Thesis submitted to the Jawaharlal Nehru University for the award of the degree of

Wave propagation and diffusion in

active and passive random media

S. Anantha Rarnakrishna

Thesis submitted to the Jawaharlal Nehru University for the award of the degree of

Doctor of Philosophy

Raman Research Institute Bangalore - 560 080

February

200

1

DECLARATION

I hereby declare that the work reported in this thesis has been independently car- ried out by me a t the Raman Research Institute, Bangalore, under the supervision of Prof. N. Kumar. The subject matter presented in this thesis has not previously formed the basis of the award of any degree, diploma, associateship, fellowship or any other similar title of any other University or Institute.

Raman Research Institute Bangalore 560 080.

CERTIFICATE

This is t o certify that the thesis entitled Wave propagation and diffusion in active and passive media, submitted by S. Anantha Ramakrishna for the award of the degree of DOCTOR OF PHILOSOPHY of Jawaharlal Nehru University, is his original work. This has not been published or submitted to any other university for any other degree or diploma.

$'Kfi/w

^{rof. N. Kumar}

(Center Chairperson) Raman Research Institute

,k$$.r

Prof. N. Kumar (Thesis Supervisor)

Bangalore 560 080.

Preface

Our work reported in this thesis concerns wave propagation and transport i11 active and passive disordered media. The work can essentially be divided into two parts:

The first deals with coherent wave transport and localization in spatially random media and the associated statistics of fluctuations, and the second is concerned with the development of stochastic models t o describe the wave propagation in terms of an incoherent energy transport in stochastic media. In the first part, we also discuss the use of an imaginary potential as a quantum 'clock' for the sojourn time of a wave in a scattering potential.

In the first part (Part-A), consisting of Chapters-2, 3 and 4, we study wave prop- agation in spatially random media. It is well known t h a t the interference associated with coherent multiple scattering of waves (for quenched or static disorder) causes physical quantities such as the reflection/transmission or the conductance/resista~ice of a disordered sample t o become non-self-averaging (i. e., the fluctuations grow faster than the mean) regardless of the sample size. Thus, one needs the entire probabil- ity distribution over an ensemble t o describe such quantities. In low dimensions.

the effect of disorder on wave propagation is more drastic resulting in the localiza- tion of all states in one and two dimensions. This is particularly true for light, for which localization in higher dimensions can be accomplished only by a combinatio~l of strong Bragg scattering and large refractive disorder. Moreover, the Bosonic na- ture of light, which allows for coherent amplification and absorption, leads t o new phenomena such as random lasers. In this part of the thesis, we study the statistics of the non-self-averaging quantities such as reflection and quantities describing t h e dy- namical aspects such as the delayldwell times of scattering from a random ~ n e d i u m . One of the main ideas in this thesis is that the coherent amplification/absorption.

which preserves the temporal coherence of the wave, will also cause a concorriitarit scattering of the wave. Thus, we have a synergetic interplay of Anderson localizatio~i and coherent amplification/absorption, where the localization could be caused by the

very scattering concomitant with a spatially fluctuating arriplification/absorytiori iri a medium. In Chapter 3, we define a 'clock' for the quaniturn-mccha~iical sojourn ti11ic in a scattering potential using coherent amplification/absorption as a niathcrriatical artifice, but

find

that we have to correct for the extra scattering conicomitarit with it.

In the second part (Part-B), consisting of Chapters-5, 6 and 7, we develop ricw models for the problem of photon migration in turbid media, where we describe the wave propagation in a random medium as an incoherent energy transport in a stochastic (time varying random) medium. The underlying idea is t h a t under conditions of weak, but multiple scattering, the transport of a wave becomes almost diffusive and the entire process can be described as a random walk of a particle.

This approach, originally developed in the context of radiative transfer in stellar atmospheres, results in a Boltzmann transport equation for the specific intensity or the particle flux in the phase space. However, the general analytic solutions of this equation are unknown even for the simplest geometries, and the diffusion equation, which is a good approximation for the transport equation a t long length scales (L >>

I*, where I* is the transport mean free path) is most-often used. But, t h e diffusion approximation, which is a Wiener process for the spatial co-ordinates of a particle. is physically unrealistic, and accounts neither for a finite mean free path nor for a finite and constant speed(c) of the particle, which is characteristic of light propagation in a random medium. It has also been shown experimentally t o fail t o describe phenomena at short length-scales (L

<

101*) and short time scales (t

<

lot* where t* = l * / c ) . It is precisely these length- and time-scales t h a t are involved in medical imaging and diagnostics using laser light, and it is of importance t o develop better and alternative schemes t o the diffusion approximation. We develop here simple models of photo11 migration in a stochastic medium, where the photon propagates with constant speed in between the scattering events. This imposition of the constraint of constant speed effectively incorporates a finite mean free path into the problem as well as persistence in the phase space.

Below, we present a summary of the problems studied and the results obtaincd chapterwise. Each chapter has a self-contained introduction, which sets the back- ground for the material presented in that chapter.

Chapter 1: This is an introduction t o the thesis, pertinent t o the work reported in

Preface ^lr it. We review the important ideas and previous literature that are directly related to the problems addressed in the thesis.

Chapter 2: ,4 mismatch in the imaginary part of the refractive index (i~riagi~iary potential) always causes a concomitant scattering in addition to amplification or ab- sorption. Pure imaginary (amplifying) potentials can cause resonant emliancement of the scattering coefficients, which is important for Anderson localization of light in higher dimensions. The probability distribution of the reflection coefficient for light reflected from a one-dimensional random amplifying medium with cross-correlated spatial disorder in the real and the imaginary parts of the refractive index is derived using the method of invariant imbedding. The statistics of fluctuations have been obtained for both the correlated telegraph noise and the Gaussian white noise mod- els for the disorder. In both cases, an enhanced backscattering (with a reflection coefficient greater than unity) results because of coherent feedback due t o Anderson localization and coherent amplification in the medium. The results indicate that the effect of randomness in the imaginary part of the refractive index on localization and reflection is qualitatively different.

Chapter 3: The delay time for scattering is the most important quantity regarding the dynamical aspect of scattering in quantum mechanics, and one of the common measures for this quantity is the Wigner phase(q5) delay time (hdq5ldE). Th' is quan- tity, however, has certain deficiencies and alternative clocks such as precession of a spin in a magnetic field have been proposed. Here, we discuss a non-unitary clock, in- volving absorption/amplification by an added infinitesimal imaginary potential(iT.',) and find it not t o preserve the positivity of the conditional sojourn times, in gen- eral. The sojourn time is found t o be affected by the scattering concomitant with the mismatch, however weak, due t o the very clock potential(iV,) introduced for this purpose. We propose a formal procedure, separately for the cases of wave propaga- tion (non-tunneling a t above-the-barrier energy) and tunneling (at below-the-barrier energy), by which the sojourn time can be clocked ideally using the non-unitary counter by correcting for these spurious scattering effects. We further find that the conditional sojourn time for reflection is positive definite only if we agree to consider only those partial waves that have traversed the region of interest. This is justified in that the sojourn time should causally relate to the region of interest. The resulting

Preface vi time is then positive definite for an arbitrary potential and has the proper high- and low-energy limits. We also discuss why the spurious effects effectively cancel out for a random potential.

Chapter 4: In this Chapter, we consider the distribution of sojourn/delay tirrics for wave reflection from a one-dimensional random potential. We show that the so- journ time distribution for the reflection is related directly to that of the reflectiori coefficient, derived with an arbitrarily small but uniform imaginary part added to the random potential. The sojourn time distribution in the weak disorder-high en- ergy limit then follows straightforwardly from the earlier results for the reflection coefficient, and coincides with the distribution for the Wigner delay time obtained recently by other workers. All the moments of the distribution are divergent. The sojourn time distribution for a random amplifying medium is then derived. In this case, however, all the moments work out to be finite. We also correct an earlier cal- culation of the distribution of the Wigner delay time, where a slightly different for~ri of the distribution had been obtained. Further, our numerical simulations using the Tight Binding Hamiltonian indicate that the probability distribution obtained for the sojourn time using the imaginary potential and the distribution for the Wigner delay times coincide for both weak and strong disorder. For energies very close to the bandedge, however, the Wigner delay time distribution begins t o differ and beconlcs non-zero for negative delay times indicating a strong deformation of the incident wave packet. The sojourn time distribution obtained from the imaginary potential method displays no such behaviour, indicating that this clocks the 'literal' sojourn time of the wave in the potential.

Chapter 5: Here we adapt use of the Ornstein-Uhlenbeck process of Brownian nlo- tion to describe photon migration in turbid media. The Ornstein-Uhlenbeck process of Brownian motion is able to incorporate the finiteness of the mean free path and a well defined root-mean-squared (rms) velocity but assuming, of course, a distribution of speeds. We show by a path integral approach that the finite r.m.s speed defined by the fluctuation-dissipation theorem for this process is a stronger global constraint than a weaker average constraint implemented recently by others. We have devel- oped approximate analytic solutions based on the mirror-image method for absorbing boundaries for this process. The results have been compared to Monte-Carlo sirriu-

Preface ~ 1 1 . .

lations and good agreement is found even a t short time-scales. The simplicity of the solution for absorbing boundaries makes it an important and useful alternative to tlic diffusion approximation.

Chapter 6: Here, we develop a model in which the propagation of light in a scat- tering medium is described as the motion of a special kind of a Brownian particle on which the fluctuating forces act only perpendicular to its velocity. This strictly and dynamically enforces the constraint of constant speed of the photon in the medium. A Fokker-Planck equation is derived for the probability distribution in the phase space assuming the transverse fluctuating forces to be a white noise. Analytic expressions for the moments of the displacement

<

xⁿ

>

along with an approximate expression for the marginal probability distribution function P ( x , t ) are obtained. Exact numerical solutions for the phase space probability distribution have been obtained for infinite media, semi infinite media and slab geometries with absorbing boundary conditions.

The results show that the velocity distribution randomizes in a time of about eight times the mean free time (8 t * ) , only after which the diffusion approximation becorries valid. This factor of eight is a well known experimental fact. A persistence exponent of 0.4350zt.005 has been calculated for this process in two dimensions by numerically studying the survival probability of the particle in a semi-infinite medium. \Ve also study the case of a stochastic amplifying medium.

Chapter 7: In this Chapter, we discuss a generalization of the Telegrapher process to higher dimensions, which describes the diffusion of inertial particles in a model phase space. In one-dimension, it is long since known that the Telegrapher equation describes exactly the probability distribution function in one dimension for a particle undergoing random scattering events and moving with constant speed between the scattering events. Here only two values of the velocity f c are allowed. We have generalized this process to higher dimensions (d 2 2 ) rigorously, where the particle can move only along the 2d directions of the diagonals of a d dimensional hypercube.

Using a stochastic approach, a closed set of 2d coupled, linear, first-order, partial differential equations for the probability distribution function is obtained. This ad- mittedly artificial phase space incorporates considerable persistence in the photon random walks. We discuss several aspects of this model including the effects of the angular non-symmetry of the model that has been missed out in similiar other studies.

Acknowledgments

I am greatly indebted t o my supervisor, Prof. N. Kumar for the constant encourage- ment, education, help and support a t every stage of my work. Every discussion with him has been a great education for me. I thank him for his patience (which he has needed a lot with me), and friendliness (I have come out of every meeting with him feeling happy).

I thank my collaborators Prof. A.M. Jayannavar, Dr. G.V. Vijayagovindan and Prof. A.K. Sood for fruitful collaborations and useful discussions. I would like t o specially thank Prof. Jayannavar for several discussions regarding many aspects of my work which have benefitted me a lot. I thank Prof. Rajaram Nityananda for encouragement and several enlightening discussions. Thanks are due t o Prof. T.V.

Ramakrishnan for encouragement. I am also thankful t o Dr. Divakara Rao for support and encouragement from afar.

I wish t o thank Vidhyadhiraja for patiently listening t o many of half-baked ideas and also useful discussions. Discussions with Sandeep Joshi, Ahsan, Giri, Vishwanath and Srivatsa have been beneficial t o me. The Optics group meetings and lectures have taught me a lot. I thank all the faculty a t R.R.I. for their friendliness and help (both academic and non-academic) .

I thank the administrative staff a t R.R.1, particularly Raghavachar, Krishnama- raju, G.V. Srinivasa and Manjunath for all their help and making my stay a t R.R.I.

very comfortable. The library staff deserve a special mention for their help and friend- liness. The canteen staff have always cheerfully obliged my requests for tea and coffee a t any time of the day.

I will remember the enjoyable times that I have spent with Raja, Prachi, Pani:

Vishwanath, Pandey, Shreenivas and several others. I am grateful t o my wife, Kanchan, for her constant encouragement, patience, affectionate care and support throughout this work. I thank my family and her family for their support.

Acknowledgments ix Finally, I would like t o acknowledge the Abdus Salarri ICTP, Trieste, for irivitirig and supporting me to attend the Winter College on Opt'ics and Photonics, Feb. 2000, and the Institute of Physics, Bhubaneswar for supporting my visit there in ,4ug. 2000.

Contents

Introduction

. . . 1.1 Scattering systems

1.1.1 A single scatterer . . . 1.1.2 Collection of scatterers: Single scattering and multiple scattering 1.2 Random media, stochastic media and models of disorder . . . . . . 1.3 Wave propagation, diffusion and energy transport

1.3.1 Random phases: connection between wave propagation and dif- . . . fusion

. . . 1.3.2 Radiative transfer and diffusion

. . .

1.3.3 Inadequacy of the diffusion approximation

1.3.4 Connection between the specific intensity and the mutual co- herence function . . .

. . . 1.4 Wave interference and Anderson localization

1.4.1 Coherent backscattering . . . . . . 1.4.2 Anderson localization

. . . 1.4.3 Localization of light

. . . 1.5 Active random media

1.5.1 Random lasers . . . . . . 1.6 Sojourn time in a scattering potential

. . . 1.6.1 Quantum clocks

. . . 1.7 Fluctuations and statistics

Super-Reflection of light from a random amplifying medium with

disordered complex refractive index 35

2.1 Introduction . . . 35 2.2 Time-independent Maxwell's equations and amplifying media . . . 38

CONTENTS xi 2.3 Random amplifying medium with disordered complex refractive index 42 2.3.1 The Gaussian 6-correlated (white-noise) disorder . . . ⁴³

. . .

2.3.2 Correlated telegraph disorder 46

. . .

2.4 Conclusions 50

3 Correcting the quantum clock: The sojourn time in a scattering

potential 52

. . .

3.1 Introduction 52

3.2 Imaginary potential as a counter of sojourn time . . . 54 . . .

3.2.1 The average dwell time 55

. . .

3.2.2 The case of unitary reflection 55

. . . 3.2.3 Negativity of the conditional sojourn times 57

. . .

3.3 Correcting the 'non-unitary' clock 59

. . . 3.3.1 The case of propagation (non-tunneling) 59

. . .

3.3.2 The case of wave tunneling 61

. . . 3.3.3 The conditional sojourn time for reflection 63

. . . 3.4 The reflection delay time in the WKB approach 64

. . .

3.5 The case of the random potential 66

. . .

3.6 Conclusions 66

4 Distribution of sojourn times for wave reflection from a random

potential 69

. . .

4.1 Introduction 69

4.2 The sojourn time for wave reflection from a random potential . . . . 71 . . . 4.2.1 The case of a passive random medium 72

. . . 4.2.2 The case of an active random medium 74

. . . 4.3 Distribution of Wigner delay time for reflection 75 4.4 Strong disorder and a periodic background: Numerical results . . . . 77

. . . 4.4.1 The phase distribution for the reflected wave 78

. . . 4.4.2 Distribution of delay and sojourn times 80

. . .

4.5 Conclusions 83

CONTENTS xii 5 Adapting the Ornstein-Uhlenbeck process to describe photon mi-

grat ion 86

5.1 Introduction . . . 86

5.1.1 The path integral approach t o photon migration . . . 87

5.2 The Ornstein Uhlenbeck process and light diffusion . . . 89

5.2.1 Adapting the O-U process to light . . . 90

5.2.2 How strong is the 'weak constraint' of fixed speed in the O-U process ? . . . 91

5.2.3 Comparision between the O-U process and the diffusion ap- proximation . . . 91

5.2.4 Approximate solution t o the O-U process in the presence of absorbing boundaries . . . 92

. . . 5.3 Conclusions 98 6 Diffusion-at-a-constant-speed of photons: The 'strong' constraint of constant speed 99 . . . 6.1 Introduction 99 6.2 The modified Ornstein-Uhlenbeck process . . . 100

6.3 Solutions in unbounded media . . . 104

6.3.1 Moments of the displacement and the cumulant expansion . . 104

6.3.2 An approximate solution for strong, isotropic scatterers . . . 106

6.3.3 Numerical solutions in the phase space . . . 107

6.4 Solutions in bounded media with absorbing boundaries . . . 111

. . . 6.4.1 A semi-infinite medium 111 . . . 6.4.2 A finite slab 113 . . . 6.5 Random Amplifying Media 116 . . . 6.6 Conclusions 117 7 Diffusive transport with inertia: the generalization of the Telegra- pher process to higher dimensions and a model phase space 119 7.1 Introduction . . . 119

7.2 The Telegrapher process in one dimension . . . 121

7.3 Generalization of the Telegrapher process t o higher dimensions . . . 123

CONTENTS xlll ...

7.3.1 The equations for the generalized Telegrapher process . . . .

.

¹²³

7.3.2 Absorbing boundary conditions

. . . .

. .

. . .

. . . . 125 7.3.3 Projected motion along any axis and angular non-symmetry of

the model . .

. . . .

. . . .

.

126 7.4 Kubo-Anderson like stochastic processes

. . .

. . .

.

. . . . .

. . .

. 127 7.5 Conclusions

. . .

. . . .

. . . . .

.

. . . .

.

. . .

. 127

A The method of invariant imbedding 129

B The Novikov theorem 133

C The "formulae of differentiation" of Shapiro and Loginov 135