1. Introduction

Speckle field measurement is, now, a usual tool to characterize the dynamic behavior of scattering media like, for example, the flow or Brownian movement of scatterers. Indeed, well-known techniques such as Dynamic Light Scattering [1], Diffusing Wave Spectroscopy [2], LAser Speckle Contrast Analysis (LASCA) [3] based on the temporal characterization of fluctuations in speckle intensity are now widely investigated for further applications in biology and medicine [4].

Nevertheless, researchers in biomedical optics are equally interested in the determination of the optical parameters of a scattering medium, i.e. the scattering, absorption and anisotropy coefficients denoted µ_s , µ_a and g, respectively [5]. So, in a previous study of transmission through weakly-scattering media, we showed that the speckle size is affected not only by µ_s , but also by the scatterer dimensions [6].

Here, the issue being strongly-scattering media, e.g. milk, blood, skin,…. we chose a backscattering configuration to investigate the effects of either circular, or linear, incident polarization on the size of backscattered speckles. We also analyzed the transmitted- or cross-polarization states so as to assess changes in speckle size according to the scattering parameters. Though speckle polarization analysis has often been used to study metal surfaces or turbid media [7, 8, 9], the investigations reported here are of a different nature since they use the spatial properties of speckle. Thus, Section II will detail the polarization analysis we made and describe our experimental conditions. Section III will report on the experimental data collected from different media before discussing them. Our conclusions will be drawn in section IV.

2. Experimental method

The size of a speckle can be determined from calculations of the normalized autocovariance function of the intensity speckle pattern got in the observation plane (x,y). This function, denoted c_I (Δx, Δy), corresponds to the normalized autocorrelation function of the intensity; it has a zero base, and its width provides a reasonable measurement of the “average width” of a speckle [10]; c_I (Δx, Δy) is calculated from the intensity distribution of the measured speckle, I, as described in [6]:

where FT is the Fourier Transform, < > is a spatial average, c_I (Δx,0) and c_I (0, Δy) are the horizontal and the vertical profiles of c_I (Δx, Δy), respectively.

Let us term dx the width of c_I (Δx, 0) so that c_I (d_x /2, 0)=0.5 and dy the width of c_I (0, Δy) such as c_I (0, dy/2)=0.5.

Figure 1 illustrates the experimental set-up. The 7-mW HeNe laser used emits a 1.12 mm-wide polarized (linear) beam at I₀ /e ² where I₀ is the maximum laser intensity at the 632.8 nm wavelength with a coherence length of about 20 cm.

A CCD camera records the medium-produced speckle field. The CCD imager contains 788×268 pixels of size 8 µm×8 µm. When a newly acquired image is digitized, the analog-to-digital converter assigns an intensity value (gray level) in the range of 1 to 1024 (10-bit precision).

To record the speckle, one should be aware of the Brownian motion of particles in the medium. These movements induce a random agitation of the speckle, which is termed “boiling speckle”. The time scale of the speckle intensity fluctuations is given by the correlation time, which must be longer than the image acquisition time to avoid recording a blurred speckle [4]. The correlation times measured for our samples exceeded 0.1 ms; we also used a CCD camera with a time exposure in the range 0.1–60 ms to acquire images at 0.1 ms.

Another consideration of importance in recording the speckle data is the size of speckle on the CCD array: the speckles must be large as compared to the pixel size so as to resolve variations in speckle intensity [11]. Moreover, for a meaningful statistical evaluation each image must contain many speckles. These conditions were fulfilled by setting the surface of the medium under study at 15 cm (D) from the CCD camera.

The laser, sample and CCD constituting the experimental set-up were all placed in the horizontal plane with an angle θ=45° between the CCD camera and the optical axis (see Fig. 1). The value of θ was chosen to prevent the measurement from being spoilt by specular reflections from the surface of sample like skin. As previous measurements had taught us that varying θ affected dy (vertical speckle size) by slight changes contrarily to those undergone by dx (horizontal speckle size), we decided to use dy as the speckle size-characterizing parameter. It is worth noting that with a diffusing surface, dx is increasing with θ, while dy remains constant between 0° and 90°. Indeed, for a circularly-illuminated surface, we have dy=1.22λ D/D_e and dx=1.22λ D/(D_e cosθ), where λ is the wavelength of the light source and D_e is the diameter of the circularly-illuminated area [12]. When a scattering medium is illuminated the variations of dx and dy mentioned above are roughly verified by applying these relations while considering that D_e corresponds to the diameter of the surface of the illuminated scattering volume as seen by the CCD camera.

These considerations led us to place the CCD camera so that we got the highest number of pixels, 788, along the vertical direction.

 

Fig. 1. Top view of the experimental set-up, P1 and P2 are linear polarizers, QWP1 and QWP2 are quarter-wave plates. Sample dimensions are 1×1×4 cm.

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To carry out our experiments the scattering media were prepared with deionized water mixed with either semi-skimmed milk, or skimmed milk, or whole milk or solutions of polystyrene-microspheres of three different diameters (0.20, 1.44 and 3.17 µm) from Polyscience Inc.; these microspheres are sold in solutions of deionized water, and their number per cubed meter, N₀ , is known (see Table 1). We investigated three kinds of milk because of differences in their scattering properties after removal, by skimming, of large fat particles; furthermore, milk contains also small particles of casein (0.02 to 1 µm). The scattering coefficient, µ_s , being affected by any change in milk and solution of microspheres concentrations, it was measured with respect to the concentration of milk or microspheres solution in the medium, c, expressed in percent. For solutions of polystyrene microspheres or milk, at the experimental wavelength, the absorption coefficient is insignificant compared to the scattering coefficient, and, µ_s versus c is therefore determined from the Beer Lambert law in single scattering regime [13]. It was found to be equal to 0.42, 1.40 and 3.00 c in cm^-1 for skimmed, semi-skimmed, and whole milks, respectively. Table 1 gives the scattering coefficients of the polystyrene-microspheres measured with respect to c, with a ±5% precision; it also lists the particle diameters, d, as well as the anisotropy factor, g, issued from Monte Carlo simulations; g is the mean scattering cosine, g=<cos α> where α is the scattering angle; g=1 corresponds to total forward scattering, whereas g=0 is indicative of isotropic scattering. The refractive indices used were 1.59 and 1.33 for the spheres and medium, respectively. One should note, that as already done in a previous report, we consider scatterers to be small when g≤0.3 and large for g≥0.7 [14]. In this study, the 0.2-µm microspheres were thus considered as small scatterers.

Table 1. Scattering coefficients of polystyrene-microspheres measured and anisotropy coefficient calculated (Mie calculation).

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From two monodisperse solutions of 0.20- and 1.44-µm microspheres, whose concentrations had been adjusted so that both exhibited the same µ_s , we prepared two scattering samples by mixing them at the ratios 1:1 and 1:5 corresponding to g=0.676 and g=0.851, respectively.

Each of these samples was, first, illuminated by a linearly polarized laser beam, then by a circularly polarized one. As stated by Morgan and Ridgway, Fig. 2 shows that, with a linear polarization, the light emerging from a medium is composed of photons in mixture; some of them have still their original polarization state because of the little number of scattering events they have undergone, whereas the other ones have a random polarization further to numerous scattering events [15]. On the other hand, a scattering medium illuminated with a circularly-polarized light gives rise to three types of photons: some of them are helicity-flipped by mirror reflection, others keep their original polarization state as previously and the last ones are depolarized by many scattering events [15].

 

Fig. 2. Different types of photons emerging from a scattering medium for an incident light linearly or circularly polarized [15].

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For each sample under study, dy was measured four times under different incident polarizations and polarization analysis conditions:

i) when the laser beam was linearly polarized (vertical polarization), the quarter wave plates were removed, and dy^pl was measured in the case of parallel linear polarizers, P₁ and P2; on the other hand, for crossed P1 and P2, the measured parameter was dy^cl .

ii) with a circularly-polarized beam, the quarter-wave plates were kept and dy^tc was determined when the incident polarization was transmitted to the CCD, and dy^fc when the polarization transmitted to the CCD was circular and helicity-flipped. Whatever the experimental conditions, the values of dy are given at the nearest 0.1 µm.

In order to compare our measurements, we calculated the degree of polarization D_p . It allows one to assess the predominant type of photons. It is worth recalling that the measurement of depolarization properties is a well known method of analysis to probe scattering media like, for example, skin [16, 17].

D_p can be determined from the measurement of the speckle mean intensity through the polarization analysis as follows:

where I_p is the mean intensity measured at the CDD surface when the light of the incident polarization is fully transmitted, and I_c is the mean intensity when the light totally transmitted results from either a cross linear, or helicity-flipped circular polarization. D_pl and D_pc the linear and circular polarization degrees, respectively, were thus determined at the nearest 0.01.

3. Results and discussion

Table 2, where samples are ranked in the ascending order of g, lists the speckle sizes found for the three polystyrene-microspheres of diameters 0.20, 1.44 and 3.17 µm and the two mixtures. For the first two series of microspheres, the measurements were made at two different µ_s . Table 2 also gives the reduced scattering coefficient ${\mathit{\mu }}_s^{’}$ for all samples; ${\mathit{\mu }}_s^{’}$=µ_s (1-g) is introduced as the equivalent isotropic scattering coefficient of an otherwise anisotropically-scattering medium.

Table 2. dy in µm and polarization degree obtained for all the samples. Mix (1:1): 1:1 volume ratio mixtures of 0.20-µm microspheres (µ_s =140 cm^-1) and 1.44-µm microspheres (µ_s =140 cm^-1), Mix (1:5): 1:5 volume ratio mixtures.

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At µ_s =140 cm^-1, the speckle size measured for the small scatterers is larger than the one got for large scatterers. Figure 3 depicts the speckle pattern obtained for both sizes of scatterers in linear polarization analysis and clearly shows that dy^pl (d=0.20 µm) > dy^pl (d=1.44 µm).

 

Fig. 3. (a) and (b) illustrate the speckle produced by the 1.44-µm microspheres and 0.20-µm microspheres (linear polarization) when µ_s =140 cm^-1.

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Moreover, the values got at other scattering coefficients (µ_s =930 cm^-1 for 1.44-µm microspheres and µ_s =42 cm^-1 for 0.20-µm microspheres) highlight an increase in the speckle size with higher scattering coefficient.

When µ_s is kept constant, any decrease of the anisotropy coefficient g (Table 2) goes along with an increase of dy^tc . Finally, the speckle size is affected by the scattering and anisotropy coefficients. One should note that, whatever the sample under study, we always found dy^pl > dy^tc and dy^fc > dy^tc for 0.20-µm microspheres, and dy^fc < dy^tc for 1.44-µm microspheres. So for monodisperse polystyrene-microspheres, the comparison between dy^fc and dy^tc gave information about the scatterer size when the scattering coefficient was unknown: dy^fc < dy^tc means large-size scatterers, and dy^fc > dy^tc indicates small ones. On the other hand, in linear polarization dy^pl was always greater than dy^cl , whatever the scatterer. Figure 4 summarizes the results produced by the 0.20- and 1.44-µm microspheres.

 

Fig. 4. Scheme summing about the evolution of data produced by small and large microspheres (g=0.323 and 0.928, respectively).

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In a mixture of small and large scatterers, the latter decrease the size of speckle, even though the mixture behaves like the monodisperse sample composed of small scatterers with dy^fc > dy^tc . Nevertheless, the anisotropy coefficient shows that the light is forward scattered alike scattering by monodisperse large microspheres. Thus, depending on the distribution in the scatterer size, two samples with the same scattering and anisotropy coefficients may produce different speckle sizes. Indeed, a comparison of the experimental values got for the mixture 1:5 and the monodisperse 3.17-µm microsphere sample (similar scattering parameters) gives dy^fc < dy^tc for the latter (large particles-like behavior) and dy^fc > dy^tc for the former (small particles-like behavior).

As previously mentioned in section II, the polarization of photons emerging from the medium can be assessed from D_p : indeed, under circularly-polarized light conditions, D_pc >0 indicates that many photons have kept the same polarization state; when D_pc < 0 most of them are helicity-flipped (mirror effect: for a mirror, D_pc =-1). With large particles, the polarization of a circularly-polarized light was kept after scattering events more numerous than those undergone by linearly-polarized photons. Table 2 data agree with those available in the literature: for large particles: we, indeed, measured D_pc >0 and D_pl =0. With small particles, the light being scattered with the same probability in any direction, the linear polarization is favored contrary to the circular one; in this case, the linear character of linearly-polarized light is not affected by backscattering, which reverses the helicity of circularly-polarized light and randomizes it more rapidly [15, 18]. So, for small particles, we measured D_pc <0 and |D_pc | < D_pl .

Explaining how a speckle size will evolve according to the anisotropy and scattering coefficients is difficult. It is worth recalling that the size of speckle in reflection on a diffusing metal surface depends on the dimension of the light spot responsible for the surface illumination according to the Van Cittert and Zernike theorem [10]; moreover, illumination of a small area produces a speckle of large size. Our experimental set-up permitted us to measure dy=90 µm for a metal surface of about 1 µm in roughness, and this speckle size is the largest one that we can measure. With scattering media, the expected mean speckle size in the far diffraction zone is determined by the characteristic size of the scattering volume [19]. Particularly the speckle size depends on the surface of the backscattered-light spot i.e. the surface of the scattering volume as seen by the CCD camera (in application of the Van Cittert and Zernike theorem). Moreover, from solutions of monodisperse-sized polystyrene spheres Hielscher, Mourant, and Bigio evidenced an increase of spatial pattern size of backscattered light with increasing g and decreasing µ_s [20]. Consequently, as previously shown in this study for monodisperse-sized polystyrene-spheres and in agreement with the Van Cittert and Zernike theorem, the speckle size is becoming smaller for higher g and lower µ_s . Moreover, any increase of ${\mathit{\mu }}_s^{’}$ causes a reduction of the backscattered spot width [21]. The plot of dy^pl , dy^cl , dy^tc and dy^fc evolution versus ${\mathit{\mu }}_s^{’}$ (Fig. 5) highlighted a roughly linear increase of dy^cl and dy^tc with ${\mathit{\mu }}_s^{’}$ in agreement with the Van Cittert and Zernike theorem. On the other hand, when ${\mathit{\mu }}_s^{’}$=24.3 cm^-1 and ${\mathit{\mu }}_s^{’}$=66.9 cm^-1 the evolution of dy^pl and dy^fc versus µ_s ’ shows drops in the values of dy^pl and dy^fc . Indeed, ${\mathit{\mu }}_s^{’}$=24.3 cm^-1 and ${\mathit{\mu }}_s^{’}$=66.9 cm^-1 were respectively obtained with 3.17- and 1.44-µm microspheres and, in these two cases, contrarily to small microspheres and mixtures, dy^fc was found to be less than dy^tc together with close values for dy^pl and dy^cl . In other words, dy^pl and dy^fc are not governed by only ${\mathit{\mu }}_s^{’}$ since they are affected by the polarization analysis and, thus, by the size of particles.

 

Fig. 5. Evolution of speckle size versus the reduced scattering coefficient for all samples.

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Thus, the speckle size is indicative of the scattering characteristics of a medium, and a polarization analysis evidences “the polarized scattering volume”, but this parameter depends on the type of polarization as well as on the scattering medium under analysis. For example, Fig. 2 shows that the incident polarization-keeping photons have traveled at a depth closer to the surface than the depolarized ones. This property induces a small polarized volume (large speckle size) for the former, and a large depolarized volume (small speckle size) for the latter. The noticeable difference observed with the 0.20-µm microspheres between the sizes of the cross linear and linear polarized speckles (46.3 and 61.0 µm, respectively, for µ_s =140 cm^-1) evidences that the incident polarization-maintaining light traveled nearer the surface than the depolarized light; the relation dy^fc > dy^tc indicates a flip in helicity for the circular polarization in agreement with the measurements of polarization degrees (Table 2).

Table 3 gives the size of speckles (dy^pl , dy^cl , dy^tc and dy^fc ) and the polarization degree in the case of skimmed, semi-skimmed and whole milk. Whatever the milk, the small particles of casein induce dy^fc > dy^tc . Moreover, when µ_s =42 cm^-1, the more the milk is skimmed, the larger the speckle is; a higher concentration in large particles reduces the speckle size in agreement with the experimental data reported above for mixtures of small and large microspheres.

Table 3. dy in µm for different milks, µ_s =42 cm^-1.

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Figure 6 illustrates the variations of speckle size (dy^pl , dy^cl , dy^tc and dy^fc ) versus µ_s and the polarization degree versus µ_s for semi-skimmed milk. As shown above, the speckle size is increasing with the scattering coefficient. Moreover, the evolution of the polarization degree is limited when the scattering coefficient is increasing (see Fig. 6 b). So, our experimental set-up highlighted that any change in scattering coefficient affected more the speckle size than the polarization degree.

 

Fig. 6. Evolution of speckle size a) and polarization degree b) versus scattering coefficient for semi-skimmed milk.

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Figure 7 depicts the evolution of dy^pl - dy_cl versus the scattering coefficient for semi-skimmed milk and evidences the small and random fluctuations of dy^pl - dy^cl .; the difference between dy^fc and dy^tc was found to vary more than the one between dy^pl and dy^cl (data not shown). Thus, a variation of the scattering coefficient will have no effect on the difference between dy^pl and dy^cl only because it depends on the scatterer size-distribution. Table 2 exhibits similarity in the behaviors of the 0.2-µm microspheres at µ_s =42 cm^-1 and µ_s =140 cm^-1 (dy^pl - dy^cl =14.7 µm).

 

Fig. 7. Variations of dy^pl - dy^cl versus the scattering coefficient for semi-skimmed milk.

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Moreover, for the samples containing scatterers of different sizes (milk and mixtures of polystyrene-microspheres), Table 4 shows a decrease of dy^pl - dy^cl with increasing concentration of large scatterers. According to Table 2, for big scatterers (for example, 3.17-µm microspheres and the 1.44-µm ones at µ_s =140 cm^-1 and µ_s =930 cm^-1), the differences between dy^pl and dy^cl are small and around 2 (2.4 and 2.1).

Table 4. dy^pl - dy^cl in µm for sample with distribution in size of scatterers. Mix (1:1): 1:1 volume ratio mixtures of 0.20-µm microspheres (µ_s =140 cm^-1) and 1.44-µm microspheres (µ_s =140 cm^-1), Mix (1:5): 1:5 volume ratio mixtures.

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Finally, Table 5 gives the experimental data got with samples of human blood and pig skin. Optical characteristics for blood at λ=633 nm are typically µ_a =25 cm^-1, µ_s =400 cm^-1 and g=0.98 [5]. The high value of g and the dominance of large scatterers in blood both explain why dy^fc is less than dy^tc in agreement with previous data. Conversely, the skin biopsies gave dy^fc > dy^tc because of the wide size distribution of scatterers.

Table 5. dy in µm for human whole blood and pig skin

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4. Conclusion

We studied the size of backscattered speckles produced by strong scattering media. We showed its dependence with scattering and anisotropy coefficients of the analyzed scattering medium. For a given anisotropy coefficient, any rise of the scattering coefficient widened the speckle size. Similarly, at a given scattering coefficient, the size of the speckle grew when the anisotropy coefficient was lowering.

We also investigated the effect of linearly-polarized laser beam on the speckle size by analyzing the linear, or cross linear, polarization as well as the circularly-polarized one through analysis of transmitted circular or helicity-flipped circular polarization. The comparison between the “transmitted circular” speckle size (denoted dy^tc ) and the “flipped helicity circular” speckle size (denoted dy^fc ) permitted us to discriminate the scattering media with respect to the size of scatterers, even in the case of an unknown scattering coefficient. Indeed, with large particles dy^tc > dy^fc , whereas for small ones, dy^tc < dy^fc .

The speckle measurements carried out on mixtures of large and small microspheres produced results similar to those of a monodisperse sample of small microspheres though the anisotropy coefficient of mixtures was found to be alike that of a sample of large monodisperse microspheres. These results agree with those reported by Gosh et al. further to experiments made in transmission [14]. These authors, indeed, concluded that the depolarization behavior of light in a medium containing scatterers of various sizes would differ from that of a matched monodisperse scattering sample of similar anisotropy and scattering coefficients.

Moreover for samples characterized by a scatterer size-distribution, the difference between the size of “linearly polarized” speckle and that of the “unpolarized” one (dy_pl and dy_cl , respectively) will remain constant when the scattering coefficient is varying, and will decrease at higher concentration of large scatterers. So, the measurement of a polarized speckle size should allow one to discriminate among media, even when the scattering coefficient is unknown.

Further investigations are still required to gain much insight into our experimental data, and theoretical studies are needed on media of known optical parameters to understand the evolution of speckle size.

Such measurements of speckle size in polarization could be used to evidence any scattering change, i.e. modification in the number, size, shape or structure of scatterers, and applied to biomedical studies for gaining valuable information beneficial for diagnosis. We envision carrying out measurement on irradiated samples of skin.