and far-field - Imperial Spiral - Imperial College London [PDF]

U niversity of London Im perial College of Science and Technology The Blackett Laboratory Applied Optics Section

Correlation scale effects in light scattering from rough surfaces Alan S. Harley

Thesis subm itted for the degree of D octor of Philosophy of the U niversity of London (1989)

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To my parents

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ABSTRACT The scattering of coherent radiation by random media and surfaces is a subject that has increasingly attracted the attention of mathematicians and experimental physicists alike. A key to their earlier success has been due to the application of the Kirchhoff approximation in calculating the low-order scattering statistics from phase screens and surfaces for cases in which it is valid. In this thesis, the validity of the approximation is discussed and it is applied to single-scale and two-scale phase screens, as well as to an interesting class of structures, known as fractals, which have a hierarchy of scales sizes and which are self-affine. Being non-differentiable, and hence purely diffracting, random fractal scatterers exhibit qualitatively different intensity statistics than do smoothly varying scatterers. Solutions for their first intensity moment are given for particular values of the fractal index and also, in the asymptotic limit of large angles, for arbitrary index. In order to test these theoretical predictions, experiments have been conducted using specially prepared surfaces made in photoresist, whose statistical parameters can be characterised by a mechanical profilometer. The scattering distributions of the single-scale surfaces so produced are shown to agree well with theory when /3 A, A — where ft is the correlation length and cr^ is the sur­ face height standard deviation. If, however, (3 > A and (3 ~ oq* (such that the average slopes are close to 40 degrees), the surfaces exhibit enhancement in the backscattered (anti-specular) direction and significant depolarisation, which leads to an interesting four-fold symmetric scattering pattern. Although this behaviour can be qualitatively accounted for by multiple scattering and shadowing effects, there are, currently, no analytic solutions to this particular problem and it must be examined by way of numerical solution of the scattering integrals. The angular scattering distributions from the synthesised fractals, in transmission, are found to agree with the theoretical predictions although in reflection, multiple scattering and shadowing, once again, appear to account for the strikingly different results. Sim­ ilarly, ground-glass surfaces, which exhibit fractal characteristics in transmission, are shown to behave quite differently in reflection. The far-field second intensity moment, or speckle contrast, is calculated numer­ ically for weak single- and two-scale phase screens, and they are both shown to ex­ hibit the ‘peak shift’, whose direction of motion is dependent on the phase variance. In contrast, numerical computations of the second moment for one-dimensional frac­ tals show a monotonic rise to saturation at the Gaussian limit provided the fractal index v < 1.5. For v > 1.5, however, there appears to be a broad, low contrast peak.

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ACKNOWLEDGMENTS I would like to extend my particular thanks to my supervisor, Professor Chris Dainty, and to my ‘mentor’ at R.S.R.E, Professor Eric Jakeman, for the support and encouragement they have given me over my period of study. I am also especially grateful to Professor Walter Welford for his patience in listening to and helping with my obscure mathematical problems. My thanks also to Fred Reavell and Tony Canas, for having designed and built much of the scattering rig, and to Mike Channon for having built the photoresist spinner. I would also like to mention Eugene Mendez and Kevin O’Donnell for their part in our collaborative efforts, Ari Friberg for many stimulating discussions, Kaveh Bazargan for helping with T^X and, with Neil Haigh and Mike Channon, in providing mutual crutches and entertainment over the past few years. In the face of what must have seemed, at times, like an uphill struggle to ‘get the thing finished’, I wish to thank my parents and Seema dearly for their enduring support and for having suffered with me — mostly in silence!

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TABLE OF CONTENTS A B S T R A C T _____________________________________________________________ 3 A C K N O W L E D G M E N T S_______________________________________________ 4 C H A PT E R 1 — INTRODUCTION________________________________________9 1.1 Review of Thesis Contents----------------------------------------------------------------- 10 1.2 The Mathematics Of Stochastic Processes_____________________________ 11 1.2.1 Random variables, samples and ensembles_________________________ 11 1.2.2 Probability distribution and density functions______________________ 12 1.2.3 Moments of a random variable____________________________________ 13 1.2.4 Joint moments of random variables________________________________ 13 1.2.5 Characteristic functions__________________________________________ 14 1.2.6 Stationarity and ergodicity_______________________________________ 15 1.2.7 The autocorrelation function and the Wiener-Khinchin relationship____________________________________________________ 15 1.3 Methods of Numerical Integration_____________________________________17 1.4 Fractals_____________________________________________________________ 19 1.4.1 Some mathematical properties of fractals__________________________ 20 1.4.2 The fractal power spectrum_______________________________________23 1.4.3 The fractal correlation function___________________________________ 25 1.4.4 The topothesy----------------------------------------------------------------------------- 26 1.5 The Theory of Speckle Formation_____________________________________ 26 1.5.1 Gaussian speckle_________________________________________________ 27 1.5.2 Non-Gaussian speckle____________________________________________ 28 REFERENCES_________________________________________________________ 29 C H A P T E R 2 — THE SCATTERED MEAN INTENSITY__________________32 2.1 Physical Optics Diffraction Theory____________________________________ 32 2.1.1 The application of Physical Optics theory to random rough surfaces________________________________________________________ 33 2.1.2 Discussion of the Beckmann Theory approximations________________ 37 2.1.3 The application of Physical Optics theory to random phase screens___.________________________ 41 2.1.4 The calculation of the scattered mean intensity from a ran­ dom phase screen_______________________________________________ 44 2.2 The Boundary Condition Method using the Rayleigh Hypothe­ sis _______________________________________________________________ 46 2.3 The Extended Boundary Method using the Ewald-Oseen Ex­ tinction Theorem_________________________________________________ 46 2.4 The Angular Intensity Distribution from Single-scale Scatter­ ed ------------------------------------------------------------------------------------------------48 2.4.1 Single-scale scattering from random phase screens_________________ 49 5

2.4.2 Single-scale scattering from random rough surfaces-------------------------52 2.4.3 Discussion and results----------------------------------------------------------------- 53 2.5 The Angular Intensity Distribution from Two-scale Scatterers---------------53 2.5.1 Two-scale scatteringfrom random phase screens____________________ 56 2.5.2 Discussion and results____________________________________________57 2.6 The Angular Intensity Distribution from Fractal Scatterers_____________61 2.6.1 Calculation of the scattered intensity from random fractal phase screens-----------------------------------------------------------------------------61 2.6.1.1 The expression for the mean intensity in one-dimension-----------------62 2.6.1.2 The expression for the mean intensity in two-dimensions__________ 63 2.6.1.3 The finite-aperture mean intensity for the Brownian fractal, v = 1 -----------------------------------------------------------------------------64 2.6.2 Large-aperture approximation solutions for the mean in­ tensity -------------------------------------------------------------------------------------- 66 2.6.2.1 The solution for v = 1 in one-dimension--------------------------------------- 67 2.6.2.2 The solution for v = 1/2 in one-dimension________________________68 2.6.2.3 The solution for v = 1 in two-dimensions_________________________69 2.6.3 The mean intensity from fractals with arbitrary index_____________ 69 2.6.3.1 The mean intensity for one-dimensional fractals_________________ 70 2.6.3.2 The mean intensity for two-dimensional fractals---------------------------72 2.6.4 Discussion and results-------------------------------------------------------------------74 REFERENCES_________________________________________________________ 75 C H A PT E R 3 — AN EXPERIMENTAL EXAMINATION OF THE FIRST INTENSITY MOMENT________________________________ 81 3.1 The Experimental Apparatus-------------------------------------------------------------- 81 3.1.1 The scattering equipment geometry------------------------------------------------81 3.1.2 The detector response-------------------------------------------------------------------84 3.2 The Preparation and Analysis of Photoresist Surfaces__________________86 3.2.1 The fabrication of rough surfaces in photoresist------------------------------ 86 3.2.2 The analysis of random rough surfaces_____________________________87 3.2.2.1 The effect of the stylus on the recorded profile___________________ 87 3.2.2.2 The effect of the sample length on the recorded profile------------------87 3.2.2.3 The fitting of the mean-line to the profile data___________________89 3.2.3 Statistical parameters determined from the Talysurf profile data___________________________________________________________ 89 3.3 Light Scattering from Single-scale Surfaces____________________________92 3.3.1 The synthesis of single-scale surfaces______________________________ 92 3.3.2 The analysis of the single-scale surfaces____________________________97 3.3.3 The single-scale experimental reflection results____________________ 101 3.4 Light Scattering from Multi-scale Diffusers and Surfaces _______________107 3.4.1 Spectral synthesis using combinations of annular and circu­ lar apertures___________________________________________________108 3.4.2 Spectral synthesis using Gaussian apertures_______________________ 113 6

3.4.2.1 The analysis of the Gaussian-aperture surfaces---------------------------117 3.4.2.2 Light scattering from the Gaussian-aperture samples____________127 3.4.3 Spectral synthesis using circular apertures_______________________130 3.4.3.1 The analysis of the circular aperture surfaces___________________ 132 3.4.3.2 Light scattering from the circular aperture samples______________139 3.5 Light Scattering from Naturally Rough Surfaces--------------------------------139 3.5.1 The preparation of the ground glass surfaces--------------------------------- '41 3.5.2 The Talysurf analysis of thegroundglass surfaces__________________144 3.5.3 Light scattering from ground glass diffusers in transmission _________ 157 3.5.4 Light scattering from ground glassdiffusers inreflection_____________ 158 REFERENCES________________________________________________________ 165 APPENDIX 3 .1 _______________________________________________________ 167 APPENDIX 3 .2 _______________________________________________________ 169 C H A P T E R 4 — THE INTENSITY SECOND MOMENT_________________ 172 4.1 Historical Review----------------------------------------------------------------------------- 172 4.2 Derivation of a General Expression for the Intensity Second Moment------------------------------------------------------------------- ------------------ 175 4.3 The Single-scale Intensity Second Moment___________________________ 177 4.3.1 The computation of the single-scale second moment_______________ 178 4.3.2 Discussion of the single-scale second moment results_______________ 179 4.4 The Two-scale Intensity Second Moment_____________________________183 4.4.1 The computation of the two-scale second moment_________________ 184 4.4.2 Discussion of the two-scale second moment results_________________ 185 4.5 The Second Intensity Moment for Corrugated Brownian Frac­ tals ____________________________________________________________ 190 4.6 The Second Intensity Moment for Corrugated Fractals with Arbitrary Index---------------------------------------------------------------------------193 4.6.1 Examination of the function symmetries__________________________194 4.6.2 Normalisation of the second moment expression___________________194 4.6.3 Integration of the second moment expression_____________________ 196 4.6.4 Discussion of the fractal second moment results___________________199 REFERENCES_______________________________________________________ 201 APPENDIX 4 .1 ________________________________________________________ 203 APPENDIX 4 .2 ________________________________________________________ 209 APPENDIX 4 .3 ________________________________________________________ 212 APPENDIX 4 .4 __________ I_____________________________________________214 APPENDIX 4 .5 ________________________________________________________ 217 APPENDIX 4 .6 ________________________________________________________ 219 C H A P T E R 5 — THE MEAN INTENSITY SCATTERED FROM FINE SCALE ROUGH SURFACES________________________________ 222 7

5.1 Fabrication of the Fine-scale Surfaces------------------------------------------------222 5.2 The Scattering Equipment Geometry_________________________________224 5.2.1 Experimental results using a He-Ne laser light source-------------------- 230 5.2.1.1 Incident s-polarisation________________________________________ 230 5.2.1.2 Incident p-polarisation________________________________________ 231 5.2.2 Experimental results using an Argon-ion laser light source________231 5.2.3 Discussion of experimental results________________________________237 5.2.3.1 The polarisation effect_________________________________________241 5.2.3.2 The backscattered intensity enhancement effect_________________ 242 5.3 A Simple Multiple-Scattering Theory________________________________ 242 5.3.1 Derivation of the far-held scattered amplitude, to secondorder, from a one-dimensional surface profile____________________244 5.3.1.1 Evaluation of the singly-scattered contribution, 'Fl5(r) __________ 246 5.3.1.2 Evaluation of the doubly-scattered contribution, ^F2s(r) _________ 247 5.3.1.3 Evaluation of the (second-order) total scattered held, $ ( r ) __________________________________________________________250 5.3.2 Discussion of the simple scattering m odel________________________ 250 REFERENCES_______________________________________________________ 252 C H A P T E R 6 — CONCLUSIONS_______________________________________255 6.1 Summary of results------------------------------------------------------------------------- 255 6.2 Conclusions_______________________________________________________ 258 R E FER E N C E S (for entire thesis)______________________________________ 260 PA PE R S A N D P U B L IC A T IO N S _______ ________________________________

TABLE OF PLATES PLATE 1: SEM micrograph of gold-coated, ground-glass surface__________ 164 PLATE 2: SEM micrograph of a Gaussian-correlated, gold-coated photoresist surface________________________________________________ 164 PLATE 3: The cross-polarised scattered intensity from the s-polarised illumination of a very hne scale surface_____________________________ 232 PLATE 4: Photomicrograph of a very hne surface showing the cross-polarised scattered intensity for s-polarised (vertical) illumination______________________________________________________ 232

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CHAPTER 1 CHAPTER 1 INTRODUCTION Scattering and wave propagation play an essential role in our everyday lives, be­ ing responsible for our ability to see and hear things happening around us. Yet, the complexity of the scattering process is such that for many decades, it has concen­ trated the minds of physicists and mathematicians as they have tried to determine the properties of waves scattered from rough surfaces or transmitted through turbu­ lent media. This generalised problem is of considerable interest in many branches of science, being of particular importance in optics, radio science, acoustics, astronomy, remote sensing and in radar, to mention just a few. Its importance lies not only in that scatter-induced noise may degrade the performance of the various signal detection systems, but also that the statistics of the scattered fields may provide useful information about the scatterer’s physical structure. Since the introduction of the laser, considerable attention has been focused on the role of optical experiments in the study of scattering, an area previously dom­ inated by studies of radio wave scintillation phenomena. This is due not only to the fact that the performance of many optical systems using coherent laser sources can be severely affected by scattering-related speckle noise but also because opti­ cal wavelength experiments can be readily performed in the laboratory, making the verification of the various scattering theories an attainable goal. A key to the early success of these theories was the application of the Kirchhoff approximation to the Physical Optics diffraction theory in order to study the scattering from randomly rough objects. Although it is inherently a scalar wave theory and is subject to a variety of other constraints (see Chapter 2), the version of the theory developed by Beckmann and Spizzichino1-1 in their seminal work on the subject has provided the basis for many of the more important results, also confirmed experimentally, which have followed. It is to their credit that despite the two-and-a-half decades which have elapsed since the first edition was published, it is only recently that concerted efforts have been made to systematically explore the limitations of their theory through controlled laboratory experiments. In order to obtain mathematically tractable results, the early theoretical work usually made convenient assumptions regarding the statistical properties of the scatterer. Since little attempt was usually made to test these assumptions and, moreover, the experiments were largely confined to measuring the scattered intensity statistics from ‘naturally’ abraded scatterers, the empirical data was more important in con­ firming the general trends of the theoretical results rather than establishing a basis for verifying the theories. As the theories became more complex, however, the need for sophisticated experimental verification became all the more important so that with the emergence of a technique for producing scatterers with known statistical properties, a way of testing the theories has finally been established. The surface parameters which are most important in determining the scattered statistics are related to the characteristic size of a surface feature, the correlation 9

CHAPTER 1 length. Thus the ratio of the vertical roughness to the correlation length, or average surface slope, was shown by Beckmann and Spizzichino to be important in deter­ mining the scattered intensity. In addition, the intensity contrast, which is related to the signal-to-noise ratio and is therefore of great relevance for signal detection, is intimately related to the number of illuminated correlation cells on the scatterer. In this thesis, the relationship of the correlation length to the other scattering param­ eters is explored for a variety of types of scatterers. In particular, we consider both very large and very small correlation lengths (relative to the probing wavelength), illuminated areas much greater than or of the same size as a correlation cell and surfaces for whom a correlation length cannot, strictly, be defined.

1.1 REVIEW OF THESIS CONTENTS

Following this review, the remainder of Chapter 1 serves to provide the back­ ground information common to all of the following chapters and basic to the ensuing discussions. The mathematics of random processes is central to the characterisation of the randomly rough surfaces and diffusers and is important in the understand­ ing of the experimentally measured ensemble average quantities. A section of this chapter is, therefore, devoted to describing the statistical parameters used to charac­ terise stochastic processes and it is followed by a discussion of numerical integration techniques, which are used extensively throughout this work. A further section is given to a discussion of fractals, which are used to describe a particular class of rough surfaces whose scattering properties are discussed at some length throughout this work. The chapter concludes with the theory of speckle formation, which is a common theme linking all chapters, as both the theoretical and experimental results are concerned with examining the intensity moments of the mean speckle field. Chapter 2 examines the theoretical basis and assumptions inherent in the Phys­ ical Optics Theory, placing particular emphasis on calculating the scattered field in the far-field region of the scatterer. In addition, some of the more rigorous scatter­ ing theories are reviewed. Based on the approach developed for phase screens, the mean scattered intensity is calculated for a variety of different scatterers, ranging from the most ‘simple’ single-scale diffuser to the fractal phase screens described in Chapter 1. New results include predictions of the angular intensity distribution for two-scale scatterers and for fractals having indices given by v = 1 and v = 1/2, as well as for arbitrary index in an asymptotic limit. The connection between theory and experiment is made in Chapter 3, which sets out to experimentally examine the theoretical predictions of Chapter 2 for the single-scale and fractal scatterers. Following the description of a technique for fabricating randomly rough surfaces with prescribed statistics, the results from the Gaussian-correlated surfaces so produced, which are statistically characterised by their profilometric data, are thought to represent the first tests of Beckmann’s theory using well-characterised surfaces. A new technique is also described whereby surfaces having fractal-like power spectra can be synthesised. These more complex surfaces are tested both in transmission and in reflection, where their scattering behaviour is shown to be markedly different. Finally, in the most systematic, known tests of the widely used diffuser, ground glass, the angular intensity distributions 10

CHAPTER 1 are measured and fitted to the most appropriate surface model. Chapter 4 tackles the more complicated theoretical problem of calculating the far-field speckle contrast. Employing the multinomial approach as used by Mercier, infinite series solutions of the contrast both for one and two-scale diffusers are pre­ sented. These are evaluated numerically as a function of the illuminated area for higher phase variances than before (without using a deep screen approximation), although the two-scale work is new, and they are compared with existing results. The observation of the ‘peak shift’ phenomena, which appears to be unknown in the literature, is commented on. Much less is known about the behaviour of the far-held speckle contrast from fractals, for which the only existing result is a mod­ ification of a solution derived for a different surface type and applicable for u = 1. Consequently, the latter part of this chapter concentrates on numerically solving the three-dimensional integral for arbitrary index, giving details of the steps of the calculation. An alternative aspect of the effect of the correlation scale on the scattering statistics is examined in Chapter 5, which looks at what happens when the corre­ lation length approaches the same dimension as the illuminating wavelength and the surface slopes are steep. The scattered mean intensities from these fine-scale surfaces, produced using the method of Chapter 3, are thought to show the first cases of enhanced backscatter from characterised, rough surfaces. The results are compared with known theories and a phenomenological account of the scattering and polarisation behaviour is given. Because the steep surface slopes are likely to make multiple scattering and shadowing effects significant, a second-order, scalar scattering model is derived which enables the scattered intensity to be calcula od. Finally, Chapter 6 presents the main conclusions of the thesis and suggests interesting topics for further experimental and theoretical research.

1.2 THE MATHEMATICS OF STOCHASTIC PROCESSES

Random processes play an important role in light scattering because of the ran­ dom nature of real surface roughness and refractive index fluctuations of turbulent media. As a result, usually only the statistical properties of the scatterer can be known from which, in turn, the appropriate statistics of the scattered field must be calculated. While the following discussion of the statistics of random processes is completely general, its use in the majority of this thesis is concerned with describing the statistical properties of the random surface height or phase distribution of the scatterer.

1.2.1 Random variables, samples and ensembles

For every possible outcome, A, out of a set of outcomes, {A}, of some notional experiment whose result cannot be known in advance, we assign a real number u(A). A random variable, U, is then defined as consisting of all possible u(A) together with an associated measure of their probabilities. A random process is defined by assigning a further and independent variable, t, to each event such that the collection of possible ‘sample’ functions, u(A, £), together with their associated probabilities constitutes the random process. Usually the explicit reference to the set of possible outcomes, {A}, is omitted so that the random process is represented 11

CHAPTER 1 by U(t) and the specific sample functions by u(t). Thus the random process consists of the entire ensemble of the possible sample functions. In practice we are usually able only to measure properties of the sample function and so are frequently interested in estimating the properties of the ensemble, or process, from the sample. However, in the limit of the number of samples being very large, the sample statistics tend to the ensemble values. We denote ensemble (population) average properties by < ... > and shall assume their use throughout this thesis unless specified otherwise.

1.2.2 Probability distribution and density functions

Given a random variable, U, the probability distribution function, which defines the probability that the random variable, U, has a value less than or equal to some specific value, u, is written as Pu(u) = Prob{U < u}. Of more practical importance, however, the probability density function, or pdf, is defined by V P iu ) =

Writing this as pu(u) = lim Au then

0

Pu(u) - P jj(u - Au) Au

pu(u)Au Pu(u) — Pu(u —Au) = Prob {u — Au < U < u}, for sufficiently small Au. Thus pu(u)Au gives the probability that the random variable U lies in the range u —Au < U < u. In addition, the pdf is subject to the constraints that f oo pu(u)du = 1, /_

and pu(u) > 0. The (first-order) probability density function which we shall most frequently refer to throughout the thesis is the Gaussian, or normal, density, which is written as 1 exp ( (u— < u > )2 pu (u ) = \ 2cr2 V2-,tircr* and which describes the familiar bell-shaped curve. This distribution is particularly important as, through the application of the central limit theorem1-2, it represents the limiting function of the sum of many independent random processes. The sig­ nificance of the mean, < u >, and standard deviation, cr, are discussed below. This approach may be extended to apply for two or more variables and we define the joint probability distribution function by1,3 P[/(ui,U 2 , . . .u n) = Prob{£7 < u i,U < u 2,.-.U < un}, and the corresponding joint probability density function by

Pu{uu u2,

= d-^ y : dUnp u ( ^ u 2, ... o . 12

CHAPTER 1 We shall shortly show that if the nth-order joint probability density function is completely known, then all of the statistical properties of the random process can be determined.

1.2.3 Moments of a random variable

One of the ways in which a stochastic process can be described is in terms of its statistical moments. The nth (ensemble) moment of a random variable U, if it exists, is defined as oo unpu(u)du.

/ / /

-oo

Of particular importance is the first moment, or mean, which is simply given by oo upu(u)du, -oo

and the second moment by <

oo

u2pjj(u)du.

-oo

If we are concerned with the fluctuations of the variable about its mean value we examine the central moments. The most important of these is the variance, defined by oo (u— < u > )2pu{u)du,

/

-oo

whose square root, cr, is known as the standard deviation. This gives a measure of the distribution, or spread, of the random variable about the mean. For a Gaussian probability density function (see §1.2.2), it can be shown that whereas there is a 68% likelihood of finding the random variable within ±1(7 (one standard deviation) from the mean, the likelihood increases to 95% for ±2cr.

1.2.4 Joint moments of random variables

Higher order moments are also of considerable importance, as they relate the fluctuations of a random variable at one point to those of a similar, or different, variable at another point, or points. When examining rough surfaces, for example, it is not sufficient simply to know the ‘point’ (first-order) statistics, that is the mean and standard deviation, as we have no indication of the coarseness of the lateral structure. By a simple extension of the previous discussion the joint moments of the random variables U and V, which are jointly distributed with probability density Puv{u-,v)-, are given by oo < unvm > = J J unvmpu v(ui v)du dv.

For our purposes, the most important joint moments of U and V are the correlation, which is related to the second-order probability density function by oo c \j v

= < uv > =

J J

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uvpuy(u,v)dudv,

CHAPTER 1 and the covariance, given by Covuv = ((u— < u > )(v— < v >)) = cuv~ < u > < v > • In addition, the autocorrelation, which expresses the correlation of a function with itself at two points x\ and X2 and, consequently, measures the structural similarity of U(xi) and U(x 2) over the ensemble, is given by 00

C12 = < u(xi)u(x2) > = J J u(xl )u(x2)puu u2(u{x i ) i u (x 2 ) ) d u i d u ‘4

We note, in passing, that for jointly Gaussian random variables U\, U2 , . . . , Un, joint moments of order higher than two can always be expressed in terms of the firstand-second-order moments1,2. Because of this property Gaussian random variables are particularly simple to use and assumptions of normality are therefore used ex­ tensively (although not without some justification) in describing surface height or phase statistics.

1.2.5 Characteristic functions

Another way of calculating the moments of a random variable, U, is by using the characteristic function, which is equal to the expected value of exp(zo;u). Thus / oo exp(iuu)pu(u)du, (1.1) -00

where i = y/—l. As this is a Fourier integral1,4, it can be inverted to yield the probability density function, Pu{y) =

1

7T

°° IJ f—00 Mu(u) exp(-iLJu)du,

so that the characteristic function is seen to contain all of the same statistical infor­ mation as the first-order probability density function. If the exponent in equation 1.1 is expanded as its infinite series, then the characteristic function can be re­ written as (za;)n v n\ < unn >, M u {y > ) = n=0

where < un > is the nth-order moment. If, further, the nth moment of U exists, it can be found from 1 dn M u ( uj) < un > = in dujn w=0 It can be readily shown that for a Gaussian-distributed random variable, the char­ acteristic function is given by Mu(u>) = exp

crV 5

2

14

exp(zo; < u >),

(1.2)

CHAPTER 1 a result which we shall use in Chapter 2. As for the joint moments, we can also define a joint characteristic function of U and V, given by oo

M uv^u^v)

=/ / exp(i[ui/u +w yv])puv(ui v)du dv.

1.2.6 Stationarity and ergodicity

Having already made a distinction between the statistical properties of an en­ semble, or population, and those of a sample from that ensemble, we can occasionally simplify our results by making further assumptions about the nature of the random process. A random process is called strictly stationary if the kth-order joint probability density function pu(uu u2->• • • u k\ ^2 • • • t k ) ls independent of the choice of time origin for all k. Thus the first-order density is independent of time and can be written as pu(u), and the second-order density depends only on the time difference A t = t 2 ~ 11 and can be written as Pu(ui->u2i A£). A process is also known as widesense stationary if < u(t) > is independent of time and if < w(^i)w(^2) > depends only on the time difference A t = t 2 —t\. Although every strictly stationary random process is also wide-sense stationary, the converse need not be true. The most restrictive type of random process is known as ergodic. This defines a process where each sample function from the entire ensemble has statistical prop­ erties which are identical to those derived from the infinite ensemble. In effect, it states to what extent a sample function is representative of the ensemble. Thus for an ergodic process, the time average (denoted by overbars) of any one of the possible sample functions is equal to the ensemble average, u(t)u(t + At) = < u(t)u(t-\-At) >. Note that while all ergodic processes are strictly stationary, as before, the converse need not be true.

1.2.7 The autocorrelation function and the Wiener-Khinchin re­ lationship

We have already shown that the autocorrelation function, or acf, of a random process is given by oo

:(xi,x2) = J u(xi)u(x 2 )p{u(xi), u(x 2 ); £ 1 , X2 )du(xi)du(x 2 ) —00 = < u(a;i)u(a;2) > .

If the random process is wide-sense stationary, then the correlation depends only on the coordinate difference, A x = xi —x2, and may be written as c(Arc) = < u(x)u(x + Arc) > . For most cases of interest in this thesis we assume that the random processes are wide-sense stationary. Further references to stationarity therefore apply to this particular case unless explicitly stated otherwise. The correlation function can then 15

CHAPTER 1 be normalised to unity for zero lag (Ax = 0) by dividing by the variance of u(x), since c(0) = < u(x)2 > = cr2 for a zero-mean process, to give c(Ax) < u(x)u(x + Ax) > c(0) < u(x)2 > Note that p(Ax) = p(—Ax) so that the function is symmetric about the origin, and that /)(0) = 1, as required. For an alternative representation, the second-order statistics may be examined in the inverse (frequency) space by taking the Fourier transform of the autocorrelation function. This gives the power spectral density, or power spectrum1A oo c(Ax) exp(iu;Ax)dAx, (1.3)

/

-oo

a relationship known as the Wiener-Khinchin theorem1 3, which we can express in shorthand as G(w) c(Ax). (1.4) The power spectral density is so-called because it is related to the average power in the random function, or waveform, by < u(x)2 > = The term G(u>)duj then expresses the proportion of the average power of u(x) due to frequency components lying between w and cu + duj, and its connection with the correlation function may be shown by the application of Parseval’s theorem1-4. Another second-order statistic is provided by the structure function. Derived originally for the study of atmospheric turbulence1-5, its application is particularly useful for random processes which are not wide-sense stationary and may, instead, have mean values that change with, say, time or position. Instead of considering just the random function itself, u(x), the structure function depends on the increment u(xi) —u(x2). Defined as S (xi,x 2) = < [ix(xi) - u(x2)]2 >, the structure function (approximately) characterises the power of those fluctuations of u(x) having periods smaller than, or comparable with Ax = x\ —x2. It is related to the correlation function for a wide-sense stationary process by

S'(xi,x2) = < u(xi)2 > + < u(x2)2 > —2 < u(xi)u(x2) > < u(x)u(x -f Ax) > = 2 < u( x )2 > 1 < u(x)2 > = 2 < u(x )2 > [1 —/?(Ax)] = 2 = < u(x2)2 > = < u(x)2 >. If the correlation functions obeys the (memory) condition that p(Ax) — >■ 0 as Ax — ► 0 0 , then the structure function 16

CHAPTER 1 levels-off for large lags with a value 5(A z) = 2a2. The structure function is also particularly useful wherever the power spectrum has an inverse power-law form, as it remains finite even if the power spectrum has a singularity at the origin (see §1.4).

1.3 METHODS OF NUMERICAL INTEGRATION

Many integral equations have no exact analytic solution and must therefore be solved numerically. If the integrals are one-dimensional, then the procedure is usually relatively straightforward. However, the accurate numerical integration of multi-dimensional functions is not easy and it requires both care and a detailed knowledge of the integrand’s behaviour, without which it is impossible to gauge either an adequate sampling point density or the form of sampling to be used. This is best illustrated by considering a one-dimensional integral over a function containing a cusp as in Figure 1.1. Using a standard Newton-Cotes approach1,6 (e.g. Simpson’s or the trapezoidal method), the function is crudely sampled at equidistant points along the abcissa. The portion of the function around the cusp region that should contribute most to the integral will be undersampled, giving an imprecise answer. Simply increasing the point density may only slowly increase the accuracy, as the gains of only a small proportion of the new points sampling this area may be counter-balanced by a corresponding increase in round-off errors. An alternative approach is to use a variable sampling point separation. Having the freedom to choose both the weighting coefficients and the abcissa points at which the function is to be evaluated can be shown to be more accurate1,7 for many types of function than the equidistant sampling strategies. The particular method used in this thesis is Gauss-Legendre integration, which has the form1,8

L

b

f(x )d x =

N

i= 1

where the integral is expressed as a sum over the N sampling points evaluated at the unevenly spaced abcissa points, and multiplied by the appropriate weighting functions, to,-. If the boundary of the integrand is simple, then further dimensions can be integrated by breaking the function into repeated one-dimensional integrals. Thus, r r r / /

J J J

r x* dxdydzf(x,y,z)

=

/

‘22(i,y)

rv*(x) dx

Jxi

dy

Jy\(x)

Jzi(x,y)

f(x ,y ,z)d z.

In Gauss-Legendre integration, the sampling density is highest at either end of the sampling range thereby suggesting an efficient way of integrating over the cusp of figure 1.1. For example, two sampling ranges could be used, each starting outwards from its midpoint. In this way, the point density will be high where the function changes most rapidly so that the answer should quickly converge. As before, further accuracy is obtained by increasing the number of abcissa points. This is done either by increasing N directly, or by breaking the integrand up into several regions so that the integrand is smooth in each of them and then integrating them separately. An upper limit to the number of sampling points that may be used is given by the amount of computer time required to perform the integration. For example, as the IT

CHAPTER 1

Fig. 1.1 Illustration of the sampling point distributions for Newton- Cotes and Gauss-Legendre quadrature.

Fig. 1.2(a)

Fractal function with v — 1.98, (D=1.01). 0.1 -

0.0

-

0.1

-

0.2

-0.3 -0.4 0.30

Fig. 1.2(b)

0.31

0.32

0.33

0.34

• 0.35

Magnified section of Fig. 1.2(a), v — 1.98, (D =1.01). 18

CHAPTER 1 number of dimensions of integration increases to N, so the number of function eval­ uations increases as the Nth power of the number required for the one-dimensional evaluation. If, say, a function requires only 30 evaluations in one-dimension, then for the same sampling density in three-dimensions, some 27000 function evaluations will be required! It is essential, therefore, that wherever possible, ways of reducing the integration time such as by exploiting function symmetries should be sought.

1.4 FRACTALS

Before the recent ‘discovery’ of fractals and fractal geometry by Mandelbrot1,9, many of their distinguishing characteristics had long been recognised in one form or another throughout the diverse branches of science. Because of their different origins, however, these properties were described by a wide variety of terms making common comparisons difficult. Mandelbrot’s contribution was to recognise the common elements of these areas and to systematise the mathematics that could be used to describe them. Thus the Kolmogorov turbulence spectrum1,5, the underside of sea ice1,10, the radial mass density of colloidal aggregates1,11, mechanically engineered surface profiles1,12 and many other examples can all be classified using the same ‘fractal’ nomenclature. The need for the existence of an essentially new branch of geometry was first highlighted by a number of mathematical developments around the turn of the cen­ tury. Weierstrass’s continuous, but non-differentiable, function1,9,1,13 and Peano’s space-filling, continuous curve1,9 are examples of the dilemmas which challenged the accepted notions of Euclidean dimensionality. The problem was partially re­ solved by the creation of the (real) Hausdorff-Besicovitch dimension, D , which was related to the (integer) topological dimension, D t , through the Szpilrajn inequal­ ity, D > D t • Based on this work, Mandelbrot defined the fractal as being the set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension, i.e. D > D t , and he was able to equate ‘his’ fractal dimension with D. The concept of fractal dimensionality is best illustrated by considering the case of a zigzag line. While Euclidean geometry defines lines as being uni-dimensional (D t = 1), the zigzag, at least in some sense, ‘fills’ a plane with the result that in fractal parlance, its fractal dimension, D, should lie somewhere between one and two. The lack of a tangent is fundamental to a fractal’s properties and is exemplified by the well-known (fractal) coastline example. Suppose we try to measure the length of a coastline with a measuring stick and we obtain a particular result. If the length of the stick is shortened, then it will be able to follow the finer detail around the rocks and bays and so the perceived length will increase. If the stick is made still shorter, the length of the coastline will continue to increase without bound, the perceived length being intimately related to its fractal dimension, D. In contrast, if the stick is shortened for a rectifiable (differentiable) curve such as the circle, the perimeter will tend to a constant value (27rr). Thus the need to be able to characterise a fractal curve other than by a length scale, namely by its fractal dimension, is important. The real relevance of fractals to this work lies in their ability to describe irreg­ 19

CHAPTER 1 ular geometries more adequately than can conventional geometry, as these are the very shapes that occur in nature. Examples of natural fractals abound and include clouds1,14, landscapes1,15, percolating clusters1,16 and thin films1,17. If we try to draw tangents at any points on these fractals, there is always yet finer detail which prevents us. The difference between the light scattering properties of ‘convention­ ally’ correlated models and fractal models centres around the non-differentiability of the latter. This is discussed in detail in Chapters 2, 3 and 4.

1.4.1 Some mathematical properties of fractals

In this section, the appropriate mathematical properties of fractals are discussed for use in the following chapters. A fractal function, h(x), satisfies the condition that its increments behave as1,9 \h(x + Ax) — h(x)\ ~ Arc17/2, 0 < v < 2,

(1.6)

where v is the fractal exponent (related to the Lipschitz exponent by v — 2a), and from which we can derive its relevant statistical properties. The function, h(x), can be shown to be continuous since it satisfies the continuity condition lim Ax — ► 0

|h(x -f Ax) — h(x)\ = 0,

and can also be shown to be non-differentiable if the limit h(x -f Ax) — h(x) lim Ax — y 0 Ax does not exist. In this case, its derivative h'(x) ~ lim Ax — ► 0

Ax1'/2-1,

is finite only for v = 2. Thus the restriction to 0 < v < 2 specifies a function which is both continuous and non-differentiable (the function fails to be continuous if v = 0). Figures 1.2 to 1.4 show plots of the fractal function h(x) for different values of the exponent, z/, which can be shown to be related to the fractal dimension, Z), b y 1 -9 *1-18

0 = 1 r -

Note that as v decreases from 2 to 0, the curves become much rougher and consequently increasingly ‘space-filling’, since the fractal dimension is increasing from 1 (the marginal fractal) to 2 (the extreme fractal). The particular case when v = 1 (Z) = 1.5) is known as the Brownian fractal, as it describes the path followed by particles undergoing Brownian motion1,9. The fractal definition, equation 1.6, is clearly related to the expression for the structure function, S'(Ax), given in §1.2. Thus the fractal structure function is written as 5(Ax) = < [h(x + Ax) — h(x)]2 > ~ A x1', 0 < v < 2. 20

CHAPTER 1

Fig. 1.3(a)

Fig. 1.3(b)

Fractal function with u = 1.00, (D=1.50).

Magnified section of Fig. 1.3(a), u = 1.00, (D=1.50).

CHAPTER 1

Fig. 1.4(a) Fractal function with v = 0.02, (D=1.99).

Fig. 1.4(b)

Magnified section of Fig. 1.4(a), u = 0.02, (D = 1 .99 ).

22

CHAPTER 1 Figure 1.5 shows the fractal structure function for v — 1 plotted together with the structure function of a negative-exponentially correlated function, p(Ax) = exp (-|z|/{)Whereas for the former the structure function increases without bound as the lag is increased, the presence of finite scale sizes causes the structure function of the latter to level off at a value of 2 cr2 for large lags (1.5). However, since ‘natural’ fractals must possess finite inner and outer scales, their structure functions will also plateau for large lags after increasing as A xv for possibly several decades.

1.4.2 The fractal power spectrum

It is an interesting problem to consider how fast the power spectrum must decay in order for the function to be non-differentiable. Mandelbrot1,19 shows that the spectrum falls off as an inverse power law, ~

’ 0), and

rc(fA x) = r < h(fx)h(fx') >*=> g \ j ),

must be identical. Thus f v < h(fx)h(fx') > = < h(x)h(xt) >. Making use of the similarity theorem for Fourier integrals1,4, this may also be written as r " - lG ( j )

=

g (u

).

Finally, if we substitute the inverse power-law spectrum (1.7) for G(cu), the equality is satisfied as required, verifying that the form of spectrum in equation 1.7 satisfies the fractal characteristics of self-affinity. 23

CHAPTER 1 S(Ax)

Ax Fig. 1.5 Examples of structure functions for an unmodified power-law power spectrum, S(A x) = (A x /L )v, and for a modified negative-exponential correlation function, S(A x) = [1 —exp(—A x/L )] (a 2 = 1 ).

p (A x)

Fig. 1 . 6 Asymptotic behaviour of the autocorrelation function for small lag, Ax, for (a) v — 0.5, (b) v = 1.0 and (c) v — 1.5.

24

CHAPTER 1 For small wavenumbers, the power spectral density increases without bound so that the variance, as defined by a2 —

is infinite and thus the variation of h(x) is unbounded. It is precisely for these reasons that the structure function, defined (using the Wiener-Khinchin relationship) by /*oo

S( x) = 2 I

(1

—cos ujx)G(uj)dujJ

is useful, as it remains finite, even when the spectrum has a singularity at the origin (see §1.4.4).

1.4.3 The fractal correlation function

Whilst the correlation function, p(Ax), cannot be defined for large lags (equiva­ lent to low wavenumber) as the variance is infinite, its behaviour for small lags can be determined from its relationship with the structure function, S(Ax). Since S'(Ax) = < [h(x + Ax) — h(x )]2 > = < h(x -f A x ) 2 > -f < h(x ) 2 > —2 < h(x + Ax)A(x) >, the (un-normalised) correlation function, c(Ax), can be written as c(Ax) = < h{x + A x)h(x) > = < h(x ) 2 > —^Li < [h(x + Ax) — h(x )]2 >, where we have assumed stationarity in order that < h(x + A x ) 2 > = < h(x ) 2 >. Further, if we let \h(x + Ax) — h(x)\ ~ A x 17/2 = y/cAxu/2, where y/c is a constant, then the correlation function becomes c(Ax) = < h(x ) 2 > —-A x 17 = c(0 ) -

(1.8)

Figure 1.6 shows a plot of the correlation function for small lags for v — 0.5, v — 1.0 and v — 1.5. Physical situations can occur (discussed in Chapter 4) in which a fractal scatterer behaves like a scatterer possessing a ‘normal’ correlation function, i.e. one which obeys the memory condition (lim Ax — > oo, c(Ax) — > 0), by virtue of their similar behaviour for small lags. This is illustrated by noting that as v ________ i— VCIVIIV-O (the max^inal nciv^tciiy, me iraclai correlation iunction decays with lag as c(Ax) ~ 1 — A x2, which is similar to the decay of the single-scale (Gaussian) correlation function for small lag, c(Ax) ~ 1 — A x 2 (Note that when v = 2 , the function is trivial and its increments remain correlated for all Ax - see below). 25

CHAPTER 1 It is also interesting to consider the correlation of the increments of h(x). Con­ sider three points at x = —t, x = 0 and x = t. The correlation of the increments can then be derived from the structure function, < ([h{t) - h{0 )] - [h(o) - h { - t ) } ) 2 > = < [h(t) - A(0)]2 > + < [MO) - M - * ) ] 2 > —2 < [h(i) —M0)][M0) — M—01 > * It is not difficult to show that if S(x) = L 2 v A xv (see §1.4.4) and if stationarity is assumed, then the normalised correlation can be written as < [h(t) - h(0)P(0) ~ K ~t)} > r - i _ , < [h(t) - fe(0)]2 > This remarkable result shows that not only is the correlation independent of the separations of the points, \t\, but that for v — 0 , the increments have a negative correlation of -1/2, for v = 1, the correlation vanishes (the Brownian case) and for v = 2 (the marginal case), the function is positively correlated, p(t, —t) = 1 (these effects are clearly shown in Figures 1.2 to 1.4). Thus the past and the future are correlated independently of the point separation, \t\.

1.4.4 The topothesy

We have already shown that the roughness of a fractal cannot be expressed as a variance (because the mean-square slope of h(x) is infinite) so that an alternative means of describing vertical roughness must be used. Since the mean-square slope of the chords joining h-values remains finite, Sayles and Thomas1,20 define a parameter which they call the topothesy, L, (derived from the Greek word TonoOecria, meaning a description of a place or region) which defines the distance over the surface over which a chord separating two h-values has an r.m.s. slope of one radian, thus < [h(x -f- L) — h(x )]2 > (1.9) Z2

From this definition, Berry1,18 was able to evaluate the structure function constant, c (equation 1 .8 ), so that the structure function becomes S(Ax) = L2- v|A *r.

1.5 THE THEORY OF SPECKLE FORMATION

1 10)

( .

Speckle has been familiar to scientists for a considerable time, having been reported since the time of Newton as the scintillation, or twinkling, of starlight through the turbulent atmosphere1,21. It was investigated by Exner who, using a candle as a light source, sketched the radial granular structure of the diffraction pat­ tern produced by breathing on a glass plate, and he attributed the pattern to the non-monochromaticity of the light source, a theory which was later confirmed by De Haas1,22. Much later, the theory of speckle was developed for the microwave region of the electromagnetic spectrum 1,23 but it was only ‘rediscovered’ for optical wave­ lengths following the invention of the laser. Nowadays, speckle is a comparatively 26

CHAPTER 1 well understood phenomenon known to result from the interaction of coherent ra­ diation with a surface. Not only present throughout the electromagnetic spectrum, it is also frequently significant for acoustic waves1,24. While in many applications it manifests itself as a granular noise1,25-1,27, it can also provide information about the properties of the rough surface or medium which caused it1,28,1,29. Speckle theory usually assumes that a monochromatic wave is incident upon a scatterer and interacts in such a way that the polarisation of the incident wave is preserved following scattering. The speckle is generated by the coherent addition of a large number of independently scattered amplitudes at the observation point and its complex amplitude is approximately described by the discrete summation N

A(x,y,z) = ^ 2 ak(x,y,z) k=i N

= ^ |a jb |e x p ( i< ^ ) ,

(L11)

k=l where a and represent the amplitude and phase of the kth phasor. The the­ ory is then developed assuming that the a and (/>). are statistically independent of each other and of the amplitudes and phases of all the other phasors. In addi­ tion, it is usually assumed that the phases {,} are uniformly distributed in the interval [—7r, 7r], equivalent to the surface being rough compared with the incident wavelength.

1.5.1 Gaussian speckle

The two-dimensional random walk problem (equation 1.11) was first solved by Lord Rayleigh1,30, who showed that as the number of contributions to the field becomes very large (N — ► oo), the application of the central limit theorem (§1.2) shows that the joint probability density of the real and imaginary parts of the field, N

A(r) = &{A(x, y, z )} = Y l«*l cos k

k=i N

= 3{A (x, y, z )} = Y la * lsin

k=l asymptotically approaches a circularly complex Gaussian random process, so-called because the function describes contours of constant probability density in the com­ plex plane. That is, p (A ^ ,A ^ ) =

^ 2

exp

[4 W ]2 + [4 W ]2 2a2

where cr2 = lim

— 'S T ' < N ^fc=i 27

2

>

?

CHAPTER 1 This type of speckle is known as Gaussian speckle, as the statistics of the phase follow a circularly complex Gaussian random process. In optics, however, we are more usually concerned with the statistics of the intensity, I, rather than of the phase. Related to the real and imaginary complex amplitudes by a coherent sum, I(x,y) = A(x,y)A *(x,y) = (A 2 + AC')2), the probability density function of the intensity alone (not including its phase) can be shown to be 1,31

w^ = 7 7 7 exp( _7 7 > )’ 7- °

(U2)

where

< I > = 2 a2, is the ensemble average or mean intensity. Thus the intensity at a point in a po­ larised speckle pattern obeys negative exponential statistics with the most probable intensity being zero. Higher order statistical moments of the intensity are also of interest and of these, the second moment is usually the most important as it is related to the contrast, C 2, of the speckle pattern. For a Gaussian speckle pattern, the nth moment of intensity can be shown to be 1,26 < I n > = ri!(2 n,

and the second moment and variance to be < I2 > = 2 < / > 2 erf = < I 2 > - < I > 2= < I > 2 . The speckle contrast (see Chapter 4), C, is defined in many different ways in the literature but throughout this work it is defined by C=

< I2 > < I> 2 < /> ’

- 1

(1.13)

which, for a fully-developed, or Gaussian, speckle pattern, has a contrast of unity, ( 7 = 1 . Whilst we have only described the point statistics of the speckle pattern, in Chapter 3 some of its two-point, or second order, statistics are described together with a discussion of the statistics of the intensity produced by the sum of a number of speckle patterns on an intensity basis.

1.5.2 Non-Gaussian speckle

We have described how gaussian speckle results from the addition of many contri­ butions to the scattered field so that the central limit theorem can be invoked. NonGaussian speckle occurs when the number of contributions to the field, A (x,y,z), •28

CHAPTER 1 is small so that the theorem cannot generally be used and, as a result, the complex amplitude is not necessarily normally distributed1,28. Although there are no readily derivable results for p/(7), Kluyver1,31 derived an expression which can be used to calculate it1,32, 1 f°°

P l(I) = r / UJ 0 {uVI) < J 0(C/|a| > * dU. * Jo In practice, however, the pdf of |a| is required so that unlike Gaussian speckle, the analysis of non-Gaussian speckle explicitly depends on the statistics of the scattering surface. Due, however, to the greater complexity in calculating the statistics of nonGaussian speckle, comparatively few results have been presented for either the in­ tensity probability density or the speckle contrast. In Chapter 4, however, we present methods for calculation of the contrast for a number of scatterers, each characterised by having a different correlation function.

REFERENCES [1.1] P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon Press, London, 1963). [1 .2 ] A. Papoulis, Probability, Random variables and Stochastic processes (McGrawHill, New York, 1965). [1.3] J. W. Goodman, Statistical Optics (Wiley, New York, 1985). [1.4] R. Bracewell, The Fourier transform and its applications (McGraw-Hill, New York, 1965). [1.5] V. I. Tatarski, Wave propagation in a turbulent medium (McGraw-Hill, New York, 1961). [1.6] P. A. Stark, Introduction to numerical methods (Collier Macmillan, London, 1970). [1.7] W. H. Press, B. P. Flannery, S. A. Tevkolsky and W. T. Vetterling, Numerical Recipes - The Art of Scientific Computing (C. U. P. , Cambridge, 1986). [1 .8 ] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions (Dover, New York, 1972). [1.9] B. Mandelbrot, The fractal geometry of nature (W. H. Freeman, San Fran­ cisco, 1982). [1.10] D. A. Rothrock and A. S. Thorndike, “Geometric properties of the underside of sea ice”, J. Geophys. Res. 85(C 7), 3955 (1980). [1 .1 1 ] D. W. Schaefer, J. E. Martin, P. Wiltzius and D. S. Cannell, “Fractal Geom­ etry of colloidal aggregates”, Phys. Rev. Lett. 52,2371 (1984). [1.12] E. L. Church, H. A. Jenkinson and J. M. Zavada, “Relationship between surface scattering and microtopographic features”, Opt. Eng. 18, 125(1979). 29

CHAPTER 1 [1.13] M. V. Berry and Z. V. Lewis, “On the Weierstrass-Mandelbrot fractal func­ tion”, Proc. Roy. Soc. Lond. A 370, 459 (1980). [1.14] S. Lovejoy and D. Schertzer, “Generalised scale invariance in the atmosphere and fractal models of rain”, Water Resources Res. 2 1 , 1233 (1985). [1.15] R. F. Voss, Random fractal forgeries (Proc. N.A.T.O. A.S.I. Fundamental Algorithms in Computer Graphics, Ilkley, 1985). [1.16] Y. Gefen, A. Aharony, B. B. Mandelbrot and S. Kirkpatrick, “Solvable fractal family, and its possible relation to the backbone at percolation”, Phys. Rev. Lett. 47, 1771 (1981). [1.17] J. E. Yehoda and R. Messie;, “Are thin film structures fractal?”, Appl. Surf. Sci. 22/23, 590 (1985). [1.18] M. V. Berry, “Diffractals”, J. Phys. A:Math. Gen. 1 2 , 781 (1978). [1.19] B. B. Mandelbrot, Fractals: Form, Chance and Dimension (W. H. Freeman, San Francisco, 1977). [1.20] R. S. Sayles and T. R. Thomas, “Surface topography as a nonstationary random process”, Nature 271, 431 (1978). [1.21] I. Newton, Opticks (reprinted by Dover, New York, 1952), Book I, Part I, Prop. VIII, Prob. II (1730) [1.22] W. J. de Haas, Koninklighe. Acad, van Wetenschager (Amsterdam) 2 0 , 1278 (1918). [1.23] S. T. Wu and A. K. Fung, “A Noncoherent Model for Microwave Emissions and Backscattering from the Sea Surface”, J. Geophys. Res. 77, 5917-5929 (1972). [1.24] J. C. Bamber and R. J. Dickinson, “Ultrasonic B-scanning: a computer simulation”, Phys. Med. Biol. 25, 463 (1980). [1.25] J. W. Goodman, Introduction to Fourier optics (McGraw-Hill, San Francisco, 1968). [1.26] J. W. Goodman, “Statistical properties of laser speckle patterns”, in Laser Speckle and Related Phenomena, Ed. J. C. Dainty (Springer-Verlag, Berlin, 1984). [1.27] J. C. Dainty, “Introduction”, in Laser speckle and related phenomena, Ed. J. C. Dainty ( Springer-Verlag, Berlin, 1984). [1.28] E. Jakeman and P. N. Pusey, “Non-Gaussian fluctuations in electromagnetic radiation scattered by a random phase screen”, J. Phys.A: Math. Gen. 8 , 369 (1975). [1.29] J. Ohtsubo and T. Asakura, “Measurement of surface roughness properties using speckle patterns with non-Gaussian statistics”, Opt. Commun. 25, 315 (1978). [1.30] J. W. Strutt (Lord Rayleigh), Phil. Mag. 10, 73 (1880). 30

CHAPTER 1 [1.31] J. C. Kluyver, “A local probability problem”, Proc. Roy. Acad. Sci., Amst. 8 , 341 (1905). [1.32] E. Jakeman and P. N.Pusey, “A model for non-Rayleigh sea echo”, I. E. E. E. Trans. Ant. Propgn. A P —24, 806 (1976).

31

CHAPTER 2 CHAPTER 2 THE SCATTERED MEAN INTENSITY In this chapter we discuss three approaches which have been used to describe the field amplitude resulting from the scattering of an incident wave by a rough surface or phase screen. Each of these is based upon the integral equation approach, and they differ in their level of sophistication both regarding the details of the surface interaction and in the constraints they impose upon the surfaces to which they may be validly applied. Considering only single scattering, the application of the Kirchhoff boundary conditions leads to the Physical Optics Theory 2 '1 - 2 '3 , which is later used to derive the expression for the mean scattered intensity for a number of different types of ran­ dom phase screen, each characterised by a different power spectral density. We shall examine this theory in some detail, as its predictions are widely used throughout this thesis. Despite the successful predictions both of ‘exact’ electromagnetic theories and of the ray theories which have been ‘improved’ by the inclusion of diffraction eff­ ects2,4,2,5, their application has been limited to simple, and perfectly conducting, deterministic, or quasi-deterministic surfaces2,2. It is only comparatively recently, therefore, that significant advances have been made in finding more rigorous, and general, approaches to the rough surface scattering problem. Of these, we shall only briefly describe the Boundary Condition M ethod2"^, which is based upon the Rayleigh Hypothesis, and the Extended Boundary Method, which is based upon the Ewald-Oseen extinction theorem2,7-2,9, as they lie outside the scope of this work and have currently met with only limited analytical success. Both approaches take multiple scattering effects into account and can apply, in principal, when the surface features are of the same order of size as the incident wavelength. Each of these theories is examined in turn and their approximations and limita­ tions are discussed.

2.1 PHYSICAL OPTICS DIFFRACTION THEORY

The first theoretical steps taken to explain the diffraction phenomenon accu­ rately reported by Grimaldi in 1665 were made by Huygens who, in 1678, proposed the idea that the disturbance at a point could be considered as the superposition of the secondary spherical wavelets arising from notional secondary sources on the wavefront at a previous instant2,2. In 1818, Fresnel supplemented Huygens’ con­ struction with Young’s principle of interference and, making somewhat arbitrary assumptions about the amplitudes and phases of the secondary sources, was able to calculate the light distribution in the aperture diffraction patterns with surprising accuracy2,1. In 1882, these ideas were given a firmer mathematical foundation by Kirchhoff, who showed that Fresnel’s assumptions were the natural consequence of the wave na­ ture of light2,1. However, due to the particular choice of boundary conditions which were used, this approach was later shown by both Poincare (1892) and Sommerfeld 32

CHAPTER 2 (1894) to be subject to certain internal inconsistencies which rendered it inappli­ cable close to the diffracting aperture210,2-11. These problems were later solved by Sommerfeld’s modification of Kirchhoff’s theory, through the application of a differ­ ent choice of Green’s functions. Following microwave experiments which concluded that the predictions of the Fresnel-Kirchhoff diffraction theory were closer to the experimental results than those of the Rayleigh-Sommerfeld theory2-12, the relative accuracy of both approaches is still in some doubt. For observation points some distance from the aperture, however, both theories appear to converge2-10. Due to its inherent simplicity and wide range of applicability, the Physical Op­ tics theory, which is based on the application of the Kirchhoff boundary conditions to the Helmholtz integral, has received a considerable amount of attention in the literature2-3,2-13-2-15. While we do not intend to present a full derivation of the Kirchhoff scattering equations for rough surfaces and phase screens, instead, we shall outline their formulation and approximations so that their likely range of ap­ plicability may be discussed. We start by examining the application of the Kirchhoff approximations to the rough surface scattering problem using the approach first described by Beckmann2-3. In the second section, §2.1.3, we contrast this approach with that developed for ran­ dom phase screens, which represent an important model for describing transmissive scatterers. In both cases, we treat light as a scalar phenomenon. Thus we con­ sider only the scalar amplitude of one transverse component of either the electric or magnetic field, and implicitly assume that other components can be treated inde­ pendently in a similar fashion. Although this is not strictly valid, as the components of the electric and magnetic fields are coupled through Maxwell’s equations2-16, the assumption is reasonable provided both that the linear dimensions of the diffracting aperture are large compared with the wavelength and that the diffracted field is not observed very close to the aperture. We discuss these restrictions further in §2.1.1 and 2 . 1 .2 .

2.1.1 The application of Physical Optics theory to random rough surfaces

The scattering of light or other wave phenomena by rough surfaces is of consider­ able importance in optics2-3, microwave and radar physics 2-17>2-18? acoustics2-19 and in many other areas of scientific endeavour. Much of the progress made in these areas has been based on the Beckmann theory, which is a modification of the boundary conditions originally applied by Kirchhoff in his study of the diffraction of electro­ magnetic waves by an aperture in an infinite plane. We present here an outline of the derivation of the Beckmann expression for the scattered field amplitude. We shall use Cartesian coordinates x, y and 2 with origin 0 and unit vectors x, y and z. Referring to Figure 2.1, we define a radial vector r = ix + y y -f zz which, for points on the surface, has the form, r = xx + y y + h(x, y)z, where h(x, y) is the local surface height. The mean level of the surface is the plane z — 0. We assume the incident wave to be plane and monochromatic, i]/*nc = exp (zki.r —iu t) , where ki is the incident propagation vector in the xz plane and k = cu/c. We define 33

CHAPTER 2 the angle of incidence, being the angle between the propagation vector ki and the z-axis, by Oi, and the scattering angle, the angle between k 2 and the z-axis, by 6 2 , where

|k2| = M = * = y .

For out-of-plane scattering, we also define the angle 93 . Suppressing further time dependence, the field must satisfy the time-independent wave equation2,2, ( 2 . 1) (V 2 + k2) # ‘nc = 0, which is also known as the Helmholtz equation, and V 2 is the Laplacian, d2 d2 d2 V 2~ = — + + dx 2 dy 2 d z 2 Following standard treatments2,1,2,2, the Helmholtz integral equation can be derived. We merely present the result2,2,

22

( . )

where the integral is over the surface, S. This describes the scattered field at xQ,y 0 ,z Q, i£(ro), in terms of the field and its normal derivative on the surface, 'F and = 2 k(x\ +

35

CHAPTER 2 and

(2.5)

— J = (1 - R c) ^ inck 1.n

where n is the normal to the surface at the considered point and Rc is the Fresnel reflection coefficient of a smooth plane. Although we have not taken the incident field polarisation into account, its effect can be included by assuming a suitable form for the Fresnel reflection coefficients. These, of course, only refer to the local polarisation which will not, in general, be identical to the polarisation with respect to the xy plane. As a result, the incident field should be resolved into its two perpendicular components whose reflection char­ acteristics will be determined by the appropriate Fresnel coefficient2,3. However, for most cases of interest in this work the scattered field is more strongly influenced by the spatial, rather than the electrical, properties of the surface. As a result, the restriction of the theory to perfectly conducting surfaces, where \RC = 1 |, can be used to simplify the integrals. Defining the scattering coefficient, p, by

where $ 0 is the field that is reflected by a smooth, perfectly conducting plane of the same dimensions as S and assuming the same angle of incidence etc., then the coefficients outside the integral for ^(ro) can be removed. With the approximations described above, Beckmann then showed that the resulting scattered field from a two-dimensional surface is given by 2,3 p( 0 1 ^ 2 , #3 ) =

1

-f cos 9\ cos $2 —sin sin 92 cos 93 1 cos #i(cos 9\ -f cos $2 ) S(x, y)

J J

exp(iv.r) dx dy

S{x,y)

S (x,yY

( 2. 6)

where v = &[(sin 9\ —sin $2 cos $3 )xi —sin 9Q2,Qz) p *{9\,92,9 z) > 5 (2.8) where p* represents the complex conjugate of p. Since the subsequent treatment of this derivation of the intensity scattered from a rough surface closely follows that from a phase screen, we present the expressions for both cases in §2.1.4. Finally, we give the expression for the phase retardation imposed on a wave both after reflection from a rough surface and following transmission. Referring to Fig. 2.2, the phase difference, A^i>, between ‘rays’ 1 and 2 following reflection is A= kh(\Jn2* —nam sin2 9 —nam cos 9), which, for normal incidence, becomes A (f>= kh(nm — 1),

(2.10)

where nm is the refractive index of the transmitting material and we assume that the ambient medium has unit refractive index, nam = 1. Thus a surface causes approximately twice the phase retardation to a wave reflected from it than it does for a transmitted wave (see Chapter 3). Note that while (2.9) assumes that the variation of A with changes in the local incidence angle is negligible, (2.10) assumes that the local angle of incidence on the rough surface is given by 9m. These approximations will be valid for both cases only if the surface has gentle slopes.

2.1.2 Discussion of the Beckmann Theory approximations

Having outlined the derivation of the expression for the scattered field (2.7), we now discuss the restrictions imposed by the many approximations and assump­ tions in order to establish the range of applicability of the theory. The following assumptions have been made: f In the original formulation of the Kirchhoff theory, the scattered field was calculated from a plane conducting screen with a finite aperture2 3. Kirchhoff assumed that the field and its normal derivative are both zero on the observer’s side of the screen and equal to the incident field and its normal derivative within the aperture. 37

CHAPTER 2

Fig. 2.3 Phase retardation of a wave following tranmsission through a medium with refractive index nm.

Fig. 2.4

Illustration of incidence shadowing.

Fig. 2.5

Illustration of scatter shadowing.

38

CHAPTER 2 • The surface is perfectly conducting (eqns. 2.4,2.5). This is equivalent to as­ suming that the Fresnel reflection coefficients are independent of position on the surface and that the surface has unit reflectivity. As we have discussed, this restric­ tion is not severe in our case as the spatial properties of the surfaces in which we are interested generally have more bearing on the scattered field than its electrical properties. • There is no depolarisation of the scattered wave. It can be shown that for perfectly conducting surfaces, there is no depolarisation of the scattered wave pro­ vided that the surface radius of curvature is much larger than the wavelength of the incident radiation (an assumption implicit in the Physical Optics Theory). If the surface has features on the same scale as the wavelength, then the second term in (2 .6 ) is no longer negligible and is reponsible for the ensuing depolarisation2,3. • Shadowing effects are ignored. Assuming a ray description for the way in which a wave interacts with a surface, two types of shadowing may be described. Incidence shadowing occurs if, because of the surface profile, the incident ray strikes the surface at point 1 of Figure 2.4 and, by casting a shadow over the lower points on the surface, is unable to strike the surface at point 2 . Scatter shadowing (see Fig. 2.5) occurs if the scattered, or reflected, ray is blocked by the surface at point 2 and is consequently prevented from freely propagating to the observation point, P. In this case, the ray will be scattered again, possibly several times, before leaving the surface. This is an example of multiple scattering and is discussed in more detail below. Several attempts have been made to try and modify the Kirchhoff theory to take account of shadowing2,13,2,21,2*22. They mostly assume that either the surface current or the surface height may be modified by a ‘shadowing function’, which equals unity if a particular part of the surface is illuminated or zero, if not. However, not only are these approaches somewhat ad-hoc, but the modifications fail to take account of the ‘blocked’ energy and so do not include any multiple scattering effects. As these are expected to be important for precisely those surfaces for which the shadow correction is most necessary, it leaves severe doubts as to the utility of the approach. This conclusion is supported by the findings of Brown2,21 and Thorsos2,23, who both examine the shadow-corrected Kirchhoff theory. In conclusion, it is apparent that the shadowing assumption will be violated if the incident or scattered rays are blocked. This will occur if the surface slopes are high or if the angles of incidence and/or detection are large. • Multiple scattering effects are ignored. Multiple scattering occurs when a wave has more than one surface interaction. This will occur, for example, if a wave incident on a surface undergoes more than one ‘scatter’ before either propagating away from the surface or being ‘lost’ within it. This may happen if the surface has steep slopes, sharp edges or undercuts which can ‘trap’ the radiation, and it is often the result of scatter and/or incidence shadowing (see above). • The radii of curvature of surface irregularities is large compared with the wavelength. As we have shown, the Kirchhoff theory is a single-scattering theory due to the nature of the approximations made in equations 2.4 and 2.5. By assuming that the field may be approximated by the field that would be present on the tangent 39

CHAPTER 2 plane at that point, we are effectively limiting the application of the theory to those surfaces for which the radius of curvature of the irregularities is very large compared with the wavelength, the very situation in which multiple scattering and shadowing effects will be negligible.! Nevertheless, note that this theory will be exact for flat and tilted, planar surfaces. If, however, the local radius of curvature is comparable to the wavelength, the Kirchoff assumption will break down as multiple scattering and shadowing effects become significant. Because many natural surfaces may be considered rough on the scale of an optical wavelength, this assumption would appear to impose a serious limitation on the range of applicability of the Kirchhoff theory. It is, however, a source of constant surprise just how widely applicable the theory proves to be, as its application for surfaces lying outside its expected scope frequently reveals its powerful predictive abilities. From the preceeding discussions of shadowing and multiple scattering it is ap­ parent that, in many cases, one gives rise to another. With the development of more sophisticated theories (discussed later) which take these effects into account, attempts have been made to try and define conditions for which the Kirchhoff theory is valid, as this uncertainty remains as one of its limitations. In a paper comparing the Kirchhoff theory with the extinction theorem (see §2.3), Nieto-Vesperinas and Garcia2,20 conclude that the former will apply provided that the inequalities Y»l, ^

and

P

< 0.05,

(2.11,2.12)

hold, where cr^ and /? represent the surface height standard deviation and correlation length respectively. Whilst (2.11) suggests that the theory will break down if the correlation length and the wavelength are similar, equation 2 . 1 2 refers to the surface slopes since, for a Gaussian-correlated surface, the average slope, m, is given by 2,24 V2 = / / / / A (x 1 ,y 1 )A(x 2 ,y 2) exp(— h(x). Most criticisms of this approach are based on the argument that within the corrugations, where Zimin < z < Amax, there must also be incoming waves due to sources on the surface lying above the point of observa­ tion. The main restrictions on its use appear to be if the surface is not analytic, for example, by having corners, or if there are large surface slopes2-6. Analyticity is not, however, a sufficient property to ensure that the Rayleigh Hypothesis applies. Few conditions have been established for its validity in the case of random rough surfaces and, as a result, this uncertainty ultimately limits its use.

2.3 THE EXTENDED BOUNDARY METHOD USING THE EWALDOSEEN EXTINCTION THEOREM

Techniques based on the Ewald-Oseen extinction theorem2-7,2-35-2-38, known also as the null-field method, appear to provide the most promise for predicting the 46

CHAPTER 2 exact scattering properties of rough surfaces as they are not subject to the limitations of either the Kirchhoff theory or the boundary condition method described above. Instead, the approach used is valid without restriction for any type of surface, so that surfaces with vanishingly small correlation lengths and infinite slope variances may be considered. As a result, the theory correctly takes account of multiple scattering, shadowing and depolarisation effects. The only limitation is that the analytic solutions are perturbative, and hence are limited in that sense. In its simplest form, the extinction theorem is commonly interpreted as express­ ing the extinction of the incident field within a perfectly conducting medium by the field that originates from induced dipoles on the surface2*7. These dipoles ra­ diate both within the medium, cancelling the externally applied field, and above the surface, to provide the scattered component of the field, although the precise details of this interpretation have been shown to be incorrect2*36. In the absence of the medium, Lalor2*39 shows that the extinction theorem correctly reduces to the Helmholtz-Kirchhoff integral theorem for the incident field. In practice, the scattered field inside the perfectly conducting medium must be equal to the negative of the incident field in order to ensure that the total field is zero. Using the expression for this field, the surface source density, or surface current, may be found. At least in principle, knowledge of the surface current enables the exact prediction of the scattered amplitude. Before the resulting expressions can be solved, however, the unknown surface field normal derivative must be described. Analytic solutions are then treated by perturbation using either of the Field Perturbation or the Phase Perturbation methods. Both scalar2*20,2*40 and vector treatments2*9 of the Field Perturbation method have been used to study the scattered intensity from rough surfaces. In this case, the surface field normal derivative is described by the normal derivative which would exist if the surface were flat multiplied by a function which is related to the amplitude of the surface roughness. Nieto-Vesperinas and Garcia2*20 have found, however, that the expressions for the field (for a jointly Gaussian, Gaussian-correlated surface) for terms of higher than second order in the surface height standard deviation are so cumbersome as to be effectively unusable. As a result, the expressions which they did obtain are valid only for very weakly scattering surfaces, 07 Using an expansion first suggested by Shen and Maradudin2*8, the Phase Pertur­ bation method2*41,2*42 multiplies the normal derivative by a function closely related to the complex phase of the surface field. The advantage of this approach over the field perturbation method is that due to the non-linear relation between the surface height and the phase, the scattered fields can be calculated in cases where the surface relief is significant compared with the wavelength of the illuminating radiation. Employing this approach, Winebrenner and Ishimaru2*41,2*42 have exam­ ined the scattering from one-dimensional rough surfaces and obtained results for the moderate scattering case, =

Jj A(x!)/l(x2)exp(— = f j exp —OO

x exp Integrating over d 2 1 2 ’. / +°° / 2 t 2 - 2 ta. t 2

L

exp l

W 2

F

- P 4 ,( ti) ])

ikH. ti

+ 2 t^ —2 ti. t 2 W2

F

- f

t 2 exp

2t |

d? t i c P t2.

W 2) 0 (, - Qh ) exp 2 |til|t 2 |cos( )1).

< 1(0) > = tt2W 2 exp(-crj) ^ 2

K )"1

i

2.4.2 Single-scale scattering from random rough surfaces

We have already mentioned that the Physical Optics intensity predictions from rough surfaces and phase screens having similar correlation functions are very closely related and have similar derivations, the main difference being due to the inclusion of angular factors for the rough surface both within the integrand and externally as the F-factors (eqn. 2.6). The reason for this is due to the fact that in the Beckmann formulation, the rough surface is treated in a similar way to the phase screen with its height being translated into a phase change through eqn. 2.92-53. We shall not present a derivation of the corresponding expression for the scattered intensity, but will, instead, simply quote the result derived by Beckmann from eqn. 2.7 (see chapter 5 of Ref. 2.3). Thus, for a single-scale surface of correlation length /? and height standard de­ viation cr/j, the in-plane ($3 = 0 ), diffuse component of the mean scattered intensity is given (for hard-edged aperture illumination of area A) by < ^(#1 , 02 ) >d=

7T{32 F 2 A

E mlg mm exp

(2.26)

where yfg = kcr^cos $i + cos d2), vxy — fc(sin2 9\ —2 sin 9\ sin 0 2 + sin 2 0 2 ) 1 ^2 and the inclination factor, F — (1 + cos(0j -f 6 2 ))/(cos #i(cos 9i + cos #2 ))For normal incidence, F = 1, vxy = fcsin# and y/g = k a ^ l + cos#), so that eqn. 2.26 reduces to ^fc2cr^(l + cos 0 )2) < I{9) >d = ~~A -e x p (—fc2 * ( t i ) )~1 +< 7^(ti)+O (crfal) , wheret h eh i g h e ro r d e rt e r m s ,0 ( < r ^ P ^ ) ,mayben e g l e c t e df o r = *2W2 exp (-^ ) f > 2 )m £ Am' ^ ln^ m2 m= 0

x

mj

1

Z

ti^ fc tjsin e je x p j- (m x + ^

i r 2P 2W 2

,

*i

2 N m V ^ A m i( l - A ) ^

= — 5— e*p(-*3) E ( TD m =0

(m i

m 2/C ) +1 /2 N

k2fi2 s i n 20 \ 4 ( 7 7 1 !j -m 2/ C +1 / 2/ V y’ ( 2 . 2 9 )

w h e r e ,a sb e f o r e ,N =W 2/ ft2 i st h enumbero fi l l u m i n a t e ds c a t t e r e r s( r e f e r r i n gt o t h el a r g e ro ft h etwos c a l e s ) . Bys e t t i n gA = 1 ,s ot h a tt h esecondtermi nt h e c o r r e l a t i o nf u n c t i o nd i s a p p e a r s ,s i n c em 2 =0andhencemi=m,e q n .2 . 2 9r e d u c e s t ot h es i n g l e s c a l emeani n t e n s i t y( e q n .2 . 2 5 )a se x p e c t e d ,p r o v i d i n gac h e c kont h e r e s u l t .

2.5.2 Discussion and results

Asf o rt h es i n g l e s c a l ephases c r e e n ,i ti smorei n f o r m a t i v et oc o n s i d e rt h el i m i t s o fbothv e r yweakanddeepphases c r e e n sr a t h e rthant oexaminee q u a t i o n2 . 2 9

57

CHAPTER 2 d i r e c t l y .F o rt h ev e r yweakc a s e( c r ^< C1 ) ,i ti sn o td i f f i c u l tt oshowt h a tt h e i n t e n s i t yh a st h eform .. < m

>~

2 n2 , ox f A a f «P(-^){(1 + 1/2JV)«P

+

1-A

{1 /C + 1 /2 N )

exp

(

k 2(d2 sm 2 9 \

(-4(i-+i/2JV)J k2fl2 s i n 29 \ 4 ( l / ( 7+l /2 N ) J

wheret h ec o r r e l a t i o nexponenth a sbeenexpandedt os econdo r d e ra sb e f o r e( s e e §2.4.3). T h i se x p r e s s e st h ef a c tt h a tt h er e s u l t i n gi n t e n s i t yi sg i v e nbyt h ei n c o h e r e n t sumo ft h ei n t e n s i t i e st h a twouldbeproducedbytwos i n g l e s c a l es c r e e n sh a v i n g t h ec o r r e l a t i o nl e n g t h s/ ?andy/C/3, e a chtermh a v i n gt h esameforma se q u a t i o n 2.27, w i t hm =1 .S i m i l a r l y ,t h ei n t e n s i t yi nt h edeeps c r e e nl i m i ti sfoundbyl e t t i n g ~ 1 -( A+ (1 -a)/ C ) t\!P2, s ot h a t ~

________________ 7 T ^ ________________

2(

whichr e d u c e st oJakemanandMcWhirter’ se x p r e s s i o n 2-62 f o rt h es i n g l e s c a l ec a s e i fwes e tA — 1and( 7=1 .I nt h i sc a s e ,t h et w o s c a l ed i f f u s e rbehavesi nmucht h e samewaya st h es i n g l e s c a l ed i f f u s e r( § 2 . 4 . 3 )i nt h a tw h i l s tt h ec o r r e l a t i o nf u n c t i o n canbefoundi nt h eweaklys c a t t e r i n gc a s e ,o n l yt h er a t i oo f/ 9 / y / c r 2 (Af(1 —A)/C) canbefoundfromexaminingt h es t r o n g l ys c a t t e r e da n g u l a ri n t e n s i t yd i s t r i b u t i o n , i tb e i n gi m p o s s i b l et oi s o l a t et h ei n d i v i d u a lv a l u e so fA ,C and/ 9 . F i g u r e s2 . 1 0 ( a ) ( c )demonstratet h edependenceo ft h es c a t t e r e dmeani n t e n s i t y f o raweakd i f f u s e ront h ec o r r e l a t i o np a r a m e t e r s ,A and< 7 ,t h es i n g l e s c a l ei n t e n s i t y a l s obei n gp r e s e n t e df o rc o m p a r i s o n .I nF i g s .2 . 1 0 ( a )and( b )t h ei n t e n s i t ya x i si s p l o t t e dl o g a r i t h m i c a l l yi no r d e rt oh i g h l i g h tt h ed i f f e r e n c e sbetweent h es i n g l e ,and t w o s c a l er e s u l t s . Notet h a tt h et w o s c a l en a t u r eo ft h es c a t t e r e ri sp a r t i c u l a r l y markedf o rt h ec a s e swhenA =0 . 1 ,C =0 . 1andA =0 . 9 ,C — 0 . 1 . I ti simportantt ounderstandt h a tt h ea c t u a li n t e n s i t yd i s t r i b u t i o nd o e sdepend c r u c i a l l ybothont h eparameterA and on t h ephasev a r i a n c e . Fore x a m p l e ,w i t h t h ec o r r e l a t i o nf u n c t i o no fF i g .2 . 9 ( c )(A = 0 . 1 ,C = 0 . 1 ) ,t h eweakphases c r e e n g i v e sana n g u l a rd i s t r i b u t i o nmarkedlyd i f f e r e n tfromt h a tproducedbyt h es i n g l e ­ s c a l emodel. I f ,h o w e v e r ,t h ephasev a r i a n c ei sl a r g e ,theno n l yt h a tp a r to ft h e c o r r e l a t i o nf u n c t i o nc l o s et ot h eo r i g i nh a sanys i g n i f i c a n te f f e c tont h es c a t t e r e d d i s t r i b u t i o n . Thusi nt h i sc a s e ,t h es c a t t e r e dd i s t r i b u t i o nw i l lbes i m i l a rt ot h a t producedbyas i n g l e s c a l ed i f f u s e rhavingac o r r e l a t i o nl e n g t he q u a lt ot h a to ft h e s h o r t e rc o r r e l a t i o nl e n g t h ,y/C(3. I nc o n c l u s i o n ,t h ephasev a r i a n c ew i l ldeterminewhethert h ei n d i v i d u a lpa­ r a m e t e r so ft h ec o r r e l a t i o nf u n c t i o ncanbei s o l a t e d .I nmanyc a s e s ,h o w e v e r ,i t i si m p o s s i b l et odeterminewhethert h es c a t t e r e da n g u l a ri n t e n s i t yd i s t r i b u t i o ni s producedbyo n e ,o rmore,l e n g t hs c a l e s .

58

CHAPTER 2 F i g .2 . 1 0 ( a ) ( c ) Thes c a t t e r e dmeani n t e n s i t i e sfromtwo—s c a l ephases c r e e n sf o r d i f f e r e n tv a l u e so fA andC t o g e t h e rwitht h ep r e d i c t i o n sf o ras i n g l e s c a l ephase s c r e e n . < m

>

< m

>

59

CHAPTER 2 < m

>

l o g

F i g .2 .1 1 Thes c a t t e r e dmeani n t e n s i t yfromacorrugated-Brownianf r a c t a l(v 1 )p hases c r e e n .

60

CHAPTER 2

2.6 THE ANGULAR INTENSITY DISTRIBUTION FROM FRACTAL SCATTERERS I ns e c t i o n s§ 2 . 4and§ 2 . 5wed i s c u s s e dhowl i g h ti ss c a t t e r e dbys i n g l eandtwos c a l erandomphases c r e e n s .Botho ft h e s ec o r r e l a t i o nmodelsd e s c r i b ephases u r f a c e s whicha r ed i f f e r e n t i a b l e ,ar e s u l to fwhichi st h a tg e o m e t r i c a l( r a y )o p t i c sp l a y s animportantr o l ei nd e t e r m i n i n gt h ed i s t r i b u t i o no ft h es c a t t e r e di n t e n s i t y ,which wasshownt or e f l e c tt h es l o p ed i s t r i b u t i o no ft h es c a t t e r e dw a v e f r o n tf o rs t r o n g s c a t t e r e r s . Them a j o r i t yo fn a t u r a l l yo c c u r r i n groughs u r f a c e so rs c r e e n sdon o t ,h o w e v e r , havesuchs i m p l efo r m so fc o r r e l a t i o nf u n c t i o nanda r ef r e q u e n t l yfoundt op o s s e s sa rangeo fl e n g t hs c a l e sw i t has i m p l epower-lawr e l a t i o n s h i pd e t e r m i n i n gt h ee n e r g y o fe a c hs c a l es i z e( s e eChapter3 ) .I n§ 1 . 4t h e s em ulti-scale, or fractal 2 , e 3 ,s u r f a c e s wereshownt oc o n t a i ns t r u c t u r edownt oa r b i t r a r i l yf i n el e n g t hs c a l e ss ot h a te v e n thought h e ya r ec o n t i n u o u s ,t h e yp o s s e s sno( s u r f a c eh e i g h to rp h a s e )t a n g e n t s . Buts i n c er a yd i r e c t i o n sa r ed e f i n e dbyw a v e f r o n tn o r m a l s ,t h en o n d i f f e r e n t i a b i l i t y o ft h ef r a c t a li m p l i e st h a tt h ea p p l i c a t i o no ft h ec o n c e p to fr a y st od e s c r i b ei t ’ s s c a t t e r i n gw i l lbei n a p p r o p r i a t e .Theconsequenceo ft h i si st h a tf r a c t a l sa r ep u r e l y d i f f r a c t i n gs t r u c t u r e sanddon o te x h i b i tg e o m e t r i c a lo p t i c se f f e c t s 2 , 6 4 .I ti st h e s c a t t e r i n gp r o p e r t i e so ft h e s erandomf r a c t a ls t r u c t u r e swhichformst h eb a s i so ft h e i n v e s t i g a t i o n so ft h eremaindero ft h i sc h a p t e r .

2.6.1 Calculation of the scattered intensity from random fractal phase screens Althought h ec o n c e p to ff r a c t a l swasn o tf a m i l i a rt oe a r l yw o r k e r si nt h es c a t ­ t e r i n gf i e l d ,someo ft h e i re s s e n t i a lp r o p e r t i e shadbeenr e c o g n i s e dandsomep r e ­ d i c t i o n smader e g a r d i n gt h e i rs c a t t e r i n gs t a t i s t i c ss e v e r a ld e c a d e sa g o .Perhapst h e e a r l i e s ta p p l i c a t i o no f‘ f r a c t a l s ’wasi nt h em o d e l l i n go ft h et u r b u l e n ta t m o s p h e r e , wheret h ewell-known( i n e r t i a ls u b r a n g e )power-laws p e c t r a lmodelsemployedby Kolmogorov2,65 ( t h e‘ 5/3-spectrum’ )andT a t a r s k i 2,43 a r es i n g u l a ra tt h eo r i g i no f t h epowerspectrum( s e e§ 1 . 4 ) .I no r d e rt oovercomet h i su n p h y s i c a lb e h a v i o u r , VonKarman2,47 i n t r o d u c e das p e c t r a lmodelp o s s e s s i n gaf i n i t ei n n e rs c a l es i n c e when,numerousa d a p t a t i o n so ft h e s eb a s i cpower-lawmodelshavebeenu s edhaving bothi n n e rando u t e r s c a l ec u t o f f s 2,48,2,49 andv a r i a b l epower-lawexponents2 , 6 6 .I n o t h e ra r e a s ,t o o ,t h en e g a t i v e e x p o n e n t i a l l i k ec o r r e l a t i o n sc h a r a c t e r i s t i co fBrow­ n i a nf r a c t a lmodelshavebeeno b s e r v e dt omorea c c u r a t e l yd e s c r i b et h es u r f a c e r o u g h n e s so fmanyn a t u r a l l yabraded2,67-2,69 andm a c h i n e f i n i s h e ds u r f a c e s 2,70,2,71 thant h es i n g l e s c a l emodelsu s e dp r e v i o u s l y .Thep o t e n t i a lf o rt h e i ra p p l i c a t i o nt o manyo t h e rbr a n c h e so fo p t i c s ,ands c i e n c e ,remainsenormous. I no r d e rt oc a l c u l a t et h es c a t t e r e di n t e n s i t yfromf r a c t a lphases c r e e n sweu s et h e s m a l l a n g l eP h y s i c a lO p t i c sapproacho u t l i n e di n§ 2 . 1 . 3 .Theassumptionsi n h e r e n t i nt h i sapproacha sa p p l i e dt of r a c t a l sa r eq u e s t i o n a b l e ,h o w e v e r ,a st h ep r o p e r t i e s o ff r a c t a l sv i o l a t et h eb a s i cr e q u i r e m e n t so ft h et h e o r yt h a tt h es u r f a c ebel o c a l l y f l a t ,andhenced i f f e r e n t i a b l e .N e v e r t h e l e s s ,t h en e a r s p e c u l a rs c a t t e r i n gc h a r a c t e r ­ i s t i c sw i l lbel a r g e l yu n a f f e c t e dbyt h i si n c o n s i s t e n c yp r o v i d e dt h a tt h er o u g h n e s s o ft h es m a l ls c a l es t r u c t u r ehavingt h esames p a t i a ldimensiona st h ewavelength

61

CHAPTER 2 i smuchl e s sthant h ew a v e l e n g t h ,s ot h a tt h el o c a ls u r f a c eremainsa p p r o x i m a t e l y f l a t .S i n c et h et o p o t h e s y( e q n .1 . 9 ) ,L ,i sameasureo ft h er a t ea twhicht h es u r f a c e h e i g h tc h a n g e s ,i fL i so ft h esameo r d e ra st h ew a v e l e n g t h ,thens u r f a c ee l e m e n t s s m a l l e rthanL w i l lbeweaklys c a t t e r i n gs ot h a tt h eP h y s i c a lO p t i c sa p p r o x i m a t i o n s maybev a l i d .T h i sh a sa l s obeens t a t e dbyJordanet a l 2 , 6 8 , 2 , 6 9 ,whofoundt h a t t h ep r e d i c t i o n so ft h eP h y s i c a lO p t i c st h e o r yappearedt obec o n s i s t e n twitht h e i r i n f r a r e de x p e r i m e n t a ls c a t t e r i n gr e s u l t sfromas u r f a c ef o rwhichI< A .F u r t h e r e m p i r i c a ls u p p o r tf o rt h i sh y p o t h e s i si sp r e s e n t e di nChapter3 . Becauseo ft h ee a r l yc o n n e c t i o nw i tht h er a d i of i e l d ,r e l a t i v e l ymorei sknown aboutt h en e a r f i e l ds c a t t e r i n gp r o p e r t i e so fpower-lawphases c a t t e r e r sr a t h e rthan o ft h e i rf a r f i e l dp r o p e r t i e s ,w i t ht h er e s u l tt h a tfewr e s u l t sf o rt h el a t t e rc a s ee x i s t . Theo n l ya n a l y t i c a lr e s u l tf o rt h es c a t t e r e di n t e n s i t yo fwhicht h ea uthori saware i sf o rt h e( l a r g ei l l u m i n a t e da r e a )Brownianf r a c t a l(v — 1 ) ,whichwasd e r i v e d i n d e p e n d e n t l yi nt h epro p a g a t i o n2,43 ands u r f a c es c a t t e r i n gf i e l d s 2 , 6 8 , 2 , 7 0 ,andwhich i sp r e s e n t e dh e r ef o rc o m p l e t e n e s si n§ 2. 6. 2 .1 and§ 2. 6. 2. 3 . 2.6.1.1 The expression for the mean intensity in one-dimension

F o l l o w i n gt h eapproachd e s c r i b e di n§ 2 . 1 . 4 ,t h ef a r f i e l ds c a t t e r e dcomplexam­ p l i t u d efromao n e d i m e n s i o n a l ,o rc o r r u g a t e d ,randomphases c r e e nmaybew r i t t e n

whereA {x) i st h e( a m p l i t u d e )a p e r t u r ef u n c t i o nandt h ep r e f a c t o r shavebeenomit­ t e d . Theu s u a lproceduret oc a l c u l a t et h es c a t t e r e di n t e n s i t yi st h ent or e l a t et h e ensemblea v e r a g eo ft h ee x p o n e n t i a lo ft h ephased i f f e r e n c e ,exp(iA^),i nt h eex­ p r e s s i o nf o r< 4 > ( r ) \ P * ( r )> t ot h ec o r r e l a t i o nf u n c t i o n , byassumingt h a tt h e phasei snormallyd i s t r i b u t e dwithz e r omean. Aswehaved i s c u s s e di nChapter1 ( § 1 . 4 ) ,h o w e v e r ,t h eu s eo ft h ec o r r e l a t i o n f u n c t i o ni si n a p p r o p r i a t ef o rs u r f a c e s ,o rm e d i a ,withpower-laws p e c t r a ld e n s i t i e s s i n c et h ev a r i a n c ei si n f i n i t e ,duet ot h epowerspectrumb e i n gs i n g u l a ra tt h eo r i g i n . I n s t e a do ft h ec o r r e l a t i o nf u n c t i o n ,t h e r e f o r e ,t h es t r u c t u r ef u n c t i o n ,5 ^ ,s h o u l dbe useda si tremainsf i n i t e .Equation2 . 2 0showshowt h es t r u c t u r ef u n c t i o ni so b t a i n e d fromt h eensemblea v e r a g eo ft h ecomplexe x p o n e n t i a lphased i f f e r e n c e , ( e x p( * [ ^ ( * i )-(x2)])) = exp

^,

( 2. 20)

wherewehaveassumed,a sb e f o r e ,t h a tt h erandomp r o c e s si sw i d e -senses t a t i o n ­ a r yandc o n s e q u e n t l ydependso n l yont h ec o o r d i n a t ed i f f e r e n c e ,Ax. Assuminga Gaussianformf o rt h ea p e r t u r ef u n c t i o nA(x)( s e e§ 2 . 4 ) ,

A (x ) = exp(-^rj), thent h emeani n t e n s i t ycanbew r i t t e na s 00

I(r)>=JJ (exp(*'[0 (x)-4>(y)])) exp^ k x o(x + v) _ L_±JL)

<

— OO

62

( 2 . 2 3 )

CHAPTER 2 Bymakingt h ev a r i a b l ec h a n g e sx = (x 1 +y, ) / 2andy = (x f —y ' ) / 2 ,t h i st h e n becomes oo

<

I(r)> = \ JJ =

___ yoo

w

VTk

J

/

u2 t 2 - v

c o s (k x s in 9 )e x p ^----xv —

( 2 . 3 0 )

x2 \

j

( 2 . 3 1 )

whichi st h ee x p r e s s i o nf o rt h ei n t e n s i t ys c a t t e r e dfromac o r r u g a t e df r a c t a lphase s c r e e n . 2.6.1.2 The expression for the mean intensity in two-dimensions

Usingt h ee x p r e s s i o n( w i t h o u tp r e f a c t o r s )f o rt h ef a r f i e l ds c a t t e r e damplitude fromatwo-dimensionalphases c r e e n(frome q n .2 . 1 8 ) , +oo

tf(ro)

JJ A(x, y) exp !*>(*, y)

dx d y ,

wemayw r i t et h ee x p r e s s i o nf o rt h es c a t t e r e dmeani n t e n s i t ya s oo

< / ( r 0) > =

JJ A(i i , yi)A(x2,y2)e x p ( - —

A^ )

— OO

x exp

^ 2 )^0 + ( y i - V 2)yo]

dx 1 dx 2 dy 1 dy 2

00

=

J J .4(xi) A(x2) exp

(~ ~ ^ ~ 2

—1)

—OO

x e x p i - if c R - (X; ~ X 2 ) ) ^ x 1 (f 2x 2 ,

wherewehaveused( 2. 2 0 )anda ssumed,a sb e f o r e ,t h a tt h ephasei sc i r c u l a r l y s y m metric. (Then o t a t i o ni st h esamea st h a tu s e di n§ 2 . 4 ) .Asb e f o r e ,aGaussian a p e r t u r et r a n s m i t t a n c ef u n c t i o ni su s e d ,

A(xi) = exp 63

( 2 . 2 3 )

CHAPTER 2 andbymakingt h ev a r i a b l es u b s t i t u t i o n sX i= ( i q+r 2) / 2 andX2 = ( r i—T2 ) / 2, thent h ei n t e n s i t ybecomes 00

=

_ (rf + 1*2) JJ exp ik R.2 r2 _ ^(r2) 2 2W 2

d21*1 d2r 2-

S i n c et h eri-dependenti n t e g r a t i o n ,whichi snows e p a r a t efromt h eo t h e rv a r i a b l e , r2 ,y i e l d so n l yac o n s t a n tf a c t o ro f2 n W 2, i ti se x c l u d e dfromt h eremaindero ft h e d e r i v a t i o n .Thus,t h ei n t e n s i t yi sg i v e nbyt h ei n t e g r a l

r exp( J -0 0 \ I n t e g r a t i n g 2,52 o v e rt h ea n g l e ,dOr2 ( r o )>= < /

j

f 2ir

_

z

2 W 2)

2

d21*2 -

f ik \H \\r 2 \ cos0 r 2 \ . exp^---------J dOr2 =27rr2Jo{kr2 s in0 ) ,

wherewehavemadet h es m a l la n g l ea p p r o x i m a t i o n ,s i n #« ( R /z ) .F i n a l l y ,t h e i n t e n s i t yi n t e g r a lf o rs c a t t e r i n gfromatwo-dimensionalphases c r e e nbecomes =2 tt

r 2Jo(kr2 sin^)exp

_J jL - j d r2l

w h i c h ,f o raf r a c t a ls t r u c t u r ef u n c t i o n( e q n .2 . 3 0 ) ,becomes

J

=27 T

( k2L 2~v r 2Jo(kr2 s i n9) exp^------r v2 -

r2 \ )dr2.

( 2 . 3 2 )

Havingd e r i v e dt h eg e n e r a le x p r e s s i o n sf o rt h es c a t t e r e dmeani n t e n s i t yfromf r a c t a l phases c a t t e r e r s ,e q u a t i o n s2 . 3 1and2 . 3 2 ,t h ei n t e g r a l sa r ee v a l u a t e di nt h ef o l l o w i n g s e c t i o n sf o rs p e c i f i cc a s e so ft h ef r a c t a li n d e xandi l l u m i n a t i o nd e t a i l sa n d ,i n§ 2. 6. 3, f o rt h ec a s eo fa r b i t r a r yi n d e x .Theser e s u l t sa r ed i s c u s s e di nd e t a i landcompared wi t ht h o s efromt h es i n g l e ,andt w o s c a l er e s u l t si n§ 2 . 6 . 4 . 2.6.1.3 The finite-aperture mean intensity for the Brownian fractal,

u=

1

I ti so n l yf o rt h e‘ s i m p l e s t ’f r a c t a l s ,t h o s ew i t hu n i tf r a c t a li n d e x(v = 1 )and i no n l yo n e d i m e n s i o n ,t h a tt h emeani n t e n s i t ycanbeo b t a i n e da n a l y t i c a l l yf o ra f i n i t ea p e r t u r e . Byo m i t t i n gt h ep r e f a c t o r sfrome q n .2 . 3 1 ,t h eg e n e r a le x p r e s s i o n f o rt h ef a r f i e l ds c a t t e r e di n t e n s i t yi sg i v e nby

< 1(0) > =- f I

*0

cos(kx,W s'mO) exp

k2L W

JV

XI

/2 ------ 7T

d x ',

( 2 . 3 3 )

wherewehaves e tv — 1 andmadet h ev a r i a b l echangex — ►W x 1 .F o r t u n a t e l y , t h i sh a st h eformo fastandardi n t e g r a l 2,52

64

CHAPTER 2 where$(x)i st h ep r o b a b i l i t yi n t e g r a l ,o re r r o rf u n c t i o n 2 , 6 0 ,andi = y / — l . I no r d e rt oc a l c u l a t et h es c a t t e r e di n t e n s i t y ,i ti sf i r s tn e c e s s a r yt os e p a r a t et h e r e a landimaginaryt e r m s .L e t t i n g/3 = 1 / 2 ,b = k W s i n ( 0 )and7 = k 2 L W / 2 , and i g n o r i n gt h ee x t e r n a lm u l t i p l i c a t i v ec o n s t a n tf o rs i m p l i c i t y ,t h en

< m >-

( 7’ ~ 7 atT)

“ P( l ! ~ 7 2,t7)

where Expandingt h ecomplexe x p o n e n t s ,t h e n 3 2 =exp/72 -/

(1 — $_) ^cos(&7) —isin(&7)^

+exp 1 2 ~ P

(1 —$ +) ^cos(&7) + i sin(&7)^

= 2cos(&7)exp(2 i^ ! V e x p ^ 2- ^ 2 x ^cos(& 7 ) ($ _ + $ + ) + i sm (by) ($+ — S i n c eacomplexnumber,2 ,h a st h ep r o p e r t i e st h a tz +z* = 2 3 £ { z }andz — z* = 2i £ y { z } ,wherez* i si t scomplexc o n j u g a t eand5 R { z }and^{2 }i t sr e a landimaginary p a r t sr e s p e c t i v e l y ,t h en -f i y ) + 3>(z — i y ) = 23^{$(x + i y ) } , $(a; + i y ) — 4>(x — i y ) = 2 i9 :{ $ ( x + i y ) } -

Asar e s u l t ,t h es o l u t i o nf o rt h es c a t t e r e dmeani n t e n s i t ycanbew r i t t e na s =2 c o s ( & 7 )e xp x ^2 cos(& 7 ) 3£ { $ + } + 2z2 sin(& 7 )^{ 4.}^ =2COS (by) exp( ^ 1 ) (l"*{*( ^ f ) }) +2 s i n ( i > 7 )e xp

g& 3

(^jp)}‘

I fwee x p r e s st h i si ntermso fk , L , W ands i n # ,t h ent h emeani n t e n s i t yi sf i n a l l y g i v e nby , k2W 2 sm2 9 \ ( f k 2 W 2L s in f lV , < 1 ( B ) > „ = i = exp ( ----------------------J jc o s l --------- ---------- J (1 - 3M>+) . / k 2 W 2L sin 9 \ + ™ ( -----------g-----------)

65

"I

J>

(2.34)

CHAPTER 2 where

k2L W

.k W s i n6 1 1

2V2

+ *

V2

S'

Althought h eg e n e r a lbehav i o u ro ft h ei n t e n s i t yc annotbededucedbys i m p l ye x ­ amininge q n .2 . 3 4 ,i tcanbecomputedu s i n gt h ecomplexe r r o rf u n c t i o napprox­ imation2 6 1 , e x n f —x 21 4>(rr+ i y ) = $(x) H--------((1 — cos(2xy)) + i sin(2:ry)) 2nx

6 +

/ 9\

exp (— / 4 ),r t

- e x p ( — a; ) ^

x

(n 2 + 4a.2) [/»(*> S') +

, x , , x V)]

+

2/),

f n{x-) y) = 2x — 2x cosh(ny)c o s ( 2a ; ? / )+n s i n h ( n y )s i n ( 2x ? / ) , gn(x , y )=2 xc o s h ( n ? / )s i n ( 2a ; y )fn s i n h ( n ? / )c o s ( 2 x y), |e(z,2/)| w 10_16|$(:r + i y ) \ .

Ther e a lp a r to ft h ee r r o rf u n c t i o n ,4 > ( x ) ,maythenbec a l c u l a t e dfromi t ss e r i e s expansion2 5 2 , ... 2 ^ (-l)nI2+ l X

^

^

«

! (2 n + 1) '

Ther e s u l t so ft h en u m e r i c a lcomputationo fe q u a t i o n2 . 3 4a r ed i s c u s s e di n§ 2 . 6 . 4 .

2.6.2 Large-aperture approximation solutions for the mean in­ tensity

Forf r a c t a li n d i c e so t h e rthanu n i t y ,e x a c ta n a l y t i c a ls o l u t i o n sf o rt h es c a t t e r e d meani n t e n s i t ycannotbeo b t a i n e df o rf i n i t ea p e r t u r ei l l u m i n a t i o na si n§ 2. 6. 1. 3. I n s t e a d ,h o w e v e r ,s o l u t i o n scanbeo b t a i n e di fweassumet h a tt h ea p e r t u r ei se f ­ f e c t i v e l yi n f i n i t e ,byl e t t i n gt h ei n t e g r a l sconv e r g e n c ebec ausedbyt h edecayo f t h ef r a c t a lexponentt e r m .T h i sd o e sn o tr e s t r i c tu s ,h o w e v e r ,t oan‘ u n p h y s i c a l ’ i n f i n i t ea p e r t u r e ,a si ta p p l i e sp r o v i d e dt h a tt h ea p e r t u r etermd e c a y smuchmore s l o w l ythant h ef r a c t a le x p o n e n t .Ratherthanb e i n gr e s t r i c t i v e ,wes h a l lshowt h a t t h i sapproximationa l l o w st h enat u r eo ft h ei n t e n s i t yd i s t r i b u t i o nt obec l e a r l ys e e n andg i v e smuchg r e a t e ri n s i g h ti n t ot h es c a t t e r i n gp r o c e s sthant h ep r e v i o u sr e s u l t ( e q n .2 . 3 4 ) . P h y s i c a l l y ,t h ena t u r eo ft h eapproximationi se q u i v a l e n tt oi l l u m i n a t i n gal a r g e a r e ao ft h es u r f a c er e l a t i v et oat y p i c a ls c a t t e r i n g‘ c e l l ’s i z es ot h a tt h er e s u l t i n gf a r h e l d( G a u s s i a n )s p e c k l ew i l lbev e r ys m a l l( § 1 . 5 ) .Asar e s u l t ,t h ei n t e n s i t ye n v e l o p e w i l lbedeterminedbyt h es u r f a c es t r u c t u r eandn o tbyt h ei l l u m i n a t i o np r o f i l e( s e e a l s o§ 2 . 4 . 1 ) . Notet h a tt h es c a t t e r i n g‘ c e l l ’s i z e ,b e i n gt h el a t e r a ls i z eo fat y p i c a ls c a t t e r e r , i snormallyd e s c r i b e dbyt h ec o r r e l a t i o nl e n g t ho fap a r t i c u l a rs u r f a c em odel. As wehavea l r e a d yshown,h o w e v e r ,t h ec o n c e p to fac o r r e l a t i o nf u n c t i o nf o rf r a c t a l si s i n a p p r o p r i a t eduet ot h ev a r i a n c eb e i n gi n f i n i t e .N e v e r t h e l e s s ,ane f f e c t i v es t r u c t u r e s i z ed o e se x i s t ,andi tcanber e a d i l yfoundfromt h eh e l dcoher e n c ef u n c t i o n 2-46 t obe\ 2!v j L 2!v~^ w h i c h ,f o rv =-l 2 6 9 ,becomesA2 / Z . Thismatteri sd i s c u s s e d

66

CHAPTER 2 f u l l yi nChapter4( § 4 . 2 ) ,wherei t simportancei sc e n t r a li nt h estudyo ft h es p e c k l e c o n t r a s t . S t a r t i n g ,a sb e f o r e ,fromt h ee x p r e s s i o nf o rt h es c a t t e r e dmeani n t e n s i t yfroma one-dimensionalphases c r e e n( e q n .2 . 3 1 ) ,wehave < 1 (9 )

( k 2 T 2~ v W v c o s ( k x ,W s m O ) e xp( ------ ----x ,v

>= J

—

x^\

^x>■>

( 2 . 3 5 )

wherewehavemadet h es u b s t i t u t i o nx — ►W x 1 andomittedt h ep r e f a c t o r s . By l e t t i n g k2L 2- v W ’' > 1 , ( 2 . 3 6 ) wee f f e c t i v e l ymaket h ea p e r t u r etermc o n s t a n ts ot h a tt h ei n t e g r a l ’ sc o n v e r g e n c ei s ca u s e dbyt h es t r u c t u r ef u n c t i o nexponenta sr e q u i r e d .Asar e s u l t ,t h es l o wdecayo f t h ea p e r t u r etermw i l ln o ta f f e c tt h er e s to ft h ei n t e g r a n ds ot h a ti tcanbee x c l u d e d fromt h ei n t e g r a l .Thuswecanapproximatet h emeani n t e n s i t yby <

1(9)

cos^x'W'sin#) exp ^---- ---- x ,v \

>= J

2.6.2.1 The solution for

v~

d x '.

(2.37)

l in one-dimension

I nt h en e x tt h r e es u b s e c t i o n swep r e s e n td e r i v a t i o n so ft h ee x p r e s s i o n sf o rt h e i n t e n s i t yd i s t r i b u t i o n sr e s u l t i n gfromt h es c a t t e r i n gfromf r a c t a lphases c r e e n sf o ra l l c a s e so ft h ef r a c t a li n d e x ,knownt ot h ea u t h o r ,whichcanbea n a l y t i c a l l ys o l v e d .I n p r a c t i c e ,mosto ft h e s es o l u t i o n s ,a sf o rt h eBrownianf r a c t a lc a s ewitha r b i t r a r ya p e r ­ t u r e( § 2 . 6. 1 . 3 ) ,a r er e s t r i c t e dt oc o r r u g a t e d( o n e d i m e n s i o n a l )phases c r e e n s .T h i s d o e sn o tl i m i tu s ,h o w e v e r ,a swes h a l lshowt h a tt h eb ehaviouro ft h ei n t e n s i t yd i s t r i ­ b u t i o nfromtwo-dimensionalphases c r e e n si sr e l a t e dt ot h a tfromo n e d i m e n s i o na l s c r e e n sbyas i m p l echangeo ft heexponento ft h ei n t e n s i t yd e c a y . R e c a l l i n gt h el a r g e a p e r t u r eanalogueo ft h ec a s es t u d i e di n§ 2 . 6 . 1 . 3 ,t h eex­ p r e s s i o nf o rt h es c a t t e r e dmeani n t e n s i t yfromaBrownianf r a c t a l(v = 1 )i s( f r o m e q n .2 . 3 7 )g i v e nby <

1(6)

>=

j

r°o

/

k 2L W

cos^z'Wsin#)exp^--- -— x

\

j d x '.

Havingt h eformo ft h estandardi n t e g r a l 25 2 , [OQ

]_

/ exp(—px) c o s (qx +A )= —-- ( p c o sA—q s i nA ) ,

Jo

p 2+ r

t h ei n t e g r a lcanber e a d i l ys o l v e ds ot h a tt h ei n t e n s i t yd i s t r i b u t i o ni sg i v e nby < m

> , =1=

L /2 W k2L 2 /4fs i n 2

( 2 . 3 8 )

ar e s u l ti n i t i a l l yd e r i v e dbyT a t a r s k i 2 , 4 3 ,anda l s omuchl a t e rbybothChurchet ai2,70 ( i nas m a l l a n g l el i m i t )andJordanet a i 2 , 6 9 .

67

CHAPTER 2 2.6.2.2 The solution for i/= 1/2 in one-dimension

Therei so n l yoneo t h e rf r a c t a li n d e xo fwhicht h ea u t h o ri sawarewhichmaybe u s e dt og i v eac l o s e dformf o rt h er e s u l t a n tmeani n t e n s i t y ,t h ec a s ewhenv =1 / 2. Frome q n .2 . 3 7 ,t h ee x p r e s s i o nf o rt h e( l a r g e a p e r t u r e )meani n t e n s i t yi sw r i t t e na s

< 1(0) >= j

ro o

J0

cosykx'W smO) exp

I fwemaket h ev a r i a b l echangez=V a 7 ,t h e n < /($)>=J

yo°

/ k2rZ/2W lj2 \

z c o s ( k z 2W s m 6 ) e xpI --------z j d z .

Makingf u r t h e rs i m p l i f i c a t i o n sbyl e t t i n ga = (k 2L zl 2W ll 2)l2 and/ ?= kfFsin#, t h e n roo

=JoI

z cos({3z2)e~ az dz.

T h i scanbei n t e g r a t e dbyp a r t st og i v e = ^sin(/3z2 ) e x p ( o ; z ) Pa

a = 7^

o

Jo

oo s i n ( / ? z 2 )

l

exp(—a z ) dz

jo

sin((3z2) exp(—az)d z,

wherewehavei g n o r e dt h em u l t i p l i c a t i v ec o n s t a n t sf o rs i m p l i c i t y .I fwemaket h e f u r t h e rs u b s t i t u t i o n s ,y =\ffdz and7 =ct/yfp , t h en < A 0) > =

s in t/ 2 e x p (—71/) d y ,

7^02 J

w h i c h ,f o l l o w i n gaf u r t h e ri n t e g r a t i o n( a g a i nn e g l e c t i n gc o n s t a n tp r e f a c t o r s ) ,g i v e s <

1 (0 ) > =

' '

7

(■?(») e x p ( - j/)

2

g +7

f

7

‘?(t/)exp(- j/)|

dy

roo

0:7 2^ 3/2 j

0 s y eM (

)

7y d v ,

-

)

whereS (y ) i st h eF r e s n e li n t e g r a l 2 , 6 0 .T h i snowh a st h eformo fastandardi n t e g r a l ( aLaplacet r a n s f o r m )w i thas o l u t i o ng i v e nby2,52 k ~

<

+s i nT

i

2

V2

whereC (x ) i st h eo t h e rF r e s n e li n t e g r a lf u n c t i o n 2 , 6 0 .I fwes u b s t i t u t ebackf o ra ,(J and7 ,t h emeani n t e n s i t ycanf i n a l l ybew r i t t e na s < W ) > * = 1/2 =

k 2L z!2

f /PT3\ [ l 2( f c s i n 0) 3 / 2\ \8 s i n 0/ 2 . /k3L 3 \

+ BmUriT?J 68

1

2

C ( m

f (k L )V 2Y \4 Vsin0 J 3/ 2 X

y4 s i n9 )

)

( 2 . 3 9 )

CHAPTER 2 I np r a c t i c e ,t h i se x p r e s s i o ni se v a l u a t e dbyu s i n gt h ei n f i n i t es e r i e se x p a n s i o n sf o r t h eF r e s n e li n t e g r a l s 2 60 Q ( x ) _ V ' ( ~ 1 ) ’’( ,r/ 2 ) 2" 4n+ l

W

£j(2n)!(4n + l)*

’

and _ V

(-1 )" (V 2 )^ »

4n+3

W _ n'=W02v n + l)!(4n ' v + 3) 7 Ther e s u l t sa r ed i s c u s s e di n§ 2 . 6 . 4 . 2.6.2.3 The solution for

v—

'

l in two-dimensions

Onlyt h es i m p l eBrownian{v = 1 )c a s ea p p e a r st ohaveas o l u t i o ni ntwod i m e n s i o n s .Using( 2 . 3 2 )t os o l v ef o rt h ef a r f i e l ds c a t t e r e dmeani n t e n s i t y ,wehave < m

>- f

r J o (k r W sin#)exp

k2L W

d r,

wherewehavemadet h es u b s t i t u t i o nr— ►W r ^ ,andassumedt h a tk 2L 2~vW v 1 ( e q n .2 . 3 6 )s ot h a tt h eGaussiana p e r t u r etermcanbeo m i t t e d .I fwemakeu s eo f t h estandardi n t e g r a l 2*52

i:

exp(—ax)J]c({3x)xk+1dx =

2a(2/?)*r(Jb + 3/2) y /T r( a 2 + /3 2) 3/ 2

andi g n o r et h em u l t i p l i c a t i v ec o n s t a n t s ,t h ent h ea n g u l a rbehaviouro ft h es c a t t e r e d i n t e n s i t yg o e sa s „=!=(^£2 /4 +s in 2 0 ) 3 / 2 -

( 2' 4° )

Althought h i sr e s u l tw i l lbef u l l yd i s c u s s e di n§ 2 . 6 . 4 ,i ti si n t e r e s t i n gt on o t ei t s s i m i l a r i t yt ot h ec o r r u g a t e dBrownianr e s u l t ,e q u a t i o n2 . 3 8 .

2.6.3 The mean intensity from fractals with arbitrary index

Becauset h er e s u l t so b t a i n e di nt h ep r e v i o u ss u b s e c t i o n sa r ev a l i do n l yf o rp a r ­ t i c u l a rv a l u e so ft h ef r a c t a li n d e x ,t h e ydon o tg i v eac l e a rp i c t u r eo ft h ea n g u l a r d i s t r i b u t i o no fl i g h ts c a t t e r e dfromf r a c t a l swithana r b i t r a r yi n d e x .I na d d i t i o n ,t h e a n g ulardependenceo ft h e s er e s u l ti sn o ta l w a y sc l e a rs ot h a tmanyo ft h ee x p r e s s i o n s mustbebee v a l u a t e dn u m e r i c a l l ybycomputer. F o l l o w i n gt h eo r i g i n a ld e r i v a t i o n so ft h ei n t e n s i t ys c a t t e r e dfromBrownianf r a c ­ t a l s( e q n s .2 . 3 8and2 . 4 0 ) ,t h ei n t e g r a le x p r e s s i o n sf o rt h ei n t e n s i t y( 2 . 3 1 ,2 . 3 2 )were n u m e r i c a l l yi n t e g r a t e d( § 1 . 3 )i no r d e rt oexaminet h es c a t t e r e dd i s t r i b u t i o nf o ro t h e r f r a c t a li n d i c e s .L a t e r ,awaywasfoundo fa n a l y t i c a l l yp r e d i c t i n gt h ebeha v i o u ro f t h e s ed i s t r i b u t i o n swhichr e n d e r e dt h ei n i t i a lapproachl a r g e l yr e d u n d a n t .I nt h e f o l l o w i n gs u b s e c t i o n s ,wep r e s e n td e r i v a t i o n so ft h e s ea n a l y t i ce x p r e s s i o n s ,which a r ev a l i di nt h el i m i to fl a r g ek W s i n0 ,bothf o rc o r r u g a t e dandt w o -dimensional phases c r e e n s .

69

CHAPTER 2 2.6.3.1 The mean intensity for one-dimensional fractals

Asb e f o r e ,t h ee x p r e s s i o nf o rt h emeani n t e n s i t yfromao n e -dimensionalphase s c r e e nassumingl a r g ea p e r t u r ei l l u m i n a t i o n ,

k2L 2~ vW v > 1 ,

( 2 . 3 6 )

i sg i v e nby( 2 . 3 7 ) , fO O

< 1(0) >= / cos(k x W s i n0) exp

k2L 2~v W v

Jo

x v ) dx,

whichh a st h eformo faF o u r i e rc o s i n et r a n s f o r mw i t ht h er e s t r i c t i o n ,f o rf r a c t a l s , o f0< i/ < 2 .T h i se q u a t i o nwasf i r s ts o l v e dbyLevyi n1925a sar e s u l to fh i s examinationo fs t a b l e( L e v y )d i s t r i b u t i o n s 2 , 6 3 , 2 7 3 ,f We s o l v et h i se q u a t i o nby f o l l o w i n gt h eapproacho fM o n t r o l l& West2,73 who,f o rr e a s o n swhichs h a l ls h o r t l y becomea p p a r e n t ,s u b d i v i d et h ei n t e r v a l0 < v < 2 i n t ot w o ; 0 < v < 1and 1 > == / co s(/3 x )ex p (—a x v) dx.

( 2 4 1 )

Jo' o

I n t e g r a t i n gt h i se x p r e s s i o nbyp a r t s , 00 a v f ° °

1

i

= —s i n ( / ? x )exp( —a x u) +“ 77"/ xU 1 s i n ( / ? : r )exp( —otxv) dx P 0 P Jo /•OO

/ x- 1 s i n ( / ? x )exp(-ai")dx, P Jo andmakingt h es i m p l es u b s t i t u t i o n ,v =f3x, t h e n =

< m

av

/»oo

"- 1 s i n ( v ) > = JE+l / v exp V W ' j Jo

dv‘

T h isi n t e g r a lcanbeg r e a t l ys i m p l i f i e dbymakingt h eapproximationf$ —* o o( t h e s i g n i f i c a n c ei sd i s c u s s e dbelow),s i n c et h ee x p o n e n t i a ltermcanthenbes e tt ou n i t y ,

f Stable distributions satisfy the chain co n d itio n , the property that the product of their characteristic functions retains the form of the individual characteristic function. Well-known examples are the Gaussian distribution, where exp(—x 2/(T i)e x p (—x 2/= — / c o s ( u ; )exp( —T]au) d u —-/ an p

oo

L

uj cos( uj) e xp(—rjauj)

c o s( a ; )exp( —7/ a u ) duj

duj f0 (rj2a 2),

where0 ( a 2r]2) r e p r e s e n t st h eh i g h e ro r d e rtermsi np owerso fa27/ 2 ,whichcanbe n e g l e c t e di nt h el i m i to fl a r g ej3. Thei n t e g r a l sa l lh a vestandardfo r m ss ot h a tt h e i n t e n s i t ycanbew r i t t e na s 1 / na

\

1a fr(v+1 )c o s [ ( f+1 )tan 1 (1 / t j q t ) ]

J \ \ + n 2a 2) ~ T \

(1 + ,2a2)(^+l)/2

- r(2)cos[2tan: y / H \ + 0( (1 + r)l a l )

J

wherewehaveu s e dt h es t a n dardi n t e g r a l 2-52 r

x a 1 exp(—r]x) c o s(ax) dx =

Jo

T(a)

( t) 2 + a 2) a ! 2

c o sa t an- l

Now,i nt h esamel i m i ta sb e f o r eo f/3 — >o o ,t h e n77 — >0 ,s ot h a tt h ei n t e n s i t yi s g i v e nby ~

+1 )s i n(^) a U T'f \ ' ( V1T\

= ^ r r W slH T ) (k W s m e y + ^

1<

V

< 2 ,

( 2 . 4 3 )

s i n c eY ( v +1 )=v T ( v ) 2-Ql andtan-1 ( l / a ; 77)— > 7r / 2. S i n c ee q u a t i o n2 . 4 3i ss i m i l a rt o( 2 . 4 2 ) ,whichwasd e r i v e df o rt h ec a s e0 2 maybecomen e g a t i v ef o rsomev a l u e so f(5 = k W s i n # ,p r o v i n g t h a ti tcannotr e p r e s e n tav a l i dp r o b a b i l i t yd i s t r i b u t i o n . Ther e l e v a n c eo ft h i st o o urworki st h a tf o rv > 2 ,t h ep r e d i c t i o n sf o rt h es c a t t e r e di n t e n s i t ymighta l s obe n e g a t i v e ! ;t h u s ,t h er e s t r i c t i o no f0 < v < 2 f o rf r a c t a l si samplyd e m o n s t r a t e d . 2.6.3.2 The mean intensity for two-dimensional fractals

F o l l o w i n gt h ep r e c e e d i n gt r e a t m e n tf o rt h eone-dimensionalasymptotics o l u ­ t i o n s ,t h ed e r i v a t i o no ft h es c a t t e r e dmeani n t e n s i t yfromtwo-dimensionalphase s c r e e n smaybeexaminedbys t a r t i n g ,t h i st i m e ,frome q u a t i o n2 . 3 2 , „ rW sm0 «)exp(I-kW--- W--r rJTo (k

72

CHAPTER 2 where we have made the large aperture approximation, k2L2~VW V 1 (eqn. 2.36). Letting (3 = kW sin #, a = k2L2~vW v/2, 77 = (3~v and making the further substitu­ tion lj = /?r, then the expression for the scattered intensity becomes <

1

I(0 )> = — J

u J q( u ) e xp(-0770/)du.

However,duet ot h eproblemsa s s o c i a t e dw i t hs o l v i n gt h ei n t e g r a l/0 ° °ujjQ(uj)du, i t i sn o tp o s s i b l et os o l v ef o rt h es c a t t e r e di n t e n s i t ybyd i v i d i n g ,a sb e f o r e ,t h ei n t e r v a l 0 2andv musta l s oc h a n g e . Equations2 . 3 8and2 . 4 0 ,whichd e s c r i b et h es c a t t e r e di n t e n s i t yd i s t r i b u t i o na r i s ­ i n gfromoneandtwo-dimensionalBrownianf r a c t a lphases c r e e n s ,a r es i m i l a re x c e p t f o rt h eexponentso ft h ea n g u l a rt e r m ,whicha r e-1 and3 / 2r e s p e c t i v e l y .I n i t i a l e x p e c t a t i o n st h a tt h er e l a t i o n s h i pbetweent h etwowass i g n i f i c a n twasv a l i d a t e dby t h er e s u l t sfromt h en u m e r i c a li n t e g r a t i o no fe q n .2 . 3 2 ,whichr e v e a l e dt h a twhereas t h eone-dimensionalr e s u l tb e h a v e sl i k e ^ (k W sin e y + 1’

<

( 2 . 4 2 )

i ntwo-dimensionst h ec o r r e s p o n d i n gr e s u l twass i m p l yg i v e nby ~ (k W s i n0)"+2'

^ 2' 45^

Ther e l a t i o n s h i pbetweent h ee x p o n e n t ,v, andt h ed i m e n s i o n s ,D andD y ,i sd i s ­ c u s s e df u r t h e rbothbyMandelbrot2,63 andbyAdler2 , 7 4 .

2.6.4 Discussion and results

I nt h i ss e c t i o n ,wed i s c u s st h ea n a l y t i c a lr e s u l t sfromt h ep r e v i o u s( f r a c t a l )sub­ s e c t i o n s .Aswehavea l r e a d ym e n tioned,i ti sn o timmediatelya p p a rentfrommost o ft h e s ea n a l y t i c a le x p r e s s i o n sj u s thowt h es c a t t e r i n gd i s t r i b u t i o nc h a ngesa st h e f r a c t a lin d e xi sc h a n g e d ,althought h edependencei sc l e a rf o rt h e( 1 -D)Brownian f r a c t a lmodel(v = 1 ) ,e q u a t i o n2 . 3 8 ,

^’

k 2L 2/4 + sin2 0

Thea n g u l a rdecaywhicht h i sd e s c r i b e si sc l e a r l yt h esamea st h a tp r e d i c t e dbyt h e a r b i t r a r yi n d e xs o l u t i o n ,e q u a t i o n2 . 4 3 ,

whichshowst h a tt h ef r a c t a lind e xmaybedeterminedbyexaminingt h ef a l l o f fo f t h es c a t t e r e dmeani n t e n s i t ywitha n g l e . Fi g u r e2 . 1 1i l l u s t r a t e st h i sp o i n tbyp l o t t i n gl o g< 1(6) > a g a i n s tl o g ( s i n 0 ) frome q u a t i o n2 . 3 8 .I nt h el i m i to fl a r g es c a t t e r i n ga n g l e ,t h ei n t e n s i t yf a l l s o f f

74

CHAPTER 2 linearly withagradiento f—( 1/f -1 ) .Thei n t e r c e p to ft h i sl i n ew i thah o r i z o n t a l onedrawnfromt h ei n t e n s i t ya t9 =0 °i swheres i n #= k L /2o r ,i ng e n e r a l ,where s i n9 = ( f e X ) 2/ —1 ,s ot h a ti na d d i t i o nt of i n d i n go u tt h ef r a c t a li n d e x ,t h ev a l u eo f t h et o p o t h e s ycana l s obef o u n d .Thusa l lo ft h epara m e t e r so faf r a c t a lphases c r e e n canbefoundfromanexaminationo ft h ea n g u l a rd i s t r i b u t i o no ft h es c a t t e r e dmean i n t e n s i t y .T h isi si nc o n t r a s twitht h esmoothlyv a r y i n g( s i n g l e s c a l eandt w o s c a l e ) modelsd i s c u s s e dp r e v i o u s l y ,whosepowerspectrumcanbefoundo n l yfromv e r y weaks c a t t e r e r i n gw h i l ef o rs t r o n g e rs c a t t e r e r i n g ,o n l yt h er a t i oo ft h ec o r r e l a t i o n l e n g t ht ot h ephasev a r i a n c e ,( 3 / < 7 ^ ,canbedetermined( s e e§ 2 . 4 . 3 ) .Thust h e r ei sa s i m p l eandq u a l i t a t i v ed i f f e r e n c ebetweent h ei n t e n s i t i e ss c a t t e r e dfromt h esmoothly v a r y i n gandf r a c t a lm o d e l s . T h i sa b i l i t yt od e terminet h epara m e t e r so ft h ef r a c t a lphases c r e e nh o l d sgood f o ra l lf r a c t a li n d i c e sandf o rl a r g e ,ands m a l l ,i l l u m i n a t e da r e a s .ThusF i g u r e2 . 1 2 showst h es c a t t e r e di n t e n s i t yf o rv = 1 / 2(f r o me q n .2 . 3 9 ) ,v — 1(frome q n .2 . 3 4 ) andf o rv — 3 /2(fromt h er e s u l t so ft h en u m e r i c a li n t e g r a t i o nd i s c u s s e de a r l i e r ) ,f o r one-dimensionals c r e e n s .Fora l lo ft h e s ec a s e s ,andr e g a r d l e s so ft h ec o m p l e x i t yo f t h ea c t u a li n t e n s i t ye x p r e s s i o n s ,t h e ya l lshowt h esamepower-lawa n g u l a rdecayo f —(v f -1 )a se x p e c t e d . I nt w o d i m e n s i o n s ,e q u a t i o n2 . 4 0i l l u s t r a t e st h eg e n e r a lp r i n c i p a lt h a tt h eangu­ l a rdecayi snowgovernedbyt h eexponent—(v f2 ) .Thatt h i sr e mainst r u ef o ra l l vi sshownbyF i g u r e2 . 1 3 ,whichshowst h edecayf o rt h esamei n d i c e sa si nF i g u r e 2 . 1 2 ,witht h er e s u l t sf o rv — 1 / 2andv = 3/2b e i n ge v a l u a t e dbyt h en u m e r i c a l i n t e g r a t i o no fe q u a t i o n2 . 3 2 . Becauset h ef r a c t a lmodel,u n l i k et h esmoothlyv a r y i n gm o d e l s ,i sp u r e l yd i f f r a c t ­ i n g ,t h es c a t t e r e di n t e n s i t yw i l le x h i b i tas t r o n gwavelengthd e p e n dence.Although n o to fe x p l i c i tr e l e v a n c et ot h es t u d i e sb e i n gd i s c u s s e dh e r e ,a(now)s t a n d a r d wayo fmeasuringt h ef r a c t a ldimensiono fs m a l ld i f f r a c t i n gs t r u c t u r e s ,s u c ha sc o l ­ l o i d a la g g r e g a t e s ,i st oexaminet h es c a t t e r e di n t e n s i t ya tanumbero fd i f f e r e n t wavelengths2 , 7 5 .T h i st e c h n i q u ec o u l dpresumablybej u s ta se a s i l ya p p l i e dt orough s u r f a c e sandphases c r e e n s .

REFERENCES [ 2. 1] J .W. Goodman, Introduction to F o u r i e rO ptics (McGraw-Hill,SanFran­ c i s c o ,1 9 6 8 ) . [ 2 . 2 ] M.BornandE .W o l f ,Principles o f O ptics (PergamonP r e s s ,O x f o r d ,1 9 8 0 ) . [ 2 . 3 ] P .BeckmannandA .S p i z z i c h i n o ,The Scattering o f E lectrom agnetic Waves from Rough Surfaces (PergamonP r e s s ,London,1 9 6 3 ) . [ 2 . 4 ] J .B .K e l l e r ,“ GeometricalTheoryo fd i f f r a c t i o n ”,J. O pt. Soc. A m . 5 2 ,1 1 6 ( 1 9 6 2 ) . [ 2 . 5 ] V .Twersky,“ M u l t i p l es c a t t e r i n go fwavesando p t i c a lphenomena”,J .O pt. Soc. A m . 52,145( 1 9 6 2 ) .

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CHAPTER 2 l o g

F i g .2 .1 2 Thes c a t t e r e dmeani n t e n s i t i e sfromcorrugatedf r a c t a lphases c r e e n s withf r a c t a li n d i c e s( a )v — 0 . 5 ,( b )v =1 . 0and( c )v = 1 . 5 .

l o g

F i g .2 . 1 3 Thes c a t t e r e dmeani n t e n s i t i e sfromtwo-dimensionalf r a c t a lphase s c r e e n swithf r a c t a li n d i c e s( a )v =0 . 5 ,( b )v = 1 . 0and( c )v = 1 . 5 .

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CHAPTER 2 [ 2 . 6 ] R .F .M i l l a r ,“ TheR a y l e i g hH y p o t h e s i sandar e l a t e dl e a s t s q u a r e ss o l u t i o n t os c a t t e r i n gproblemsf o rp e r i o d i cs u r f a c e sando t h e rs c a t t e r e r s ”,Rad. Sci. 8 ,7 85( 1 9 7 3 ) . [ 2 . 7 ] D .N .PattanayakandE .W o l f ,“ GeneralFormandaNewI n t e r p r e t a t i o no f t h eEwald-OseenE x t i n c t i o nTheorem”,O pt. Commun. 6 ,217( 1 9 7 2 ) . [ 2 . 8 ] J .ShenandA .A .Maradudin,“ M u l t i p l es c a t t e r i n go fwavesfromrandom roughs u r f a c e s ”,Phys. R ev. B 2 2 ,4234( 1 9 8 0 ) . [ 2 . 9 ] M.N i e t o V e s p e r i n a s ,“ D e p o l a r i z a t i o no fe l e c t r o m a g n e t i cwavess c a t t e r e dfrom s l i g h t l yroughrandoms u r f a c e s :astudybymeanso ft h ee x t i n c t i o ntheorem”, J .O pt. Soc. A m . 7 2 ,539( 1 9 8 2 ) . [ 2 . 1 0 ]E .WolfandE .W.Marchand,“ Comparisono ft h eK i r c h h o f fandt h eR a y l e i g h Sommerfeldt h e o r i e so fd i f f r a c t i o na tana p e r t u r e ”,J .O pt. Soc. A m . 5 4 , 587( 1 9 6 4 ) . [ 2 . 1 1 ]E .W.MarchandandE .W o l f ,“ C o n s i s t e n tf o r m u l a t i o no fK i r c h h o f f ’ sd i f f r a c ­ t i o nt h e o r y ”,J. O pt. Soc. A m . 5 6 ,1 7 12( 1 9 6 6 ) . [ 2 . 1 2 ] M.J .E h r l i c h ,S .S i l v e randG .H e l d ,“ S t u d i e so ft h ed i f f r a c t i o no fe l e c t r o m a g ­ n e t i cwavesbyc i r c u l a ra p e r t u r e sandcomplimentaryo b s t a c l e s :t h enearz o n e f i e l d ”,J. A pp. Phys. 26, 3 36( 1 9 5 5 ) . [ 2 . 1 3 ]F .G.BassandI .M.F u k s ,Wavescattering from sta tistica lly rough surfaces (PergamonP r e s s ,London,1 9 7 9 ) . [ 2 . 1 4 ] R.P .M e r c i e r ,“ D i f f r a c t i o nbyas c r e e nc a u s i n gl a r g erandomphasef l u c t u a ­ t i o n s ”,Proc. Cam b. Phil. Soc. 5 8 ,382( 1 9 6 2 ) . [ 2 . 1 5 ]E .JakemanandP .N .P u s e y ,“ Thes t a t i s t i c so fl i g h ts c a t t e r e dbyarandom phases c r e e n ”,J. Phys. A:M ath., Nucl. Gen. 6 ,L8 8 ( 1 9 7 3 ) . [ 2 . 1 6 ]B .I .BleaneyandB .B l e a n e y ,E lectricity and M agnetism ( 0 .U .P . ,O x f o r d , 1 9 7 8 ) . [ 2 . 1 7 ]A .K.Fung,“ I n v e r s eMethodsi nRough-SurfaceS c a t t e r i n g ”,i nInverse M eth­ ods in E lectrom agnetic Im aging ,Part 2, Ed.W.M.Boernere ta l( D .R e i d e l P u b l i s h .C o . ,1 9 8 5 ) . [ 2 . 1 8 ]S .T .Wu andA .K.Fung,“ ANoncoherentModelf o rMicrowaveE m i s s i o n s andB a c k s c a t t e r i n gfromt h eSeaS u r f a c e ”,J .Geophys. Res. 7 7 ,5917( 1 9 7 2 ) . [ 2 . 1 9 ]J .C .BamberandR .J .D i c k i n s o n ,“ U l t r a s o n i cB s c a nning: acomputer s i m u l a t i o n ”,Phys. M ed. Biol. 25, 463( 1 9 8 0 ) . [ 2 . 2 0 ] M.N i e t o V e s p e r i n a sandN .G a r c i a ,“ Ad e t a i l e dstudyo ft h es c a t t e r i n go f s c a l a rwavesfromrandomroughs u r f a c e s ”,O pt. A cta 28, 1 6 5 1( 1 9 8 1 ) . [ 2. 2 1 ]G .S .Brown,“ Thev a l i d i t yo fshadowingc o r r e c t i o n si nroughs u r f a c es c a t ­ t e r i n g ”,R ad. Sci. 19, 1 4 6 1( 1 9 8 4 ) . [ 2 . 2 2 ]P .Beckmann,“ S c a t t e r i n go fl i g h tbyroughs u r f a c e s ”i n Progress in O ptics , Vol.VI, Ed.E.Wolf(Nortli-Holland,Amsterdam,1 9 6 7 ) .

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CHAPTER 2 [ 2 . 2 3 ]E .I .T h o r s u s ,“ Thev a l i d i t yo ft h eK i r c h h o f Fapproximationf o rroughs u r f a c e s c a t t e r i n gu s i n gaGaussianr o u g h n e s sspectrum”,J. A coust. Soc. A m . 8 3 , 7 8( 1 9 8 8 ) . [ 2 . 2 4 ]H .E .BennetandJ .0 .P o r t e u s ,“ R e l a t i o nbetweens u r f a c er o u g h n e s sand s p e c u l a rr e f l e c t a n c ea tnormali n c i d e n c e ”,J. O pt. Soc. A m . 5 1 ,1 2 3( 1 9 6 1 ) . [ 2 . 2 5 ] M. N i e t o V e s p e r i n a sandJ .M. S o t o C r e s p o ,“ M u l t i p l eL i g h tS c a t t e r i n g FromP e r f e c t l yConductingRoughS u r f a c e s ”,Proc. 14th Conf. I. C. 0 ., E d .H .H .A r s e n a u l t ,Quebec( 1 9 8 7 ) . [ 2 . 2 6 ]C .L .R i n o ,“ A powerlawphases c r e e nmodelf o ri o n o s p h e r i cs c i n t i l l a t i o n ”, Rad. Sci. 14,1135( 1 9 7 9 ) . [ 2 . 2 7 ]S .T .McDaniel,“ P h y s i c a lO p t i c st h e o r yo fs c a t t e r i n gfromt h ei c ecanopy”, J .A coust. Soc. A m . 8 2 ,2060( 1 9 8 7 ) . [ 2 . 2 8 ]E .Jakeman,G.P a r r y ,E .R .P i k eandP .N .P u s e y ,“ Thet w i n k l i n go fs t a r s ”, C ontem p. Phys. 19,127( 1 9 7 8 ) . [ 2 . 2 9 ]J .G.W a l k e r ,M.V .BerryandC .U p s t i l l ,“ Measuremento ft w i n k l i n gex­ p o n entso fl i g h tf o c u s e dbyrandomlyr i p p l i n gwater”,O pt. A cta 3 0 ,1 0 0 1 ( 1 9 8 3 ) . [ 2 . 3 0 ]R .B r a c e w e l l ,The Fourier transform and its applications (McGraw-Hill,New Y o r k ,1 9 6 5 ) . [ 2 . 3 1 ]P .Beckmann, “ S c a t t e r i n gbyNon-GaussianS u r f a c e s ”,I. E. E. E. Trans. Antenn. Propgn. AP—2 1 ,169( 1 9 7 3 ) . [ 2 . 3 2 ]S .0.R i c e ,“ R e f l e c t i o no fe l e c t r o m a g n e t i cwavesfroms l i g h t l yroughs u r f a c e s ”, Com m un. P ure A ppl. M ath. 4 ,3 5 1( 1 9 5 1 ) . [ 2 . 3 3 ] K.A .M i t z n e r ,“ E f f e c to fs m a l li r r e g u l a r i t i e sone l e c t r o m a g n e t i cs c a t t e r i n g fromani n t e r f a c eo fa r b i t r a r yshape”,J .M ath. Phys. 5 ,1 7 76( 1 9 6 4 ) . [ 2 . 3 4 ]U .F a n o ,“ Thet h e o r yo fanomalousd i f f r a c t i o ng r a t i n g sando fq u a s i s t a t i o n a r y wavesonm e t a l l i cs u r f a c e s(Sommerfeld’ swaves)”,J. O pt. Soc. A m . 3 1 ,2 1 3 ( 1 9 4 1 ) . [ 2 . 3 5 ]P .C .Waterman,“ S c a t t e r i n gfromp e r i o d i cs u r f a c e s ”,J. A coust. Soc. A m . 5 7 ,7 9 1( 1 9 7 5 ) . [ 2 . 3 6 ]J .J .S e i n ,“ A Noteont h eEwald-OseenE x t i n c t i o nTheorem”,O pt. Com mun. 2 ,170( 1 9 7 0 ) . [ 2 . 3 7 ]G .S .Agarwal,“ S c a t t e r i n gfromRoughS u r f a c e s ”,O pt. Com mun. 1 4 ,1 6 1 ( 1 9 7 5 ). [ 2 . 3 8 ]G .S .Agarwal,“ TheG e n e r a l i z e dEwald-OseenE x t i n c t i o nTheoremandt h e I n t e g r a lEquationi nE l e c t r o m a g n e t i cS c a t t e r i n g ”,O pt. Com mun. 1 8 ,2 38 ( 1 9 7 6 ). [ 2 . 3 9 ]E .L a l o r ,“ An o t eont h eLorentz-Lorentzformulaand t h eEwald-Oseen e x t i n c t i o ntheorem”,O pt. Com m un. 1 ,5 0( 1 9 6 9 ) .

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CHAPTER 2 [ 2 . 4 0 ]N .G a r c i a ,V .C e l l iandM.N i e t o V e s p e r i n a s ,“ Exactm u l t i p l es c a t t e r i n go f wavesfromrandomroughs u r f a c e s ”,O p t .Commun. 3 0 , 279( 1 9 7 9 ) . [ 2 . 4 1 ]D .P .WinebrennerandA .I s h i m a r u ,“ A p p l i c a t i o no ft h ep h a s e p e r t u r b a t i o n t e c h n i q u et orandomlyroughs u r f a c e s ”,J. O pt. Soc. A m . 2 ,2285( 1 9 8 5 ) . [ 2 . 4 2 ]D .WinebrennerandA .I s h i m a r u ,“ I n v e s t i g a t i o no fas u r f a c ef i e l dphasep e r ­ t u r b a t i o nt e c h n i q u ef o rs c a t t e r i n gfromroughs u r f a c e s ”,R ad. Sci. 2 0 ,1 6 1 ( 1 9 8 5 ) . [ 2 . 4 3 ]V .I .T a t a r s k i ,W ave propagation in a turbulent m edium (McGraw-Hill,New Y o r k ,1 9 6 5 ) . [ 2 . 4 4 ]J .A .R a t c l i f f e ,“ Somea s p e c t so fd i f f r a c t i o nt h e o r yandt h e i ra p p l i c a t i o nt o t h ei o n o s p h e r e ”,Rep. Prog. Phys. 1 9 ,1 8 8( 1 9 5 6 ) . [ 2 . 4 5 ]E .E .S a l p e t e r ,“ I n t e r p l a n e t a r ys c i n t i l l a t i o n s ”,A strophys. J. 1 4 7,4 3 3( 1 9 6 7 ) . [ 2 . 4 6 ]E .JakemanandJ .G.McWhirter,“ C o r r e l a t i o nf u n c t i o ndependenceo ft h e s c i n t i l l a t i o nbehindadeeprandomphases c r e e n ”,J. Phys. A:M ath. Gen. 1 0 ,1 5 9 9( 1 9 7 7 ) . [ 2 . 4 7 ]J .W.Goodman,S tatistical O ptics ( W i l e y ,NewY o r k ,1 9 8 5 ) . [ 2 . 4 8 ]B .J .U s c i n s k i ,H .G.BookerandM.M a r i a n s ,“ I n t e n s i t yf l u c t u a t i o n sduet o adeepphases c r e e nw i thapower-lawspectrum”,Proc. Roy. Soc. Lond. A374, 5 03( 1 9 8 1 ) . [ 2 . 4 9 ]H .G .BookerandG .M a j i d i A h i ,“ Theoryo fr e f r a c t i v es c a t t e r i n gi ns c i n t i l l a ­ t i o nphenomena”,J. A tm os. Terr. Phys. 4 3 , 1 1 9 9( 1 9 8 1 ) . [ 2 . 5 0 ]B .J .U s c i n s k iandC .M a c a s k i l l ,“ I n t e n s i t yf l u c t u a t i o n sduet oad e e p l ymod­ u l a t e dphases c r e e n ”,J. A tm os. Terr. Phys. 4 5 , 595( 1 9 8 3 ) . [ 2 . 5 1 ]J .G .WalkerandE .Jakeman, “ Non-gaussianl i g h ts c a t t e r i n gbyar u f f l e d waters u r f a c e ”,O pt. A cta 2 9 , 313( 1 9 8 2 ) . [ 2 . 5 2 ]I .S .GradshteynandI .M.R y z h i k ,Table o f Integrals, Series and P rodu cts (AcademicP r e s s ,NewY o r k ,1 9 8 0 ) . [ 2 . 5 3 ]E .JakemanandP .N .P u s e y ,“ Non-gaussianf l u c t u a t i o n si ne l e c t r o m a g n e t i c r a d i a t i o ns c a t t e r e dbyarandomphases c r e e n ”,J .Phys. A:M ath. Gen. 8 , 369( 1 9 7 5 ) . [ 2 . 5 4 ] M.V .B e r r y ,“ Thes t a t i s t i c a lp r o p e r t i e so fe c h o e sd i f f r a c t e dbyroughs u r ­ f a c e s ”,Phil. Trans. Roy. Soc. 2 7 3 , 6 1 1( 1 9 7 3 ) . [ 2 . 5 5 ]H .F u j i i ,“ C o n t r a s tv a r i a t i o no fnon-gaussians p e c k l e ”,O pt. A cta 2 7 , 4 0 9 ( 1 9 8 0 ) . [ 2 . 5 6 ]P .J .Chan d l e y ,“ Determinationo ft h ea u t o c o r r e l a t i o nf u n c t i o no fh e i g h ton aroughs u r f a c efromc o h e r e n tl i g h ts c a t t e r i n g ”,O pt. Q uant. Elect. 8 ,3 29 ( 1 9 7 6 ) . [ 2 . 5 7 ]J .M.ElsonandJ .M.B e n n e t t ,“ R e l a t i o nbetweent h ea n g u l a rdependence o fs c a t t e r i n gandt h es t a t i s t i c a lp r o p e r t i e so fo p t i c a ls u r f a c e s ”,J .O pt. Soc. A m . 6 9 , 31( 1 9 7 9 ) .

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CHAPTER 2 [ 2 . 5 8 ]P .RocheandE .P e l l e t i e r ,“ C h a r a c t e r i s a t i o n so fo p t i c a ls u r f a c e sb ymeasure­ mento fs c a t t e r i n gd i s t r i b u t i o n ”,A pp. O pt. 2 3 , 3564( 1 9 8 4 ) . [ 2 . 5 9 ]G .P a r r y ,P .N .P u s e y ,E .JakemanandJ .G .McWhirter, “ Thes t a t i s t i c s andc o r r e l a t i o np r o p e r t i e so fl i g h ts c a t t e r e dbyarandomphases c r e e n ”,i n Coherence and Q uantum O ptics Vol.IV, Ed.L .MandelandE .Wolf(Plenum P r e s s ,NewY o r k ,1 9 7 8 ) . [ 2 . 6 0 ]E .JakemanandJ .H .J e f f e r s o n ,“ S c i n t i l l a t i o ni nt h eF r e s n e lr e g i o nbehinda s u b f r a c t a ls c a t t e r e r ”,O pt. A cta 3 1 ,8 53( 1 9 8 4 ) . [ 2 . 6 1 ] M.AbramowitzandI .A .S t e g u n ,H andbook o f m athem atical functions ( D o v e r , NewY o r k ,1 9 7 2 ) . [ 2 . 6 2 ]E .JakemanandJ .G .McWhirter,“ Non-Gaussians c a t t e r i n gbyarandom phases c r e e n ”,A ppl. Phys B 2 6 , 125( 1 9 8 1 ) . [ 2 . 6 3 ]B .Mandelbrot,The fractal geom etry o f nature (W.H.Freeman,SanFran­ c i s c o ,1 9 8 2 ) . [ 2 . 6 4 ] M.V .B e r r y ,“ D i f f r a c t a l s ”,J. Phys. A:M ath. Gen. 1 2 ,7 8 1( 1 9 7 8 ) . [ 2 . 6 5 ]A .Kolmogorov,Turbulence, Classic Papers on S ta tistica l Theory, Ed.S .K. F r i e d l a n d e randL .Topper( W i l e y ,NewY o r k ,1 9 6 1 ) . [ 2 . 6 6 ]C .L .R i n o ,“ Numericalcomputationsf o raone-dimensionalpowerlawphase spectrum”,R ad. Sci. 1 3 ,4 1( 1 9 7 9 ) . [ 2 . 6 7 ]R .S .S a y l e sandT .R .Thomas, “ S u r f a c etopographya san o n s t a t i o n a r y randomp r o c e s s ”,N ature 2 7 1 , 4 3 1( 1 9 7 8 ) . [ 2 . 6 8 ]D .L .J o r d a n ,R.C .H o l l i n sandE .Jakeman,“ Experimentalmeasurementso f non-gaussians c a t t e r i n gbyaf r a c t a ld i f f u s e r ”,A ppl. Phys. B 3 1 , 1 7 9( 1 9 8 3 ) . [ 2 . 6 9 ] D.L .J o r d a n ,R .C .H o l l i n sandE .Jakeman,“ I n f r a r e ds c a t t e r i n gbyaf r a c t a l d i f f u s e r ”,O pt. Com mun. 4 9 ,1 ( 1 9 8 4 ) . [ 2 . 7 0 ]E .L .Church,H .A .J e nkinsonandJ .M.Za v a d a ,“ Measuremento ft h ef i n i s h o fdiamond-turnedmetals u r f a c e sbyd i f f e r e n t i a ll i g h ts c a t t e r i n g ”,O pt. Eng. 16 , 3 60( 1 9 7 7 ) . [ 2 . 7 1 ]E .L .Church,H.A .JenkinsonandJ .M.Z a v ada, “ R e l a t i o n s h i pbetween s u r f a c es c a t t e r i n gandmicrotopographicf e a t u r e s ”,O pt. Eng. 18,125( 1 9 7 9 ) . [ 2 . 7 2 ] D.E .BourneandP .C .K e n d a l l , Vector A nalysis and Cartesian Tensors ( N e l s o n ,Sunbury-on-Thames,1 9 7 7 ) . [ 2 . 7 3 ]E .W. M o n t rollandB .J .Wes t ,“ On ane n r i c h e dc o l l e c t i o no fs t o c h a s t i c p r o c e s s e s ”,i nStudies in S tatistical M echanics Vol.7: F luctuation Phenom ena ,E d.E .W.M ontrollandJ .L .Lebowitz(NorthH o l l a n d ,NewY o r k ,1 9 7 9 ) . [ 2 . 7 4 ]R .J .A d l e r ,T he geom etry o f random fields ( W i l e y ,NewY o r k ,1 9 8 1 ) . [ 2 . 7 5 ] D.W.S c h a e f e randJ .E .M a r t i n ,“ F r a c t a lgeometryo fc o l l o i d a la g g r e g a t e s ”, Phys. R ev. L ett. 5 2 , 2371( 1 9 8 4 ) .

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CHAPTER 3 CHAPTER 3 AN EXPERIMENTAL EXAMINATION OF THE FIRST INTENSITY MOMENT I nt h i sc h a p t e r ,t h ee x p e r i m e n t a ls c a t t e r i n gappar a t u sf o rmeasuringt h es c a t ­ t e r e dmeani n t e n s i t yfromat e s ts u r f a c ei sd e s c r i b e dt o g e t h e rw i t hamethodf o r producingrandomroughs u r f a c e swithp r e s c r i b e ds t a t i s t i c s .F o l l o w i n gat r e a t m e n t o ft h ep r o f i l o m e t r i ca n a l y s i so froughs u r f a c e s ,at e c h n i q u ef o rf a b r i c a t i n gs i n g l e ­ s c a l es u r f a c e si sp r e s e n t e dandi ti sf o l l o w e dbysomee m p i r i c a l l ydetermineds u r f a c e s t a t i s t i c s . Theses u r f a c e sa r et h e nexaminedbyl i g h ts c a t t e r i n gandt h ee x p e r i ­ mentalandt h e o r e t i c a l l yp r e d i c t e dr e s u l t sa r ecompared.Basedont h es i n g l e s c a l e f a b r i c a t i o nt e c h n i q u e s ,s e v e r a lmethodsf o rs y n t h e s i s i n gm u l t i s c a l e( f r a c t a l )s u r f a c e s a r ea l s oexaminedande x p e r i m e n t a ls c a t t e r i n gr e s u l t sa r eg i v e n .F i n a l l y ,some‘ nat­ u r a l l y ’o c c u r r i n groughs u r f a c e sa r eexaminedbothbys u r f a c ep r o f i l i n gandbyt h e i r s c a t t e r i n gs t a t i s t i c s ,andt h er e s u l t sa r ecomparedwithknowns c a t t e r i n gm o d e l s .

3.1 THE EXPERIMENTAL APPARATUS I nt h i ss e c t i o nt h ee x p e r i m e n t a la p p a r a t u s ,u s e dthroughoutChapters3and5 , andi t sgeometryi sd e s c r i b e d .Thed e t e c t o rr e s p o n s ei sd i s c u s s e di n§ 3 . 1 . 2 .

3.1.1 The scattering equipment geometry

Thes c a t t e r i n gi n s t r u m e n td e s c r i b e di nt h i ss e c t i o nwasb u i l tbyF .R e a v e l land t h ec o n t r o le l e c t r o n i c swered e s i g n e dandb u i l tbyA .C a n a s . Botht h eauthorand E .Mendezwrotet h ei n i t i a lassembly-coder o u t i n e st oc o n t r o lt h edataf l o wand werei n v o l v e di nt h eo r i g i n a ld i s c u s s i o n sands p e c i f i c a t i o n so ft h es y s t e m .TheF o r t h l i b r a r yc o n t r o lr o u t i n e su s e dt oc o n t r o lt h eexperimenta tt h emacrol e v e lweremostly w r i t t e nbyE .Mendez. R e f e r r i n gt oF i g u r e3 . 1 ,t h ei n c i d e n c e p l a n es c a t t e r i n gapparatusc o n s i s t s ,e s ­ s e n t i a l l y ,o fasamples t a g et oh o l dt h es u r f a c eunderexaminationandad e t e c t o r arm.Botht h es t a g eandt h earma r ea r r a ngeds ot h a tt h e ycanr o t a t ei n d e p e n d e n t l y aboutacommon,v e r t i c a la x i s . I l l u m i n a t i o ni sp r o v i d e dbyal i n e a r l y p o l a r i s e d5mW S p e c t r aP h y s i c sHe-Ne l a s e r( 6 3 2 . 8nm). Forsomeo ft h eexpe r i m e n t sa( t u n a b l e )Argon-ionl a s e rwithan ouputa t4 5 7 . 9nmi sa l s ou s e d .L i g h tfromt h el a s e ri sf e dv i aap e r i s c o p earrangement o ftwom i r r o r s ,M2 andM3,o n toanotherm i r r o r ,M4,fromwhicht h ebeamt r a v e l s h o r i z o n t a l l yt ot h es u r f a c eundert e s t .A l lo ft h em i r r o r sa r ef r o n t s i l v e r e danda r e a n g l e da t±45°withr e s p e c tt ot h et a ble-normali no r d e rt op r e s e r v et h el i n e a r p o l a r i s a t i o no ft h el a s e rbeam.Toi n c r e a s et h ea r e ai l l u m i n a t e dont h es u r f a c e( s e e b e l o w ) ,t h ebeami sexpandedandc o l l i m a t e du s i n gstandardt e c h n i q u e s . Fort r a n s m i s s i o ne x p e r i m e n t st h i sarrangementi sa d e q u a t e ,butf o rr e f l e c t i o n e x p e r i m e n t s ,andp a r t i c u l a r l ywhend e t e c t i n gb a c k s c a t t e r e dr a d i a t i o n( s e eChapter 5 ) ,i ti simportantt h a tt h em i r r o r ,M4,o b s t r u c t snomoreo ft h ed e t e c t o ra p e r t u r e thani ss t r i c t l yn e c e s s a r y .T h isi sa c h i e v e df o rt h e s ec a s e sbymakingt h em i r r o r M4s onarrowt h a tt h ebeamm u s t ,bef o c u s e do n t oi t .Whilet h i sw i l lminimiset h e

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F i g .3 . 1 Experimentals c a t t e r i n gapparatusshowingthem i r r o r sM1-M4andt h e d e t e c t o rarm.

F i g .3 . 2 Close-upo ft h ed e t e c t o rarm,i l l u s t r a t i n gthea n g l eandp o l a r i s a t i o n t e r m i n o l o g y .

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CHAPTER 3 o b s t r u c t i o no ft h ed e t e c t o rf o rs m a l ld e t e c t i o na n g l e si ts u f f e r sfromt h ed i s a d v a n t a g e t h a td u s to ro t h e rd i r tont h em i r r o rs u r f a c ew i l laddn o i s et ot h ei l l u m i n a t i o n beam.I na d d i t i o n ,i tmeanst h a tt h ebeami sexpandingwheni tr e a c h e st h es u r f a c e , althought h i sh a sbeenfou n dt ohavenoa p p r e c i a b l ee f f e c tont h es c a t t e r e di n t e n s i t y e n v e l o p e( s e eChapter5 ) . T y p i c a l l y ,ac i r c u l a ra r e aont h es u r f a c eo fabout2cmd i a m e t e ri si l l u m i n a t e d i no r d e rt ominimises p e c k l en o i s ei nt h ed e t e c t o r( s i n c et h es p e c k l es i z ei si n v e r s e l y p r o p o r t i o n a lt ot h ei l l u m i n a t e da r e a— s e e§ 3 . 3 . 1 ) .A d d i t i o n a l l y ,f u r t h e ra v e r a g i n g o ft h es p e c k l e si sa c h i e v e dbyp l a c i n ga15mmd i a m e t e rf i e l dl e n s ,whiche f f e c t i v e l y i n t e g r a t e so v e raf i n i t es o l i da n g l e ,i nf r o n to ft h ed e t e c t o r .T h i sa l s og u a r a n t e e s t h a tt h ed e t e c t o r ,whichi sp l a c e da tt h el e n sf o c u s ,v i e w st h ee n t i r ei l l u m i n a t e d a r e ai r r e s p e c t i v eo ft h ea n g l eo fd e t e c t i o n .Thea c t u a ls p e c k l es i z e( d i a m e t e r ) ,2 / 5 , a tt h ed e t e c t o rp l a n eproducedbyaGaussiani n t e n s i t yp r o f i l eont h et e s ts u r f a c e ( s e e§ 3 . 3 . 1 )i sg i v e nby n / 2A 2(3 = ( 3 . 8 ) 7r tan0 ’ wheret a n6 i st h eh a l f a n g l esubtendedbyt h ei l l u m i n a t e da r e aa tt h ed e t e c t o r .Us­ i n gt h eparameterso ft h ee x p e r i m e n t ,about1000 s p e c k l e sf i l lt h ef i e l dl e n sa p e r t u r e . S i n c et h es p e c k l en o i s ei sp r o p o r t i o n a lt ol / \ / / V ,whereN i st h enumbero fs p e c k l e s , t h ec o n t r i b u t i o no fs p e c k l en o i s et ot h es i g n a lw i l lbes m a l l .Notet h a ti n t e g r a t i n g o u tt h es p e c k l en o i s ei nt h i swayt r a d e so f fa n g u l a rr e s o l u t i o nwitha c c u r a c y ;i ti sob­ v i o u s l yd e s i r a b l et h a tt h ere d u c e da n g u l a rr e s o l u t i o n ,cau s e dbyu s i n gt h ef i e l dl e n s , i ss u f f i c i e n tt ominimiset h en o i s ey e ts t i l lb e i n gc a p a b l eo fr e s o l v i n ganys t r u c t u r e p r e s e n ti nt h es c a t t e r e df i e l d .F i n a l l y ,w i t ht h ed i s t a n c ebetweent h esamples t a g e andt h ed e t e c t o ra p e r t u r eb e i n g62cm,t h ed e t e c t e di n t e n s i t yr e a d i n g sa r ee n s u r e d o fb e i n gi nt h ef a r f i e l dr e g i o n 3,1 o ft h es u r f a c em i c r o s t r u c t u r e . A Hammamatsu R647 head-onp h o t o m u l t i p l i e ri susedt od e t e c tt h ev i s i b l e r a d i a t i o n .I t soutputi sf e dv i aa1 2 b i tADC11 a n a l o g u e / d i g i t a lc o n v e r t e ri n t o anMDB LSI11/23computerf o rs t o r a g eanda n a l y s i s . Thecomputer,whichi s programmedi nt h eForthl a n g u a g e ,i sa l s ou s e dt oc o n t r o lt h er o t a t i o no ft h esample s t a g eandd e t e c t o rarma sw e l la st h eo r i e n t a t i o no ft h el i n e a rp o l a r i s e rp l a c e di n f r o n to ft h ed e t e c t o r .T h i sa l l o w sf o rt h ed e t e c t i o no fe i t h e rt h ep a r a l l e lo ro r t h o g o n a l p o l a r i s a t i o ncomponentso ft h es c a t t e r e dl i g h t .Thet e r m i n o l o g y ,s andp ,u s e dt o d e s c r i b et h el i n e a rp o l a r i s a t i o no ft h ei n p u to rd e t e c t e df i e l di sshowni nF i g u r e 3 . 2 .S p o l a r i s a t i o nr e f e r st ot h ec a s ewhent h ee l e c t r i c f i e l do s c i l l a t e sp e r p e n d i c u l a r t ot h ep l a n eo fi n c i d e n c eandp p o l a r i s a t i o nt owheni to s c i l l a t e sp a r r a l l e lt ot h e i n c i d e n c ep l a n e . Thei n c i d e n c ea n g l eo ft h ei l l u m i n a t i o nbeami ss e t ,i np r a c t i c e ,bya p p r o p r i a t e l y r o t a t i n gt h esamples t a g et ot h ed e s i r e da n g l ew h i l s tt h ei n c i d e n tp o l a r i s a t i o n ,so rp , i ss e tbyr o t a t i n gt h eHe-Nel a s e r .T h isi sc h e c k e dbyt e s t i n gt h eoutputp o l a r i s a t i o n withana n a l y s e ri nf r o n to ft h esamples t a g e .Thes c a t t e r e dp o l a r i s a t i o n ,o fwhich al i n e a rcomponentmaybed e t e c t e de i t h e rp a r a l l e lo rp e r p e n d i c u l a rt ot h ei n c i d e n t p o l a r i s a t i o n ,i sdeterminede i t h e rbycomputerc o n t r o l(whichr o t a t e st h ea n a l y s e r ) o r ,i fa l lp o l a r i s a t i o n sa r et obed e t e c t e d ,byp h y s i c a l l yremovingt h ep o l a r i s e rfromi n f r o n to ft h ed e t e c t o r .Thed e t e c t o rarmt h enr o t a t e saboutt h esamples t a g et a k i n g

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CHAPTER 3 r e a d i n g so ft h emeani n t e n s i t ye n v e l o p e . Eachr e a d i n gs t o r e di nt h ecomputeri s , i na d d i t i o nt ot h es p e c k l ea v e r a g i n g ,a v e r a g e do v e rt e nc o n v e r s i o n sfromt h eA/D c o n v e r t e r .

3.1.2 The detector response F o rt h ee x p e r i m e n t sd e s c r i b e di nt h i st h e s i s ,i ti ss u f f i c i e n ts i m p l yt oknowt h a t t h ed e t e c t o rr e s p o n s ei sapproximatelyUnearo v e rt h eran g eo fi n t e n s i t i e su s e d , r a t h e rthant oknowt h ea b s o l u t ed e t e c t e di n t e n s i t y .I nt r a n s m i s s i o ne x p e r i m e n t s , h o w e v e r ,wheret h ei n t e n s i t ydynamicr a n g ei sl a r g e( s i n c ei tmustc o v e rt h er a n g e fromn e a r s p e c u l a rt oh i g h a n g l es c a t t e r e dl i g h t )i ti sp a r t i c u l a r l yimportantt o knowo v e rwhatran g eo fi n t e n s i t i e st h ed e t e c t o rmaybet r u s t e d . Thel i n e a r i t yo ft h ed e t e c t o rr e s p o n s et oani n p u tl i g h ti n t e n s i t ywasexamined o v e rt h ef u l ldynamicr a n g ea f f o r d e dbyt h e1 2b i tA/Dc o n v e r t e ru s i n gKodak‘ Wratt e n ’n e u t r a ld e n s i t yf i l t e r s .Althought h e i ro p t i c a ld e n s i t i e swerec a hbratedw i t ha p h o t o d e n s i t o m e t e r ,t h e‘ e f f e c t i v e ’t r a n s m i t t a n c eo ft h e s ef i l t e r si nt h eo p t i c a lsystem i nwhicht h e ya r et obeu s e dmayd i f f e rbya smucha s2t o3%3 , 2fromt h e i rc a l i b r a t e dv a l u e sa sar e s u l to fs c a t t e r i n go ft h et r a n s m i t t e dl i g h tandi n t e r r e f l e c t i o n s . Others o u r c e so fe r r o rmaya l s obei m p o r t a n t .Forex a m p l e ,l o w l e v e lf l u c t u a t i o n s ca u s e dbyanu n s t a b l el a s e routputo rfromd e t e c t o rn o i s ew i l lbei n s i g n i f i c a n tf o r h i g hl i g h tl e v e l sbutf o rl o wi n t e n s i t i e s ,t h e ymayr e p r e s e n tas i g n i f i c a n tp e r c e n t a g e o ft h et o t a ls i g n a l .I na d d i t i o n ,i twasfoundt h a twithnoi n c i d e n ti l l u m i n a t i o n , as l i g h t l yf l u c t u a t i n gd . c .b i a swasd e t e c t e d .I ti sr e a d i l yapparentt h a tt h ee r ­ r o ro b t a i n e dfromt h es u b t r a c t i o no ft h emeans i g n a lfromt h ed e t e c t e ds i g n a lw i l l i n c r e a s e sa st h es i g n a ls t r e n g t hi sd e c r e a s e d . L i n e a r i t yt e s t i n gwasdonebyi n s e r t i n gj u s ts u f f i c i e n to p t i c a ld e n s i t yi n t ot h e beamt ob r i n gt h ed e t e c t o routputc l o s et os a t u r a t i n gt h eA/D c o n v e r t e r .T h i s p r a c t i c ed o e sn o thaveany‘ b l e a c h i n g ’e f f e c tont h ep h o t o m u l t i p l i e rwhichmight bee x p e ctedt or e d u c ei t ss e n s i t i v i t y . Thec a l i b r a t e df i l t e r sweret h eni n s e r t e d , i n d i v i d u a l l yo ri nc o m b i n a t i o n ,andt h ed e t e c t e di n t e n s i t ymeasured. I no r d e rt o re d u c el o c a le f f e c t sc a u s e dbyanys c r a t c h e so rd i r tont h es u r f a c e ,al a r g ea r e ao f t h ef i l t e rwasi l l u m i n a t e d . Eacho ft h er e a d i n g swasa v e r agedo v e r8s a m p l e s ,and t h emeansandt h e i rd e v i a t i o n sr e c o r d e d .I fal i n e a rd e t e c t o rr e s p o n s ei sassumed t h e n ,f o rag i v e nf i l t e rd e n s i t y ,t h ec a l i b r a t i o no ft h ef i l t e r sa l l o w sane s t i m a t et obe madeo ft h eexpectedd e t e c t e di n t e n s i t ywhichcanbecomparedwitht h a ta c t u a l l y o b t a i n e d .Inp u t o u t p u tp l o t scant h enbedrawn( F i g s .3 . 3and3 . 4 )t oc h a r a c t e r i s e t h er e s p o n s e . F i g u r e3 . 3showst h a tt h ed e t e c t o rr e s p o n s ei sapproximatelyl i n e a ro v e rt h e1 2 b i trangeo ft h eA/Dc o n v e r t e r .Forlowi n t e n s i t i e s ,h o w e v e r ,F i g .3 . 4showsas l i g h t l y n o n l i n e a rr e s p o n s ewhichappearst or e d u c et h es e n s i t i v i t yo ft h ed e t e c t o r .W h i l s t t h i smaybeag e nuinef e a t u r eo ft h er e s p o n s e ,s e v e r a lf i l t e r shadbeen‘ sandwiched’ t o g e t h e ri no r d e rt or e d u c et h ei n t e n s i t ys u f f i c i e n t l y .I n c r e a s e ds c a t t e r i n gmightt h e n bee x p e c t e dt oaccountf o rsomeo ft h eo b s e r v e dd i p si nt h ed e t e c t e di n t e n s i t y .I n c o n c l u s i o n ,i twasassumedt h a tt h ed e t e c t o rhadal i n e a rr e s p o n s eo v e rt h erangeo f i n t e n s i t i e sf o rwhichi twast obeu s edt h u so b v i a t i n gt h eneedf o r‘ look-upt a b l e s ’ .

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Fig. 3.3

Input/output response of the 12-bit detector for high light levels.

Output

Fig. 3.4 Input/output detector response for low light levels.

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CHAPTER 3

3.2 THE PREPARATION AND ANALYSIS OF PHOTORESIST SUR­ FACES

In order to enable meaningful comparisons to be made between theory and exper­ iment, it is necessary that the experimental conditions match as closely as is possible the requirements of the theory. One problem in the field of light scattering, and one which appears to have been recognized only comparatively recently3,3-3,5, is that the surfaces used in previous experiments have usually failed to fit the surface models most commonly used in the theories. For example, experiments have been conducted using ground glass surfaces3,6 and gold-coated sandpaper3,7. In both cases, serious discrepancies were reported between the experimentally observed results and the theoretical predictions which were based on commomly made assumptions about the surface statistics. In only a few cases3,4,3,8,3,9 have the statistical properties of the surfaces actually been determined; in most, they have been assumed3,10,3,11. As an important step in overcoming this problem, Gray3,12 describes a technique for producing random rough surfaces which, at least in principle, can have prescribed statistical characteristics. Exposing photoresist-coated plates to laser speckle pat­ terns, he successfully produced jointly Gaussian surfaces with Gaussian autocorrela­ tion functions and controlled correlation lengths. Once made, the surfaces are then characterised by a mechanical stylus device, whose operation is described in §3.2.2.

3.2.1 The fabrication of rough surfaces in photoresist

Using the method first described by Gray3,12, and used successfully since then by Levine Sz Dainty 3,4 and later by Berry3,9, rough surfaces were made with prescribed statistical characteristics for the experiments of Chapters 3 and 5 . The method is essentially this: following a stringent cleaning procedure (a de­ tailed description of the entire process is given in Appendix 3.1), (50mmx50mm) glass plates are spin-coated with high resolution Shipley 1375 Microposit positive photoresist. Left to dry for approximately 24 hours, they are baked at 90° Celsius for 20 minutes. Depending on the coating thickness required (one photoresist ap­ plication gives a thickness of about 5 microns), the plates are either given a further coating or are ready to be used. Most of the plates used in this work were given either two, or three, coatings which resulted in a net layer thickness of, typically, about 1 0 /zm. In order to avoid interference fringes being recorded in the photoresist by the in­ tense exposing beam as a result of internal reflections within the plate, the uncoated face is painted matt black. The prepared plates are then exposed to speckle patterns using the 457.9nm green line of an Argon-ion laser (note that the photoresist sen­ sitivity increases with frequency through the visible spectrum). Typical exposure times were about two hours, but the precise details of the exposures together with the exposure geometry are discussed in detail in later sections. Following exposure, the plates are developed in Shipley photoresist developer and washed gently under warm water. This yields a surface whose height is, ideally, linearly proportional to the exposure received. It should be stressed that although this technique appears relatively straightforward, it requires both skill and patience if good results are to be obtained. Some of the problems that may lead to imperfectly 86

CHAPTER 3 coated surfaces are described in Appendix 3.1.

3.2.2 The analysis of random rough surfaces

Having described a method for producing random rough surfaces with prescribed statistics, a mechanical stylus profiling instrument, the Talysurf, is used to examine the exposed photoresist surfaces. It consists of a small (approximately 1/im diame­ ter) truncated-pyramid diamond stylus and a transducer, which converts the vertical oscillations of the stylus as it is dragged over the surface into an analogue electrical signal. This signal is amplified and fed, via a 12-bit analogue/digital converter, into a PDP11 computer where the digitised profile is stored for later analysis. In order for an analysis of the recorded profile to be carried out, it is generally assumed that the instrument faithfully records the surface features. This assumption and its implications are examined below. 3.2.2.1 The effect of the stylus on the recorded profile

There are three main reasons why the stylus may lead to an unfaithful repre­ sentation of the surface: • The finite size of the stylus imposes a bandwidth limit on the measurements and acts to filter out high frequencies. This is caused, for example, by poor valley penetration, especially if surface slopes are high (see Figure 3.5). • Because the stylus is constrained to move in only the vertical direction, it will be unable to record reentrant features (undercuts etc.). This is of lesser importance for the photoresist surfaces, as the nature of their construction all but eliminates this type of feature. It is important, however, for ground glass surfaces (see §3.5), whose method of manufacture tears up the glass surface to produce a highly reentrant, and characteristic, ‘Autumn leaves’ profile when viewed by SEM (Scanning electron microscope). Plates 1 and 2 show SEM micrographs of a smooth surface and ground glass respectively. • There may, in addition, be the problem of surface damage caused by the very act of dragging the diamond-tipped stylus across the surface. This can either knock surface asperities off or else ‘plough’ through the surface top-layer, leaving a straight-line trail. In both cases, the recorded profile will be altered. The latter effect is apparent both with photoresist and with ground glass surfaces3,13 although there does not appear to be any way of eliminating it completely. 3.2.2.2 The effect of the sample length on the recorded profile

It might, naively, be thought that the longer the sampling length on the sur­ face, the more accurate will be the statistically determined surface parameters after analysis. Whilst there will certainly be more data, it should be realised that these parameters are generally functions of the bandwidth of the measurement (discussed above) and of the subsequent analysis, rather than being intrinsic properties of the surface. In practice, once the bandwidth of interest has been decided on, the appro­ priate sampling length for analysis may be chosen. In order to understand this more intuitively, it is important to consider how the surfaces are analysed and, in particular, how the mean-line is fitted. This will then show how the higher frequency components of the surface may be increasingly masked as the sampling length itself is increased. This is discussed further below. 87

CHAPTER 3

Fig. 3.5

Illustration of poor valley penetration by a stylus profilometer.

Fig. 3.6 Probability distributions in the presence of positive (S > 0) and negative (S < 0) skewness. Also shown is the normal distribution, for which 5 = 0.

8 8

CHAPTER 3 3.2.2.3 The fitting of the mean-line to the profile data

Before any analysis of a surface profile can be done, a mean-line must be fitted to the trace in order to remove the effects of tilt etc. The most commonly used method is to equate the areas above and below the mean using a least-squares technique. This will remove any gross tilt effects or waviness, but may still be unsatisfactory. For example, if we are interested in some small-scale roughness which has been imposed on a low-frequency sinusoidal wave, the best mean will follow the gentle curve of the sinusoid and will not be the straight least-squares mean-line. In this case, the fitting of a higher order mean by, for example, a cubic spline technique would be more appropriate than using a straight line. This is equivalent to highpass filtering the data. Thus the way in which the data is to be analysed will be determined by the bandwidth of interest and the type of surface substrate. An alternative approach is to divide the sample length into a number of equallength portions for each of which a separate mean-line is calculated, such that the resulting mean-line segments need not join up. The individually determined surface parameters are then simply averaged. This is known as (high pass) graphical filtering, and it is similar to the higher-order method described above. In the case of the surfaces examined in this thesis, both the ground glass and the photoresist-coated surfaces lie on flat, glass substrates. Although we are primarily interested in the high frequency undulations on these surfaces, the flat substrate largely eliminates the need for the higher-order analysis described above. The sam­ ple may still be subject to tilt, but this will be correctly compensated for by the straight mean-line (least-squares) approach outlined above. If an examination of the high frequencv part of the spectrum of a genuine, ‘naturally’ rough surface is re­ quired, however, it would be prudent to apply one of these higher-order techniques. In our examples, 3000 points laterally separated by 2pm were used for each profile. In conclusion, the successful analysis of random profiles requires both that the sampling length be sufficiently long so as to be representative of the process and, in addition, that the portion to be analysed is of the correct length so that bandwidth of interest is included. These effects are discussed in some detail by Ward3-14.

3.2.3 Statistical parameters determined from the Talysurf profile data

Using a suite of programs developed by the author to analyse the Talysurf profile data, the surface can be characterised by its height variance, probability density function (pdf) and autocorrelation function (discussed in Chapter 1 ). In addition, the structure function and the power spectrum are also sometimes used. Their calculation, together with two additional parameters, the skewness and the kurtosis, is described below. • The probability density function (pdf) (defined in §1.2.2) is the probability density that the height of the function, h(x), lies in the interval x to x -f dx. The interval between the height separations is termed the class interval, the number of which are used being a trade-off between resolution and accuracy3*15. The results presented in this work use 20 intervals. In addition to the empirically determined pdf, a theoretical normal distribution is also fitted for easy comparison. 89

CHAPTER 3 • The skewness is the normalised third order moment of the pdf and is defined by 5 =

/

< x* > < x 2> y *

oo £3

p(x)dx / (3.2)

/

< x 2> 2

-oo

where a and p(x) are defined as above. The Gaussian distribution has a kurtosis of K = 3. Those surfaces for which K < 3 are called pla.tykurtic, whilst those having K > 3 are known as leptokurtic (see Figure 3.7). • Both the autocorrelation (acf) and structure functions are calculated exactly as in §1.2.7. The power spectrum, G( uj), which is simply the Fourier transform of the acf, c(x), and is defined by

/

oo

c{x) exp(?u;:r) dx,

(1*3)

-oo

is, in practice, evaluated by Fast Fourier Transform (FFT) methods. Care must be taken, however, in order to ensure that the spectrum obtained is representative of that part of the input in which one is interested. By way of illustration, Figure 3.8 shows a typical autocorrelation function ob­ tained from a Talysurf trace of a piece of ground glass, which shows how, after a lag many times greater than the correlation length (« 6 /xm), the correlation function still hasn’t decayed to zero. This is usually a consequence of low frequency roughness and/or the non-stationarity of the rough surface. From the perspective of light scattering, however, we are usually more interested in the initial decay of the correlation function, as this gives the largest contribution to the scattering integrals (see §2.4.3), particularly if the phase variance, 3) probability distributions.

Fig. 3.8 Autocorrelation function of a ground glass sample from a Taylsurf pro­ file. 91

CHAPTER 3 function3,17. Instead, the acf is multiplied by a smooth function, or -lag window, which decays to zero at a suitable point. This should be chosen so that it will not introduce ‘ringing’ effects into the region of the spectrum in which we are interested. A particularly simple form of lag window has been used to produce the spectra pre­ sented in this work. Straightforward both to apply and to analyse, the cosine ‘bell’ can be shown (see Appendix 3.2) to have no effect on the higher frequency part of the spectrum (which corresponds to small acf lags). Finally, it remains to be decided where to truncate the correlation function. This is largely a matter of judgement and is decided separately for each function that is to be transformed. The FFT algorithms allow the number of sampling points to increase in powers of two, i.e. 16, 32, 64 points etc. Since the points making up the acf are separated by 2 ^m intervals (the minimum horizontal Talysurf step size used in this work), the correlation-space sampling range will correspond to either 32, 64 or 128/zm respectively.! Figures 3.9(a)-(c) show three spectra, of 16, 32 and 64 points, produced from the same acf and using the cosine ‘bell’ window. Bearing little similarity to one another, only prior knowledge that the correlation length for this example was approximately 30/im would suggest that Figure 3.9(a) most closely resembles the spectrum of the initial region of the acf.

3.3 LIGHT SCATTERING FROM SINGLE-SCALE SURFACES I nt h i ss e c t i o n ,t h ef a b r i c a t i o nands c a t t e r i n gfroms i n g l e s c a l e ,G a u s s i a n c o r r e l a t e ds u r f a c e si sd e s c r i b e d . Usingt h et h e o r yd e velopedi nChapter2 ,t h ee x ­ p e r i m e n t a lr e f l e c t i o ns c a t t e r i n gr e s u l t sa r ecomparedwitht h ep r e d i c t i o n so fBeck­ mann’ st h e o r y ,u s i n gf o rt h es t a t i s t i c a ls u r f a c ep a r a m e t e r s ,t h ev a l u e so b t a i n e dfrom t h eT a l y s u r fa n a l y s i so ft h es u r f a c e s .T h i sworko r i g i n a l l ys t a r t e da sac o l l a b o r a t i v e v e n t u r ebetweent h ea u t h o r ,E .MendezandK.O’ Donnell(andwasp r e s e n t e da tt h e 0 .M.P .S .A .c o n f e r e n c e )f ,acommoni n t e r e s tb e i n gt ocomparet h ep r e d i c t i o n s o fBeckmann’ st h e o r ywithe x p e r i m e n t a lr e s u l t so b t a i n e dfromw e l l c h a r a c t e r i s e d s u r f a c e s . Althought h ec o n c l u s i o n sdrawnfromt h i sworkmust,n a t u r a l l y ,o v e r l a p witht h o s eo ft h eauthor’ sc o l l e a g u e s ,t h e yr e p r e s e n ts o l e l yt h eauthor’ sowne f f o r t s . Theremaindero ft h eworkp r e s e n t e di nt h i ss e c t i o nw a s ,h o w e v e r ,pursuede n t i r e l y i n d e p e n d e n t l y .

3.3.1 The synthesis of single-scale surfaces

As we have already discussed in §3.2, many of the experiments used to examine the scattering statistics from rough surfaces and diffusers have made assumptions about the statistical properties of the surfaces. With few exceptions3,18, they usu­ ally assume joint Gaussian statistics and, in many cases, a Gaussian autocorrelation function (acf). The former assumption is made both because it is mathematically f In principle, it is possible to interpolate between the acf sampling points so that the number of spectrum points can be made arbitrarily large. Using polynomial interpolation, however, Berry3,9 discovered that the resulting spectrum was too sensitive to the polynomial order to be of practical use. X International Conference on O p tic a l an d M illim eter W ave P ro p a g a tio n a n d S c a tte rin g in th e A tm o sp h e re , Florence, Italy, May 27,-30, 1986 92

CHAPTER 3 Figs. 3.9(a)-(c) Examples of power spectra, G(co), taken from the acf of Fig. 3.8 using respectively (a) 16, (b) 32 and (c) 64 points and a cosine ‘bell* lag-window.

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CHAPTER 3 1

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