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dop histograms are measured in the off-specular far field speckle of disordered media under polarized and unpolarized illumination. Three surface samples with increasing roughnesses, and three bulk samples with different absorption levels, are investigated. Results show that both rough surfaces and absorbing bulks hold the incident polarization, while transparent bulks allow to depolarize or to enpolarize the incident light. Hence we provide a first experimental evidence of such effects.
© 2014 Optical Society of America
1. Introduction
Recent and new efforts have been devoted to the study of light polarization [1–5] in the speckle of disordered media. In particular electromagnetic theories and phenomenological models have allowed to predict specific properties of complex media such as spatial depolarization [6], temporal depolarization [7] and enpolarization effects [7–9]. Such effects may occur or not depending on the scattering origins (surface or bulk), and they may dominate the polarization process, depending on their microstructure [6–9].
However whereas most papers address theoretical analysis on these topics, few are concerned with polarization measurements at the speckle size [10–14]. Filling this gap was the main motivation of this work since a number of phenomena have not yet been confirmed by experiment. This work is devoted to the analysis of polarization degree histograms measured for different kinds of surfaces and bulks. It is shown how these dop histograms:
- confirm most theoretical predictions in connection with the microstructure of the scattering samples
- provide additional signatures to identify the origins of scattering within a series of surface or bulk samples
The experimental set-up is not presented since it has been the focus of a recent paper [10] with a first validation step. To summarize, the set-up allows to measure the polarization state and polarization degree [15–17] at the speckle size within a far field speckle pattern at visible wavelengths. Here we use this set-up to investigate a series of samples (surfaces and bulks) under polarized and unpolarized illumination.
2. Surface scattering
We use a series of 3 surface samples shown in Fig. 1 and 2. Their scattering patterns decrease from the first to the third (Fig. 2). Sample S1 is a metallic (Au) surface which delivers a lambertian pattern, which means that the whole incident light is scattered by reflection with a cosine law. Samples S2 and S3 are opaque (black) glasses respectively grounded and moderately polished; the incident light is mainly absorbed (around 96%) in the visible while the amount of scattering approaches some % (grounded sample) or much less (polished sample).
Fig. 1 Photographs of the 3 surface samples: lambertian (left), grounded (middle) and polished (right).
2.1 Incident polarized illumination
The samples were first illuminated with a fully polarized (45° linear) and coherent quasi-monochromatic laser beam (He-Ne, 633nm, 2 mm spot size) at normal incidence, and the scattering data were recorded around 15° in the far field (80cm) on a CCD camera (106 pixels with 13μm size). Each speckle grain is resolved with 256 pixels. A least mean square (LMS) procedure [10] was used to extract both polarization degree and polarization states within all speckle patterns.
The resulting 3 sample histograms are given in Fig. 3. A 4th histogram is given as a reference, which was measured for the direct (specular) beam [10]. Strictly speaking the reference histogram would be a Dirac curve so that its width characterizes all uncertainties in the measurement process (bias, noise, LMS procedure…).
Fig. 3 Normalized dop histograms measured for the lambertian (in red), grounded (black) and polished (in purple) surfaces- Case of polarized illumination.
Concerning the scattering samples now, their histograms in Fig. 3 are wider than that of the reference. The largest root-mean-square is observed for the rougher (lambertian) sample (S1). The average dop is 0.94 and the mean deviation is 0.08 for this sample S1. Then for lower roughnesses (samples S2 and S3), the dop curves are very similar.
To be complete, the polarization states are also plotted in Fig. 4 and 5 (Poincaré spheres) for the 3 surface samples. In Fig. 4 we considered 3 speckle grains to check that polarization is quasi-constant within one grain, and also from one grain to another. Notice on the spheres that the color indicates the maximum grey level on the pixel where the dop is measured [10].
Fig. 4 Polarization states on the Poincaré sphere for the Au lambertian sample (S1)- Polarized illumination.
Fig. 5 Polarization states on the Poincaré sphere for the grounded (left, S2) and the polished (right, S3) black surfaces - Polarized illumination.
2.2 Incident un-polarized illumination
The same procedure was used with the samples illuminated with an un-polarized He-Ne laser beam. Results are plotted in Fig. 6, with all the histograms around the origins (dop ≈0). Again we observe that all root-mean-squares are larger than that of the reference, and the largest one is again for the rougher sample S1. The polarization states are plotted in Figs. 7, 8, and 9 where a sphere section is introduced to emphasize the non-unity dop.
Fig. 6 Normalized dop histograms measured for the lambertian, grounded and polished surfaces- Unpolarized illumination.
Fig. 7 Polarization states on the Poincaré sphere for the Au lambertian surface sample (S1). Unpolarized illumination.
Fig. 8 Polarization states on the Poincaré sphere for the grounded surface sample (S2). Unpolarized illumination.
Fig. 9 Polarization states on the Poincaré sphere for the polished surface sample (S3). Unpolarized illumination.
2.3 Conclusion for the surface samples
These results show that whatever the surface roughness, the polarization degree of the speckle pattern remains close to that of the incident light. Furthermore, the polarization states rely in the vicinity of the incident state. Notice however that we could observe at high roughness (Au sample) a slight departure from the incident polarization. To conclude, surfaces approximately hold the polarization state of the incident light.
3. Bulk scattering
The second investigation concerned 3 bulk samples (Bi). Bulk samples are usually known not to hold the polarization, and vector singular optics have shown that even within a speckle area the polarization state could move throughout the whole Poincaré sphere [18,19].
The bulk photographs are given in Fig. 10. For each sample, the whole incident light is scattered or absorbed. From B1 to B3, and due to absorption properties, the scattering level decreases from 1 (white sample) to 10% (grey sample) and less (10−3 black sample). The ARS curves are given in Fig. 11.
Fig. 10 Pictures of the 3 bulk (Bi) samples. From left to right: white (B1), grey (B2) and black (B3).
3.1 Incident polarized illumination
Figure 12 gives the histograms of the 3 bulk samples under full polarized illumination. The largest departure from the reference, and the largest root-mean-square, is obtained with the white sample (highest scattering level). Otherwise we observe that the histogram root-mean-square decreases when absorption increases (from B1 to B3). Such effect can be explained by the weight of multiple reflections which is absorption dependent. In case of high-absorption level (sample B3), the bulk behavior is similar to the surface one. On the other hand, in case of low-absorption (samples B1 and B2), and in regard to the surface histograms of Fig. 3, the bulk histograms are clearly wider. These higher bulk root-mean-squares of the dop can be seen as the result of depolarization effects [7].
Fig. 12 Normalized dop histograms of the 3 bulk samples -Polarized illumination. B1: white bulk, B2: grey bulk, B3: black bulk.
The Poincaré spheres are also given in Figs. 13, 14, and 15 for the samples. We considered 3 speckle grains in Fig. 13 (lambertian sample B1) to show that polarization strongly varies from one grain to another, which is a key difference with the surfaces. For this sample B1, the sphere would be completely covered if we considered all the speckle grains on the CCD area. On the other hand, when absorption increases (samples B3), the bulk behaviour again mimics the surface one.
3.2 Incident un-polarized illumination
The 3 bulk samples now are under un-polarized illumination. The results are given in Fig. 16. As previously, higher absorption levels (sample B3) make the bulk samples mimic the surface ones, that is, a narrow dop histogram around zero. On the other hand, the white and grey samples (B1-B2) emphasize a great root-mean square characteristic of an enpolarization effect [8,9]. To our knowledge it is the first experimental evidence of such polarization process. More information is plotted with the Poincaré spheres in Fig. 17, 18, and 19.
Fig. 16 Normalized dop histograms of the 3 bulk samples –Unpolarized illumination B1: white bulk, B2: grey bulk, B3: black bulk.
3.3 Conclusion for the bulk samples
High absorption bulk samples reveal a polarization behavior similar to those of surfaces, what can be attributed to the weight of multiple reflections or mean-free path which is absorption dependent. On the other hand, transparent bulks clearly emphasize specific effects such as depolarization [7] and enpolarization [8,9].
Such effects (depolarization and enpolarization) were recently predicted [7–9] and are again emphasized in Fig. 20. To calculate these last data the spectral correlation length of the scattering coefficients and the laser band-pass were assumed to be of the same magnitude order. The green curve in Fig. 20 is calculated for a bulk sample under polarized illumination, and its dop histogram shows a depolarization effect similar to the grey bulk sample measured in Fig. 12. The blue curve of Fig. 20 is calculated for a bulk sample under un-polarized illumination, and its dop histogram emphasizes an en-polarization effect similar to the white and grey samples of Fig. 16. Notice that though en-polarization effects in bulks still occur for achromatic scattering coefficients [8,9], depolarization effects require a strong chromatic behavior [7] of the scattering coefficients within the source bandpass.
Fig. 20 pdf of the polarization degree calculated for a bulk sample under polarized (green curve) and un-polarized (blue curve) illumination.
4. Conclusion
To our knowledge these are the first dop histograms measured in the far-field speckle of surface and bulk media under polarized and un-polarized illumination. All results clearly show that surfaces approximately hold the polarization of the incident light, in the form of a memory effect [20, 21]. In the same way, absorbing bulk samples emphasize a behavior similar to those of surfaces. On the other hand, transparent bulk samples reveal a very specific signature in the sense that they can depolarize or enpolarize the incident light. This is again the first experimental evidence of such enpolarization effects predicted in [7–9].
Acknowledgments
This work was supported by the French National Research Agency (ANR) through the funding of TraMEL Project.
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