1. Introduction

Light scattering probes of the random media play an important role in various applications in modern material science and biomedicine. The vast variety of physical phenomena accompanying elastic, quasi-elastic, or inelastic light interactions with the probed medium serves as the physical basis for these probes. These interactions modify all or part of the properties (amplitude, frequency, initial phase, and polarization state) of the light waves propagating in the medium. Accordingly, the changes in these properties occurring due to multiple scattering of the probe light in the medium contain the necessary information about the structure, or dynamics, or chemical composition of the examined substance.

The coherence domain probes of the random media are associated with elastic or quasi-elastic light-medium interactions; these terms mean that the propagating waves do not change or insignificantly change the frequency in the sequences of scattering events, whereas random variations in the propagation direction from one scattering event to another can be significant. This physical picture illustrating the features of such interactions corresponds to the so-called discrete scattering model [1, 2] previously applied for the phenomenological description of coherence phenomena in the multiple scattering (see, e.g., [3–7]).

The diffusing wave spectroscopy pioneered by G. Maret with P.-E. Wolf [8] and D.J. Pine et al. [9] is the most popular coherence domain technique used to characterize the scatter dynamics in the complex multiple scattering systems at the spatial scales of the order of the used wavelength. This technique applied in probing of various scattering systems (from biological tissues and soft media [10–15] to sand flows [16]) in the past three decades has undergone essential modifications in part of instrumentation and data processing procedures from the classical single-point detection schemes to the multispeckle arrangement [17–19].

In the “frozen” media with the stable, time-independent scatter positions, the multiple scattering of the coherent light is not accompanied by the temporal decorrelation of the scattered radiation, which exhibits the static speckle modulation. Such media can be considered as multiple-beam interferometers with stochastically distributed path differences between the interfering beams. Speckle patterns modulating the outgoing multiple scattered light are associated with stochastic interference of partial contributions to the light field in the medium. Thus, the probing techniques based on the statistical analysis of such stochastic interference patterns can be identified as the reference-free stochastic interferometry of random media. Additionally, application of the field correlation techniques (see, e.g., [20, 21]) provides promising results in the area of coherence-domain diagnostics of random scattering systems characterized by the stable structure.

Typically, the multiple scattered speckle-modulated light is characterized in the weak scattering limit by the Rayleigh statistics of intensity fluctuations, which results from the superposition of two fully developed non-correlated speckle patterns with equal values of average intensity. These speckle patterns correspond to the independent orthogonally polarized modes of the propagating light in the medium [22–24].

The transition from coherent to partially coherent illumination of multiple scattering random media can cause a remarkable suppression of stochastic interference modulation of the outgoing multiple scattered light in the case when the average path of light propagation in the medium is comparable or exceeds the coherence length of the light. Consequently, varying the coherence length of the probe light or changing the light propagation conditions, and evaluating the speckle contrast or oscillation index in the outgoing light, we can characterize the optical transport properties of the probed medium or visualize its inhomogeneous structure. In particular, the imaging of macroscopically heterogeneous random media using speckle contrast as the visualization parameter and illumination source with the fixed coherence length (a He-Ne laser) was considered in [25]. Evaluation of the optical transport properties (the mean transport free path and the scattering anisotropy parameter [1],) of the random media using the probe light with the tunable coherence length and speckle contrast estimates for the outgoing multiply scattered light was discussed in [26]. In this case, the coherence length of the probe light was varied by changing the pumping current in the light source (the laser diode operating below the generation threshold). The polarization discrimination of the outgoing light was applied for additional characterization of the light depolarization length in the medium. An alternative approach to the coherence length tuning in the reference-free stochastic interferometry is based on frequency modulation of the light sources. In particular, the periodic binary modulation of the output frequency of the laser diode with the tunable modulation depth in combination with the long-exposure capture of the speckle patterns ensures the necessary effect [27].

Another way to the reference-free low-coherence interferometry of the random media can be related to the broadband sources of the probe light and the narrow-band spectral selection of the outgoing multiple scattered radiation. In this case, the coherence length of the detected radiation is determined by the selection conditions $l_c\approx {\lambda }_c^2/\Delta \lambda$ ($\Delta \lambda$ is the width of the spectral window used for the selection, and ${\lambda }_c$ is the central wavelength). Note that such broadband light source can be embedded in the probed medium. In particular, this approach can be provided by doping the probed medium with the fluorescent dye with follow-up pumping by the CW laser radiation [28, 29].

Interpretation of the empirical data obtained by the above cited modifications of the reference-free low-coherence interferometry is usually based on the approximate analytical or numerical inversion of integral transformation of the probability density function of the path differences $\Delta s$ for partial contributions to the optical field in the medium. This integral transformation contains the square of the coherence function module (or spectral density) for the probe light and establishes the relationship between the second-order statistical moment of the outgoing light intensity or its derivative such as speckle contrast and coherence length of the probe light for the used illumination and detection conditions. It is worth stressing that higher-order statistical moments of the detected speckle intensity must exhibit higher sensitivity to the variations in the optical transport properties of the examined medium compared to with the generally applied normalized second-order moment $〈I^2〉/{〈I〉}^2$ and its derivative speckle contrast $V={\sigma }_I/〈I〉=\sqrt{〈I^2〉-{〈I〉}^2}/〈I〉$. In particular, intensity statistics in the case of the narrow-band spectral selection of fluorescence radiation outgoing from the laser-pumped dye-doped random medium exhibits a strong influence of the wavelength of detected light on the skewness of intensity distributions [28]. The presumable interpretation of this effect is related to a strong wavelength dependence of the ratio of average $\Delta s$ value to the coherence length associated with amplification of spontaneous fluorescence emission for the long-path partial contribution of optical field in the medium under condition of high dye concentrations [28, 29].

The aim of this work is to consider a novel approach to the analysis of empirical data obtained using the reference-free low-coherence interferometry of the random media in application to the case of position-dependent narrow-band spectral selection of fluorescence radiation from the CW laser-pumped dye-doped random media. In our opinion, this case could be of great interest for numerous practical applications beginning from material science and ending biomedical optics. Additionally, this analysis can be useful for better understanding of statistical properties of partially coherent light multiple scattered by the random media.

2. Second- and third-order statistical moments of intensity fluctuations of multiple scattered partially coherent light: from the discrete scatter model to continuous path length distributions

We will consider the statistical properties of partially coherent light propagating in the multiple scattering random medium using the following approach.

In the case of superposition of $N$ partially coherent waves emitted by the continuous-wave source, the resultant time-averaged intensity in the arbitrarily chosen detection point can be expressed as

It should be taken into account that partial waves propagate into the probed multiple scattering medium along various random paths, and undergo random sequences of statistically independent scattering events. Each scattering event leads to random transformation in the amplitude of the propagating wave $E_i^{m+1}\left(t\right)=A_i^m{E}_i^m\left(t\right)$ with the factor $A_i^m$ dependent on the random scattering vector $q_i^m=k_i^{m+1}-k_i^m$, where $k_i^m$ and $k_i^{m+1}$ are wave-vectors of the $i$-th propagating wave before and after the $m$-th scattering event. The products $A_i^1\cdot A_i^2\cdot \cdot \cdot A_i^M$ and $A_{i^{\prime }}^1\cdot A_{i^{\prime }}^2\cdot \cdot \cdot A_{i^{\prime }}^{M^{\prime }}$for $i$-th and $i^{\prime }$-th waves are statistically independent random magnitudes. Thus, we can assume statistical independence of the values $E_i\left(t\right)=\sqrt{I_i\left(t\right)}$, $E_{i^{\prime }}\left(t\right)=\sqrt{I_{i^{\prime }}\left(t\right)}$, ${\phi }_i\left(t\right)$, and ${\phi }_{i^{\prime }}\left(t\right)$ for each pair of interfering partial waves. A similar assumption is one of the basic points of the discrete scattering model, which was successfully used to describe the various interference-based phenomena in the random multiple scattering media (see, e.g., [3, 5–8, 26, 27]).

Therefore the each time-averaged term $\overline{\sqrt{I_i\left(t\right)}\sqrt{I_{i^{\prime }}\left(t\right)}\cos \left[k\left(s_{i^{\prime }}-s_i\right)+\Delta \varphi \left(t,s_i-s_{i^{\prime }}\right)\right]}$ can be transformed in the following way

Let us consider the following scheme of the medium probing: the random values $I$ are sequentially measured at various positions of the detection point, and the obtained set of random values $\left(I_1,\mathrm{....},I_P\right)$ is used to calculate the second- and third-order statistical moments of spatial intensity fluctuations. In this case the averaging procedures should be carried out over the ensemble of all the possible path lengths $s_i$. The second-order moment of the spatial intensity fluctuation can be written as

Appropriateness of replacing the random terms $E_{0i}$ in Eqs. (5) and (6) by the deterministic unit value can be validated considering the general statistical properties of sums of the random phasors $E_{0i}\exp \left(j{\phi }_i\right)$. In the case of statistical independence of the random variables $E_{0i},{\phi }_i$ and large values of the phase terms ${\phi }_i\left(〈\left|{\phi }_i\right|〉>>\pi \right)$, the sum ${\displaystyle \sum _i{E}_{0i}\exp \left(j{\phi }_i\right)}$ of a large number of random phasors is characterized by identical Gaussian distributions of probability densities of real and imaginary parts. These parts are independent on statistical properties of amplitude terms $E_{0i}$. This results from the central limit theorem (see, e.g [31], section 1.5.6). Therefore we must expect that probability density distributions of the random values $I={\left|{\displaystyle \sum _i{E}_{0i}\exp \left(j{\phi }_i\right)}\right|}^2$, which correspond to the various probability density distributions of $E_{0i}$, are identical if their mean values $〈I〉$ are equal. This conclusion is illustrated (Fig. 1) by the results of statistical modeling of the probability density distributions $\rho \left(I/〈I〉\right)$ for the two random sums $I={\left|{\displaystyle \sum _i{E}_{0i}\exp \left(jk{s}_i\right)}\right|}^2$. The first sum is characterized by the uniform probability density distribution of $E_{0i}$ with the average value $〈E_{0i}〉$ equal to 1.0, and the standard deviation ${\sigma }_{E_{0i}}$ equal to 0.57. The second sum is characterized by the “deterministic” probability density $\rho \left(E_{0i}\right)=\delta \left(E_{0i}-1.0\right)$. Regarding the distributions $\rho \left(I/〈I〉\right)$, we can see that these two cases are identical. We also took into account the suppression of interference due to partial coherence of the contributions $E_{0i}\exp \left(jk{s}_i\right)$ by introducing the factors $g\left\{\left(s_i-s_n\right)/l_c\right\}$ into the products $E_{0i}{E}_{0n}\exp \left(jk{s}_i\right)\exp \left(-jk{s}_n\right)$. The coherence function $g\left\{\left(s_i-s_n\right)/l_c\right\}$ was taken in the form $g\left\{\left(s_i-s_n\right)/l_c\right\}=\sin \left\{\pi \left(s_i-s_n\right)/l_c\right\}/\left\{\pi \left(s_i-s_n\right)/l_c\right\}$, which corresponds to the conditions of our experiment (see below the section 4).

 

Fig. 1 The simulated distributions $\rho \left(I/〈I〉\right)$ in the cases of the deterministic value $\left|E_{0i}\right|=1$ (1, 3) and random uniform distribution of the phasor amplitudes (2, 4). The number of statistically independent phasors used in the simulation procedure is equal to 500; the sample size applied for the reconstruction of $\rho \left(I/〈I〉\right)$ is equal to 10000. The cases (1, 2) correspond to coherent summation of the random phasors ($〈s_i-s_m〉/l_c=$1.0⋅10⁻³) and the cases (3, 4) correspond to the partially coherent summation ($〈s_i-s_m〉/l_c=$1.0).

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Note that the simulated distributions $\rho \left(I/〈I〉\right)$ exhibit the expected peculiarities of the coherence-dependent statistical properties of $I$ (the negative exponential form of the probability density in the case of total coherence of contributions and the gradual narrowing of the probability density distribution $\rho \left(I\right)\to \delta \left(I-〈I〉\right)$ with transition to the incoherent summation of contributions). It should be mentioned that independence of phasor amplitudes from time is the necessary condition for the above consideration. This condition is met in the case of application of the intensity-stabilized CW source. Also, it can be provided using the appropriate detection time $T>>2\pi /\Delta \omega ~{\tau }_c,$ there ${\tau }_c$ is the correlation time of amplitude fluctuations. In the case of amplitude fluctuations with the correlation time comparable to $T$, the probability density distributions $I/〈I〉$ will deviate from the shapes presented in Fig. 1. As it is shown below (see section 4), the condition $T>>2\pi /\Delta \omega$ is reliably fulfilled in our experiment. Thus, in our case we can assume validity of the above consideration and application of the deterministic value $E_{0i}=1$ instead of the randomly distributed amplitudes of contributing phasors should not challenge the results of further analysis.

Equations (7) and (8) are transformed into the following forms

Finally, considering the transition from discrete to continuous distribution of $s_i$values under the condition $N\to \infty$, we obtain the following expressions for the normalized second- and third-order moments in case of stochastic interference of the polarized partial waves

Let us consider stochastic interference of non-polarized partial waves; in this case we introduce the statistically independent local random values of the intensity $I_{\perp }$ and $I_{II}$ for orthogonally polarized components of the propagating light, which are characterized by equal statistical properties. Thus, the normalized second- and third-order moments of the total intensity $I=I_{\perp }+I_{II}$ can be expressed as

3. Universality of the ratios of pathlength-dependent functionals

The numerical analysis of the functionals

 

Fig. 2 The plot of values $J_1\left(l_c,〈\Delta s〉\right)$ and $J_2\left(l_c,〈\Delta s〉\right)$ for the various shapes of $\rho \left(\Delta s,〈\Delta s〉\right)$; a – open triangles; b – closed circles; c – open circles. The $〈\Delta s〉/l_c$ratios are equal to: i – 0; ii – 0.318; iii – 0.637; iv – 0.955; v – 1.273; vi – 1.592; vii – 2.228; viii – 3.183. Dashed line – the power-law approximation of the relationship between $J_1\left(l_c,〈\Delta s〉\right)$ and $J_2\left(l_c,〈\Delta s〉\right)$.

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Universality of the ratios $J_1\left(l_c,〈\Delta s〉\right)/J_2\left(l_c,〈\Delta s〉\right)$ can be described with adequate accuracy by the power-law approximation

The established relationships between $M_2,M_3$ (Eq. (13) for the non-polarized light or Eqs. (9) and (10) in the case of polarization discrimination of the detected multiply scattered light) and the integrands $J_1\left(l_c,〈\Delta s〉\right),J_2\left(l_c,〈\Delta s〉\right)$ (Eqs. (14) and (16)) allows for quantification of the general tendencies relating suppression of the stochastic interference modulation of the quasi-monochromatic light in the random multiply scattering media. The suppression can be caused by broadening the emission spectrum of the probe light or/and broadening of the path length distribution for the light propagating in the medium. These relationships can be used for rapid recovery of the characteristic scales of light propagation in the medium ($〈\Delta s〉$ and $〈s〉$) using the measured values $M_2$ and $M_2$ depending on the experimental conditions (e.g., the used wavelength). This approach is discussed below in sections 4 and 5.

4. Experimental data

The empirical data used in the further analysis were obtained using experimental techniques and model samples similar to those previously described in [28, 29]. Arrangement of the experiment is shown in Fig. 3. The fluorescent coarse-grained layers were pumped by the CW laser radiation at 532 nm. The layers were composed by randomly distributed coarse silica grains of irregular shape. The average size of grains was equal to 156 μm and their volume fraction in the layers was approximately equal to 0.35. Thickness of the layers $L$ was equal to 2.0 mm. Fluorescence of the probed layers was provided by their saturation with the high-concentration Rhodamine 6G water solution (the weight fraction of Rhodamine 6G in the solution was equal to 0.005). The laser beam was expanded using the concave lens to cover the whole surface of the sample; the estimated pumping fluence rate at the sample surface was about 80 mW/cm².

 

Fig. 3 Arrangement of the experiment. 1 – the CW pumping laser (${\lambda }_1=$532 nm, the output power is 50 mW); 2 – the concave lens with the focal length of −200 mm; 3 – the sample under study; 4 – the confocal system; 5 – the pinhole diaphragm; 6 – the monochromator; 7 – the processing unit (PC). The fluorescence radiation with the wavelength ${\lambda }_2$ is detected while scanning along the $x$direction; $\Delta z$ is the scan depth. Units 4 −7 are parts of the Horiba Jobin Yvon LabRam HR800 assembly.

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The fluorescence radiation was detected from the various points of the probed layer and from the various depths ranging from 50 μm to 200 μm with respect to the layer surface using the confocal detection scheme with the narrow-band spectral selection of the captured light. The Horiba Jobin Yvon LabRam HR800 assembly was used as the detection system; the width of the spectral window was equal to 0.052 nm and the signals were detected in the spectral range from 560 nm to 700 nm. The transverse size of the sampling volume in the confocal scheme was not larger than 2 μm. The choice of the detection zones within the layers was provided using transversal scanning of the probed samples with respect to the axis of the detection system with the scan step equal to 10 μm; the length of the scan traces at various depths was equal to 1000 μm.

The intensity acquisition time $T$ for the chosen wavelength ${\lambda }_2$ at the fixed detection position $x$ was equal 0.1 s; this value significantly exceeds the parameter $2\pi /\Delta \omega ~$10⁻¹¹ s, which is the measure of quasi-monochromaticity of the detected radiation. Note that the sequences of the intensity values acquired under the conditions ${\lambda }_2=const$, $x=const$ for the non-overlapping time intervals $T$ exhibit the appropriately high stability in time (the normalized standard deviation ${\sigma }_I/〈I〉$ of the intensity values in these sequences does not exceed 0.02).

As an example, the inset in Fig. 4 displays typical distribution of the fluorescence intensity along the arbitrarily chosen scan trace located at the depth of 150 μm below the layer surface. After collecting the raw data sets $I_{i,j,h,\lambda }$ ($i$ denotes the sampling point position at the scan trace $j$ for the given scan depth $h$; $1\le i\le 100;1\le j\le 10$; $h=$1 (50 μm), 2 (100 μm), 3 (150 μm), 4 (200 μm); $\lambda$ denotes the chosen vavelength) the first-, second-, and third-order moments of spectrally-selected fluorescence intensity were calculated using the averaging procedure over the scan traces and scan depths:

 

Fig. 4 The values of second- ($M_2$) and third-order ($M_3$) moments of fluorescence intensity versus the wavelength; 1 - $M_2$; 2 - $M_3$. Inset: the distribution of fluorescence intensity at $\lambda =$ 575 nm along the arbitrarily chosen scan trace; the scan depth is 150 μm.

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Figure 4 displays the obtained dependencies of the normalized second- and third order moments $M_2\left(\lambda \right)=〈I_{\lambda }^2〉/{〈I_{\lambda }〉}^2$, $M_3\left(\lambda \right)=〈I_{\lambda }^3〉/{〈I_{\lambda }〉}^3$ on the wavelength; the error bars correspond to the confidence level 0.9. The remarkable feature is that the spectral positions of $M_2\left(\lambda \right)$ and $M_3\left(\lambda \right)$ minima are significantly red-shifted with respect to the fluorescence maxima positions varied from one detection point to another in the range from 580 nm to 610 nm. Additionally, the degree of the linear polarization $P_L=\left(I_{II}-I_{\perp }\right)/\left(I_{II}+I_{\perp }\right)$ of the outgoing fluorescence radiation was evaluated for the used detection geometry. The value $P_L$ is insufficient and does not exceed 0.05 in the whole range of the examined wavelengths (from 560 nm to 700 nm). Therefore the fluorescence light can be considered as the totally depolarized and the expressions (10) are appropriate for the follow-up analysis of intensity moments.

5. Discussion of the results

The probability density distributions $\rho \left(\Delta s,〈\Delta s〉\right)$ of fluorescence radiation in the analyzed systems were simulated using the Monte-Carlo procedure similar to the described in [26–29]. The surface source of fluorescence photons was considered because of high absorbance of the pumping laser light by the saturating dye solution. Thus, we have the shallow CW source of fluorescence radiation in the medium; spectrally selected light waves emitted by this source interfere into the detection zone after passing through the medium along the various random paths $s_i$. We used the principle of weighted photon packets for the simulation procedure; in concordance with this principle, the single photon wandering in the probed layer is considered as the photon packet with the initial weight 1, which propagates along the same trajectory in the model layer. The model layer has the same thickness, effective refractive index, and scattering properties (the scattering coefficient ${\mu }_s$ and the reduced scattering coefficient ${{\mu }^{\prime }}_s$) as the simulated layer, but is characterized by zero absorbance. The influence of the non-zero absorption coefficient can be taken into account at the stage of $\rho \left(s\right)$ evaluation via summation of the unit weights of photon packets with the propagation paths within $s$ and $s+\Delta s$($s>>\Delta s$) with the follow-up renormalization of the sum by the Bougier factor $\exp \left(-{\mu }_{a}s\right)$, where ${\mu }_a$ is the absorption coefficient of the probed system.

We used the following assumptions for the Monte-Carlo based numerical analysis of the shapes of path length distributions for detected fluorescence radiation in the examined samples:

The shapes of the simulated path length probability density distributions $\rho \left(s\right)$ (see inset in Fig. 5) significantly depend on the factor $L{{\mu }^{\prime }}_s$; in the case of $L{{\mu }^{\prime }}_s>>1$ the distributions exhibit the power-law behavior $\rho \left(s\right)~s^{-\alpha }$ in the region of small $s$ values and exponential decay in the case of large propagation paths. The exponent $\alpha$ is close to 1.5 and crossover between the power-law and exponential regimes of the probability density decay occurs around the values $s\approx$5 ÷ 10 mm. In general, this behavior corresponds to predictions of diffusion approximation for the temporal response of highly scattering slabs probed by short light pulses in the backscattering mode (see, e.g., [32]). The cases of “optically thin” layers with $L{{\mu }^{\prime }}_s\le 1$ are featured by the expressed probability density peaks associated with fluorescence waves back-reflected from the bottom boundary of the probed layer. A strongly suppressed rudimentary peak marked by the arrow is caused by the double reflection from the bottom boundary; the peaks associated with higher-order reflections (triple, quadruple, etc.) are blurred due to path length randomness and relatively small reflectivity of the boundaries. Appearance of the single- and double-reflection features on the probability density distributions is caused by specific conditions of the fluorescence light propagation (the large scattering anisotropy and small values of $L{{\mu }^{\prime }}_s$) applied in the simulation procedure.

 

Fig. 5 Distributions of the probability density of path length difference $\rho \left(\Delta s\right)$; 1 - $L{{\mu }^{\prime }}_s$ = 12; 2 - $L{{\mu }^{\prime }}_s$ = 1.6; 3 - $L{{\mu }^{\prime }}_s$ = 0.4; 4 - $L{{\mu }^{\prime }}_s$ = 0.16. Inset: distributions of the path length probability density $\rho \left(s\right)$; 1 - $L{{\mu }^{\prime }}_s$ = 12; 2 - $L{{\mu }^{\prime }}_s$ = 0.16.

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Here, we also see the exponentially decaying long-path tail related to the components of the fluorescence field undergoing multiple reflections from the layer boundaries and propagating inside the layer over distances significantly exceeding the layer thickness. Note that this feature is characteristic for the case of backward detection of the diffusing components propagating in the optically thin random layers [33].

Figure 5 displays the retrieved probability density distributions $\rho \left(\Delta s\right)$ for the case of “optically thick” layer ($L{{\mu }^{\prime }}_s>>1$; 1) and cases of “optically thin” layers ($L{{\mu }^{\prime }}_s\le 1$; 2, 3, 4). The latter cases are characterized by the exponential decay of $\rho \left(\Delta s\right)$ in the wide range of $\Delta s$ values. This behavior results from the relationship between $\rho \left(s\right)$ and $\rho \left(\Delta s\right)$: $\rho \left(\Delta s\right)=2{\displaystyle \underset{0}{\overset{\infty }{\int }}\rho \left(s\right)\rho \left(s+\Delta s\right)ds}$; in the case of the exponentially decaying path length probability density $\rho \left(s\right)~\exp \left(-Ks\right)$, where $K$ is the normalization constant related to the average path length density, and the probability density of the path length difference can be written as $\rho \left(\Delta s\right)~\exp \left(-K\Delta s\right){\displaystyle \underset{0}{\overset{\infty }{\int }}\exp \left(-2Ks\right)ds}$.

The simulation results allow us to interpret the above mentioned experimentally observed ergodicity and statistical homogeneity of intensity distributions obtained from the various scan depths in the range from 50 μm to 200 μm. These statistical properties (ergodicity and statistical homogeneity) manifest themselves in the close values $M_2\left(\lambda \right)$ and $M_3\left(\lambda \right)$ for the intensity samplings captured at the fixed wavelength $\lambda$ from the different depths. Indeed, the expected average values $\Delta s$ for the probed system and the used illumination and detection conditions significantly exceed the used scan depths.

Figure 6 displays the relationship between the second- and third-order statistical moments of fluorescence intensity for the examined system compared to the theoretical dependence $M_3\left(\lambda \right)=f\left\{M_2\left(\lambda \right)\right\}$for exponentially decaying probability densities of the path length differences and the rectangle spectral window with the fixed width. The latter dependence was retrieved using Eqs. (13) and the values $J_1$ and $J_2$ from the data set (b) in Fig. 2. A good agreement between the empirical and simulation data is obvious. Figure 7 shows the dependence of the average propagation path $〈s〉$ of fluorescence radiation in the examined layers on the wavelength $\lambda$, which was recovered using the above mentioned intensity moments analysis. The theoretical values $M_3=f\left(M_2\right)$, which are nearest to the average empirical values marked by the open circles in Fig. 6, were used to retrieve $〈\Delta s〉/l_c$ and, correspondingly, $〈s〉$ magnitudes.

 

Fig. 6 The values of third-order moments of fluorescence intensity versus the values of second-order moment; 1 – empirical data presented in Fig. 3; 2 – theoretical dependence for the exponential form of $\rho \left(\Delta s\right)$. The ratios $〈\Delta s〉/l_c$ are equal to: i – 0; ii – 0.318; iii – 0.637; iv – 1.273; v – 2.546.

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Fig. 7 The recovered values of the average path length propagation of fluorescence radiation in the examined layers versus the wavelength. 1 – the recovery using the considered intensity moments analysis; 2 – the recovery using the numerical inversion of the functional $J_1\left(l_c,〈\Delta s〉\right)$ (in accordance with [28, 29]).

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It should be noted that such recovery procedure based on the a-priori knowledge of the shapes of $\rho \left(\Delta s\right)$ distributions is sufficiently simpler and faster than the numerical inversion of the integral transform ${\displaystyle \underset{0}{\overset{\infty }{\int }}{\left|g\left(\Delta s,l_c\right)\right|}^2\rho \left(\Delta s\right)d\left(\Delta s\right)}$ previously used in [27–29] and based on the inverse Monte-Carlo procedure. For reference, some $〈s〉$ values obtained by means of this numerical inversion technique, are also plotted in this figure.

The error bars for the data set (2) correspond to the confidence level 0.9 and are related to uncertainties in the empirical data $M_2$ and in the simulated path length distributions used for the recovery procedure. In our opinion, consistency between the two recovered data sets is reasonable; some discrepancies can be caused by a slightly inappropriate choice of the input parameters (the mean transport free path, the scattering anisotropy parameter, and the effective refractive index) in the numerical (Monte-Carlo) inversion of ${\displaystyle \underset{0}{\overset{\infty }{\int }}{\left|g\left(\Delta s\right)\right|}^2\rho \left(\Delta s,〈\Delta s〉\right)d\left(\Delta s\right)}$ in the second case.