TIME-DOMAIN LIGHT SCATTERING
AND
STUDY OF LIQUID-GLASS TRANSITIONS
by
Yong-Xin Yan
B.S. Physics, University of Science & Technology of China (1982)
SUBMITTED TO THE DEPARTMENT OF PHYSICS
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY IN PHYSICS.
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
April 1988
Massachusetts Institute of Technology 1988
Signature redacted
Signature of Author
Department of Physics
April 29, 1988
Signature redacted
Certified by
_
Dr. Keith A. Nelson
Thesis Supervisor
Signature redacted
Accepted by
George F. Koster
Chairman, Graduate Committee
(MAY 2 A
TIME-DOMAIN LIGHT SCATTERING
AND
STUDY OF LIQUID-GLASS TRANSITIONS
by
Yong-Xin Yan
Submitted to the Department of Physics on April 29, 1988 in partial fulfillment of the
requirements for the degree of Doctor of Philosophy in Physics.
ABSTRACT
Theoretical analysis of the time resolved Impulsive Stimulated Light Scattering
(ISS) method is presented. A general theoretical framework is developed to describe
ISS experiments on any type of material mode which is active in light scattering and
conforms to linear response theory. ISS experiments permit time-resolved observation
of material motion through the dielectric response function GI E (q,t). In the simplest
case of ideal time and wave vector resolution, ISS signal gives |G e(qt)j 2 directly.
Various consequences of limited t- and q-resolution are discussed in detail. ISS
experiments on acoustic and optic phonons, Debye relaxational modes, and some
combinations of modes are treated explicitly.
A detailed comparison between time-domain impulsive stimulated light scattering
and frequency-domain spontaneous light-scattering spectroscopy is carried out in both
theoretical and practical terms. In some cases, the two experiments probe different
material responses. In many cases the information content of ISS and LS data is
identical in principle. The results can be related to each other through the time- and
frequency-dependent response function GE E (q,t) and GE E(q,w), or through the timecorrelation function CE E (q,t). Simulated ISS and LS data from vibrational and Debye
relaxational modes are compared in view of experimental considerations, including
wave vector and time or frequency resolution and range, and sources of "noise". In
many cases, one or the other experimental approach offers significant advantages in
practice. The complementary nature of the techniques is illustrated.
A time-domain light scattering study of acoustic and Mountain modes in glycerol
is carried out. By using light-scattering angles between 0.89* and 88.90. a wide range
of acoustic frequencies is sampled. The data also yield information about timedependent density responses to stress and to heat (the latter is the time-dependent
thermal expansion). These responses are associated with the Mountain mode and
provide additional information about structural relaxation dynamics. A theoretical
framework is presented which can treat these experiments as well as ultrasonics and
specific heat spectroscopy. The time or frequency dependences of the elastic modulus,
heat capacity, and pressure response to temperature change are all accounted for and
appear to be significant. The experimental results are fit best with a distribution of
relaxation times which is somewhat less asymmetric than a Cole-Davidson distribution.
-
2
The width of the distribution (on a logarithmic frequency scale) does not change
significantly in the 200 - 300K temperature range.
Similar study of acoustic behavior during the liquid-glass transition process of
60%KNO 3-40%Ca(NO 3 )2 shows that, unlike glycerol. the width of the distribution of
relaxation times shows significant narrowing when the temperature of the material
changes from 380K to 510K. This difference may be attributed to differing
temperature-dependent behavior in organic and ionic glass-forming liquids.
Surface acoustic waves can also be studied by impulsive stimulated light
scattering. We show some preliminary experimental results.
Dr. Keith A. Nelson
Associate Professor of Chemistry
- 3
-
Thesis Supervisor:
Title:
ACKNOWLEDGEMENT
I would like to thank all members of our research group. They together
provided me a friendly and collaborative environment. Professor Keith A. Nelson,
my advisor, has been a rich source of guidance, inspiration and encouragement. No
matter how busy he is, he always has time for his students. He is a model of
dedication and hard work for us. Lap-Tak Cheng has been my closest collaborator
for the last four years. From him I learned how to finish an experiment. I
probably will never be able to do as well as he can in making the best use of
existing hardware and software. Margaret R. Farrar, Leah R. Williams and Edward
G. Gamble have shared with me the hopes and frustration of the "early years".
They helped me in many respects like sisters and brothers. And they also tolerated
my ignorance and innocence.
Bern Kohler and Tom Dougherty have written
powerful utility computer programs for our group. I benefited from their programs
and more importantly, their help and advice have helped me to change from a
computer idiot to a computer amateur.
Sanford Ruhman and Alan Joly
experimentally confirmed one of my theoretical predictions. It's hard to discribe
how excited I was when I looked at the data. The new members of our group, Ion
Halalai, Scott Silence, Anil Duggal, Gary Wiederrecht and postdoctor Mark
Trulson, add new vigor to our group. Their joining of our group showed, among
other things, their appreciation of the work of us earlier members and their
confidence in the future of our approach.
I thank Professor T. D. Lee of Columbia University and everyone involved in
making CUSPEA program work, for giving me the opportunity to study in the US.
I would like to express my gratitude to the authors and editors of
4-.
-
4
-
tj :
, this book series was the single most important factor for my
early interest in science. I hope I can contribute to the'future editions of this series.
DEDICATION
To Wei Jing-Sheng (4t
), a courageous advocate of democracy. He
I
is currently held in prison by the Chinese government.
-
- 5
TABLE OF CONTENTS:
ABSTRACT
ACKNOWLEDGEMENT
DEDICATION
LIST OF ABREVIATIONS
2
4
5
8
1. INTRODUCTION
9
2. BASIC IDEA AND EXPERIMENTAL SETUP
2. 1 Basic idea
2.2 Experimental setup
11
11
19
3. GENERAL THEORY OF IMPULSIVE STIMULATED LIGHT SCATTERING
3.1 Introduction
3.2 General
3.3 Impulsive limit
3.4 ISS experiments on optic and acoustic phonons, relaxational
modes, and coupled modes
3.5 Nonideal situations
3.6 Summary and concluding remarks
Appendix A
Appendix B
Appendix C
22
22
23
29
- 6
-
4. COMPARISON TO FREQUENCY-DOMAIN SPONTANEOUS
LIGHT SCATTERING
4. 1 Spontaneous light scattering
4.2 Comparison of ISS and LS
4.3 Simulations of ISS and LS data from vibrational and
relaxational modes
4.4 Comparison of ISS and LS methods
4.5 Summary
33
38
66
70
71
73
75
76
79
82
86
94
5. FORWARD ISS
5.1 Excitation process
5.2 Probing process
95
96
97
6. ISS STUDY OF LIQUID-GLASS TRANSITIONS IN GLYCEROL
6.1 Introduction
6.2 Theory
6.3 Experimental
6.4 Results and discussion
6.5 Concluding remarks
105
105
108
115
118
137
7. ISS STUDY OF ACOUSTIC BEHAVIOR IN KNO 3 -Ca(NO )
3 2
DURING LIQUID-GLASS TRANSITION
7.1 Introduction
7.2 Experimental
7.3 Theoretical
7.4 Results and analysis
7.5 Summary
138
138
138
139
141
147
8. ISBS OF SURFACE WAVES
8.1 Theory
8.2 Results and discussion
149
149
9. COMMENTS
158
REFERENCES
161
- 7
-
151
LIST OF ABREVIATIONS
Impulsive Stimulated Light Scattering
Impulsive Stimulated Brillouin Scattering
Impulsive Stimulated Raman Scattering
Impulsive Stimulated Thermal Scattering
Spontaneous Light Scattering
- 8
-
ISS
ISBS
ISRS
ISTS
LS
CHAPTER 1.
INTRODUCTION
Light scattering has long been an important method for
studying properties of condensed matter (Berne & Pecora, 1976;
Hayes & Loudon,
1978).
There are several
commonly used
methods of doing light scattering:
1) Frequency-domain spontaneous light scattering.
In
these experiments, a single CW light beam passes through the
sample and the material properties are inferred through
spectral analysis of light spontaneously scattered by thermal
excitations.
At present,
the
lower limit of the applicable
frequency range as well as the best frequency resolution of
this method is tens of MHz.
2)
Time-domain spontaneous light scattering,
called correlation spectroscopy.
usually
The experimental setup is
similar to the frequency-domain method, except that instead of
frequency spectrum,
the time-correlation function of scattered
light is recorded.
The best time resolution at present is
about 1 /is.
3)
Frequency-domain stimulated light scattering, usually
called CW four-wave mixing.
In these experiments,
two CW
light beams of different frequencies are overlapped spatially.
The interfering light field thus formed provides a driving
force to the material modes at the difference frequency of the
two beams.
The scattering efficiency-of the third probe light
beam as a function of the driving force frequency is recorded
and from the
resulting spectrum material information
is
deduced.
The advent of short pulsed lasers brought with them the
birth of a new light-scattering method, called impulsive
stimulated light scattering (ISS).
It is similar to CW fourwave mixing except that the three laser
pulses instead of CW light.
beams consist of short
The combination of being time-
domain and stimulated enables ISS to occupy a unique position
-9-
At present, the time
in light scattering spectroscopy.
resolution of ISS extends all the way down to femtoseconds.
In the first part (Chapters 2 -
5) of this thesis, we
shall give a detailed analysis of the ISS method.
The basic
principles and experimental setup are described in chapter 2.
It is followed by a theoretical analysis on how to relate
experimental data to the underlying material properties
(chapter 3).
In chapter 4 we compare the ISS method with
frequency-domain spontaneous light scattering (abbreviated as
LS in this thesis).
The reason we choose to compare with LS
is because the majority of light-scattering experiments in
literature
in the author's field of
interest (and over all)
are carried out with this method.
In chapters 1 -
4 the ISS excitation process is
accomplished by two crossed laser pulses.
excite with just a single pulse.
It is possible to
In chapter 5 we shall show
why it can be done and discuss how to make use of this option.
This has additional
importance since
it means that ISS
excitation occurs whenever a very short pulse passes through
many types of materials.
The second part of this thesis is on the study of liquidglass transitions.
In chapter 6 we first give a
short
introduction to the subject of liquid-glass transition and
develop a phenomenological theoretical framework as a basis
for analyzing experimental data.
Then we present an
experimental study of the liquid-glass transition in glycerol
In
using impulsive stimulated Brillouin scattering (ISBS).
chapter 7 we present an ISS experimental study of the liquid-
scattering to study surface acoustic waves.
3
-
glass transition in an ionic glass former, Ca(N0 3 )2 -KNO
It is possible to use impulsive stimulated light
In chapter 8 we
show some experimental data from a preliminary investigation
of surface waves using ISBS method.
A discussion of main experimental
obstacles and
suggestions for further work in ISBS and liquid-glass
transitions is presented in chapter 9.
-10-
CHAPTER 2.
BASIC IDEA AND EXPERIMENTAL SETUP
2.1 Basic idea
The impulsive stimulated scattering experiment is
illustrated schematically in Fig.
2.1.
pulses derived from the same laser,
Two short excitation
of central
frequency and
wave vectors (wLkl), (wLk2), are overlapped spatially and
temporally inside a sample to exert a spatially periodic (due
to the
interference pattern formed by two excitation beams),
temporally impulsive driving force on the material modes.
The pulse duration must be
time scale of interest.
shorter than the shortest material
For example,
to study acoustic or
optic phonons, the laser pulse durations must be short
compared to a single oscillation cycle of
mode.
the vibrational
The material response to this impulsive driving force
is the impulse response function of dielectric constant
characteristic of the material.
This spatially periodic
response acts like a time dependent volume "grating" which can
(i.e., "diffract") a "probe" laser beam
coherently scatter
which is incident at the phase-matching angle (Bragg angle)
for diffraction.
efficiency,
function.
From the time dependence of the diffraction
one finds the dielectric constant response
From it,
information is extracted on dynamical
properties such as frequencies and dec-ay rates of the material
modes.
Time-resolved detection is usually carried out in
either of two ways:
(1)
a cw laser beam is used as probe and
the intensity of scattered light is time-resolved
electronically; or 2) a variably delayed, short laser pulse is
used as probe,
repeating the excite-probe process at gradually
increasing delays of the probe pulse,
the total scattered
intensity of each pulse being recorded as a function of delay.
(It's like to make a film of someone jumping down a building
and unable to make one hundred exposures during one jump.
-11-
We
PULSE SEQUENCE
ISS
DIFFRACTED
PROBEN\
PULSE
\
SAMPLE
\
INDU CED
STAN DING
WA V E
--
X
D ELAYEDXA
P ROBE
P ULSE
P
EXCITATION
PULSES
Figure 2.1. Schematic diagram of the ISS experiments. The crossed excitation pulses "impulsively"
excite a material response in the sample the time evolution of which is monitored by diffraction of
variably delayed probe pulses.
-12-
and take one snapshot
just ask the jumper to jump 100 times,
each time at a different delay relative to the start of the
jump.
time,
If the jumper
jumps with exactly the same motion each
the 100 snapshots put together should be the same as
The latter method is
taking 100 exposure during one jump.)
necessary when the coherent material motion is too fast to be
resolved by available electronic means.
pulsed probe is assumed,
In this thesis, a
although many theoretical conclusions
are also valid for CW probing.
To
illustrate the ISS technique, we show several examples
of data.
Figure 2.2 shows impulsive stimulated Brillouin
scattering
(ISBS)
optical glass.
data from coherent acoustic phonons in
These data first demonstrated mode-selective
optical excitation of coherent longitudinal and shear
ultrasonic waves in bulk media.
In Fig.
2.2a,
time-dependent
diffraction from a longitudinal ultrasonic wave is shown.
The
two picosecond excitation pulses which generated the acoustic
wave were polarized parallel to each other and vertically (VV)
relative to the scattering plane.
The incident and
diffracted probe pulses were also parallel (V-V) polarized.
ISBS signal from a transverse acoustic wave is shown in Fig.
2.2b.
In this case the excitation pulses were polarized
perpendicular to each other, vertically and horizontally (V-H)
relative to the scattering plane.
The incident and diffracted
probe pulses were also perpendicularly (V and H,
polarized.
respectively)
These ISBS experiments are impulsive stimulated
analogs of spontaneous polarized (V-V). and depolarized (V-H)
Brillouin scattering.
From the time-dependent data and
(
measurement of scattering geometry, the speeds of sound, v. =
&g/q 0 , where o, is the angular frequency of acoustic mode
and q 0 is the scattering wave vector,
shear elastic
calculated.
constants Co = pv 2 where p is the density, were
The the apparent decay of signal and rise of
baseline are artifacts due
resolution.
and longitudinal and
to insufficient wave vector
Detailed discussion of this problem are presented
in chapter 3.
-13-
PICOSECOND ISBS INTENSITY
BK-7 GLASS
LONGITUDINAL
425 MHZ
I
I
I
266 MHZ
(a)
-j
I
II
I
TRANSVERSE
0
I
I
I
I
2
3
4
I
I
(b)
I
I
I
I
5
6
7
8
TIME (NS)
Figure 2.2. Impulsive stimulated Brillouin scattering data from ultrasonic waves in BK-7 optical
glass. excited and probed with 80 ps pulses. The time-dependent oscillations correspond to acoustic
standing-wave oscillations in the glass. (a) V-V polarized excitation pulses generate a longitudinal
acoustic wave. (b) V-H polarized excitation pulses generate a transverse acoustic wave.
-14-
Figures 2.3 and 2.4 show ISBS data from crystals of KDP
(KH2 PO 4 ) and KD*P
(KD 2 PO 4 ), respectively,
structural phase transition (spt)
near
their
temperatures Tc.
Although
the results of ISBS studies of spt's will not be detailed in
this thesis, the data illustrate some of the capabilities of
the technique.
In Fig.
acoustic mode as T ->
with 7*
2.3, the softening of the shear
Tc is measured.
The data were recorded
scattering angle and V-H excitation pulses.
In KD*P,
similar data are recorded with several scattering angles.
At
small angles (10*) mode softening is similar to that in KDP.
At large angles,
less mode softening occurs because wT->l,
where -r is the order parameter relaxation time.
attenuation also becomes very strong.
several
These data illustrate
important capabilities of ISS.
low frequencies
(by light
characterized.
Second, a wide
be used.
The acoustic
First,
scattering standard)
range of
modes of very
can be
scattering angles can
This permits investigation over a wide range of
acoustic frequencies which typically bridges the gap between
ultrasonics and conventional Brillouin scattering (
5 GHz
).
Third, very heavily damped or overdamped vibrational modes
which pose problems
for ultrasonics or conventional
LS methods
can be characterized by ISS.
In Fig.
2.5, impulsive stimulated Raman scattering (ISRS)
data from coherent optical phonons in a crystal of terbium
vanadate (TbVO 4 ) is shown.(Farrar et al.
1986b)
Femtosecond
excitation and probe pulses were needed to excite impulsively
and time-resolve the motion of the 122. cm- 1 mode.
data from molecular vibrations is shown in Fig.
In principle any type of Raman-active mode,
electronic,
scattering intensity in ISS is
I (q, t)
5.3A.
including
rotational, spin and other excitations,
coherently excited and probed.
cc
IG c (q, t) 12
-15-
Similar
can be
'The general expression
for
the
ISBS
KDP
-
DATA
24
131.7 K
167 MHz
D
2400
123.0 K
93MHz
z
z
122.4 K
58MHz
U
<~
0
-11000
3
6
9
TIME (ns)
12
15
18
Figure 2.3. Temperature dependent ISBS data from the C 66 acoustic mode in KDP. Scans are taken
near the phase transition temperature, Te = 122 K. Perpendicularly polarized 1.06 pm excitation
beams were crossed at an angle 9 = 7.060 giving an acoustic wavelength X = 8.64 pm.
-16-
K
D*P
I
e0-o.o
S BS
E - 31.1*
AT = 19.69K 1
W = 3.05 nsy = .02 ns- 1
D AT A
8- 60.0*
AT = 19.04K
W 2 9.38 nsN
.11 ns-
=
AT - 20.84K 1
w = 17.6 ns-1
= .30 ns-
7.1y
60
'V
640
AT
to
y
2.17K
- 2.07 ns- 1
-
t
* IL
6
260
f
3.65K
7.09
ns-
1
0
I
2
3
.86 ns-I
560
A
x1o
.2 ns
640
AT - .01K
y
-
-1800
1.21 ns-1 1
X 100
---AT - .20K
w a 23.71ns-I
1500y - . 7 7 ns- 1
-4420
=
ns
100
X10
w
5
4
2.04K
13.6 nS5.7 ns-
AT
a
y
-
AT
.27 ns-
AT
2
5
XI0 0
y
A
-. 08K
.18.0
u7.1
z2.6
x 5 2500
2
3
4
5
0
1
2
T
I
M E
4
3
(N
5
6i
2
.3 .4
.5 .6
.7 B
.9 1.0
S
)
0
Figure 2.4. Temperature dependent ISBS data from the coupled C 66 acoustic and P3 polarization
modes in KD*P. Data scans are shown for three scattering angles (E is the angle between the
excitation pulses) and at various temperatures near T,.
-17-
TbVO 4 ISRS
122 cm-I
OPTIC
8.6 P 5
DATA
PHONON
DEPHASING
TIME
(/)
z
LI
z
I
I
I
I
2
TIME
1s
(p S)
4
5
Figure 2.5. Impulsive stimulated Raman scattering data from optic phonons in TbVO 4 crystal excited
and probed with 70fs pulses. The 3.65 THz oscillations correspond to optic phonon standing-wave
oscillations in the crystal. (Data taken by L.R. Williams.)
-18-
where Gec is the dielectric response function of the material.
In comparison, for frequency domain spontaneous light
scattering,
the expression is
Im[---CI([G E
)
KBT
I (q, w)
This means the information content of data of ISS and LS is in
principle the same.
The detailed derivation of these
equations and comparison of time domain and frequency domain
light
scattering methods are presented in chapters 3 and 4.
2.2 Experimental
Setup
The principle
many ways.
Fig.
illustrated in Fig.
2.1 can be
realized in
2.6 shows one version of the experimental
setup we have used for the picosecond time
This is discussed further in chapter 6.
scale experiments.
A CW Nd:YAG laser is
Q-switched and mode-locked to produce a 1.0641pm light
wavelength output consisting of about 30 pulses separated from
each other by about 9 ns. The pulses are 85 ps in duration and
the largest ones contain about 80pJ of energy.
train" repetition rate is typically 500 Hz.
separated out of the pulse
The "pulse-
Three pulses are
train for use in the experiment.
The two excitation pulses are overlapped spatially and
temporally inside the sample.
The mechanical delay line
provides a maximum delay of
~ 20 ns for the probe pulse, but
total delays of >200 ns can be achieved by selecting later
pulses in the pulse-train for the probe.
For small scattering
angles,
all three beams are focused to the sample using one
lens, as shown in Fig. 2.6.
separate lenses are used.
For large scattering angles,
The chopper on one of the
excitation laser beams and lock-in amplifier are needed to
remove elastically scattered light.
The light spot sizes must be chosen with some care.
While
tightly focused spots improve signal/noise by increasing the
-19-
SINGLE PULSE
SELECTORS
EXCITATION PROP
1~iti
1.064,4m
PULSES
'"
-
YAG LASER
MODE-LOCKED
--SWITCHED
PULSE
SAMPLE
-CP
ISHG
0.53
LOCK-IN
AMPLIFIER
DELAY LINE
COMPUTER
Figure 2.6. ISBS experimental setup. See text for details: SHG=second
harmonic generator;
PD=photodiode; CP=chopper.
-20-
excitation pulse intensity,
uncertainty.
they result in large wave vector
To gain good wave vector definition while not
significantly reducing signal/noise, we use cylindrical lenses
to focus the excitation pulses to oval spots measuring about
0.2mm high x 2mm in the direction of q 0 .
sufficiently well-defined wave vector.
This results in a
In thick samples,
the
reduced excitation-pulse intensity due to the use of wide
beams does not lead to reduction in signal intensity because
it
is compensated by the
increase
in grating length.
However,
bigger spot sizes are more susceptible to sample
inhomogeneities
and surface imperfections and therefore the
signal is more noisy.
The diffracted signal
is usually detected by a photodiode
detector whose output is averaged by a lock-in amplifier and
stored in an on-line computer.
Scans such as those shown in
Fig.
2.2
consist of about 105
repetitions of the excitation-
probe pulse sequence (at 500Hz), with probe pulse delay
gradually varied.
The entire data acquisition process is now
computer controlled.
Accurate measurement of angles between laser beams is
carried out as follows:
a mirror with positional and angular
adjustments is placed at the point where the laser beams
cross, i.e., at the sample position.
The mirror angles are
adjusted such that first one beam,
then the other,
reflected back along its incident path.
is
The difference
between the readings give the angle between the beams. At
present, the accuracy of angular measurements is limited by
the accuracy of the rotation stage, about 0.3% in most cases.
The beam quality should be carefully maintained, It is
helpful to avoid tight focusing along the pathway to the
sample.
The overlap of three beams should be carefully
maximized, making sure they lie in the same plane.
especially important when oval spots are used.
-21-
This is
CHAPTER 3.
GENERAL THEORY OF IMPULSIVE STIMULATED LIGHT SCATTERING
3.1. Introduction
Early reports on impulsive
stimulated Brillouin and Raman
scattering experiments included rather limited theoretical
descriptions which applied to the specific material modes
under investigation (Nelson et al, 1981; Nelson,
1982).
Little or no consideration was given to experimental
limitations in time or wave-vector
resolution or
complications which can be significant.
to other
While detailed
presented,(Shen & Bloembergen,
Mukamel,
1965;
Shen,
1985a; Mukamel & Loring, 1986)
with the unique
excitation.
consequences of
1984;
Loring
&
theories of stimulated light scattering have been
these have not dealt
temporally impulsive
In this chapter we give a more detailed
theoretical treatment of the ISS method than has been
presented previously,
in the belief that such information is
important to explore the method's full potential.
This
chapter presents a consolidation and unification of earlier
theoretical descriptions of ISS, as well as significant new
results.
In the next section, general theoretical background is
presented.
In Sec.
3.3,
the
ISS experiment under conditions
of ideal time and wave vector resoluti-on is treated.
3.4, the results of Sec.
3.3 are applied to calculate the
forms of time-dependent ISS
phonons,
Sec.
3.5,
In Sec.
signal from optic and acoustic
simple relaxational modes, and coupled modes.
In
several important results concerning experiments
under nonideal conditions are treated.
The limitations in
time resolution due to finite laser pulse durations are
calculated,
and the influence of experimental geometry on time
resolution in the femtosecond regime is discussed.
We show
that the probe pulse duration can affect not only the
-22-
time-dependence, but also the frequency content, of coherently
scattered light.
Our results indicate that by monitoring
selected spectral components of the diffracted light,
significant improvements in the time resolution of detection
should be possible.
to the
Limitations
in wave vector
resolution due
focussing of excitation and probe pulses to finite spot
sizes are treated in detail.
These are especially important
in cases of dispersive material modes, and the consequences
for acoustic phonons are treated explicitly.
The ISS experiment is a time-domain,
frequency-domain,
spectroscopy.
stimulated analog of
spontaneous light scattering
(LS)
Most of the theoretical treatment that follows
is closely analogous to theories of spontaneous light
scattering.
In the next chapter,
a detailed comparison
between ISS and LS is carried out.
illustrate
Simulated data are used to
situations in which either the time
or frequency
domain approach may be advantageous.
3.2
General
In this section, we give a simple macroscopic treatment of
the ISS experiment.
The sample is considered a continuous
medium,
and all material quantities are understood as ensemble
averages.
A. Excitation Process
The Hamiltonian for the medium under the
influence of the
excitation pulses can be expanded in powers of net electric
field of the light pulses:
H = H0 + H1 + H2 +
.-.
(3.2.1)
.
H 0 is the Hamiltonian of the isolated system. H 1 represents
the first-order interaction between the optical electric
-23-
field,
E, and the linear part of the dipole moment represented
by the operator P:
H1
=
E Pi(r) Ei(r,t) d 3 r
-
(3.2.2)
.
i
This term describes the absorption of light.
Transient
&
grating and four-wave mixing experiments based on this effect
have been discussed extensively(Eichler et al. 1986; Loring
Mukamel 1985b; Fayer 1982).
Since we are concerned with
stimulated scattering processes,
term in this chapter.
experiments,
The
we will
not focus on this
This effect is important in some of our
and will be discussed in chapter 6.
second-order nonlinear interaction between light and
matter is described by
H2(t)=
- -1
-
=
d3r
E 6cij(r)Ei(r,t)Ej(r,t)
d 3 r E Scij(r)
Fij(rt)
(3.2.3)
ii
where
-Ei(r,t)E
Fij(rt)
and Scij(r)
(r,t)
(3.2.4)
is the dielectric operator.
describes light scattering processes.
order terms.
H 2 (t)
__
(2n) 3
Eci(q)Fi(-q,t),
ij
where the dielectric operator,
and Fij(q,t)
We will ignore higher
H 2 (t) can be expressed in wave vector space as
= -
Scij(q)
It is this term which
=
fd 3 r e-iq-rSe
(3.2.5)
Scij(q),
j(r),
is defined similarly.
is defined by
(3.2.6)
In our treatment we shall
assume that the changes in properties of excitation and probe
pulses due to interaction with the medium are small enough to
be ignored. We therefore need not solve the coupled electric
-24-
field and material response problem, and can treat ISS
excitation and probing processes separately.
Linear
basic
response theory (Reichl,
result for the dielectric
response,
Se(r,t):
E GCCi(r-r',t-t')Fkl(r',t'),
dt'd3r'
=
Seij(r,t)
1980) yields the following
(3.2.7)
or,
in wave vector space,
8sij(q,t)
=
dt'
--
E GC~ijkl(qt-t')Fkl(q,t')
kl
where Gee is the impulse
for
(3.2.8)
,
response function (Green's Function)
the dielectric tensor.
Causality requires that Gce(t-
t') = 0 for t<t'.
Eq.(3.2.7) or
(3.2.8)
is a general
result in that the
material modes which give rise to Seij(t) have not been
specified.
Of course our ultimate concern is description of
material dynamics, and this requires knowledge of the
connection between material displacements and the dielectric
tensor
components.
In the following,
we will consider only
modes which are active in first order light scattering,
i.e.
modes whose displacements are linearly coupled to the
dielectric constant.
Seij
In such cases,
= E axijQ,
(3.2.9)
where Q" is the displacement operator of normal mode a.
dielectric constant derivatives aa
.('gj
a)
The
can in some
cases be related to single molecule polarizability derivatives
with local field corrections(Shen, 1984).
media,
For isotropic
ae/8Q can be related to the total spontaneous
scattering cross section (Kaiser & Maier,
1972).
that when nonlocal effects are not important,
Note also
the Green's
function in r-space can be written as
G(r-r',t-t')=G(r,t-t')6(r-r').
-25-
(3.2.10)
Substituting Eq.
and
for Sci
(3.2.9)
into Eq.
repeating the steps leading to Eq.
Q'(t)
where G01
yields
(3.2.8)
E Gcxl(t-t'r) Fl( t,')
dt'I
=
(3.2.3) or (3.2.5)
response functions
are the
(3.2.11)
for the material modes
and FO are the forces exerted by the excitation pulses on the
material modes:
Fa(t)
=
E a i *Fij(t).
(3.2.12)
1J
The dielectric response function is related to material
response functions
Gc~ijkl(t) =
Equations
through
(3.2.13)
E ami a~k Ga(t).
1
(3.2.1l)-(3.2.13) hold for either (r,t)- or
(q,t)-
The results of this section are general
dependent quantities.
in that the temporal and spatial characteristics of the
excitation fields and the material modes which are excited
In sections Sec.
have not been specified.
3.3,
3.4 and 3.5
specific forms of excitation fields and specific material
modes will be considered.
B. ISS probing process
From Maxwell's equations, we get the general equation
governing the scattering process:
E[-k2iij+kik
CO ij a2
-
c0 2
j
where,
-
]E
(k t)
at 2
E is the total field, c 0
vacuum, cc
4n
32
c0 2
at 2
-
Pi(k,t),(3.2.14)
is the speed of light in
is the unperturbed dielectric constant tensor and P
is the polarization due to deviations in dielectric constant,
Sc(t),
which resulted from ISS excitation.
In this chapter,
we take the small scattering efficiency limit, which is valid
for most
ISS experiments.
At this limit,
-26-
=
P(r,t)
Since E
Se(r,t)-E
-
is the electric field of the probe pulse and
Eq.
satisfies homogeneous equation,
C
E[-k26ij+kikj-
ij 92
4n
2
]Esj(k,t)
-
(3.2.14) becomes:
c0 2 at 2
where Es is the scattered field.
Eq.(3.2.15)
Pi(kt)
Pi(kit),(3.2.16)
=-
c 0 2 at 2
j
Fourier transformation of
gives:
-
-
Z
(3.2.17)
d3q Si(q,t)Epj(k-qt).
(2n) 3
j
From Eq.
(3.2.15)
(r,t).
(3.2.16), the scattered field Es is given by
Esi(k,t)= 4n
E fdt'
GE..(k,t-t')
c0 2 j
a Pj(kjt'),
at' 2
where the Green's function for the electric
(3.2.18)
field,
GE,
is
given by the solution of the wave equation
E [-k28ij+kikj-
2
c 02
(3.2.19)
GE jk(k,t)-8ikS(t).
-O
a
at 2
(3.2.18) by parts yields
Integration of Eq.
Esi(kt)=
ij
c 02
tdt'
j
-c
a
at 2
GEi
(ktt')Pj(kt').
(3.2.20)
In getting this result, we used the fact that GE(t-t')=0 for
t'>t and that P--+0 when t'---.
is retained in Eq.
(3.2.20).
Also,. only the radiation part
For isotropic media, the Green's
function is given by
GE(krt>
-_)=
)sin
where I is a unit tensor,
medium, and w(k)=ck.
(3.2.21)
(k)t.,
c is the speed of light in the
For anisotropic media, GE contains two
terms, corresponding to two optical normal modes.
In practice,
only one term will be involved in a single experiment,
write the Green's function containing only one mode:
-27-
so we
GE(kt>0)= -
C
T(k) sinw(k)t,
w(k)
(3.2.22)
where the existence of non-unity tensor T(k) means that in
anisotropic media, the polarization of the radiation is in
general not the same as the direction of the radiating dipole.
Appendix A
gives more details on the calculation of T(k).
Substitution of
Eq.
(3.2.22) into Eq.
E5 (kt)= 4n dt'D(k,t-t')-P(k,t')
D(kit)= w(k)
CO
Eq.
(3.2.23)
(3.2.23) together with Eqs.
form the basis for
For some applications,
calculating the
The equation corresponding
E i(r
4n
to Eq.
(r-r',t-t')
Odt'd3r'GE
(3.2.8) and
scattered field.
it is more convenient to work
r-space.
t)=
;
sin[w(k)t]T(k),
where C 0 =(c 0 /c) 2 .
(3.2.17)
(3.2.20) yields
(3.2.18)
a2Pj(r',t').
in
is
(3.2.24)
For isotropic media,
-
GE(r-r',t-t')=
4nr
,
_
-
r'|).
(3.2.25)
For anisotropic media, this is no longer true. Along different
directions, there are different sets of optical normal modes,
with different polarizations and speeds.
For our
applications, the light beams are reasonably collimated, and
the difference in speed and polarization within the angular
divergence of a single beam are small enough to be neglected.
We can thus use the Green's function above for one normal
mode:
GE(r-r',t-t')= -
4nTr T r'
where TO=T(k0 ), and k 0
Es(rt)
(3.2.24)
= -
1
and (3.2.26) give
d 3 r'dt'
rI),
the central wave vector.
(t-t'-
)
(3.2.15),
is
6(t-t'- Ir -
c
4nc2
-28-
(3.2.26)
Eqs.
To
a
Ir-r'I
t'r2
2
x
Since
(3.2.27)
[Se(r',t')-E(r',t')]
Se varies with time much more slowly than the
of light, we can replace 82 /9t'
2
by
-
w2,
frequency
with w=ck 0 , to
yield
E5 (r,t)
=
2
4nc 0
x
Eq.
f
d 3 r'dt'
jr-r'
8(t-t'-
r
- r'|)
(3.2.28)
-[8c(r',t')-E(r',t')]
(3.2.28) will be used in the next section.
Coherent scattering or
"diffraction" from phase coherent
material excitation has been treated extensively(Eichler et
al.
1986).
Our treatment is distinguished from others in that
the
fact that material excitation is excited and probed by
short pulses which travel in space and time is treated
explicitly.
The general formalism given here allows us to
treat complicated situations.
3.3 Impulsive Limit
When the time scale of excitation is much shorter than the
time scale of material modes of interest, we can approximate
the excitation force as a 8-function in time compared with
material response function.
In addition, when the laser spot
sizes used are much larger than the wavelength of material
excitation, we can approximate them as plane waves.
the ideal situation for the ISS method.
experiments,
This is
In most ISS
one tries to be as. close to this ideal situation
as possible.
The excitation force is then a 6-function in
time and wave vector:
-
Fkl(q,t)=Akl8(t)[8(q
Akl= (2n) 3
22 cOSe
q0 ) + S(q + qO) +
Ukl ,
2
8 klS(q)]
;
(3.3.1)
(3.3.2)
-29-
where q 0 equals the wave vector difference of the two
excitation pulses and I is the total energy per excitation
If the excitation pulses have unequal energies
pulse.
12,
Il and
U is a tensor
then I should be replaced by (I1I2)1/2.
determined by the polarization state of the two excitation
pulses and Se is the excitation pulse spot area, defined below
by Eqs.
(3.5.7) and (3.5.15), respectively.
of light in vacuum.
c0
is the speed
The propagation time of the excitation
pulses through the sample has been neglected in Eq.
but
is treated in Sec.
3.5.
proportional to Skl 8 (q)
The term in Eq.
(3.3.1),
(3.3.1)
represents a spatially uniform
excitation force which gives no contribution to coherent
scattering when probed at the Bragg angle.
include this term below.
However,
We will not
this term does give rise
to
a spatially uniform response which can be detected in other
ways,
it is discussed in chapter 5.
Substitution of
Eq.
(3.3.1)
into Eq.
(3.2.8) yields the
dielectric response:
88ij(q,t)= E Akl[Gccijk(qot)S(q-qo)
kl
+ GCeijkl(-qOt)8(q+qO).
(3.3.3)
To find the scattered field in the impulsive limit is
difficult in k-space,
(3.3.3).
i.e. with Eqs.
(3.2.17),
(3.2.23), and
This is because although the 8-function
approximation in wave vector is valid in the sense that the
spot size is big compared with the wavelength of material
excitation, the spot size is still small compared with the
distance
from the sample to the detector.
make approximations with Eq.
(3.2.28)
It is easier to
that are consistent with
the 8-function approximation made in Eq.
(3.3.1).
In Sec.
3.5, when we treat nonideal situations in which the finite
sizes of the laser pulses are explicitly written out, we shall
use the k-space formalism which furnishes the convenience of
integration over all space.
In Eq.
(3.2.28), the probe field can be written as:
-30-
Ep(r',t')=E(r'-ct')exp[i(kp-r'-&t')]
(3.3.4)
+ C.C.
where c=ckp and E(r'-ct')
is the probe pulse profile.
The
probe can be a short or long pulse, or continuous wave.
Choosing the coordinate system such that the scattering volume
is centered at
Ir
-
r'I =
Ir -
Irl
= r -
r'
r 0 -O,
and using the standard approximations
= r in the denominator of Eq.
r-r'
(3.2.28) and
in the argument of the exponentials, we
find
Es(rjt)=
W22
4nco r
d3r'TO-8c(r',t p)exp(-iwt p- iq-r')-E(r'-ct P)
+ C.
where q=ks-kp,
ks=kpr,
C.
and t p is
(3.3.5)
the time when scattered light
detected at location r and time t arrived at
t-tp=r/c.
the sample,
i.e.
Since the scattering volume is finite, there is a
range of arrival times.
To approximate it as a single time as
we have done here is valid when the time of light passage
through the scattering volume is sufficiently short compared
with the time scale of coherent material motion (e.g., short
compared with a single vibrational period).
More careful
analysis appears in Sec. 3.5 where we consider nonideal
situations.
We further make the approximation that E(r'-ctP)
varies with r' slowly compared with SE(r',tp)
words,
.
In other
the laser spot size is much larger than the excitation
interference fringe spacing. Then
(3.3.6)
E(r'-ctp)= E(r 0 -ctP),
where
r0
is the center of the scattering region.
chosen r 0 =O.
Substituting Eq.
(3.3.6)
into Eq.
We have
(3.3.5), we
get
Es(rt)=
(A2
4nco2r
exp(-iwt ) TO-8c(q,t
f
is
The diffraction efficiency is
-31-
)-E(r 0 -ct
) + C.C.
(3.3.7)
Srk
4
)-E(ro -ct
d-TO-S(q,t
n2
)] 2
SOIE(r 0 -ct)12
0_]2__
= ke
[d-TO-Sc(q,t p)-ep2
4ncor)2
k 2 L2
=2
Xr
)2
SOA
3
2
(3.3.8)
(4co)A
where d is the tensor characterizing the polarization
selectivity of the detection system, defined from
Sr is the spot area of
E(transmitted)=d-E(incident);
scattered light at distance r from scattering region, and So
is the spot area of the probe beam at the scattering region.
X is the wavelength of probe light in the medium, ep=Ep/IEpl,
Se' is equal to Se divided by the
scattering volume, and L is
the path length of the probe beam in the scattering region, Se
= d-TO-cf'(q,tp)-ep is the relevant projection of the
dielectric tensor. Eq. (3.3.8) is just the well known grating
diffraction efficiency formula (Siegman, 1977).
In the
derivation leading to Eq. (3.3.8), we neglected the difference
in the speed of light between incident and scattered probe
beams. This can lead to a small modification of Eq. (3.3.8)
but we will not delve into it here. Substitution of Eq.
(3.3.3) into Eq. (3.3.8), assuming that the probe beam is
incident exactly at the Bragg angle, and using the relation
S(q-q0)=V/(2n)3 qO
with V the scattering volume, gives
k2 L 2
I
(-)2
Y1=
2cOSe
(4c0)2
x
1 [ E dnmTomiGccijkl(qtp)Uklepj)2.
n mijkl
(3.3.9)
Equation (3.3.9) relates the time dependence of ISS signal
to the time-dependence of Gec, which relates to the material
dynamics through Eq. (3.2.11) and (3.2.13). It also shows that
-32-
ISS signal intensity, I(q,t), depends quadratically on
excitation pulse energy.
In many ISS experiments, only one
independent component of G88 is probed.
In general,
n(q,t)
can be viewed as proportional the square of a projection of
the Gce tensor.
For different experimental situations
(polarizations of excitation,
projections are taken.
scalar symbol GCE
I(q,t)
probe and detection),
In the following, we will use the
to denote the projection sampled,
Gee(qt)1
ISS signal
(3.3.10)
square of a
impulse response function.
that heterodyne detection methods
to GCC
In the ideal
is proportional to the
projection of the material
applied to ISS,
i.e.
2
This is the main result of this section.
situation,
different
(Eichler et al.
1986)
We note
can be
in which case the signal would be proportional
itself.
3.4 ISS Experiments on Optic and Acoustic Phonons,
Relaxational Modes, and Coupled Modes
ISS experiments carried out to date
Ruhman et al.
(Yan et al.
1988;
1987) have involved optic and acoustic phonons,
intramolecular vibrations,
orientational motions of liquids,
and several combinations of these modes. Here we derive the
time-dependent forms of ISS signal for some of these cases,
assuming ideal excitation and probe conditions.
Our purpose
is to illustrate the application of the general theory to
various specific cases of immediate interest.
A. Optic Phonons
For a nondispersive optic phonon mode a,
motion is
-33-
the equation of
P(a
Qt + 2yC at
;t 2
at
Q( 0
2
)+
OCO
ij
(
aQ()
(3..0
is the corresponding inertia density,
where po
(3.4.1)
)oFij
wa0
is the
natural frequency, and y, is a phenomenological damping
constant.
Since we have assumed that the mode is
Q(0)
nondispersive,
can be in either q-
(3.4.1)
The corresponding Green's function is
r-space.
space or
in Eq.
and Fij
determined from
a( + 2
P
2 0 )G(O)(t) = 8(t)
a +
The solution for underdamped modes
GINx)(t>O)
= e_(t
s n wX
(w2
Q
(3.4.2)
.
-
2
= (
2 >
0 ) is
)(3.4.3)
shows that impulsive excitation produces a damped
This
standing-wave oscillation.
(XO 2
-
The
solution for overdamped modes
Y 2 < 0) can be written as
G(()(t>0)
=
e Ycxt
e-Y2t
-
(3.4.4)
l)
-
where
Yax2,cl = Yx
In this case,
(Yx2
-
Wao2)1/2
impulsive excitation leads to an increase in
Q(0) after t=0 followed by monotonic, nonoscillatory return to
equilibrium.
In either case, Eq.
(3.2.13) shows' that Gec = G(0).
time-dependent ISS signal is given by Eq.
I (q, t)
GIx) (q..t) 12
G
The
(3.3.10) as
.(3.4.5)
Thus ISRS signal from a single underdamped optic phonon mode
oscillates at twice the phonon frequency and decays at twice
the dephasing rate.
ISRS signal from overdamped optic phonons
rises after t = 0 to a maximum,
-34-
then decays monotonically.
Simulations of ISRS data are presented and compared to LS
spectra in the following chapter.
B. Acoustic Phonons
The wave equation for acoustic displacement, u,
driven by
ISBS excitation can be written in the form
; 2 u,
2 ul
E
1
jkl
8
a
n Pklji axi DkD
jkl
jkl ijkl axjaxk
-t2
Kklji
jkl 8naJ
a
EkEl
(3.4.6)
,
where p is the mass density, Cijkl are elastic stiffness
constants,
Dk =
E
Pijkl are photoelastic constants,
and
ekmEm
m
is the electric displacement.
constants
We have defined new coupling
K to relate the acoustic
electric field.
response directly to the
The coupling constants are defined in terms
Sij-1/2(aui/axj+auj/8xi),
of acoustic strain,
inverse dielectric tensor,
B,
or
and in terms of
the dielectric tensor,
e,
respectively:
Pijkl = aBij/aSkl
(3.4.7)
Kijkl = aeij/aSkl
For an acoustic phonon with wave vector along an arbitrary
direction of a crystal, there are thre'e eigenmodes,
one
longitudinal or quasilongitudinal and two transverse or
quasitransverse.
We
label these eigenmodes by a (a=1,2,3)
and
the corresponding eigen-displacements by u":
um =
E b
ui
(3.4.8)
,
i
where the unitary matrix {bi}
of q.(Auld, 1973)
is a function of the direction
The equation of motion for these eigenmodes
is
-35-
a2 uc=
-2uo
=
t
1
b
ijkl 8n Kklji
axE
is the distance measured along the direction of q.
where
term 2pycSx,
S-=aux/a& and adding a damping
Introducing
the
equation above becomes
9 2 so
2 +2pycSc
p
atBE
a2 Sa
Ck(
i axkl(
92
kl
a2
2
a&
(3.4.9)
)Fkl
where
kli Kklji bni qj
a
(3.4.10)
=
The equation of motion and its solutions for the acoustic
vibrational Green's function are
identical to Eqs.
(3.4.2)-
(3.4.4) with po replaced by p/q 2 and with the natural
frequency given by wC0
2 =COq 2 /p.
Thus the ISBS signal from
underdamped or overdamped acoustic phonons has the same
form
as that for ISRS signal from optic phonons.
C. Debye relaxational modes
For nondiffusive relaxational modes, the driven equation
of motion is
jQ
+
1 Q(C)
=
(
1
)oFi
,
(3.4.11)
where Xu is the susceptibility and T, the relaxation time.
The Green's function is
G()(q,t>0)
In this case,
= X
e t/.cx
(3.4.12)
impulsive excitation results in an instantaneous
rise of ISS signal followed by exponential decay back to
equilibrium at twice the decay rate of the material response.
Eq.
(3.4.12) is often used to describe approximately the
motion of a non-oscillatory mode
(such as an overdamped
vibrational mode) whose initial response to an impulse driving
force is not actually instantaneous, but is rapid compared to
-36-
the subsequent return to equilibrium.
chapter
that the ISS method may be
It is shown in the next
better suited than
frequency-domain LS spectroscopy to resolving such short-time
dynamics which reveal the inertial,
relaxational,
rather than purely
character of the mode.
Time-domain ISS observations of relaxational modes have
been carried out by many investigators on time scales ranging
from picoseconds to seconds
(Eichler et al.
1986).
These
experiments have often been labeled "forced" Rayleigh
scattering,
time-delayed four-wave mixing,
etc..
We
distinguish between these and other dynamic grating
experiments on relaxational modes which have involved first
order
excitation processes,
heating
i.e.
optical absorption and
(often called "forced thermal Rayleigh scattering")
(Eichler et al. 1986).
D. Coupled modes
As an example of coupled modes, we treat a system with
bilinearly coupled, LS-active acoustic and relaxational modes.
Coupling of this kind is typical of piezoelectric solids and
many other condensed materials.
For simplicity, we will use S
and Q to denote acoustic and relaxational mode responses,
respectively, and all the indices will be dropped.
The
equations of motion are thus:
(2
P (--
at
aQ +
at
C 82
a2
)S + b _7Q
ax
a
a
+ C
+ 2y-2
W
Q
-
IT-
bS = A
-r
(ac
aQ
2
at2
+2y
2
at
Denoting
C 2+
)
q
(3.4.13)
a, T, etc.
are the
(3.4.9) and (3.4.11) with subscripts and
superscripts omitted.
LS
2 aF;
a
)0 F
where b is the coupling constant and X,
same as in Eqs.
a2
-37-
+
I(
LQQ
and
LSQ = LQS
=
-b
the components of the Green's functions G40(q,t)(superscripts
take on values S and Q) are determined from:
where
= I 8(t)
(3.4.14)
,
LG(q,t)
I is a unit matrix.
The solution of Eq.
(3.4.14)
is
either
= A
G04(qt)
e yt
+ B
+
sin(i't
,eY2t
(3.4.15)
ao)
or
e Yt + B
Ie
y2 t+ Dace Y3t
(3.4.16)
,
Gx((qt) = A
depending on whether or not the vibrational part of the
remains underdamped.
response
etc.,
can be
related to the parameters in Eq.
analytic expressions.
=
GEC(qt)
The quantities yi,
K2 G5 S
Eq.
(3.2.13)
Y2,
(3.4.13) through
gives
+ K(Be/3Q) 0 (GSQ + GQS)
+ [(ge/gQ) 0 ] 2 GQQ
(3.4.17)
The square of this expression gives ISS signal which,
for underdamped modes,
I(q,t)
=
takes the
[Ae yit + Be-Y2tsin(w't+
*
where A,
B, and
(3.4.15)
and the coupling constants,
(3.4.17).
)]2
for
form
(3.4.18)
,
example
Aap,
depend on the amplitudes and phases in Eq.
ISS signal
from coupled
K and
(9c/9Q)O,
acoustic and
in Eq.
relaxational
modes shows both damped oscillatory and relaxational features.
ISS data of this type have been reported (Yan et al.,
1988,
Farrar et al. 1986) and simulated data is shown in the next
chapter.
3.5 Nonideal situations
-38-
A. Qualitative discussion
In any real experiment, the light pulses used have finite,
not
infinitesimal,
infinite,
durations.
The
spot sizes are finite,
so the fields are not pure plane waves.
not
In fact,
limited time and wave vector resolution are the major limiting
factors in ISS time-domain light scattering,
just as limited
frequency and wave vector resolution are the major limiting
factors in frequency-domain LS spectroscopy.
A clear
understanding of how these factors influence the result of an
ISS experiment is important for designing experiments with
requirements and for the correct interpretation of
conflicting
experimental
of
results.
We begin with a qualitative discussion
the complications arising from finite spot sizes and pulse
durations.
We see from Eq.
(3.3.9)
that the scattering efficiency is
inversely proportional to the excitation spot area.
reason,
For this
given some limited laser pulse energy, one would often
like to decrease the spot sizes in order to increase signal
intensity.
However, with smaller spot sizes,
the range of
wave vectors of material modes excited is larger.
When
studying material modes whose temporal behavior is wave vector
dependent,
some loss of resolution will occur.
Acoustic phonons comprise one such example.
The
excitation force excites a standing wave packet (see previous
section on ISBS), the dimension of which along the wave vector
direction is about the size of the
excitation spot size.
The
standing wave forms a time-dependent grating which gives rise
to signal whose intensity oscillates at twice the acoustic
oscillation frequency. However,
the standing wave is a
superposition of two counterpropagating traveling waves which,
during the course of a vibrational period, alternately add
constructively (to yield the maximum standing wave amplitude)
or destructively
(to yield zero net strain).
Gradually,
the
two waves travel apart from each other and cannot add
effectively or cancel each other completely.
-39-
This gives rise
to a reduction of peak ISS signal and a rising baseline, as
can be seen in Fig.
3.la.
Furthermore,
the
two traveling wave
packets gradually propagate out of the probe
region, giving
rise to additional overall decay of signal.
These effects are
the counterparts of the spectral line broadening and
distortion of Brillouin lines which can occur due to finite
collection angles and spot sizes
in Brillouin scattering.
These effects can reduce the accuracy of acoustic attenuation
measurements.
Second, the pulses have finite pulse durations.
lead to
reduced excitation efficiency and time
Several additional factors can become
femtosecond regime.
This can
resolution.
important in the
When the excitation pulses overlap for a
time which exceeds their own duration,
the time resolution
depends not only on pulse duration but also on experimental
geometry. Also, when the spatial length of the excitation
pulses is smaller than the spot sizes,
the region of overlap
between them may become significantly smaller than the spot
size.
This geometrical effect limits the scattering angle one
can use.
In this section we treat the effects of finite time and
wave vector
resolution on ISS signal.
In connection with
finite pulse durations we also treat explicitly the changes in
the
frequency content of the probe pulse upon diffraction(i.e.
upon Stokes and anti-Stokes coherent scattering).
This
treatment leads to a prediction of an oscillatory timedependence in the spectrum, as well as. in the intensity, of
diffracted light.
Similar predictions made for forward ISS
have recently been confirmed experimentally(see next chapter).
B. A simple example
We first consider how the material response deviates from
that excited by a perfectly impulsive driving
force when the
excitation time scale is finite. Other characteristics are
still assumed to be ideal.
The excitation force from pulses
of duration Te is
-40-
SIMULATED ISBS DATA
SMALL We AND Wp
SMALL We LARGE Wp
LARGE We
0
10
TIME (NS)
C
20
Figure 3.1 Simulated ISBS data showing the effects of acoustic-wave propagation on signal with
various excitation and probe spot sizes. The effects of finite probe pulse durations are also illustrated.
a) Small excitation and probe spot dimensions in the direction of acoustic wave propagation (x-axis).
we = we = wX = wPy = 50pm., wz = 0.1cm, wPZ = 3.Ocm, v = 3.Okm/s, a = 5*,
X(= X = .0pm. The rise in "baseline" is due to the two counter-propagating acoustic wave packets
propagating away from each other, giving rise to incomplete interference between them. This happens
rapidly because of the small excitation spot sizes. The overall decline in signal is due to acoustic wave
packets leaving the region monitored by the probe beam. which is also small in size. The "baseline"
in signal even near t = 0 is due to the probe pulse duration, which is a significant fraction of the
vibrational period. b) wpz = 0. 1cm = w = w,, other parameters as in a). The "baseline" in
signal near t = 0 is gone because the pro.e pulse duration is now very short compared to the
vibrational period. The baseline rises rapidly after t = 0, because as in a), the excitation spot size is
small and the acoustic wave packets travel apart from each other rapidly. The probe spot dimension.
w . is larger in b) than in a). so the separated wave packets are still monitored by the probe pulse.
T Iis gives rise to a persistent DC signal at long times. c) wex = wPX = 200pm, wPZ = 0. 1cm, other
parameters same as in a). The excitation spot size. wex, is larger than in a) or b) and so separation of
the counter-propagating acoustic waves takes far longer. The apparent attenuation is greatly reduced.
-41-
=
F0
exp(-
t2
-
)
F(t)
Te
Since we are
concerned at this point only with the
consequences of finite pulse durations, we need not indicate
explicitly the spatial variations of the excitation force or
the material response.
example,
G(t)
The
Taking a vibrational mode as our
the Green's function in the underdamped case is
= Go e-Yt sinwt
response is
Q(t)
=
G(t-t')F(t')
=
dt'
T2
-e (W2
GOFOn Te exp[-
+ y 2 )] e-y(t-ts)
sinw(t -
ts)
(3.5.1)
where
G(t)
Q(t)
Y-2
YTe
-
ts =
= Go
is a time
( e Ylt
shift.
e-y2t )
-
For overdamped modes,
and
T2
Te
= GOFOn Te exp(- T-YlY2)
x {exp[-yj(t-ts)1-exp[-Y 2 (t-ts)]}
where
(3.5.2)
,
the time shift
(1+Y2)te
ts
2
Y
4
In deriving Eqs.
2
(3.5.1)
integrations from -
and (3.5.2),
o to + c.
we have carried out the
The results are only valid for
times following excitation, i.e'. t >> Te.
Equations (3.5.1) and (3.5.2) show that, because of the
finite time duration of the excitation force,
vibrational
response deviates from the ideal
response (the Green's function GO)
the coherent
impulsive
in two respects:
there is a reduction in excitation efficiency;
is a phase shift.
-42-
first,
second, there
C.
Excitation Force
In order to treat more general
situations, we first take a
close look at the excitation force.
Here and in later
derivations, we shall assume that all the laser pulses are
gaussian in all dimensions.
In order to have analytical
expressions for the final results, we shall assume that all
the laser beams cross at their beam waists and that the beam
overlap
region is shorter than the Rayleigh range
(a length
over which the spot size remains not too much larger than that
at the beam waist),
so that the spot sizes can be assumed
constant over the overlapping region.
derivations,
To simplify the
an isotropic sample of infinite size is assumed.
The expression for the excitation force is [Eq.
(3.2.4)]
1
Fij(r,t)= 1Ei(r,t)Ej(r,t).
The electric field is the sum of electric fields of the two
crossing pulses,
e'e
E(1)=El0exp[-
20
x1 2
-
y1
E=E( 1 )+ E(
), with
(zl-ct) 2
2
2
2
2
-e
]
2
(e iklo(zl-ct) + e- ikl0(zl-ct)
x
E( 2 )E
i.e.
exp[- x 2
Y22
2
2
x
(z 2 -ct)
2
2
wez 2
(eik20(z2-ct)+ e-ik20(z2-ct).)
;
(3.5.3)
E 1 0 and E 2 0 are the amplitudes of the two pulses whose central
wave vectors are kl 0 and k 2 0 .
For each pulse, wex and wey are
spot sizes in transverse directions (wex9wey for an elliptical
spot),
and wez is the pulse length which is related to the
pulse duration through Te = wez/c.
x 1 , yi, zj and x 2 , Y2, z 2
are measured in their respective coordinate systems, which are
defined as follows(refer to Fig.
3.2):
zi is in the direction
of kio, Yi is perpendicular to the plane defined by the two
-43-
qO
k 10
k2 0
kpo
Z
kso
x
Figure 3.2. The (xyz) reference system and the relations between the central wave vectors of
excitation beams. kj0 and k2 0, probe beam, k 0 . coherently scattered beam , ksO, and material
response. q0 . Other reference frames defined in the text but not shown in the figure are defined such
that z, Z1. zp and zS are along k 10 , k20. kPO and k, 0 . respectively. All the y axes are parallel and
point into the page. The angles of incidence of the excitation and probe beams are Ot and 0.
respectively.
-44-
beams, and YilI IY21(k 2 oxkio).
(3.5.3)
We set kl 0 -k 2 0 -ke 0 -
Eq.
is a valid approximation when the beam divergence
within the overlapping region is small.
(3.5.3) into Eq.
Substitution of Eqs.
(3.2.4) yields the excitation force, which
can be separated into two parts F(l) and F( 2 ) as follows:
2 2 2 2
1 1 20 1 0 20,)e L x1 +x2 Y +Y2
F(')
F(llij(r,t)=
*
(,,
~E2
~
+El0 E2
Wexp[
wex22
Wy2
wey2
wez 2
(3.5.4)
F( 2 )=(high frequency terms and q=0 terms)
Since the high-frequency terms
material
motion)
response
in F(
2
) drive no significant
(we are not interested in purely electronic
and the q=0 material
response has no contribution to
diffracted signal when the probe pulse is incident at the
Bragg angle, we consider only F(l)
hereafter.
Fig.
3.2,
and drop the superscript
Choosing the coordinate system (xyz) as shown in
we have the following transformation:
xl= x cosa -
z sina
zl= x sina + z cosa
(3.5.5)
x 2 = x cosa + z sina
z2= -x
sina + z cosa
Yl= Y2= Y
where a is half the angle between the two excitation pulses.
Substitution of these relations into Eq.
Fij(r,t)=F 0 ijexp[-
X2
4 mx
z2
4mz
2.zoa
Z
4my
c)
zcosa(ct)-t
2
Wez
ez
x
[e iqx+e~ iq"x
where
-45-
(3.5.4) yields
2 (ct) 21
w 2
ez
(3.5.6)
,
El 0 E 2 0 U
Foij= 4 (E10iE20+20E10)=
(3.5.7)
qo=2keOsinot,
and
4mx
1
sin 2
(cos 2 2a
1
Wez 2
Wex
(3.5.8)
=2( sin 2 a + cos 2 C
Wex 2
4mz
1
2
4m y
Wey 2
2
2Wez
Written in another form, we can
see more clearly the evolution
of the excitation force in space and time:
F2
F=Foexp[- 4m,
4m 2
4mx
4my
x
(z-Qct) 2
m14[e
4mz
[e iqxi+e
(ct) 2
4R
iqox
-i
q+e~q~
qX],
(3.5.9)
with
Wex 2 cosa
w ez 2 sin 2 CX +
wex 2 cos 2
1
=2sin
4R
(3.5.10)
2c
Wez 2 sin 2 X + wex 2 cos 2 a
One can see from Eq.
(3.5.9)
that the excitation force
a traveling ellipsoid of sizes Sx=(4mx)
in the x, y and z dimensions,
Qc in the z-direction.
(F(
2
, Sy=(4my)
respectively,
) is in this
is
, Sz=(4mz)
and with a speed
respect the
same).
The dimension Sy is independent of angle a, while Sx and Sz
are determined by angle-dependent combinations of wex and wezIn the limiting cases when o is. close to 0 or 900, Sx and Sz
reduce to
simple forms.
At
intermediate angles,
the
smaller
of wex and wez primarily determines mx and mz.
When wex/wez
is large, as in femtosecond experiments where wez=cTe may be
less than 10pm, small excitation angles are necessary so that
the
"pancake"-like excitation pulses can overlap efficiently.
-46-
The factor Q, which determines how fast the excitation
ellipsoid moves forward, approaches a maximum of 1/cosa for a
given angle a when wezsinc
when the opposite is true.
happen
<< wexcosa, and approaches zero
The former case is likely to
in ISS experiments in the
femtosecond region.
The
latter happens when excitation pulses are longer than the
region over which the two excitation beams overlap.
6t = (4R)
c
c
of the excitation force is
[(1/2)(wez 2 +wex 2 ctg 2 a)
(
c,
c
.
"strike time"
The total
(3.5.11)
which is longer than that determined solely by pulse duration.
This is because wez/c is the time
a single
laser pulse
spends
passing through a point in the sample, while wexctga/c is the
extra time two excitation pulses
spend traveling
through different points in the sample.
together
This extra
time does
not reduce time resolution as long as the probe pulse is of
the same color, so that the Bragg angle for the probe pulse is
the
same as
the excitation pulse angle of
incidence and the
probe pulse sweeps through the excited region with the
speed as the excitation pulses did.
same
In this case, the probe
pulse "sees" the same vibrational distortion everywhere as it
progresses from front to back of the sample.
If different
excitation and probe wavelength are used, time resolution can
be reduced.
For example,
and probe a 100-cm-1
if 100-fs pulses are used to excite
mode,
then as little as 30pm of
pathlength difference between excitation light and probe light
can cause the probe to sample significantly different
vibrational distortions at different regions in the sample.
This leads
one
to a reduction in the
time resolution.
can use a thin sample to avoid the problem,
Of course,
at the expense
of signal intensity.
The Fourier transform of Eq.
(3.5.6) gives the excitation
force in wave vector space as F(q,t)=f(q,t)+f*(-q,t), where
2
2
f(q,t)=f0 exp[-mx(qx-q 0 ) -myqy -mzqz2-iGqz(ct)-
(ct) 2
(3.5.12)
-47-
with fo=F0(4nmx4nmy4nrm)
vector space,
1/ 2
.
Eq.
(3.5.12) shows that in wave
the excitation force has two Gaussian
distributions around tq 0 =(
The distributions reflect
q 0 ,0,0), with a small
the fact that the
z-component.
focused excitation
pulses are not perfectly collimated but rather contain a range
of wave vectors.
When St=(4R) /c
is much
smaller than the time scale
of the
material response, the excitation force can be approximated as
a delta function in time:
F(r,t)=Fl 0 exp[-
x4
4x
-
4
4m y
Z
4,u
e][i +e 8]6(t); (3.5.13)
f(q,t)=floexp[-mx(qx-qo)2-myqy2_#9z2]8(t),
(3.5.14)
with
Wex2
2sin 2 c
IL
wez
2 c
flo=F1o(4nmx4nmy4 np)'
When the excitation spot size is much larger than the
interference fringe spacing, we can sometimes approximate
it
as a delta function in wave vector space also:
f(q,t)=AS(q - q0 )S(t),
(3.5.15)
where A=(2n) 3 2
cOSe U and I is the total energy per excitation
pulse,
Se=Twexwey is the excitation spot area and U is defined
in Eq.
(3.5.7). This expression has been used in Sec.
Ly,
The derivation given here assumed sample dimensions Lx,
Lz larger than the dimensions of the scattering region,
(4mx)
, (4m )0 and (4h), respectively.
is smaller than (4pu)
If, for example, Lz
, we can still make use of the
expressions above by substituting Lz 2 /4 for p.
D.
3.3.
Probing process, general
-48-
As mentioned in the introduction, there are usually two
ways of probing:
1)CW laser used as probe with time resolution
provided by fast electronics; 2)variably delayed laser pulses
In the following, we shall treat only the
used as probe.
second method which is necessary for highest time resolution.
Most of the
results obtained here also apply to the CW probe
method.
Here it is convenient to carry out the treatment in kspace,
since the pulses have finite spatial sizes.
Substitution of Eqs.
(3.2.8) and (3.2.17)
into Eq.
(3.2.23)
yields
Sd 3 k
Esifksft)=
t
dkIp
tdt'
2 n)3
-cc
x
E
Dij(ksft-t')Epk(kpft')
jklm
t'r
dt"Gc.jklm(q
t'-t")Flm(q
t")
.
(3.5.16)
where q=ks-k . To simplify the algebra, we shall carry out the
derivation only for isotropic media.
that
in r-space all
functions are
In addition, we
real.
require
This means that D,
Ep,
Es, GC8 and F all have the following form:
flkjt)=
From Eq.
(kjt)+
*(-kjt).
(3.5.17)
(3.2.23), D(kst) = d(kst) + d*(-ks,t), and
Ws
d(ks,t)=(--)iexp(-iwst)I
(3.5.18)
,
where ws=c(ks)=cks.
The probe beam is assumed to lie in the same plane as the
excitation beams and the probe pulse is
assumed to be
a
gaussian pulse described in wave vector space as
Ep (kp,t)=ep(kp,t)+ep*(-kp,t),
with
ep(kprt)=Epoexp[-apxkpx 2 -apykpy 2 -apz(kpz-kpo) 2 -iwp(t-tp)]
(3.5.19)
-49-
kpx,
kpy and kpz are components of the wave vector kP measured
in the medium in the probe-coordinate system (indicated by
subscript p).
This system is defined such that zp|jkpO and
ypIy (refer to Fig.
are
3.2).
The parameters apx,
related to spot sizes wpx,
wpy,
apy and apz
and pulse length wpz
through
W 2 px,y,z
apx,y,z =
The pulse duration Tp is
Tp=wpz/c
-
(3.5.20)
related to wpz through
,
(3.5.21)
and tP is the delay time between probe pulse and excitation
pulses.
Note
that the probe pulse,
like the excitation force,
consists of two Gaussian distributions
kpo)p
-
kpo=(0,0,
centered around
Continuation of the treatment in an explicit manner
requires specification of Gec.
material
in Sec.
The Green's
function for
response can have many different forms,
3.4.
the
as discussed
For our purpose of illustration we consider a
vibrational response described by
GEc(q,t)=Geosinwat=gc(q,t)+gc*(-qlt)
where ge(q,t)=GCO
-
exp(iwat)
.
,
(3.5.22)
The scattered field should
also have the form:
ES(k5st)= es(k5st)
Substitution of Eq.
+ es*(-ks,t).
(3.5.17)
into Eq.
(3.5.23)
(3.5.16) yields(see
Appendix B)
e s ( kstt)=es(ks,a)-es(k
s
,--a),
with
-50-
(3.5.24)
es(ks'a)=C 2 exp[-iwS(t-tp)+iaOtp]
d3 k
x
3
2
6(ksz-kpz+p)exp[-ak-mq-R(p+Qqz) +i Satp]
(3.5.25)
WaO2=-a (q0)
P=OaO/c,
8wa=wa (q) -wa (q)
,
where
(3.5.26)
and
ak=apxkPX 2 +a pyk py 2+apz(kpz-kpo)2,
2
+myqy 2 +mzqz 2
,
mq=mx(qx-qQ)
C2
=
f
s0n(4itR)
40
C2
The tensor product above
[A-(BC)]i =
In deriving Eq.
(3.5.27)
is defined such that
Ejkl AijklBklCj.
(3.5.28)
(3.5.25), the probe beam was assumed to be
kpo = q 0 /(2sino)
0, i.e.
(3.5.29)
.
incident at the Bragg angle,
Since q-ks-kp, we introduce ks 0 , defined by q0 -ks 0 -kp 0 . The
diffracted light therefore also consists of two gaussian
distributions, es(ks,t) centered near kso=(0,O,kpo)s
(subscript s indicates that it is measured in the s-reference
system) and es*(-ks,t) centered'near -ks 0 . Eq. (3.5.25) will
be used as the basis for subsequent derivations.
The diffraction efficiency is defined as
Is
VI=
Ip
(3.5.30)
I
-51-
incident probe pulses,
The total energy of the
respectively.
probe pulse is
so
=
o
2
-Ep(kpft)1
P(2n)P
J
[jle
dgn
P
(kp,t)12+le *(-kp,t)12+2Re((e
(kp,t)e *(-kg
t
)
Ip=
d3 k
2c0
=-
d3kt)1
P lep(kp,t)2
n (2n)3
(3.5.31)
The cross term has no contribution because the distributions
in k-space of ep(kp,t)
width is far smaller
total energy of the
2s0
1s= 4
d 3 ks
4n
are
far apart.
than their separation.
scattered pulse
esgkstn 2
t)n2
|*sks
d3 k
4
=
and ep*(-kp,t)
Their
Similarly,
the
is
.
(2n)3
where Is and Ip are the total energies of the scattered and
Ies(ks,&a)1
2
+ les(ks,-wa)1
2
e(2n)3ka)
(3.5.32)
-2Re[es(ksrwales*(ksr-wa)]}
The diffraction efficiency is
fd 3 ks(Ies(kswa)1
2 +es(ks,-wa)1
2 -2Re.[
es(ksrwa)es*(ks1~wa)]}
fd3kplep(kp,t)12
The denominator can be found from Eq.
(3.5.19):
fd 3 kptep(kpt)1 2 =lEp I2 (n/2)3/2
0
(apxapyapz) 1/2.
E. Perfect time resolution limit
-52-
(3.5.33)
(3.5.34)
Since the general integration for Eq.
(3.5.25) is still
quite cumbersome, we start from simple limiting cases.
0
we approximate waO=
and Swa=O,
i.e.
First,
we assume perfect time
resolution and no dispersion in the vibrational frequency.
This allows us to derive effects due exclusively to finite
spot sizes.
Under these approximations, Eq.
(3.5.25)
reduces
.
(3.5.35)
to
es(ksrwa)=C2exp[-ios(t-tp)+iaOtp]
d 3k
s(ksz-kpz)exp[-ak-mq-RQ 2qz 2 ]
x f
(2n)3
Note
that mz+RQ 2 =p which was defined earlier.
We now change
to the new variables:
Sq=q-qo=(qx-q,
qy'
q
qz)
Skp=pkpo=(kpx, kpy,
kpz-ko)p
6ks=ks-kso=(ksx, ksy,
ksz-ko)s
The subscripts p and s denoting the coordinate system used.
Since q=ks-k and q 0 =ks 0 -kp 0 , we have Sq=Sks-Skp. In terms of
components,
Sqx=-Skpxcoso+8kpzsino+Sksxcos0+Skszsin,
Sqy=-Skpy+Sksy ,
(3.5.36)
Sqz=-Skpxsino-Skpzcoso-8ksxsino+Skszcos.
Substitution of Eqs.
(3.5.36) into (3.5.35) and carrying out
of the integration yields
-53-
es(ks'(a)=C 3 exp(-byySksy 2 -bxxSksx 2 -bxzSksxSksz-bzzSksz
x exp(-iws(t-tp)+ioa~tp)
2)
(3.5.37)
where
byy=apymy/cyy
cyya
py+my
,
bxz=b'xz/cxx
bzz=b'zz/Cxx
(3.5.38)
,5
b'xz= 4 singcos~mx(apx+2psin 2 O)
,
b'xx=apx(psin2 3+m cos 2 3)+4mxpsin 2 Ocos 2
,
,
cxx=apx+psin2o+mxcos2o
b'zz=apxapz+4apxmxsin2o+apz/lsin2l3+apzmxcos2l3+4mxpsinds
C3 =
T
C2
We see from Eq. (3.5.37) that unlike the probe pulse, the
direction of propagation of the scattered pulse is not one
axis of the ellipsoidal pulse shape in k-space.
It is rotated
by an angle 9, given by
tg(2e)= b'zz-b'
(3.5.39)
It is shown in Appendix C that in the limit of long pulse
duration and small scattering angle,
Eq.
(3.5.37)
reduces to a
result obtained by Siegman (1977).
From Eq.
(3.5.33), the diffraction efficiency as a
function of delay time tp is
-54-
n(tP)=Vl
(3.5.40)
2(1-cos2wa0tp),
where
41C31 2 (apyapxapz) 1 / 2
(3.5.41)
ni2/4)]1/2
EpO 2[b yy(bxxbzz-bxz
We see from Eq.
to the
(3.5.40) that the diffraction efficiency due
standing wave oscillation can be described as a sum of
two parts, a DC part and an oscillatory part of frequency
2 wa0-
Since the two parts have identical amplitudes, the
diffraction efficiency exactly vanishes periodically.
This is
the
result of
the approximation of perfect time
is consistent with Eqs.
resolution and
(3.4.3) and (3.4.5).
Siegman (1977) noted the reduction in diffraction
efficiency when wp/we is too small and gave an explanation.
Here we
Eqs.
look into this problem in more detail.
(3.5.38)
l=YI
0 /(
into Eq.
Substituting
(3.5.41), we find
(3.5.42)
1 2)l/2
where no is the same as n in Eq.
(3.3.8), and
apy
my
2=1+
+
(4psin2
apX
(4psin4
a PZ
+
cos 2
p 2 sin 4 o
mx 2 cos 2 o
We see from Eq.
apzI mx >>
o
+ 42sin
mxcos 2
+
)+a1(
_
1
a
+ 4apxpsin 4 i3
4
mxcos o
mx 2 cos 4 o
mxcos2o
4p 2 sin 6
psin 2 I
)
(3.5.43)
mxcos 4
2
+ 6psin o
mxcos 2
(3.5.42) and (3.5.43)
apx >> p
that only in the limit
(3.5.44)
and
-55-
,
my >> apy
does the diffraction efficiency approach the limiting value
00. Recalling the definitions of m's and ap's in Sec. 3.5.C,
this means the grating dimensions and pulse length are larger
than the probe spot sizes and the probe spot dimensions exceed
the grating thickness.
For all other situations
diffraction efficiency is smaller.
the
The explanation lies in
the fact that the grating and probe pulse have finite sizes
both r-space and k-space.
If the probe pulse is too big
compared with the grating in either of
diffraction efficiency will be low.
wpx
(and therefore apx)
is too big,
pulse misses the grating and ni
small,
in
these two
spaces,
the
When the probe spot size
some portion of the probe
thus drops.
When wpx is too
the corresponding dimension in k-space becomes too big
and some portion of it misses the grating in k-space which
also reduces 01.
In other words, when wpx is too small,
divergence of the probe beam is too great,
probe light is outside of the Bragg angle
the
some portion of
tolerance (Bragg
angle tolerance is inversely proportional to the grating
thickness),
and therefore is not diffracted effectively.
requirement Eq.
(3.5.44)
The
ensures that the Bragg angle
tolerance is always larger than the probe beam divergence.
This discussion also applies to the CW probe case.
When the probe length wpz is too small
or less than p),
(apz is comparable
the diffraction efficiency also drops.
This
result is unique to short pulses and cannot be derived from
the CW limit.
The reason for the effect is similar to that
for the drop in diffraction efficiency due to small wpx:
probe pulse has too big a range of wave vectors,
the zp direction.
Eq.
the
this time in
(3.5.43)' shows that this problem is
reduced with small scattering angle 0.
The reduction in apz
also causes b'xz to become comparable with b'zz, which means
the scattered light in k-space becomes a
Eq.
(3.5.39) shows.
rotated ellipsoid,
as
This is because the grating ellipsoid and
probe ellipsoid are oriented differently.
-56-
When the axes of
the two ellipsoids do become parallel,
b'xz=O and 9=0,
i.e. 0=0 or O=n/2,
then
and no rotation occurs.
In practice, wpx can be optimized quite easily.
When the
probe beam spot size is too small, the diffracted spot far
away from the scattering region becomes noticeably elliptical
relative to the transmitted probe spot.
The probe spot size
is adjusted until the diffracted spot has about the same shape
as
the transmitted spot.
F.
Imperfect time
resolution
We now turn to the case in which LaO*O and
Swa=0 .
This
is
the case for optical phonons or molecular vibrations with no
dispersion.
Since waO 0
, perfect time
Carrying out the integration in Eq.
(3.5.25) yields
= C 3 exp(-p 2 b0
)
es(ksca)
resolution is lost.
x exp(-byySksy2-bxxSksx2-bxzSksxSksz-bzzSksz 2 -pbxksx-pbzksz)
x exp(-ios(t-tp)+iwaotp)
(3.5.45)
where C 3 , bxx,
bzz, bxz cxx,
cyy are the same as those in Eqs.
(3.5.37), and
b' 0
bo = cxx'
b'o=(apz+R)cxx+myp-Q2R2sin2o
-
2QRcoso(apx+mx)
b =b'g/Cxx
,
+apx(mxsin 2 ++pcos 2 o)
,
bz=b'z/Cxx
b'
=2[2mxpsinocoso+apx(mx+p)sin~coso-RQ(apx+2mxcos2o)sin0I
-57-
.
b'z=2[apxapz+apz(mXcos20+pjsin2o)+2mx(apx+p-RQCOSO)sin2o)
(3.5.46)
The total diffraction efficiency is now
r(tp)=njexp[-2p2(b0 -h)]
[1 - exp(-2p 2 h)cos2waOtp]
(3.5.47)
where nj is the same as defined in Eq.
h-
(3.5.41), and
bxxbz 2+bzzbx 2-bxbzbxz
4bxbzz - bxz2
(3.5.48)
Comparing Eq. (3.5.40) with Eq. (3.5.47), we see that the
effects of finite time resolution are 1) a reduction in total
scattering efficiency, by a factor exp[-2p 2 (b0 -h)]; and 2) the
oscillating term is now smaller than the DC part, by a factor
exp(-2p 2 h), so that the diffraction efficiency remains finite
at its periodic minima. The two exponential factors start to
become significantly smaller than one when b0 and h are
comparable with p 2 , i.e. when the experimental time scale is
comparable with the period of oscillation. From Eqs. (3.5.46)
and (3.5.48), we see that in the femtosecond regime the
experimental time scale is not solely determined by pulse
durations, spot sizes and experimental geometry all play their
important parts.
We now show how the experimental geometry can affect the
time resolution. Assuming a very shor.t pulse, we neglect
terms proportional to apz in evaluating h and get
h=h'/(cxx) 2
h'=(apxp2sin2o+apx2p+2apxmxpcos20)(cosa-cosO)2
-2apxmxp[cososin2o(cosa-coso)+sind6]
+mxp(mxcos 2 o+psin 2 0) (1-cosacoso)
-58-
2
+apxmxsin 2 P(apx+mxcos 2 o)
(3.5.49)
We
see that h reaches a minimum
resolution)
near cosa-coso=0,
(corresponding to best time
i.e.
when the excitation and
probe scattering angles are the same.
This is possible when
the excitation and probe pulse frequencies are the same.
is what we have anticipated in the discussion in Sec.
This
3.5 C.
Note that this effect is not due to optical dispersion in the
sample, which has been neglected.
It is due to the different
propagation times of excitation and probe pulses through the
grating thickness if these pulses have different angles of
incidence.
G.
Frequency spectrum of scattered pulse
It is interesting to see how the spectrum of the scattered
.
light behaves as a function of probe pulse delay time,
t
Instead of finding the total energy of the scattered pulse by
integrating over d 3 ks, as is done in Eq.
integrate
over ksx and ksy only.
=I(ksz)
I(Ws)
=
a
Since ws=cksz,
2 +Ies(ksf-wa)1 2
2Re[es(kswa)es(ks-wa) 11
= aO[I1+I2- 2 I3cos2waOtp]
(3.5.50)
,
where
=
2(bxxbyy)1 / 2
I=exp(- b 36cs 2 + pb 2 8os + p 2 bj)
,
a0
,
2
12 =exp(- b 3 6ws 2 - pb 2 6ws + p b1 )
with
we get
les(kszft)12
fdksxdksyfles(kswa)
-
(3.5.32), we
1 3 =exp(-b 3 8ws 2 ) f
s=s =W - WsA f
-59-
3bzz-4 b x
=b~ (
b=C2
2
'
4b
Txx
2
xz__
b2=-(bz +
'
2b
2b2bx
bx 2
bj =
2bxx'
We see from Eq.
components.
and
(3.5.50)
il is up-shifted in frequency,
13cOs( 2 watp)
is unshifted.
Il and 12
shifts for
that the scattered light has three
12 is down-shifted,
The magnitude of the
spectral
is given by
b2
A
=
2b3 )
P(
~
a0-
(3.5.51)
A is independent of excitation and probe pulse
Substitution of the definitions of b 2 , b
Eq.
3
energies.
and p = waO/c into
(3.5.51) shows that the approximate relation A = wao holds
as long as wz 2
> w
2
sin 2 a/cos 2 a and wz 2
>> wx 2 sin 2 a/cos 2 p.
This obtains in almost every case of interest.
The spectral
shift is therefore equal to the vibrational frequency
(neglecting vibrational damping), as in other forms of
stimulated or spontaneous scattering.
When the probe pulse duration,
tp,
is much shorter than
the period of vibrational oscillation, Ta- 2 n/a,
the terms
proportional to p in the expression for I, and 12 are not
important. The spectral shift between the components is much
smaller than their spectral width.
Th-e three components have
essentially identical spectra and the amplitudes 213 and I+I2
are identical, so 2 13cOs( 2 wa0tp) can periodically cancel 11+12
The
completely.
frequency
2 wa
intensity of diffracted signal oscillates at
and vanishes periodically.
This case, with
Tp<<Ta, corresponds to perfect time resolution as discussed in
Sec 3.5.E.
However, when the probe pulse duration is a
significant fraction of the vibrational
period,
the terms
proportional to p in the expression for Il and 12 become
important. The spectral width of the three components may only
-60-
slightly exceed the spectral shifts between them and the
cancellation of the terms is
therefore not complete
(i.e.
so the diffracted signal remains nonzero at it
< Il+I2),
periodic minima.
213
This behavior was discussed in Sec 3.5.F.
The new feature shown by Eq.
(3.5.50) is that the spectrum of
the diffracted probe pulse, as well as the intensity,
oscillates as a function of probe pulse delay at the
vibrational frequency.
Similar behavior was predicted for
coherent forward-scattering of a variably delayed probe pulse
following forward-ISS,
and this behavior has
confirmed experimentally
Fig.
recently been
(See chapter 5).
3.3 (solid curves)
shows the simulated spectra of the
diffracted probe pulse at different probe pulse delays,
with
the pulse duration comparable to the vibrational period,
Ta-
The top curve is the spectrum of the diffracted signal when
5Ta/ 4 , etc.
3 Ta/ 4
The bottom curve shows the spectrum of diffracted
signal at its minimum intensity,
course,
i.e. t = Ta/ 4 ,
,
the signal intensity is at its maxima,
i.e.
t =0,
Ta/2,
Ta,
etc.
(Of
signal is nonzero at these delays because the probe
pulse duration is significant.)
spectrum at
intermediate delays.
The middle curve shows the
The dotted curve
shows the
spectrum of the incident probe pulse.
Fig.
3.3 suggests a way to improve the effective time
resolution in an ISS experiment.
If instead of collecting all
the scattered light, one were to collect only the central
portion of the spectrum,
then the depth of modulation in the
time-dependent signal must improve.
The ratio of AC to DC
contributions to signal, which is unity for neglegibly short
probe pulse duration, was shown in the previous section to
decrease as exp(-2p 2 h), where h increases with probe pulse
duration.
This relation obtains if one measures all of the
diffracted signal.
However, if only the central portion of
the spectrum of diffracted light is collected, Eq.
shows
that this
(3.5.50)
ratio is
2I3
11+12
=
=6"s=0
exp(- p 2 b1 ).
-61-
(3.5.52)
TIME-DEPENDENT SPECTRUM
OF ISS SIGNAL
300 (CM
0
1
)
-300
Figure 3.3. Simulated spectra of time-dependent diffracted ISS signal with various probe pulse delays,
t. The probe pulse duration is 60fs and the vibrational period Ta is 167 fs (i.e. a 200cm 1 mode).
The dotted curve is the spectrum of the incident pulse. Top curve: probe pulse is delayed such that
signal intensity is at a maximum, i.e. t = Ta/ 4 , 3 Ta/ 4 , etc. Bottom curve: minimum signal intensity,
i.e. t = 0, Ta/2, Ta, etc. Middle curve: intermediate delays, t = Ta/ 8 , 3 Ta/ 8 , etc. The total signal
intensity (integrated area under the curves) varies only by 32% from maximum to minimum since the
probe pulse duration is a substantial fraction of the vibrational period. However, at the center of the
spectrum, the maximum intensity is more than 100x greater than the minimum intensity.
-62-
Since bl is usually much smaller than 2h,
the AC modulation in
time-dependent signal can be enhanced significantly by
collection of only the central spectral region.
In effect,
the time resolution of the probing process can be enhanced.
Of course,
there is a limit to how much one can achieve with
this method,
since it entails discarding much of the
diffracted light.
In addition,
as shown by Eq.
(3.5.47), when
the excitation pulse duration is also significant,
excitation efficiency and therefore
the
the total diffraction
efficiency are reduced.
For strongly Raman-active modes,
doubling or tripling the
resolvable frequency range seems
possible.
H.
Acoustic waves
As discussed in Sec.
3.5.A,
one major limiting factor in
accurate measurement of acoustic wave properties
is the
problem of the counterpropagating acoustic wave packets
traveling apart from each other.
The effect on the light
scattering signal can be analyzed by substituting
8w = v
qx + v qz 2 +y
2q 0
2
(3.5.53)
where v is the speed of sound, into Eq.
(3.5.25).
The first
term accounts for the wave packets traveling apart, and the
second term includes the effect of acoustic wave divergence.
Carrying out the integration, we get
e s (kswa)=C4 exp(-dO)
x exp(-dyysksy 2 -dxxSksx 2 -dxzsksx~ksz-dzzsksz 2 -dxksx-dzksz)
x exp(-ios(t-t p)+ioa0tP).
(3.5.54)
This is essentially the same as Eq.
parameters:
-63-
(3.5.45) but with modified
p
I
dx=d' /fx
dz=df'/fx
11
I
do=d' /fx
fxx~cxx
-
ivt pasifl 2 o
yy
-
ivt p a
fyy=
if
)
a=1/( 2q0
C4=(n/(fxxfyy )1/2 )C2
d' XXbrx
-
iv
I
~p+mxo~~i~
dfzzbrzz- ivt pa(apz+4mxsin2 o)sin 2 i3
doXZbfxz-
drX~rx-
ivtpa8coslsin 3 o
iv
-2a(vt p) 2 cos~sin 2 o
~p+psnpcs
d' Z=bf z
-
-
-ivtpap2 ( ap+2mx) cospsin 2 o IF
ivt p2 (apx+mzsin 2 o)
-2a(vt p )2sin3o
ivtpap2(apz+2mxsin 2 o)sin 2o
-64-
A,
,
d' yybyy - ivtpaapy
d'o=b' 0 p2+(vtp)2 1 cos 2 0 -
ap(vtp) 2 sinO -ivtpp(pu+apx-QRcoso)
-ivtp Tp 2 (mx+apXcos2 0+apzsin 2 +Rsin 2
)
(3555
To concentrate on the the problem of the wave packets
leaving each other,
we consider only the perfect time-
resolution limit and neglect the acoustic wave divergence,
that is,
we neglect terms with p and a.
Since picosecond
pulses are sufficiently short for ISBS experiments, we assume
that the laser pulse length wz=c-r is much larger than its spot
size and that the probe beam is
excitation beam,
i.e.
a=o.
The
incident at same
angle as the
scattering efficiency is then
n=nl(1/2)exp[-(vtp)20C1{1-exp[-(vtP)2 a2cos2watp}
(3.5.56)
where
2
1
cos 2 o
(apx+aex)
2 1s
2 cos
2
(3.5.57)
aex
aex-
w2
ex
4
We see from Eqs.
(3.5.56) and (3.5.57) that both the DC part
and the AC part suffer artificial decay due to finite spot
sizes.
The decay factor exp[-al(vtp) 2 '] affects both the DC
and AC parts.
It is caused by the acoustic wave packets
leaving the region monitored by the probe beam.
The decay
time is proportional to the geometrical sum of probe and
excitation spot sizes and inversely proportional to the sound
velocity.
The factor exp[-a 2 (vtp) 2 ] is the extra decay
suffered by the AC part of the signal due to the wave packets
leaving each other.
Since the size of the acoustic wave
packets is determined by the excitation spot size only, the
-65-
decay time is just the time for acoustic wave to travel one
wave packet size.
The results here can be expected from the
qualitative description in Sec.
3.5.A.
Figure
3.1 illustrates
the effects discussed in this section.
One
important result of the above
analysis is that the
gaussian decay of signal with time due to finite spot sizes is
only related to the spot sizes in the x-direction.
Mathematically this is because in the expression for the
scattered field
[Eq.
(3.5.54)], there
in the exponential.
is no linear term in ksy
This is why, throughout the derivation,
we have not assumed round excitation or probe spots.
ISS excitation
is a nonlinear process,
Since
higher scattering
efficiency can be achieved through tighter focusing of the
excitation beams.
However,
vector definition,
one must increase the spot sizes.
in order to increase the wave
analysis above tells us that we only need to
sizes in the x-directions.
can be
The
increase
the spot
For thick samples the x-spotsize
increased with no reduction
in scattering efficiency
because of the compensating effect of increased grating
thickness.
Of course, the spot size in the y direction cannot
be made too small.
At some point the divergence of
the
acoustic wave packets, which we have neglected in getting Eqs.
(3.5.56) and (3.5.57), will start to matter. From the
expression
vtp ~
for fyy and d'yy, we see that this occurs when
a my
=
2
2
Xa
way
a
,
(3.5.58)
where Xa is the acoustic wavelength and way is the acoustic
wave spot size.
We see that, just as with light waves, once
the acoustic wave packets
travel out of their Rayleigh
range(nw2y/Xa), their divergence becomes significant.
The
best choice for the spot sizes is such that the acoustic wave
Rayleigh range is larger than the excitation spot size wex.
3.6 Summary and Concluding Remarks
-66-
We have presented a general theoretical framework for
analysis of impulsive stimulated scattering experiments.
theory provides a direct connection between the
observable,
I(t),
The
ISS
and the dielectric constant response
function, Gcc(t), which is in turn related simply to the
dynamics of LS-active material modes though Eq.
connection between I(t)
and Gee(t)
(3.2.13).
The
becomes especially simple
when experimental conditions of near-ideal
vector resolution can be realized.
time and wave
In such cases, the simple
relation
I(t)
c
JGEe(q.t)1
holds, where Gcc(q,t)
2
(3.3.10)
is a projection of the Ges(q,t)
which depends on the light polarizations used.
tensor
In many cases,
the polarizations can be arranged such that Gcc is
a single
independent component of Gee.
ISS experiments on acoustic and optic phonons (and
molecular vibrations)
have been carried out,
are treated explicitly in Sec.
Eq.
(3.3.10)
mode,
can be used.
labeled a,
and these
cases
3.4 under the assumption that
In cases where only one material
is excited coherently, Gcc(q,t)
= GO(q,t) and
so ISS data takes an especially simple form (e.g. damped
oscillations).
Cases of single and multiple mode excitation
are discussed in Sec.
3.4.
A substantial part of this chapter is directed toward
quantitative treatment of nonideal experimental conditions.
The
relation between I(t)
complicated than Eq.
and Gcc(q,t)
becomes more
(3.3.10) when limitations in time
resolution (due mainly to finite laser pulse durations) and
limitations in wave vector resolution (due mainly to focussing
of laser beams to finite spot sizes)
are taken into account.
If time resolution is limited then convolution of the material
response with the probe pulse duration must be calculated to
reproduce I(t) accurately.
For underdamped modes, this
becomes necessary when the pulse duration is a significant
-67-
fraction (i.e.
is Eq.
> 1/10) of the vibrational period.
(3.5.47).
The result
The effects of finite excitation pulse
duration, discussed in Sec.
3.5.B,
are simply a reduced
vibrational amplitude and a phase shift.
An additional result related to finite probe pulse
duration is that the spectral content of the pulse is altered
upon diffraction.
The spectrum, as well as the intensity, of
light diffracted by the vibrational standing wave is predicted
to show an oscillatory time-dependence.
Improved time
resolution may be realized by selectively monitoring only the
central portion of the spectrum of the diffracted
The main consequence of
ISS
limited wave vector
signal.
resolution in
experiments on nondispersive modes is reduced signal
intensity, with no effect on the temporal profile
of I(t).
The intensity effects are given by Eqs.
(3.5.40) and (3.5.41),
through which the results of focussing
of excitation and
probe beams can be calculated.
focussed to a certain spot size,
If the excitation pulses are
then the probe pulse must be
focussed comparably to avoid "missing" the grating in space.
However,
the
vectors,
some of which will not be optimally phase matched for
focussed probe beam will contain a range
of wave
diffraction (i.e. some will "miss" the grating in q-space).
In addition, pulses of very short duration (i.e.
are short in spatial length as well,
femtoseconds)
and this introduces
additional wave vector uncertainty which further reduces
diffraction efficiency.
This is one of several examples of
interplay between experimental geometr.y and pulse duration.
The wave vector resolution can be influenced by pulse
duration,
and time resolution can be influenced by
experimental geometry.
These effects can be significant with
femtosecond pulses.
Finally, the special consequences of limited wave vector
resolution for dispersive material modes (in particular,
acoustic modes) are treated in detail.
temporal profile of I(t)
In this case,
the
is affected because the counter-
propagating acoustic waves generated through ISS excitation
-68-
can propagate in space away from each other and out of the
region monitored by probe beam.
Eq.
(3.5.51),
These effects,
summarized in
must be calculated when measurements
of acoustic
attenuation are made.
In conclusion, a detailed theoretical treatment of
impulsive stimulated scattering experiments under ideal and
nonideal conditions has been presented.
The treatment is
closely analogous to theories of conventional
(frequency-domain) light scattering spectroscopy, and
facilitates comparison between ISS and LS experiments.
comparison is presented in the next chapter.
-69-
Such a
Appendix A
Fourier transformation of Eq.
L(k,w)GE(k,w)
where
= I
(3.2.19) gives
(A.1)
,
I is the unit matrix,
and
0.
Lij(kw)
GE(k,t)
= -k28ij+kik
is thus
GE(kjt)=
.+02
.
(A.2)
found to be
C1 d
e- it GE(kwo)
(A.3)
with
GE(k,w) = L(k,w)-l
(A.4)
.
The integration path C circles the lower half of w-space,
including the real axis, in the clockwise direction.
result
The
is
GE(kt>0)
= -i
E Res{GE[kn(k)]}e
in(k)t
(A.5)
n
where Res denotes taking the residue.
The sum over n sums
over three pairs (wnt-wn) of residues of GE(k,w).
One has w-0
and is a longitudinal mode which should be dropped.
The other
two pairs satisfy the transverse wave requirement and are the
optical normal modes.
T(k)
=
o(k)
C2
For any one of these two modes,
{Res[GE(k,w(k))] - Res[GE(k,-&(k))]}
-70-
.
(A.6)
Appendix B
Substitution of Eq.
(3.5.17)
into Eq.
(3.5.16) yields a
total of 24=16 terms for Es, eight for es(ks,t) and eight for
e*(-ks,t).
We therefore need to consider only eight terms.
Since we discuss here only the cases in which material modes
respond much more slowly than the
frequency of light,
these 8 terms become vanishingly small.
The
6 of
remaining terms
give
es(ks,t)=
d3k pt
{3
f(2n)3--
t'
dt'
dt"{d(ks)gC(q)'[f(q)ep(kP)]
C
+ d(ks)'g9*(-q)'[f(q)ep(kP)]}
=es(kstwa)-es(ks'-wa)
(B.1)
1
with
es(ksta)=
t
3
t'
dt'
-C-
(2n)3
-ios(t-t')-iO p(t'-t
dt" Clexp[-ak-mq
--
p)+iwa(t'-t")-iQqz(ct")-(ct")2/4R],
ak=apxkpx 2 +apyk py 2 +apz(kpz-kpo)
2
,
d 3k
,
mq=mx(qx-q0) 2 +myqy 2 +mzqz 2
Cl = -44Ge'(fOEpO)
.
(B.2)
The result of integration of Eq.
(B.1) over t" from -o to
t'
should be expressed in terms error functions.
t'
sufficiently large, we can set the upper limit of
integration to +w,
C 1 (4nR)
/c
yielding
exp[-ak-mq-R(oa/C+Qq z)2~ist+ioptp+i(WsoP
Integration over t'
However,
from -o to +c yields
-71-
+Wa)t'
for
d3 kg
es
( ks 'Ma)= (
3
C 2 8(ks-kp+a/c)
x exp[-ak-mq-ist+i
ptp- R(wa/c+Qqz) 2 ]
(B.3)
with
C2
'
2
=4c0
(sOJT(4J.R)
c2
Again the
integration from -=
justification.
can let
t + +w.
(B.4)
-(B4
Ge'0PO
to +w
The upper limit +a
The lower
needs
some
is not a problem because we
limit -w
is an approximation,
because in integrating over t", we assumed that t' was not
small.
Let us look carefully at
the integrand.
For
< t'<t, the integrand is an oscillating function.
t'<(4R)1/2/c
,
it is very small.
That is,
the
(4R)1/ 2 /c
For
integrand,
instead of being a perfect harmonic oscillatory function from
-D
to +G
(in which case the
result would be exact)
is an
oscillatory function which rises from zero to a steady
amplitude during -(4R)
at t'=t.
1 / 2 /c<t'<(4R)1
/ 2 /c and returns to zero
As long as the number of oscillations is
sufficiently large, the delta function in Eq.
(B.3) is a good
approximation.
The requirements ws= 6 p
la and ks=kp-q restrict the probe
beam to incidence at or near the Bragg angle to get
appreciable scattering.
In the subsequent derivation, we will
assume it is incident at Bragg angle 1, i.e.
kpo = q 0 /(2sina)
.
(B.5)
Since q=ks-kp, we introduce ks 0 where q 0 =ks 0 -kp0 .
The
diffracted light therefore also' consists of two gaussian
distributions,
es(ks,t) and es*(-ks,t), centered respectively
near +ks 0 =(OO, kp 0 )s(subscript s indicates the s-reference
system).
-72-
The wa in R(wa/c+Qqz)
2
reflects the influence of the
finite frequency of the material mode in the excitation
process.
in Sec.
If we assume &a=&+iy, we can get the result derived
3.5.B.
Wa in this term can be neglected if Tp =2n/wa
is much longer than the excitation time scale.
The wa in
6(ks-kp++a/c)
reflects the same influence in the probing
process.
It can be neglected if Tp is much longer than the
probe time scale.
In both cases, we will make the
approximation wa=WaO.
That is,
although the excitation and
probe time scales may be long enough to let wao have some
influence (i.e.,
some material motion can occur during the
excitation or probe process),
the pulses are never long enough
to let Swa to manifest itself.
For acoustic phonons,
this
means that the traveling apart of acoustic wave packets during
the time when excitation or probe pulses traverse the sample
is negligible.
This is certainly a good approximation.
Another approximation made is that the pulse spot sizes are
much larger than the wavelength of light,
so that in 6(ks-
kp+(AWa/c),
ks=kszf
(B.6)
kp kpzThis is equivalent to neglecting the divergence of laser beams
in the region in which they overlap.
(B.6) into Eq.
(B.3) yields Eq.
Substitution of Eq.
(3.5.25).
Appendix C
Here we show that in the limit of long pulse durations and
small scattering angle, Eq.
obtained by Siegman.
(3.5.37)
reduces to a result
To see this, we note first that in the
long pulse limit, apz>>apx and wez>>wex.
-73-
Introducing
ae =w2 /4
etc.,
we have
mx=aex/(2cos 2 a)
bzz=apz
b'xz
In the
'
p=aex/(2sin 2 x)
<< b'zz-
small angle limit,
Introducing 90/a,
sinot=L,
we have,
(1977)
be
(The "s"
cos6=1.
)+2aexe2
.
2 =(aex/2)[apx(1+e 2
Substituting the above relations into Eq.
neglecting the b'xz term,
sinp=p,
for this limit,
b'xx=apx(aex/2+aexe2 /2 ) +a 2
cxx=apx+(aex/ 2 )(1+9 2 )
cosa=1,
we arrive at Eq.
(3.5.37) and
(19) of Siegman
in the factor (s+e)/Xp) of this equation may
"st".)
-74-
CHAPTER 4.
COMPARISON TO FREQUENCY-DOMAIN SPONTANEOUS
LIGHT-SCATTERING
The ISS experiment is a time-domain,
stimulated analog of
frequency-domain, spontaneous light-scattering spectroscopy
(LS).
Since the two approaches provide complementary and
often overlapping information, it is of interest to compare
their capabilities both in principle and in terms of practical
experimental considerations.
in this chapter.
Such a comparison is presented
Time-dependent ISS signal I(q,t)
frequency-dependent LS signal
their
connections to the dielectric
(Green's functions) Gec(q,t)
Since
I(q,w)
and
are discussed in terms of
response
and Gcc(q,w),
functions
respectively.
these response functions are Fourier transforms of each
other,
ISS and LS data can be
I(q,t)
and I(q,w)
are not
compared readily.
[Note
that
Fourier transforms of each other,
but are related in a manner derived below.]
Simulated ISS and
LS data from underdamped and overdamped vibrational modes are
presented and compared.
Simulations of data from Debye
relaxational modes and from combinations of LS-active modes
are also presented.
As discussed in the previous chapter,
between I(q,t)
and Gcc(q,t)
the connection
is especially simple in the limit
of ideal wave vector and time resolution.
(Much of the
previous chapter is devoted to conside.ration of nonideal
conditions.)
Gcc(q,o)
Similarly, the relation between I(q,w) and
takes
on a simple form in the limit of
vector and frequency resolution.
below,
ideal wave
In the simulations
the idealized limits are' assumed.
presented
However, the
comparisons between the two techniques for various cases are
made in view of realistic experimental resolution,
signal/noise
ratio,
and competing contributions to
signal.
have attempted to present the relative merits of the two
approaches for various cases in an even-handed manner.
-75-
We
However, our emphasis is on those situations in which the
time-domain approach offers real advantages.
this chapter is twofold.
The purpose of
Primarily, we hope to make clear the
complementarity between ISS and LS methods in general and for
a variety of specific cases of importance.
illustrate, via comparison,
Second, we wish to
some of the particularly useful
capabilities of the time-domain approach.
In the next section,
ISS
some theoretical background on LS and
is reviewed and connection between the
demonstrated.
data.
A
Sec.
two techniques is
4.2 presents simulations of
ISS and LS
comparison of LS and ISS experiments and their
capabilities is presented in Sec.
summarized in Sec.
4.3.
The
results are
4.4.
4.1 Spontaneous light scattering
Thermal fluctuations of the dielectric tensor give rise to
spontaneous light scattering,
treated extensively
1978).
the theory of which has been
(Berne & Pecora,
1976;
Here we treat some aspects of the
Hayes & Loudon,
theory analogous to
the treatment of ISS in the previous chapter.
We assume ideal
wave vector resolution
Discussion of frequency resolution requires specification
of the experimental arrangement for analysis of scattered
light.
For purpose of illustration, we assume that scattered
light is passed through a Fabry-Perot.(FP) interferometer.
Very similar results are obtained from consideration of a
grating monochromator.
The light field transmitted through FP,
the field before FP,
E'(t,t)
=
(1-R)
E', is related to
E, by
E RME(t-mt)
(4.1.1)
m=0
where
-r is the time required for light to bounce
back and
forth once inside the FP and R < 1 is the intensity
reflectivity.
The quantity directly measured is the timeaverage intensity of the transmitted light I(T) given by
-76-
I(t)
=<IE'(t,T)I
(1-R) 2
=
E
2
>t
R(m+n)<E(t-mT)-E(t-nT)>t
m=0
n=0
= 1 R
dt'<E(O)-E(t')>e
1+R
-0
(1+R) -r
Rjnj
n=-o
r2i
dt'[1+2
1-R
E 8(t'-n-r)
-
E
cos(2nnt')]
n=1
x exp(-
-' ) <E(O)-E(t')>e
(4.1.2)
where a is defined through the relation R = e~4 and the
subscripts
t and e indicate time and ensemble average,
respectively.
Hereafter the subscripts will be dropped and
ensemble average is always assumed.
We see from Eq. (4.1.2)
that the detected signal is a sum of equally spaced Fourier
components of the correlation function of the scattered light
field.
Different values of n correspond to different orders
of the transmitted light.
If the spectral distribution of
interest is narrow compared with the spacing between orders,
then only one order, say n=m, can have a significant
contribution.
I(W)
c
The expression then simplifies to
dt cos(wst)exp(-a
)<E(O)-E(t)>,
(4.1.3)
where
=
2nm/u.
(4.1.4)
This is the basic formula for frequency domain light
scattering.
To connect the scattered light spectrum to the
material properties,
we use Eq.
(3.2.28)
derived in the
previous chapter, assuming the incident probe beam is ideal
monochromatic light, linearly polarized along the unit vector
Op, then
-77-
Ep(r',t')
= ApEpo cos(k
-r'
-
pt')
(4.1.5)
If we assume further that the frequency resolution is perfect,
o =
then
0 and
= I(T)
I(q,w)
= bV S(q,w),
(4.1.6)
where V is the volume of the scattering region, q -
b = [k2Ep 0 /r]
2
S(q,w).
and S(q,w)
kp,
(4.1.7)
,
r is the distance between the center of
detector,
ks -
the sample and the
is a projection of the
scattering tensor
The projection sampled in a given experiment depends
on the light polarizations inside the sample:
=
S(q,w)
fd 3 rdt<[6d'TO-6e(0,0)-
-[TO-Sc(r,t)*Ce
PI]
1> cos(q -- t)
edhTohjediToilepmsjklm(,&),
-
(4.1.8)
hijklm
where Od is a unit vector which designates the polarization
direction of detected signal.
The scattering function S(q,W)
in the equation above is directly related to the dielectric
constant time-correlation function CCC'(r,t) and CCC(q,w):
Sjklm(q'w)
d3r
=
-
where
{
dt Csklm(r,t)
dt CEklm(q,t)
cos(q-r -
ot)
cos(wt),
C]Clm(r,t) = <Sejk(OO) 6 elm(rt)> and
-78-
(4.1.9)
Ccflm(q,t) =
d3r e-iq'r Ccim(r,t)
= <Scjk(
-qO)&elm(q,t)>.
For optically isotropic media, To
(4.1.8)
S(q,cw)
(4.1.10)
is a unit tensor and Eq.
reduces to
= {d3r
{dt <Scdp(0,0)Sedp(rt)>cos(q-r -wt),
(4.1.11)
where
Scdp =
gives the
S(q,w)
^d'Sce-p.
The fluctuation-dissipation theorem
following relation between the spectral
function
and the frequency-dependent response function GEc(q,w):
211i
mGEl
~),
S~qo)[1-exp(-hw/kT)]
(4.1.12)
Gcc(q,w) is the Fourier transform of Gcc(q,t), the same
impulse response function used in Sec. 4.1.A to describe ISS
signal:
From Eqs.
kl(qo)]
13kl~tw)]
=
fo
dt G
1
kl(q,t)
sinwt.
(
Im[G
(4.1.6) and (4.1.11), we have,
temperature limit (tlw/kT << 1),
4.1.13)
in the high
the frequency-dependent LS
signal in the familiar form
I(q,&)
-
WA
where GCe
Im[Gcc(
(4.1.14)
is understood as
the projection of Gee determined
experimentally by the polarizations of
light. Equation (4.1.14),
derived assuming ideal
incident and detected
analogous to Eq.
(3.3.10), was
frequency and wave vector
resolution.
Simulations of LS data shown below were generated using Eq.
(4.1.14).
4.2 Comparison of ISS and LS
-79-
We see from Eqs.
(3.3.10) and (4.1.12) if the same
projection of Gcc is examined in ISS and LS experiments,
then
in principle the information content of ISS and LS data is
identical.
Since Gcc(q,t)
and Gcc(q,o) are related by Fourier
transformation, the results of each experiment could be
predicted from the results of the other without knowledge of
the material modes involved.
Equations (3.3.10) and (4.1.14)
will form the basis for comparing ISS and LS in subsequent
sections.
We will see that even when in principle the data
provide equivalent information,
advantages
there may be
in practice to use either the
significant
time-domain or
frequency-domain approach.
LS
spectra are often analyzed in terms of time-correlation
functions Ccc(q,t)
rather than response functions G".
photon correlation spectroscopy, Ccc(q,t)
microsecond or longer time scales.
CEc(q,t)
i.e.,
Cijkl(q,t)
(4.1.9)
and
is measured on
Faster components of
are determined by Fourier
from Eqs.
In
inversion of LS
spectra,
(4.1.14),
dw Sijkl(q,w)coswt
0
=
CD
2kT
f
d&
Im
[
Gijkl(q,&)]cosot.
(4.2.1)
0
It
is therefore useful to compare LS and ISS data
Cee as well as G".
in terms of
Of course, ISS data can be used to
determined Gcc(q,w) and Cee(q,t) through the Fourier
transforms indicated in Eqs. (4.1.9)-(4.1.14).
However, a
more direct route is available.
Using the Kramers-Kronig
relation,
one can show
Gce(qt) = 9(t)
d&
Im[Gee(qw)]sinwt,
TEf0
where 0(t)
is the step function.
and (4.2.2) yields
-80-
Comparison of Eqs.
(4.2.2)
(4.2.1)
We see
that GCC(q,t)
differentiation.
response
and CCC(q,t)
are
related through simple
ISS and LS data can be compared in terms of
functions through Eqs.
(3.3.10) and
(4.1.14),
terms of time-correlation functions through Eqs.
or in
(4.2.1)
and
(4.2.3).
We mention briefly some situations in which the two
experiments are not equivalent even in principle.
First,
it
is possible through ISS to examine some components of G~E
which cannot be accessed directly in LS.
Although the
same
Green's function is studied in both cases, different
projections of
two methods.
the Green's function tensor are taken in these
For a given q,
the number of independent linear
combinations of components of
the Green's
be probed in LS and ISS is different.
3,
For LS,
the number is
usually they come from VV, VH and HH scattering geometries.
If we define G'
G'
function which can
of the
= TO-G,
form G'ijij.
then one can only study components of
Since the ISS method has no such
restriction,
components such as G' 1 1 2 2 or G' 1 2 2 2 can be
examined directly.
This capability, which is common to any
four-wave mixing experiment has been exploited recently.
(Ruhman et al. 1987b)
A second distinction between ISS and LS is that, in the
former, the excitation and probe frequencies may differ.
When
either
frequency approaches an electronic transition,
resonance enhancement of ISS signal intensity can occur.
In
addition, if the excitation pulses are near resonance, they
may produce coherent vibrational motion in the electronic
excited state through "impulsive" absorption, as has been
demonstrated experimentally. (Rosker et al. 1986; Ha et al.
1986; Williams & Nelson 1987)
This motion will affect the
time-dependence as well as intensity of the observed signal.
Third, the possibility exists for ISS excitation of
sufficient vibrational amplitude to leave linear response
regime probed by LS.
This has not yet been explored.
Finally, the coherent vibrational motion induced by
(nonresonent) ISS excitation may be probed not only by
-81-
coherent scattering, but by other optical measurements as
well.
Most important is the possibility of probing by time
resolved absorption spectroscopy.
This would permit recording
of absorption spectra of well-defined, vibrationally distorted
species (e.g.,
stretched or bent molecules) with various
vibrational displacements.
4.3 Simulations of ISS and LS data from vibrational and
relaxational modes
In this section, we compare LS and ISS for the
important
cases of underdamped and overdamped vibrational modes, Debye
relaxational modes, and some combinations thereof.
time- and
frequency-dependent impulse
these modes,
From the
response functions for
the dielectric constant Green's functions
GCC(q,t) and GCC(q,w) are calculated using Eq. (3.2.13). Based
on these, ISS and LS data are simulated using Eqs. (3.3.10)
and (4.1.14), which imply "ideal" experimental conditions.
To
make the comparison precise, we assume that in the ISS
experiment the excitation and probe pulses are of same
frequency, thus no dispersion of dielectric constant exists.
We assume
further that the two excitation pulses are linearly
polarized in the same directions as the incident and scattered
probe pulses,
so that the
same dielectric tensor
components
Glcij are sampled as in LS.
A. Scattering from single modes-underdamped and overdamped
vibrational modes and relaxational modes
The time-dependent impulse response functions Ga(q,t) were
derived in Sec. 3.4 of the previous chapter.
Omitting
subscripts of tensor components,
reduce
I(q,t)
Eqs.
(3.2.13) and (3.3.10)
to
tGCe(q,t)1
2
=
(ax)
-82-
4
[Gc(q,t)]
2
(4.3.1)
when only one material mode is observed.
Here we will simply
state
For underdamped
the
results for I(q,w)
and I(q,t).
vibrational modes with natural frequency wo
and dephasing rate
Y,
Im[ GEC(w)]
kT
1
(W-Wa)2 + y 2
W
I(t)
IGEC(q,t)1
=
where wo(q)
y2 -
modes,
(wa -
2
Y2)
=(e
y
+
-
-Yt
(4.3.2a)
2
(4.3.2b)
For overdamped vibrational
1
2 + y
(4.3.3a)
-y2t
(4.3.3b)
where Y1,2 = Y T (Y 2
)
> 0.
1
w6 > 0 and
k2
I(
1
(&+o)2 + y 2
= (e-yt sin& t)
kT (
I~t)
-
,
kT
1
kT(
W2 +
-
-2
(06)h.
)
For Debye relaxational modes,
(4.
,
I~o)
I(t) c (e-rt) 2
3. 4a)
,
(4. 3.4b)
where r = T-1 is the relaxation rate. Note that purely
relaxational behavior can be considered a limiting case of
overdamped vibration as y -+ c, in which case Eqs.
reduce Eqs.
(4.3.3)
(4.3.4) with the relaxation rate given by
r = Wa/2 y.
(4.3.5)
In Eqs.
(4.3.2)-(4.3.4), the dependence of I(w) or I(t) on q
is implicit, since in general, wo, y, and r are q dependent.
Figure 4.1 shows simulated LS data (left-hand side) and
ISS data (right-hand side)
for various cases of vibrational
-83-
SIMULATED DATA
LSs
I(t)
ISS,
1(w)
UNDERDAMPED VIBRATIONAL MODE#
WEAKLY DAMPED,
(o)
WEAKLY OAMPEDs
.
'-Mw/SO
1 "wo/50
1
2
01
4
w.t /2-f
2
6
8
10
-
0
NEARLY OVEROAMPEDs
NEARLY OVEROAMPEOS
(b)
-. 75wo
\ -. 75wo
x 600
1
w / w g-
2
3
0. 0
0.5
1. 0
1. 5
wat /2r
-
0
OVEROAMPED VIBRATIONAL MODEs
SLIGHTLY OVEROAMPEDt
Y1. 06wo
SLIGHTLY OVEROAMPEDs
(c)
1-1. Ow6
0
1
2
3
\
OVERDAMPEDI
(d)
0
1-1. Swo
0
1
2
0.0
4
2
OVERDAMPEDs
1-1. 5wo
2.
5
5. 0
Figure 4. 1. Simulated LS spectra (left-hand side) and ISS data (right-hand side). The simulations
illustrate the utility of ISS for characterization of heavily damped modes. For overdamped modes, even
qualitative distinction from Debye dynamics (dashed curves) is very difficult in LS. (a) Weakly damped
vibrational mode. (b) Heavily damped vibrational mode: relaxational mode (dashed curve). (c) Slightly
overdamped vibrational mode; relaxational mode. (d) Heavily overdamped vibrational mode;
relaxational mode. Only one side of the LS spectra is shown. For the relaxational modes. I was
selected such that the I(w) curves cross at o = 0 and at the half-maximum. In each case, r ~
2y/wo
0
-84-
and relaxational modes.
Figure 4.1(a)
shows the familiar
result that a lightly damped mode appears as a rather narrow,
well resolved peak in the LS spectrum (only the Stokes side of
the spectrum is shown).
oscillations appear.
In ISS data, well-defined, damped
Figure 4.1(b) shows simulated data from
a heavily damped,
but still underdamped, vibrational mode.
the frequency-domain spectrum, this gives rise to a broad
feature centered at
domain,
zero frequency
(solid curve).
In the time
rapidly decaying oscillations are observed.
Figure
4.1(c) and 4.1(d)
show simulated data from overdamped
vibrational modes.
The frequency-domain LS spectra
curves)
(solid
appear as near-Lorentzian features centered at
frequency.
a gradual
The ISS data no longer
In
show oscillations,
zero
but show
rise to a maximum (corresponding to maximum
vibrational displacement)
followed by monotonic return to
zero.
The dashed curves in Figs.
4.lb-4.1(d)
illustrate LS and
ISS data for purely relaxational modes, whose decay rates were
chosen such that the LS spectra for relaxational and
vibrational modes would overlap at
at w =
0)
and half-maxima.
the intensity maxima
(i.e.,
This permits comparison between
the vibrational spectrum and the Lorentzian spectrum resulting
from purely relaxational mode.
As the vibrational mode
becomes more heavily overdamped,
the limiting value r = wd/2y
is approached and the data become more and more similar.
.
B. Scattering from multiple modes
We consider two cases which are
to our actual ISS results.
illustrative and
relevant
We treat scattering from uncoupled
acoustic and relaxational modes,
as discussed in Sec.3.4.D of
previous chapter, and from two uncoupled optical phonon modes.
For acoustic and relaxational modes,
A
I(W)
2Byw&
+
w2+r2
(4.3.6a)
[(w+o,)2+y2
[(w-
-85-
2+y23
c
(A
e-rt
2
+ B e-ytsin,t)
( 4.3. 6b)
,
I(t)
where A and B are amplitudes and all other parameters are as
defined in Sec.4.3.A.
Note that for the purpose of
illustration, we need not consider explicitly coupling between
the two modes, since comparison to Eq.
(3.4.18) of chapter 3
shows that the response take on essentially the same time
dependent form.
For two underdamped phonon modes,
Bjyjwj
I~o
xkT{
[(
w+ wl)) 2
l )2
+ y J ][( ( w -
+ y J]
B2y2w2
[(+A(
I(t)
(B1 e
+
2
2 )2 + y
+ y l][ ( w -
-yit
sinwlt + B 2 e
-y
2
(4 .3 .7a
]
)
)2
t
sinw 2 t
)2
(4.3.7b)
where the subscripts label the two modes.
Simulated data from acoustic and relaxational modes are
shown in Fig.
4.2 With the parameters chosen,
the LS
spectrum
shows broad central peak which overlaps the Brillouin peak.
In the ISS data, the signal from the relaxational mode decays
rapidly. After sufficient decay of this feature, damped
vibrational oscillations become apparent.
Simulations from two underdamped phonon modes are shown in
Fig.
4.3.
The LS spectrum shows two distinct peaks.
data show a "beating" pattern.
Note that in Fig.
4.2,
The ISS
the
scattering features from two modes oveIrlap extensively in
frequency domain, but not in the time domain.
In Fig.
4.3,
the features are well separated in the frequency domain but
overlap in the time domain.
4.4 Comparison of ISS and LS methods
A. Experimental considerations
-86-
VIBRATIONAL
+
RELAXATIONAL MODES
SHORT TIMES
I (w)
I (t)
0
1
LONG TIMES(X30.000)
0 . 25
2
2
I
3
4
(/)--
Figure 4.2. Simulated LS spectra (left-hand side) and ISS data (right-hand side) from uncoupled
vibrational modes described by Eqs. (4.3.6) with w = r. y/r = 0.12, and D/B = 0.01. The
scattering features merge in LS, but are largely separated in ISS.
TWO VIBRATIONAL MOQES
I (w)
I (t)
J
0.0
0.5
V
W(/
1.0
2-
VAA
1.5
0
4
A
8
W2t/2Tr
12
J A.
16
20
--
Figure 4.3. Simulated LS spectra (left-hand side) and ISS data (right-hand side) from uncoupled,
=0.3 7 w,. and B =B2 . The
underdamped modes described by Eqs. (4.3.7) with W, =0.77w,. y
2,=y
scattering features are well separated in LS, but overlap in ISS.
-87-
Here we consider briefly experimental factors in ISS and
LS
including sources of noise,
wave vector
resolution and
range, and frequency or time resolution and range.
The
discussion below concerns ISS and spontaneous LS. A brief
discussion of coherent frequency-domain LS methods is
presented in Sec.
4.4.C.
Al. Wave vector resolution and range
In any light-scattering experiment involving dispersive
modes,
In
the
wave vector
spontaneous LS,
extent of
resolution and range are
the wave vector
to be
optimized.
resolution is limited by
focusing of the incoming laser beam and by the
angular range of the scattered light which is collected.
To
get better wave vector resolution, one should have less
focusing and use a.smaller collection angle.
The
former will
lead to a bigger spot size therefore "see"
more parasitic
scattering thus reduce the signal/noise, and the latter will
reduce the total signal.
The problem of parasitic scattering,
which is unshifted in frequency from the incident light,
made worse by the decreasing
small scattering angles.
is
frequency of acoustic wave at
This can result in loss or
distortion of Brillouin peak due to parasitic "noise."
In ISS, wave vector resolution is also limited by the
focusing of excitation and probe laser pulses as discussed in
detail in the previous chapter.
Since ISS usually have better
signal/noise, one can sacrifice some s.ignal to gain better
wave vector resolution. A rather wide range of wave vectors
(at present state of art, more than two orders of magnitude in
good samples) is accessible.
Scattering angles of less than
50
can almost always be used and angles of less than 10 have
been used.
Beams can be sufficiently collimated such that
acoustic dephasing times as long as y-l = 100 ns can be
measured,
even at small scattering angles.
This
is analogous
to measurement of a Brillouin line width of only 3 MHz.
-88-
A2.
Frequency and time resolution and range
Frequency resolution in LS depends on the spectral
bandwidth of
the incident laser light and the spectral
resolution of the detector, which is usually a grating
monochromator or an interferometer.
The total range of
accessible frequencies depends on monochromator or
interferometer design.
The minimum frequency shift at which
reliable measurements can be made
the
frequency range)
device to
depends on
(i.e.,
the
lower limit of
the ability of the
resolving
reject light from the laser source.
For Raman spectroscopy of molecular vibrations and optical
phonons,
the usual experimental arrangement
involves a double
monochromator with resolution of the order of 0.2 cm-1 and
total frequency range from about 5 cm- 1 to values exceeding
the largest vibrational frequencies of
-
3000 cm~
1
.
For most
vibrational modes of moderate or high frequencies, this
arrangement permits convenient measurement of frequencies and
linewidths.
cm-
1
Very low dephasing rates (i.e., linewidth <
) can be determined,
0.2
but with much more difficulty, by
combining an interferometer with a monochromator in a tandem
arrangement.(Ouillon et al. 1984)
Brillouin spectroscopy of acoustic phonons (and the study
of low-frequency optic phonons)
requires the use of
interferometers, which can reject probing light well enough to
permit accurate measurements of scattering features within
about 1 GHz (0.03 cm~-) of the probe l.ight frequency.
"Typical" spectral resolution and range are hard to quantify,
since many schemes involving multiple-pass or tandem
interferometers have been developed to meet different
experimental needs.
The best
resolution
(-10MHz)
can be
achieved at the expense of very restricted spectral
range.
Typical resolution (especially for studies of solids)
GHz, with spectral
range on the order of 50 GHz
is 1
for single
pass interferometer and up to 500 GHz for multipass or tandem
arrangements.
Better resolution can be achieved at the
-89-
expense of reduced range.
Detailed discussions of
experimental problems in Brillouin spectroscopy and various
possible solutions have been presented.(Cummins & Levanyuk,
1983; Dil,
1982)
In ISS, the time resolution is determined almost
exclusively by the laser pulse duration. (For very short pulse
durations,
by the
i.e.,
< 100 fs,
time
resolution can be influenced
scattering geometry as discussed in chapter 3)
ISS has
been carried out routinely with 60fs pulses and shorter pulses
(-30 fs) could be used without much difficulty.
contrast to LS,
in which instrumental
This is in
linewidths due to
monochromator or interferometer capabilities are far broader
than the linewidths of narrow-band lasers.
The upper limit of
the temporal range in ISS is determined by the method by which
temporal delay is introduced between excitation and probe
pulses.
(i.e.,
Mechanical delays can be used
distances up to -10 meters).
for times up to -30
For longer delays,
ns.
other
optical methods, electronic delays, or electronic gating of a
CW probe laser are practical.
Thus the temporal range can be
made as large as necessary.
This makes ISS well suited for
the study of modes of very low frequency (e.g., <100 MHz) such
as long wavelength acoustic phonons which are difficult to
observe in LS.
Similarly, the upper limit of the frequency
range accessible to LS can be extended as far as needed so
that LS is better suited for the study of high frequency modes
> 500 cm- 1 ) which are inaccessible to ISS.
In
practice, ISRS has been carried out with 60 fs time resolution
(e.g.,
and delays up to 1 ns, and ISBS has been carried out with 80
ps time resolution and delays up to 250 ns.
"Forced" Rayleigh
scattering, which can be thought of as impulsive stimulated
scattering of relaxational modes, has been carried out on time
scales range from picoseconds to many seconds.(Eichler et al.
1986)
The overall resolution of FP interferometers is
characterized by finesse, which is essentially the total
number of independent data points one can take in a spectrum.
Typically it is in the range of 40-70, best ones about 100.
-90-
A3.
Signal/noise considerations
Noise in LS spectra appears mainly around zero frequency
and as a "baseline" at higher frequencies.
The low-frequency
noise arises because there is always some elastically
scattered light due to parasitic scattering from sample
surfaces,
impurities,
monochromator or
resolution.
and other inhomogeneities,
interferometer used all have
and the
finite
Since the elastic scattering can be very strong,
even the
far wing of this elastic peak can overwhelm the real
signal.
The effect of this source of noise is to make the
intensity of low-frequency scattering artificially high and to
mask or distort the low frequency spectrum.
The
baseline at
higher frequencies may be due to intrinsic scattering from
fast material motions,
fluorescence,
(e.g., dark current).
Inaccuracy in baseline subtraction can
or electronic noise
lead to further distortion of the spectrum.
additional
source of noise
Finally, an
is light from unwanted orders of
diffraction from or transmission through the device.
Noise can appear in ISS data from parasitically scattered
excitation or probe light and from electronic sources,
LS.
However,
there
is a important difference:
as in
the noise in
ISS can be reduced by signal averaging, unlike the LS case in
which it appears as part of the spectrum.
This prospect
arises since coherent scattering of the probe pulse occurs
only because of the material response to the excitation pulses
and not from pre-existing sample surfaces or defects or other
scattering sources.
Typically,
pulses are of different color,
scattered excitation light
the excitation and probe
so that filtering prevents any
from reaching the detector.
The
intensity of elastically scattered light from probe beam can
be measured with the excitation pulse blocked and subtracted
from the total signal intensity to determine the intensity of
real signal.
this.
The chopper and lock-in in Fig.
2.6 accomplishes
The remaining noise is equally likely to be positive or
negative,
so they can be averaged away.
-91-
In connection with signal/noise considerations, we note
that LS signal intensity depends on the population of
thermally excited modes and therefore decreases with
temperature as indicated by Eq. (4.1.14).
ISS signal
intensity is unaffected by thermal populations, so ISS is well
suited for low temperature spectroscopy. Note also that for
the same reason, there is not much one can do to increase the
signal to noise of LS experiments while for ISS one can always
increase the excitation pulse energy to increase the signal to
noise.
I believe that with improvements in laser energy and
stability, the signal to noise of ISBS experiments can have
more than two orders of magnitude in the next ten years.
Chapter 8 discusses some possible improvements.
B. Comparison of simulated ISS and LS data
We now return to Figs. 4.1 and 4.2 for a comparison of ISS
and LS spectroscopies of vibrational and relaxational modes.
Figure 4.1(a) shows that lightly damped vibrational modes can
be characterized accurately by either method. The main
criterion for preference is the vibrational frequency. As
discussed in the last section, high-frequency modes such as
many molecular vibrations are best characterized in the
frequency domain, while low-frequency modes such as longwavelength acoustic phonons are most easily characterized in
the time domain. Vibrational excitations in the frequency
range 2 GHz-200cm~ 1 can be readily examined by either method.
Of course, conventional Raman spectroscopy is more easily
carried out than femtosecond time-resolved ISRS, so in cases
where the former faces no special difficulty, it remains the
method of choice. On the other' hand, spontaneous Brillouin
scattering and picosecond time-resolved ISBS are probably
comparable in experimental difficulty.
Figures 4.1(b)-4.1(d) show simulations for heavily damped
vibrational modes and Debye relaxational modes. The
relaxation rates r for the purely relaxational Debye modes
-92-
were chosen such that the LS
intensities at
the maxima
(i.e.,
at w = 0) and at the half-maxima would coincide with the
intensities from the damped vibrational modes.
The LS spectra
from these modes become increasingly similar as the damping
rate increases, and for modes which are even slightly
overdamped [Fig. 4.1(c)], it is usually impossible to
distinguish vibrational from relaxational responses.
For the
parameters selected in Figs. 4.1(c) and 4.1(d), the
vibrational and relaxational spectra overlap very closely
where
signal is strongest and differ most in the wings,
where
the signal/noise ratio is usually poor and baseline
subtraction is problematic.
Of course,
selection of
longer
relaxation times would yield better agreement at the wings and
more difference near zero frequency.
Given the problems
involved in accurate characterization of
zero frequency and in the wings,
the LS
spectrum near
it is not surprising that
very few overdamped vibrational modes have been characterized
unambiguously through LS.
It is usually possible to fit the
spectra within experimental uncertainty with a Debye model.
Since even discerning the vibrational character of heavily
damped modes is difficult, accurate determination of natural
frequencies wo
and dephasing rates y is extreme rare.
ISS data from heavily damped vibrational modes are often
more promising since the short-time responses of the
vibrational and relaxational modes are totally different.
Even the overdamped mode gives rise to signal which increases
gradually after t=O, reaches a maximum, and relaxes
monotonically back to zero.
The
relaxational mode reaches its
maximum instantaneously at t=0 and then relaxes.
As long as
the ISS experiment is conducted with sufficient time
resolution to observe the
initial
rise,
complete and
reasonably accurate characterization of the overdamped mode is
possible.
If only longer-time decay is resolved, then as in
LS,
the
ISS data can be fit
reasonably well with a simple
exponential decay (dashed curve).
example.
-93-
Figure 4.2 shows another
In essence, what discussed above reflects the fact that
events which occur occasionally in the time domain are spread
out in
the frequency domain,
thus for such events,
time-domain
approach is usually the best one.
Figure 4.3 shows simulated data from two underdamped
vibrational modes.
In this case, scattering features well
separated
4.5
in the frequency domain overlap in the time domain.
Summary
The frequency domain spontaneous light scattering and
time-domain ISS are complimentary methods.
When the same
components of the dielectric tensor are sampled,
experiments
the
can be equivalent in principle in terms of
information content of the data they produce.
ISS is easiest
for slow responses and frequency domain techniques are easiest
for large frequency shifts.
In addition, the time-domain
approach is advantageous for the study of heavily damped or
overdamped vibrational modes.
-94-
CHAPTER 5.
FORWARD ISS
ISS excitation can also be carried out with a single
ultrashort excitation pulse
instead of two overlapping pulses.
In this chapter we shall show that impulsive Raman scattering
will occur with no laser intensity threshold even when only
one ultrashort laser pulse passes through many types of media.
In other words, ISS is a ubiquitous process through which
excitation of coherent lattice or molecular vibrations will
take place whenever a sufficiently short laser pulse passes
through a Raman-active solid or molecular liquid or gas.
Impulsive stimulated scattering is therefore a generally
important aspect of ultrashort-pulse interactions with matter.
ISS may be important in a number of situations of current
interest,(Auston & Eisenthal,
1984)
including femtosecond
pulse propagation in crystals and optical fibers and
femtosecond time-resolved experiments on semiconductors and
metals.
In the following section, we present a theoretical
treatment of ultrashort-pulse excitation of coherent
vibrational waves through ISRS.
We then discuss time-resolved
probing of the vibrations by coherent scattering of a second,
time-delayed ultrashort pulse which is incident at the
(collinear) phase-matching angle.
Here, too, we find unique
effects due to short pulse duration.
We show that, depending
precisely on its delay after the excitation pulse, the probe
pulse spectrum undergoes either
a red shift or a blue
shift,
or for certain delays no shift at all.
5.1 Excitation process
As in the case of crossed beam experiments,
the excitation
pulse exerts a temporally impulsive force on Raman-active
-95-
modes of the medium.
fringes,
However,
since there are no interference
the excitation is essentially uniform in the
direction perpendicular to the direction of excitation pulse
propagation.
For a linearly polarized,
excitation pulse,
the excitation force
collimated, ultrashort
can be written as
(see
chapter 3)
Fkl(qft')
where
511
= Akl~kl8(t')S(q),
the propagation time of the pulse through the
been accounted for by defining a "local
-
sample has
time" variable t'
= t
zn/c where z is the direction of light propagation and c/n
is the speed of light pulse inside the
of the medium is given from Eq.
sample.
The
(3.2.8) by
jkk(q,t')S(q)
6eij(qt') = AkkG
response
(5.1.2)
where we have taken the excitation-pulse polarization to lie
in the k direction.
Scij(q,t')
Scij(r,t)
~
For a single damped vibrational mode,
~
Gijkk(q,t')
=
Q0 e
-
Qa(q,t')
sinxt'
8(q).
(5.1.3)
Qx(r,t)
-Y(X(t-zn/c)
=0
sin[c,(t -
e
zn/c)]
(5.1.4)
The equation above describes a traveling-wave oscillation of
frequency w. and wave vector q'= z&,n/c.
The vibrational
wave vector is not exactly zero because of the finite time
required for the excitation laser pulse to propagate through
the
sample.
The pulse first strikes
the front of the sample,
then the middle, and finally the back, and so the vibrational
phase varies linearly as a function of depth in the sample.
-96-
The vibrational wavelength is simply the distance that light
travels
inside the sample during the time
excitation vibrational cycle.
excited,
of one material
The phase speed of the mode
oc/qc, is equal to the speed of light,
the sample.
c/n, inside
modes whose dispersion relation is such that
phase speed cannot equal to the speed of light in the sample
cannot be effectively excited by forward ISS.
forward-ISRS
Note that
is simply Raman scattering in the forward
direction which is
stimulated because the fundamental and
Stokes-shifted frequencies are contained within the excitation
pulse bandwidth.
5.2 Probing process
The traveling-wave excitation produced by forward ISRS may
be probed by a variety of methods.
variably delayed probe pulse
Coherent scattering of a
(which is phase matched optimally
when collinear with the excitation pulse) is possible, but
unlike the crossed beam case, the coherently scattered signal
is not separated spatially (i.e.,
transmitted probe pulse.
"diffract") from the
The probe pulse velocity or
polarization may be affected, and so the vibrational motion
can be monitored by measurement of the time
required for
passage of the probe pulse through the sample (Halbout & Tang,
1982) or of the polarization of the transmitted probe pulse
(i.e.,
"optical Kerr effect" experimental configurations).
Alternatively,
the spectral properties of the probe pulse are
influenced by coherent scattering, as discussed below,
and
through their measurement, the vibrational motion can be
monitored.
It is easy to see why the probe pulse undergoes delay
dependent spectral changes.
When the probe pulse is incident
collinear with the excitation pulse,
the ultrashort probe
pulse "rides" the crest or null of the wave through the sample
since the phase velocity of the material vibrational wave
-97-
The probe pulse
equals the speed of light in the medium.
each region of the sample with the
"sees"
distortion and velocity, i.e.,
same vibrational
sees a spacially uniform
Depending on the phase of the vibrational velocity
sample.
(which is determined by the probe delay relative to the
excitation pulse), the impulsive force exerted by the probe
pulse either accelerates or slows down the vibrational motion.
This is analogous to applying successive driving forces (which
may be in or out of phase)
to a pendulum.
When acceleration
occurs, the probe pulse loses energy to the vibrational mode
and therefore is red shifted.
the
When slowing down occurs,
probe pulse gains energy and therefore is blue shifted.
(Applying the same argument to the excitation process,
one
concludes that the excitation pulse should always red shift.)
In the
following we support the arguments above with a
mathematical derivation.
For simplicity, we assume that the medium is isotropic.
(3.2.14)
Equation
a2 E
_
az 2
n2
92 E
c2
2
at
can be written as
02
(QEL),
c2
(5.2.1)
at2
where E is the total field, EL is the probe field,
c is the
speed of light in vacuum, n is the index of refraction of the
medium, and 0 = N(aa/aQ) 0 is the dielectric susceptibility
derivative
(a is the polarizability per molecule and N is
number density).
z is the chosen to be in the direction of
propagation and the pulse is approximated as a plane wave.
The scattered field is assumed to be small enough to be
neglected in the right-hand side of Eq.
To solve Eq.
(5.2.1).
(5.2.1) we neglect depletion
probe field and the vibrationalwave.
of the
The results therefore
apply only for the case of weak a probe pulse.
the local time variable t' = t -
(or gain)
Introducing
zn/c, we rewrite Eq.
(5.2.1)
as:
a2 E
az 2
a2 E
2
_2n
c
azat'
~2
_4n
2-
_2
c2
[Q(t')EL(t')]
at' 2
-98-
.
(5.2.2)
This can be solved to yield
E(z,t';tD)
= EL(t')
-
z
EL(t
where the boundary condition E(t'=O,
been used.
Equation
(5.2.3)
shows
(5.2.3)
z=O)=EL(t=O,
z=O)
has
that when depletion of the
incident field is negligible, the scattered field grows
linearly as a function of distance in the sample.
For convenience,
in this
t = 0 as the
section we define
time at which the (center of the
) probe pulse strikes the
front of the sample,
preceded by the excitation pulse which
struck at time -tD-
To determine
the spectral content of the
emerging field, we choose
Q(t')
= Q 0 sin(&xt'
+ tD)
to describe the vibrational mode.
Eq.
Fourier transformation of
(5.2.3) yields
E(z,w)e-znw/c
= EL(&)
+ Bw[e
t DEL(w+o)
where B =(n/nc)OlQ 0
from Eq.
(5.2.4)
-
e
at D EL(C
-
1,
wa)
and 1 is the sample length.
that the spectral
It is clear
content of the emerging
pulse differs from that of the incident pulse.
pulse consists of three parts:
(5.2.4)
The emerging
the unshifted field,
ELUW), and
the red- and blue-shifted fields EL(w -w) which arise due to
Stokes and anti-Stokes scattering,
pulses (w0tL
respectively.
For long
1), the three spectrally narrow fields
oscillate at distinct, well-separated frequencies and the
observed intensity spectrum simply consists of three lines at
For short pulses, the spectral width of each field
wL' wL w(.exceeds the frequency separation between them, and
interference occurs among all three overlapping fields.
Figure.
5.1 illustrates this point.
The results of this
interference depend on the probe pulse delay,
tD,
since the
phases of the shifted fields depend sinusoidally on tD.
-99-
For
Incident probe pulse spectrum
Stokes
Anti-Stokes
/ /
/
Exit probe
pulse spectrum
K
Figure 5. 1. The exiting probe pulse is a sum of incoming probe pulse and Stokes and anti-Stokes
shifted pulses. The dashed curve in the bottom sketch is the unshifted curve,' same as the top one.
-100-
example, a short in-phase probe pulse (watD=0, 2n, 4n, etc.)
is red shifted while an out-of-phase probe pulse ((atD = n,
3n, etc.) is blue shifted.
The dependence of coherent scattering on probe pulse delay
can be illustrated explicitly by calculating the
experimentally measured intensity spectrum, I(w) =
(nc/4n)IE(w)I 2 . For clarity we calculate I(w) for a Gaussian
incident field described by
EL=Ae-(t-zn/c)2/ 2 TL2csAL(t-zn/c)],
(5.2.5)
where A is the electric field amplitude, TL the pulse
duration, and wL the central frequency of the Gaussian
frequency spectrum whose width is TL 1
Transverse spatial
variation has been neglected.
This yields
I(w)
)2
a e
=
+ 2aBw e
-
e
02 L2 /4coswatD { e~Ew~(wL~wa/2)2
2
L2
(wL+/ /2)] 2 TL 2 I
+ a(Bw)2{e[~AI
-
L2
-
(wL~wa)]2 TL 2 + e- '
coswO tD e-x2 TL 2 e_(
_L) 2TL 2
},
-
(wL+w(x)] 2 TL 2
(5.2.6)
where a - ncA 2 TL 2 /16n.
Since BwL << 1 (incident light is not
2
depleted), the (Bw) terms are negligible in most cases.
The
first term represents most of the probe intensity which passes
through the sample unaffected. The second and third terms in
Eq. (5.2.5) show that the spectrum is red or blue shifted
depending on delay.
We note that for long pulses (caTL
1), the second,
third and sixth terms of Eq. (5.2.5) vanish and we find the
standard result of both Stokes and anti-Stokes shifted light
in equal amounts, independent of delay.
Experimental observations of the predictions above have
been successfully carried out by S. Ruhman et al.(Ruhman et
al. 1987c).
Fig. 5.2 shows red- and blue-shifted spectra
together with the unshifted spectrum. Fig. 5.3 shows red and
-101-
blue shifting of the transmitted probe intensity as a function
of delay (B) together with data in the crossed excitation
pulse geometry (A).
-102-
CH 2 Br 2 Spectral Shifts
4-)
CT)
-J-
.6
610
618
622
length
(nm)
614
W ave
626
) spectra. taken at
-), and blue-shifted ( - Figure 5.2 Unshifted (solid line), red-shifted (is shown in
material
this
for
data
ISS
beam
different delays. The sample is CH 2Br2 liquid. Crossed
Fig. 5.3A.
-10 3-
ISRS of CH2Br2
A) Crossed Excitation
Pulses
U)
C
a1)
C
C
0
B)
Forward ISRS
Single Excitation Pulse
609nm
620nm
E
C:
-
C-
~0
L
n~
.
_C
3
0 .3
0.9
1.5
Time in Picoseconds
Figure 5.3 ISRS data from the 173 cm 1 Br-C-Br "bending" vibrational mode in CHBr,. (A)
Crossed excitation pulses were used, and the time-dependent intensity of diffracted probe pulse light
was measured. The oscillations in the data correspond to coherent vibrational oscillations of the
molecules. (B) A single excitation pulse (Forward ISS) was used, and the time-dependent intensities
of red and blue frequency components of the transmitted probe pulse were measured. The spectrum of
the transmitted probe alternates from red to blue shifting at the vibrational frequency. This causes the
intensities of the two spectral components to undergo antiphased oscillations.
-104-
CHAPTER 6.
ISS STUDY OF LIQUID-GLASS TRANSITION IN GLYCEROL
6.1
INTRODUCTION
When a liquid is heated or subject to a change of applied
pressure
or field,
it relaxes toward a new equilibrium state.
On a microscopic level, the response includes some changes
which also occur in solids,
e.g. changes in the average
distance between molecules or the average vibrational energy.
Unlike ordinary solids, liquids can also undergo structural
relaxation which involves changes
topology.
In this sense,
in intermolecular bonding
relaxation in liquids is more
extensive than in solids.
The
large heat capacity,
thermal
expansion coefficient and stress compliance of liquids
relative to solids all reflect this fact.
relaxation slows down as
temperature
crystallization is somehow avoided
example),
structural
Structural
is lowered.
If
(by rapid cooling,
for
relaxation becomes very slow and an
amorphous solid is formed.
Macroscopically,
structural
relaxation manifests itself through relaxation of properties
such as elastic modulus,
etc.
By studying the
specific heat, dielectric constant,
relaxation of these macroscopic
properties, one may be able to infer some aspects of the
underlying microscopic relaxation process.
One characteristic of structural relaxation in liquids is
the wide range of time scales involved.
The relaxation time
changes from picoseconds in the high temperature "simple
liquid" state to hours or longer in the low temperature
"glass"
state.
To elucidate the relaxation spectrum at even a
single temperature,
measurements usually must be
carried out
over a temporal range of several decades.
This presents an
enormous experimental
to infer a
challenge.
In order
broadband relaxation spectrum from narrow bands of data,
"Master plot" method is frequently used.
-105-
The underlying
the
assumption is that the relaxation spectra at different
temperatures differ only by a translation on a logarithmic
frequency scale.
has been suggested (McDuffie & Litovitz, 1962; Angell
Torell,
1983) that at least in some materials the relaxation
spectrum becomes narrower at high temperatures.
the
&
This is not necessarily true and in fact it
In addition,
choice of the temperature-dependent translation parameter
is subjective to some extent.
Finally,
necessary
normalization constants (such as limiting moduli) are often
not available and must be estimated through extrapolations.
The motivation
fact that
for the present experiment stems from the
there is a general
shortage of accurate experimental
data for viscoelastic liquids in the frequency region from
100MHz to
5GHz.
Ultrasonics usually provides accurate
acoustic data in the 1 -
100 MHz range.
usually provides data in the
5 -
20 GHz
Brillouin scattering
range.
The
development of ISS have made small-angle light scattering
experiments possible on a
reasonably routine basis.
Thus the
frequency gap between ultrasonics and spontaneous Brillouin
scattering can be bridged, yielding four continuous decades of
data for the elastic modulus.
Besides electrostriction, which was discussed in chapter
3,
laser heating also played an important
experiments we performed in glycerol.
role in the
If the pulses are
absorbed by the sample and the absorbed energy is thermalized
rapidly, then impulsive heating occurs at the optical
interference maxima and stress is exer-ted.
The density
response to this stress will also scatter light.
To conform
to our nomenclature, we shall call this process impulsive
stimulated thermal scattering
(ISTS).
For ISTS,
the
dielectric constant response is' slightly different from what
was presented in chapter 3.
We shall treat this in the theory
part of this chapter.
In impulsive stimulated Brillouin or thermal
scattering,
the time-dependent density response to impulsive stress or
heating,
respectively, is observed.
-106-
These responses
(essentially, the elastic or thermal expansion response
respectively) may have slow components in addition to the
fast
components that contribute to transient acoustic wave
generation.
The slow density response, often called the
Mountain mode,(Mountain,
1968)
can also contribute to
diffraction of the probe pulse.
The behavior of this mode has
been studied on the nanosecond time
scale
(with some
difficulty) by spontaneous light scattering
spectroscopy.(Carrol and Patterson,
1984)
Finally,
thermal
diffusion (the Rayleigh mode) can be observed in ISTS at long
times.
Both Rayleigh and Mountain Modes provide important
information about the underlying structural relaxation and
other properties of the material.
This aspect of
ISBS and
ISTS is shared by spontaneous light scattering but not
ultrasonics.
We shall see below that in many cases ISTS
is
better suited than spontaneous Rayleigh-Brillouin scattering
for the study of these modes.
Here we present experimental study of glycerol using
impulsive stimulated light scattering with scattering
wavelengths between 68.8pim and 0.76pum.
acoustic waves observed range
with ultrasonic data
from 20 MHz to
(Jeong et al.
Brillouin scattering data,
The frequencies of the
1986)
3 GHz.
Together
and spontaneous
(Pinnow et al. 1968) this provides
nearly four continuous decades of acoustic data from 2MHz10GHz.
Based on this elastic modulus information and other
results, we are able to conclude that the width of the
distribution of relaxation times does-not change significantly
for the temperature range from 200K to 300K.
In addition,
Mountain mode and scattering intensity measurements
indicate
that the thermal expansion coefficient also shows relaxation
similar to the elastic modulus and specific heat.
A
phenomenological theory is developed to describe the observed
data.
The acoustic frequency and damping, Mountain mode time
dependence,
relative acoustic mode and Mountain mode
amplitudes, and thermal diffusion are all accounted for.
theoretical framework developed here also applies to
ultrasonic and specific heat spectroscopy.
-107-
The
In the next section,
6.3,
the theory is presented.
the experimental arrangement is described.
data are presented in Sec.
6.4.
In Sec.
ISBS and ISTS
A discussion of qualitative
features of the data is followed by more detailed quantitative
analysis.
The
results are summarized in Sec.
6.5.
6.2 THEORY
In this section we present a phenomenological theoretical
framework for the interpretation of light scattering and other
experimental data in terms of macroscopic material parameters
and
relaxation functions.
We start from the following
linearized equations:
Mass conservation:
Newton's law:
p @
Thermal diffusion:
where
+
pV-v = 0
,
(6.1)
-Vp + VF,
T a
at
=
(6.2)
KV 2 T + Qe,
(6.3)
p is the mass density, p is pressure, S is entropy per
unit volume,
K is thermal conductivity, Qe is the heating
rate per unit volume by external heat sources
heating),
light.
(e.g.
laser
and F is the electrostrictive pressure due to laser
We have neglected shear stress in Eq.
(6.2).
We now need to express p and S in 'terms of density p and
temperature T.
If local equilibrium is assumed,
then we
should have
sp
-
(
- ) ST
ap ) T Sp + ( aT
P
TSS = Cv8T + T(
(6.4)
P)T sp
(6.5)
These relations are not valid away from local equilibrium.
therefore introduce memory functions M(t), N(t), C(t), and
D(t), such that
-108-
We
Sp=
dt'
TSS
M(t)
M(t-t'
)
%S)-
dt'C(t-t')
=
can be understood as the
time-dependent specific heat,
(6.6),
dt'
+
N(t-t')
dt'D(t-t')
;
(6.6)
.(6p
(6.7)
time-dependent modulus, C(t)
etc.
(6.7) and (6.1) into Eqs.
the
Substitution of Eqs.
(6.2) and (6.3) yields (for
one spatial dimension coincident with the scattering wave
vector)
32p
2
r
1
-Jdt'M(t-t')
at 2
aP
-
9
a2
2
-fdt'N(t-t')
ax 2 at'
(@
F)
=
-2
ax 2
8X2
t
F
(6.8)
and
dt'
C(t-t) a2
-
dt'D(t-t') a2
K
Laplace-Fourier transformation of Eqs.
--
+ m(s)s-
n(s)s
2
Qe
(6.9)
(6.8) and (6.9) gives
p(qs)
F(q,s)
-
-
d(s)s 2
c(s)s 2
+
q2
K
T(qs)
(6.10)
Qe(qs)
where
=
*(q,s)
f dt dx exp(-iqx-st)
(x,t).
* denotes any transformed function.
n,
I-n Eq.
(6.11)
(6.10) we used m,
c and d to denote the Laplace transformations of M(t),
N(t),
C(t)
letters M,
and D(t)
because we want to reserve the
capital
N, C and D for the corresponding s-dependent
quantities with the same physical meaning as their timedependent counterparts.
This is done by defining M(s) =
m(s)s,
N(s)
= n(s)s,
C(s)
= c(s)s and D(s)
= d(s)s.
We shall
freely use phrases such as "the frequency dependence of M" or
"the time dependence of M" in this chapter.
It is easy to see
that relations such as M(t->0) = M(s->o) = M, M(t->m) =
-109-
M(s->0) = M0
hold for M,
N, C and D.
We shall follow the
convention to use subscripts a and 0 to denote high frequency
(short time)
limit and low frequency (long time)
limit,
respectively.
The matrix equation (6.10) can be expressed in abbreviated
form as
HX =
Y.
(6.12)
with
S2 +1
+
q
H=
2
-N
N
P
(6.13)
Cs + q
Ds
-
2
K
Note that although the shear modulus was neglected in Eq.
(6.2),
as
it can be included in Eq.
just the bulk modulus but as
includes the shear contribution.
(6.13)
if we consider M not
the longitudinal modulus which
The inversion of Eq.
(6.12)
gives
X(q,s)
where
= G(q,s)Y(q,s)
(6.14)
the components of the Green's
G,,(q,s)
GpT(q, s)
function are:
Cs + q2K
A
-N
= A
(6.15)
Ds
A
GTp (q, s)
GTT(qs)
s292
+ M/p
A
and
A
=
det(H)
=
(-
q2
+
P
M)(Cs + q 2 K) + NDs
-110-
(6.16)
Experimentally, one tries to apply a particular force Y and
measure the
response X to infer M,
or frequency domain.
N,
C and D in either time
The corresponding Green's functions are
defined by the expressions
X(qt) = {dt'G(qt-t')Y(q,t')
X(xs)
(6.17)
,
X(xt) = fdx'dt'G(x-x',t-t')Y(x',t')
I
(6.18)
= fdx'G(x-x',s)Y(x',s)
(6.19)
These Green's functions are found through appropriate LaplaceFourier
transformation of G(q,s).
G(q,t),
of special
importance in time-domain scattering experiments,
is
determined by
a+iw
G(qt>0) =
S
ds G(q,s)est
a-ic
lim(s-si)G(q,s)esi
t
+ 2
i s-+si
1
E
2j
ds G(q,s)est
(6.20)
C
where si are positions of discrete poles, and Cj are contours
around branch cuts.
Mathematically,
summarizes many experiments.
the above formalism
In spontaneous light scattering
experiments,
I(q,w)
= (kBT/W)Im[Gpp (q,s=-iW)],
where kB is the Boltzmann constant, is measured.
I(qit)
I(q,t)
oc
In ISBS,
(6.22)
= sGur(q,t)2
is measured.
(6.21)
In ISTS,
IGpT(q,t)12
(6.23)
-111-
is measured.
heating
Sometimes both electrostriction (ISBS)
and
contribute to signal in an ISS experiment.
(ISTS)
such cases the total signal can be found from Eqs.
In
(6.12) and
(6.14) to be
I(q,t)
where F 0
heating
=
[6p(qt)]
and Q0
rate,
are
2
= [F 0 GoP(qt)
+ QoGpT(q,t)]
2
(6.24)
integrated electrostrictive pressure and
respectively.
In many ultrasonic measurements,
the roots on the complex q-plane of
A(q,s)
= 0
(6.25)
for specified s= -i&
are measured.
In specific heat
spectroscopy measurements,
GTT(x,s=-iw)
(6.26)
is measured.
We first consider ISS experiments at the thermal
equilibrium limit,
i.e. experimental time scales much faster
or slower than the range of relaxation times in the sample.
In this case, M, N, C, and D are time independent and given by
the expressions
2)T
M = p( ap
T ,
N =
(")
8T p
C = CV
,
(isothermal compressibility)
(6.27)
,
(6.28)
(constant volume heat capacity)
(6.29)
D = -T(I-S)
ap T
=
(ap)
=
p aT p
p
N .
(6.30)
We further assume that the thermal relaxation time is much
longer than the experimental time scale,
in Eq.
(6.13) can be neglected.
so that the Kq 2 term
The Green's function in this
case has no branch cut, and the eigenmodes are solved from
-112-
M)C + ND ]s = 0
+
S[(
(6.31)
.
P
q2
The s = 0 root corresponds to the thermal relaxation mode.
Using thermostatic relations,
we have,
for the adiabatic
elastic modulus Ms:
SpND
Ms = M +
C
The roots, given by
_q2Ms=
p
=
s2
_.2
correspond to the acoustic mode.
Green's
The components of the
function relevant to light scattering are
Gop(q,s)
q2
s-iw)(s+iw)
=
G PP (q,t) =
9
v sin(wt)
(6.32)
,
where v = w/q is the speed of sound;
GpT(qs)
= Cs(s-iN)(s+io)
GpT(q,t)
=
N [1 v2C
cos(wt)]
(6.33)
Thus the signal for ISBS is
I(q,t)
IG ,(q,t)I
2
=
sin(wt)
2.
(6.34)
We see that under the assumptions made', only the acoustic mode
is observed in ISBS.
The response of
the material in ISBS
is
analogous to that of a mass on a spring driven by an impulsive
force.
It starts with zero displacement and maximum speed,
and oscillates about the original equilibrium position.
For
[N [1 - cos(wt)]
.
(6.35)
I(t)
ISTS,
c
IGpT(q,t)1
2
N
Lv2C
-113-
I
We see that unlike Gpp, GpT has a non-oscillatory part in
addition to the oscillatory part.
The Rayleigh mode
(s=Q)
is
excited and produces a density response whose amplitude is
equal to that of the acoustic mode.
The ISTS excitation
pulses produce a sudden temperature grating to which the
steady state thermal expansion response is a time-independent
density grating.
The transient response
is an acoustic
standing wave in which the instantaneous density at any point
oscillates about the steady-state value.
Note that the
intensity of ISTS has no strong q-dependence.
ISTS is
therefore
small angle
scattering.
Note also that I(q,t)
of N/(v 2 C) =
is proportional to
the
square
CPC
and v. are usually available, we can obtain N. by
comparing the signal
(3p/aT)p/C .
Since
(ap/aT)po,
Cp0
,
(<5*)
somewhat better suited than ISBS for
intensities at the high and low frequency
limits.
Eqs.
(6.32) and (6.33) hold for both high and low
frequency limits.
For ISTS,
Eq.
(6.35)
shows that the
signal
depends on GpT which in the high-frequency limit is
proportional to N./vC.
In the low-frequency limit GpT
No/v6Co.
These limits are realized experimentally at very low
and very high temperatures,
respectively.
In the intermediate
cases, i.e. when structural relaxation occurs on the same time
scale as the collective motion, we cannot approximate M, N, C
and D as constants.
The non-oscillatory part of GpT will
assume the high-frequency limiting amplitude, N./v.C., at
short times and reach the low-frequenc.y limiting value,
No/vaC 0 , in a time longer than that required for structural
relaxation.
The exact time evolution of
time dependences
M, N,
C and D.
(or equivalently,
signal depends on the
frequency dependences)
of
Although there are theories predicting their
time-dependent forms, we will consider them adjustable
functions to be found through comparison to experimental
results.
f(s)
We write them in the general
=
.
+
( 0 -
.)h(s),
form
(6.36)
-114-
with
h(s)
=
1
(6.37)
( 1+ ( s-r 1)
and with 0 < a,
0 < 1.
This relaxation function can be
described in terms of a distribution of relaxation times f(t):
h(s)
=
where f(T)
f(-r)
dln(t)(T)
,
(6.38)
is given by (Titchmarsh, 1948)
=
) -
[h(e
)]
h(-e
(6.39)
Several distributions with different values of
shown in Fig.
6.1.
When 0
Cole-Cole distribution.
-
1,
f(t)
(Cole & Cole,
reduces to the
1941)
becomes the Cole-Davidson distribution.
The
combination of parameters
extent of asymmetry as well as
The parameter
T 0 in h(s)
in h(s)
x and 0 are
symmetric
When o = 1,
f(T)
(Davidson & Cole 1951)
allows us to vary the
the width of
the distribution.
can be different for M,
N,
C,
and D.
In the simulation of data, we assumed that the relation of Eq.
(6.30) between D and N holds for all times,
the number of unknown functions.
functions are found from Eqs.
C of the form of Eq.
by Eq.
(6.37).
thereby
reducing
The time-dependent Green's
(6.12) and (6.14) using M, N and
(6.36) with the relaxation function given
The Green's function G(q,s) has two poles
(complex conjugate of each other) in the left half of the splane,
corresponding to the damped aco'ustic mode,
and a branch
cut on the negative real s axis, which gives rise to the
Rayleigh and Mountain modes.
For the high and low frequency
limits, the branch cut reduces to a single pole,
to the Rayleigh mode.
of Eq.
corresponding
If Debye relaxation is assumed instead
(6.37), then the branch cut reduces to two poles, one
mostly Rayleigh mode and the other mostly Mountain mode.
6.3.
EXPERIMENTAL
-115-
(1)
(1)
(2)
(3)
(4)
0.8
0.4
0.7
0.9
1.0
1.0
0.8
0.4
(4)
- -(3)
(2)
-2.0
-1.0
0.0
1.0
2.0
3.0
LOG OF -r/-r,
Figure 6. 1. Distributions of relaxation times corresponding to the relaxation function given by Eq.
(6.37) for different values of (x and 0.
-116-
The ISTS experimental setup is illustrated schematically
in Fig.
2.6.
A
cw Nd:YAG laser is acoustooptically mode-
locked and Q-switched (at a 500-Hz
repetition rate)
to produce
pulse trains of 1.064-pm, 85-ps pulses with up to 60 pJ of
energy.
Three of the pulses are isolated from the pulse
trains by electrooptic Pockels'
cells.
Two of the pulses are
overlapped spatially and temporally inside the sample for ISTS
excitation.
Since glycerol has weak vibrational
overtone
absorption at 1.064 pm, heating effects dominate the signal.
The third pulse is frequency-doubled, variably delayed along a
DC-motorized delay line,
and used to probe
at the phase-matching angle.
the excited region
The diffracted signal is
detected by photodiode and a lock-in amplifier whose output is
stored by a personal computer.
The optical delay line is
double-passed to provide a maximum delay of about 20ns.
To
probe longer delays, later pulses from the pulsetrain are
selected.
In this manner, delays of more than 150ns are
possible.
In such a long time,
propagate about 400 pm.
the acoustic wave can
In order
to minimize the apparent
decay of acoustic signal due to acoustic wave packets leaving
each other and leaving the probed region, cylindrical lenses
were used to
focus the excitation and probe
laser beams so
that the excitation region was an ellipse whose long axis
(about 2mm long) was aligned along the direction of acoustic
wave propagation.
The sample, obtained from Mallinckrodt, was
pumped under vacuum for 48 hours and s.ealed into a 20-mm
pathlength fused quartz cell with flat windows.
Experiments
were carried out with scattering angles of 0.890, 2.640,
5.210,
9.830,
wavelengths
and 0.76pum.
18.30,
30.70,
and 88.90,
of 68.8pm, 23.1pm,
yielding scattering
1l.7pm, 6.21pm,
3.35pm,
2.0lpm,
Note that "scattering angle" is defined here to
mean the angle between excitation pulses outside the sample.
For most scattering angles,
data were recorded at temperatures
from 360K to 200K with 5K steps.
with
~
Collection of one data scan
150-ns temporal range took about one hour,
small scattering angles.
-117-
longer at
The ISBS experimental setup was similar to that used for
ISTS.
In this case the entire Nd:YAG pulse train was
frequency-doubled to yield 532-nm pulses.
used for ISBS excitation,
Two of these were
which in this case produced a larger
response than heating since the sample absorption at this
wavelength is very weak.
The rest of the pulse train was used
to synchronously pump a tunable picosecond dye laser whose
cavity-dumped 590-nm output was used for the probe pulse.
The
ISBS experiments were carried out before data acquisition was
under computer control.
were used,
Round
laser spots of 150-pm diameter
and acoustic wave propagation effects were
apparent.
delay.
Data were recorded with only about iOns total
Due to these limitations, only acoustic frequencies
(and acoustic damping when sufficiently strong)
measured with reasonable accuracy.
could be
ISBS experiments were
carried out with scattering wavelengths 10pm,
5.Opm,
2.0pm.
(5.0pm)
experiments were
14.6K.
Although sample
For one
scattering wavelength
carried out for temperatures as low as
cracking was extensive at
and
temperatures below 130K, we were
able to find small regions in which data could be taken with
the small laser spot sizes used.
Scattering angles were measured with an uncertainty of
+1%,
yielding similar uncertainties in the absolute speeds of
sound.
At each scattering angle, relative values of the speed
of sound are usually reliable to better than 1%.
6.4. RESULTS AND DISCUSSION
A. Qualitative features of the data
We
start this section by discussing some of the data scans
to get a qualitative understanding of what we observe and what
information can be extracted.
Fig.6.2 shows
three ISBS scans
scattering wavelength.
recorded with a 5pm
Since ISBS data were recorded under
poorer experimental conditions than ISTS data,
-118-
almost all of
ISBS GATA
T=405K
///\
T=290K
x25
x5
x2. 24
2
,xll.
x55. 9
T=95. 6K
0
2
4
6
8
T IME (NS)
Figure 6.2. ISBS data from acoustic mode in glycerol. taken with acoustic wavelength X = 5.Op m.
The apparent damping and baseline rise in the top data scan are due to counterpropagating acoustic
wave packets travelling away from each other and out of the probed region. In the middle data scan.
each successive oscillation (up to the sixth) is magnified by the factor 2.24 = 5'12. The dashed curve
is a simulation generated using the theory presented in this paper and assuming a small (2.5%) ISTS
contribution to signal amplitude.
-119-
the quantitative results presented below were extracted from
ISTS data.
However,
ISBS data.
In addition,
some acoustic
speeds were determined
from
it is of value to see how various
scattering features manifest themselves in the ISBS
experiment.
The top sweep shows data recorded at T=405K,
structural
relaxation spectrum lies well
at which the
above the
acoustic
frequency of 327 MHz. This corresponds to the low-frequency
limit discussed in Sec.
6.2.
From Eq.
(6.32), we expect the
signal to show oscillations with little or no damping.
practice,
signal
In
some distortion is apparent (the baseline rises and
strength diminishes gradually)
propagation effects,
due to acoustic
as discussed in the previous section.
A
small amount of sample heating occurs even with 532-nm
excitation wavelength, and so there is also a slight
contribution to signal through ISTS.
from Eq.
(6.24),
The effect,
is to make the second,
fourth,
apparent
sixth,
etc.
oscillation appear slightly stronger than would be the case
from ISBS excitation alone.
At
T=290K,
acoustic
the
structural
frequency.
relaxation spectrum overlaps the
The acoustic wave
is therefore heavily
damped and its frequency is higher than that at 405K.
In this
case,
the decay of signal is due almost entirely to real
effects inside the sample.
In the figure, each successive
oscillation (up to the sixth peak) is magnified by the factor
2.24 = 5 .
The oscillations in the figure clearly do not
undergo simple exponential decay.
The. data are reproducible
and show this behavior for temperatures between 250K and 330K.
The alternation in oscillation
additional
"Mountain Mode"
therefore to signal.
intensities arises
from an
contribution to GoP(q,t)
To see this,
and
consider the the signal
which arises from a superposition of Debye and damped
oscillatory terms:
I(t)
=
{a e-Yt + b e-Y'tsin[w(t-t0 )]} 2
-120-
,
(6.40)
where a/b = sin(wt 0 ) << 1.
term alternates in time,
Since the sign of the acoustic
the acoustic peaks in the data are
alternately increased or decreased by the Debye term whose
sign does not change.
In the 290K data of Fig. 6.2, the
Mountain mode term decays more rapidly than the acoustic mode.
In
fact,
decay.
the Mountain mode does not show a
single exponential
The dashed line in the figure is a simulation
generated using the theory and relaxation function discussed
in Sec.
6.2 and material parameters given below.
apparent that the main features of the data are
the
simulation.
It is
reflected in
Simulations of data are discussed in more
detail below.
At still
lower temperatures,
structural
relaxation occurs
much more slowly than the acoustic oscillation period,
the
i.e.
relaxation spectrum does not overlap with the acoustic
frequency.
This corresponds to the high-frequency limit
discussed in Sec. 6.2 and described by Eq. (6.34).
The data
resemble the high-temperature data except for higher acoustic
frequency.
This reflects the fact that M. > M 0 . In the high-
frequency limit,
the acoustic wave
is essentially decoupled
from structural relaxation while in the low-frequency limit
the local structure fully "follows" the acoustic oscillations.
Thus the acoustic damping passes through a maximum at
intermediate temperatures and the acoustic frequency increases
monotonically as temperature is reduced.
Figures 6.3 and 6.4 show ISTS data at various temperatures
along with simulations.
Since much la.rger excitation and
probe spot sizes were used in these experiments,
recorded out to longer delays
for Fig.
(note the change of
time scale
6.4) without serious problems from acoustic wave
propagation.
The
top sweep shows data
recorded at
Again we are in the low-frequency limit,
Eq.
data can be
(6.35).
temperatures.
360.5K.
described for ISTS by
The acoustic damping is weak at high
In addition to acoustic oscillations,
a steady-
state density grating (the Rayleigh response) also contributes
to signal as described earlier and in Eq.
-121-
(6.34).
This term
SIM ULATION
ISTS 2.6o DATA
360 - 6K
306. 4K
X2
291. 3K
X2
X3
276. 3K
X3
X4
266. 5K
X4
0
X4
261 . 2K
X6
239 . 6K
20
40
60
80
X6
120 6
100
TIME
20
40
60
80
100
12?0
(nsec)
Figure 6.3. ISTS data scans recorded at several temperatures with a scattering angle of 2.60. The
simulations were generated using the theory and procedures described in the text.
-122-
ISTS 18.30 DATA
S I MU
T0 N
360. 5K
X2
300. 8K
X2
X3
286.6K
X3
X3
275. 9K
X3
X4
XK
256. OK
X6
0
5
X6
241 .5K
10
15
20
25
30
75
TIME
120 0
10
15
20
25
30
75
120
(nsec)
Figure 6.4. ISTS data scans and simulations for a scattering angle of 18.30 at various temperatures.
Note the change of scale at t = 30ns.
-123-
decays eventually due to thermal diffusion. Qualitatively,
impulsive heating produces a immediate rise in pressure.
The
steady-state density response gives rise to the constant
signal which remains at long times.
response
The transient density
"overshoots" the steady-state level and this launches
the counter-propagating acoustic waves.
Note that if the
pressure rise did not occur very rapidly relative to the
acoustic period,
then excitation of the acoustic mode would be
less efficient.
If the pressure increased much more slowly
than the acoustic period then only the steady-state
would be observed.
6.5,
response
This is illustrated by the data in Fig.
which were recorded with the excitation pulse duration
lengthened to 500ps.
Since the pulse duration exceeds
the
acoustic oscillation period for the large scattering wave
vector used, no transient acoustic response is observed.
Fig.
6.5a,
In
the signal rises with the pulse duration and showed
only thermal decay thereafter.
Turning to the temperature-dependent data in Figs.
6.3 and
6.4, we note first the monotonic increase in acoustic
frequency as temperature
is reduced and the damping maximum at
intermediate temperatures.
mode at intermediate
dependence
contribution of the Mountain
temperatures
is also apparent as a time-
(i.e. a slow increase)
in the non-oscillatory part
of the signal.
as well.
The
This behavior can be seen clearly in Fig. 6.5b
What is being observed is the time-dependent density
response to impulsive heating, i.e.
expansion.
time-dependent thermal
This time-dependent response becomes progressively
slower as the temperature is reduced.
However it eventually
reaches the same steady-state value at long times,
the lowest temperatures.
In Figs. 6.3 and 6.4,
except at
an additional
temperature-dependent feature is the amplitude of acoustic
oscillation.
Although the steady-state signal intensity
reached at long times remains constant as the temperature is
reduced,
the intensity of the oscillatory part decreases.
To understand these effects, the time dependences of the
elastic modulus, M, the specific heat, C, and the off-diagonal
-124-
(()
0
1
2
3
TIME
5
4
6
7
8
(ns)
Figure 6.5. ISTS data scans recorded with a scattering angle of 88.90. The laser pulses were
lengthened to - 500 ps to avoid excitation of the acoustic mode. Curve (a) was recorded at 360K.
The rise time is due entirely to laser pulse duration. The decay is due to thermal diffusion whose
effects appear rapidly because of the large scattering angle used. Curve (b) was recorded at 285.9K.
The slow rise of the signal reflects gradual thermal expansion. This is a manifestation of the
Mountain mode. Curve (c) was recorded at 210.9K. As in (a). the rise time is due entirely to laser
pulse duration. The reduced scattering intensity at low temperature, as in Figs. 6.3 and 6.4. is
explained in the text.
-125-
element, N, must all be considered.
Each of these consists of
a short time (before structural relaxation occurs) limiting
value and a time-dependent part
(determined by the structural
relaxation dynamics) which becomes progressively slower as the
temperature is reduced.
At intermediate and low temperatures,
impulsive heating leads to an instantaneous temperature rise
(determined by C.)
followed by a more gradual change due
largely to the time-dependent part of C.
pressure
Similarly,
the
response to temperature and the density response to
pressure have slow components due to the time-dependent parts
of N and M,
respectively.
All these effects contribute to the
gradual density change following impulsive heating which is
observed in Figs. 6.2-6.5.
oscillatory part of ISTS
amplitude
limit
Earlier we noted that for the non-
signal,
the short-time
limiting
is proportional to N./v2C., while the
is proportional
to N 0 /v6C 0 .
The
rise of
long-time
signal from
.
short times to long times tells us that N./v.2C. < N /v6C
0
0
The amplitude of the transient acoustic response is also
temperature-dependent since this response
the
short-time
N, and M.
(relative to the acoustic period)
parts
of C,
The steady-state density response is not affected
until the structural
thermal
is only driven by
relaxation time is longer than the
relaxation time for the wavelength
lowest temperatures in Figs.
6.3 and 6.4,
chosen.
the
At the
structural
relaxation time is longer than the experimental time scale and
the steady-state response is not observed.
This is close to
the high-frequency limit in which the.signal is described by
Eq.
(6.35).
B. Quantitative data analysis and discussion
To analyze data quantitatively, we first extract the
acoustic speed and decay as functions of acoustic wavelength
(determined by scattering angle) and temperature.
are plotted in Figs.
by ultrasonics
The results
6.6 and 6.7 together with data obtained
(Jeong et al.
1986) and spontaneous Brillouin
light scattering (Pinnow et al.
-126-
1968).
Strictly speaking,
the
3.4-
o
C-)
w
**,0 ,00
0 S00,
0
K
*
0 0
x
0
u-i
2.2-
VO
*
am
0
0
%
x
V.
*
a
0
0.76 Mm
0.30 Am,
*
Urn
LL-
..
,
a
*
.4
Y
e
2.6
0
*
x"",
MHz
Am
ym
Am
Am
3.35 Am
2.01 Am
-
.
i
X*
# ,.00/,
amy
-
0
3.0
-
(D
(U)E
x
o
x
a xg,C
.
0
J
#
2.0 MHz
10.0
68.8
23.1
11.7
6.21
a
x
-4
C-
x
.0
a
Cf,
a
1.8180
220
260
300
TEMPERATURE
340
S
-4
380
420
(K)
Figure 6.6. Acoustic speed as a function of temperature for various wavelengths or frequencies. The
2 and 10 MHz data are from Jeong et al. (1986) The X = 0.30pm data are from Pinnow et al.
(1968). The straight lines are v 0 and v, given by Eqs. (6.41) and (6.42).
-127-
(U)
C
o
3.0-
0
#
E
x
a
2.5-
a
]
*
CD
68.8 Ym
23.1 Mm
11.7 Mm
6.21 MM
3.35 Mm
2.01 ym
0.76 Mm
0.30 pm
2.0
Lii
LLJ
1.5-
1.0CD
0.0
210
240
270
300
TEMPERATURE
3k
360
(K)
Figure 6.7. Acoustic damping multiplied by wavelength as a function of temperature for various
wavelengths. The lines are guides to the eye.
-128-
results obtained from light scattering and ultrasonics are not
exactly equivalent since in the former, (real) wave vector is
specified and (complex) frequency is measured while in the
latter, (real) frequency is specified and (complex) wave
vector is measured. For most cases, the difference in
calculated speed of sound is not significant. The limiting
speeds of sound, v0 and v., can be extrapolated from Fig. 6.6.
We find
vo = 2.518-2.045x10-3 T (km/s)
v. =
4.675-5.69x10-3 T
(km/s)
,
(6.41)
.
(6.42)
They are plotted as straight lines in Fig. 6.6.
The
expression for v. is not very reliable because the data range
for the extrapolation is small. At lower temperatures, the
speed actually increases less than indicated by Eq. (6.42).
Fig. 6.8 shows the speed of sound determined from ISBS
experiments extending to lower temperatures.
In the following
analysis, we only apply Eq. (6.42) for temperatures above
200K.
Figures 6.9 and 6.10 show the speed and damping as
functions of acoustic frequency. As pointed out in the
introduction, it has been suggested that the width of
distribution of relaxation times may change from broad at low
temperatures to narrow at high temperatures. Comparing the
data shown in Figs. 6.9 and 6.10 with the results of shear
modulus measurements at low temperatures (Knollman & Hamamoto,
1967; Jeong 1987) and light correlation spectroscopy,
(Demoulin et al., 1974) it appears tha-t the distribution of
relaxation times does not change significantly from 200K to
300K. Better data are needed to find possible small changes.
Higher frequency data are needed to draw conclusions about
higher temperatures.
We find, within signal/noise, that the long time signal
intensity level remains unchanged from 360K to 260K. From Eq.
(6.35), this means that (3P/8T)p 0 /Cp 0 =N0 /v6C0 is essentially
temperature independent. This agrees with reported data in
the literature. Both (ap/aT)p0 (McDuffie, 1969) and CpO
-129-
4.0
00 0 0 0 0 0 a 0
0
%o
000000
00o
000a
0
0
3.5E
a
0
0
000
0%
*00
00
0
00
00
3.0
I-
-)
0,
0
0,
0
0
2.5
00
Lij
000
Lii
Ln
0n
*0
0.
2.0I
I
0
50
100
150
200
TEMPERATURE
250
300
350
(K)
Figure 6.8. Acoustic speed as a function of temperature for acoustic wavelength X = 5.0pnm,
measured through ISBS.
-130-
3.4-
UX
D 3.0
0
E
2.6o
Z
D#
0
2.2-
#
0
LU
LU
0.0
i.0
3.0
2.0
LOG OF FREQUENCY
4.0
5.0
(MHz)
Figure 6.9. Acoustic speed as a function of acoustic frequency at several temperatures. The solid
curves are fits based on the theory of section 2. as discussed in the text. The corresponding
temperatures are, from top, 241.0K, 258.2K, 263.9K, 269.7K, 275.4K, 281.2K, 286.9K, 309.9K.
332.8K. 355.8K.
-131-
'')
3.0
C
-f
-0
E
0
'000V
CD
2.0
Z
LUJ
LUJ
3:
1.0
II,,
-J
C
0.0
1.5
2.0
3.0
2.5
LOG OF FREQUENCY
3.5
4.0
(MHz)
-
Figure 6.10. Acoustic damping as a function of acoustic frequencies at several temperatures. The
curves are generated together with the curves in Fig. II, using the same parameters as discussed in the
text. The temperature corresponding to each curve and set of points is as follows: * and
286.9K: Oand 269.7K; V and - - -,
258.2K: and ---.
-.
241 .0K: # and 355.8K.
-.
309.9K; t and -.
-132-
(Birge & Nagel, 1985) have been measured to be essentially
temperature independent.
The other frequently quoted source
of Cp 0 ,(Gibson & Giauque,
1923) however,
shows significant
temperature dependence.
We also find the ratio of the steady-state signal
intensities in the low frequency and high frequency limits,
i.e.
10
-- = 9 + 2
According to Eq.
No
~
(6.35), this gives:
NO
2
2
=
(6.43)
3.0 + 0.4
Taking CPO/Cp_ = 2,17 y=1-12,
y.=1 and v 0 2/v'
2
= 0.37,
we get
No
-
(6.44)
+
0.3
2.0
=
Finally,
6.2
we turn to see
if the
theory presented in Sec.
correctly describes the observed data,
temporal behavior at
especially the
intermediate temperatures,
including the
Mountain mode time-dependence and the acoustic amplitude.
The
right columns of Figs. 6.3 and 6.4 show simulations of the
temperature-dependent ISTS data based on the theory presented
in Sec.
6.2.
In the following, we detail the procedures used
to generate these simulations.
Eqs.
(6.12)-(6.24) together with Eqs.
were used to generate the simulations.
(6.36) and (6.37)
Besides the
experimentally determined parameters q and T these equations
need a
TM,
total
TC,
TN,
of 13
parameters:
a and 0.
The
p,
K,
Mo,
No,
Co,
Mc,
Nco,
Cco,
first eight parameters were
determined using data found in the literature and our limiting
value measurements (Table I and Eq.
(6.43)) and the following
relations:
MO =
pv0 2 /Y 0
No
,
1 +
)
{Tv 0 2 [(ap/aT)P 0 ] 2 }/(PCP0
yo =
= V0 2 (ap/aT)PO/yo
-133-
Table
I.
Partial
list of parameters used for the simulation
parameter
value
Ref.
g/cm 3
a
p
1.2723-6.55x10-
K
0.29
W/(m-K)
b
3.0
J/(K- cm 3
c
Cp 0
4
units
(T-273.2)
Cp0/ 2 .0
c
2.518-2.045x10-3 T
Km/s
d
v0
4.675-5.69x10-3 T
Km/s
d
(ap/aT)pO
6. 55E-04
g/(K. cm 3
a
(8p/9T)p
PC
(ap/aT)po/6.0
a)
McDuffie et al.
b)
Rastorguev & Gazdiev,
c)
Birge
d)
Present work.
&
Nagel,
)
v0
1969.
1970.
1985.
-134-
d
CO
=
CPO/YO
y
=
=
1 +
Mo
=
pv0 2 /y.
(Tv=2[(
p/aT)PO]2}
pCPO)
,
I
Cc, = Cp0/y0.
(If
shear modulus is considered,
therefore y.
should be modified.
the value for
(9p/aT)P,, and
The actual numerical change
for parameters used in the simulations is quite small).
these parameters,
the fits to acoustic speed
attenuation
6.10)
and
TM in
(Fig.
6.9)
Since
rather insensitive to C and N,
TM in this part of the analysis.
held fixed at all temperatures,
optimum values of a,
O,
the acoustic
we set TN = TC
=
The values of a and 0 were
and TM was adjusted at each
temperature to optimize agreement with experiment.
manner,
and
data were optimized by adjusting a,
the distribution function h(s).
parameters are
(Fig.
Using
and TM(T) were
0,
In this
chosen.
The
temperature-independent values
a=
0.7
0=
0.8
were used throughout the subsequent analysis.
TM vary slowly on a logarithmic scale.
convenience,
For
The values of
computational
we fit them with the second-order polynomial,
lg(TM) = 30.35 -0.1737T + 2.327x10- 4 T 2
.
(6.45)
This is the form used to generate TM for simulations
6.3 and 6.4)
(Figs.
in the temperature range 220K -360K.
Although the influence of TN and TC on the speed of sound
is small,
their effects on the acoustic and Mountain mode
amplitudes
and Mountain mode time-dependence are quite
significant.
By comparing several simulations with different
combinations of T's to experimental data, we set TN = 5TM and
TC =
2 TM
at all temperatures.
We did not attempt an
exhaustive optimization.
Comparing the simulations with experimental data,
that the general trends are matched quite well.
we see
Simulations
to ISBS data and ISTS data at many temperatures and acoustic
-135-
wavelengths were generated based on the selected values of a,
0,
and T(T)
for M, N and C.
The temperature-dependent
acoustic speed and attenuation,
acoustic and Mountain-mode
amplitudes, and Mountain mode dynamics are reproduced
qualitatively and to a substantial degree quantitatively.
We have also tried to fit the acoustic speed and damping
data (Figs. 6.9 and 6.10) with other choices of a and S,
including the
limiting Cole-Cole and Cole-Davidson
distributions,
as well as other distributions.
The acoustic
speed data is fit better by a slightly more asymmetric
distribution than the one we actually used.
However the
damping calculated by such a distribution would be too
asymmetric.
The choice we made is a compromise.
No
relaxation function that we tried yielded as much "total"
damping as found experimentally,
i.e. the breadth and strength
of the observed damping as a function of frequency exceed
theoretical predictions.
The Rayleigh-Mountain mode has been studied extensively by
Allain et al.
(Allain et al.,
1980; Allain
&
Lallemand,
1979b;
Cowen et al., 1976) at longer time scales and lower
temperatures, and they observed similar trends.
Allain and
Lallemand (1979a) argued in favor of a dynamic matrix with no
relaxation in the off-diagonal elements.
(6.43),
No and N,. are not equal.
As we saw in Eq.
The assumption of
frequency-
independent off-diagonal matrix elements is inconsistent with
our experimental result.
Their argument was based on the
assumption that the time dependence ca.n only be put in the
transport coefficients.
For the diagonal elements,
there are
transport coefficients defined, viscosity and heat
diffusivity,
respectively.
So it doesn't make any real
difference whether one chooses to use a time-dependent modulus
and heat capacity ( our approach ) or choose to use timeindependent elastic modulus and heat capacity plus timedependent viscosity and heat diffusivity.
For the off-
diagonal elements, there are no such transport coefficients
defined, so there is a real difference.
-136-
There is no clear
physical basis to support the suggestion that the time
dependence can only be put in the transport coefficients.
6.5 CONCLUDING REMARKS
A study of the acoustic and Mountain modes in glycerol
covering a rather wide range of frequencies and temperatures
has been carried out.
A theoretical
results has been presented.
framework
to analyze the
All of the qualitative trends in
the data are matched by the theory,
and some but not all
quantitative results are matched well.
The
results indicate
that the width of distribution of
relaxation times remains
essentially constant in the 200 -
300K temperature range.
find that a time-dependent "thermal pressure
necessary to explain the observations,
i.e.
coefficient"
We
is
the pressure
response to an instantaneous temperature jump would not be
entirely instantaneous.
Finally,
we find that the
distribution of relaxation times is described reasonably well
by a distribution function which is somewhat less asymmetric
than the Cole-Davidson distribution
-137-
function.
CHAPTER 7.
ISS STUDY OF ACOUSTIC BEHAVIOR IN K(N0 )-CA(NO3)
3
2
DURING LIQUID-GLASS TRANSITION
7.1.
Introduction
60%K(NO 3 )-40%Ca(NO3 )2
(mol%) mixture
is an interesting
material because it is one of very few known ionic materials
which is also an easy glass-former.
More important,
results of ultrasonic
1970)
(Weiler et al.,
light scattering (Torell,
unlike glycerol,
1982)
the
and Brillouin
experiments indicated that,
the distribution of relaxation times changes
from very broad at low temperatures to very narrow at high
temperatures.
Numerous theoretical analyses (Ngai et al.,
1984; Angell and Torell, 1983; Mezei et al., 1987; Campbell et
al.,
1988)
works.
have been carried out based on the
However,
two experimental
the frequency range covered by the ultrasonic
work was 1-185MHz, and the Brillouin work covered 90" and 1400
scattering angles,
Also,
giving frequencies of about 9 and 13 GHz.
in both of these experiments there is a complete lack of
data in the 1301C -
165"C temperature
range because the
samples used tend to crystallize easily in this temperature
range.
This temperature range is important because most of
the change in width of the distribution of relaxation times is
believed to take place in this temperature range.
In addition
the average relaxation rate in this te-mperature range happens
to fall into the 200MHz -
8GHz frequency range which is just
the range difficult for both ultrasonics and spontaneous
Brillouin scattering.
The purpose of our experiment is to
find out whether there is indeed a change in width of the
distribution of relaxation times and if so how the change
takes place.
7.2.
Experimental
-138-
The experimental
setup is the same as
for the
ISTS
experiment in glycerol, although in the ionic liquid the
heating effect is not dominant.
60%KNO 3 and 40%Ca(NO3)2-4H 2 0
(mol%, Mallinckrodt) is weighed and mixed together in air.
The mixture is dehydrated by heating in air for many hours.
The sample so prepared is then filled into a glass cell with
flat windows (Hellma 225-PY, 20mm pathlength).
In order to
ensure
complete dehydration,
cell.
The sample chamber is pumped in vacuum and the unsealed
we did not seal off the
sample
sample cell is heated at 650K for more than 24 hours in the
chamber before taking data.
low.
The water content should be quite
We have not tried any purification or dust-removing.
The samples so prepared can usually stay at the "dangerous"
130-1651C temperature region for several hours without
crystallizing.
The scattering wavelengths studied are:
2.84,
1.51, 1.06, and 0.78 pm.
31.8,
15.2, 7.63,
Temperature is measured by a
copper-constantan thermocouple dipped inside the sample just
above where laser beams pass through.
Temperature
control is
usually good to about +0.1K.
7.3.
Theoretical
We shall use the theory developed for glycerol experiment
to analyze data here.
However,
since no ISTS data is
available and other basic parameters s.uch as frequencydependent heat capacity are also not available as for
glycerol, we shall use a simpler form to interpret the
observed data.
We shall consider only the acoustic speed and
attenuation, not the overall signal form.
The speed and attenuation are
of
obtained from the
roots of s
the equation
A(q,s)
= 0,
(7.1)
-139-
with A given by Eq.
(6.16).
Neglecting thermal conduction,
which is not important in our experiment, Eq.
Eq.
(6.31)
(It
dependence.)
2 +N(s)
is valid even when M,
We then rewrite
Eq.
N,
(7.1)
(7.1) reduces to
C and D has sin the form
0
=
(7.2)
P
q2
M here has the meaning of adiabatic elastic modulus.
the roots giving
rise to
the acoustic wave have the
Since
form S1,2
= -y
iw, we have speed v = w/q and attenuation y.
Besides
the acoustic roots, there is also a branch cut in Eq. (7.2)
which gives rise
to the Mountain mode.
In the present
experiment, we observed data similar to those shown in Fig.
6.2.
We found that the data can be
Cole-Cole distribution, i.e.,
fit adequately with the
0 = 1 in Eq.
distribution of relaxation times can be
(6.37).
The
found using Eq.
(6.39)
to be
f(t)
sinan
=
n(ea& + e~a
where E = ln(T/To).
,
We see f(T)
the average relaxation time.
corresponding to
x = 0.4
is symmetric in & and T0
Fig.
is
6.1 shows two distributions
and to a = 0.8.
The low-frequency-limit speed of
vo = 2.215 -
(7.3)
+ 2cosan)
0.00083T (km/s)
sound we used is
(km/s):
(7.4)
.
It is obtained from fitting to data.
The high frequency
elastic modulus used are taken from Torell
(1982):
1/M.=(2.41+2.860x10-2(T-273 .15))x10-11
(N-lm2)
.(7.5)
The density used is taken from Weiler et al.(1969)
p = 2.23 -
0.793x10- 3 (T-273.15)
-140-
(g/cm 3 ).
(7.6)
7.4 Results and analysis
To obtain speed and attenuation values, we fit
the data
scans with a damped harmonic oscillation plus some heating
effects. The effects of incomplete time resolution are also
taken into account according to chapter 3. As discussed in
the
chapter on glycerol,
Mountain mode
This,
however,
can be quite
numerical
overall
signal
the
strong when damping is strong.
has not been included in the
because
at
even in ISBS experiments,
fitting procedure
integration would be needed to get the
form.
For this reason,
the attenuation values
the peak of attenuation have large uncertainties,
+8%.
At other places,
the fits are usually very good and
independent of fitting routines used or
free parameters.
and attenuation,
about
Figs.
7.1 and 7.2
respectively,
various scattering angles.
initial choices of
show the speed of sound
as functions of temperature
for
Figs. 7.3 and 7.4 plot them as
functions of acoustic frequency,
along with theoretical
fits.
The data for Figs.
7.3 and 7.4 are interpolated from Figs. 7.1
and 7.2 using smooth curves.
The curves in Figs. 7.3 and 7.4
are fits based on the equations presented in Sec. 7.3.
These
fits are
"manual" fits in which the adjustable parameters are
the average relaxation time t 0 and the width parameter a.
fit data at each temperature,
To
we first generate a set of data
based on a guess of T 0 and a, plot the generated data together
with the actual data, compare the two and then make another
guess.
We repeat the process until no further visible
improvements can be made.
10 so obtained is plotted in Fig.
a Vogel-Fulcher form
-0
= A
7.5.
It can be fit with
exp[B/(T-T0 )]
with A=1.287x10-31
T 0 =338K,
B=422.4K.
Our value of To
is
close to that obtained from viscosity measurements by Weiler
et. al. (1969).
Their value for T 0 is 334K for the
-141-
K+-Ca++-NO3~
DATA
3.2
0.95 MHz
15.6 MHz
31.8Mm
15.2 pm
A 7.63 pm
02.8
2.84 Mm
o
1.51 Mm
1.06 Mm
x
# 0.78 pm
o 0.24 Mm
U
+
v
o
8
CD
0.18 Pm
2.4-
m 2.0-
1.6
340
380
420
460
TEMPERATURE
500
540
580
(K)
Figure 7. 1. Acoustic speed as functions of temperature for various acoustic wavelength or frequency.
The lines in the figure are guides to the eye. The 0.95 MHz and 15.6 MHz data are from Weiler et
al. (1970). The 0.24 pm and 0.18 pm data are from Torell (1982).
-142-
:
2.0.
5
K+
0.84
0.2
350
400
550
Temperature
600
(K)
Figure 7.2. Acoustic attenuation times wavelength as a function of temperature for various
wavelengths. The lines in the figure are guides to the eye. The acoustic wavelengths are. from left
peak to right peak, 15.2, 7.63, 2.84, 1.51, 1.06. 0.78 and 0.24 pm. The 0.24 pm data are from
Torell (1982).
-143-
3.0
-13V
~2.6
'00
-
U)2.2
-3
-2
-1
0
1
LOG 1 0 [frequency(GHz)]
Figure 7.3 Acoustic speed as functions of common log of frequency for various temperatures. The
curves are theoretical fits. Data from Weiler et al. (1970) and Torell (1982) are also included. The
temperatures of the data sets are, from top down: 380. 390. 400, 410, 420, 440, 460, 480, and 510
K. The scatter of data on the 380-400K data curves are actually discrepancies between our data and
those of Weiler et al.. In the fits, we favored our data.
-144-
(I)
K+-Ca++-NO 3 - DATA
2.0
C
4-50K
E
3 +-oK
.4340K
-7oK
0
1.5HLUJ
-J
LUJ
0 39KKZ
i 0~~-/0
7D0
3SOK,
*4x
0.5-
z
0
0.0-2 .0
I
I
-1.5
-1.0
I
-0.5
0.0
0.5
i.0
1.5
LOG 1 0 [frequency(GHz)J
Figure 7.4. Acoustic damping as functions of common log of frequency for several temperatures.
The curves are theoretical fits.
-145-
V-F plot
Cn
2.0-
0.0C
4-J-2.0-
-4.0-
0.6
1.2
I/
1.8
(T-338)
2.4
(K)
a function of
Figure 7.5. The relaxation time parameter To is plotted on the natural log scale as
text.
the
in
338K). The straight line is the Vogel-Fulcher fit described
alpha
1.0
0.80.6-
.
0.4
0 .2-
0
400
450
T
500
(K)
Figure 7.6. Parameter ot as a function of temperature.
-146-
temperature range To to To + 76K.
Angell and Torell
(1983)
give a value of 321K base on experiments which span a wider
temperature
range.
Figure. 7.6 plots the best fit parameter a obtained as a
function of temperature.
The increase in value of a from 0.4
to 1 as temperature rises means that the width of
relaxation
time distribution changes from -2.5 decades to 0 decades.
(i.e.,
7.5
single
relaxation time at high temperature).
Summary
Experimental
study of 60%KNO 3 -40%Ca(NO
this material the
relaxation
3
)2
confirms that in
structural relaxation changes from single
time at high temperatures to a broad distribution
toward glass transition.
It is possible that the ionic glass behaves very
differently then glycerol because
temperature-dependent.
its
"chemical" makeup is
It is known that the coordination
numbers of the ions change with temperature, and so in a sense
the elementary "molecular" unit is temperature-dependent.
The
relaxation dynamics of different "molecular" are different,
and understanding a single ionic liquids may require
understanding what are
rather different
"molecular" liquids at
different temperatures.
These experiments have permitted study of structural
relaxation in glass forming liquids over a rather wide dynamic
range, especially when the results can be compared with those
of ultrasonics and Brillouin scattering.
The
relaxation
dynamics in the crucial temperature range during which liquid
behavior changes from "simple" to "viscoelastic" can now be
characterized.
Ultimately, we seek a microscopic understanding of liquidglass transition.
This goal
remains rather distant.
To
approach it will require significant progress on several
fronts.
First,
experimental data like that presented and from
-147-
other techniques spanning very wide frequency and temperature
ranges must be amassed on a variety of materials so that a
broad data base is available.
progress is needed.
Second,
substantial theoretical
At present only crude microscopic
theories of the liquid-glass transition proposed.
1986 and
reference quoted there)
We expect
(see Juckle
that experimental
results will lead to refinements and improvements of
microscopic theories.
-148-
CHAPTER 8.
ISBS OF SURFACE WAVES
The bulk
ISS methods are useful only when the samples
studied are transparent.
materials,
For many reflective or absorbing
this condition is not satisfied.
In this case we
can still use ISS method to study the elastic properties by
studying surface acoustic waves.
The surface properties are
also interesting in their own right.
This chapter illustrates the possibility of studying
surface acoustic waves with the
ISS method.
methods, with which the author has
Unlike bulk
ISBS
studied structural phase
transitions in crystalline solids and liquid-glass transitions
in viscoelastic fluids, surface ISBS method has not yet been
used to
solve problems in condensed-matter dynamics.
We
believe it should find such use in the future.
THEORY
The theory of surface acoustic waves and waves in thin
layers has been treated extensively (Farnell 1970; Farnell
Adler,
1972).
Here we only mention some
&
8.1
results.
The equation of motion for surface acoustic waves is the
same as that for bulk waves, but the boundary condition is
different.
The surface wave propagati.ng on plane surface
between an isotropic medium and vacuum is called a Rayleigh
wave.
The speed of sound of a Rayleigh wave is, depending on
the Poisson ratio of the material, in the range of 0.87-0.96
of the
speed of sound of the transverse bulk wave.
The
amplitude of the surface wave decays rapidly with depth,
usually on the order of one wavelength.
Figure 8.1 shows the
displacement as a function of depth for a typical Rayleigh
wave.
Also,
the Rayleigh wave is non-dispersive,
speed is independent of wavelength.
-149-
i.e., the
08
0 6
-
U
U
-
A
A3
a
02
I
05
O
5
2.0
2.5
Depth (wavelengths)
-0 2
Figure 8.1. Variation of vertical (u 3) and longitudinal (u,) displacements with depth for Rayleigh
wave. Isotropic material with p = 18.7g/cn 3, C = 5.126x1011, C, 2 = 2 058x10''n/m 3. u 1 is in
the direction of propagation, u3 is perpendicular to the surface, pointing out the material. (Adapted
from Farnell 1970).
4
R4
3
R-
E
R2
R,
Velocity
V, -
0
-
-Group
3
2
5
4
kh
Figure 8.2 Phase velocity for the first five Rayleigh modes. calculated for gold layer on fused quartz
substrate. Vt is the bulk transverse wave speed of sound of the substrate. V, is the bulk transverse
wave speed of sound of gold. Broken curve is group velocity for the second Rayleigh mode. R,. The
triangles etc. are not data points and have no relevance here. (Figure adapted from Farnell & Adler
1972).
-150-
Often a thin layer on a substrate is of interest.
surface
is harder
substrate,
there
(i.e.,
elastic modulus
is greater)
is only one Rayleigh mode.
is softer than the acoustic mode,
If the
If the
than the
top layer
then depending on the ratio
of thickness to wavelength, the number of possible surface
modes varies.
When the top layer
is very thin,
one Rayleigh mode, same as no layer.
there is only
As the thickness
increases,
the number of possible modes increases,
related phenomena in electromagnetic wave guides.
similar to
Figure 8.2
shows the calculated results for a gold layer on a fused
quartz
substrate.
thickness),
Because of the
added length scale
(layer
the modes are dispersive.
The impulsive excitation mechanism can be either heating
or stimulated
is heated,
scattering.
When a very thin layer of surface
the thermal expansion on surface is coupled to the
surface wave.
If the depth of heating is deeper, bulk wave
will also be excited.
The surface ripple due to
surface wave
forms a surface grating which diffracts probe light.
8.2
RESULTS AND DISCUSSION
Figure 8.3 shows data from a glass IR filter which
strongly absorbs excitation light.
Since it is also
transparent at 0.532,um which is the optical wavelength of the
probe pulse used, we studied both reflected and transmitted
signal.
It turned out that the reflected signal is the same
as the transmitted signal, and the speed of sound corresponds
to the bulk
longitudinal mode
instead of the surface mode
whose speed should be -90% that of bulk transverse wave.
This
means that although the filter is highly absorbing, the
penetration depth of excitation light is still far deeper than
the acoustic wavelength.
Calculation shows that the
excitation pulse penetration depth is about 150pm,
ten times
the acoustic wavelength (15.2pm) used for this experiment.
Bulk waves near the surface can also make surface ripple, and
-151-
IR filter,
0
trans and
12
6
Time
ref
18
(ns)
Figure 8.3 Data from infrared filter with 4.00 scattering angle (X= 15.24prn). Top curve:
transmitted signal; bottom curve: reflected signal.
-152-
The speed of sound is 5.80km/s.
that's why we see signal in a reflection geometry in this
case.
Figure 8.4 shows data from a gold mirror (gold coated on a
glass substrate)
at 29.250
wavelength X=2.107,um).
scattering angle
(acoustic
The top row are data recorded when the
excitation and probe beams hit the gold-air interface.
The
bottom row were recorded with light beams hit the sample on
the gold-glass
column)
interface.
From the Fourier transforms
of the signal, we see that there are two modes
involved.
The excitation efficiencies of the two modes are
different in the two situations.
for this
(left
scattering angle
From Fig.
(which determines
8.2,
we know that
the wave vector
k),
the thickness h of the gold film is such that kh is above
the threshold for the second Rayleigh mode but below the
threshold for the third Rayleigh mode.
The speeds of sound
are 2.8km/s
for the first Rayleigh mode and 4.8km/s for the
second Rayleigh mode.
also show beating,
Data taken at a 4* scattering angle
indicating that k is still not small enough
so that kh is below the threshold for the second Rayleigh
mode.
But the speeds of sound are 1.92 km/s and 2.89 km/s for
first and second Rayleigh modes respectively.
Considerable
dispersion occurs between 4* and 29.25* scattering angles.
This is to be expected from the calculation of Fig. 8.2.
Since Fig. 8.2 is for fused quartz substrate and not for the
glass substrate used in our experiment, we have not made any
quantitative comparison.
Note that this method can be used as
a way to estimate the thickness of thi-n films.
Note also that there is a relaxational mode with a decay
time of about a few nanoseconds in the air-gold excitation
data (top row), but not in the glass-gold excitation data.
Aluminum mirror data show the
not know what is
same
the nature of this
result.
At present, we do
relaxational
feature.
It
is not likely to be caused by elastic behavior, because the
relaxation time of this feature is the
angle.
same at 4*
scattering
It may be related to heat transport perpendicular to
the surface.
-153-
(a)
(b)
(c)
0
1,0
time
20
(d)
3
23
0
frequency
(n s)
(GHz)
Figure 8.4 ISBS data of gold mirror with scattering angle 29.250 (X=2.1 1pm). (a) gold-air
interface. (b) Fourier transformation of (a). (c) gold-glass interface. (d) Fourier transformation of
(c). The two broad peaks in (b) at f, = 0.91 GHz and f, = 1.37 GHZ are fundamental frequencies.
The peaks at 0.46 GHz and 2.28 are difference and sum frequencies. The peaks at 1.82 GHz and
2.75 GHz(barely seen in (b)) are second harmonics. The two speeds of sound are 1.92 and 2.89
km/s for the first and second Rayleigh mode, respectively.
-154-
Figure 8.5 shows data (29.250 scattering angle) from
poorly polished surface of a block of brass
film).
(not a coated
The coherent acoustic wave damps away very rapidly due
to rough surface.
The relaxational feature remains there.
Similar data are observed on poorly polished gold and platinum
surfaces.
metal
We have not been able to make a good non-thin film
surface.
Figure 8.6 shows data from the very well polished (we
don't know exactly how it was made smooth)
end of micrometer.
The fast oscillations from Rayleigh mode yield a speed of
sound of 3.67 km/s, which is faster than the known Rayleigh
wave speed for stainless steel
(3.0km/s or lower).
This
indicates that the material either has had some surface
hardening treatment or is made of some special alloy.
is also a
of the
slow oscillating feature,
fast one.
about 1/10 the
There
frequency
The fact that the overall signal form is
basically independent of the scattering angle indicates that
the slow feature here, unlike the slow feature we see in gold
and aluminum mirrors,
is elastic in nature.
The fact that the
signal has some dependence on the orientation of sample
indicates that the material
is not isotropic on the length
scale larger than several pm.
At present we do not
fully
understand the data.
We believe surface ISBS can be used profitably to study
the elastic and thermal properties of material surfaces.
It
is more convenient than transducer method, since it does not
require coating the material with transducers or require the
material itself to be piezoelectric.
A data scan usually
takes a few minutes to half an hour.
The sample mounting and
optical alignment is easy.
-155-
6
6
3
time
(ns)
Figure 8.5 See text for details.
-156-
(1)
100
60
20
time
(C)
(ns)
6
6
time
12
18
(ns)
Figure 8.6 ISBS data of micronmeter end surface. (a) 4.00 scattering angle (X= 15.24prn). The
speed of sound is 3.67km/s. (b) and (c): 29.25 scattering angle (X=2.1 1pm). The speed of sound is
3.63km/s. (b) and (c) are taken with the sample rotated for - 900.
-157-
CHAPTER 9.
COMMENTS
In this thesis, we have studied in detail the impulsive
stimulated light scattering
(ISS) method,
and compared it with
frequency-domain spontaneous light scattering
(LS).
We showed
that ISS method has some significant advantages and potential
advantages over LS method.
We can say quite confidently now
that we understand the ISS method.
is still at an early stage.
Its application,
however,
In the following, we discuss some
of the experimental limitations of ISBS.
We shall not discuss
femtosecond experiments here because they lie beyond author's
direct experimental
experience.
For small scattering angles,
there are three major
limiting factors:
1) Reduction of ISBS intensity.
that the material
order of magnitude
magnitude
response
From Eq.
(6.34) we see
response depends on wave vector as q 2 .
reduction in q brings two orders
reduction in response.
One
of
Although this reduction in
can be compensated by an increase in excitation beam
overlapping length (thus
in the transient grating thickness)
if the sample is sufficiently thick,
it also makes it
difficult to avoid sample inhomogeneities.
2)
Increase in elastically scattered light.
Elastically
scattered light is usually strongest at small angles.
3)
Decrease in acoustic wave attenuation.
us to use bigger excitation spot sizes
This requires
(to make sure observed
attenuation is not caused by acoustic waves leaving the
excitation
region)
and longer delay of
the probe pulse.
Bigger spot sizes will reduce the signal intensity as well as
bring in more noise.
Longer delay is difficult to achieve
with mechanical delay lines.
changes,
the probe
As the optical path length
laser spot at
changes in size and divergence.
the sample usually undergoes
It is impossible to collimate
the laser beam at the delay line perfectly, so for very long
-158-
delays some problems would be unavoidable.
For delays of more
than 20 ns, we selected later pulses in the pulse train.
However, different pulses in the pulse-train have different
energies.
We solved this problem by normalizing the signal by
the excitation and probe pulse energies.
is not perfect,
and there is some error in normalization.
For large scattering angles,
duration.
But pulse selection
the main limitation is pulse
With the 85-ps pulse duration of a CW pumped mode-
locked YAG laser, we can resolve clearly oscillations of less
than 1.7 GHz for ISBS and 3.4 GHz for ISTS.
durations of less than 10 ps
We need pulse
to cover the frequency range
covered by frequency domain Brillouin light scattering.
Although the technology for such a system is available, we
have not yet incorporated it.
is solved,
If the pulse duration problem
ISS could almost totally
replace spontaneous
Brillouin scattering.
The experiments on liquid-glass transitions presented in
chapters 6 and 7 show that at present,
we can only get the
gross features of acoustic and Mountain mode behavior.
To
better test the theories we would like the accuracy in speed
values to be better than 0.1% and in attenuation values to be
better than 1% for over three decades of frequency range and
for more than 100K in temperature range.
Improvements should result from the
1) Build a high power system.
following steps:
This is already underway.
The new system consists of two lasers instead of one.
Cavity
damped pulses of 1.6mj/pulse will be used which provide 10
fold increase in energy. This should result in 100 x increase
in signal/noise.
New laser technology developments should
make further improvements possible.
With a 2-laser system,
the delay between the excitation and probe pulses is
controlled electronically,
not by mechanical delay.
delay beyond 200 ns will be possible,
useful
for
Thus
this is particularly
studying Mountain modes and long wavelength
acoustic waves.
-159-
2) Heterodyne detection.
Heterodyne detection will not
only make the method more sensitive but also make the
interpretation much easier,
since
the Green's
function G
.
itself will be the observable instead of G 2
With the above improvements, we hope that we will be able
to surmount the problems just discussed.
Additional
possibilities such as being able to see transverse acoustic
waves in glass-forming materials, may also be realized.
The understanding of liquid-glass transitions requires
measurement of many properties on nearly every time scale.
So
far, most measurements are carried out by small groups with
limited resources and restricted fields of expertise.
Measurements by different groups
frequently cannot be compared
because of small variation in samples.
Faster progress
certainly requires a more collaborative approach.
A
research
center in a national lab or university can be a appropriate
form.
The existing techniques,
including ISS,
long way toward better characterization of
transitions.
-160-
can carry us a
liquid-glass
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