Spatial depolarization of light from the bulks: electromagnetic prediction

Myriam Zerrad; Hervé Tortel; Gabriel Soriano; Ayman Ghabbach; Claude Amra

doi:10.1364/OE.23.008246

Introduction

Light polarization and coherence [1–7] have become the focus of numerous works, and were accompanied by the emergence of unified theories [8, 9]. Additional complexity was brought by the introduction of random media, hence mixing spatial and temporal disorders [10–12]. Specific effects were recently emphasized at the speckle size, like the local (temporal) en-polarization of light [13–15], and the multi-scale (spatial) depolarization [11, 12, 16]. Most effects were confirmed by experiment [16–19], which justifies a motivation to go further in these fields.

Far field light scattering has been a powerful tool for decades to analyze complex or random media. For small disturbed media, perturbative theories have shown how the emitted pattern was driven by the Fourier transform of the surface roughness [20–22] or the bulk heterogeneity [23], a result which allowed to directly solve inverse problems and characterize substrates and optical coatings [22, 24]. Arbitrary scattering regimes are much more complex to analyze, due to the presence of integral or coupled equations which connect the fields at different spatial frequencies. In all situations the behavior of the far field pattern is intimately connected with the sample microstructure, which drives the angular, spectral and polarization properties. Recent examples give a detailed analysis of the polarization degree at the speckle size in the far field [11, 12, 16, 25], and emphasize spatial depolarization [11, 16, 26] and temporal repolarization [14, 15] of light, all phenomena which strongly depend on the samples microstructure (roughness, inhomogeneity). Applications were also found for biological tissues and others.

Indeed prediction of the polarization process in disordered media often involve a phenomenological approach, similar to that of Goodman’s phasor model [27] with partially or fully developed speckles. Such description of the scattering process is very practical and allowed to reveal unexpected phenomena such as the en-polarization process [14, 15]; also, it allowed to predict a spectral depolarization [3, 28] responsible for strong modification of the polarization degree (dop) histograms [13], as well as the modification of time coherence [29] in disordered media. However this phenomenological model is valid for highly heterogeneous samples and fails at intermediate electromagnetic regimes, whereas these regimes occur in numerous situations. As an example, electromagnetic theories predict the scattering coefficients to be strongly inter-dependent, except for extreme situations where they can be considered uncorrelated; these last situations are those of highly heterogeneous samples currently described with the phasor model.

Within this framework there is a current need to introduce exact values of the scattering coefficients in the calculation of the polarization process, that is, to provide a more exact link between the sample microstructure and the scattering coefficients. Such data would allow to analyze the gradual transition of polarization degree from a slightly heterogeneous sample to a highly heterogeneous one, hence providing a signature of the complex media together with a key information for optical imaging through these media.

We already performed [11] this kind of study for rough surfaces, that is, for surface scattering. For that we used an exact electromagnetic model based on the frontier integral [30]. Numerical results allowed to emphasize the multi-scale nature of the spatial polarization degree of the speckle emitted by surfaces of increasing slopes or root-mean-squares [11]. In other words, depending on the surface characteristics, it is possible to know from which receiver aperture full polarization starts to fail, due a spatial average process. The theoretical model was one-dimensional since time consuming is prohibitive for high slope 2D surfaces.

However to our knowledge the similar bulk problem has not yet been addressed, even though the analysis of bulk media constitutes a relevant topic for imaging in diffuse environment [31–33]. One question concerns the number of speckle grains whose integration cancels polarization, and the result will depend on the statistics and amplitude of the heterogeneity. To solve this point, an exact electromagnetic bulk model is required to calculate the volume scattering coefficients with accuracy, and to introduce them within the calculation of polarization degree (dop). Compared to the surface problem, this requirement is much more difficult to satisfy, because the basic bulk problem is necessarily at least bi-dimensional, and because the bulk/light interaction involves a larger region (not limited to the surfaces).

Hence this paper is an extension to the bulks of previous works [11] devoted to surface scattering. Here we use an exact electromagnetic model of bulk scattering (based on finite elements) to emphasize the gradual relationship between the sample bulk microstructure and the spatial polarization degree at the speckle size. Diagrams are plotted to emphasize the dispersion of polarization states on the Poincaré sphere, and to calculate the spatial dop versus the bulk heterogeneity. Results allow to calibrate the spatial depolarization process versus sample microstructure (thickness, index heterogeneity), with applications in most polarimetric techniques involved in imaging and diffusive media, spatial and defense, biomedical and biophotonics...

Principles of spatial depolarization

We start with a brief recall on spatial (global) depolarization, not to be confused with temporal (local) depolarization [3, 11, 19]. Indeed the incident light is assumed to be collimated, fully polarized and coherent (perfectly monochromatic), a situation where temporal depolarization does not occur. The analytical signal of the electromagnetic field is described by two complex polarization modes $E_{S}^{0}$ and $E_{P}^{0}$ . With two non-zero modes (non-linear polarization), the full polarization (temporal and local) of the incident field is equivalent to a unity modulus of the complex correlation coefficient $μ_{0}$ given by:

µ_{0} (ρ) = \frac{1}{α_{0}} {〈 E_{S}^{0} (ρ, t) E_{P}^{0}^{*} (ρ, t) 〉}_{t}

with

< >_{t}

the temporal average, * the complex conjugate, ρ the space coordinates and

α

the normalization coefficient:

α_{0} {(ρ)}^{2} = {〈 {| E_{S}^{0} (ρ, t) |}^{2} 〉}_{t} {〈 {| E_{P}^{0} (ρ, t) |}^{2} 〉}_{t}

Therefore the incident light here follows: |μ₀| = 1 = > dop = 1, with dop the polarization degree.

As for the scattered field denoted $(E_{S}, E_{P})$ , it can be written as the product of the incident field by the scattering matrix $(ν_{i j})$ :

(\begin{matrix} E_{S} \\ E_{P} \end{matrix}) (ρ, t) = (\begin{matrix} ν_{S S} (ρ) & ν_{P S} (ρ) \\ ν_{S P} (ρ) & ν_{P P} (ρ) \end{matrix}) (\begin{matrix} E_{S}^{0} \\ E_{P}^{0} \end{matrix}) (ρ, t)

with ν_ij the scattering coefficients [19]. This last relationship is valid under the assumption of achromatic scattering coefficients within the laser band-pass [13], in accordance with the assumption of a perfectly monochromatic source.

In a second step the complex correlation coefficient of the scattered field can be classically written as:

µ (ρ) = \frac{1}{α} {〈 E_{S} (ρ, t) E_{P}^{*} (ρ, t) 〉}_{t}

with:

α^{2} = {〈 {| E_{S} (ρ, t) |}^{2} 〉}_{t} {〈 {| E_{P} (ρ, t) |}^{2} 〉}_{t}

Previous works [11, 13, 15] have shown that because the illumination is perfectly monochromatic and fully polarized, full temporal polarization is guaranteed for the scattered field at any space location, so that temporal averages vanish in Eq. (4) and mutual coherence of the scattered field can be turned as:

µ (ρ) = \frac{E_{S} (ρ) E_{P}^{*} (ρ)}{\sqrt{{| E_{S} (ρ) |}^{2} {| E_{P} (ρ) |}^{2}}} \Rightarrow | μ | = 1

In other words, as illustrated Fig. 1, a fully polarized monochromatic illumination creates a fully polarized speckle pattern. As a consequence, any depolarization effect will be the result of spatial (not temporal) depolarization, that is, the result of an average process of all full polarization states collected within the receiver aperture. In Fig. 1 for instance, the resulting polarization degree will approach zero, due to the random variation of polarization states from one speckle grain to another. However in the general case the correlation length of polarization measured over the speckle grains will depend on the bulk microstructure, what we calculate in the next sections.

Fig. 1 Dispersion of polarization state within a spatially depolarizing speckle pattern- Schematic view

Abstract

Introduction

Principles of spatial depolarization

The electromagnetic bulk scattering model

Numerical considerations

Bulk and scattering characteristics

Dispersion of the polarization states

Spatial depolarization from the bulks

Parametric study

Conclusion

Acknowledgments

References and links

Supplementary Material (1)

Cited By

Figures (15)

Equations (24)

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