ADAPTIVE CONTROL OF THE PROPAGATION OF ULTRAFAST LIGHT
THROUGH RANDOM AND NONLINEAR MEDIA
MARK DAVID MOORES
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
I would like to thank my three parents for all of their support and encourage-
ment over the years, my success certainly builds on the foundation that they have
My thanks go to my advisor, Dr. David Reitze, for his help, advice and guidance
during my graduate career at the University of Florida. I learned a great deal, and
had fun doing it too.
I am grateful to Dr. Antoinette Taylor of Los Alamos National Laboratories for
the chance to work in her laboratory and to Dr. Fiorenzo Omenetto for his patience
and shared expertise during the time I spent in New Mexico. Dr. Benjamin Luce's
fiber modeling simulations were invaluable.
This dissertation is dedicated to Joy Kloman, for everything.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS . . . . . . . . . . .ii
ABSTRACT . . . . . . . . . . . . . viii
1 INTRODUCTION . . . . . . . . . . . . 1
1.1 Overview .............. .... ............. 1
1.2 Ultrashort Pulses ............................ 2
1.3 Scattering by Random Media .................. ..... 3
1.4 Adaptive Control of the Effects of Optical Nonlinearity . . . 7
1.5 The Layout of the Dissertation ..... . . . . .. 11
2 GENERATION, MANIPULATION AND CHARACTERIZATION OF FEM-
TOSECOND PULSES . . . . . . . . . . . 12
2.1 Outline ............. ..... .............. 12
2.2 Pulse Terminology .......... ......... ....... 12
2.2.1 Pulse Duration and Full Width at Half Maximum . . 12
2.2.2 Time-Bandwidth Product ..... . . . . . ..... 13
2.3 Titanium:Sapphire Laser .................. .... .. 16
2.3.1 Optical Properties of Titanium Sapphire . . . .... 17
2.3.2 Cavity .................. ........... .. 19
2.3.3 Modelocking. .................. .. .... .. .. 20
188.8.131.52 Kerr lensing. .... . . . . . .... 20
184.108.40.206 Kerr lens modelocking (KLM) . . . . .... 21
2.3.4 Dispersion and Self Phase Modulation . . . . .... 22
2.3.5 Typical Operating Characteristics . . . . . . 24
2.3.6 OPO ........... .... ... ...... 24
2.4 Pulse Shaping .............. . . . . . ...... 26
2.4.1 Overview .................. ......... .. 26
2.4.2 Overview of Pulse Shaping . . . . . . ..... 27
2.4.3 Fourier Domain Pulse Shaping .. . . . . . 27
2.4.4 Spatial Light Modulator ...... . . . . .. 29
220.127.116.11 SLM construction and operation . . . ... 29
18.104.22.168 SLM alignment ....... . . . . .. 32
2.5 Low Energy Coherent Pulse Detection .. .....
2.5.1 Why Coherent Pulse Detection is Necessary
2.5.2 Common Techniques .............
22.214.171.124 Cross and auto correlation .....
126.96.36.199 FROG .. .. .. ... .. .. ..
188.8.131.52 Spectral interferometry .......
2.5.3 Pros and Cons .. . ............
2.6 Random Media Sample ...............
3 ADAPTIVE LEARNING ALGORITHM .. ....
3.1 O utline . . . . . . . . . . . . .
3.2 Overview of the Genetic Algorithm .........
3.3 Specific Details of the GENESIS Implementation
3.3.1 A 3D Example .. . ............
184.108.40.206 Parameterization of the problem to
220.127.116.11 Initializing the population .....
18.104.22.168 Setting the environment, paradise or plague? . .
22.214.171.124 Evaluation . . . . . . . . . . .
126.96.36.199 Application of operators, creation of the next gen-
eration .................. .....
3.3.2 Graphical Results of the Example Simulation .. ......
188.8.131.52 Result 1, crossover=0.7, mutation=0.02 .. ....
184.108.40.206 Result 2, no Crossover; no mutation .. ......
220.127.116.11 Result 3, Changing the rates for crossover and mu-
tation . . . . . . . . . . . . .
3.4 Practical Implementation of the GA ..................
3.4.1 Experimental Time Limit .. . ..............
3.4.2 Size of the Experimental Parameter Space ..........
3.4.3 Considerations When Using a GA ..............
4 INVESTIGATIONS OF PULSE PROPAGATION AND CONTROL IN
RANDOM MEDIA .. . . . . . .. . .. . ..
4.1 Outline ................... .............
4.2 B background . . . . . . . . . . . . . . . .
4.3 Experimental Overview .. . . ...................
4.4 R results . . . . . . . . . . . . . . . . .
4.4.1 Unshaped Pulses .. . . ..................
4.4.2 Phase Compensated Pulses ...................
4.4.3 Spectra . . . . . . . . . . . . . . .
4.4.4 M material Dispersion .. . ..................
4.4.5 Boundary Reflectivity . . .
4.5 Discussion . . . . . . . .
. . . . . . . 8 2
. . . . . . . 83
5 RANDOM SAMPLE SIMULATION .. ....
5.1 Outline ......................
5.2 T heory . . . . . . . . . . . .
5.2.1 Dielectric Boundaries ...........
5.2.2 Dielectric Stack ..............
18.104.22.168 Notation ............
22.214.171.124 Derivation ...........
126.96.36.199 Boundary matrix, H ......
188.8.131.52 Layer matrix, L .........
184.108.40.206 Reverse propagation ......
5.3 Modeling .....................
5.4 Simulation Results .. .............
5.4.1 A analysis . . . . . . . . .
5.4.2 Comparison With Experimental Results
5.5 D discussion . . . . . . . . . . .
6 INVESTIGATIONS OF PULSE PROPAGATION AND CONTROL IN
NONLINEAR MEDIA .. . . . ........... .
6.1 Outline .................. .............
6.2 B background . . . . . . . . . . . . . . . .
6.3 Silica F iber . . . . . . . . . . . . . . . .
6.3.1 O verview . . . . . . . . . . . . .. .
6.3.2 Single M ode Fiber .. . ................ .
6.3.3 Dispersion and Nonlinear Lengths ............... .
6.4 Experimental Details .. . ................. .
6.4.1 Genetic Algorithm Parameters .................
6.4.2 Pulse power . . . . . . . . . . . .. .
6.5 R results . . . . . . . . . . . . . . . .. .
6.6 D discussion . . . . . . . . . . . . . . .
6.7 C conclusion . . . . . . . . . . . . . . . .
7 CONCLUSION .. . ..... ..
. . . . . 131
A WAVE PROPAGATION . . . . . . . . . . 135
A.1 Maxwell's Equations in Insulators ....... ............ 135
A.2 Fourier Transforms .................. ........ 137
A.3 Dispersion Considerations .................. .. .139
A.3.1 Dispersionless Material ...... . . . . . .. 139
A .3.2 Dispersion . . . . . . . . . . . . .. 140
B SECOND HARMONIC GENERATION . . 141
B.1 Maxwell Equations in Nonlinear Media . . . . . . 141
B.2 Second Harmonic Generation .................. ... .144
B.2.1 Spatial Phase Matching ................ . .145
B.3 Pulse Generated Second Harmonic . . . . . 146
B.3.1 Correlation ............ . . . . . . 147
B.3.2 Noncollinear Pulse Correlation . . . . . . ..... 148
C OPTICAL PARAMETRIC OSCILLATION . . . 150
C.1 Overview. .................. . . . . . ... 150
C.2 Derivation .................. . . . . . 150
D PHASE ONLY PULSE SHAPING . . . . . . . . 153
D.1 Overview. .................. . . . . .. 153
D.2 Gaussian Pulse .................. .......... 153
D .2.1 Derivation . . . . . . . . . . . . .. 153
D.2.2 Applied Phase .................. ..... . 155
D.3 Linear Phase-Temporal Shift .............. .. ... 156
D .3.1 Derivation . . . . . . . . . . . . .. 156
D.3.2 Discussion ........... . . . . . . .. 157
D.4 Quadratic Phase-Compression/Expansion . . . . . . .. 157
D .4.1 Derivation . . . . . . . . . . . . .. 157
D.4.2 Discussion .......... . . . . . . .. 158
D.5 Higher Order Terms-Deformation of Envelope . . . ... 159
D.5.1 Numerical Application ....... . . . . . ..159
D .5.2 D discussion . . . . . . . . . . . . .. 160
E RANDOM SAMPLE DISPERSION MEASUREMENT . . . 161
E.1 Outline. ................... . . . . .. 161
E.2 Calculation . . . . . . . . . . . . . .. 161
F NONLINEAR FIBER OPTICS . . . . . . . . 165
F.1 Nonlinear Refractive Index . . . . ..... ..... 165
F.2 Calculation of the Envelope Function A(z,T) . . . . . .. 168
F.3 Anomalous Dispersion and Solitonic Behavior . . . . .... 169
REFERENCES . . . . . . . . . . . . 171
BIOGRAPHICAL SKETCH . . . . . . . . . . 177
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
ADAPTIVE CONTROL OF THE PROPAGATION OF ULTRAFAST LIGHT
THROUGH RANDOM AND NONLINEAR MEDIA
Mark David Moores
Chairman: D. Reitze
Major Department: Physics
Ultrafast light sources generate coherent pulses with durations of less than one
picosecond, and represent the next generation of illuminators for medical imaging
and optical communications applications. Such sources are already widely used
experimentally. Correction of temporal widths or pulse envelopes after traversal of
optically non-ideal materials is critical for the delivery of optimal ultrashort pulses.
It is important to investigate the physical mechanisms that distort pulses and to
develop and implement methods for minimizing these effects.
In this work, we investigate methods for characterizing and manipulating pulse
propagation dynamics in random (scattering) and nonlinear optical media. In
particular, we use pulse shaping to manipulate the light field of ultrashort infrared
pulses. Application of spectral phase by a liquid crystal spatial light modulator
is used to control the temporal pulse shape. The applied phase is controlled by
a genetic algorithm that adaptively responds to the feedback from previous phase
profiles. Experiments are detailed that address related aspects of the character of
ultrafast pulses- the short timescales and necessarily wide frequency bandwidths.
Material dispersion is by definition frequency dependent. Passage through an
inhomogeneous system of randomly situated boundaries (scatterers) causes addi-
tional distortion of ballistic pulses due to multiple reflections. The reflected rays
accumulate phase shifts that depend on the separation of the reflecting boundaries
and the photon frequency. Ultrafast bandwidths present a wide range of frequencies
for dispersion and interaction with macroscopic dielectric structure. The shaper
and adaptive learning algorithm are used to reduce these effects, lessening the
impact of the scattering medium on propagating pulses.
The timescale of ultrashort pulses results in peak intensities that interact with
the electronic structure of optical materials to induce polarization that is no longer
linear. This leads to modification of the pulse characteristics through nonlinear
effects such as self phase modulation. Changing the temporal intensity profile of a
propagating pulse modifies the nonlinear interaction. A linear application of phase
is used to control the nonlinear self shaping effects of propagation of a twenty five
milliwatt pulse over forty nonlinear lengths in a single mode optical fiber.
We show the strength of adaptive learning techniques for arriving at experi-
mental solutions to problems with little hope of direct analytical solution. Linear
control of nonlinear propagation of guided waves is demonstrated, with broad ap-
plicability in fundamental science and is a step towards ultrafast optical telecom-
munications. Reduction of the optical effects of a scattering material demonstrates
successful adaptive control of the effects of a non-ideal optical material. Correlat-
ing the applied phase to a modeled dielectric stack gives insight into the random
internal structure for the purpose of characterization.
Random (turbid) and nonlinear media each present different physical mecha-
nisms which alter coherent optical pulse propagation. In the case of turbid media,
the interaction of the light fields with scattering sites results in phase random-
ization and destruction of the coherence of the pulse on length scales comparable
to the mean free path of the photons. For nonlinear optical media, the presence
of large amplitude electric fields alters the polarization of the media, thereby in-
ducing self-propagation effects. In this work, we investigate the propagation of
sub-picosecond light pulses through random and nonlinear optical media via cross
correlation (XC), frequency resolved optical gating (FROG), and spectral interfer-
ometry (SI). By manipulating the spectral phase of the pulse using a liquid crystal
spatial light modulator (SLM), we show that it is possible to substantially mod-
ify the propagation dynamics through randomly scattering and optically nonlinear
A pulse shaper, controlled by a feedback driven adaptive learning algorithm,
is used to modify the phases of the field's Fourier frequency components. This
results in manipulation of the temporal pulse envelope. Reduction of temporal
pulse distortion by random media is used to characterize the combined material
and macroscopic structural properties of the scatterer. Reconstitution of light field
envelopes distorted by propagation through optical fiber demonstrates control of
the nonlinear interaction between the pulse electric field and the material properties
of silicon dioxide.
1.2 Ultrashort Pulses
Ultrashort pulses have durations measured in femtoseconds (Ifs=1015s). Even
though the energies per pulse directly from a femtosecond laser oscillator are low
typically in the nanojoule range-the brevity of the pulses means that the peak
powers can easily be of the order of megawatts. For a beam diameter of one mil-
limeter peak intensities of gigawatts/cm2 or higher are generated and significantly
high local electric fields result. The conventional description of pulse propagation
developed from Maxwell's equations under the assumption of a linear material
response needs then to be extended to include higher order field terms
P(r, t) = Co [ Ej(r, t) + XjkEj(r, t)Ek(r, t) + ... (1.1)
For ultrafast pulses, these terms are no longer negligible by comparison with the
linear terms that adequately describe the behavior of low field electromagnetic
waves, and result in an intensity dependent nonlinear refractive index
n(w, IE) = no() + n2E(t) 2 (1.2)
where n2 is a material dependent constant, and for bulk silica (the major con-
stituent of optical fiber) has a value of 2.3 x 1022. (or 3.2 x10-162 if pulse
intensity is to be used). The low value of n2 makes high electric fields necessary
for the manifestation of nonlinear effects.
Time and frequency are conjugate Fourier variables. Wide bandwidths are
therefore necessary to support the short timescale of ultrashort pulses. While the
broad spectral width is advantageous in that it permits manipulation of the phases
of a wide range of Fourier components, photonic interactions with bulk materials
(dispersion) and macroscopic structures (scattering) are strongly frequency de-
pendent. The shortest pulses currently are around 5fs centered at 800nm, which
requires a staggering 190nm of bandwidth (see equation 2.13). The large range of
frequencies present in the pulse make careful selection of optical materials critical
if the modification of pulse characteristics by traversal of optical components is to
1.3 Scattering by Random Media
The scattering of light occurs constantly in our day to day lives, sometimes use-
fully, sometimes not. The wavelength dependent scattering of sunlight by particu-
late matter in the atmosphere causes beautiful sunsets; scattering of the same light
in the turbid media of our eyelids allows us to ignore its image content and sleep
if we wish. Multiple scattering of ambient light by early morning mist presents a
considerable hazard to motorists. Useful image information is smeared by the inte-
gration of photons that have travelled many different optical paths. The perceived
image is a foggy average of all these paths.
Experiments and applications using coherent and quasi coherent light are sub-
ject to the same phenomenon [5, 16]. Multiply scattered diffuse photons acquire a
complicated distribution of phases due to the various different path lengths trav-
elled from source to detector. As the detector integrates over this distribution,
interference effects sum the average to a much lower value than would be obtained
if all the photons were minimally scattered and remained "ballistic' . The most
useful characteristic of the majority of the distribution of photons (their coherence)
is effectively negated.
Control or minimization of the effects of scattering by traversed media has
particular relevance to those areas where some form of imaging through optically
inconvenient media is necessary. This includes, but is not limited to, optical co-
herence tomography [26, 17], detection of phantoms in tissue using ballistic pho-
tons [22, 12] and long haul atmospheric traversal . Imaging through random
media requires the use of some technique (such as interferometric, optical or tem-
poral gating) to discriminate between the unwanted diffuse and required ballistic
The inherent scattering length scales in a material are determined by its in-
trinsic properties (spatial distribution, size, and geometry of scatters), and by the
wavelengths of the light used. Scattering lengths can be as short as tens of microns
in the case of biological tissue or tens of meters in the case of sea water. After
traversing many scattering lengths, phase coherences are exponentially attenuated.
This results in a very small fraction of the photon population maintaining similar
properties that are characteristic of their state at the source and the subsequent
path traversed. The minimal scattering of these photons means that their optical
paths from the source to the detector must be similar. Given similar paths, the
material and structural effects should be alike for all the detected ballistic photons.
The same path integrated scattering effects will then be applicable to all the bal-
listic and near ballistic light, and image information is preserved (see figure 1.1).
Figure 1.1: 2D OCT Reflectogram of a retina, taken through the pupil of a sedated
glaucomatous beagle . Image size is 3 x 0.564 mm. Bright areas are more
reflective. The upper curved surface of the retina is distorted at the right side
by the progressive effects of glaucoma. Reflectivity of the tissue layers below the
retina surface is non destructively measured by collection of ballistic photons that
travel down into the tissue and are reflected back out to the detector.
Ballistic photons are detectable, and phase shaping allows us to manipulate
the phase of a pulse as a function of frequency. This presents a method for pos-
sible compensation of frequency dependent phase arising from multiple scattering
and may reduce the effects of the scatterer on the ultrafast pulses. The compen-
sating phase profile would be indicative of the material and structure traversed.
The character of the detected light might be improved (even if only dispersion
compensation is achieved, shorter transmitted pulse durations mean higher inten-
sities), but information might also be obtained about the macroscopic structure
of the scatterer. This would be useful for optical characterization of the traversed
For minimally scattered light, the path taken through the medium must be
close to a straight line and is essentially one dimensional in character. Under the
assumption that the interaction time of the pulse with a particular scattering site
is short (i.e., the dynamics of the scattering are slow with respect to the pulse
duration), the traversal of the inhomogeneous material can then be modeled by
propagation through a dielectric stack or etalon. The result (for the transmitted
light) is a combination of frequency dependent phase shifts and amplitude mod-
ulation. Tailored stacks are used for pulse shaping and compensation of ultrafast
intracavity dispersion . Application of a frequency dependent phase to a scat-
tered ballistic pulse will then compensate at least in part for the effect of the
The complexity of the light scattering process poses a potential problem at this
point. The number of degrees of freedom of the interaction between a scattering
medium and a large number of incident photons prevents an analytical solution for
the required compensating phase. The internal structure of the examined material
is by definition not a priori known. Necessary boundary conditions for a diffusion
equation  are unavailable were an analytical solution to be attempted.
All is not lost however. A small fraction of the incident photons are transmit-
ted with well defined coherence and are detectable. The phase of the input pulse
can be adjusted using a pulse shaper in order to improve some characteristic of
the detected signal (for example, the intensity or perhaps pulse shape). The sub-
sequent iteration of the applied phase toward a profile that optimally shapes the
incident pulses can be computer controlled. Adaptive learning algorithms [2, 9]
are particularly adept at solving this kind of problem-a feedback process effec-
tively produces an experimental solution to the required intractable mathematical
This approach to generating a solution is somewhat the reverse of the expected
order of events. The pulse is preshaped and the sample corrects the input pulse
to produce the required optical characteristics after transmission. This treatment
as a so called "inverse pi-l.dl i. is necessary since the input beam to the shaper
needs to be collimated (details of which will be given in 2.4.3). This is obviously
not the case for the majority of the photons that exit the sample.
Application of shaping apparatus and adaptive learning to scattering of ultra-
short pulses by random media then achieves two things. The effects of dispersion
and scattering on the pulse are reduced, and the applied phase profile gives some
insight to the macroscopic structure of the scatterer.
1.4 Adaptive Control of the Effects of Optical Nonlinearity
The scattering phenomena discussed in the previous section are essentially lin-
ear (although quite complicated) in character. However, when intense electric fields
interact materials, new realms of nonlinear interactions arise. The conjugate as-
pect of ultrafast pulses-their short timescale and resulting high intensity electric
fields-is now addressed.
In order to observe these effects at least one of two things is necessary. Either
extremely high intensity pulses are required, or a material needs to be used that
presents an optical path length much longer than the length over which nonlinear
interaction becomes noticeable. The length scale of nonlinear interaction, LNL, is
defined in terms of the nonlinearity coefficient 7 (see equation F.23) which has a
value of 1.365km 1W 1 for standard communication fiber, and the peak power Po
of the pulse which is around 2.7kW for the experiments performed here (calculated
in equation 6.15)
Silicon dioxide demonstrates a weak optical nonlinearity (n2=3.2x10-162)
and is the basic material component of fiber optic cable. As such, understand-
ing and controlling nonlinear interactions in guided wave silica structures are of
tremendous importance. The molecule is symmetric, so only odd electric field
terms, E, appear in the higher order expansion of the polarization, P, and the
third order susceptibility X(3) effects are the lowest order nonlinear effects [1, chl].
P = C (X(1)E + x(3)E.E.E + ...) (1.5)
The superb optical properties of telecommunications grade fiber means that the
nonlinear intensity dependent effects can be observed separately from those effects
due to traversed macroscopic structure or optical inhomogeneities. The effects of
dispersion will still be apparent, on a length scale given by LD where
LD 4nWHM (1.6)
= 0.69m (1.7)
Here TFWHM is the full width at half intensity maximum of the pulse, and /2 is the
dispersion of the material, and is equal to -21.6- for the fiber used. The pulse
duration is 0.204ps. Comparison of equation 1.3 and 1.6 show that propagation in
this case is dominated by the nonlinearity.
Fiber is available in near infinite lengths (hundreds of kilometers) by compari-
son with the characteristic micrometer lengths of optical science. The total number
of nonlinear lengths traversed may be easily adjusted, either by cleaving a fiber
to the desired length or modifying the intensity of the pulse from the laser with
neutral density filters. Experimental investigation of the control of nonlinearities
in fiber demonstrates the manipulation of a nonlinear phenomena by application
of linear phase and represents proof of principle for a technique that extends the
usefulness of existing fiber networks. This lays the groundwork for possible devel-
opment of a system for delivering high intensity pulses. Higher pulse intensities
mean longer distances between repeater stations (which receive attenuated dis-
persed signals and electronically retransmit amplified compressed replica signals
on long fiber links).
Modern society's communication is supported almost entirely by optical fiber
networks transmitting picosecond pulses. Data transmission rates could in prin-
ciple be improved by a reduction to femtosecond duration pulses, with shorter
pulse widths allowing adjacent pulses to be spaced closer together. A practical
limit to communication speed would still be present in the form of the speed of
the detection and transmission electronics, which is around a picosecond response
time for the fastest optical detectors even without the requirement of time for any
processing. A more realistic scenario is the development of a technique to maintain
pulse characteristics after delivery via fiber optic for medical or micro-machining
The higher order nonlinear field terms distort both the temporal and spectral
profile of a propagating pulse. Nonlinear interactions are capable of broadening the
bandwidth of pulses, resulting in true temporal compression. Fibers are used in
this manner for short pulse generation . From a fiber transmission standpoint
instability of the pulse shape is typically undesirable. Pulses that traverse a length
of fiber while maintaining their shape require either that their intensities are below
the threshold levels at which the nonlinear terms become apparent over the entire
length of the fiber (and are still subject to dispersive broadening), or that a delicate
balance is maintained between the dispersive and nonlinear effects in order that
they cancel each other and solitonic propagation  is observed (the shape of the
pulse is self-sustaining).
While operation close to the onset of nonlinearities would normally be avoided
due to the possibility of unwanted signal distortion, the capability to manipulate
the pulse envelope makes this an highly desirable regime for operation of a pulse
shaper. Perturbations to the optical interaction (which can be achieved by the
application of spectral phase) can have drastic effects on the output pulse from the
fiber. Adjustment of the temporal pulse shape is required to control the number
of nonlinear lengths travelled by a pulse as it traverses a given length of fiber. The
temporal pulse profile is also modified by this nonlinear interaction, so that the
application of a very simple phase profile in this regime will be observed to have
very complicated effects.
Pulses are used whose peak power remains capable of generating nonlinear
effects over the length of the fiber. We use gentle adjustment of the nonlinear
interaction to tailor the output pulse so that it displays the same temporal profile
as an unshaped pulse that is not subject to the nonlinear interaction. The result
demonstrates the strength of the genetic algorithm for solution of problems in
complicated parameter spaces where many non global extrema exist. The most
important point though, is that linear application of phase is being used to control
the behavior of a nonlinear system.
1.5 The Layout of the Dissertation
Background information for the experimental apparatus is provided so that the
reader can follow the text and understand the purpose of the components used.
Wherever possible, mathematical digression has been relegated to appendices, and
referenced at the relevant points within the main body of the text. We begin
with an overview of the generation of ultrashort pulses, their characterization and
phase manipulation in chapter 2. Details of the operation of the adaptive learning
algorithm that orchestrates the various actively controlled components are given
in chapter 3. Chapters 4 and 5 discuss the random media experiments, results
and simulation. The adaptive control of nonlinear propagation and theoretical
comparison are presented in chapter 6. Finally we present our conclusions and
discuss future work in chapter 7.
GENERATION, MANIPULATION AND CHARACTERIZATION OF
Before we begin a discussion of our investigations of pulse propagation in ran-
dom and nonlinear media, an overview of the techniques and apparatus used will
be given. This chapter details the physical components used to perform the ex-
periments in the work. As an introduction, nomenclature will be explained in
2.2. Then the three distinct experimental areas are considered-the light source
in section 2.3, pulse shaping in section 2.4, and pulse characterization in 2.5.
2.2 Pulse Terminology
For a clear description of a pulsed optical system, some common terms need to
be explained. The concept of pulse duration is of pivotal importance defining the
realm of applicability of this work, and is mathematically parameterized in 2.2.1.
The duality between frequency and time has some important consequences that
are explicitly pointed out in 2.2.2.
2.2.1 Pulse Duration and Full Width at Half Maximum
To define the pulse width for an arbitrary pulse, the full width at half max-
imum (FWHM) in intensity is used. Using the FWHM as a specification allows
a single number to unambiguously describe the properties of the curved envelope
(see fig 2.1). Assuming a Gaussian pulse shape, and writing the pulsed electric
field centered at wo as
E(t) = [Eoe 2 i0
Detectors respond to intensity, which is defined as
When the intensity is at half its maximum value,
2.2.2 Time-Bandwidth Product
Time bandwidth product (TBP) is a dimensionless number given by multiply-
ing the FWHM and the spectral width (SW). Different classes of pulse shape have
different TBP , some of which are shown in table 2.1. The spectral field is
-T 0 T
Figure 2.1: Full width at half maximum. FWHM=2T.
given by the Fourier transform of the temporal field (2.1). Note that it is the fields
that are transformed, and not the intensities.
E(w)= E(t)e wtdt (2.9)
= VIEoTe [2 ] (2.10)
Comparison with equations 2.1 and 2.8 shows that
Spectral Width =- 2n(2) (2.11)
Table 2.1: Time bandwidth products for Gaussian, hyperbolic sech and Lorentzian
pulse shapes. Time in seconds, frequency in Hertz, TBP is dimensionless.
Pulse shape Form TBP AtAf PWR
Gaussian e 0.4413 0.7071
Hyperbolic Sech sech2(x) 0.3148 0.6482
Lorentzian 1 0.2206 0.5000
The SW in equation 2.11 is measured in radians/second. The dimensionless TBP
(time in seconds, frequency in Hertz) is thus given by
TBP = (2) (2.12)
= 0.4413 (Gaussian Pulse) (2.13)
The TBP is a consequence of the behavior of curves under transform and inverse
transform, and is the optical equivalent of the Heisenberg Uncertainty Principle in
quantum mechanics. It is the absolute lower bound to the product of the SW and
FWHM. For a given SW, the absolute shortest temporal pulse that may be gen-
erated is specified. Ultrashort pulses clearly require large bandwidths. Pulses are
usually measured via their auto or cross correlations. The width of the correlation
is wider than the width of the generating pulse. Pulse width ratio (PWR) is given
PWR FWHMpuse (2.14)
2.3 Titanium:Sapphire Laser
The titanium doped sapphire laser is used as the light source in these experi-
ments. The advent of Ti:sapphire in the early 1990's vastly improved the experi-
mental capability of femtosecond spectroscopists, particularly with regard to pulse
durations. Pulsewidths prior to its introduction were around 100fs; the Ti:sapphire
system reduced this to sub 5fs.
The generation of sub picosecond pulses has evolved over the last 35 years from
a frustrating process requiring specialized techniques and exotic materials  to
one that can be achieved with the press of a button. The Ti:sapphire laser is the
system of choice for most short pulse ultrafast research environments for a number
of reasons related to ease of operation, gain bandwidth, and emission cross section.
Commercial turnkey systems are available that produce pulses with a FWHM as
short as 35 femtoseconds. Custom built state of the art systems will produce pulses
as short as 4.8 fs , merely 2 cycles of the optical field!
The mode-locked titanium sapphire laser used for the experiments detailed
in this dissertation was "home" built using the layout described by Murnane et
al. , and will produce pulses as short as 13.8 femtoseconds, though more typ-
ically around the 20 fs mark for day to day work. The operation of the system
requires only a marginally more involved startup procedure than a pre-built unit,
and the adjustability of the operating parameters (specifically the center wave-
length and pulse duration) more than makes up for the inconvenience of construc-
tion and maintenance. A schematic of the layout is shown in figure 2.3, and will be
referred to in the following sections. Note that the scale of the diagram is vastly
distorted for clarity.
The discovery of titanium sapphire as a pulsed gain medium was extremely
fortuitous, and in some sense represents an elegant closure of a cycle of discovery.
The first constant wave (CW) laser, demonstrated in 1960 by Maiman  used
ruby, Cr3+ :Al203, as the gain medium. Replacement of the chromium dopant ion
with titanium, Ti3+, creates Titanium:sapphire.
2.3.1 Optical Properties of Titanium Sapphire
One glance at the absorption and fluorescence curves in figure 2.2  explains
why titanium:sapphire is so useful as a gain medium. Its broad absorption peak
makes pumping possible with a variety of visible diode sources or lasers. The wide
bandwidth of the fluorescence is necessary to support short pulse generation. Ap-
proximating the fluorescence curve as a Gaussian, the shortest possible supported
pulse is given using the TBP from 2.13, with co the vacuum speed of light
SW = o (2.15)
675 x 10-9 865 x 10-9
= 9.76 x 1013 Hz (2.16)
FWH 1,, ..0= (2.17)
S 9.76 x 1013
=4.52 fs (2.18)
The Ti:sapphire laser can theoretically generate pulses as short as approximately
p 0.4 -
400 500 600 700 800 900
Figure 2.2: Absorption and Fluorescence Properties of Ti:Sapphire for 7r polarized
light. The 190nm fluorescence bandwidth can support pulses as short as 4.5 fs.
Data courtesy of Dr. Peter Moulton.
P \ OC
S1 P2 CW Ar
Figure 2.3: Schematic of the Ti:Sapphire laser.
Our Ti:sapphire crystal is pumped by a Coherent 310 argon-ion laser that can
deliver up to 8W of power in the 457-515nm range. For day to day operations 4.6W
of pumping power is used and produces approximately 300mW of output infrared
from the Ti:sapphire laser. In figure 2.3 the pump beam is shown entering from
the right through lens L1. This is used to increase the intensity of the pump beam
in the crystal and to match the CW mode of the ion laser to that of the cavity
for efficient energy transfer. Ti:sapphire has a thermal conductivity comparable to
metals; even at high pump powers there is little chance of thermal damage to the
For lasing to occur, some of the fluorescence from the crystal must be confined
in the folded X shaped cavity. The outermost mirror pair (output coupler, OC, and
high reflector, HR) represent the external boundaries of the cavity. The inner pair
of high curvature mirrors (Ml and M2) are a sub-resonator, and focus the infra red
light within the crystal. This focus increases the intensity in the gain medium and
is crucial for modelocking and the subsequent long term stability of pulsed output.
Due to the folded layout, adjustment of the separation of the resonator mirrors
also changes the length of the outer cavity. This selects the permitted modes for
which standing waves are supported. Losses in both cavities must preferentially
favor pulsed operation over constant wave (CW).
For stable pulsed operation of the laser, three criteria must be satisfied. First,
the cavity must support a wide range of frequencies; second, the associated longi-
tudinal modes must all be in phase; finally, modes that are outside of the necessary
spectral range or out of phase with the main pulse should be selectively discrimi-
nated against over time. In order to achieve operation under these conditions the
cavity length is set so that the required frequencies needed for pulsed operation
are supported. There is no reason though that there be any specific phase rela-
tionship between the different modes within the cavity, and it is at this point that
something really quite elegant is made to occur.
220.127.116.11 Kerr lensing
Kerr lensing is a spatial, nonlinear intensity dependent effect which allows self
focussing to modelock the laser. It is pivotal to the stable pulsed operation of the
Ti:sapphire laser. The refractive index of a 3 material (such as Ti:sapphire) is
given by 
n(r) = no + -n2I(r) (2.19)
An ideal mode has a Gaussian profile as a function of transverse coordinate; the
center of the beam has higher intensity than the edges. For n2 > 0 the mode
self focuses, and propagates as though travelling through a series of increasingly
powerful Gaussian profile positive lenses. The focus makes the intensity even
higher, causing yet stronger Kerr lensing (see figure 2.4). The process continues
until the self-focussing is balanced by ordinary diffraction due to the small beam
Net Optical Path
o --- ---- L
Figure 2.4: Kerr lensing. As the intensity profile propagates, it experiences a
positive lens with the same spatial profile as the beam intensity. At A, the Gaussian
beam is focused by a Gaussian shaped lens. The intensity at the center of the
beam increases, and the lensing gets even stronger at B, and again at C. The net
optical path is a sum of increasingly distorted Gaussian profiles. Kerr lensing is
limited by linear diffraction-when the spot size becomes so small that it is self
apertured. Note that this figure is not a numerical simulation.
18.104.22.168 Kerr lens modelocking (KLM)
If all the modes in the cavity have random phases, the intensity profile shows
less structure and lower peak values than for a mode distribution with a well
defined phase shift between each mode. This is shown in figure 2.5, where the
following sum of 21 modes is used to illustrate the effects of their individual phases
(2.29873 x 1015 ra corresponds to a wavelength of 820nm)
I(t) = cos(wit + bi) (2.20)
w, = (1 + 2 2.29873 x 1015 (2.21)
random, non modelocked
; = (2.22)
The phases in the cavity may be changed by the application of a spatial per-
turbation (achieved by moving one of the internal prisms, P2, or tapping a mirror,
Ml). Intensity peaks are generated as the phases of the modes shift relative to
each other and pass through configurations for which the phases of different modes
have favorable relationships. Kerr lensing initiates and becomes increasingly dom-
inant with each round trip through the cavity. These Kerr lens modelocked (KLM)
modes are thus perpetuated; the cavity geometry having been adjusted to favor
these modes over those that do not contribute to the short pulse regime. The slit,
S1, is adjusted in order to further discriminate against unwanted frequencies.
2.3.4 Dispersion and Self Phase Modulation
The same nonlinear mechanism that causes Kerr lensing also generates a time
dependent refractive index within the gain crystal, since the pulse intensities are
a function of time. In the same way that phase shifts in the spectral domain
alter the temporal profile of the pulse, time dependent phase shifting (as a result
of the refractive index, n(t), changing the optical path) causes modification of
the spectral envelope. Quite surprisingly, this self phase modulation (SPM) has
a beneficial effect-the spectrum becomes wider, and the pulsewidth decreases.
Neglecting absorption and dispersion, and writing a power normalized electric field
0 500 1000 1500 2000
Figure 2.5: Intensity Distribution for Phase Locked
and Randomly Phased Wave
amplitude U= in the retarded frame T=t--; equation F.19 becomes
-U -U 2U = U(z = 0, T)iNLC(,)
U(z, T) = U(z = O, T)eiONL (zT)
ONL(Z, T) = U(, T)|2
where LNL is defined in equation 1.3. The temporally varying phase is interpreted
as a change, Aw, in the pulse optical frequency
Aw = aN (2.26)
= II 2 (2.27)
Here dispersion was neglected, so all frequency components travel at the same
speed. The temporal change in pulse frequency means that new frequency com-
ponents must be added to the spectrum-which broadens-resulting in temporal
Again, there is a limit to this effect; in this instance, it is the point at which
(back to the real world) cumulative dispersion of the optical components within the
cavity balances the SPM. The intracavity prisms, P1 and P2, are used to introduce
negative dispersion  which adjusts the balance between the effects of material
dispersion and SPM. Their separation is empirically set in order that the pulses
generated by the laser are as short as possible while the output of the laser remains
stable over long enough time periods for experiments to be performed.
2.3.5 Typical Operating Characteristics
Details of the typical day to day operational state of the laser are shown in
table 2.2. The spectrum and corresponding temporal autocorrelation using a 100
/mi KDP crystal are shown in figure 2.6. The FWHM of the pulse is around 23fs.
It is useful to be able to shift the center frequency of the pulse. The bandwidth
of the Ti:sapphire crystal allows tunability from 700nm to 900nm. For wavelengths
outside this range an optical parametric oscillator may be used. Experiments using
Table 2.2: Typical operating parameters for the Ti:sapphire laser.
740 760 780 800 820 840 860 880 900
-100 -80 -60
-40 -20 0
20 40 60 80 100
Figure 2.6: Spectrum and autocorrelation from the laser. FWHM is 23fs, band-
width around 65nm.
Central Wavelength 800nm
Repetition Rate 91MHz
Pulse Duration 23fs
Pulse Energy 3.08nJ
Average Power 280mW
silica fiber require pulses centered at 1550nm in order that the fiber has optimal
optical characteristics. An explanation of the parametric signal generation process
is given in appendix C.
2.4 Pulse Shaping
Active phase control is achieved in this work with a phase only spatial light
modulator (SLM) operating in the frequency domain. The phase is optically ma-
nipulated by placing the pixelated SLM between a pair of diffraction gratings. The
refractive index of each pixel is separately controlled; allowing different Fourier
components to traverse different optical path lengths between the gratings. After
diffracting off of the last grating this results in an imposed phase shift as a function
Since it may not be particularly obvious that phase shifting the Fourier com-
ponents appreciably changes the temporal shape of the pulse envelope, the math-
ematics of phase manipulation in the spectral domain is presented in appendix D;
in addition this has been pictorially demonstrated earlier (see figure 2.5).
An overview of the background of pulse shaping is given in section 2.4.2. The
optical layout used to separate the frequency components in order that they may
be manipulated is shown in 2.4.3, and the apparatus that performs the actual
phase shifting is explained in 2.4.4.
2.4.2 Overview of Pulse Shaping
The use of parallel diffraction gratings for optical pulse compression and expan-
sion was explicitly detailed in 1969  and proposed as much as 4 years prior .
The topic grew from considerations of pulse chirp for the efficient generation of
radar waves .
The shortening of temporal widths by successive improvements in laser designs
is in some sense the simplest (though by no means the least significant, nor the
easiest) improvement to the shape of optical pulses. As durations are reduced the
effects of material dispersion become increasingly significant and it is no surprise
that compensation of phase effects has evolved and become increasingly important
as short pulse laser systems have developed.
Several techniques can be implemented to shape optical pulses in a control-
lable fashion. Modification of the electric field envelope can be achieved by apply-
ing phase shifts or amplitude modulation either in the time or frequency domain
or both simultaneously. Actively controlled (spatial light modulators, acousto-
optic modulators and deformable mirrors) or passive methods (gratings, prisms,
phase masks, dielectric stack mirrors) may be used. Materials used for optical
components also passively shape via their frequency dependent refractive indices,
transmissivities, and reflectivities. A mathematical treatment of the application
of phase is demonstrated in appendix D.
2.4.3 Fourier Domain Pulse Shaping
For Fourier domain pulse-shaping we apply phase shifts in the spectral domain.
Separation of the frequencies that comprise the temporal pulse is achieved using
a 4f grating layout (figure 2.7). This configuration is theoretically dispersionless;
phase shifts are imposed only by the shaper but not by the component geometry.
The first grating, G1, diffracts the incident temporal pulse into a fan of Fourier
components. A focussing element, L1, then images these rays into a collimated and
spectrally dispersed horizontal ribbon. This ribbon passes through the confocal
plane of LI and L2, in which sits a phase shifting element (not shown). The ribbon
of frequencies is then imaged by L2 onto a second diffraction grating, G2. The last
diffraction produces a collimated temporal phase shaped pulse.
The grating G2 is carefully placed so as to present the same diffracting and
incident angles as the first grating, but their order is reversed
0ntput 0_ output (2.28)
input output (2.29)
That a collimated output beam is generated can be seen from the following geomet-
rical argument. The traversal of the gratings and focussing elements is symmetric
under time reversal, neglecting diffracted rays of order higher than 1 and grating
blaze direction (for efficiency). If the photon travels back along the right hand
beam and diffracts off of G2 to be imaged onto the focal plane this is identical to
the behavior of the input photons from the left. A collimated output beam-of
reduced amplitude-is thus reconstructed. The net effect is to transform the pulse
from time to frequency, traverse a phase shifting device, and then inverse trans-
form back into the temporal domain. Examination of the G1 angle and G2 angle
for the ray labelled "a" shows that the gratings cannot be parallel for dispersion
free shaping with this configuration.
L1 Plane L2
0, b Od
fl fl f2 f2
Figure 2.7: 4f grating schematic. Gl, G2 are 300 lines/mm. LI, L2 are f=25cm.
A phase shifting element (not shown) sits in the focal plane.
Our experimental setup uses mirrors rather than lenses in order to avoid mate-
rial dispersion. The 4f layout was folded so that the SLM and gratings are coplanar.
It is, however, difficult to see from this layout that the exiting beam should be col-
limated and that the last grating angle is set correctly so the schematic (figure 2.7)
is shown instead.
2.4.4 Spatial Light Modulator
The optical manipulation of the phase occurs in the spatial light modulator,
and is a result of the controllable birefringence of liquid crystal.
22.214.171.124 SLM construction and operation
The benefit of using liquid crystal over a solid birefringent material is that the
directions of the apparent ordinary and extra-ordinary axes are controllable with
the application of an external electric field'. The SLMs used for these experiments
were commercially produced units (one built by Meadowlark Optics, and a second
unit from CRI). The first incarnation of this device  was developed at Bellcore
in the late 1980's. Most commercial units are small variations on this basic design.
10m O Liquid Crystal
Polarization No Voltage
Cell e axi / Voltage Applied
Cell o axis Transparent
Figure 2.8: A cell of nematic liquid crystal.
A single cell of LC is shown in figure 2.8. The inner surfaces of the cell are
directionally brushed; the chains against the perimeter key into these microgrooves
and order the entire cell when no voltage is applied (top half of the diagram). The
internal front and rear faces of the windows are coated with indium tin oxide
(ITO), a transparent conductor. A high voltage (shown in the lower half of the
diagram) across these contacts causes the longitudinal axis of the LC molecules
to rotate towards the direction of the applied field. This motion is subject to the
constraint that those members at the edges must remain parallel to the container
1Liquid crystal materials were discovered in 1888 by an Austrian botanist named F.Renitzer,
and demonstrate the same optical properties as uniaxial crystals such as calcite and KDP. The
contemporary class of chemical compounds loosely termed "liquid crystal" were discovered in
1973 at the University of Hull . Aside from its application in pulse shaping the use of liquid
crystal in polarized displays has had immense impact on our daily lives, and is essentially the
reason for the existence of the laptop computer. For his research into liquid crystal in the 1970's
and 1980's Pierre-Gille de Gennes received the 1991 Physics Nobel Prize.
walls. The direction of the individual molecule's axes changes, and results in the
required modification of the integrated index of refraction. The extraordinary and
ordinary axes of the cell are indicated.
The polarization of the incoming E field must be parallel to the orientation of
the long axis of the liquid crystal molecules for use of the cell as a phase modulator.
If the polarization of the E field is at 450 to the long axis of the chains (rotated
about the direction of beam propagation) the cell elliptically polarizes the outgoing
beam. Crossed polarizers before and after the cell changes the mode of operation
to an intensity modulator.
For spectral shaping, a long cell is used. The ITO contacts are separated into
128 pixels that are 2mm high, 97/m wide, 15/m thick, and separated by 3/m gaps
(figure 2.8). The phase shift applied to pixel i by the change of its refractive index
is equal to
27rAni(A) x 15 x 10-6
AA = (2.30)
Changing the phase as function of pixel number therefore allows spectral phase
to be applied at the focal plane of the 4f grating setup detailed in section 2.4.3.
For a given change in refractive index, the phase shifts at each pixel depend upon
the wavelength of light passing through that portion of the SLM. The change in
refractive index against voltage response of the LC is not a linear function, so that
a calibration curve is also required along with the spatial dispersion caused by the
lens/grating system. Using x as the coordinate in the transverse direction parallel
to the front of the SLM, f for the focal length of the lens, 0O the diffraction angle
and D the grating line spacing, this is given by
cos 0 (2.31)
Since the phase shifts applied rely on changes in the refractive index of the liquid
crystal, causality cannot be violated-it is not possible to linearly shift a pulse
forward in time past the point where the speed of light in the medium would be
3Pm, 4f Focal
Figure 2.9: The 128 pixel SLM sits at the focal plane of the 4f grating setup.
126.96.36.199 SLM alignment
The transverse alignment of the SLM in the shaper is critical. The phase
shift applied by the liquid crystal is wavelength dependent. Each pixel must be
associated with a known frequency in order that the corresponding voltage can be
applied. The shaper may be accurately positioned by applying large phase jumps
at selected pixels (here numbers 40, 64, and 90). These phase jumps are visible on
the spectrum, allowing the lateral position of the SLM to be set while watching
the spectrometer response. Here pixel 64 corresponds to 810nm. A spectrum
through the shaper is shown (figure 2.10), with the associated fencepost phase
profile applied to the SLM. Calibration curves for phase against drive voltage are
shown at three wavelengths for the CRI SLM (figure 2.11). For values that do not
lie on one of the three curves, interpolation is used.
1.0 1' -
740 760 780 800 820 840 860 880
Figure 2.10: Unphased, and phased
60 80 100 120
spectra and the corresponding ( vs pixel
2.5 Low Energy Coherent Pulse Detection
Pulse detection and characterization techniques have necessarily evolved as
the durations of the pulses they are designed to measure have decreased. When
pulse widths moved into the sub-picosecond regime (which marks an approximate
practical limit to the response times of purely electronic detectors), self referential
techniques became necessary. These methods require the interaction of one or
more beams with a nonlinear medium .
24- ----- 850 nm
22 --- --. 800 nm
0 500 1000 1500 2000 2500 3000 3500 4000
Drive Level (mV)
Figure 2.11: Phase as a function of drive voltage for the CRI SLM. Three curves,
750nm, 800nm, 850nm are shown. Interpolation is required for arbitrary enclosed
2.5.1 Why Coherent Pulse Detection is Necessary
Since the application of a spectral phase to a pulse modifies the temporal
envelope, a diagnostic is required that can take account of these experimentally
imposed changes. The majority of the photons that pass through the scatterer
will have lost their coherence. The small fraction of (coherent) ballistic photons
is the signal of interest. A measurement method is used that requires light in the
sample arm to be coherent with photons in the reference arm, avoiding the task
of separating the signal of interest from a far larger background. The detection
scheme is also part of a feedback loop that may have to iterate for several thousand
trials, so must be capable of acquiring a signal quickly without requiring human
2.5.2 Common Techniques
The majority of contemporary pulse diagnostics use nonlinear second harmonic
generation (see appendix B) as a foundation to indirectly measure the temporal
profile (and phase in the case of FROG) of the pulses of interest.
188.8.131.52 Cross and auto correlation
A schematic of a cross correlator is shown in figure 2.12. A second harmonic
signal depends upon the product of the intensities of two coherent spatially and
temporally coincident beams that cross in the second harmonic crystal (details of
this process are given in appendix B)
ISHG(T)= (t) 2(t + -)dt (2.32)
If the two beams are not coherent, individual photons can still interact to create
second harmonic signals. This random generation will be a very weak background
DC signal; pulses are not observed.
The detector can be any device with sensitivity at 2w and a response time faster
than the period of oscillation of the delay line. A photomultiplier tube (PMT) is
used for maximum sensitivity in the experiment detailed here. The delay causes
the two beams, I1 and 12, to sweep through each other. The detector plots the
correlation as a function of their temporal separation, T. From the diagram, T = 2.
If Ii and 12 are replicas of the same beam (after passing through a beam splitter
perhaps) the setup is called an autocorrelator.
I(T, 2w) 0
_v Ii(t+r, o)
Figure 2.12: Second Harmonic Correlator.
Frequency resolved optical gating (FROG)  is a spectrally resolved corre-
lation setup and associated phase retrieval algorithm. For this case, the detector
shown in figure 2.12 is a spectrometer. The apparatus simultaneously measures
spectral and temporal information in a spectrogram. The intensity, ISHG, is mea-
sured as a function of cw and the delay 7
IGFRO = E(t)E(t T)e idt
This equation can be rewritten with
Eig(t, T) = E(t)E(t- T)
The new field may be expressed as a Fourier transform with respect to T in 2 : T
space; and the frog equation 2.33 is transformed into
The spectrogram measures the magnitude of Esg, and knowing that the mathemat-
ical form of Esig(t,7) is given by equation 2.34  is enough information to solve
the two dimensional problem. This requires an iterative numerical solution that
accurately fits the magnitude of E(t,Q) and its phase profile to the spectrogram,
from which E(t) and the phase profile are recovered.
184.108.40.206 Spectral interferometry
Spectral interferometry (SI) is used to measure the phase differences between
two beams. The beams are collinearly shone into a spectrometer. The result-
ing spectral interference pattern between the two pulses (which will usually be
separated by a time 7, due to optical path length differences) is given by
Isi(w) = 1(w) + 12(w) + 2 II(W) 12()cos (01(w) 2(w) WT) (2.36)
Measurement of two reference spectra, one for each of Ii(w) and I2(w), enables the
argument of the cosine to be calculated as a function of w. The linear component
gives the value of 7; higher terms give 1(cw) b2(w). SI can not be used in the
feedback loop (because of the requirement for the spectral measurement), rather it
is used to calculate the refractive index profile of the samples used in the random
media experiments. Attempts were made to use SI for measurements of the phase
profiles of the scattering media, but this was unsuccessful. The resolution of our
150mm- grating is too low, while the alternate 1200mm- grating does not image
a wide enough range of wavelengths into the CCD camera of the spectrometer.
2.5.3 Pros and Cons
All of the techniques above have pros and cons. The random media experiments
use cross correlation as a detection scheme. The nonlinear fiber experiments use
a photodiode coupled with a nonlinear crystal for simple peak intensity measure-
ment, followed by post experiment characterization with a FROG apparatus.
The cross and autocorrelations are convenient as they are acquired at twice the
scanning rate of whatever method is used to move the mirror, typically around 20
to 100 Hz for a loud speaker type device. The mass of the mirrors restricts the
upper frequency limit to well below the aural range. Their optical layout is very
simple, and re-alignment is easy to perform. The autocorrelation is typically a
more stable signal than cross correlation, as the beam may be split very close to
the point at which it is to be measured. The effects of changes in beam pointing
are thus minimized, but measurement of weak signals is not particularly easy even
with the huge signal amplification provided by photomultiplying tubes.
Cross correlation is less pointing-stable particularly for the random media ex-
periments. The two different beams will tend to have quite different optical paths.
Being able to use one high intensity beam does means that the weak signal through
the scattering sample is in some sense amplified before detection. In this applica-
tion this is the difference between a detectable and an undetectable second har-
monic signal. As with cross correlation, acquisition is very fast. For these reasons
it is the method of detection selected for use in the random media experiments.
The hardest part of detecting a signal is actually finding the point at which
the optical path from the beam splitter and the reference arms are equal once a
sample has been introduced. Not only must these lengths be set correctly but
the two beams in the nonlinear crystal must also cross. The introduction of the
scattering sample modifies the "ideal" initial alignment. This then needs to be
adjusted for the change in optical path length, and the beam pointing.
FROG is a very useful technique, particularly as temporal and spectral informa-
tion are simultaneously measured, allowing phase reconstruction The downside
is that spatially inhomogeneous beams do not produce reliable traces , which
prevents its use for the random media experiments. A CCD is used to collect the
spectrogram, and the update time (over two seconds) required for the detection
of weaker signals makes the initial realignment after the introduction of a scat-
tering sample a daunting prospect. This technique is not well suited for use in a
feedback loop; the reconstruction of the temporal and phase information from the
spectrogram requires some time (around 1 minute) and occasionally expertise in
the selection of reconstruction parameters. It is not a task that could be easily
automated. For post experiment characterization in the nonlinear fiber experi-
ments, FROG is an excellent choice. The beam is still well collimated, and the
measured phase profile demonstrates the existence of the effects due to nonlinear
2.6 Random Media Sample
The samples used for the random media scattering experiments are made with
clear epoxy resin (Devcon 5 Minute Clear Epoxy, purchased at Wal-Mart). The
epoxy is mixed 50/50 with hardener. If scattering centers are required, rapid
stirring introduces large bubbles that become broken up as the mixing continues.
The aerating process is more efficient if the mixing instrument has sharp edges; the
viscosity being high enough that the imparted turbulence generates more bubbles.
Conversely, for reference measurement "blanks", slow stirring with smooth edged
mixers introduced fewer bubbles.
The epoxy is poured into a small sample holder. A -20 nut with microscope
slide covers on the front and back sides makes a conveniently small container. The
resin hardens very quickly and after 2 to 3 minutes is too polymerized to pour,
which limits the amount of possible stirring.
Figure 2.13 shows a CCD image taken in reflection through a microscope. The
white bar top left is 1mm high. The bubbles with circles inside them are against
the front face of the microscope slide, the second circle is where the plane of the
slide cuts the spherical volume of the bubble. The larger 0.5mm diameter bubbles
are visible within the sample. At the top of the image and slightly left of center
the slide cuts the volume of a large bubble. The smallest bubbles are around 6/m.
Figure 2.13: CCD Image of random media glue sample, taken with a CCD cam-
era through a microscope. The white bar is 1mm high. Larger bubbles (0.5mm
diameter) can be clearly seen.
The averaged beam power before the sample is measured to be 11mW, and
4/W after the sample. The thickness of the sample is 5.3mm. The samples are
clear with a slight amber tint indicating that blue light is being absorbed. The
measured spectra through the sample are almost identical to the laser spectra
(figure 4.9). Assuming that the absorption is negligible, the scattering coefficient,
ps for the sample is
1 In( i011 (2.37)
5.3 4 x 10-6
As a beam propagates through the sample, a speckle pattern and distribution
of diffuse light is observed at the back side of the sample. A CCD image of the
exiting beam was taken approximately 20cm behind the sample. A logarithmic
contour plot of the intensity levels is plotted in figure 2.14. The central portion of
the beam saturated the CCD camera. A speckle pattern is seen arranged around
the beam almost concentrically.
-10 -08 -06 -04 -02
00 02 04 06 08 10
ADAPTIVE LEARNING ALGORITHM
This chapter details the adaptive learning algorithm that controls the experi-
mental optimization of the applied phase profile. An overview of what is meant
by adaptive learning and genetic algorithms is given in 3.2; an example is demon-
strated and explained in depth in 3.3 and experimental considerations are detailed
3.2 Overview of the Genetic Algorithm
The term adaptive learning is a broad classification given to a number of algo-
rithms that find solutions by gaining experience about the nature of the problem
they are attempting to solve. Application of this feedback-driven artificial un-
derstanding improves the convergence rate of the algorithm from the initial trial
solutions to the optimal final solution.
Genetic algorithms accumulate machine intuition using a process that is mod-
eled after the evolution of a desirable genetic trait. Multiple potential solutions are
attempted, and trends that appear to give good results are iteratively exchanged
between the current set of trial solutions. This exchange efficiently explores the rel-
evant volume of the parameter space. To prevent stagnation at local N-dimensional
minima or maxima, random parameter changes or mutations are introduced. The
program we use is GENESIS v5.0 [25, 24], a public domain genetic algorithm
written by J. J. Grefenstette.
These algorithms are good at finding extrema for problems in complex param-
eter spaces where there are many non-global maxima and minima. If a physical
quantity can be measured, and affective physical parameters varied then adap-
tive algorithms will optimize the experiment for the conditions) specified. No
knowledge of the exact mechanisms or interactions in the underlying system to be
optimized is required.
This is very useful from the perspective of an experimental scientist, and to
some extent mimics the procedure used to perform final adjustment to a compli-
cated experiment. As a topical example, one might spend many hours calculating
the required positions of the end mirrors in a laser cavity-it is far more efficient
to approximately place and adjust them until lasing is achieved. This is not to say
that a blind experimental approach is replacement for knowledge of the system in
question. Rather, that a set of tools exists that allow experimental examination of
problems for which theoretical analysis cannot generate a solution in a reasonable
amount of time.
3.3 Specific Details of the GENESIS Implementation
The GENESIS program mimics the evolution of genetic code. The adaptation
of the optimal solution to the problem in question is viewed as a survival of the
fittest scenario. The parameters that give the best result are interpreted as the
genetic code for the super-being in the microcosm of the problem space. Growth of
desirable traits is achieved through the use of operators mimicking the perpetuation
of parental characteristics (termed crossover in the algorithm), the introduction of
new genetic traits (mutation), and natural selection (survival and domination of
the so called fittest solution in the next generation). A flow diagram outlining the
program's operation is shown in figure 3.1.
Yes Has optimization or
Figure 3.1: Flowchart showing the operation of the Genetic Algorithm.
3.3.1 A 3D Example
To illustrate the operation and algorithms employed in the GENESIS program,
and the effects of the different parameters a 3D example is used. This enables an
evaluation surface, z(x,y), to be plotted over the complete 2D parameter space in
order to demonstrate that it has local minima and maxima. The progress of the
parameters through a series of iterations can be pictorially represented.
As a reminder, this is an artificial mathematical model. The function space is
chosen to be given by
(x, ) cos(x)sin(y)
1 + 0.1(x- 2)2 + 0.1(y + 2)2
This surface is shown in figure 3.2. It is an egg-crate surface distorted to have a
slight maxima, that occurs at coordinate (2.96839, -1.63639, 0.88791). The GA
will be required to find the maximum value in the space.
220.127.116.11 Parameterization of the problem to be solved
The first step when using an adaptive algorithm is to parameterize the variables
that are assumed to control the problem to be solved. For the example, this is
straightforward, positions along the x and y axes will be varied. The GA's internal
representation of the chosen variable is a binary substring (gene). These genes
are concatenated to form individuals (see table 3.1) to which the genetic operators
are applied. This requires that every variable range is subdivided into an integer
power of 2 increments, in order that a binary string can represent this required
Figure 3.2: The equation z(x,y)= c0(x2)2 (y+2)2
For x and y in equation 3.1 the ranges will be from -4 to +4, with 64 increments,
requiring a pair of 6 bit binary numbers (varying from 0 to 63) for each trial
solution. Notice that the smallest numeric step is then = 0.1270; this is a
fencepost problem, there are one more posts (values) than panels (increments). In
a symmetric value range (i.e. -4 to +4), 0 is excluded from the search space as
this would require an odd number of total values. Shifting the value range by half
an increment includes the origin, if the operator thinks that this is necessary. For
this arbitrary example, the range will remain symmetric.
The ranges for each parameter are set and the substrings for each variable
are concatenated to create complete genetic strings that represent one possible
solution vector for the problem to be solved. This complete string will be one
member of a population of individuals that all experience a process of evolution
over a series of generations. For the demonstrated problem, 10 individuals will
make up one generation, and run for 50 generations (500 total iterations). These
small values are used in order to display clear graphs. To demonstrate the internal
representation, an example binary string of x and y values are shown in table 3.1,
and their numerical values are calculated in equations 3.3 to 3.5.
x = -4 + 23 x (3.2)
y = -4 + 3 x (3.4)
Table 3.1: Two variable strings, x and y, comprising a vector. Their string values
are 23, 3 respectively. Conversion of these strings to numerical values is shown in
equations 3.3 to 3.5.
18.104.22.168 Initializing the population
The initial population may be wholly or partially initiated with specific values,
or filled entirely randomly. In this instance, no initial population was specified.
The GA automatically generates the initial 10 members to be tested.
22.214.171.124 Setting the environment, paradise or plague?
Once the size of the population and the individuals therein are specified, the
artificial world in which the evolution of the population will occur must be set up.
This entails setting the rates for mutation and crossover, which for the example
are set at 0.02 (2%) and 0.7 (70%) respectively. Higher crossover and mutation
fractions (a more hostile environment) mean that the algorithm has a greater
tendency to hop around in the parameter space. While fast sampling of the greatest
fraction of volume is desirable, the routine needs to be stable enough that some
fraction of the current trial solutions that are close to the global solution remain
close enough so that the extremum can be located.
For the mathematical example, evaluation is straightforward, and comprises
substitution of the two current variable values into the known formula. A maximum
value is being searched for. The measure of success of a given parameter is merely
the returned value of z(x,y), and is to be maximized. This somewhat masks the
strength of the GA in that for an experimental setup the evaluation can be a
measured quantity (we use pulse shape) that depends on many factors. This
dependence may also rely on factors that are not directly controlled by the input
parameters (such as phase), in which case the inputs will be adjusted so as to also
compensate for the uncontrolled effects (even misaligned optics), to the best of the
ability of the algorithm.
126.96.36.199 Application of operators, creation of the next generation
After all the members of the current generation have been evaluated, their suc-
cesses are ranked in order and the next generation is created. Each parent member
of the current generation creates duplicate children whose total number is propor-
tional to the parent's relative current evaluation strength, until the population is
filled (see table 3.2). Although some fraction members of every generation may
be completely randomly generated, this option was not used. Once the set of new
strings is formed, and randomly ordered (so that crossover does not always occur
between the same set of individuals) they are then subject to the genetic operators.
Table 3.2: A population of 10 example parent strings, their evaluations, and the
number of children (round up) they will spawn in the next generation. The total
evaluation for the whole population is 20.
The mutation setting subjects 2% of the individual zeros or ones (chromosomes)
in each gene to having their values randomly set. This means that the final value
Parent 1 10 5
Parent 2 3 2
Parent 3 1 1
Parent 4 1 1
Parent 5 1 1
Parent 6 1 0
Parent 7 1 0
Parent 8 1 0
Parent 9 1 0
Parent 10 1 0
Number of Children
of a mutated chromosomes may not change. On average, half are flipped to their
original value, while the remaining half are exchanged for their complement.
The crossover proportion means that the first 70% of the randomly ordered
population has a fraction of its genetic substring exchanged with the adjacent in-
dividual. Crossover is shown pictorially in table 3.3. Two crossover points are
randomly chosen within the chains, and the binary bits between them are ex-
changed. For this procedure, the chains are treated as rings. The exchanged bits
can wrap from the low bits around to the higher bits. Two new child strings are
created for the next generation.
Table 3.3: Two parent strings, and the children they spawn after ring crossover.
The crossover points are between the 2nd and 3rd bits from the right, and the 2nd
and 3rd from the left.
Parent 1 110000000011
Parent 2 001111111100
Child 1 000000000000
Child 2 111111111111
When the creation of the next generation is complete, the new members are
then evaluated and another generation is created. The cycle iterates until either the
algorithm converges (all the members of the population are identical), stagnates
(members of the population are different and no new genetic material is created
by crossover or mutation), or some limiting number of iterations is reached.
3.3.2 Graphical Results of the Example Simulation
Graphs and surfaces of the results are shown. The mutation and crossover rates
are varied to demonstrate their roles in the process of locating the set of variables
that give the optimal solution within the parameter space. Each trial uses the
same sequence of pseudo-random numbers.
188.8.131.52 Result 1, crossover=0.7, mutation=0.02
The initial settings are presented, crossover at 0.7 and mutation at 0.02. The
algorithm is run for 500 trials. The results are shown in figure 3.3. The top graph
shows the evaluation as a function of trials number. The incremental improve-
ment of the algorithm can be seen, the value of z increases until the extremum is
found. The middle and bottom graphs show the evolution of the variables in the
x range and y range axes as a function of trial number. They converge toward
the optimal values as these genes begin to dominate the population. The outlying
values for the higher trial numbers are the effects of the mutation parameter which
prevents the algorithm from stagnating. The optimized, analytic coordinates, and
the quantization of the parameter space are shown in table 3.4.
Table 3.4: Results for the example GA experiment, with mutation=70%,
crossover=2%. The optimal coordinate is located to within the accuracy of the
quantization of the parameter space.
Optimized coordinate (2.98413, -1.71429, 0.88458)
Analytic maximum (2.96839, -1.63639, 0.88791)
To within the given step-size in x and y, the analytic maximum is successfully
located. The algorithm locates the extremum at iteration 365. Note that the choice
of 0.02 and 0.7 as values for mutation and crossover were completely arbitrary
for this problem. One of the greatest strengths of the GA is its simplicity and
functionality "out of the box".
184.108.40.206 Result 2, no Crossover; no mutation
The effects of setting crossover to 0 with 2% mutation, and then 70% crossover
and 0 mutation are shown in figure 3.4. A lack of crossover (upper panel) slightly
slows the convergence. The optimal value 0.8804 is found at iteration 385. Lack
of mutation (lower panel) causes the algorithm to stagnate, and halt prematurely
at a z value of 0.248.
220.127.116.11 Result 3, Changing the rates for crossover and mutation
If the crossover rate is reduced from 0.7 to 0.35 (figure 3.5 panels A, B re-
spectively) the spatial distribution of the parameters is reduced, as expected. The
program stagnates and prematurely exits. The groupings, particularly the open
box at coordinate(3, -1.5), are far less clustered. This clearly demonstrates the ap-
plication of the crossover, producing very local changes around the best evaluation
points as desirable numerical traits are exchanged between different individuals.
For a mutation rate reduced from 0.02 to 0.01 (figure 3.5 panels C, D respec-
tively), the spatial distribution of parameters is reduced. The algorithm once
again stagnates and exits. The reduction in tested parameters is uniform across
"F XX XX X X >X< X
X X X X X X X
0 100 200 300 400 500
2 ...... 2
SI I I
0 100 200 300 400 500
4 x x
2 xxx xx X
XX XX X XX X X XXX X
-4 xX x x
0 100 200 300 400 500
Figure 3.3: Results for the example GA run, with crossover rate of 70% and
mutation rate of 2%. Top: z value vs trial. Center: x value vs trial. Bottom: y
value vs trial. Convergence to the optimal value can be clearly seen.
\ X X K X / X.
XX Xf< X
X X X
W X XO XX X X
XXXX X .
XX X >0<
Figure 3.4: The effects of only mutation (top) and only crossover (bottom). Lack
of mutation causes the experiment to stagnate and prematurely end.
-2 0 2
Figure 3.5: Spatial coverage, variation with parameters (M=mutation,
X=crossover). A, M=0 X=0.7. B, M=0, X=0.35. C, M=0.02, X=0. D, M=0.01,
the space. There is no evidence of an increase in spatial clustering from panel D
to panel C.
The use of a GA to solve the given example demonstrates that an extremum
can be found quite quickly even on a relatively bumpy surface. Approximately
10% of the parameter volume needed to be tested before a solution was found
(365 trials of a possible 642 possible values). For more complicated problems, this
fraction is far lower. Some small number of iterations are required (200 to 300)
for the algorithm to gain an idea of the basic behavior of the parameters in the
3.4 Practical Implementation of the GA
In an ideal experiment, mutation and crossover would be set to very low values,
and the algorithm allowed to iterate away quite happily for a few days until a
solution is found. There are, however, practical limits to the length of time that the
algorithm can run in an experiment-particularly when these limits are mandated
by factors inherent to the experimental apparatus.
3.4.1 Experimental Time Limit
For the experiments in this work the upper time limit is the expected period
for long term laser mode stability (which usually is of the order of 4 to 6 hours).
This must be balanced against the time required for physical parameters to be
changed, and measurements taken (5 seconds approximately). Divide the former
by the latter and an upper limit to the number of iterations of the GA is obtained
(in the region of 3600 iterations).
3.4.2 Size of the Experimental Parameter Space
With a time constraint in mind, the searched parameter space needs to be
considered. The SLM has 128 pixels, which may be varied over 256 grey levels.
This results in a gargantuan parameter space (256128 possible values). To reduce
the searched volume, the pixels are grouped in pairs, and allowed to vary over 64
values from -Tr to +Tr, a possible 3.94x 10105 points.
3.4.3 Considerations When Using a GA
If the parameter space is smooth with a single extremum the use of a GA is
inefficient, as the algorithm hops around in the parameter space. For these cases
gradient searching should be used. The GA will still find the correct answer-it
will simply take a larger number of iterations.
The GA needs some initial level of signal in order for the optimization to
proceed. With no feedback on the effect of parameter changes it is not possible for
the algorithm to gauge its progress and stagnation results.
Lastly, it must be remembered that the algorithm optimizes for exactly the
characteristic that the researcher specifies and this may not necessarily be the
effect on the experiment that is intended.
INVESTIGATIONS OF PULSE PROPAGATION AND CONTROL IN
A significant fraction of our work will use adaptive learning in which the pro-
gression of a pulse shaping experiment is guided by the response of the detected
pulse shape to changes in applied phase. This feedback is used to iteratively drive
the shaping system towards an optimal solution. With this in mind, and building
upon the background information presented in chapters 2 and 3 the experimental
layout and procedure are discussed. This chapter is concerned with the adaptive
control of pulse propagation through random media. Adaptive feedback controls
the effects of the random media upon ultrafast pulses. The effects of material
dispersion and scattering are reduced, and the optimal phase profile provides in-
formation about the optical structure of the media.
The core components are the light source, the pulse shaping apparatus, a de-
tection system, and the adaptive feedback algorithm used to control the active
elements. Details of the subject background are given in 4.2. Section 4.3 outlines
the apparatus and its operation. Results are presented and discussed in 4.4.
The interaction of optical fields and random or scattering media is an area
that has attracted much attention from scientific research. A significant fraction
of the optical research in this area is aimed at perfecting techniques for noninva-
sive detection or selective destruction of (malignant) inhomogeneities in biological
material [16, 63]. With increasing public awareness of the risks of skin, breast,
and prostate cancer, medical detection topics are extremely high profile and their
research is well funded.
Ultrafast Ti:sapphire laser sources are centered in the near infrared, and the
reduced scattering coefficient of tissue at these wavelengths are much larger than
the absorption coefficients. For human muscle tissue at 760nm, P = 0.0176mm 1
whereas (1- g)ps = 0.",,",, 1, with the cosine of the average scatter angle
g=0.95 . Imaging or quasi-imaging through the scattering media may be at-
tempted. Even though the turbidity causes the photons to be strongly scattered,
it conveniently remains essentially transparent.
The underlying physics of the interaction between large numbers of photons
and any random material is complicated. A single photon entering a scattering
material can expect either to be absorbed (to be thermalized or reradiated at a
later time ) or scattered (see figure 4.1). If the photon leaves the medium, its
state will depend on the total number of times it scatters within the medium.
The photons of interest in this work are the ballistic  photons that are
minimally scattered and retain their coherence with the light in the original pulse.
These photons traverse similar paths and exit with similar phases. The detected
optical character of this light is representative of the average path taken by the
Absorbed Photon Multiply Scattered
Figure 4.1: Possible photon paths in a scattering medium. Multiple scattering
leads to diffuse light. The ballistic photons retain their coherence. Absorption is
indicated by the premature ending of the photon path.
ballistic photons. The intensity of the ballistic light is described by Beer's Law
I = Ioe (^S+A)L (4.1)
where Io is the incident intensity, L the optical path length (approximately equal to
the thickness of the medium, since scattering is minimal) and Ps is the scattering
coefficient and is equal to the inverse of the mean free path for scattering, PA is
the absorbtion coefficient.
The multiply scattered diffuse photons have phase shifts that are dependent
upon their individual optical paths. The overall distribution of diffuse phases is
so complicated as to appear random with respect to the light prior to entering
the sample. While these are not the main focus of this work, mention should
be made of their treatment. Rather than treating the photons individually, a
statistical approach is employed, and transport theory may be used to describe
the propagation as a scalar photon-diffusion equation 4.2 
au- vDV2U + vaU = qo (4.2)
where U(r,t) is the total photon density, qo is the source term for the multiply
scattered light, v is the speed of light in the scattering medium. The absorption
and scattering rates are v/f and vp, respectively, and D is the diffusion constant,
defined in terms of the reduced scattering coefficient. The cosine of the average
preferred scattering angle is given by g
D = (4.3)
3/at + 3/t(1 g)
Use of this form of equation is valid only in strongly scattering media, far from
any boundaries or sources, and where qo is isotropic.
Experimental techniques that obtain measurements via the passage of light
through scattering media fall into two categories, those that use the diffusely scat-
tered light for imaging and those that rely on the smaller ballistic component (and
possibly the snakelike photons that straddle both regimes) with some discrimina-
tion technique to filter out the diffuse signal. For the random media experiments,
we use strictly ballistic photon detection.
It is our purpose to extend these investigations into ultrashort, ultrabroad
bandwidth pulse propagation in random and turbid media; exploring the applica-
tion of pulse shaping as a method for controlling pulse propagation dynamics in
these systems. It has been previously shown that material dispersion can be com-
pensated for using phase only pulse shaping . We show that phase only pulse
shaping will also reduce the effects of macroscopic changes in the optical structure
of a material.
Since the photons we detect photons are ballistic, their t_. i. I. It -s through the
random sample are all similar and of low dimensionality. It is our conjecture that
this low dimensionality propagation across the series of air/glue boundaries may
be treated as a one dimensional traversal of a multilayer dielectric stack. If this is
the case, the accumulated frequency dependent phase due to the optical path in
the medium can be corrected for, reducing the impact of the scattering media on
the wide bandwidth of ultrashort pulses.
These investigations have both fundamental and technological significance. Ma-
nipulation of the effects of multiple scattering by the application of phase demon-
strates exertion of control over a fundamental interaction between light and opti-
cally inconvenient materials. The phase profile applied contains information about
the optical structure traversed by the optimized pulses. From a technological per-
spective dispersion compensation alone will increase sensitivity of techniques that
rely on intensity dependent detection. OCT is an interferometric technique, new
units will also use high (100nm) bandwidths. Writing IT for the total detected
interference intensity, and I1 and 12 as the individual reference and weak ballistic
intensities, with a phase shift s, between them
IT = 1 + 12+ c2 cos() (4.4)
Arbitrarily assuming that 12 = 0.00111. If 12 doubles (due to phase compensation of
material dispersion), the DC component changes by approximately 0.1%, whereas
the amplitude of the interference term increases by 41%. Improvement of the
character of pulses transmitted through random media increases the sensitivity of
optical imaging techniques, even without considering the possible reduction of the
effects due to multiple scattering.
4.3 Experimental Overview
To investigate the effects of phase control on the scattering of ultrafast pulses in
random media, the feedback system shown schematically in figure 4.2 is employed.
Short pulse coherent infrared light is used directly from the titanium sapphire laser.
The FWHM is around 25fs, the central wavelength of the laser is at 810nm.
The beam is split with a beamsplitter (BS) into two portions. The arm contain-
ing the sample is described first. The beam passes through an optical pulse shaper,
propagates through the scattering sample and arrives in a second harmonic gener-
ation crystal coincidentally with the pulse from the second (reference) arm. The
second arm is necessary for detection of pulses via second harmonic generation.
Much of the light incident upon the sample is scattered out of the beam line,
or loses its coherence with the beam in the reference arm. Only that small fraction
(-T-) of minimally scattered ballistic photons that remains coherent with the ref-
erence pulse is of interest. The chosen detection technique is used as it specifically
discriminates against incoherent multiply scattered light.
The reference arm from the beam-splitter travels a variable length delay line,
used in order that the optical path length matches that of the beam in the sample
arm. The delay line also includes a pair of perpendicular mirrors that oscillate
back and forth parallel to the beam at 30Hz. This sweeps the reference pulse
Figure 4.2: Experimental Schematic.
through the sample pulse. The oscillation is produced by mounting both mirrors
on a loud-speaker like device (Pasco Corporation mechanical vibrator). The mass
of the mirrors restricts the frequency of operation to well below that of the aural
Detection of the temporal pulse shape is by second harmonic cross correlation.
The measured pulse is at frequency 2w, and the generated beam bisects the crossing
angle of the reference and sample beams. Coherence is required between the pho-
tons in the sample arm and those in the reference arm for 2w pulses to be detected
along this bisecting beam. The minimally scattered coherent ballistic photons are
thus detected. The multiply scattered diffuse (and incoherent) photons are not.
I2 () X Isample(t)Iref(t + 7)dt (4.5)
The reference probe beam is of far higher intensity (by approximately 104 times)
than the ballistic beam transmitted through the sample. If the ultra-low intensity
sample beam were to be split and self referenced, the signal would be too small
to be measurable. The product of the high intensity beam with the sample beam
amplifies the signal to the point where it is detectable.
As the sample beam passes through the pulse shaper, a spectral phase profile
is applied. For those experiments where no phase is applied, the SLM is disabled
and the shaper acts merely as a zero dispersion delay line. The adaptive learning
algorithm gauges the effectiveness of the current profile by assigning the pulse
profile a numerical value based upon its FWHM (the lower the better) and the
fraction of the total area under the curve that lies outside of the FWHM points
(the lower the better). This steers the solution towards shorter sample arm pulses
with less structure in the "wings" of the correlation.
The current generation of phase profiles is applied, and using their evaluations
to gauge fitness of each trial a new generation is then created using the genetic
operators. The new generation attempts to improve the pulse shape further and
the cycle continues. After the optimization converges to an optimal solution, or
the algorithm stagnates, the process stops. Comparison of the unoptimized and
optimized cross correlations demonstrates the imposed improvement. Two pickoff
beams are used for spectral measurements.
Results are presented for the adaptive phase control for two different experi-
ments. Experiment 1 is the data from the very first run demonstrating that the
adaptive learning compensated for the effects of the sample and used the Mead-
owlark Optics SLM1. Experiment 2 uses the replacement CRI SLM. The FWHM
details given assume a Gaussian profile which multiplies the widths seen on the
cross correlation by 0.7071 (see 2.1). The FWHM in figure captions and the text
are corrected by this factor, so they represent the widths of the pulses rather than
the cross correlations. The correlations will appear slightly wider than the quoted
The results are separated into two sections. The effects of the random sam-
ples for each experiment are shown first, followed by the compensation with the
application of the optimizing phases.
4.4.1 Unshaped Pulses
We begin by showing the effects of the random samples on the pulse cross
correlations. The first set of data (figure 4.4) is a cross correlation taken through
3.5mm of chicken breast. This was initially to be the material studied. The
chicken is not physically stable enough for the adaptive optimization experiments,
over hour long timescales its physical properties change too drastically because of
moisture loss. These changes in shape alter the optical path taken by the beam
1This SLM was inadvertently damaged, leaving some dead pixels and the device leaking liquid
crystal. The voltage to phase characteristics of the SLM needs to remain fixed for the duration
of an experiment, and ideally from day to day. It was necessary to replace this element to ensure
that dynamic changes were avoided due to drifts of the calibration curve of the shaper
making the experiments too dynamic. For this reason the epoxy is used, it has
good optical properties and is physically very stable once it has cured.
The qualitative features of the effect of scattering on ultrashort pulses is nicely
illustrated by the chicken data. This data is shown to demonstrate that the fea-
tures observed in the epoxy data are representative of the effects of random media
Results for the random epoxy samples are shown for experiments 1 and 2 in
figures 4.5 and 4.6 respectively. Material dispersion (see appendix D) broadens
the correlations, and results in the increase of the FWHM shown on all three
curves. The solid lines are the correlations with the sample inserted, which are to
be compared with the dashed reference curves from the laser. Different amounts
of broadening are seen. The data for the chicken 4.4 shows different dispersion
from figures 4.5 and 4.6 since it is a different material with different thickness and
The two epoxy experiments, even though they use the same sample have two
different points of incidence at the front face. The different optical paths taken
results in different relative amounts of air and epoxy. The material dispersion
(neglecting any effect due to scattering) will then not be the same for these two
It should be pointed out that while the graphs of the unphased propagation
results are presented together, the experiments were performed separately. The
data for graphs 4.5 and 4.7 was taken sequentially without moving the sample
relative to the beam; similarly for the second set in figures 4.6 and 4.8.
The effect on the correlation shape characteristic of the scattering caused by
these random samples is the tail that appears at the back side (at positive time)
of the trace. This is labelled in figures 4.4, 4.5 and 4.6. The tail is caused by the
ballistic components of the pulse that are multiply reflected by the boundaries.
The light travels a longer optical path and so arrives later relative to the peak of
the pulse (which itself is slightly shifted). In frequency space this corresponds to
optical path length phase shifts, generated by the boundaries. This causes an effect
analogous to that which makes the SLM function. Figure 4.3 shows two photonic
paths, 1 and 2. The phase shift A4 between photons 71 and 72 with frequencies
cw, c2 respectively, c is the vacuum speed of light and n(w) the material refractive
index is given by
n(w)wiD n(2)2( + d + d + d4) (4.6)
A0 = (4.6)
It is these path and frequency dependent phase shifts that shape (see appendix D)
the pulse envelope resulting in the scattered tails.
Figure 4.3: The different scattered paths for 71 and 72 lead to a frequency depen-
dent phase shift given by equation 4.6.
Distortion of the front edge (negative time) of the curve can also be seen, with
the peak being shifted toward positive time. This makes the front edge of the
curve appear to bow upward. The bowing can clearly be seen in figure 4.5 at -25fs
and in 4.6 at -50fs. The shift of the peak is caused by the same mechanism that
generates the scattering tail. Few photons pass straight through the material, the
distribution of path lengths travelled by those that are multiply reflected mean
that the peak of the pulse is no longer as close to the leading edge as for the case
with no sample. The result is that the peak is shifted back to more positive time.
The chicken trace is ambiguous, the notch at close to zero time makes it difficult
to accurately locate the position of the peak.
The cross correlation detects only those photons which travel along the ballistic
or near ballistic path. The phase matching angle discriminates against photons
with k vectors not well phase matched (see appendix B). Only the coherent pho-
tons are detected, not those that diffuse out of the sample. With such large and
well separated bubbles (5pm to 50pm across typically, with average separation of
centers visually estimated at approximately 200-300pm) the validity of a mathe-
matical description using a diffusion approximation is questionable. Weak speckle
is observed with an infrared viewer behind the sample (a contour plot displays
the approximate pattern in figure 2.14), as is diffuse light. Recall that -20 nuts
are filled with the epoxy. There are matte metal surfaces forming a tube coaxial
with the beam-line; these surfaces will generate Lambertian reflections. Painting
the interior of one of a pair of nuts before filling them with epoxy from the same
whipped batch, then comparing the amount of diffuse light would be one way to
test whether this contributes to the number of diffuse photons.
4.4.2 Phase Compensated Pulses
Details are now presented of the application of the adaptive algorithm for lo-
cation of optimal phase profiles in the random media scattering experiments. The
effects of applying the optimal phases found by the successive runs of the adaptive
feedback algorithm are shown in figures 4.7 and 4.8. The aim of the compensation
is to generate pulses that are as narrow as possible with the added requirement
that they have as little fraction of their area outside of the FWHM range as pos-
sible. This is done to prevent iteration toward a narrow spike feature sitting on a
pedestal. The chicken experiment was not subject to optimization.
Figure 4.7 shows the compensation for experiment 1, figure 4.8 details the
results from experiment 2. The graphs are plotted again on both linear and log-
arithmic scales. Here the optimized curves are of interest and are shown with a
solid line. In each graph, the reference from the laser is dotted and the effect of
the sample with no compensation is shown as a dashed line for comparison.
In both cases the material dispersion is well compensated for, the FWHM of
the optimized 29fs pulse in figure 4.7 is almost as narrow as the 27fs reference. For
experiment 2, the 29fs FWHM of the compensated trace improves on the FWHM
of the 39fs reference, indicating that there was some residual phase from the optics
that is also being compensated for. It is not possible to reduce the FWHM of a
transform limited pulse (equation 2.13), but the converse is possible. The phase
plot in figure 4.9 shows a complicated shape that is not easily identifiable as any
obvious polynomial of frequency. Figure 4.10 shows a strong negative quadratic
shape over the region of the spectrum where the laser has appreciable optical
weight; this is responsible for the compensation of the material dispersion, the
extra structure (where the phase plots deviates from the quadratic) is postulated
to be the compensation for the effects of the random media (to be discussed further
in chapter 5).
The adaptive phase compensation of the scattering tail is most strongly demon-
strated in figure 4.7. The tail is reduced in magnitude, as can be most clearly seen
on the logarithmic plot, but not quite removed entirely. This experiment was per-
formed with the Meadowlark SLM. The phase compensation in figure 4.8 is not
quite as good. While close to the pulse (t=-100fs and t=+100fs) the tail is very
well controlled, outside of those regions there is considerable structure. That the
structure is pushed out to the front of the pulse (large negative time) indicates
that there is something more complicated than scattering occurring as a result of
the shaping. The CRI SLM has a different orientation (horizontal) of liquid crystal
chains to the Meadowlark unit. Where the horizontal chain ends brush those of
the adjacent pixel, scattering effects occur which might account for the spreading
of the pulse into the wings.
The spectrum for experiment one (figure 4.9) was measured after the exper-
iment had been performed, so is representative of the effect of the sample, not
definitive for that case. The modulation seen on the sample spectra (not present on
that from the laser) was most likely caused by the pixel damage to the Meadowlark
SLM. The spectrum for experiment two (figure 4.10) does not show modulation in
the same manner is that for experiment 1. These spectra were measured with a
150mm 1 grating.
Measurement of a spectrum with an high resolution 1200mm grating is shown
in figure 4.11. The top curve is a spectrum measured without a sample, the bot-
tom is measured through the sample. Both curves are shown using solid lines for
accuracy. Calculation of the characteristic length that produces the interference
patterns shown (2nm modulation and the fuzz at around 0.2nm modulation) high-
lights why the effects of the samples cannot be observed on the spectra. The 2nm
modulation is caused by length scales of the order of 300pm or around double
the thickness (because of reflection) of a glass slide, neutral density filter or beam
splitter. These components exist in both arms or the interferometer so their effects
are measured on both spectra.
The modulation depth of the fuzz is deeper on the sample curve, this corre-
sponds to a length scale of approximately 3mm, and is likely caused by two widely
separated boundaries within the random media sample, possibly by the glass slides
at the front and back which are 5.3mm apart. The effect of 30/m features is a
modulation with a period of the order of 20nm, approaching the bandwidth of
the spectrum. It is no surprise then that interference effects due to the scattering
sample are not seen, particularly since the reflectivity at each air-glue boundary is
approximately 3% (equation 4.4.5).
4.4.4 Material Dispersion
The distortions present in the cross correlated signals qualitatively display two
distinct features. First, we universally observe a broadening and distortion of the
main body of the pulse. In addition, we find that there is significant structure
in the wings, particularly in the tail of the pulses. We now consider the origin of
-- 3.5mm Chicken
r 0.8 -
S0.8 ----- Laser
0 0.0 .
-200 -100 0 100 200 300 400
-200 -100 0 100 200 300 400
Figure 4.4: Cross correlations plotted on linear and logarithmic scales, taken
through 3.5mm of chicken breast (solid line), and for the laser (dotted). The
chicken trace shows strong broadening from 20fs to 68fs due to material disper-
sion, and the characteristic scattering tail.
.- 0.8 Sample
0 .0 - '''---
-300 -200 -100 0 100 200 300
-300 -200 -100 0 100 200 300
Figure 4.5: Experiment 1. Cross correlations, plotted on linear and logarithmic
scales, taken using the Meadowlark SLM and the LiI03 crystal. Laser (dotted),
and the effects of the sample (solid). The scattered tail is labelled. The FWHM
from the laser is 27fs, this is broadened by the sample dispersion to 41fs and a
scattering tail is produced.
0.8 -- Sample
/ Scattered Tail
-300 -200 -100 0 100 200 300
a 0.01 --
-300 -200 -100 0 100 200 300
Figure 4.6: Experiment 2. Cross Correlations of the laser (dotted) and through
the scattering sample (solid) with the CRI SLM and a LiI03 crystal. Strong
broadening and deformation are seen in the dashed curve. The FWHM of laser
correlation (39fs) is broadened by the sample (61fs) and a scattering tail is again
0.8- ---- Sample
0.0- / -
-300 -200 -100 0 100 200 300
0.01 /, ,
-300 -200 -100 0 100 200 300
Figure 4.7: Experiment 1. Cross correlations, plotted on linear and logarithmic
scales, taken through sample A using the Meadowlark SLM and the LiI03 crystal.
Laser (dotted), sample (dashed) and the sample with the compensating phase
applied (solid). The scattered tail is labelled. The broadening and deformation of
the correlation are corrected for. The FWHM through the sample is 41fs and is
reduced to 29fs. The scattering tail is significantly diminished.
0.8 ,- Sample
-300 -200 -100 0 100 200 300
rithmic scales are shown. Sample A, CRI SLM, 3 crystal. Strong broadening
^ 0.01 -_ \ \ f A ; / \ '\ j
:' \ ,,lo I I, ' I* rl^ i i '\^ \
l E \ ,/ ., ^ ..,,.i" t '
Si I I '
-300 -200 -100 0 100 200 300
Figure 4.8: Experiment 2. Cross Correlations of the laser (dotted), through the
scattering sample (dashed) and with the optimizing phase (solid), linear and loga-
rithmic scales are shown. Sample A, CRI SLM, LiI03 crystal. Strong broadening
and deformation are seen in the dashed curve. The FWHM of the compensated
correlation (29fs) improves on that of the sample (61fs) and is narrower than that
directly from the laser (39fs, not shown). The scatter tail remains present but is
700 720 740 760 780 800 820 840 860 880 900
700 720 740 760 780 800 820 840 860 880 900
Figure 4.9: Experiment 1. spectra of laser (dotted) and through the sample (solid)
at an arbitrary position. The optimizing phase (bottom panel) is shown as a func-
tion of wavelength. The phase profile does not resemble any obvious polynomial
700 720 740 760
780 800 820 840 860 880 900
700 720 740 760 780 800 820 840 860 880 900
Figure 4.10: Experiment 2. Spectra of laser (dotted) and through the sample
(solid). The optimizing phase (bottom panel) is shown as a function of wavelength.
A negative quadratic trend can again be seen from 770 to 830nm.
1.2 No Sample
780 785 790 795 800 805 810 815
Figure 4.11: High resolution spectra measured with and without sample using a
1200mm1 grating. The modulation seen corresponds to reflections from bound-
aries separated by 3mm and 300/m. A modulation with a wavelength of 20nm
would correspond to 30pm features, too large a fraction of the laser bandwidth to
these effects. In particular, it is considered whether or not the tails seen in the cor-
relations are due to scattering or merely material dispersion (see appendix E). To
do this, the dispersion of the epoxy is measured using a "blank" without bubbles.
The effects of this dispersion are numerically applied to a 26fs FWHM Gaussian
pulse. The shaping effects of the epoxy alone can then be seen.
Comparison of the modeled output correlation due to the dispersive effects of a
5.3mm glue blank (solid) with the input correlation of the test Gaussian (dashed)
are shown in figure 4.12. The trace is slightly asymmetric in positive time (the back
side of the pulse), but the distortion caused by the dispersion of the epoxy does not
account for the magnitude of the experimental tail effect seen at positive times in
the cross correlations (figures. 4.5 and 4.6), nor the distortion of the leading edge of
the pulses at negative time. In addition, the fact that the correlations through the
samples display slightly different structure at the positive time side supports the
argument that the effects are due to the inhomogeneity of the sample rather than
to just the material's refractive index. These effects are caused by the multiple
reflections and ensuing phase shifts accumulated between the bubble boundaries,
and will be modeled in chapter 5.
4.4.5 Boundary Reflectivity
Using the mean value for the refractive index, calculated in equation E.4, the
reflectivity at an air/glue boundary is given by
p = (4.7)
1.405 + 1
= 0.1684 (4.8)
-150 -100 -50 0 50 100 150
Figure 4.12: Modeled cross correlations showing material dispersion. The dashed
line is the reference trace, the solid line has material dispersion applied. The
temporal trace is slightly asymmetric, but a scatter tail is not seen.
R = 0.16842 (4.9)
= 0.0284 (4.10)
This low value for the reflectivity, coupled with the bandwidth considerations for
the structural feature size (section 4.4.3) explains why the effects of the cavities in
the random samples are not observed on the spectra.
We have demonstrated that adaptive phase control can compensate for the
two effects, dispersion and low dimensionality scattering, of random media. The
material dispersion is very well corrected for, dispersed pulse widths are reduced
back to values that are comparable or lower than the reference values measured
without the presence of the samples in the beam-line.
The scattered tails seen on the cross correlations, caused by multiple reflections
in this low dimensionality scattering regime can be reduced but not completely
eliminated. This may be due to one of two causes. The CRI SLM may be scattering
light that is preventing good reduction of the effects of scattering. The genetic
algorithm may simply need more time to locate the optimal phase profile.
Since the problems are occurring in the wings of the cross correlations, modi-
fication of the evaluation function would seem to be necessary. The effects in the
wings need to weighted so that the optimization is more sensitive to changes there.
Taking a logarithm of the cross correlation intensity profiles will achieve this. A
detection system with higher dynamic range than that used will also improve the
The term scattering may not be entirely accurate for a description of the in-
teraction between the ballistic photons and the boundaries within the scattering
medium. This will be discussed further in chapter 5.
RANDOM SAMPLE SIMULATION
This chapter details the simulation of the interaction between the ballistic pho-
tons and the equivalent dielectric stack for the traversed sample. The goal is to
be able to correlate the experimental results to a theoretical structural layout that
may be used to classify the nature of the individual samples.
The photons of interest that propagate through the random media to be co-
herently detected at the second harmonic crystal are ballistic, or quasi-ballistic
in nature. Their interaction with the microcavities caused by the included air
bubbles is of low dimensionality. In each separate experiment the path traversed
through the sample is similar for all the detected photons. These photons retain
their coherence, and the temporal pulse structure remains detectable prior to and
The path through the sample need not be a straight line (figure 5.1). This path
can be unfolded, and represented as a 1D propagation through a dielectric stack
(figure 5.2). The incident angles to the surfaces are all assumed to be perpendicular
to the optical boundary. This assumption is made to reduce the number of variables
in the fit of a theoretically modeled stack to experimental data; the model will
therefore at best be an approximation.
Figure 5.1: Projection of a possible 3D path ABCDE through a random sample.
A C E
Figure 5.2: 1D ABCDE path through a dielectric stack.
Mathematical background for propagation through the interfaces is developed
in 5.2. Details of the modeling algorithm are given in 5.3, the results of which are
shown in 5.4. Results are analyzed, compared to experimental values and discussed
Optical properties of cavities and interfaces are discussed in the frequency do-
main, since frequency dependent optical effects and phase shifting of the Fourier
pulse components are the topics of interest. Traversal of material is mathemati-
cally expressed as a phase shift. The relevant optical properties of dielectric cavities
are the reflective and transmissive boundary effects detailed through the Fresnel
relations, and phase shifts caused by the effects of optical path length.
5.2.1 Dielectric Boundaries
The Fresnel relations [30, P278] for normal incidence give the transmitted and
reflected amplitudes in terms of the incident field amplitude. A minus sign (for
the case of n1 < nT) indicates a Tr phase shift that will be retained. The layout
of the boundaries is shown in figure 5.3.
EI nI + nT
ER I nT
EI nj + nT
Amplitudes of perpendicularly incident, transmitted and reflected
Traversing a length of optical medium (which contains no boundaries, shown
in figure 5.4) invokes a frequency dependent phase shift. With El(w) the initial
beam, E2(w) the beam after traversing a length d of material, n(w) the refractive
index of the slab, Co the vacuum speed of light
E2(w) = E(w)e w)
( ) =-
1E(n) E(c )
Figure 5.4: Beam E2, is phase shifted by O(w) after traversing a length d of mate-
For calculation of the effects of stacked slabs of dielectric, a formalism is neces-
sary that can be applied for a variable number of boundaries and thicknesses. An
elegant analytic method is shown in 5.2.2 [34, PP297-300], that exactly specifies the
amplitude modulation and phase shift as a function of frequency. The amplitude
reflection and transmission coefficients calculated will be complex, incorporating
the phase shifts due to material.
5.2.2 Dielectric Stack
The introduction of a single parallel interface into a beamline generates an in-
finite cascade of rays of decreasing amplitude (figure 5.5, note that the angle of
incidence has been increased from zero to create a fan of rays). For the case of mul-
tiple boundaries of varying thickness, the situation becomes far more complicated.
Each cascading ray exiting a single slab causes similar cascades in an adjacent
slab. Tracking of terms in a series solution for this type of problem becomes too
laborious to be useful.
To build the required multi-layered solution, a completely general argument will
be used. The properties of the stacks are detailed, and a time-reversed transfer
E, E b E5
E7 -6 d- E,
Figure 5.5: Cascade of rays from single slab multiple reflection.
matrix written that specifies the input and reflected field amplitudes in terms of
the transmitted "final" beam. The lack of a beam reflected back from infinity on
the transmitted side enables this approach to give solutions in a form that may be
easily inverted and solved.
There are four fields of interest for each individual slab (i or j). For the front
and back faces, there are left and right traveling beams. In order to distinguish
between these fields, unprimed variables are used for those beams on the left side
of a slab, and primed for those at the right side of a slab (see figure 5.6). Left
traveling and right traveling beams are indicated by L,R subscripts respectively.
Using this notation
ERj = E on left side of layer j, travelling right (5.5)
E'j = E on right side of layer i, travelling right (5.6)
Note that the prime/unprime coordinates refer to the slabs, and not the boundary.
Slab i Slab j
Figure 5.6: Primed coordinates on the right side of a slab, unprimed on the left
side of a slab.
To write the four E fields on either side of a boundary in terms of each other,
Tij and pij are used for the amplitude transmission and reflection coefficients. The
subscripts indicate the direction of travel of the ray-"ij" indicates a ray initially
in slab i, interacting with the boundary with slab j.
ij = beam incident from i, transmitted into j. (5.7)
pij = beam incident from i, reflected at boundary with j. (5.8)
Writing equation (5.2) in this form,
ni + nj
ni -n nj
ni + nj
Pij = -pji (5.11)
The assumption will be made that the medium is lossless for the wavelengths of
TjiTij jiPij = 1
Using the specified notation, the arbitrary amplitudes, ERj and E'i, shown in
figure (5.6) are given by
ERj = TjERi + jiELj (5.13)
ELi = PijER' + TEL (5.14)
Rearranging (5.13) to give
Ei ER jiEL (5.15)
and substitution of (5.15) into (5.14) gives
E = Pj(ERj pjiELj) + TELJ (5.16)
(TJ-T -J P- p) EL + -PiiER (5.17)
Using (5.12) in (5.17), and equation (5.11) in (5.13) respectively, gives
ELi = + ERj (5.18)
ERi ER- + PiL ELj (5.19)
These two coupled linear equations give the field amplitudes to the left of
an interface in terms of those fields to the right of the interface (explicitly, the
equations give the fields on the right side of one slab in terms of those on the left
of the next slab, an equivalent statement). This is useful, since the last boundary