The evolution of the intensity and polarization state of radiation transmitting through a medium is often treated in terms of a $ 4 \times 4 $ Mueller matrix $ {\bf M}(z) $ that evolves according to the differential equation [1–3]

Ossikovski and Arteaga found the physical interpretation of the matrix $ {\bf m} $ to be related to the non-depolarizing mean properties $ \langle {\bf m}\rangle $ and the depolarizing variances $ \langle \Delta {{\bf m}^2}\rangle $ of the material properties [7]. Using elementary fluctuation theory and assuming that the material is homogeneous along the propagation direction (long coherence length limit), it was found that the differential Mueller matrix is given by Refs. [7,8]:

 

Fig. 1. Schematic illustrating the stacking of layers in transmission: (a) traditional view, where only forward transmission is considered, and (b) layers interacting through reflection or scattering.

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Several studies have indeed observed quadratic evolution of depolarization, at least for small thicknesses [9–13]. However, in a number of these studies [9–11,13], the materials being studied, or how their thickness was varied, would not have been expected to be in the long coherence length regime. Charbois and Devlaminck recognized this issue and developed an approach by which a linear or quadratic dependence can be observed [10]. They constructed measurements that showed an initial quadratic effect for thin samples and a linear effect for thick samples, in expectation of the transition from a thickness shorter than the coherence length to one greater than it. Agarwal et al. recognized that the results of their measurements were incompatible with a finite thickness of material being represented by the product of Mueller matrices for subdivided thin layers [9]. Yet, different sample thicknesses were created by stacking $ {\text{TiO}_2} $-impregnated polyvinyl chloride blocks in Ref. [9] and by layering adhesive tape in Ref. [11]. Within fluctuation theory, if each layer of thickness $ {z_0} $ is homogeneous along the propagation direction, the Mueller matrix of $ n $ independent layers (i.e., long coherence within a layer, but no coherence between layers) would be

In this Letter, we suggest an alternative approach that explains the observed growth of depolarization with thickness, even in the short coherence length regime. As mentioned above, all of the previous analyses have assumed that radiation is passing only sequentially through the medium, as illustrated in Fig. 1(a). If the material is diffusely scattering, some radiation reflects backward, so that an approach illustrated in Fig. 1(b) would be more appropriate. The approach we take is a polarimetric extension of the theory of Kubelka and Munk (KM) for reflectance (but applied here in transmittance) and represents the simplest approximation to the radiative transfer equation [14,15].

We begin by assuming that there are two streams, one propagating forward and one propagating backward, and that there is coupling between the two due to scattering. Following KM, but generalizing the absorption and scattering coefficients with matrices, we have

As an illustrative example, we consider a material with absorption coefficient $ \alpha $ and birefringence $ r $, so that

We further choose $ \alpha { = 10^{ - 6}} $ (non-zero to prevent singular intermediate matrices), $ r = 0.1 $, $ s = 1 $, and $ c = 0.7 $. Non-zero birefringence ($ r \ne 0 $) is included here because some of the past measurements [11,12] included samples with birefringence. Figure 2 shows the non-zero elements of the matrix $ {{\bf L}_{\text{t}}} = \log ({{\bf M}_{\text{t}}}/{M_{00}}) $. One immediately notices the curved nature of the depolarizing elements $ {L_{\text{t},11}} $, $ {L_{\text{t},22}} $, and $ {L_{\text{t},33}} $, which appear quadratic near zero thickness and become linear at larger thicknesses. This behavior was observed by Charbois and Devlaminck in their measurements of turbid liquids [10]. The non-depolarizing elements $ {L_{\text{t},23}} $ and $ {L_{\text{t},32}} $ are approximately linear with thickness, as has been observed in Refs. [11,12].

 

Fig. 2. Non-zero elements of the logarithmic decomposition $ {{\bf L}_{\text{t}}} $ of the Mueller matrix transmittance $ {{\bf M}_{\text{t}}} $ as a function of dimensionless thickness. The left frame shows the depolarizing elements $ {L_{\text{t},11}} $, $ {L_{\text{t},22}} $, and $ {L_{\text{t},33}} $, while the right frame shows the non-depolarizing elements $ {L_{\text{t},23}} $ and $ {L_{\text{t},32}} $. The curves for $ {L_{\text{t},11}} $ and $ {L_{\text{t},22}} $ are nearly identical.

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Multiple scattering is the root of why depolarization behaves quadratically in this model. In order for scattering to affect radiation in transmission, there must be at least two scattering events: one to transfer radiation from the forward direction to the backward direction, and a second to transfer radiation back to the forward direction. The scattering parameter $ s $ is the product of the scatterer concentration and the mean backscattering cross section. The KM model described here exhibits a depolarization that also depends quadratically on $ s $. This behavior was also found in Ref. [9], although it was expressed as the model-extracted standard deviation of the birefringence [$ {\langle \Delta {{\bf m}^2}\rangle ^{1/2}} $, determined from Eq. (5)], found to be linear in particle concentration.

At this time, we do not know of a specific measurement that would distinguish between the behavior predicted by fundamental fluctuation theory [Eqs. (4) and (5)] and that resulting from scatter. It is expected that the measurements performed by stacking media [11,12] would have achieved different results if the layers were spaced apart, since diffuse back reflections would be reduced. Furthermore, different results would be observed if the incident radiation were well collimated and only that radiation transmitted in the forward direction were collected, since contributions from scatter would be reduced. However, some of the artifacts that were studied, specifically those where materials were stacked to create different thicknesses (blocks of $ {\text{TiO}_2} $ in polyvinyl chloride [9] or strips of adhesive tape [11]), would be expected to have correlation lengths shorter than the thinnest layer measured. That is, the polarimetric properties of the material could not have been coherent in the direction of propagation as blocks were stacked. Yet, those measurements observed the quadratic onset of depolarization.

References [9,11] acknowledged that, as layers were added, the radiation was attenuated. The KM theory described here not only predicts the polarization properties, but also net attenuation, which is found to be linear (i.e., $ {M_{00}} $ is exponential) for small thicknesses. Attenuation can be included in the fluctuation theory through addition of an isotropic term in $ \langle {\bf m}\rangle $, but attenuation does not arise naturally out of the fluctuation terms themselves.

One apparent paradox is solved by the present model: it was found that the results were not consistent with the multiplication of successive Mueller matrices implied by Fig. 1(a). Our resolution of that paradox is that scattering in the material causes radiation to propagate backward as well as forward. That is, the addition of a layer affects the radiation pattern in previous layers.

The polarimetric extension of the KM model we use here is highly simplified and does not represent a quantitative approach towards solving the radiative transfer equation [15]. Monte Carlo (MC) simulations better simulate the three-dimensional nature of radiative transfer. In fact, quadratic depolarization was predicted by MC simulations reported in Ref. [9] using scattering phase functions from Mie theory. Like the KM model described here, MC simulations do not propagate radiation coherently, demonstrating that quadratic depolarization can be a scattering phenomenon, rather than a coherent one, like the fluctuation theory. We presented the KM approach, because it illustrates how simply adding scattering into the calculation results in a behavior similar to that observed in the measurements.

One of the experimental studies measured transmitted depolarization for different thicknesses sliced from bio-mimetic skin equivalents and focused on birefringent dermal regions [12]. These materials had thicknesses in the range of 2–15 µm. Because the different samples were each obtained subtractively from blocks of material (in contrast to those studies that investigated different thicknesses by additively stacking layers), the condition of homogeneity of the polarimetric properties in the direction of propagation may have been satisfied. Thus, part or all of the quadratic behavior observed in the depolarization may indeed have been a result of the unidirectional approach that the authors used to interpret their data.

In this Letter, we have shown that one does not need to assume longitudinal homogeneity (long coherence length) to observe quadratic evolution of depolarization in transmittance. Instead, the presence of scattering prevents application of a unidirectional application of Mueller matrices, illustrated in Fig. 1(a). By simply considering backward propagating radiation and scattering, quadratic evolution of depolarization can occur in the short coherence length limit. These results are important for interpreting depolarization in turbid media. Because there are two competing mechanisms that predict quadratic depolarization, extracting physical information from data may be difficult. It should be borne in mind that, because both scattering and depolarization result from inhomogeneities in a turbid media, scattering and depolarization are necessarily intertwined phenomena.