Adaptive control of the propagation of ultrafast light through random and nonlinear media

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Adaptive control of the propagation of ultrafast light through random and nonlinear media
Moores, Mark David, 1969- ( Dissertant )
Reitze, David H. ( Thesis advisor )
Place of Publication:
Gainesville, Fla.
University of Florida
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Subjects / Keywords:
Ballistics ( jstor )
Cross correlation ( jstor )
Electric fields ( jstor )
Electric pulses ( jstor )
Electrical phases ( jstor )
Lasers ( jstor )
Light refraction ( jstor )
Phase shift ( jstor )
Photons ( jstor )
Signals ( jstor )
Dissertations, Academic -- Physics -- UF
Physics thesis, Ph. D
government publication (state, provincial, terriorial, dependent) ( marcgt )
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )


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 additional 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 experimental solutions to problems with little hope of direct analytical solution. Linear control of nonlinear propagation of guided waves is demonstrated, with broad applicability in fundamental science and is a step towards ultrafast optical telecommunications. Reduction of the optical effects of a scattering material demonstrates successful adaptive control of the effects of a non-ideal optical material. Correlating the applied phase to a modeled dielectric stack gives insight into the random internal structure for the purpose of characterization. ( ,, )
ultrafast, femtosecond, laser, coherent control, adaptive learning, genetic algorithm, nonlinear
Thesis (Ph. D.)--University of Florida, 2001.
Includes bibliographical references (p. 171-176).
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by Mark David Moores.

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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.


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

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 Kerr lensing. .... . . . . . .... 20 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 SLM construction and operation . . . ... 29 SLM alignment ....... . . . . .. 32

2.5 Low Energy Coherent Pulse Detection .. .....
2.5.1 Why Coherent Pulse Detection is Necessary
2.5.2 Common Techniques ............. Cross and auto correlation ..... FROG .. .. .. ... .. .. .. Spectral interferometry .......
2.5.3 Pros and Cons .. . ............
2.6 Random Media Sample ...............

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 .. . ............ Parameterization of the problem to Initializing the population .....

be solved Setting the environment, paradise or plague? . . Evaluation . . . . . . . . . . . Application of operators, creation of the next gen-
eration .................. .....
3.3.2 Graphical Results of the Example Simulation .. ...... Result 1, crossover=0.7, mutation=0.02 .. .... Result 2, no Crossover; no mutation .. ...... 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 ..............

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

5.1 Outline ......................
5.2 T heory . . . . . . . . . . . .
5.2.1 Dielectric Boundaries ...........
5.2.2 Dielectric Stack .............. Notation ............ Derivation ........... Boundary matrix, H ...... Layer matrix, L ......... 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 . . . . . . . . . . .

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.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.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.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



Mark David Moores

May 2001

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.


1.1 Overview

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

be minimized.

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' [15]. 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 [14]. 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 [7]. 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 [53]. Application of a frequency dependent phase to a scat-

tered ballistic pulse will then compensate at least in part for the effect of the

scattering materialss.

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 [11] 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)

LNL (1.3)
S0.23m (1.4)

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 [18]. 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 [21] 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.


2.1 Outline

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


Ele -



When the intensity is at half its maximum value,







T 21n(2)






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 [48], some of which are shown in table 2.1. The spectral field is




0.4- FWHM


-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 [8] 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 [52], 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. [42], 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 [36] 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 [41] 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

4.5 fs.



0.8 -

^ 0.6

p 0.4 -

0.2 -
400 500 600 700 800 900
Wavelength (nm)

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.

25fs 800nm

P \ OC
S1 P2 CW Ar

[M2 TiSaf

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


2.3.2 Cavity

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).

2.3.3 Modelocking

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. 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 [23]

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. 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)

10 2
I(t) = cos(wit + bi) (2.20)

w, = (1 + 2 2.29873 x 1015 (2.21)

random, non modelocked
; = (2.22)
0, modelocked

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

Time (fs)

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(,)


with solutions

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 [13] 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.

2.3.6 OPO

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
Wavelength (nm)



o 0.4

0.0 -,
-100 -80 -60

-40 -20 0
Time (fs)

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
Bandwidth 65nm
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

2.4.1 Overview

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

of frequency.

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 [55] and proposed as much as 4 years prior [54].

The topic grew from considerations of pulse chirp for the efficient generation of

radar waves [33].

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)
d (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.

E(tFocal E(t)

L1 Plane L2

0, b Od

c :G2

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. 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 [61] 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
ITO Contacts

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 [20]. 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

dA D
cos 0 (2.31)
dx f

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

exceeded [6].

3Pm, 4f Focal

97m.. Plane


Figure 2.9: The 128 pixel SLM sits at the focal plane of the 4f grating setup. 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
Wavelength (nm)



20 40

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 [59].


24- ----- 850 nm
22 --- --. 800 nm
750 nm
18 -
t 14-
10 -
2- --'-------------------

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. 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.


(t, o0)

I(T, 2w) 0

_v Ii(t+r, o)

Figure 2.12: Second Harmonic Correlator. FROG

Frequency resolved optical gating (FROG) [57] 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

E(t, Q))e-i(wt-QT)dtdQ

FROG (c,)=


The spectrogram measures the magnitude of Esg, and knowing that the mathemat-

ical form of Esig(t,7) is given by equation 2.34 [46] 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. 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 [56], 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
1.49mm (2.38)

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

y i

00 02 04 06 08 10












3.1 Outline

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

in 3.4.

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
adequate convergence

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. 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)
-1.079 (3.3)
y = -4 + 3 x (3.4)
-3.619 (3.5)

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.

x y
010111 000011 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.

 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. Evaluation

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.

 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. 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)
Ax 0.1270
Ay 0.1270
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". 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. 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

-1 -
< x

0 100 200 300 400 500

4 -

2 ...... 2
S -


-41 x


0 100 200 300 400 500

4 x x

2 xxx xx X
-2 xx

-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



xxx xx

x X
XX X >0<


Crossover=0.00, Mutation=0.02




Crossover=0.70, Mutation=0.00






Figure 3.4: The effects of only mutation (top) and only crossover (bottom). Lack
of mutation causes the experiment to stagnate and prematurely end.


0.5 -

o .


-0.5 -k


0.5 -


M=0.00, X=0.70

M=0.02, X=0.00


-2 0

M=O.0, X=0.35

M=0.01, X=0.00



-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

problem space.

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.


4.1 Outline

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.

4.2 Background

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 [44]. 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 [45]) 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 [15] 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
Diffuse Photon

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 [11]

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 [10]. 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

T i
Diagnostic Pickoff

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.

4.4 Results

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

optical properties.

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.

I I71


SId4 72

n(cw) d3

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.

4.4.3 Spectra

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

Scattered Tail

0 0.0 .
-200 -100 0 100 200 300 400


S0.1- 20fs


-200 -100 0 100 200 300 400
Time (fs)

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

d 0.6-
S- 41fs
0.4 -
Scattered Tail

0 .0 - '''---
-300 -200 -100 0 100 200 300

Time (fs)

-s- 27fs



0.01 -

-300 -200 -100 0 100 200 300

Time (fs)

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.


--- Laser
0.8 -- Sample


/ Scattered Tail

0.0 -
-300 -200 -100 0 100 200 300

Time (fs)


a 0.01 --

1E-3 -.

-300 -200 -100 0 100 200 300

Time (fs)

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


-------- Laser
0.8- ---- Sample
S- 29fs


0.0- / -
-300 -200 -100 0 100 200 300

Time (fs)

0.1 -

0.01 /, ,

-300 -200 -100 0 100 200 300

Time (fs)

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.


S-------- Laser
0.8 ,- Sample
S --Optimized


0.2' '

0.0 -

0.01 -
-300 -200 -100 0 100 200 300

Time (fs)

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



- 0.4

Z no

700 720 740 760 780 800 820 840 860 880 900
Wavelength (nm)




700 720 740 760 780 800 820 840 860 880 900
Wavelength (nm)

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



, 0.4


700 720 740 760

780 800 820 840 860 880 900

Wavelength (nm)

700 720 740 760 780 800 820 840 860 880 900

Wavelength (nm)

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



3 0.8

Fuzz 3mm

0.4 -
780 785 790 795 800 805 810 815
Wavelength (nm)

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
be apparent.

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

1.405- 1
p = (4.7)
1.405 + 1
= 0.1684 (4.8)


S- Reference

5 06-



-150 -100 -50 0 50 100 150
Time (fs)

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.

4.5 Discussion

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

experimental result.

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.


5.1 Outline

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

after optimization.

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.


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

in 5.4.1.

5.2 Theory

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

ri -^i


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.

ET 2n,
EI nI + nT

EI nj + nT


*4 ---




Figure 5.3:

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
1 2

E7 -6 d- E,
no no

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. Notation

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.


E'Ri ERj


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,

= (5.9)
ni + nj
ni -n nj
Pij (5.10)
ni + nj

Pij = -pji (5.11)

The assumption will be made that the medium is lossless for the wavelengths of

interest, mathematically

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)
Tij Tij

Using (5.12) in (5.17), and equation (5.11) in (5.13) respectively, gives

ELj .
ELi = + ERj (5.18)
Tij Tij

ERi ER- + PiL ELj (5.19)
7ij 7ij

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