Fractality of speckle intensity fluctuations

Fractality of speckle intensity fluctuations

Dmitry A. Zimnyakov and Valery V. Tuchin

Coherent-light diffraction on random phase screens with fractal properties leads to the formation ofspeckle patterns with peculiarities in correlation characteristics in the small-scale region. Such pecu-liarities are manifested in asymptotic behavior in intensity autocorrelation and structure functions in thevicinity of the zero values of their arguments. Intensity fluctuations in the far and the near diffractionzones are also characterized by values of fractal ~Hausdorff–Besicovitch! dimensions DHB, differing fromthe corresponding Euclidean dimension. Relationships between the exponential factors of the structurefunctions of boundary field phase and scattered-light intensity fluctuations as well as between values ofDHB have been obtained as a result of speckle-formation analysis for different conditions. Their depen-dencies on the illumination and observation conditions obtained in experiments with fractallike scatter-ers ~rough glass plates! are in satisfactory agreement with theoretical results. © 1996 Optical Societyof America

Key words: Diffraction, speckles, intensity structure function, Hausdorff dimension.

1. Introduction

Coherent- and incoherent-light diffraction on differ-ent types of fractal structure has been a subject ofsignificant research in scattering theory and experi-mentation in the last two decades.1–3 Coherent-beam scattering by objects with only phasetransmittance and special properties of boundaryfield phase distributions ~nondifferentiability, thepower-law character ofWiener spatial spectra, etc.! isone of the most attractive and important directions inthis field of investigation. One reason for such anassertion is that this problem is closely connectedwith many applications in measuring technologiesincluding noncontact profilometry of smooth and ul-trasmooth surfaces.4 More widely, coherent-lightanalysis of the evolution of surface- and volume-scat-tering structures ~e.g., aggregation of single elementsthat make up the structure! can also be an example ofthe practical application of the research.5 On theother hand, investigations of relationships betweenthe fractal properties of the scattered field and thecharacteristics of phase distributions can be usefulfrom the viewpoint of understanding better differentnonlinear processes ~e.g., simulation techniques innonlinear dynamic and oscillation research!.One most general approach to the description of

The authors are with the Department of Optics, Saratov StateUniversity, 83 Astrakhanskaya, Saratov 410071, Russia.Received 4 October 1995.0003-6935y96y224325-09$10.00y0© 1996 Optical Society of America

spatially distributed random values of phase f~x, y!with fractal properties is using a corresponding struc-ture function6:

Df~tx, ty! 5 ^@f~x 1 tx, y 1 ty! 2 f~x, y!#2&, (1)

where averaging ^ & is carried out over the possiblerealizations of phase distributions; it is proposed thatthese distributions are statistically homogeneous andisotropic. The phase structure function is closelyconnected with the autocorrelation function Rf~tx, ty!of phase fluctuations f̃~x, y!:

f̃~x, y! 5 f~x, y! 2 ^f~x, y!&, Df~tx, ty!

5 2Rf~0, 0! 2 2Rf~tx, ty!. (2)

All the topological peculiarities of boundary fieldphase distributions ~such as self-similarity or fractal-ity! are manifested in the asymptotic behavior ofDf~tx, ty!. If the fractal ~Hausdorff–Besicovitch! di-mension DHB of the one-dimensional ~1D! realizationof f̃~x, y 5 const! is equal to a corresponding Euclid-ean dimension ~unity!, a conventional differentiateddistribution of f̃~x, y! takes place and Df~tx, ty! forsmall values of ur̄u 5 ~tx

2 1 ty2!0.5 can be expressed as

Df~ur̄u! ' ur̄u2.For fractal distributions a more generalized form

with a similar expression takes place6:

Df~ur̄u! < LT22nur̄un, (3)

where exponential factor n changes between 0 and 2,parameter LT, called topothesy, in combination withthe second term provides the quadratic character of

1 August 1996 y Vol. 35, No. 22 y APPLIED OPTICS 4325

Df~ur̄u!. The value of n depends on DHB of the arbi-trarily selected 1D phase realization @the arbitrarysection of a two dimensional ~2D! distribution# as

n 5 2~2 2 DHB!. (4)

Power-law spatial power spectra are characteristicof distributions with fractal properties7:

S~ fx! 5 Kafxa (5)

for one-dimensional or corrugated distributions and

S~ f̄ ! 5 [email protected]~a 1 0.5!G21~0.5!G21~ay2!# (6)

for 2D isotropic distributions.In these expressions fx is the spatial frequency cor-

responding to the arbitrarily selected direction andG~a! is the g function. Parameter a changes be-tween 21 and 23 and is directly connected with thevalue of exponential factor n: uau 5 n 1 1.Note that pure fractal distributions demonstrating

these properties for the entire range of spatial scales~from zero to infinity! are rather ideal ~mathematical!objects. All natural as well as artificial scatteringstructures with self-similarity must be examined asbandlimited or physical fractals.8 The propertiesdiscussed above are observed for similar objects in afinite region of spatial scales ~or for a given band ofspatial frequencies!.8 The width of such an intervalof fractality in the high-spatial-frequency region com-pared with the used spatial resolution determines thepeculiarities of structure and autocorrelation func-tion behavior in the vicinity of the zero values of theirarguments.The relationships between the parameters of the

structure functions of boundary field phase distribu-tions and coherent-scattered-light intensity fluctua-tions have been investigated for random phasescreens demonstrating the fractal properties in theregion of small spatial scales. Different conditions ofphase-screen illumination and observation of scat-tered-light intensity fluctuations have been takeninto consideration. Examples of natural objectsshowing fractallike properties in coherent-light scat-tering ~such as some biological tissues! are discussed.

2. Analysis of Speckle Pattern Intensity Fluctuations inthe Paraxial Region of the Diffraction Zone

Let us consider the moving random phase screen~RPS! illuminated by a coherent Gaussian beam; thedirection of the screen movement is perpendicular tothe beam axis. Intensity fluctuations are detectedin the paraxial region of the observation plane. Thescattering object ~the RPS! is proposed with Gaussianfirst-order statistics of the boundary field phase fluc-tuations f~x, y! and is characterized by certain typesof phase-fluctuation autocorrelation Rw~tx, ty! andstructure @Df~tx, ty!# functions. As mentioned aboveaveraging is carried out over the ensemble of all pos-sible realizations of f~x, y!. It is proposed that theRPS be statistically homogeneous and isotropic.The complex amplitude of scattered light in the ob-servation on-axis point, depending on the current

4326 APPLIED OPTICS y Vol. 35, No. 22 y 1 August 1996

RPS position, can be expressed as9

U~j, h! 5 *2`

1`

dy *2`

1`

F~x 2 j, y 2 h!A~x, y!dx, (7)

whereF~x, y! is a spatial distribution of the boundaryfield complex amplitude and A~x, y! is the aperturefunction. In the case of RPS illumination by a broadcollimated beam with a uniform amplitude distribu-tion, F~x, y! is equal to exp@ jf~x, y!#. An analyticalform of the aperture function A~x, y! is determined bythe complex amplitude distribution of an illuminat-ing beam in the screen plane and the observationconditions ~particularly, by distance Z between theRPS and the observation plane!. If speckles are an-alyzed in the far zone and the RPS is placed in thewaist plane, A~x, y! has the simplest form: A~x, y! 5A0 exp@2~x2 1 y2!yw0

2#, where A0 is the on-axis am-plitude value and w0 is the waist radius.The current intensity I~j, h! in the detector plane is

equal to U~j, h!U*~j, h!, where * denotes the complexconjugation. In a further analysis the asymptoticproperties of structure function DI~tx, ty! and spatialpower spectrum WI~vx, vy! of the intensity fluctua-tions are taken into consideration. The paraxial re-gion intensity I~j, h!, depending on the current RPSposition ~j, h!, can be studied as a 2D random sta-tionary process, and the main question is: If phasedistribution f~x, y! demonstrates the fractal behav-ior, how are the asymptotic characteristics of DI~tj,th! and WI~vj, vh! connected with the similar char-acteristics of f~x, y! for given observation and detec-tion conditions? Using the properties of Fouriertransformation, we can express the spatial Fourierspectrum of I~j, h! as

FI~vj, vh! 5 @FF~vx, vy!FA~vx, vy!#

^ @FF~vx, vy!FA~vx, vy!#*, (8)

where Ff~vx! denotes the Fourier transform of func-tion f ~x! and R denotes the convolution.So the behavior of WI~vj, vh! in the region of high

spatial frequencies @as well as asymptotic propertiesof DI~tj, th! for small values of ur̄u 5 ~tj

2 1 th2!0.5# is

determined by the relationship between the charac-teristics of the spatial spectra of F~x, y! and the ap-erture functionA~x, y! ~particularly their halfwidths!.Three typical extreme cases are examined below:

~1! Observation in the far diffraction zone of inten-sity fluctuations for fully developed speckle patterns.In this case illumination of the scatterers by thebroad Gaussian beam is used and DuFfu .. DuFAu.~DuFfu and DuFAu are halfwidths of the spatial spectraof the boundary field and the aperture function, re-spectively.!

~2! Placement of the RPS in the waist plane of theilluminating beam with a small value of the Rayleighrange. The detector is positioned in the Fraunhoferzone. The formation of partially developed far-zonespeckle patterns takes place and DuFfu ,, DuFAu.

~3! Observation of the intensity fluctuations of thescattered field in the paraxial region of the Fresnelzone for Gaussian-illuminating beams. In this caseDuFAu is much larger than DuFfu if Z 3 0. ~Z is thedistance between the RPS and the observationplane.!

A. Far-Zone Speckle Formation for Narrowband ApertureFunctions

The extreme case of broad coherent-beam illumina-tion mentioned above, allowing the representation ofthe spatial spectrum of the aperture function as a 2Dd-function ~comparedwith the broadband spectrum ofboundary field fluctuations!, leads to the next equa-tion for FI~vj, vh!:

FI~vj, vh! 5 @FF~vx, vy!d~vx, vy!#

^ @FF~vx, vy!d~vx, vy!#*. (9)

To determine how the fractal properties of f~x, y!are connected with the similar characteristics of thefar-zone intensity fluctuations for this particularcase, the asymptotic behavior of FI~vj, vh! can bestudied analytically, but we can take into consider-ation another approach based on the direct analysisof scattered-light correlation properties.10The relationship between exponential factors nI

and nf can be obtained by using the relationshipbetween the structure function of the phase fluctua-tions of the boundary field and its transverse coher-ence function and, on the other hand, the relationshipbetween the last function and the autocorrelationfunction of the far-zone intensity fluctuations. Thecase of fully developed speckles in the far diffractionzone ~which corresponds to a Gaussian unboundedphase screen illuminated by a collimated broadbeamwith a uniform amplitude distribution! is proposed.The first relationship can be written in form10

Gv~r̄! 5 [email protected]~r̄!#, (10)

where Gv~r̄! 5 ^v~r̄1!v*~r̄2!& is a transverse coherencefunction of the boundary field. It is proposed thatthe amplitude of the illuminating beam be equal tounity. Following Ref. 10, we can express the auto-correlation function of the far-zone intensity fluctua-tions as

RI~r̄! 5 uGv~r̄!u2 2 uv0u4, (11)

where v0 is the mean value of the boundary fieldamplitude. Using the relationship between DI~r̄!and RI~r̄!, we can obtain

DI~r̄! 5 2@uGv~0!u2 2 uGv~r̄!u2#. (12)

In the case of the Gaussian phase screen, substitut-ing Eq. ~10! into Eq. ~12! and analyzing the asymp-totic behavior of DI~r̄! for small values of ur̄u, we cansee that

DI~r̄!u ur̄ u30 < 2Dw~r̄!. (13)

So, Eq. ~13!, which is obtained in such a way, alsoshows the equality of exponential factors nf and nIcharacterizing the spatial distributions of the bound-ary field phase and the far-zone intensity fluctua-tions, respectively:

nIb 5 nf, (14)

where superscript b corresponds to broadbeam illu-mination. Such equality is the manifestation of theidentity of the fractal properties ~particularly theHausdorff–Besicovitch dimension! of both spatial dis-tributions f~x, y! and I~j, h! in the small-scale region.In the intermediate case of the partial overlap of

FU~vx, vy! by the aperture function spectrum FA~vj,vh!, the power spectrum of intensity fluctuationsWI~vj, vh! is determined in general by the propertiesofWA~vx, vy!, and for smooth amplitude distributionsin the illuminating beam the increase in nI with re-spect to nf ~to values of 2! is characteristic.

B. Far-Zone Speckle Formation with a BroadbandAperture Function

If a sample is placed in the waist plane and the illu-minating spot size decreases, broadening takes placein the spatial spectrum of the aperture function. Inthe extreme case of the pointlike illuminating aper-ture ~which can be interpreted as a 2D d-function!,the spatial spectrum of the boundary field is totallyoverlapped by one of the aperture functions with auniformly distributed spectral density. So, the Fou-rier spectrum of intensity fluctuations in the paraxialregion of the far zone can be written as

FI~vj, vh! 5 @FF~vx, vy!# ^ @FF~vx, vy!#*. (15)

Let us consider the case of a Gaussian RPS withthe power-law spatial spectrummodule of phase fluc-tuations in the region of high spatial frequencies.Parameter a of the corresponding power spectrum@see, e.g., Eq. ~5!# is proposed to satisfy the basiclimitations for fractal distributions7 ~23# a # 21 forthe 1D spectrum!. Using the relationships betweenthe 1D and 2D power spectra of isotropic fractal dis-tributions and a corresponding structure functionand taking into account the relationship between thestructure functions of the complex amplitude of theboundary field and its phase for Gaussian scatterers~see, e.g., Subsection 2.A!, one can obtain the follow-ing equation for the exponential factor of intensityfluctuations in the dependence on nf in the case dis-cussed:

nIn 5 2nf 2 2. (16)

Taking into account the equality of nf and nIb, we can

obtain the following equation describing the relation-ship between nI

b and nIn:

nIn 5 2nI

b 2 2. (169)

Using Eq. ~16!, we can obtain a similar relationshipbetween the values of DHB for the far-zone intensity


and the boundary field phase fluctuations:

DHB~I! 5 2DHB~f! 2 1. (17)

So, increasing the Hausdorff–Besicovitch dimen-sion of intensity fluctuations for an illuminating ap-erture of small size is a direct manifestation of theexperimentally observed broadening of the speckleintensity spectrum if focused illuminating beams areused.

C. Scattered Intensity Fluctuations in the FresnelDiffraction Zone

This configuration, illustrated in Fig. 1, correspondsto the next form of the normalized aperture functionA~x, y!:

A~x, y! 5 exp$@2w22~Dz! 1 jky2R̃~Dz!

1 jky2~L 2 Dz!#~x2 1 y2!%, (18)

where Dz denotes the distance between the illumi-nating beamwaist plane and scatterer, R̃~Dz! is thebeam wave-front curvature radius, and w~Dz! is thebeam radius. In the case of strongly scattering ob-jects with relatively small effective correlationlengths of phase fluctuations that are discussed be-low, the contribution of the phase-distribution termof the aperture function beam component into theformation of the high-frequency part of intensity fluc-tuations is not taken into account. The reason forsuch a simplification is that the phase fluctuationsacross the illuminated region caused by the wave-front curvature are characterized by relatively smallspatial frequencies, and their influence on the asymp-totic characteristics of the high-frequency part ofWI~vj, vh! @and on the behavior of DI~tj, th! for thesmall values of its arguments# is insignificant. Inthis case the aperture function consists of two terms,one of which describes the Gaussian distribution ofthe amplitude in the illuminating beam, and theother can be characterized as a transfer function ofthe Fresnel transformation for zero spatial frequen-cies. To provide the possibility of analyzing inten-sity fluctuations for small distances between themoving screen and observation plane H2, we used ahigh-resolving objective O ~Fig. 1!; so the intensity

Fig. 1. Near-zone observation of paraxial-region intensity fluctu-ations: H1, illuminating beam-waist plane; RPS, random phasescreen; H2, observation plane; O, reimaging lens; H3, detector po-sition plane.


distribution of the scattered light in plane H2 is re-imaged with magnification on detector plane H3.Taking into account the dependence of illuminat-

ing beam radius w on the defocusing parameter Dz~which is equal to the distance between the RPS andthe waist plane! for large values of Dz,

w~Dz! < w0DzyR (19)

~where R is the Rayleigh range of the illuminatingbeam! and substituting the expression for Z in thecase of fixed distance L between the waist and theobservation plane, we go to the next equation forthe normalized spatial spectrum module of the ap-erture function uFA~vj, vh!u:

uFA~vj, vh!u 5 k21@R2~Dz!24 1 ~L2 Dz!22#20.5

3 exp$20.5~vj2 1 vh

2!@Rk~Dz!22

1R21k~Dz!2~L2 Dz!22#21%. (20)

Broadening of the spatial spectrum FA~vj, vh!~which leads to a decrease in nI and to a rise in theHausdorff dimension of the speckle intensity fluctu-ations for fractal scatterers! takes place

~1! for the RPS position in the waist plane ~such aconfiguration is discussed in Subsection 2.B!,

~2! with an increase in the defocusing parameterwhen the contribution of the second term of theexponent argument in Eq. ~20! caused by near-field-formation effects is prevalent. Generalized param-eter V can be introduced for a description of thespectral-broadening effects in the Fresnel diffractionzone:

V 5 @Rk~Dz!22 1 R21k~Dz!2~L 2 Dz!22#0.5. (21)

3. Experiments with Fractal Phase Screens

A. Choice and Preliminary Characterization of FractallikePhase Scatterers

Ground glasses with only surface scattering werechosen as the phase screens to be studied in ourexperiments. The absence of multiple-scattering ef-fects and the validity of the scalar-diffraction ap-proach for the description of speckle patternformation have been obliquely confirmed bymeasure-ments of the scattered-light depolarization degree inthe far diffraction zone.The characterization of scatterers by the Haus-

dorff–Besicovitch dimension of the fluctuations ofsurface relief heights as a classification criterion hasbeen carried out with empirical estimations of expo-nential factor nf of the phase structure function. Aswell as the actually existing scattering objects andmedia that usually demonstrate such peculiarities inthe surface topography as self-similarity and fractal-ity in a finite region of spatial scales, the asymptoticbehavior of Df~r̄! for the specimens that we studiedhas been analyzed for 0.5 mm # ur̄u # 10.0 mm.Two different experimental techniques have been

used for nf estimations:

~1! Scattering-surface profiling by a scanning dif-ferential microinterferometer. This method hasbeen applied to scatterers with relatively large-scalesurface inhomogeneities; for such scatterers only thepartial distortion of an initial diffractional structureof the interferometer probing beams takes place andmeasurements of local phase shifts are possible ~byan appropriate averaging procedure!.

~2! Modification of an angle-resolving scattering~ARS! technique11 for specimens with a small-scalescattering structure ~fine ground glasses!.

The optical scheme of scanning the differential mi-crointerferometer used in the experiments is pre-sented in Fig. 2. The Gaussian laser beam formedby single-mode He–Ne laser 1 after it passes throughthe telescopic system is spatially modulated by theMichelson interferometer unit. The existence of aregular dynamic interference pattern in the beamcross section leads to the formation of two identicalprobing beams in the focal plane of the focused micro-objective. The axes of the probing beams are sepa-rated by distance D, depending on the focal length ofthe micro-objective and the interference fringe pe-riod. The degree of overlap of the focal spots of theprobing beams is determined by the number of inter-ference fringes within the illuminating beam aper-ture.To provide a modification of the phase-shifting

technique for measuring the local phase differencesbetween the interference signals in the object and thereference channels during the specimen scanning,the piezoelectric modulator was used as part of theMichelson interferometer unit. The reference chan-nel is formed by a beam splitter and a photodetector~a photomultiplier tube! with a pinhole diaphragm.The phase difference of each object position is causedby passing probing beams through it. It consists oftwo parts: a constant or stationary phase deter-mined by the initial adjustment of the interferometer,and a phase varying with the object movement. Thepart that we obtained in our experiments by using

Fig. 2. Optical scheme of the scanning differential microinter-ferometer: 1, single-mode He–Ne laser; 2, 3, telescopic system asthe beam expander; 4–6, Michelson interferometer unit; 7, beamsplitter; 8, focused micro-objective; 9, sample being studied; 10,micro-objective; 11, 12, photodetectors ~photomultiplier tubes!; 13,piezoelectric phase-shift modulator; 14, 2D scanning device; 15,interface board.

interference-signal computer processing ~particularlylow-frequency digital filtering and a special algo-rithm for phase-difference evaluation! can be exam-ined as an estimation of the fluctuation component ofthe boundary field phase distribution. The root-mean-square error of evaluation of the phase-fluctu-ation component when described types of scanningdifferential interferometer and processing procedureare used was no more than 0.05. The analyzed spec-imens were rough glass plates with specially formedsurface microtopography ~as a result of hot pressing!.Such surfaces have enhanced adhesion properties.All the samples being studied were characterized byfollowing the parameters of surface-height statistics:effective correlation lengths lh, from 15.0 to 60 mm;standard deviations of height fluctuations sh, from0.1 to 20.0 mm. ~These values were obtained by con-tact profilometer measurements.!Figure 3 shows the experimentally obtained struc-

ture functions of the phase fluctuations for all thesamples of the type discussed, which have been stud-ied in our experiments. Corresponding fractal char-acteristics ~such as the Hausdorff–Besicovitchdimension and the topothesy LT! and the exponentialfactor nf estimated by the asymptotic behavior ofempirical-phase-structure functions in Fig. 3 are pre-sented in Table 1.As we can see from these values and Fig. 3, speci-

Fig. 3. Empirical structure functions for rough glass plates ~sam-ples 1–6!.

Table 1. Parameters of Phase Distributions for Scatterersbeing Studied

SampleN

Topothesy LT~mm!

ExponentialFactor

~nf!

Fractaldimensiona

~DHB!

1b ~5.10 6 0.5! 3 1025 1.39 6 0.06 1.312b ~1.33 6 0.15! 3 1025 1.43 6 0.06 1.293b ~4.04 6 0.5! 3 1025 1.32 6 0.06 1.344b ~2.53 6 0.3! 3 1028 1.48 6 0.07 1.265b ~1.91 6 0.2! 3 1025 1.18 6 0.05 1.416b ~4.16 6 0.4! 3 1025 1.16 6 0.05 1.427c — 1.39 6 0.06 1.3

Preliminary testing methods:aValues of DHB have been estimated from mean values of nf by

Eq. ~4!.bDifferential microinterferometry.cARS technique.


mens can be classified into two groups. The relativelylarge values of nf ~and small Hausdorff–Besicovitchdimensions! are characteristic of the first group of sam-ples ~1–3!. On the other hand, the surfaces of thesesamples also demonstrate the largest effective correla-tion lengths and values of sh as results of surface con-tact profiling. On the contrary, relatively smallvalues of lh and sh and higher values of DHBare characteristic of the second group of samples ~NN4–6!.Note the unusual behavior of sample N4. In ac-

cordance with the position of its empirical-phase-structure function ~Fig. 3!, this sample can beclassified in the second group, but one of the highestvalues of nf and anomalously small LT has been ob-tained for it. There is no correlation between thevalues of topothesy and nf and between LT and lh andsh.For strongly scattering samples with relatively

small-sized surface details ~such as finely groundglasses! a differential microinterferometric techniqueof aDf~r! asymptotic analysis is inapplicable becauseof strong distortions in the probing-beam phase dis-tributions by the speckle patterns produced. An-other method based on the study of the angulardependencies of the scattered-light average intensity~such as the ARS technique mentioned above! iswidely used for these applications. Scattering ob-jects with fractal distributions of boundary fieldphase demonstrate the power-law behavior of the av-erage scattered intensity, depending on the scatter-ing angle for certain illumination and observationconditions.7 The estimation of the slope angle forthe dependence of ^Is& on ws in logarithmic coordi-nates ~Is is the scattered light average intensity forgiven scattering angle ws; the averaging is carried outin the usual way over the ensemble of phase-distri-bution realizations to eliminate the coherent illumi-nation effects! allows one to determine theparameters of the phase-distribution structure func-tion and the power spectrum in accordance with thetheoretical background presented in Ref. 7. Theslope value is directly determined by constant a char-acterizing the power-law fall of the power spectrum ofthe phase fluctuations for fractal objects.An analysis of the angular dependence of ^Is& for

scattering angles between 5° and 40° has shown suchpower-law behavior of scattered intensity ~Fig. 4! for

Fig. 4. Angular dependence of the normalized mean value of thescattered light intensity for sample N7; light source, single-modeHe–Ne laser.


selected specimens of fine-structure ground glasses.So, fractallike behavior is characteristic of these scat-terers ~one of them is selected as a specimen to bestudied and denoted as sample N7! for spatial scalessmaller than 4.0 mm. The corresponding estima-tions of nf and DHB for sample N7 carried out by thecurve in Fig. 4 are presented in Table 1.

B. Experimental Analysis of the Intensity Fluctuations ofParaxial Speckles

The modification of a coherent scanning microscopewith a Gaussian-illuminating beam has been used forspeckle-pattern analysis ~Fig. 5!. The Gaussianbeam generated by single-mode He–Ne laser 1 andexpanded by telescopic system 2 is focused on studiedobject 4. If required, the illuminated area diameter2w ~which is determined by the beam’s Rayleighrange value R 5 pw0

2yl! is varied

~1! by the choice of focusing micro-objective 3 withappropriate characteristics ~magnification and nu-merical aperture!,

~2! by adjusting the telescopic system 2.

Different optical schemes for the image formationof speckle patterns in the paraxial region have beenapplied:

~1! With free space behind the scatterer in the caseof far-zone speckles produced for object positions inthe waist plane of the illuminating beam. Distancez between the object and the photodetector was cho-sen to provide the far-zone condition with respect tothe illuminated region size ~z .. R!.

~2! With free space or a Fourier-transforming lensfor far-zone fully developed speckles ~for off-waistsample positions with large values of the defocusingparameter!.

~3! With a highly resolving reimaging lens in thecase of near-zone speckle analysis. A similar config-uration is shown in Fig. 1. A minimal scanning stepprovided by a used scanning device was equal to 5

Fig. 5. Experimental setup for the speckle intensity fluctuationanalysis: 1, single-mode He–Ne laser; 2, telescopic system; 3,mirror; 4, focusing micro-objective; 5, 2D scanning device; 6, sam-ple being studied; 7, image-transforming system ~lens or freespace!; 8, photodetector ~photomultiplier tube!; 9, interface board.

mm. To obtain the much higher spatial resolution,different frequencies were used for the analog-to-dig-ital controller and scanning device. The samplinginterval of analog-to-digital conversionwasmuch less~20 times at least! than the driving signal period forthe scanning device; as a result, the spatial resolutionduring the recording of speckle intensity fluctuationsproduced by sample scanning was equal to or lessthan 0.25 mm. After the recording of intensity-fluc-tuations values of the empirical structure function,DI

l~r! were evaluated for each intensity realization.The intensity-fluctuation exponential factor was cal-culated with the rms method by using values ofDI

l~r!for 0 , uru , 5.0 mm.The results in Figs. 6–8 correspond to the case of

speckle patterns observed for samples 1–6 in the fardiffraction zone with broad and focused Gaussian-illuminating beams. The dependencies of the inten-sity fluctuations of exponential factor nI on thedefocusing parameter Dz are shown in Fig. 6. All ofthem demonstrate the typical behavior caused byvariations in the width of the aperture-function spa-tial spectrum with the rise of Dz. The minimal val-ues of exponential factor nI correspond to samplepositions in the waist plane. For the samples stud-ied where fractal properties are manifested in theregion of relatively large spatial scales ~as large as

Fig. 6. Dependencies of the exponential factor of the intensityfluctuations for different samples of rough glass plates ~1, sampleN1; 2, sample N4; 3, sample N6! in the case of focused ~R ' 20 mm!Gaussian-beam illumination.

Fig. 7. Relationships between the exponential factors of theboundary field phase and the scattered light intensity fluctuationsin the case of broad Gaussian-beam illumination. Objects beingstudied—samples NN1–6.

10–15 mm!, almost a total overlap of the spatial spec-tra by the aperture function takes place. Such abroadband aperture function is produced by focusingthe micro-objective with a 403 magnification. In-creasing the defocusing parameters to values of theorder of ~2–4!R leads to the formation of maxima ~oneor a few! on the curves nI 5 f ~Dz!. This condition canbe interpreted as the influence of the partial overlapof scatterers and aperture-function spatial spectra,when the high-frequency parts of the spatial spectraof the scatterers are cut off, and nonfractal spectralcomponents are dominated in intensity-oscillationformation. This produces a relatively smooth inten-sity signal and leads to an increase in nI. A furtherincrease in the defocusing parameter is accompaniedby the slow fall of nI to the values that are determinedby the corresponding exponential factors of theboundary field phase fluctuations ~as follows from theresults of the analysis in Subsection 2.A!.Relationships between values of nf ~evaluated in

preliminary experiments with a scanning differentialinterferometer, see Table 1! and exponential factornIb and nI

n for two extreme cases ~the first corre-sponds to the developed speckle patterns formed bybroadbeam illumination and the other to speckles inthe case of the small-sized illuminating aperture! arein agreement ~Figs. 7 and 8! with the theoreticalresults @see Eqs. ~14!, ~16!, and ~169!#. Notwith-standing the seeming dispersion of the experimentalpoints with respect to linear dependencies nI

b 5 nf

and nIn 5 2nI

b 2 2, experimentally estimated valuesnIb and nI

n are significantly correlated with corre-sponding values of nf. Estimations of the samplingcorrelation coefficients are values of no less than 0.73.When a decrease in distance between the phasescreen and the photodetector transition from theFraunhofer diffraction mode to the Fresnel modetakes place, the width of the spatial spectrum of theaperture function increases again. As a result thisleads to a decrease in the exponential factor of theintensity fluctuations. Figure 9 shows similar be-havior for different samples: Q denotes the waveparameter, determined as Q 5 pzRy~Dz!2. Q varieswith a decrease ~or increase! in the defocusing pa-

Fig. 8. Relationships between the values of nIb and nI

n for samplesNN1–6; nI

n have been estimated for a sharply focused ~R' 3.2 mm!illuminating beam.


rameter for a given distance z between the phasescreen and the photodetector.For a more complete experimental study of the

evolution of nI in the near diffraction zone the opticalscheme described in Subsection 2.C was used in ap-plication to sample N7 ~fine-structure ground glass!.For this phase, screen scattering regimes with speck-le-pattern formation take place even in the case ofsmall values in the Rayleigh range ~R > 5–30 mm!.The dependencies of nI and the corresponding valuesof the Hausdorff–Besicovitch dimension on the defo-cusing parameter in the near diffraction zone forsample 7 are presented in Fig. 10 for different illu-mination and observation conditions. The behaviorof these curves in the vicinity of the zero value of Dz~the presence of minima and maxima! is determinedby the same mechanism as discussed above ~the de-crease in DuFAu!. For larger values of Dz the secondterm in Eq. ~21! describing near-field effects becomessignificantly larger than the first term ~which de-scribes the influence of the illuminated region size onthe width of the aperture-function spatial spectrum!.The influence of spectral broadening on the nI andDHB parameters of the near-zone intensity fluctua-tions caused by a rise in the V value is illustrated bythe curves in Fig. 11. These dependencies were ob-tained by the transformation of experimental datapresented in Fig. 10. The decrease in nI indicates arise in the value of the Hausdorff–Besicovitch dimen-sion of the intensity fluctuations in the Fresnel dif-fraction zone. Note that the value of nf estimated bythe ARS technique ~Subsection 3.A; see Table 1! givesa smaller value of nI

n ~of the order of 0.9! than thecorresponding minimal value obtained in experi-ments with near-zone intensity fluctuations ~nI

n '

Fig. 9. Dependencies of the exponential factor nI on wave param-eter Q in the Fresnel diffraction zone for different samples ~■,sample N2; h, sample N4; Ç, sample N6!.

Fig. 10. Dependencies of nI ~1, ■! and DHB ~- -, . . .! on defocusingparameter Dz in the near diffraction zone for sample N7 ~1, L 5 1.5mm, R ' 3.2 mm; 2, L 5 2.5 mm, R ' 50 mm!.


1.05!. Such a difference can be caused by experi-mental errors as well as by some physical limitations@particularly by a cutoff of the high-frequency part ofthe boundary field spectrum as a result of the partialoverlap of Ff~vx, vy! and FA~vx, vy!#.

4. Discussion

The schemes of speckle-pattern formation with thebroadband Gaussian ~or another differentiated type!spatial spectra of the aperture function can be inter-preted as chaos amplifiers ~see, e.g., Ref. 3! because thefractal ~Hausdorff–Besicovitch! dimensions of the in-tensity fluctuations of the scattered field in this caseare not less than ~equal to or larger than! the dimen-sions of the boundary field phase distributions.Equality takes place if broad illuminating beams withd-like spatial spectra are used. Another characteris-tic of fractal-dimension equality is concerned with so-called marginal7 fractal phase distributions for whichDHB is the same as the corresponding Euclidean di-mension ~unity for 1D realizations, nonfractal phasescreens!. In other words, chaos generation in suchlinear coherent systems with nonfractal scattering ob-jects and lenses or free space as scattered-field trans-formers is impossible in the scalar-diffraction-theoryapproach. Nevertheless hypothetical and exoticcases of speckle-pattern formation for phase screenswith conventional phase distributions and nondiffer-entiated deterministic amplitude distributions withpower-law-like spatial spectra of illuminating beamscan be discussed particularly. In the extreme case ofisotropic phase screens with nf 5 1 @see, e.g., Eq. ~16!#~Brownian fractal phase distributions!, 2D random in-tensity distributions I~j, h! with nI 5 0 and DHB of 1Dintensity realizations equal to two are generated.Following Ref. 7, such intensity distributions can beclassified as extreme fractals.The method of speckle intensity fluctuation analy-

sis that is discussed can be proposed for the rapidstudy of scattering-surface microtopology peculiari-ties manifested in the asymptotic behavior of theirautocorrelation ~or structure! functions in the small-scale region or high-frequency parts of the spatialpower spectra. Using specially formed illuminatingcoherent beams ~e.g., providing a uniform amplitudedistribution with a given radius on the scatteringsurface! permits the display of these peculiarities bythe analysis of the exponential factor of speckle in-

Fig. 11. Relationships between exponential factor nI ~■, 1, p! andHausdorff–Besicovitch dimension DHB ~h! of the near-zone inten-sity fluctuations and the V parameter for sample N7 ~VN 5 1.0mm21; 1, p, R ' 3.2 mm, L 5 1.5 mm; ■, R ' 50 mm, L 5 2.0 mm!.

tensity fluctuations ~orHausdorff–Besicovitch dimen-sion! depending on the illuminating spot radius andthe distance between the scatterer and the photode-tector.One possible application of a similar analysis of

scattered laser light intensity fluctuations is charac-terized by scattering or reflecting surfaces ~especiallyafter different types of mechanical finishing or chem-ical coating or etching! in terms of fractal dimensionand topothesy. It follows from Ref. 7, that for cer-tain conditions such a surface description is moreappropriate than the conventional description basedon the evaluation of correlation length and the stan-dard deviation of surface-height fluctuations. An-other prospective direction is a generalizeddescription of the peculiarities of biotissue scatteringstructures caused by the progress of disease or drugapplications. A typical example is the correlationanalysis of speckle intensity fluctuations for samplesof normal and psoriasis human epidermis.12,13 Forpsoriasis skin compared with normal skin, larger val-ues of local estimations of the nI

n parameter in com-bination with its smaller dispersion across theanalyzed area are characteristic. This is a manifes-tation of changes in the epidermis scattering struc-ture caused by tissue impregnation with serum andother tissue fluids.14

References1. M. V. Berry, “Diffractals,” J. Phys. A 12, 781–797 ~1979!.2. C. Allain andM. Cloitre, “Optical diffraction on fractals,” Phys.

Rev. B 33, 3566–3569 ~1986!.

3. O. V. Angelsky, P. P. Maksimyak, and T. O. Perun, “Dimen-sionality in optical fields and signals,” Appl. Opt. 32, 6066–6071 ~1993!.

4. E. L. Church, T. V. Vorburger, and J. C. Wyant, “Direct com-parison of mechanical and optical measurements of precisionmachined surfaces,” Opt. Eng. 24, 388–396 ~1985!.

5. M. V. Berry and I. C. Percival, “Optics of fractal clusters suchas smoke,” Opt. Acta 33, 577–591 ~1986!.

6. E. Jakeman, “Fresnel scattering by a corrigated random sur-face with fractal slope,” J. Opt. Soc. Am. 72, 1034–1041 ~1982!.

7. E. L. Church, “Fractal surface finish,” Appl. Opt. 27, 1518–1526 ~1988!.

8. D. L. Jaggard and Y. Kim, “Diffraction by bandlimited fractalscreens,” J. Opt. Soc. Am. A 4, 1055–1062 ~1987!.

9. J. F. Benzoni, S. Sarkar, and D. Sherrington, “Random phasescreens,” J. Opt. Soc. Am. A 4, 17–26 ~1987!.

10. S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarsky, Introductionto Statistical Radiophysics. Part 2. Random Fields ~Nauka,Moscow, 1978!, p. 464 ~in Russian!.

11. J. M. Elson and J. M. Bennett, “Relation between the angulardependence of scattering and the statistical properties of op-tical surfaces,” J. Opt. Soc. Am. 69, 31–49 ~1979!.

12. D. A. Zimnyakov, V. V. Tuchin, and S. R. Utz, “Investigation ofstatistical properties of partially developed speckles for diag-nostics of human skin structural changes,” Opt. Spectrosc.USSR 76, 838–844 ~1994!.

13. V. V. Tuchin, D. A. Zimnyakov, G. G. Akchurin, A. A. Mishin,and I. L. Kon, “Coherence-domain optical methods for cell andtissue structure and function monitoring,” in Laser Chemistry,Biophysics, and Biomedicine, V. N. Zadkov, ed., Proc. SPIE2802, 152–163 ~1996!.

14. V. V. Tuchin, S. R. Utz, and I. V. Yaroslavsky, “Tissue optics,light distribution and spectroscopy,” Opt. Eng. 33, 3178–3188~1994!.


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Fractality of speckle intensity fluctuations