. INTRODUCTIONs is well known, when a broad coherent beam is scat-
ered from a random highly scattering object, developedpeckles are formed. The statistics of such speckles haveeen extensively studied [1]. When a focused laser beams scattered from a rough surface or from a thin layer ofn optically inhomogeneous medium, the diffraction pat-ern contains only a few speckles (in practice, about tenpeckles) of a large size: So-called speckles with a smallumber of scatterers [2–6] or small-N speckles [7] areormed in this case. Such speckles possess specific statis-ical characteristics. It has been shown that the statisticsf speckles with a small number of scatterers are usuallyot Gaussian in character [3,5]. These speckles may belassified as statistically inhomogeneous random fields8]. New peculiarities of the manifestation of the Dopplerffect may be observed in their dynamics [9]. As found in8,9], the spectrum of the intensity fluctuations of scat-ered light contains a high-frequency peak in the absencef a subsidiary reference wave. The frequency position ofhe Doppler peak is defined by not only the observationngle and the velocity of the scatterers but also by theumber of scatterers in the probing volume.It should be noted that the statistics of speckles with a
mall number of scatterers have been analyzed in regardnly to surfacelike scatterers or single-scattering media3]. Meanwhile, the case where a sample is highly scatter-ng, but its thickness is small (typically, about00 �m–300 �m), is very important from a practicaliewpoint. Up to the present time, such samples (includ-ng highly scattering microflows) have been considered as
set of moving deep-phase screens [6]. Attempts at ad-ressing the analysis of the statistical properties of speck-es formed in multiple-scattering media with a smallumber of scattering events have not been reflected inrevious reports in the literature. But a solution to thisroblem is essential for the diagnostics of microflows or
hin multiple-scattering samples using diffusing wavepectroscopy (DWS).
As is well known, DWS is one of the most promisingethods that can be effectively applied for the diagnostics
f highly scattering dynamic media [10,11]. In particular,WS is applied for the analysis of particle suspensions
12–14], foams, sands [15,16], and nonergodic andultilayer media [17]; the investigation of microrheology
f complex fluids [18]; the study of spatially localized ran-om flow hidden in motionless scattering media [19], etc.n traditional DWS it is assumed that light is scattered aarge number of times [10]. In [20] it has been shown thatWS approximations are on the verge of being unaccept-bly inaccurate if the ratio between the sample thicknessand the transport mean-free path l* within the range
�L / l*�20; DWS completely breaks down if L / l*�5.Nevertheless, recently DWS has been extended to the
ase of a small number of scattering events [21]. Butheory developed in [21] has one serious disadvantage. Asn the case of classical DWS, scattered intensity repre-ents an incoherent sum over the light path [10] and doesot take into account interference phenomena.Also, it is appropriate to mention the analysis of coher-
nt phenomena in strongly scattering media, based on theolution of the Bethe–Salpeter equation using the Montearlo stochastic method [22]. Such an approach allows us
o analyze coherent backscattering and the spatial andemporal correlation of the intensity; interference phe-omena are also included for consideration. Regretfully,he solution of the Bethe–Salpeter equation does not havehe analytical power of DWS and requires additional com-uter simulation. Also, it cannot be used for the study ofhe first-order statistics of scattered light.
The main purposes of this brief report are to derive theeneral analytical formulas for the first- and second-ordertatistics of the amplitude and phase of small-N specklesfor the values L / l*�20), considering them a result of the
nterference between the different fractions of multiple-cattered light, and to verify experimentally obtainedheoretical results.
. FIRST-ORDER STATISTICS OF THEMPLITUDE AND PHASE OF SCATTEREDIGHTet us consider the simplest case of light scattering onmall nonabsorbing particles. If a plane wave is scatteredrst by an isotropic scatterer and then by a field, it illu-inates the second particle and may be written in the
orm of a spherical wave [23]:
e2 �e0�
s1exp�2�is1
�� , �1a�
here e0 is the amplitude of the initial plane wave, s1 ishe distance between the first and the second scatteringarticles, and � is the wavelength. It is assumed that theistance between two neighboring particles is larger thanand that the size of the particles is less than � /3 [24],hich is valid for nearly all cases of practical interest. Asas been found in [25], if a sample has an opaque whiteppearance, and we operate in the multiple-scattering re-ime known as DWS [25], then the form of the scatteringarticles does not have a significant effect on the statisti-al properties of scattered light.
It is expedient to make all the variables dimensionless.o all the length dimensions are normalized to the lightavelength �. The amplitudes of all the fields are normal-
zed to the field amplitude e0 of the initial plane wave.hen the last equation may be rewritten in dimensionless
orm:
E2 �1
S1exp�2�iS1�, �1b�
here E2�e2 /e0 and S1�s1 /�.After scattering by two particles, the dimensionless
mplitude of the wave illuminating the third particle isefined by
E3 �1
S1S2exp�2�iS1�exp�2�iS2�, �2�
here S2 is the normalized distance between the secondnd the third particles.Clearly, if light is scattered n times, then the resulting
eld is expressed as
En �1
�i=1n Si
exp�2�i�i=1
n
Si� . �3�
s it follows from the central limit theorem, the argumentf the complex exponent in Eq. (3) (i.e., phase of multiple-cattered light) is a random Gaussian variable.
Evidently, the statistical properties of the modulus ofhe amplitude of light scattered n times are determined ineneral by the random process 1/�i=1
n Si, where each ran-om value S obeys an exponential distribution [26]:
i
p�S� = Ms exp�− MsS�, �4�
here Ms=�s� and �s is the scattering coefficient. Therocess of light absorption is ignored.If light has been involved in two independent scatter-
ng events, the probability density function of the randomalue Z=S1S2 is given by [27]
p�Z� =�−�
�
p�S�p�Z
S�dS
S. �5�
Taking into account that all paths can take only posi-ive values and substituting Eq. (4) into Eq. (5), we obtain
p�Z� = MS2�
0
�
exp�− MsS�exp�−MsZ
S �dS
S. �6�
By integrating over S [28], we find the probability den-ity function of the product of two independent pathengths between elementary scattering events:
p�Z� = 2MS2K0�2MsZ�. �7�
s can be seen from Eq. (7), this value obeys a K distri-ution (in the last equation, Ko is a zero-order Besselunction).
Let us assume that light is scattered four times. Thenhe probability density function of the product �=Z1Z2S1S2S3S4 is expressed as
p��� = 4MS4�
−�
�
K0�2MsZ�K0�2Ms �
Z�dZ
Z. �8�
or large positive values Z and � in the last expressionan be rewritten as
p��� ��0
�
exp�− 2MsZ�exp�− 2Ms �
Z�dZ
Z. �9�
erforming the integration in Eq. (9) also leads to the ap-earance of a Bessel function:
p��� � Ko�4Ms�1/4�. �10�
vidently, for n scattering events, the recurrent formulaay be derived as
p��n� � Ko�nMs�n1/n� , �11�
here �n=S1S2S3 , . . . ,Sn, i.e., is a product of n pathengths.
Because the amplitude En of light scattered n timessee Eq. (3)] is proportional to 1/�n, then the probabilityensity function for the inverse random values is de-cribed by the following formula [28,29]:
p�En� =p��n�
dEn d�n. �12�
or inverse random values, the following well-knownormula is derived [28]:
hus, the expression for the probability density functionf the amplitude of the n-times scattered amplitudeesults in
p�En� � const�n� ·1
En2 K0�nMs�En�−1/n�, �14�
here const�n� is a constant, which depends on the num-er of scattering events.Taking into account that �0
�p�E�dE=1, it is easy toefine the factor const in Eq. (14). It equals
const�n� =1
�0
� 1
E2K0�nMsE−1/n�dE
. �15�
fter integration, we obtain the expression for the factoronst as
const�n� =n�n−1�MS
n
2�n−2���n/2 �2. �16�
inally, the expression for the probability density functionor the n-times scattered amplitude takes the followingorm:
p�En� �n�n−1�MS
n
2�n−2���n/2 �2·
1
E2K0�nMs · �E�−1/n�. �17�
Let us now consider the probability density function ofhe total complex amplitude in some observation point ashe coherent sum over different paths. Finally, accordingo Eq. (3) and Bayes theorem, the formula for the totalrobability density P�E� can be expressed as
P�E� = �s
p�ES�p�S�exp�− 2�iS�, �18�
here p�E S� is the conditional probability that afterraveling the path S, the wave has the modulus of ampli-ude E.
As it has been shown in [21] that the total probabilityensity function of the path distribution equals
p�S� = �n
p�Sn� · P�n�, �19�
here p�S n� is the conditional probability density func-ion that corresponds to the case that the photon travelsength S and participates in n scattering acts. As it alsoas been shown in [21], p�S n� is described by the func-ion
p�Sn� = MS2 ·
S�n−1�
�n − 1�!· exp�− MS · S�. �20�
he number of scattering events obeys a Poissonian dis-ribution (this hypothesis has been tested in [21]):
P�n� =�n�n · exp�− �n��
n!. �21�
fter some algebra, Eq. (18) may be written in the follow-ng evident form:
p�S� = �n
p�Sn� · P�n�
= �exp�− MsS�exp�− �n��1
S���
n
�MSS�n��n
n!�n − 1�!
��exp�− MsS�exp�− �n��1
S�· �I1�MsS�n�� + 1 , �22�
here I1 is a first-order Bessel function of the secondind.Equation (22) allows us to estimate the average path.
ssuming that, in the case of a small number of scatter-ng events, I1�MsS · �n��→0, then the average pathpproximately equals
�S� =�0
�
S · p�S�dS =1
Msexp�− �n�� · �
�n�n
�n − 1�!=
�n�
Ms.
�23�
s it follows from Eq. (23), �n� / �S�=Ms. So, making theubstitution n=S�n� / �S��MsS in Eq. (17) and simplifyingq. (18), we can get the final expression for the probabil-
ty density function of the amplitude of scattered light:
p�E� = �n
n�n−2��MS�n+1
2�n−2���n/2 �2·
1
E2K0�nMs�E�−1/n�
· �exp�− �1 +2�i
Ms�n�exp�− �n���
· �I1�n�n�� + 1 . �24�
s in classical DWS [10,30], �n represents the sum overifferent paths containing different numbers of scatteringvents. But, in contrast to the approach based on diffu-ion approximation [30], Eq. (24) expresses the coherentum of many fractions of light that travel from differentaths.Calculations made on the basis of Eq. (24) show that
he probability density function p�E� is expressed as aeal function in spite of the presence of a complex expo-ent on the right-hand side of Eq. (24). Empirical analysisf Eq. (24) allows us to conclude that the probability den-ity function p�E� may be approximated by a functionimilar to a Nakagami n distribution [31]:
p�E� ��n��n� · E�n�−2
��n�� · �E��n�−2exp�−
�n�E
�E� � , �25�
here again �n� is the averaged number of scatteringvents that is approximately equal to M L, L is the di-
ensionless thickness of the scattering sample (i.e.,=L /�), and �E� is the averaged value of E.It is interesting to note that sometimes, albeit very
arely, a similar distribution (i.e., a Nakagami n distribu-ion) arises in the case of light diffracted from solidlikecatterers with a small number of scattering events [32].
Equation (25) is the most important theoretical resultf this brief report. This formula describes the first-ordertatistics of multiple-scattered speckles formed in thease of a small number of scatterers.
. SECOND-ORDER STATISTICS OF THEMPLITUDE AND PHASE OF SCATTEREDIGHTvidently, the complex amplitude of multiple-scattering
ight may be written as
U�t� = E�t�exp�i��t��, �26�
here E�t� is the amplitude of the dynamic multiple-cattered speckles and �t� is the time-varying phase.
The autocorrelation function of the complex amplitudes expressed in the formula
��� = �U�t�U�t + ��*�. �27�
ssuming that the amplitude and phase fluctuations areot correlated, expression (27) may be rewritten as [33]
ecause the phase is very sensitive to the displacement ofcatterers over a distance on the order of a wavelength,he second term in Eq. (28) plays a dominant role:
��� � factor��n���exp�i���t� − ��t + �����, �29a�
here
factor��n�� � �0
�
E2p�E�dE �29b�
s a constant, depending on the averaged number of scat-ering events, and p�E� is the probability function, whichs expressed by Eq. (25). See also Fig. 1.
As it follows from the results presented in [34], in thease of Gaussian statistics of the phase,
ig. 1. Probability density functions, expressed by Eq. (24) (cirvents, �n�=7. The relative difference between the rigorous formuess than 7%.
nd, as a result, the expression for correlation function29a) may be essentially simplified as
��� � factor��n��exp���2 �K���� − 1 �, �31�
here � is the standard deviation of the temporal phaseuctuations and K is the correlation coefficient of thehase fluctuations of scattered light.In [30] the temporal autocorrelation function has been
erived, assuming that the fields belonging to differentaths add incoherently. It is remarkable that the struc-ure of the autocorrelation function obtained in DWS alsoas the same structure as formula (31).The last relations show that the correlation function of
he scattered intensity directionally depends on the corre-ation coefficient of the temporal phase fluctuations ofcattered light. This theoretical result will be verified inhe next section for the case of a small number of scatter-ng events.
. EXPERIMENTAL STUDY OF FIRST- ANDECOND-ORDER STATISTICS OF THEMPLITUDE AND PHASE OF MULTIPLE-CATTERED SPECKLES WITH AMALL NUMBER OF SCATTERING EVENTS. Experimental Techniquehighly scattering sample has been prepared from par-
icles of barium sulfate (1/4 volume fraction) mixed withaseline oil (3 /4 volume fraction). High-viscosity oil haseen used to reduce the velocity of scattering particlesarticipating in Brownian motion. A highly scattering me-ium is fixed between two thin glass plates separated byalibrated spacers. Changes in the size of the spacers (i.e.,hickness of the scattering gel between the glass plates)llow us to control the effective number of scatterers.An optical scheme of the setup is presented in Fig. 2.
llumination of a highly scattering sample has been per-ormed using a wide collimated beam of a He–Ne laserFig. 2(a)] with a diameter of 2 mm at a wavelength of
nd Eq. (25) (boxes), in the case of a small number of scatteringressing first-order statistics, and the Nakagami n distribution is
30 nm, power of 30 mW, and longitudinal coherenceength of 15 cm. The thickness � of the scattering gelFig. 2(b)] was 200 �m.
The dynamics of speckles formed with a small numberf scatterers has been observed in the image planeFig. 2(e)] of the micro-objective [Fig. 2(d)] with a magni-cation of 95� and an N.A.=1.35. The size of specklessee Fig. 3) has been measured using a Phoenix USB 1280igital complementary metal-oxide semiconductor
CMOS) camera (MuTech, USA). As results of the mea-urements show, the speckle size is approximately 5 timesarger than the pixel size. This means that in the de-
ig. 2. Optical scheme for observation of the temporal fluctua-ions of the phase and intensities of multiple-scattered light: a,ollimated laser beam; b, scattering object; c (gray line), objectlane; d, micro-objective; e (gray line), image plane of micro-bjective; f, pinhole; g, lens; h (gray line), lens focal plane; i,omplementary metal-oxide semiconductor (CMOS) camera.
Fig. 3. Speckle pattern formed in
cribed setup, the micro-objective completely resolves thepeckle structure in the object plane [Fig. 2(c)] where thelluminated scattering specimen is placed. Because theelocity of the scattering particles is very low, the tempo-al fluctuations of the scattered intensity are also com-letely resolved by the CMOS camera.The attenuation of the intensity of light passing
hrough the scattering sample in the configuration of aollimated transmission allows us to measure the value ofhe scattering coefficient [35]. For the sample considered,s is 16 mm−1 and absorption is ignored, �a=0.Measurements of a single-scattering indicatrix allow us
o estimate the factor of anisotropy (g factor). As experi-ents show, if light is scattered from a sample with a
hickness of 30 �m, then speckle contrast C is close tonity �C=0.98� and the speckles are practically not depo-
arized. This indirectly indicates that in such a sample re-ime, single scattering is realized. Taking into accounthe evident ratio between the correlation length of theoundary field lv and the width of the angular spectrum �f scattered light [34],
�sin���� � �/lv, �32a�
he g factor then can be easily estimated as
g � 1 − �sin����2 �1 − � � · M
o · 2�2
= 0.8, �32b�
here � is averaged diameter of speckle in the 2D inten-ity distribution registered in the image plane and M ishe magnification of the microscope.
Thus, the average number of scattering events for theample with a thickness of 200 �m is
Nsc = ��s�1 − g� � 0.6, �33�
.e., less than unity.The temporal fluctuations of the phase of scattered
ight have been studied using a wavefront analysis tech-ique. A pinhole [Fig. 2(f)], combined with a single lensletFig. 2(g)], has been placed in the image plane of theicro-objective [Fig. 2(e)]. The diameter of the pinhole
Fig. 4. Illustration of the wavefront analysis. The posit
as 10 �m, and the lens focal length was 10 mm. The
amera [Fig. 2(i)] has been set in the second focal plane ofhe lenslet [Fig. 2(h)]. This camera has been used for de-ection of the 2D position of the center of the light spot;ee Fig. 4. As previously mentioned, the speckle dynamicsre very slow because the velocity of the particles partici-ating in Brownian motion are reduced in the viscous Va-eline oil. Thus, the position of the focal spot (in Fig. 4)an be detected in each frame with high precision. Regis-ration of the dynamic changes of the position of this spotllows us to retrieve the phase of speckles in the plane ofhe pinhole by using a standard algorithm of Shack–
the light spot in the focal plane of the lenslet is shown.
ion of
ig. 5. Typical realizations of the temporal fluctuations of phase (a) and normalized intensity (b) of dynamic multiple-scattered specklesith a small number of scattering events.
artmann wavefront analysis [36]. The method, based onntensity-weighed coordinates [37], has been used to in-rease the precision of phase reconstruction.
The intensity of speckles has been measured as inte-rated over the whole spot intensity in the lenslet focallane [Fig. 2(h)]. The described setup allows us to detectynchronously the intensity and phase fluctuations inultiple-scattered speckles.
. Resultssing the experimental technique described in Subsec-
ion 4.A, the temporal fluctuations of the phase ofultiple-scattered dynamic speckles have been retrieved.he large-scale trend has been removed from the phasend intensity time seria by using kernel smoothing. In-ensity fluctuations have been normalized to the standardeviation. Typical realizations are presented in Fig. 5.As the results of the correlation analysis show, the fluc-
uations of the phase and intensity of speckles are not cor-elated at all. Typically, the modulus of the cross-orrelation coefficient is about 0.01. This confirms thealidity of the theoretical assumption used for the deriva-ion of Eq. (28).
The temporal autocorrelation function of the amplitudef scattered light is a negative exponent, so the theoreti-al results are in good agreement with the experimentalata; see Fig. 6.The correlation function for the phase fluctuations, re-
onstructed from the autocorrelation function of the am-litude of scattered light using formula (31) [value foractor��n�� has been fitted], is perfectly matched with theame function, directly measured in the experiment; seeig. 7. This confirms the validity of the theoretical expres-ion for the correlation function of the amplitude of thecattered field, i.e., Eq. (31).
The probability density function of the amplitude ofcattered light obtained in the experiment is presented inig. 6. The shape of the histogram is close to a Nakagamidistribution. Nevertheless, the hypothesis that the am-
litude obeys this distribution is difficult to test, becausehe exact value of parameter �n� is unknown. Meanwhile,he hypothesis of the Raleigh distribution of the temporaluctuations of the amplitude of scattered light (which isabitual for fully developed speckles [27]) is rejected at aonfidence level of �=0.05.
ig. 6. Temporal autocorrelation function of the amplitude ofcattered light. Solid curve, theoretical curve; dots, experimentalata.
It is interesting to mention that the obtained experi-ental results are in very good agreement with the data
ublished in [38]. In the cited paper, Brownian motionithin multiple-scattering media has been analyzed us-
ng low-coherence interferometry (LCI). The probingepth of LCI was 200 �m, which corresponds to the casef a small number of scattering events. In the present pa-er, the thickness of the sample has been also chosen toe 200 �m.The first-order statistics of the intensity fluctuations
nd statistical properties of the phase of scattered lightave not been analyzed in [38]. But, as it has been shown
n [38], the spectrum of the scattered intensity has aorentzian shape, which corresponds exactly to theegative-exponent correlation function (see Fig. 6)btained using the Shack–Hartmann technique.
. CONCLUSIONSt has been shown that speckles diffracted from highlycattering media with a small number of scatteringvents are not Gaussian. They obey first-order statisticsimilar to a Nakagami n distribution.
It has been demonstrated theoretically and experimen-ally that the correlation function of the phase and inten-ity fluctuations of light scattered with a small number ofcattering events has the shape of a negative exponent, asn the case of “classical” DWS and DWS with a smallumber of scattering events. It has been shown that in-erference phenomena do not play a critical role and doot affect the second-order statistics of multiple-scatteredpeckles.
It has also been found theoretically and verified experi-entally that the temporal fluctuations of the amplitude
f scattered light depend directly on the temporal autocor-elation function of the phase fluctuations in specklesEq. (31)].
CKNOWLEDGMENTShis research has been supported by the Russia Founda-
ion of Basic Research through grant N06-04-39016.
ig. 7. Temporal autocorrelation function of the phase of scat-ered light. Solid curve, theoretical curve; dots, experimentalata.
EFERENCES1. J. C. Dainty, Laser Speckle and Related Phenomena, Vol. 9
of Topics in Applied Physics Series (Springer, 1975).2. P. J. Chandley and H. M. Escamilla, “Speckle from a rough
surface when the illuminated region contains fewcorrelation areas: The effect of changing the surface heightvariance,” Opt. Commun. 29, 151–154 (1979).
3. E. Jakeman, “Speckle statistics with a small number ofscatterers,” Opt. Eng. (Bellingham) 23, 453–461 (1984).
4. B. Saleh, Photoelectron Statistics with Applications toSpectroscopy and Optical Communication (Springer-Verlag,1991), pp. 145–149.
5. S. S. Ulyanov, D. A. Zimnyakov, and V. V. Tuchin,“Fundamentals and applications of dynamic specklesinduced by focused laser beam scattering,” Opt. Eng.(Bellingham) 33, 3189–3201 (1994).
6. S. S. Ulyanov, “Speckled speckles statistics with a smallnumber of scatterers. An implication for blood flowmeasurements,” J. Biomed. Opt. 3, 227–236 (1998).
7. M. Kowalczyk and P. Zalicki, “Small-N speckle: Phase-contrast approach,” Proc. SPIE 556, 50–54 (1985).
8. S. S. Ulyanov, “Dynamics of statistically inhomogeneousspeckles: A new type of manifestation of the Doppler effect,”Opt. Lett. 20, 1313–1315 (1995).
9. S. S. Ulyanov, “New type of manifestation of the Dopplereffect: An application to blood and lymph flowmeasurements,” Opt. Eng. (Bellingham) 34, 2850–2855(1995).
0. D. A. Weitz and D. J. Pine, “Diffusing wave spectroscopy,”in Dynamic Light Scattering: The Method and SomeApplications, W. Brown, ed. (Claredon, 1993), pp. 652–720.
1. D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E.Herbolzheomer, and P. M. Chaikin, “Dynamicalcorrelations of multiple scattered light,” in Scattering andLocalization of Classical Waves in Random Media, Vol. 8 ofWorld Series on Direction in Condensed Matter Physics, P.Sheng, ed. (World Scientific, 1990), pp. 312–372.
2. N. Menon and D. J. Durian, “Particle motions in a gas-fluidized bed of sand,” Phys. Rev. Lett. 79, 3407–3410(1997).
3. E. M. Furst and A. P. Gast, “Particle dynamics inmagnetorheological suspensions using diffusing-wavespectroscopy,” Phys. Rev. E 58, 3372–3376 (1998).
4. M. H. Kao, A. G. Yodh, and D. J. Pine, “Observation ofBrownian motion on the time scale of hydrodynamicinteractions,” Phys. Rev. Lett. 70, 242–245 (1993).
5. A. D. Gopal and D. J. Durian, “Nonlinear bubble dynamicsin a slowly driven foam,” Phys. Rev. Lett. 75, 2610–2613(1995).
6. R. Hohler, S. Cohen-Addad, and H. Hoballah, “Periodicnonlinear bubble motion in aqueous foam under oscillatingshear strain,” Phys. Rev. Lett. 79, 1154–1157 (1997).
7. F. Scheffold, S. E. Skipetrov, S. Romer, and P.Schurtenberger, “Diffusing-wave spectroscopy ofnonergodic media,” Phys. Rev. E 63, 061404 (2001).
8. T. G. Mason, K. Ganesan, J. H. Van-Zanten, D. Wirtz, andS. C. Kuo, “Particle tracking microrheology of complexfluids,” Phys. Rev. Lett. 79, 3282–3285 (1997).
9. I. V. Meglinsky and S. E. Skipetrov, “Diffusing-wavespectroscopy in randomly inhomogeneous media withspatially localized scatterer flows,” J. Exp. Theor. Phys. 86,661–665 (1998).
0. P. A. Lemieux, M. U. Vera, and D. J. Durian, “Diffusing-light spectroscopies beyond the diffusion limit: The role ofballistic transport and anisotropic scattering,” Phys. Rev. E57, 4498–4515 (1998).
1. S. Ulyanov, “Diffusing wave spectroscopy with a smallnumber of scattering events: An implication to microflowdiagnostics,” Phys. Rev. E 72, 052902 (2005).
2. V. L. Kuzmin, I. V. Meglinsky, and D. Yu. Churmakov,“Stochastic Modelling of Coherent Phenomena in StronglyInhomogeneous Media,” J. Exp. Theor. Phys. 101, 22–32(2005).
3. H. C. van de Hulst, Light Scattering by Small Particles(Dover, 1981).
4. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering,Absorption, and Emission of Light by Small Particles(Cambridge Univ. Press, 2002).
5. R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon,and D. J. Durian, “Speckle-visibility spectroscopy: A tool tostudy time-varying dynamics,” Rev. Sci. Instrum. 76,093110 (2005).
6. S. L. Jacques and L. Wang, “Monte Carlo modeling of lighttransport in tissues,” in Optical-Thermal Response ofLaser-Irradiated Tissue, A. J. Welch and M. J. van Gemert,eds. (Plenum, 1995), Chap. 4.
7. J. W. Goodman, Statistical Optics (Wiley, 1985).8. I. S. Gonorovsky, Radiotechnical Circuits and Signals
(Sovietskoe Radio, 1977).9. J. S. Bendat and A. G. Piersol, Random Data. Analysis and
Measurement Procedures (Wiley, 1986).0. D. J. Pine, D. A. Weitz, J. X. Zhu, and E. Herbolzheimer,
1. M. Nakagami, “The m-distribution—a general formula ofintensity distribution in rapid fading,” in StatisticalMethods on Radio Wave Propagation, W. C. Hoffman, ed.(Pergamon, 1960), pp. 3–36.
2. M. Yoshikawa and H. Kayano, “Analysis of non-Gaussianspeckle by Nakagami m-distribution,” Jpn. J. Appl. Phys.,Part 1 26, 974–975 (1987).
3. A. N. Korolevich and I. V. Meglinsky, “Experimental studyof the potential use of diffusing wave spectroscopy toinvestigate the structural characteristics of blood undermultiple scattering,” Bioelectrochemistry 52, 223–227(2000).
4. S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarsky, Principlesof Statistical Radiophysics. Part 2 (Springer, 1989).
5. W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of theoptical properties of biological tissues,” IEEE J. QuantumElectron. 26, 2166–2185 (1990).
6. B. C. Platt and R. Shack, “History and principles ofShack–Hartmann wavefront sensing,” J. Refract. Surg. 17,S573–S577 (2001).
7. A. F. Brooks, T.-L. Kelly, P. J. Veitch, and J. Munch,“Ultra-sensitive wavefront measurement using aHartmann sensor,” Opt. Express 15, 10370–10375 (2007).
8. D. A. Boas, K. K. Bizheva, and A. M. Siegel, “Usingdynamic low-coherence interferometry to image Brownianmotion within highly scattering media,” Opt. Lett. 23,319–321 (1998).
