.6
2.0
|F|
2
1.0
0.4
3.0
0.2
0.0
0
1
2
3
4
5
Frequency (THz)
Fig. 3.11 Filter functions for various ZnTe crystal thickness. Reproduced with
permission from [56].
The plot brings to light many practical considerations for EO sampling. We notice that
the filter function is smoothest over the largest range of THz frequencies for thin crystals
because of the minimization of phase mismatch. Above a frequency of about 4 THz
however, the sensitivity fails due to the absorption wing of the transverse optical (TO)
phonon absorption at ~5.3 THz [56]. As the crystal thickness increases, we notice that
the bandwidth of the sensitivity decreases due to increased phase mismatch, and that the
acoustic phonon absorptions at ~1.6 and ~3.7 THz further distort the filter function [54].
These distorted filter functions can cause distorted waveforms. From equation 3.4.11, we
41
also see that the EO signal will be proportional to the crystal thickness, d. Depending on
the goals of experimentation, an appropriate thickness must be chosen for a detection
crystal to optimize for signal strength, bandwidth, signal fidelity, or a combination
thereof. Of course, experimentally one would like to have the EO output be exactly
characteristic of the input, especially when the experiment involves passing THz waves
through a sample and measuring the results via THz-TDs. It has been shown [55] that
even when distortion is present, sample properties can be extracted by use of a reference.
Now that we have laid down a fairly comprehensive groundwork on the
background and relevance of THz radiation and the main tools used in our lab, we can
explore, in detail, the results of experiments I have performed.
42
4. Intense Narrowband THz Generation via Type-II DFG
As this subject is not only one of the fundamental tools used in my
experimentation, but also my first published work, the experimental results have been
given their own section for discussion here in chapter 4.
Both broadband and narrowband THz sources are important tools in our
laboratory. Depending on the type of experiment, it is beneficial to have a narrowband
source for exciting specific effects whereas a broadband source may excite a number of
effects simultaneously. A few pertinent examples of these experiments include studying
transitions among impurity states in semiconductors[57], intra-band transitions of
excitons in semiconductor nanostructures [58,59], and many-body interactions of
strongly correlated carriers [60]. However as covered in Chapter 1, until recently, there
did not exist many THz devices (sources, detectors, optics, etc). As such, there were very
few compact, tunable sources of narrowband THz radiation available. FEL’s are intense
and tunable, yet their accessibility is limited due to their size. Molecular gas lasers are
compact sources of intense THz radiation [61], yet they lack tunability. There were a few
sources available that offered both the desired compactness, and tunability. These were
mixing of chirped optical pulses in a PC antenna [62], optical rectification of shaped
pulses [63], and the optical rectification in quasi-phase-matching nonlinear crystals [64].
These all provided the added beneficial feature of easily adapting to phase-locked, timeresolved studies. However, at the time, these available sources lacked the ability to
generate the necessary intensities we needed to induce nonlinear effects. This lack of
43
intensity led us to develop our own table-top, tunable source of narrowband THz which is
outlined in Chapter 3 based on the chirped-pulse/PC antenna arrangement. Here we
discuss the capabilities of this powerful system.
4.1 Measurements and Results
The experimental arrangement for this method is shown in Figure 3.5. Some
advantages of this setup are listed in section 3.3.2 which includes minimization of
parasitic effects, and maximization of optical pump throughput. Additionally, by
replacing the PC antenna as the source of THz waves with a ZnTe crystal, we avoid some
drawbacks of the antenna. As our pump source pulse energy is ~1 mJ, PC antennas are
either incapable of sustaining this high peak power, or will become inefficient due to
saturation effects. Additionally, due to finite carrier response times of the substrate, PC
antennas suffer a large decrease in efficiency above ~1 THz [48].
The first measurements were done to determine the spectral output of the DFG
using a Michelson interferometer. The setup of the interferometer in conjunction with a
bolometer and a typical output are shown in the figures below:
44
Fig. 4.1 Michelson interferometer setup used to determine THz power spectra. An
example of a typical field autocorrelation and its corresponding power spectrum obtained
through FFT is shown below in Figure 4.2.
Fig. 4.2 Field autocorrelation of THz pulse. Data generated by the Michelson
interferometer in Figure 4.1 at pulse length 4.11 ps, and delay of 1.9 ps. Inset shows the
corresponding power spectrum.
45
Before we move on to other characterizations of this THz source, it is important to be
clear on the difference between τ and τp in Figure 4.2 so as not to confuse them. τ is the
delay introduced between the two orthogonal pulses, while τp is the pulse duration which
is controlled by a pair of diffraction gratings present in the regenerative amplifier. Figure
4.2 was generated with a pulse duration of τp = 4.11 ps, and a delay of τ = 1.9 ps. The
spectrum (and its waveform) can also be generated via THz-TDS. Figure 4.3 below
shows a comparison of a single-cycle pulse with its broadband spectrum and a manycycled pulse generated with the DFG and its corresponding narrowband spectrum.
Fig. 4.3 Comparison of Single-Cycle TDS and DFG TDS. (a) Comparison of waveforms
in the time domain, and (b) comparison of the spectra in the frequency domain. White
corresponds to the single-cycle THz pulse in both cases.
From Figure 4.3(a) we note that the pulse duration of the single-cycle pulse is
approximately 1-1.5 ps, while the multi-cycle pulse lasts around 5 ps. Additionally the
46
full-width half-max (FWHM) of single-cycle spectrum is approximately 1.5-2 THz while
the FWHM of this particular DFG spectrum is only ~0.4 THz.
As was mentioned in Chapter 3, the difference in frequencies of the two chirped
pulses is linearly dependent on the delay between them. By changing the delay, we tune
the output frequency of the setup. Tunability is clearly demonstrated in Figure 4.4 below.
Fig. 4.4 Demonstration of DFG tunability. The output of the DFG can be tuned by
varying the delay between the two chirped pulses. The inset shows the center frequency
as a function of delay. Figures generated using the Michelson interferometer setup.
47
Figure 4.4 clearly demonstrates how powerful a tool this THz generation method can be.
Since the setup is capable of fine and continuous frequency tuning, it is possible to excite
specific resonances in physical systems not possible with a broadband pulse. Exciting
specific resonances is especially important when the need arises to separate the many
simultaneous effects induced by a broadband pulse (such as a sample whose behavior is
unknown). The inset to Figure 4.4 shows that the center frequency of the THz varies
linearly with pulse delay as predicted by equation 3.3.3. The solid line represents a linear
best fit, and hence yields a chirp parameter of b = 3.85 ps−2.
Next, we examine the power dependence as a function of THz center frequency
by using several fixed pulse durations, and scanning along several values of delay.
Figure 4.5 shows the measured power spectra for several pulse durations.
Fig. 4.5 Emitted THz beam power as a function of central frequency. Data for time
delays of τp = 1.06, 2.03, 2.78, 3.35, and 4.61 ps.
48
Figure 4.5 demonstrates several features of this THz source. First, the peak power
decreases as the pulse duration increases. Additionally, for every pulse but the longest,
the peak power occurs around 1 THz, whereas for τp = 4.61 ps, the peak power is
centered near 1.5 THz.
In general, the shape of the spectrum is limited by two factors. The first is that
Type II DFG is a second-order nonlinear process. Thus, the power is a quadratic function
of the frequency, and the spectrum to falls off sharply for low frequencies because of this.
Secondly, to increase the frequency, we must increase the delay time between the two
pulses. As the delay increases, the two pulses overlap less and less in time, and the DFG
conversion quickly gets weaker. As a result, the spectrum falls off rapidly for high
frequency components as well. The only time this does not occur is for τp = 1.06 ps. We
clearly see that the spectrum does not go to zero at both low and high frequencies. As the
shortest pulse, this also corresponds to the “least chirped” pulse, or conversely the most
compressed. These low and high frequency power components can be explained by
additional THz power being generated due to optical rectification. As we see from
equation 3.2.7, the beam aligned to the [001 ] axis generates no THz intensity by these
means, thus optical rectification is due only to the orthogonal beam aligned along the
[1 1 0] axis. This alignment is not the optimal axis for THz generation from optical
rectification, so this THz is substantially weaker than the DFG radiation. However, as
Figure 4.3 demonstrates, the frequency response for optical rectification is much broader
which accounts for the nonzero low and high frequency components.
49
If we expand these measurements to include both positive and negative time
delays, we see some additional interesting structure to the power spectra. Figure 4.6
again shows several fixed pulse durations now with longer scans over positive and
negative time delays.
Fig. 4.6 Emitted THz power for both negative and positive time delays. The inset shows
optical rectification power for positive and negative time delays.
As outlined in section 3.3, E1 corresponds to the polarization along the [1 1 0] axis, while
E2 corresponds to the polarization along the [001 ] axis. Negative time delay is associated
with the case in which E1 precedes E2. Using Figure 4.6, we notice a definite imbalance
in the THz power depending on the pump pulse order which becomes stronger as the
pulse duration becomes shorter. The weaker DFG THz generation in the negative time
50
delays is due to anisotropy in the nonlinear absorption processes. To understand this
asymmetry, we look at the inset of the figure which shows the THz generation for two
orthogonal, fully compressed pulses. For fully compressed pulses, the generated THz
power is due entirely to optical rectification. For positive time delays, E2 precedes E1.
Recall again that E2 does not generate THz waves due to optical rectification. However,
nonlinear absorptions of the optical beam leads to free carriers which heavily absorb both
E1 and the THz it may generate leading to a weak THz signal for positive time delays. In
the opposite case, E1 generates THz power more efficiently through optical rectification;
hence there is less nonlinear absorption to leech away the generated THz photons.
For DFG, the effect is different with the result being opposite. In this case, THz
photons are only generated (excepting the relatively weak rectified signal for the shortest
pulse durations) when the two pulses overlap. It has been shown that some zincblende
crystals have a stronger multi-photon absorption along the [1 1 0] axis [65]. Since there is
no rectified THz generation to “share” the pump power as in the fully compressed case,
E1 will eventually have larger parasitic nonlinear absorption when it precedes E2. When
E1 arrives first, the free carriers generated by E1 will tend to absorb some of E2 and the
THz generated by the overlapping pulses resulting in a lower power for negative time
delays instead. The asymmetry tends to weaken as the pulse duration is increased.
Finally, we look at the THz power vs. Optical pump power and the Optical-toTHz conversion efficiency for several pulse durations.
51
Fig. 4.7 THz power vs. optical pump and Optical-to-THz conversion efficiency. For
pulse durations τp = 1.06, 2.13, 2.78, 3.35, and 4.61 ps: (a) THz power vs. Optical pump
power, and (b) THz conversion efficiency vs. Optical pump power.
We see that in the range of powers we used, the emitted power reached a saturated value.
Below a certain value of pump, with an average intensity of 3.8 W/cm2, the terahertz
power varies quadratically with the input power, as expected for a second-order process.
Above this pump level it varies almost linearly with input power. The saturation of the
conversion efficiency can be attributed to the parasitic nonlinear effects in the ZnTe
crystal.
In conclusion, we have demonstrated and characterized a source of intense,
narrowband THz pulses. It has several distinct advantages over other sources of
narrowband THz including, economy of space, maximization of optical throughput, and
minimization of parasitic nonlinear optical effects. Additionally, it is easily incorporated
52
into an ultrafast spectroscopy setup and ideal for ultrafast time resolved measurements as
the pulse durations are still on the order of picoseconds. In Chapter 6, we will see one
example of the power and usefulness of this setup as it is applied to semiconductor
nanostructures. This work was published in the Journal of Applied Physics in 2008 [48].
53
5. THz Spectroscopy of Nickel-Titanium (NiTi-alloy) Thin Films
NiTi-alloy, sometimes referred to as Nitinol when the proportions of each metal
are near 50%, is an intriguing material. It is what is known as a shape memory alloy
(SMA). SMA’s have multiple stable crystalline structures which depend on the
temperature of the alloy. At some critical temperature, Tc, the alloy undergoes a
transition from the low temperature “martensitic” state to the high temperature
“austentitic” state [66]. What this means is that when the crystal is annealed above Tc in
shape 1, it can be cooled below Tc, and deformed into shape 2. Then, when the material
is heated above Tc, it will spontaneously return to shape 1. Shape memory is one of many
interesting qualities NiTi alloys possess including superelasticity and electroplasticity
[67-70]. As a result of these interesting characteristics, NiTi alloys have been exploited
for a variety of applications including medical implants and microelectromechanical
systems (MEMS) [71-74], and the micromechanical properties of this alloy are well
studied [56, 75-78]. If very small devices in the nanometer to micron range are being
developed, especially those which may be electrically heated to trigger shape memory
effects, it is important to accurately characterize the electrical properties of NiTi alloys.
Accurate characterization of the electrical properties is especially important for thin metal
films which have been shown in many cases to deviate from bulk measurements of
electronic properties [79-88].
Despite this, the electrical properties of NiTi alloy thin films remains unstudied.
Electrical conductivity (or, inversely, the resistivity) has been reported as a good property
54
with which to characterize the phase transitions and micromechanical properties of NiTi
alloys [89]. We used THz imaging and THz-TDS to measure the resistivity of NiTi alloy
thin films of varying thickness (~30-120 nm), and varying concentrations of Ti (0-100%).
As is illustrated later in figure 5.10, this yielded two sets of comparable resistivity data.
One was derived from the time-averaged power transmission (bolometer, which will be
referred to as “power transmission”), and one from the field amplitude transmission (EO
sampling). We also performed four-point-probe measurements on the samples for
comparison. Although four-point-probe measurements are typically accurate, THz-TDS
has also been shown to be an effective method for characterizing the properties of metal
thin films [87, 89, 90-92], and has the added advantage of being non-intrusive when
compared to the four-point-probe method which effectively destroys the sample.
5.1 NiTi Alloy Thin Film Preparation and Composition
This section briefly outlines how the samples were created, and how the
composition (Ti% and film thickness, d) of the samples were characterized in
collaboration with the Minot group from the Oregon State University Physics department
[93], and the Koretsky group from the Oregon State University Chemical Engineering
department [94].
The samples were prepared using a fairly standard lithographic method. After
application of a photoresist layer, a shadow mask was applied to the sample in a SUSS
MJB 3 contact aligner, and exposed to UV light. This exposure provided an area of pure
silicon on each sample for THz transmission references. NiTi alloy layers were then
55
deposited via Ar-plama assisted chemical vapor deposition (CVD) with an AJA
International Orion 4 Magnetron Sputter system using parameters determined in previous
experimental work done by the Koretsky group. For more details of the capabilities of
this system, see reference [96]. The samples were not annealed, and hence the structure
of the NiTi alloy was amorphous [66].
The Ti% composition was determined using energy-dispersive x-ray spectroscopy
(EDX). When bombarded by an electron beam, an EDS detector measures the energy
and quantity of subsequently emitted x-rays. The x-ray measurements allow the atomicresolution mapping of the composition of large areas of materials [97]. Figure 5.1 below
is an example of one of the spectra used to determine the composition.
Fig. 5.1 Sample EDX spectrum used to determine Ni and Ti concentration.
56
The thicknesses, d, of the deposited films were determined by atomic force microscopy
(AFM). Figure 5.2 shows several measurement data sets. Besides being able to
determine the thickness of the film, these measurements show that beyond ~ 10 μm from
the edge, the films are uniform.
Fig. 5.2 AFM measurements of the thickness of several NiTi alloy thin films.
The model developed later in this chapter to analyze the experimental results is quite
sensitive to the thickness of the films, so, as we will see, accurate measurement of the
thickness of the NiTi alloy films is crucial to an accurate calculation of the resistivity.
5.2 Experimental Setup
The experimental setup is depicted below in figure 5.3.
57
Fig. 5.3 Depiction of NiTi sample and detection schemes. (i) Power transmission with a
Si bolometer and (ii) Field amplitude transmission with EO sampling.
The source of the THz pulse is the same as outlined in sections 3.1 and 3.2 for broadband,
single-cycle THz pulses. Each THz pulse is focused onto either the alloy or the substrate.
The beam diameter at the focal plane is ~0.5 mm. The consequent transmission is
measured either with the bolometer or the EO sampling setup depending on the type of
measurement being done. As mentioned above each sample was deposited on a Si
substrate (300 μm thick) such that a portion of the sample remained bare Si. The Si area
ensured that the reference for each film was from the same portion of wafer for
consistency. As mentioned in section 3.4.3, this reference is important for extracting the
properties of the thin film without having to worry about possible EO sampling pulse
distortions from the substrate or detection crystal.
58
5.3 Transmission Measurements
The first measurements done were THz transmission images generated by raster
scans of several samples. These earliest samples did not have reference Si available on
the same portion of wafer. Instead, a separate piece of the wafer these samples were
grown on was used a reference. Although this is slightly less rigorous than the method
adopted later, some important preliminary results were gained. All samples, including
the reference, were attached over 5 mm wide circular holes in a thin aluminum plate. An
example of these Raster scans is shown below in figure 5.4.
d = 30 nm
d = 60 nm
T
0.8
0.6
0.4
2 mm
2 mm
0.2
Fig. 5.4 Raster-scan of two NiTi alloy thin films of different thickness on Si substrate.
The pixel size in these images is 400 μm with a pixel integration time of 100 ms. These
images demonstrate the expected increased absorption by the thicker film. More
59
importantly, within each sample, the transmitted power is uniform which reveals that the
samples are spatially homogeneous on the scale of the THz wavelength. Similar power
transmission measurements were taken on all subsequent samples and references.
Although spatial homogeneity was present, we averaged over several measurements at
various locations on the sample to be thorough. With these measurements, we can plot
2
2
the relative power transmission ( RPow E NiTi Si / E Si ). An example of some data is
Rel. Trans. (|ENiTi+Si|2/|ESi|2)
shown below.
Ti %
55
36.1
29.2
26.7
21.4
10.7
0
TNi-Ti / TSi
0.1079
0.1200
0.0909
0.0752
0.0762
0.0273
0.0025
Titanium %
Fig. 5.5 NiTi alloy Relative power transmission measurement using a Si bolometer.
Figure 5.5 shows a relatively linear increase in transmission up to around 50% titanium.
We can qualitatively justify this by the following hand-waving argument. Pure metals
are likely to have a large number of free carriers, and hence absorb radiation efficiently
for frequencies below the plasma frequency. As the alloy percentage is increased, often
60
the number of carriers decreases, and the number of scattering sites increases. Lowered
free carrier concentration and increased scattering leads to less efficient absorption and a
decrease in resistivity. In fact, many pure metals often have smaller resistivity than
alloys of those metals [98].
The next data taken was THz field amplitude measurements using THz-TDS.
Figure 5.6 shows a representative set of transmission waveforms for various Ti
concentrations along with their corresponding relative transmission (RTDS = ENiTi+Si(ν)/
ESi(ν)) as a function of frequency. Despite the fact that these samples are of finite
thickness, and are not uniform, there is essentially zero difference in phase of each of the
transmitted waveforms (i.e. peaks and valleys occur at the same times).
Fig. 5.6 NiTi alloy time-domain data. (a) Transmitted waveforms and (b) Relative
transmission for various Ti concentrations.
61
As in the power transmission data, the general trend of figure 5.6(a) is for the transmitted
amplitude to monotonically increase as the percentage of Ti increases. However, further
investigation revealed a far more complex dependence on Ti concentration which will be
shown in the next section. The curves for trel in Fig. 5.6(b) were generated by performing
a Fast-Fourier-Transform (FFT) on the time domain data. This figure displays a
characteristic flat spectral response for all the samples. Some of the implications of the
flat response will be discussed in the next section.
5.4 Analysis and Results
5.4.1 Justification for the Theoretical Model
In order to evaluate the experimental data, we developed a surprisingly simple
theoretical model based upon treating the alloy film as an “optically thin” conductive
film, and the Si substrate as an “optically thick” dielectric. (By optically thick/thin, we
mean that the thicknesses of the film and substrate are respectively small and large when
compared with the wavelength of the THz radiation). We applied these assumptions
using a combination of the Drude Model DC conductivity and the Thin-Film Fresnel
coefficients. There were several justifications for using this simplified model that were
borne out by our experimental results.
First of all, from a conceptual standpoint, pure metals in general have a high
density of free carriers. Under many circumstances, these carriers act in a manner
consistent with a kinetic theory such as the Drude model. In fact, for many materials, the
Drude model can still provide useful first order information [99,100]. Despite the fact
62
that alloys will in general have a lower density of free carriers than their constituents,
they still have large number, and we can expect to see Drude-like behavior.
Another justification for this theoretical framework comes from a consideration of
the scattering times in metals. A major strength of the Drude Model in predicting
qualitative and quantitative results is that you can derive results related to scattering
without the need of specifying the scattering mechanisms themselves [101]. All that is
necessary is to assume there exists some scattering mechanism, and specify the scattering
time, τs, which is the average time elapsed between scattering events. In the case of
metals, this is usually on the order of 10-14 seconds at room temperature [101].
On the
other hand, from figure 5.6(a), we see that the THz pulse duration is much longer than
this and is given by τpulse ~ 10-12 seconds. In other words, the frequency components that
make up the THz pulse do not extend high enough such that the oscillations of the THz
field are fast enough to act on the carriers on a time scale that is relevant compared to the
scattering time. The carriers are scattered many times before the THz field changes
appreciably and therefore have no “memory” of the THz field. The relative smallness of
the scattering time with respect to the THz frequency is one indirect reason that the
transmission has a flat spectral response. The carriers in the NiTi essentially see a time
varying DC electric field, and hence we can calculate the DC resistivity. How the DC
resistivity comes about in our calculations will be developed in the next section.
As stated above, one final justification for this formulation lies in the relative
thicknesses of the alloy film and the substrate. In general, the reflectance and
transmittance of an electromagnetic wave at a conducting surface is a function of
63
frequency. However, in the case of the NiTi, we can neglect this by virtue of the fact that
the thickness of these films is many times smaller than the skin depth of this material. To
check this, it is simple to calculate the skin depth from the Four-Point-Probe values for
the conductivity of the thin films. After the initial Raster scans, the thickest films were
on the order of 80 nm. (By comparison, the peak amplitude THz frequency generated in
our single-cycle arrangement is about 1 THz. 1 THz translates to a wavelength of
min
THz
300 m ). A simple formula to calculate the classical skin depth, δ0, is given by
[102]
0
2
f
(5.4.1)
Applying values for the approximate smallest value of resistivity, a frequency of 1 THz,
and magnetic permeability we find that
0
3e 7 m
276 nm
(4 ) (1e12 s 1 ) (1e 7 s)
2
(5.4.2)
giving a ratio of d / 0 3.45 , where d is the film thickness. The ratio shows that the
films are indeed much less than the skin depth of the metal. This relationship is also
responsible for the flatness of the Relative Transmission spectrum which was observed.
The Si substrate had a thickness of 300 μm. Combined with an index of refraction that is
essentially constant at n = 3.418, the optical path length in the substrate for the THz is
OPL nL (3.418) (300m) 1.025mm
(5.4.3)
64
showing that the width of the substrate is much larger than the THz wavelength.
Additionally, there is virtually no absorption of THz in Si at these wavelengths, so
treating it as a dielectric is justified [103,104].
5.4.2 The Drude Model
The Drude Model is a kinetic theory applied to the electrons in a metal to describe
carrier dynamics. This model still yields valuable qualitative and quantitative
information about many metal systems despite its seeming naivety. The reason for the
model’s continuing viability is due to the fact that metals often have a high density of
conduction electrons that essentially act as free carriers. Alloys usually have a lower
carrier density than pure metals, but often still have a high enough density to be treated
with the Drude model. Using the Drude model, we can directly relate electrical response
properties (conductivity/resistivity) to a few simple quantities such as carrier mass (m),
charge (e for electrons), density (n), and scattering time (τs). In the Drude model, this is
expressed through Ohm’s law as [101]:
J 0E
with 0
ne 2 s
m
(5.4.4)
In order to connect this model to the Fresnel Thin-Film coefficients, we need to
relate the conductivity to the index of refraction of the metal film. The connection is
forged by starting with Maxwell’s equations (2.1.1-4). As usual, we take the curl of
equation 2.1.1 and insert equation 2.1.2. Applying a more generalized Ohm’s law for
now, J () () E() , yields
65
E
E 0 ( ) E 0 0
t
t
(5.4.5)
If we consider the incoming field as E E0 e it this expression becomes
2 E i 0 ( ) 0 0 2 E
i ( ) 2
2 E
1 2 E
0
c
(5.4.6)
A direct result of the Drude model is that the frequency dependent conductivity can be
written as [101]
( )
0
1 i s
(5.4.7)
The product ωτs is small enough in our thin films that we can neglect this portion and
approximate σ(ω) as the DC conductivity, σ0. Armed with this approximation, and some
knowledge of the wave equation, we can rewrite equation 5.4.6 as
~
i
2
2 E k 2 E 2 E n~ 2 2 E 2 E 0 1 E 0
c
0
(5.4.8)
From equation 5.4.8, we can write the complex index of refraction as a function of the
DC resistivity as follows
i
i
n~ 2 ~r 0 1
1
0 0 0
(5.4.9)
66
Equation 5.4.9 will be used in the next section to apply the Drude conditions to the
Fresnel thin film equations.
5.4.3 Thin Film Fresnel Coefficients
Since we are treating the NiTi alloy as a thin conducting film, we are primarily
concerned with what happens at the surface that defines the interface of the alloy and the
substrate. As we will see, this means that the alloy film acts as an infinitesimally thin
absorbing interface that has very little effect on the phase of the propagating THz pulse.
To calculate how the sample effects the transmitted THz, we first consider what happens
at the Si/NiTi interface using the figure below.
r( )
t( )
t12
t23ei
r23
t23e
r21
r23
i 3
t32
r21
r32
t23ei 2
r21
t23ei 4
n1
n2
n3
(a)
n4
n1
n2
n3
n4
(b)
Fig. 5.7 Illustration of the interface of NiTi alloy and silicon. The darker section on the
left (n2) is NiTi alloy, while the lighter right-hand section (n3) is silicon. (a) The total
transmission, t(ρ), through the interface, and (b) the total reflection, r(ρ), from the
interface.
67
Because of the thinness of the films, we must sum up all transmissions and reflections at
the interface. Figures 5.7 (a) and (b) show the first few in each sum respectively. Note
that the experiment is performed at normal incidence, and an angle has been added to the
incidence and internal transmissions/reflections for illustrative purposes. Each time an
internal transmission or reflection traverses the film, a phase is added to the propagating
wave defined by
c
n2 d , (d film thickness)
(5.4.10)
This phase is not negligible, but still very small. In essence, the absorptive effects of the
carriers in the film are far larger. The coefficients, tij and rij, are simply the Fresnel
transmission and reflection coefficients [105] at normal incidence given by
t ij
2 ni
ni n j
, rij
ni n j
ni n j
(5.4.11)
Using figure 5.7, we sum up the total transmission and reflection across the interface.
The two sums are given by:
t ( ) t12 t 23 e i [r23 r21e i 2 ] n
n
i 2
i 2 n
r ( ) r32 t 32 r21t 23 e [r23 r21e ]
n
n 0, 1, 2 (# of reflections )
(5.4.12)
68
Since the ratio λTHz/d ~ 3500 ~ n, the sums above can be simplified by the identity
x
n
n
1
. Applying this identity, we arrive at the following exact expressions for
1 x
the transmission and reflection
t( )
t12 t 23 e i
1 r12 r23 e i 2
, r( )
r32 r21e i 2
1 r12 r23 e i 2
(5.4.13)
By applying some approximations, we can further simplify these equations. The first is
that because of the large absorption demonstrated in the transmission measurements, we
assume that n22 n1 , n3 . Secondly, although n2 is large, (
c
n2 d ) is small because the
thickness, d, of the NiTi alloy is so small. We cannot drop the phase altogether, but we
can apply the small angle approximation through Euler’s formula. Lastly, we assume
that the absorption is by far the dominant effect of the NiTi alloy film. This assumption
manifests itself in the relationship
i
0 0
1 and thus, n~22
i
0 0
from equation 5.4.9.
By inserting equations 5.4.11, applying these approximations, and performing a bit of
algebra, we arrive at the simplified expressions
t( )
2n1 0
n1 n3 0 dZ 0
, r( )
n3 n1 0 dZ 0
n3 n1 0 dZ 0
(5.4.14)
where Z0 is the impedance of free space. Notably, as expected from our justifications for
this model as well as the experimental data, these functions are independent of the THz
69
frequency. Thus, we have expressions in our model that completely describe the
interface between the NiTi alloy and Si and are only functions of the DC resistivity.
Equipped with these formulas, we can explore the functional dependence of the
THz field transmitted out of the Si substrate which is what we are actually measuring.
We do this in a similar fashion using figure 5.8 below as a guide.
Fig. 5.8 Illustration of the THz transmitted out of NiTi sample. (a) NiTi on Si, and (b) the
Si substrate.
Again, we account for the initial transmission followed by a series of internal reflections.
By comparing the transmission through the NiTi+Si to the transmission through the bare
70
Si substrate, we can derive an expression for the relative transmission, R = tNiTi+Si/ tSi,
which is what was measured in section 5.3.
As the measurements performed by the bolometer are a time-integrated
measurement, we need to account for all the subsequent internal reflections depicted in
figure 5.8. Summing all transmissions leads to the following expressions:
t NiTi
2
t Si t Si
2
TNiTi Si t NiTi
TSi t Si
2
2
2
t NiTi
2
t NiTi
2
2
t 2 t 34
1 r 2 r342
t2 t2
t Si 31 234 2
1 r31r34
(5.4.15)
2
Dividing these two expressions by each other yields the relative transmission
R Pow
TNiTi Si t 2 1 r312 r342
2
TSi
t13 1 r 2 r342
(5.4.16)
As t and r are a function exclusively of ρ0, we can use this equation to solve for the
resistivity. The resistivity is calculated by using a least-squares approach to fit the
formula to the measured values of Rpow.
For the THz-TDS data, a similar approach is used. However, in this case, we are
mapping out the field in time, and can see the individual pulses as they arrive at the EO
sampling detector. Figure 5.9 demonstrates a few longer THz-TDS measurements which
show the initial pulse and subsequent decaying transmissions.
71
Amplitude (V)
(a)
(b)
t NiTi
tSi
t NiTi
t NiTi
tSi
tSi
Time (ps)
Time (ps)
Fig. 5.9 Long TDS measurements taken on NiTi alloy. These long scans show the initial
transmitted pulse and two subsequent internal reflections of (a) the NiTi alloy on Si, and
(b) the Si reference.
As is shown in figure 5.9, each pulse is distinct in time. Also, we notice a few effects
that demonstrate the strong optical response of the NiTi alloy film. First, each successive
pulse is heavily attenuated when compared with its constituent reference pulse. Secondly
in figure 5.9(a), we notice a π phase shift in the polarization of t” and t’’’ not present in
figure 5.9(b). This phase shift is due to the large index of refraction of the alloy film.
Since each pulse is distinct in time we can examine each one individually and
extract the resistivity from wherever we choose. Again, using figure 5.8, we can write an
expression for the nth pulse (n = 0, 1, 2, … with n = 0 corresponding to t’, etc.):
( n)
n i ( 2 n 1)
t NiTi
Si tt34 (rr34 ) e
t Si( n) t13t 34 (r31r34 ) n e i ( 2 n1)
(5.4.17)
72
Since the first transmitted pulse has the highest signal to noise ratio, we choose this set
which gives the following simple expression for the relative transmission:
(0)
RTDS
t
t13
(5.4.18)
( 0)
Again, after measuring RTDS
, we can use a least-squares best fit to determine the value of
ρ0.
5.4.4 Results
With the analysis above, we calculated the DC resistivity of several samples
ranging in Ti concentration from 0-100% using both the power transmission
measurements and the TDS transmission values. Additionally, as mentioned above, we
also performed Four-Point probe measurements. The results are below in figure 5.10.
73
Fig. 5.10 Summary of resistivity measurements made on NiTi alloy thin-film samples.
There are several peaks located near the titanium concentrations of 22%, 44%, and 62%.
The dashed line is a fit based on an effective medium treatment.
Figure 5.10 show the dependence of the NiTi alloy thin film resistivity on Ti
concentration. As can be seen, there is good agreement between all methods of
measuring ρ0, indicating that THz wave transmission is a reliable method for extracting
the carrier properties of these thin-films. The dashed line on the graph represents a first
approximation of the resistivity based upon a very basic effective medium treatment.
eff
This first approximation assumes that the effective resistivity, Ni
Ti , is purely a linear
74
combination of its constituent parts ( Ni and Ti ) , weighted by concentration (x). In the
event that the structure of the constituents is such that the pure domains of Ni and Ti are
larger than the mean-free-path of the carriers, the plot should reduce to this form. That is
clearly not the case in figure 5.10, which implies a structure much more complicated than
this. However, the measurements at either end (pure nickel, and pure titanium) agree
with this by definition.
The most striking features of this plot are the peaks at 22, 44, and 62%
respectively. These peaks indicate a drastic increase in the resistivity of the films.
Although we cannot be certain, we speculate that these peaks correspond to an increase in
the disorder of the structure at these concentrations. This speculation is supported by
figure 5.11 showing phase transitions which occur at these concentrations [106].
Fig. 5.11 Phase diagram for NiTi alloys. This figure illustrates that the material
undergoes phase transitions at Ti concentrations corresponding to increases in resistivity
in our measurements.
75
In section 5.1, it was mentioned that the samples were not annealed, and hence are
amorphous. However, the samples seem to still maintain some vestige of these low
temperature phase transitions as they near specific concentrations. If this is indeed the
case, THz spectroscopy might be used as not only a tool for extracting electrical
properties, but also as a rudimentary tool for mapping of phase transitions. Although the
wavelength of the THz pulse is too large to be sensitive to the small domains in the alloy,
it may still be sensitive to them through their effects on the carrier dynamics.
In conclusion, we have systematically studied the carrier dynamics of NiTi alloy
thin-films by examining the DC resistivity using THz imaging and THz-TDS. These
studies allowed us to map the resistivity as a function of Ti concentration. These
calculations yielded a surprisingly complex structure that indicates several concentrations
at which the resistivity increases. We speculate that at these concentrations, the disorder
of the film is greatly increased due to being near to a phase change, causing increased
scattering and hence the spike in resistivity. This work was accepted for publication in
Applied Physics Letters in May, 2011 [107].
76
6. Transient Optical Response of Quantum Well Excitons to Intense
Narrowband THz Pulses
As most semiconductors have band gaps on the order of Egap ≥ 1 eV, THz photons
do not excite inter-band transitions. However, they are often ideal for exciting intra-band
transitions as these transitions are often of the order of a few meV (1 THz ~ 4.14 meV).
Because of this, the interaction of condensed matter with THz radiation is of fundamental
interest for a variety of reasons. These include carrier transport properties such as
Coulomb effects and many-body interactions. These carrier dynamics can manifest
themselves in a variety of nonlinear effects such as the Dynamical Franz-Keldish effect
[108-110], the Quantum Confined Stark Effect [111-113], Carrier Wave Rabi Flopping
[114-117], and ponderomotive effects [118,119]. Additionally, there are important
practical research applications such as the development of ultrafast communications
beyond the GHz [120-122], and quantum information processing [123,124].
The excitation of intra-band dynamics with THz photons is particularly interesting
because several effects which normally are not of the same order now have similar
energy. These include the ponderomotive energy of the carriers, the Rabi energy of the
carriers, and the modulating photon energy. These experiments are largely accessible
because of the prevalence of high-power, ultra-short, pulsed laser systems. The THz
outputs accessible with these systems allow us to enter the regime of extreme nonlinear
optics, which bears some differences from ordinary nonlinear optics. Some important
differences include the breakdown of a perturbative treatment, the failure of the rotating
77
wave approximation (RWA), and the importance of non-resonant effects which can lead
to Rabi oscillation of the Bloch vector [125]. In addition to these differences, there are a
host of extreme nonlinear transients that can be observed. These nonlinear transients
have been demonstrated with a single-cycle, broadband THz pulse [118]. In this study,
we resonantly excite excitonic polarization and strongly couple the intra-band states with
a narrowband THz pulse. What exactly this entails will become clearer after some
background information is provided.
6.1 Quantum Wells
It is with this spirit in mind that we explored the interaction of THz photons with
excitonic polarization in an AlGaAs quantum well (QW). A quantum well is essentially
a material with a given potential energy surrounded by materials with a higher potential
energy thus creating an area of confinement in the lower potential material. The
confinement properties of QW’s make them ideal nano-scale “laboratories” in which to
discover the properties of carriers in semiconductors. Figure 6.1 shows a generalized 1Dimensional QW. As depicted in figure 6.1, the confinement of carriers leads to discrete
energy levels. Additionally, as we will see, the confinement also leads to stronger
coupling between carriers.
78
Fig. 6.1 Simple depiction of a 1-dimensional quantum well. Higher potential (U2)
barriers surround a lower potential (U1) material forming discrete bound states in the
well.
Neglecting the momentum in directions not along the Z-axis, and considering U 2 U1 ,
we can approximate the energy levels from the infinite potential well solution:
En z
2 2 2
nz
2me l 2
(6.1.1)
where me is the effective mass of the carrier. Considering the dimensions of the QW to be
the order of l = 10 nm, the first energy level nz = 1, and the effective mass to be that of an
electron, we see that this energy falls near the THz photon energies generated by our
sources (E1 ~ 3.7 meV). This simplistic approach is not the correct formulation for the
carrier energy as we will see in the next section, but it does demonstrate that we can scale
79
the energy levels of the quantum well by changing its dimensions. Specifically, we will
be more concerned with the energy spacing between adjacent energy levels.
Our specific sample used in this study was a stack of 10 high quality, 12 nm thick
GaAs QW’s separated by 16 nm thick Al0.3Ga0.7As layers. The band structure of this
QW is depicted below in figure 6.2.
Fig. 6.2 Bulk and QW GaAs band structure. Left hand side: bulk GaAs band structure.
Right hand side: GaAs QW band structure when surrounded by Al0.3Ga0.7As. Splitting of
heavy hole (HH) and light hole (LH) is caused by the strain introduced at the interfaces of
the layers.
When Ga and As are combined, the 4s and 4p shells are occupied with the outer electrons
from the elements. The p-level is three-fold degenerate, with each level containing two
electrons. These three p-levels comprise the highest levels of the valence band when the
80
GaAs is in a crystalline structure. The angular momentum configuration of the valence
bands is important to the selection rules for optically excited transitions. Spin-orbit
coupling breaks the degeneracy of these levels. The bulk GaAs on the left hand side of
the figure 6.2 shows the top two J = 3/2, heavy hole (HH) and light hole (LH), valence
bands which are degenerate at the Γ-point. However the strain introduced by surrounding
the GaAs QW with the Al0.3Ga0.7As barriers again breaks this degeneracy and shifts the
levels of the valence and conduction bands. The broken degeneracy yields two clear
transition possibilities which are evident in the absorption spectrum measured for this
sample in the range relevant to our laser system.
Fig. 6.3 Absorption spectrum of GaAs/Al0.3Ga0.7As QW used in our study.
81
Figure 6.3 demonstrates the quality of the sample in that it shows clear, sharp absorption
resonances at wavelengths of approximately 796 nm and 805 nm. We will discuss in
more detail the carrier properties that are involved in these absorption resonances. In
particular, we will see that these resonances correspond to a different effective mass for
the LH and HH.
6.2 Excitons
In the previous section, I have been careful to restrict my description of the
particles under investigation in the QW’s to the vague term of “carriers”, or to electrons
and holes. Now I will be more specific with a description of excitons, the quasi-particles
involved in these experiments. When an electron is excited from the valence band of a
semiconductor, it leaves behind an empty space. The empty space can be treated as a
positively charged quasi-particle called a hole. The oppositely charged quasi-particles
attract each other and, in short, an exciton is a bound state of the electron and hole held
together by their Coulomb attraction. Because of this negative Coulomb energy, the
excitation energy required to create an exciton is less than the band gap energy of the
semiconductor. In optical spectroscopy, this can lead to strong absorption below the
band edge [126,127].
To better understand how this quasi-particle will behave, we explore some of its
basic properties as they occur in the GaAs/Al0.3Ga0.7As QW’s described in the previous
section. We realize right away that this system bears a similarity to the hydrogen atom
82
with a hole instead of a proton. In fact, the treatment for the free exciton is
mathematically identical, and yields the following binding energy:
mr e 4 1
1
E Free Exciton
EB 2
2 2 2 2
32 n
n
(6.2.1)2
This energy level structure is the same as the hydrogen atom with a few exceptions. One
noticeable exception is the inclusion of the background dielectric constant, ε, which is
important for carriers in semiconductors, but is excluded for atomic hydrogen as ε = 1 for
a free atom. The importance of the dielectric constant will be expanded once we have
determined some exciton properties. Additionally, we use the reduced mass of the
electron-hole combination, mr, expressed as
1
1
mr
me mh
1
(6.2.2)
As we saw in section 6.1, the band structure depends heavily on whether or not one is
working with bulk GaAs or a QW structure, hence the effective mass of the electrons and
holes, me and mh, will vary accordingly. The effective masses are especially important at
k = 0, where the degeneracy of the LH and HH is broken by the QW barrier materials.
For GaAs, the dispersion, and hence LH and HH masses are given in terms of the
Luttinger parameters [128], γ1 and γ2, such that the LH and HH effective masses near k =
0 are given by
m LH
m0
1 2 2
, m HH
m0
1 2 2
(6.2.3)
83
We will use these to determine properties of the GaAs/Al0.3Ga0.7As QW’s later, but first
we need to determine the form of the bound state energy levels in our case.
Following the procedure of Peyghambarian, Koch, and Mysyrowicz [128], we
begin with the single exciton Hamiltonian:
H
2
2
e2
2h Vconfine VCoulomb
2m e
2m h
(6.2.3)
Equation 6.2.3 consists, from left to right, of the kinetic energy of the electron and hole, a
term for the potential due to the confinement barrier of AlGaAs, and a term for the
Coulomb interaction of the electron and hole. As in figures 6.1 and 6.2, we assume the
confinement direction is along the z-axis. We then separate the x-y plane coordinates
from the z-axis, and make the following coordinate transformations:
2R
2
2
2
2
2
,
r
X 2 Y 2
x 2 y 2
(6.2.4)
In which X and Y are the center of mass coordinates, and x and y are the relative
coordinates defined by
X
m h x h me x e
m y me y e
, Y h h
m h me
m h me
(6.2.5)
x xe x h , y y e y h
Additionally, we use the x-y plane effective mass, mxy and the x-y plane total mass, Mxy
defined as
84
1
1
1
m xy
me mh xy
, M xy me mh xy
(6.2.6)
Making these changes we arrive at the Schrödinger equation
2 2
2 2
2
2
e2
2
2
R
r Vconfine
2
2
2m xy
r
2me z e 2mh z h 2 M xy
( r ) E ( r )
(6.2.7)
By defining r re rh , the function (r ) can be separated into x-y plane components
and z components. The solution for the motion in the x-y plane, which can be found in
[129], yields the following expression for the bound energy levels for excitons
E n2 D E g
EB
2 2 j 2
2
2
2m r L z
1
nj
2
(6.2.8)
From left to right, these terms are the band gap energy, the additional energy due to the
confinement potential, and the exciton binding energy where EB is the excitonic Rydberg
from equation 6.2.1. Keep in mind that this solution is for an infinite potential barrier, as
indicated by the confinement energy in the second term, which is not the same as our
QW, but will serve to demonstrate some important properties of the excitons. Comparing
the last term in equation 6.2.8 to equation 6.2.1 we see that there is a factor of four
difference in the ground state energy ( E2 D 4E3D , for n = 1) caused by the reduced
dimensionality of the QW system. Using this, and rewriting the two dimensional exciton
binding energy as
85
E2 D
2
2mr a02 D
2
(6.2.9)
we can easily show that for the ground state, the Bohr radius of the 2-D exciton is half
that of the 3-D case:
a 02 D
2 2 a 03 D
2
mr e 2
(6.2.10)
With equation 6.2.10, we have quantified the intuitive property that the confinement due
to the QW introduces a tighter binding of the electron and hole and a corresponding
increase in energy.
With the above treatment we can now calculate the LH and HH binding energies,
as well as the two dimensional Bohr radius. We start by using equations 6.2.3 to
calculate the LH and HH effective masses with respect to a free electron mass. Using the
Luttinger parameters γ1 = 6.9 and γ2 = 2.4 found in [128], we find the following values
m LH
m0
m0
0.0855 m0 , m HH
0.476 m0
6.9 2 2.4
6.9 2 2.4
(6.2.11)
Using these values, the electron effective mass, me = 0.067m0, and equation 6.2.2, we
calculate the LH and HH exciton reduced masses:
mrLH 0.0376 m0 , mrHH 0.0587 m0
(6.2.12)
86
Finally, using a background dielectric constant of ε = 13.1ε0 [130], and putting these
values into the expressions for the ground state binding energy and exciton Bohr radius,
we obtain the following values:
LH
E Binding
11 .9 meV ( 2.9 THz )
HH
E Binding
18 .5 meV ( 4.5 THz )
a 0LH 9.3 nm
(6.2.13)
a 0HH 6 nm
From equation 6.2.13, we see the binding energy falls into the THz range, and that the
distance between the electron and hole is significantly larger than the GaAs lattice
spacing of ~0.5 nm.
As mentioned before, we are mainly concerned with the spacing between the
ground state and the first excited state. Using the HH, we find that the excitonic HH
Rydberg, E B
HH
E Binding
4
. Using this, we find the energy spacing between the ground and
first excited state to be
E 21 4 E B 0.44 E B 16 .45 eV
4 THz
(6.2.14)
This frequency is higher than the DFG system described earlier can produce; however,
the actual energy spacing is smaller for a few reasons. First, although this treatment
illustrates the properties of excitons fairly well, we must keep in mind that this is not a
strictly two dimensional system. Because there is a finite, but significantly smaller,
87
dimension along the growth direction, the system is known as quasi-two dimensional.
The small, but finite, size has the effect of lowering the binding energies and thus
increasing the Bohr radius. Additionally, the potential barriers of AlGaAs are not
infinite, but about 0.5 eV higher than the bandgap of GaAs. The finite barriers will result
in closer spacing of the ground state and first excited state as well. An argument for this
can be made by comparing the energy spacing of the infinite potential well in one
dimension to the spacing of the finite one dimensional potential well. In this problem,
the finite well has a potential barrier of height V0. Both wells have a carrier mass, m, and
well width, Lz. I will not solve this problem here, but will simply state the solutions for
the figure 6.3 below. The solutions for the energy levels are as follows [131]:
Infinite Potential Well :
E z( j )
2 2 j 2
2
z
2mL
j 1, 2, 3
(6.2.15)
Finite Potential Well :
mE L2
z z
Even : E z tan
2 2
mE L2
z z
Odd : - E z cot
2 2
V E
0
z
(6.2.16)
V E
0
z
The solutions to the above equations are represented graphically below in figure 6.3.
88
Fig. 6.4 Graphical depiction of the first two energy levels for both the infinite and finite
barrier quantum well systems. Black curves represent the even solutions, while white
curves represent the odd solutions. The dotted line represents the term V0 E z
common to both solutions. The asymptotes show the infinite well energy levels.
As the solutions for the finite well are in the form of a transcendental equation, they
cannot be solved for analytically. However, by plotting them in figure 6.3, we can see a
distinct difference in the energy spacing of the first two levels for the different QW’s. It
is obvious that for similar materials and dimensions, the spacing between the first two
energy levels in a finite QW will be substantially less than those for an infinite well.
6.3 Experimental Arrangement
The experimental arrangement used is depicted below. With this, some
knowledge of the QW in question, and the carriers we are interested in, we can discuss
the specifics of the dynamics we probed.
89
Fig. 6.5 Experimental Arrangement for the DFG QW experiment.
The THz wave generation for this experiment is described in detail in Chapters 3 and 4.
Using the DFG source, we had fine and tunable control over the THz center frequency.
Generating THz radiation with the DFG consumes most of the optical pump power
(~%95), with the remaining 5% used for the optical probe. However, this setup requires
us to stretch the optical pump in order to introduce the chirp necessary for this THz
generation method. As such, there is an additional compressor on the optical probe line.
The compressor consists of a lens, mirror, and high quality diffraction grating. Similar
setups, sometimes using two gratings instead of a mirror, are commonplace for
compressing chirped pulses [132]. The simple schematic below shows the compression
process.
90
Fig. 6.6 Schematic of pulse compressor. High frequency components are depicted as
gray. Low frequency components are depicted in black.
The figure above demonstrates graphically how the leading high frequencies undergo a
slightly increased path length due to the diffraction grating. The larger path length allows
the lower frequencies to “catch up” to the higher frequencies, and thus recompresses the
pulse. By tuning the distance, Δl, while observing the pulse duration with an
autocorrelator, we can optimize the compression of a range of stretched pulses.
After the probe is recompressed, the THz pulse and optical probe co-propagate
and are focused onto the QW sample. The transmitted optical probe is then focused into
a spectrometer. As the delay between the optical probe and THz pulse is changed, we
observe time-resolved modulation of the transmitted optical spectrum, T(ω), due to the
strong THz pulse.
Specifically, we observe the THz-induced modulation of the LH and HH exciton
resonances. The optical probe alone excites a resonant excitonic polarization at the first
91
excited state of the LH and HH excitons. Since the exciton energy levels are hydrogenic,
they display similar angular momentum characteristics and are referred to by the same
spectroscopic notation (s, p, d, f …). Hence, this first excited state is named the 1s state.
As the symmetry of the LH and HH states are p-type, and photons only deliver one unit
of angular momentum, transitions from these states to the first excited 2p state are not
allowed. As a result, the 2p state is known as an optically dark state. However, a two
photon process is capable of reaching this state. As described in the previous section, the
energy separation between the 1s and 2p state falls near our THz photon energy.
Coupling an appropriate narrowband THz pulse to the 1s excitonic polarizations, we can
excite a resonance between the 1s and 2p states. The resonant transition between the 1s
and 2p states is depicted below.
Fig. 6.7 Internal structure of confined excitons. This diagram illustrates the excitation of
the 1s excitonic polarization and the subsequent strong coupling of the 1s and 2p states
by an intense THz pulse.
92
The spacing of the 1s and 2p levels are exaggerated for clarity. We know from the
previous section that the levels are within a few meV of the conduction band. Therefore,
it is also easily possible to ionize the electron into the conduction band if the THz
frequency is large enough. It is important to note that this is not a weak, linear process.
There is a strong coupling between the light (THz photon) and matter (exciton) that
cannot be treated perturbatively. However, analogous with the perturbative treatment of
a harmonically driven two level system, the driving field causes a cyclic rotation between
the two energy levels known as “Rabi flopping”. Rabi flopping is characterized by the
Rabi frequency, ΩR, defined by:
R E2 p E1s (2 p 1s )
(6.3.1)
Additionally, a general result of classical mechanics indicates that strongly coupling two
oscillators will result in a splitting of the Eigen-frequencies of the system [133]. The
splitting that occurs is also observable in quantum mechanics, where strongly coupling
two well-defined quantum states will result in a splitting of the energy Eigen-states in the
following manner:
1 , 2 c1 1,2 c2 e i 2,1
(6.3.2)
Where 1 and 2 are the original states, 1,2 and 2,1 are the strongly coupled states, c1
and c2 are the probability amplitudes, and eiφ is the definite phase between the two states.
Including the coupling-induced splitting, the following figure details the dynamics of this
experiment.
93
Fig. 6.8 Depiction of exciton polarization dynamics in our QW system.
Figure 6.8 illustrates that the weak optical probe excites an excitonic polarization at the
1s frequency. The subsequent intense THz pulse strongly couples the 1s and 2p states
which results in Rabi oscillations, and a splitting of the 1s and 2p level. The symbol g
represents a weighting factor that is proportional to the detuning of the THz energy from
the Rabi energy, E2p-1s. (For example, if ΩR = ωTHz, g =1). Figure 6.8 suggests a simple
way to find if this strong coupling occurs. In the absorption spectrum of our QW (figure
6.3), we see two distinct absorptions from the LH and HH. If this strong coupling regime
is present, and Rabi Oscillations occur, we should see evidence of the absorption
spectrum splitting into two distinct sidebands.
94
6.4 Results
6.4.1 Experimental Results
The results of this experiment compare experimental observations and theoretical
calculations of strong interactions between intense THz radiation and the 1s-to-2p
transition of the excitonic polarization in resonantly driven GaAs/AlGaAs QWs. The
theoretical calculations were performed by our collaborators in the Koch group in
Marburg, Germany. These calculations, which will be briefly discussed later, are
extremely complicated many-body calculations which include microscopic Coulomb
effects, and THz effects. As we will see, there is an excellent agreement between the
observations and theoretical results.
The experiment was a time-resolved, THz-pump/optical probe experiment. The
light source was 800 nm, 90 fs pulses from a 1 kHz Ti:Sapphire regenerative amplifier
(Coherent, Inc, Legend). The THz pulse energy was in the range of 1 nJ, and the pulse
duration was ~3 ps. The maximum electric field amplitude reached approximately 5
kV/cm. Figure 6.9 (a) and (b) shows a TDS measurement of a typical multi-cycle pulse
generated by the DFG setup for this experiment and its FFT spectrum. Figure 6.9 (c)
shows the 1-T(ω) optical spectrum at a single time delay out of the entire time resolved
scan.
95
Fig. 6.9 Modulated 1-T(ω) optical transmission spectrum. (a) TDS measurement of DFG
pulse with (b) its FFT spectrum centered at ~1.86 THz. (c) 1-T(ω) optical transmission
spectra at td = 0.6 ps. The light line is the un-modulated spectrum. The dark line
represents the THz modulated spectrum.
The light line in figure 6.9(c) shows the un-modulated optical 1-T(ω) spectrum. This
spectrum represents the linear dipole transition strength from the LH and HH valence
bands to the 1s exciton state. The dark line shows the 1-T(ω) spectrum modulated by an
intense multi-cycle THz pulse . Immediately several effects are noticeable. The peak
amplitude of the spectrum is heavily attenuated, and both peaks are broadened and redshifted. In addition, as the arrows indicate, we see the onset of an additional peak on the
high-energy side of the HH peak. These effects already suggest very strong non-linear
96
effects. To further characterize these effects, we measured the transmission at several
frequencies ranging from 1.4-2.2 THz. Figure 6.10 shows a representative sample of
these modulated transmissions at various values of Δ where
THz 1s 2 p
HH
(6.4.1)
In other words, Δ represents the difference between the driving THz photon center
frequency, and frequency corresponding to the HH 1s-2p transition energy of 1.75 THz).
97
Fig. 6.10 Several 1-T(ω) spectra. (a) THz excitation below the 1s-2p HH resonance. (b)
THz excitation near the 1s-2p HH resonance. (c) THz excitation above the 1s-2p HH
resonance. The shaded areas are the un-modulated transmission spectra, and the dark
lines are the modulated spectra. Insets show the THz excitation spectrum. Δ = νTHz – ν1s2p.
These three THz excitation frequencies are representative because they correspond to
excitation below, at, or above the 1s-2p HH resonance. In all three cases, large nonlinear
modulations of the spectrum occur. In the case of Δ = -0.30, we already see a strong
98
broadening, along with a slight red-shift, and a lowering of the peak transmission. As Δ
approaches zero, these effects become even more pronounced, and we again see the
subtle arrival of a second high-energy shoulder that appears to be the onset of Rabi
splitting. As the resonance moves to Δ = 0.4, nonlinear modulations are still very strong
despite being off resonance. These strong modulations are present because the THz
spectrum can now reach higher exciton states and into the conduction band continuum.
The disassociation of excitons into a correlated electron-hole plasma leads to a rapid
decoherence of that portion of the exciton population which washes out the coherent Rabi
oscillations in the spectrum while still demonstrating a strong response.
Below we see the full time evolution of the QW experiment of which figures 6.9
and 6.10 represent “snap-shots” in time.
Fig. 6.11 Full time evolution of differential transmission. This differential transmission
(ΔT = (TOpt+THz – TOpt) / Topt) shows the THz effects on QW system with corresponding
THz pulse for reference.
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Figure 6.11 depicts the differential transmission of the system which is another very
useful way to view the data. The differential transmission, ∆T, is the THz-pulse
modulated signal minus the optical signal alone divided by the optical signal again for
normalization (See caption in figure 6.11). The plot in figure 6.11 reiterates a few
important points. First, the dashed line falls across spectral oscillations occurring at
negative time delays. These oscillations are a hallmark of coherent polarization because
the pump and probe do not overlap in time at this point. The reason we see an effect is
that the optical pulse excites a coherent interacting excitonic polarization as it arrives
first. As non-interacting two-level systems, such as atoms, do not display this behavior
[134], we know that the excitons have a definite phase relationship between their
polarization oscillation frequencies (which is what is meant by “coherent” in this case).
The coherence time of this system lasts until the THz pulse, with which it interacts,
arrives later. Additionally, we can see the time development of the splitting of the HH
into a red-shifted peak (left arrow fig. 6.11) and higher energy shoulder (right arrow fig.
6.11) that appears to be an effect of Rabi oscillations.
6.4.2 Theoretical Results and Comparison
In order to explain these nonlinear effects, we compare the experimental
observations with theoretical calculations. I will only outline a few preliminary steps and
procedures in the theoretical treatment to illustrate its basis, but I will not go into the
calculations as the complexity of the details [114,135] are outside the scope of my
expertise. We begin with the system Hamiltonian:
100
H SYS H 0 H C H PHON H THz
(6.4.2)
H0 is a non-interacting portion, HC is the carrier-carrier interaction term, HPHON is the
carrier-phonon interaction term, and HTHz is the THz light-matter interaction term. As the
experiment is performed at ~5K, we can neglect the phonon interaction contribution.
Explicit forms for the other terms are given by:
H 0 k a,k a ,k
(6.4.3 a)
,k
HC
1
Vq a,k a,k a,k q a,k q
2 , k , k , q
e2 2
H THz j ,k ATHz
ATHz a ,k a ,k
2m0
,k
Dk , ATHz a ,k a ,k
(6.4.3 b)
(6.4.3 c)
,k
In all of the above equations, a,k represents the creation/annihilation operator for an
electron at band λ, and momentum k. Equation 6.4.3 (a) is simply the kinetic energy of
each carrier at momentum k and band λ. Equation 6.4.3 (b) represents the carrier-carrier
interaction with Vq as the bare Coulomb potential of the system. Finally, 6.4.3 (c)
represents the THz interaction. The first line in 6.4.3 represents intra-band transitions (λi
= λf) which contains linear ( ATHz
ETHz
2
) and nonlinear ( ATHz
) THz effects. The
t
last line represents THz inter-band effects. These effects are negligible compared to the
intra-band effects as the THz photon energy is much smaller than the band gap [135].
101
Beginning with this Hamiltionian, one can derive the transmitted and reflected electric
fields as a function of the THz current, JTHz, from the wave equation. The transmitted
and reflected electric fields can be used to write the Heisenberg equations of motion
(EOM). The EOM’s can then be solved in the Bloch basis which allows us to extract the
exciton dynamics. Even from this rudimentary explanation, we can see that these EOM’s
involve an extremely complicated correlated, many-body calculation. However, an
important detail that can be seen from even this cursory glance at the theory is that the
dynamics are an explicit function of the THz electric field, ETHz. In fact, the theoretical
group uses the experimental form of the THz pulse in their calculations in order to
represent the dynamics as accurately as possible. Below we see a sample of theoretical
calculations that correspond to figure 6.10.
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Fig. 6.12 Theoretical calculations of 1-T(ω). Shaded areas are the un-modulated
spectrum. Dotted lines represent a calculation in which the 2p dephasing constant is
artificially reduced. Solid lines represent the full computation.
These calculations represent snapshots of the theoretical 1-T(ω) spectra at time delay td =
0. The shaded areas are the un-modulated spectra. The solid line is the full calculation of
the THz modulated spectrum which matches very well with observation, especially in
that there is not a clear Rabi splitting. Instead we see the development of a shoulder as in
the experimental data. The lack of clear splitting is due to the very rapid dephasing of the
coherent polarizations. From this data, however, we were able to extract the polarization
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dephasing constants of the excitonic 1s and 2p levels: γ1s = 0.5 meV and γ2p = 1.5 meV,
which, to our knowledge, was the first time this had been done with this particular
system. In fact, the dephasing constant for the 2p level is three times that of the 1s level.
Returning to figure 6.12, the dotted line represents a calculation in which γ2p is artificially
reduced such that γ2p = γ1s. With this reduction, we see a much clearer Rabi splitting.
This artificial splitting indicated that the onset of the shoulder is indeed due to Rabi
oscillations, but that the very fast 2p dephasing constant prevents direct observation of
the splitting. In fact, the calculations suggest that we could observe a splitting in the LH
resonance as well since its 1s-2p transition energy is also close to the THz photon energy.
In conclusion, we have demonstrated that strong THz pulses can be applied to
generate remarkably large changes to the optical response due to Rabi sidebands. Our
analysis identifies the 2p-exciton dephasing to be three times larger than that of the 1sstate. Thus, the 2p dephasing is the limiting factor for the detection of pronounced
sidebands. In order to increase resolution, it may be necessary to have samples with
narrower exciton absorption lines to accomplish the reversible Rabi-flopping regime in
GaAs-based QWs. Additionally; shorter probe pulses would be advantageous to increase
resolution. Nonetheless, the THz pulses used produce pronounced nonlinearities even
when full reversibility is not reached. This work was accepted for publication in
November 2009 by Applied Physics Letters [136].
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7. Extreme Nonlinear THz Transients in Quantum Well Microcavities
Microcavities are excellent systems in which to study the interaction between
light and matter extending from the weak-coupling regime to the strong-coupling regime.
Because of this, much work has been done with atom optics in microcavity structures
[137-143], as well as reduced dimensional structures in microcavities [144-148]. The
non-perturbative exciton-photon coupling in a high-Q microcavity leads to the formation
of two new eigenstates called exciton-polaritons which will be discussed in more detail in
the following sections. The splitting developed by the strong coupling of excitons to a
microcavity is often considered a solid-state analog of vacuum-field Rabi splitting
(2~ΩR) of an atom-cavity system. Many fundamental questions concerning quantum
optical phenomena in semiconductors have been explored by studying the optical
properties of semiconductor microcavities.
In the previous chapter, it was discussed that THz photons can excite states in
coupled light-matter systems that are energetically inaccessible to optical spectroscopy.
Indeed, the same is true for QW systems confined in microcavities.
7.1 Microcavity General Characteristics
A microcavity is simply a resonant structure that can confine photons in a
standing wave pattern. This structure is formed by having two highly reflective mirrors
spaced by a distance, L. For any resonant cavity of length L, the number of resonant
105
wavelengths, or modes, is infinite, and corresponds to the number of half wavelengths of
light that will fit between the two ends. A simple depiction of this is shown below.
Fig. 7.1 A generic resonant cavity. Available modes depend only on the optical path
length (OPL) of the cavity, n0L, where n0 is the index of refraction of the cavity.
The field distribution of the standing wave inside the cavity above shows five antinodes
(m = 5). The placement of the antinodes for any given mode is important since the field
will be maximized at these points. Therefore, the antinodes are the optimal position at
which to place a sample to observe strong coupling between the field and the material.
Waves that do not fit into the cavity in half wavelength increments develop a destructive
phase relationship, and are damped out very rapidly.
As mentioned before, there are an infinite number of available modes. As the
equations accompanying figure 7.1 indicate, these are all equally separated modes. The
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mode spacing is sometimes referred to as the free spectral range (FSR) and is given by
[149]:
FSR
c
2n0 L
(7.1.1)
The actual modes supported are determined by the overlap of the available modes and the
photons that are pumping the cavity. Since FSR is determined by the size of the cavity,
how many modes overlap with the pump bandwidth can be controlled. For a Gaussian
pump distribution, the overlap might look as follows:
Fig. 7.2 Gaussian gain profile. (a) Overlap of Gaussian pump radiation profile with cavity
modes. (b) Gain profile of cavity modes.
Part (a) of figure 7.2 shows the overlap of a few modes with the Gaussian profile of the
pump radiation. Part (b) demonstrates the actual gain profile of the modes. By
controlling either the bandwidth of the pump radiation (FWHM), or the cavity size (L), or
both, how many modes are amplified by the cavity can be controlled.
107
7.2 Polaritons
When a material polarization is strongly coupled to a photon mode inside a cavity,
a new quasi-particle is introduced called a polariton. In our case, this is an excitonpolariton. The optical photons simultaneously excite an excitonic polarization and excite
cavity modes that couple to each other. The outcome of this coupling is similar to the
situation described in chapter 6 in which the strong coupling of the 1s and 2p exciton
states can induce a Rabi splitting of these energy levels. When the cavity resonance is
near the excitonic binding energy, there is a strong coupling of the material state to the
cavity state. The enhanced field produced by the cavity confinement induces a splitting
of the original exciton state into two polariton states. An oversimplified, but illustrative,
way to view this is to look at the dispersion relations of a single photon mode, and a
single free exciton. Details of this method can be found in the dielectric treatment of
polaritons found in chapter 11 of [129].
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Polariton Dispersion
ck
n0
E(k)
HEP
2k 2
const.
2mex
LEP
k
Fig.7.3 Polariton dispersion relation. White lines show uncoupled photon mode k ,
and exciton mode k 2 for small values of k. Black lines show the coupled modes
giving rise to the lower Low Exciton Polariton (LEP) state, and High Exciton Polariton
(HEP) state.
The white lines in figure 7.3 show the uncoupled photon and exciton modes. For small
values of k, the parabolic dispersion of the free exciton is approximately flat. When the
modes couple to each other, the new quasi-particle forms two new states: the high and
low exciton polariton states (HEP and LEP). For each value of k, there are two
energetically distinct states possible. An interesting property of these new states is that
they are neither photonic nor excitonic, but a combination of the two. For example, at the
value of k where the dispersions cross, the LEP and HEP are an equal mixture of photonlike and exciton-like states. However, as you move to a higher (lower) value of k, the
LEP state becomes more exciton-like (photon-like), and the HEP state becomes more
photon-like (exciton-like). The balance of photon-like and exciton-like characteristics is
109
an important factor in considering the THz interaction with polaritons because THz
photons should not interact with the cavity modes, but do interact with excitons. Hence,
we expect the THz photons to have the largest effect on exciton-like polaritons and less
effect on photon-like polaritons.
7.3 Our Sample
The sample used in our experiment was a stack of 10 InGaAs QW’s surrounded
by an 11λ/2 cavity consisting of Bragg reflectors designed to be %99.94 reflective. The
figure below depicts the structure of the sample. Notice that the sample is wedge-shaped.
(This is highly exaggerated in the figure). As was discussed in the previous section, the
cavity resonance is only dependent on the OPL of the cavity. The wedge shape allows us
to tune the fundamental resonant frequency of the cavity with respect to both the exciton
binding energy and the exciting THz photon energy simply by moving the sample side to
side. In this way, the incoming optical excitation encounters the cavity at a different
thickness. The significance of this is discussed later.
Fig. 7.4 Depiction of QW micro-cavity sample. (a) Side view of QW stack sandwiched
between two DBR’s and set on a substrate. (b) Top view of sample demonstrating
exaggerated wedge shape and tunability of sample resonance.
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When the weak optical excitation strikes the QW micro-cavity sample, the following
reflectivity spectrum is observed.
Fig.7.5 Reflectivity spectrum of QW micro-cavity sample.
The InGaAs QW sample actually only contains one absorption resonance in this region,
however, we clearly see two sharp absorptions. These two absorption resonances are a
direct observation of the exciton-polariton splitting described above. As with the “bare”
QW sample of Chapter 6, we see very clear, sharp lines indicating a high quality sample.
7.4 Experiment
7.4.1 Experimental Setup
The experimental arrangement is shown below.
111
Fig. 7.6 Experimental setup used with single-cycle micro-cavity QW.
The apparatus for this experiment is essentially the same as that used in chapter 6 for the
bare QW experiments with a few key differences. First, DFG is not used for the
generation of THz waves. Therefore, the fully compressed output from the amplifier is
used to generate a strong, single-cycle THz pulse by the process of optical rectification
described in chapter 3. Additionally, the compressor is not needed since the probe is
already compressed. Another practical difference with this experiment is that data is
gathered in a reflective geometry instead of a transmission through the sample. The
reflection scheme caused some practical problems as to how to intercept the reflected
optical probe beam without interrupting the input. The result was a careful construction
of a sample mount that allowed us to finely tune the position of the sample with five
degrees of freedom. These are shown in the figure below.
112
Fig. 7.7 QW micro-cavity sample mount degrees of freedom.
As figure 7.7 shows, the aperture in the last parabolic mirror that the reflected optical
probe must pass through twice was only about 1 mm wide requiring fine control over the
sample position.
The last difference is the presence of a white-light generation apparatus and a
band-pass filter. The white light generation setup, shown below, is necessary because the
exciton resonance for the InGaAs QW’s occurs at approximately 830 nm.
Fig 7.8 Simple white light continuum generation setup.
113
The setup in figure 7.8 allows us to expand our narrow probe pulse centered at 800 nm to
an extremely broad range of wavelengths, and by use of a band pass filter, pick out a
range centered on 830 nm. The broad band generation, or super-continuum generation as
it is sometimes called, is accomplished by focusing the extremely short pulse onto the
sapphire crystal. Although this method has been used for decades for various
applications, the physical process is still not completely understood, though it is widely
accepted that self-focusing and various forms of self-phase-modulation are at the heart of
the process [150]. Although our oscillator is easily tunable up to central frequency of 830
nm, we opted for a white light generation setup for the following reasons: (1) The THz
pulse generation is more efficient at 800 nm, and as we are looking for nonlinear effects,
we did not want to sacrifice THz intensity which falls off as the square of the optical
pump field, and (2) since the probe must be weak, we could accept a loss in intensity of
the probe caused by the extra optics in the probe line.
Other than these few differences, the experimental details were much the same as
in the bare QW experiment. Strong, single-cycle THz pulses were generated by optical
rectification using 800 nm, 100 fs optical pulses. Weak, optical pulses centered at 830
nm were used to probe the system. These were coincident on the plane of the sample
which was held at ~5 K with liquid helium in a cryostat. The waveforms were obtained
by EO sampling in a 1 mm thick ZnTe crystal. Additionally, we measured the incident
THz power using a silicon bolometer. The THz field reached approximately 10 kV/cm.
Spectral modulations were measured by observing the reflectivity, R(ν), of the optical
114
probe with a spectrometer. By changing the delay between the THz pulse, and the optical
probe, the time evolution of R(ν) is mapped out.
7.4.2 Experimental Considerations
With the above background and setup, we can more carefully describe the
complexity of the experiment performed. To the best of our knowledge, no prior THz
studies of this kind have been done on optical micro-cavities. As such, there was some
question as to how the THz radiation would interact with the micro-cavity QW. There
are many relevant energies present in the experiment, all of which correspond to THz
photon energies: (1) As in the bare QW experiment, the exciton 1s-2p transition falls in
the THz photon range, (2) The LEP/HEP-2p transition energies are in the THz photon
range, and (3) The splitting of the HEP and LEP resonances also falls into this range. It
was not known to what degree the THz photon might couple the HEP and LEP states. A
semi-classical analysis predicted a huge enhancement of the nonlinear polariton
responses when THz radiation is tuned to the polariton splitting [151], which, however,
contradicts the quantum symmetry of the polariton wave functions, i.e., dipole transitions
between the two polariton modes are forbidden. Additionally, it was not known whether
one of the two following contradictory situations would prevail. In the first scenario,
THz photons couple the excitonic 1s and 2p states as in our previous works [118, 136],
and these effects are enhanced by the microcavity. In the second scenario, the cavity
mode couples to the exciton first forming polariton states, and these states are in turn
115
coupled to the excitonic 2p state by the THz photon. For clarity, these two situations are
depicted below in figure 7.9.
Fig. 7.9 Possible quasi-particle dynamics in micro-cavity QW. (a) THz photon only
couples to exciton, and the cavity mode couples to the Rabi-split levels. (b) THz photon
couples to polariton λ-system and can split either LEP of HEP level.
The particular ordering of when the THz photon couples to which quasi-particle is not
important so much as answering the question of whether the THz couples to the exciton
alone or the polariton. This question was answered in other work performed in our lab
with narrowband THz pulses produced by the DFG [152]. By tuning the photon to
particular transitions, we found that the THz couples to the polariton states, and that the
correct dynamics of the LEP, HEP, and 2p states are those of the lambda system depicted
in figure 7.9(b).
With this in mind, we pumped the system with a broadband, single-cycle THz
pulse as well. The single-cycle pulse provided several advantages that were worth
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exploring. First, the single-cycle setup is much less complicated. Because of its
simplicity, and because of the use of a fully compressed optical pump, the source outputs
more intense electric fields than the multi-cycle pulse is capable of producing. Also, the
pulse duration of the single-cycle pulse is ~1 ps (compared to the DFG pulse duration of
~4 ps), allowing us greater resolution in mapping of the THz effects. However, the
simplicity of the THz generation is accompanied by an increase in the complexity of the
dynamics present in the system. The broadband nature of the pulse implies that the
observed effects will no longer be dominated by the resonant excitation of a single
transition. Instead, multiple frequencies can simultaneously cause resonant excitations.
Additionally, these effects can be augmented or diminished by various non-resonant field
induced effects. These effects could include, but are not limited to, spectral variations
due to the Dynamic Franz-Keldish Effect, the A.C. Stark effect, and ponderomotive
effects. Similar considerations were necessary in past experiments with single-cycle
pulses [118]. Unfortunately, a full theoretical comparison is not yet available, as our
experimental collaborators have not completed a treatment for this system. However,
much promising and interesting experimental data has already been gathered, so we can
at least examine the preliminary results from a phenomenological point of view.
7.5 Results
One of the first experimental procedures we undertook was to verify the polariton
dispersion in the absence of a THz pulse. To verify the dispersion, we pumped the
cooled sample with the weak optical probe, and scanned the sample along the Y-axis as
117
defined by figure 7.7. As we scan along the wedge-shaped sample, the cavity resonant
frequency changes with respect to the exciton binding energy. This energy difference is
what will be referred to as the cavity detuning, c c 1s . δc = 0 is the point of
minimum energy difference between the LEP and HEP modes, and corresponds to the
point in figure 7.5 where the uncoupled photon and exciton dispersion relations cross.
Figure 7.10 shows the result of this scan.
Fig. 7.10 Microcavity QW detuning measurement. (a) Scan of the cavity detuning, δc = νc
– ν1s. (b) Individual absorption spectra with white lines a guide.
These figures show the clear splitting of the single exciton absorption into two polariton
modes. As the cavity resonance is changed, the polariton dispersion relation emerges.
Characterizing the detuning is important because the polariton binding energy determines
the LEP/HEP-2p transition energies. The simplest way to model this dispersion relation
is with a coupled oscillator model. Based on this simple model we can approximate the
polariton energies as a function of the detuning by the following equations
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E HEP
E LEP
c
w
c
2
2 2
2
c
2
w
c
2
2 2
2
(7.4.1)
2
In equation 7.4.1, δc is the cavity detuning, and w is the mode separation at δc = 0. The
plot below shows a plot of the position of the LEP/HEP resonances with respect to E1s
versus the detuning.
Fig. 7.11 Peak positions of the LEP and HEP modes vs. cavity detuning. The 1s-exciton
resonance is set to the origin. The measurements of the THz induced nonlinear effects
were made at δc = -2.3, -1.9, -1.7, +0.3, +1.3, and +1.7 meV.
As we see from figure 7.11, this coupled oscillator model fits the data very well. As a
result, we were able to create a table of wavelength versus detuning values from equation
119
7.4.1 in order to identify a particular detuning rapidly during experimentation simply by
observing the absorption line wavelengths and matching them to the table.
The next experiments performed were the time-resolved measurements described
in section 7.3.1. At several different detunings, we scanned the delay between the
optical probe and THz pump in order to map out the THz interaction with the polariton
modes. The results of this time-resolved study were several extremely interesting
observable nonlinear effects. The figure below shows a single-cycle THz pulse used in
this experiment and its corresponding spectrum.
Fig. 7.12 Single-Cycle THz pulse used in time-resolved study of QW micro-cavity. Inset
shows the pulse spectrum.
120
Scanning this pulse with respect to the optical probe revealed the following time
evolution of the THz interaction with the polariton states.
Fig. 7.13 Time-resolved differential reflectivity. (a) Birds-eye view of differential
reflectivity (b) Lighting effects applied to better illustrate interference effects.
From figure 7.13 we notice several strong effects. First we notice clear nonlinear effects
changing the optical properties of the polariton absorptions around zero time delay.
These effects are indicated by the white portions surrounded by darker areas in figure
7.13(a). These effects are caused by broadening of the absorption, and decreased
absorption at the center frequencies of the polariton mode as the THz couples the
polariton states to the 2p exciton state. One other very pronounced effect is the
development of the spectral oscillations at negative time delays. Since the single cycle
THz pulse is broadband, these oscillations are correspondingly spread out further around
the polariton resonances. As mentioned in section 6.4.1, this is a hallmark of coherent
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polarization. The coherent LEP and HEP polarizations are driven by the optical pulse
which arrives first. As these polarizations decohere, the frequency of these polaritonic
states spreads around the resonant frequency. This spreading is the cause of the
characteristic oscillatory “fanning out” at negative time delays. In this experiment,
because the THz field is very strong, and the single cycle pulse is so short, we have an
unusually high resolution. We can see a clear interference of the polariton states as they
decohere in the cross-hatched pattern developed between the LEP and HEP states. This
pattern can be seen in both parts of figure 7.13. A lighting effect was added to figure
7.13(b) in order to more clearly demonstrate the interference by bringing out “shadows”
cast by maxima over the minima. The result is an amazing physical demonstration of the
time development of a purely quantum mechanical interference effect.
As mentioned above, one of the distinct advantages of using single-cycle THz
pulses is the corresponding increase in THz intensity, which is demonstrated in
impressive fashion by another preliminary result shown below.
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Figure 7.14 Time resolved reflectivity of the microcavity sample. This scan was done for
a relatively long scan (~30 ps).
The plot shows the time evolution of the reflectivity of the microcavity sample over a
period of time long enough to capture the effects of successive THz pulses caused by
internal reflections of the optical pulse inside the ZnTe generation crystal. These
reflections are commonplace, but are seldom useful since the amplitude of each
subsequent pulse falls off sharply with respect to the prior pulse. As we are usually
concerned with the maximum THz pulse energy to excite nonlinear effects, we often only
closely examine the effects of the first pulse. However, in this case, we can see that the
effects due to the following pulses are hardly discernable from those of the first pulse.
These strong modulations indicate that the nonlinear effects are saturated even at the
relatively lower field strengths of the weaker THz pulses. The large modulations also
123
demonstrate the enhancement of these effects by the microcavity. These results suggest
the possibility of intensity dependent studies in which we might observe the onset of
nonlinear effects by changing the THz intensity.
Unfortunately, to date, we have been unable to perform the complete set of
experiments necessary for our theoretical collaborators to explore the dynamics at play
with their model, so I am unable to give concrete evidence of the particular nonlinear
effects at play and their relative strengths. Future studies will include corroborating
experiments for the existing results, as well as THz intensity dependent studies that may
reveal at what THz intensity the nonlinear effects “turn on”. However, for now I must
leave these tantalizing first results and potential experiments for future generations to
complete.
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8. Conclusions
In the several years I have spent working in Dr. Yun-Shik Lee’s lab, I have had
the opportunity to work on a variety of interesting and challenging experiments. These
have included, but are not limited to, the development of novel THz sources, the
exploration of quasi-particle dynamics in various semiconductor nanostructures, and the
spectroscopy and imaging of NiTi thin films.
Coherent THz sources are a fairly recent scientific advance. Because they are not
largely available in turnkey modules, the construction of optical setups capable of
producing intense THz radiation was a key part of my research. These sources included
the construction of setups capable of generating both broadband, single-cycle THz pulses,
and narrowband, many-cycled pulses. The latter of these two was the first project I was
involved in. In this project we designed a novel, table-top source of intense THz
radiation that is easily integrated into an ultra-fast spectroscopic setup. By using type II
difference frequency generation with two linearly chirped optical pulses, we created a
tunable source of narrowband THz waves. This source was, and continues to be,
extremely important for a variety of projects for many reasons. The intensity of the
narrowband THz waves, although less than the intensity of the single-cycle setup, allows
us continue to excite and explore nonlinear effects. The narrow-band spectrum allows us
to excite specific resonant effects as opposed to exciting many effects at once with the
broadband pulses. Although these pulses are longer than the single-cycle pulses, they are
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still extremely short (a few picoseconds), so high resolution, time-resolved spectroscopy
is still possible.
One alluring prospect of working with THz science and technology is that,
because the field is relatively new, one often gets to work with exotic and interesting
materials and study previously unexplored properties. An excellent example of this is our
work with NiTi alloy thin films. These alloys make fascinating study in a variety of
fields because of their properties of super-elasticity, electro-plasticity, and shape
memory. As its many engineering applications become miniaturized, it is important to
characterize the electrical properties. We did this by using THz time domain
spectroscopy and THz power transmission to extract values of the resistivity of NiTi
alloy thin films at various thicknesses and Ti concentrations. These values were
extracted by comparing measured values of the relative transmission to theoretical
expressions derived from a combination of the Drude DC resistivity with Fresnel thinfilm coefficients. By comparing the results of both methods with a four-point-probe
method, we found excellent agreement between all methods. In addition to the
agreement, we note that the THz wave method does not damage the sample in any way,
unlike the four-point-probe method. Not only did we extract values of resistivity that
have not been measured before with THz spectroscopy, but uncovered some very
interesting dependence of the resistivity on Ti concentration. The resistivity goes through
several peaks at specific Ti concentrations which seem to coincide with phase transitions
of the alloys. These peaks in the resistivity suggest that although the THz wavelength is
too large to pick out micron-sized or smaller details, it can still be sensitive to effects
126
these small features have on the electrical properties of the material. This work served to
illustrate the spectroscopic capabilities of THz waves and its power to extract electrical
properties from dielectrics and thin film conductors in the linear regime.
THz radiation is not only useful for linear spectroscopy, but is also a powerful
source with which to excite carrier dynamics in semiconductor nanostructures. There is
still much to be learned about the dynamics of quasi-particles in semiconductors and the
combination of reduced dimensional devices and intense THz radiation makes an
excellent laboratory in which to explore them. My work with narrow-band THz waves
and semiconductor quantum wells explored the coherent dynamics of excitons in
GaAs/AlGaAs QW's. In this project we were able to strongly couple intense narrowband THz pulses to excitonic states. We first created an excitonic polarization at the 1s
frequency with a short optical pulse. Pumping this state with a narrow-band THz pulse,
we resonantly coupled the 1s state to the optically dark 2p state, causing a cyclic
transition between the two states. This coupling manifested itself in a number of
nonlinearities. The most important observable effect was the onset of Rabi flopping
whose characteristic frequency is equal to the energy separation between the 1s and 2p
state. By comparing our results with calculations performed by theoretical collaborators,
we found that the onset of Rabi splitting is obscured by fast dephasing times. In the
process of making this comparison, we were able to extract dephasing constants for the
1s and 2p states for the first time. This experiment was an important piece of work in
demonstrating the power of THz to excite dynamics not available with optical excitations
and to extract new information from very complicated many-body systems.
127
Often in THz science there are analogs to previous studies, such as atomic optics,
that are natural extensions of semiconductor optics, yet reveal fascinating new results.
Our work studying GaAs/InGaAs QW's embedded in a micro-cavity is one such example.
These experiments introduce a strong coupling between the cavity resonances and the
excitonic states in the QW's. The strong coupling results in a splitting of the excitonic
state into two exciton-polariton states. When introducing intense THz pulses, it was
unknown how this system would respond. When using narrow-band THz pulses, another
study of ours showed that the THz photon strongly couples the polariton states to the 2p
exciton state forming a three level lambda system, akin to atomic optics. When pumped
with a broadband, single-cycle THz pulse, the carrier dynamics are more complicated,
because many resonant and non-resonant effects are simultaneously competing.
However, the resolution of the time-resolved effects is greater due to the shorter pulse
length. Although we were unable to sort these effects out to date, we have some very
interesting preliminary results that warrant further work. The first of these is an amazing
look at the interference of decohering polariton states. This clear, time-resolved picture
of a purely quantum mechanical effect is a great demonstration of the increased
resolution of this method. The second excellent preliminary result shows that
enhancement of the nonlinearities due to the micro-cavity make it possible to perform
intensity dependent studies. These could determine at what THz intensity nonlinear
effects begin.
128
Through all of these studies, and my involvement in others, I have learned that
THz radiation has vast applications in exploring spectroscopic properties of materials and
exciting and revealing the dynamics of quasi-particles. THz-TDS is a powerful tool for
characterizing material responses in its range of the electromagnetic spectrum. Timeresolved, THz pump-probe experiments are powerful methods for uncovering carrier
dynamics in semiconductor structures. From building these setups to generate and
manipulate THz waves, to using these waves to explore a variety of systems, I have
witnessed firsthand its ability to uncover new properties and give insight into unexplained
physical systems. Indeed, one of the most rewarding aspects of THz science is that it is
still in its relative infancy. Every experiment holds the possibility of uncovering new
physics. As such, my hours spent pursuing THz science has been as rich and rewarding
as the field itself.
129
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