## Abstract

Partial polarization may be the result of a scattering process from a fully polarized incident beam. It is shown how the “loss of polarization” is connected with the nature of scatterers (surface roughness, bulk heterogeneity) and on the receiver solid angle. These effects are theoretically predicted and confirmed via multiscale polarization measurements in the speckle pattern of rough surfaces. “Full” polarization can be recovered when reducing the receiver aperture.

©2008 Optical Society of America

## 1. Introduction

Polarization is a classical property of light that has led to numerous principles, components and systems in the optics community. In most situations, one can attribute a polarization behaviour to any deterministic micro-object, which describes how the incident polarization state is modified by the interaction of light and matter. However this is mainly true for specular and diffracted waves, while it remains a great range of applications where the random nature of scatterers is responsible for a “loss of polarization” [1], most often called partial polarization or depolarization. Though these phenomena were extensively studied, they still constitute a key limitation to numerous optical techniques involving ellipsometry [2], optical microscopy [3] and imaging in random media [4]…

In two recent papers [5, 6] the polarization properties of the scattered light were used to build a far field procedure for the selective cancellation of scattering sources. The technique was shown to be successful when considering low-level scattering from polished surfaces or slightly heterogeneous bulks, but a key limitation appeared for arbitrary rough or inhomogeneous samples that scatter the total incident light. Such limitation is the result of partial polarization connected with the presence of microscopic irregularities at the samples under study. Therefore the aim of this work is to:

- specify which kind of irregularities is responsible for the polarization loss
- derive a relationship able to quantify partial polarization versus the random nature of roughness or bulk
- bring solutions to generalize the cancellation procedure to arbitrary scattering samples

## 2. Basic principles

#### 2-1 Polarization of a plane wave

Within the framework of classical electromagnetism, polarization is basically defined in the case of a parallel and monochromatic light beam. Let us denote by **E**(**ρ**) the complex electric field associated with a plane wave (Fig. 1):

with **A** the vector complex amplitude, **ρ**=(**r**,z) =(x,y,z) the space coordinates, **k** the wave vector and **σ** the spatial pulsation:

In Eq. (2), λ and n designate the illumination wavelength and the refractive index of the propagation medium, and σ the modulus of spatial pulsation (σ=|σ|). The propagation angles (Fig. 1) are derived in the far field (σ<k) from the spatial pulsation as:

with θ and ϕ the normal and polar angles, respectively.

Because the vector amplitude **A** lies in a plane perpendicular to the wave vector (Fig. 2), it can be splitted into two polarization vector components:

whose tangential projections are given in the (**u**,**v**) plane as:

where S and P designate the transverse electric (TE) and magnetic (TM) modes.

At this step the real (physical) electric field is given by:

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\left[\mid {A}_{S}\mid \mathrm{cos}(\omega t-\mathbf{k}.\rho -{\delta}_{S}),\mid {A}_{P}\mid \mathrm{cos}\left(\omega t-\mathbf{k}.\rho -{\delta}_{P}\right)\right]$$

with ω the temporal pulsation and t the time parameter. Therefore at a given location **ρ** the field direction is constant when δ_{S}=δ_{P}, in which case polarization is said to be linear. In other situations (δ=δ_{S}-δ_{P}≠0), rotation of the field occurs in the plane perpendicular to the wave vector, and polarization is said to be elliptic. The case of non polarized (natural) light is traditionally addressed with δ as a random variable within the interval (0,2π).

#### 2-2 Measurements procedure

Consider now an ellipsometric procedure to measure the polarization state of a polarized plane wave. The basic idea consists in aligning the S and P modes in order to reach an algebric or interferential sum of the two polarizations. Such projection A’ is obtained when the beam passes through an analyzer at angular position ψ from the S direction:

with ψ≠0 or π/2, so that the resulting measurable intensity will be proportional to:

Now a rotating analyzer allows to record the whole I’(ψ) curve whose analysis provides the key polarization parameters that are δ and |A_{P}/A_{S}|. In the case |A_{S}|=|A_{P}|, the contrast of the curve is given by 2cosδ, which constitutes a signature of the polarized interference (Fig. 3). Other techniques allow accurate analysis of polarization, in particular those involving specific devices to modulate the incident polarization. Most often with these techniques Eq. (9) is unchanged but a time function η(t) is added to the polarimetric phase term δ, which creates several harmonics in the resulting field.

#### 2-3 Procedure limits

However such procedure does not a priori allow to detect partial polarization of light, since any reduction of the measured contrast (2cosδ) will be attributed to a lower δ’ value of the polarimetric phase. In the case where δ is a random variable, Eq. (9) provides an apparent phase term given by cosδ’=<cosδ>, with cosδ’=0 in the extreme case of natural light (no variation of I’ versus ψ angle if |A_{S}|=|A_{P}|). Mueller matrices [7] have been used to solve this point, but here we limit ourselves to a deterministic approach. One should also notice a key property of polarized light that consists in the possibility to cancel it via destructive interferences. Such extinction can be reached thanks to a retardation plate introduced on the beam, which turns Eq. (8) and (9) into:

with η=η_{S}-η_{P} the polarization phase delay of the plate. With this additional parameter, a maximum contrast can be reached since an ellipsometric zero of I” is obtained for tgψ= |A_{S}/A_{P}| provided that the retardation term η is matched such as δ+η=(2n+1)π. From the point of view of experiment, the zero value will be replaced by a minimum connected with the relative efficiency of crossed analyzers and polarizers. These points will be helpful at the end of the next section.

## 3. Polarization of a wave packet

Let us consider the case of a monochromatic spatial wave packet **E**(**ρ**) in the far field (Fig. 4):

Eq. (12) is a general expression for the harmonic field in a homogeneous medium; it is based on the assumption of the existence of a transverse (**r**=> **σ**) Fourier Transform **Ê**(**σ**,z) of the field, which is valid for waves of finite spatial extension. Therefore the field given in Eq. (12) can be the far field result of a diffraction or scattering process. Since Maxwell equations offer rigorous solutions to these optical interactions, each elementary component **Ê**(**σ**,z) under the integral (12) is a “fully” polarized plane wave provided that the original (incident) field is fully polarized. Therefore all polarization parameters δ(**σ**) and r(**σ**)=|A_{P}/A_{S}|(**σ**) can be theoretically derived from the knowledge of the sample under illumination. Strictly speaking, the phase difference δ(**σ**) will be characteristic of the elliptic or linear polarization of the Fourier component **Ê**(**σ**,z) at infinity in the far field.

#### 3.1 Measurements procedure

Here we analyze the measurement procedure to investigate the polarization state of the wave packet **E**. Measurements at a finite distance involve a solid angle ΔΩ in which the polarization parameters δ(**σ**) and r(**σ**) should not vary in order to be able to define a polarization state in a classical way. If it is not the case, the angular frequency or angular variations create an averaging process within the receiver solid angle ΔΩ, resulting in an apparent “loss” of polarization. To investigate and quantify this effect, we first write the wave packet after passing through the analyzer, as an algebric sum on the analyzer axis:

In Eq. (13) we have neglected the influence of incidence (θ,ϕ) on the analyzer plate, which limits us to slight solid angles (less than a few degrees) that are currently used in experiment.

In a second step we calculate the Poynting flux F of the packet **E*** which is carried within the receiver. For this purpose we limit the frequency support Δσ of integral (13) to spatial frequencies that give rise to scattering angles collected in the far field within the receiver solid angle. After direct analytical calculation, this flux can be expressed as:

Such formula recalls the zero contribution of the evanescent waves (high frequencies: σ>k⇔α’=0) in the energy balance within a non absorbing medium, in opposition to plane waves that carry energy in the far field (low frequencies: σ<k⇔α’≠0). Coming back to Eq. (13-b), we obtain:

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}+{\mathrm{sin}}^{2}\psi {\int}_{\Delta \sigma}\alpha \u2019\left(\sigma \right){\mid {A}_{P}\left(\sigma \right)\mid}^{2}d\sigma +X]$$

with X the interferential term given as:

In a last step we use a similarity with Eq. (9) given for a plane wave by the introduction of parameter β, so that:

Eq. (17) allows to determine the polarization parameters β and F_{S}/F_{P} of the wave packet in a way similar to that of plane waves.)

#### 3-2 Equivalent or apparent polarization

Eq. (17) is specific of the interference between the polarization states of the wave packet. It permits to analyze the polarization measurement of the packet within a given receiver solid angle. In regard to the case of a plane wave (see Eq. (9)), the difference in the variation of the resulting F(ψ) curve lies in the presence of the β parameter that replaces the cosδ term of the plane wave. However we notice, thanks to the Schwartz inequality, that:

so that measurements will provide an equivalent phase term δ*. In case of low variations of parameters within integrals (18–19), we obtain δ*=δ specific of a plane wave. But in the general case, depending on solid angle and the random nature of scatterers further discussed in the text, the equivalent phase δ*≠δ will be the result of an averaging process.

One point to discuss now concerns the existence or not of an equivalent polarization for the wave packet, with δ* as a specific signature. In section 2–3 we have seen that a key property of polarized light lies in the possibility to cancel it via destructive interferences of its polarization modes, what was performed with a retardation plate in relation (10–11) relative to a plane wave. Such property is crucial to offer selective cancellation of scattering in random media [5,6]. Hence consider that the wave packet has also passed a retardation plate similar to that of section 2. As for the analyzer, we neglect the influence of incidence on this plate, so that the unique term modified in Eq. (17–19) is the interferential term X proportional to parameter β, that must be rewritten as:

By comparison with Eq. (19), the phase term δ(**σ**) has been replaced by δ(**σ**)+η, with η the tunable phase delay. When the angular variations can be neglected in the integral (21), we obtain a relationship similar to that of a plane wave (see section 4–1), and zero values of intensity can again be obtained provided that the condition δ+η=(2n+1)π is fulfilled. In all other situations, extinction of the field cannot be reached, due to the constant η value in opposition to the variable phase term δ(**σ**). In other words, the extreme values ±1 cannot be reached by parameter β in the general case of a wave packet with high angular polarization variations. The result is that the ellipsometric zeros of the intensity curve F(ψ,η) will be replaced by minima. Therefore equivalent polarization cannot here be introduced, and the equivalent phase δ* will be said to be characteristic of partial polarization.

Such major difference (zeros replaced by minima) with polarized light constitutes the limit of our cancellation technique based on destructive interferences between polarization modes [5]. However a solution can be found in the reduction of solid angle to values ΔΩ_{0} that allow to neglect the variations of parameters δ(**σ**) and r(**σ**) within the integrals, so that “full” polarization can be recovered. The correlation length of scatterers will be responsible for the minimum value ΔΩ_{0}.

## 4- Defect-induced partial polarization

In this section we show that low-scattering levels cannot create partial polarization, in opposition to high scattering levels.

#### 4-1 The case of low-scattering levels

Low scattering levels can be predicted with perturbative theories. These theories are valid for samples that are slightly inhomogeneous at their surfaces or in their bulk. This is currently the case of polished surfaces that scatter an amount of the incident light much lower than specular reflection. Several works [8–10] have been devoted to first-order electromagnetic theories and predict the scattered field E^{d} to be proportional to the Fourier Transform ĝ(σ) of defects, with g(**r**) the surface topography (case of surface scattering) or the relative variations in the bulk permittivity (case of volume scattering). These theories have shown successful results when the ratio of roughness to wavelength, or the root mean square of bulk heterogeneity, is much less than unity [9].

Due to this proportionality relationship (E**d**~ĝ(σ)), it is immediate to show that the polarimetric phase term δ(**σ**) does not depend on the microstructure of the scattering sample:

$$=>\delta \left(\sigma \right)=\mathrm{Arg}\left(\frac{{{E}^{d}}_{S}}{{{E}^{d}}_{P}}\right)=\mathrm{Arg}\left[\frac{{C}_{S}\left(\sigma \right)}{{C}_{P}\left(\sigma \right)}\right]$$

This result has been used to cancel first-order scattering without any knowledge of the sample microstructure [6], thanks to the fact that the C_{SorP} optical coefficients are connected with polarization, refractive index, wavelength and incidence…, but not on microstructure. Moreover, these coefficients exhibit low variations versus scattering angle in the whole range (0°, 90°) [5], which can therefore be neglected within the receiver solid angle. As a consequence, Eq. (17–19) can be rewritten with a retardation plate as:

with γ(**σ**)=(4π^{2}/S)|ĝ(σ)|^{2} the roughness or permittivity spectrum of defects, and S the illuminated area on the sample.

As a consequence, the resulting flux F is now given by:

$${\phantom{\rule{.9em}{0ex}}\phantom{\rule{.5em}{0ex}}\phantom{\rule{.5em}{0ex}}\mathrm{cos}}^{2}\psi {\mid {C}_{S}\mid}^{2}+{\mathrm{sin}}^{2}\psi {\mid {C}_{P}\mid}^{2}+2\mathrm{sin}\psi \mathrm{cos}\psi \mathrm{cos}\left(\delta +\eta \right)\mid {C}_{S}{C}_{P}\mid $$

This last relation shows that cancellation of the flux can still be reached, with a polarimetric behaviour similar to that of a plane wave (see Eq. (9)). Hence the polarization can here be clearly and classically defined, for which reason we conclude that low-level scattering cannot be responsible for partial polarization.

#### 4.2 Arbitrary scattering levels

On the other hand, surfaces or bulks that scatter the total incident light (no specular reflection) may exhibit high variations of polarization parameters versus scattering angles. Such behaviour was calculated in ref [11] for surfaces and bulks of different root mean squares, and confirmed by measurements [12]. The specific phase term δ(**σ**) of these samples may be uniformly distributed within (0°, 360°) and emphasize rapid variations versus scattering angle when the correlation length is decreased. Such variations can no more be neglected in the previous integrals (17–19). In this case measurements may detect partial polarization as an average process over the receiver aperture, unless we reduce the receiver solid angle to recover full polarization. We address this point in the section 5.

#### 4.3 Comparison with Stokes formalism

Because Stokes formalism [1] is commonly used on closely related topics, it is appropriate in this section to emphasize the link with the formalism of the preceding sections. The basic relationships that define the Stokes vector components Si are given by:

With these Stokes components introduced, Eq. (9) can be rewritten as:

One can also define the linear polarization degree as:

which is one of the classically measured quantities in Stokes polarimetry [1], and it is the only one relevant to this work, if one keeps in mind that the measurement involves a retarder which brings δ to zero and thus eliminates the ellipticity described by S_{3}. In these condtiions the dynamic d=F_{max}/F_{min} that we measure in the next sections and which is obtained when ψ is varied, can be expressed as the ratio:

Finally, the parameter β of Eq. (19) can also be expressed with the Stokes parameters as:

This dimensionless parameter tends to unity when the light is totally polarized.

## 5- Experimental data on rough surfaces

We have investigated the polarization behaviour of two rough surfaces with a procedure similar to that of the previous sections. The incident light is a He-Ne laser at 633nm wavelength with 4mw power. Its polarization is linear at 45° direction from the S or P axis. The retardation plate was replaced by a rotating quarterwave plate (angle τ) that plays an equivalent role. The scattered light first passes through this plate, before it is recorded through the rotating analyzer (ψ angle). In order to have at disposal experimental data for different solid angles without modification of the measurement distance, we used a CCD camera. With this device, it was possible to investigate the signal variations F_{ij}(ψ,τ) delivered by one or several pixels (i,j), which allows to emphasize polarization averages versus solid angle. Each pixel is 5µm in length and the integration time is 2s. The illumination incidence is 56°, whereas the measurement angle is close to the specular direction (120° from the incident beam). Notice that the speckle size was approximately given by λ/a=0.011°, with a=2mm the radius of the spot size on the sample.

For each sample we recorded the far field scattering speckle versus the rotation angles of analyzer (angle ψ) and quarterwave plates (τ angle). Hence the procedure allowed us to extract maxima F_{max} and minima F_{min} for each pixel or collection of pixels of the sensor. For the sake of simplicity, we will be focused on the dynamic d=F_{max}/F_{min} of the data rather that on the phase term δ that can be derived from the whole F(ψ,τ) curve.

#### 5-1 Case of a polished surface

The first sample was a polished black glass that scatters light at its front surface. Its level of total scattering (a few 10^{-4}) is specific of first-order scattering. The scattering speckle is given in figure 5, for a 2080×2600 µm^{2} field view. Since the measurement distance is 50cm from the sample, the angular aperture Δθ is 0.30° for the whole sensor field.

The left figure was recorded when the analyzer and quarterwave plates are matched to reach a maximum average signal (over all pixels) with a grey level of 160. The right figure is approximately recorded for the minimum signal (here in the noise) whose grey level is taken to the value 10. Therefore the dynamic (d) of the measurements, defined as the ratio of maximum to minimum, is greater than 16 (d>16). This contrast value of the F(ψ,τ) curve is characteristic of polarized light, as predicted in section 4-1 for slightly inhomogeneous samples. However the investigation is here limited by the performances of the sensor; strictly speaking we should be able to reach the dynamic value that is currently measured with crossed analyzer and polarizer on a specular beam (d≈200). Actually we have at disposal 256 grey levels and a noise level of 10, so that the dynamic is limited to 25 in the best case. Varying the integration time did not allow to improve this point.

One key point in Fig. 5 is that all pixels are simultaneously cancelled, for which reason we did not investigate additional average effects. This is in accordance with relation (24) specific of first-order scattering, which predicts the same reduction for all pixels when the two plates are rotated. Such result recalls why selective imaging can be performed in a large solid angle with a unique matching [6] of the analyzer and quarterwave plates.

#### 5-2 Case of an arbitrary rough surface: multiscale polarization data

The second sample scatters all the incident light with an angular lambertian distribution. It is a metallic (Au) etalon used for calibration, whose scattering originates from roughness [12]. The speckle pattern was again measured at 50 cm and is given in figure 6, with a field view of 2500×2020 µm^{2}.

For this high scattering sample the optimal conditions (ψ_{min}, τ_{min}) for minimization are different for each pixel, in opposition to the previous case of low-level scattering. In other words, from one pixel to another the variations are independent when the 2 plates (analyzer and quarterwave) are rotated. As discussed in the previous section, this is due to the fact that the phase term δ(**σ**) is now microstructural dependent and exhibits high variations versus scattering angle (here versus pixel). For this reason we studied different zones of the CCD area that are fitted into each other:

- The second zone (2) is included in zone (1), and appears at the middle of Fig. 7. The field view is 165×300 µm
^{2}, which corresponds to an angular aperture (Δθ) of 0.034°. - The third zone (3) is included in zone (2), and appears at the right of Fig. 7. It is related to one pixel (5×5µm
^{2}), which corresponds to an angular aperture (Δθ) of 5.7 10^{-4}°.

Such procedure permits to address a multiscale study of polarization, that is, polarization versus solid angle.

For each zone (1, 2 and 3) we have measured the average signal variations I(ψ,τ) versus the rotation angles of analyzer (Δψ=2°) and quarterwave plates (Δτ=2°). The results are given in figure 8.

From the data of Fig. 8 it was possible to extract the maxima and minima of each zone, and therefore the dynamic (I_{max}/I_{min}) of the I(ψ,τ) curve. The results are given in table 1, together with the corresponding value of solid angle. As predicted in the previous sections, we observe a noticeable increase of the dynamic (from 3.7 to 26.6) when the angular aperture is reduced, which indicates a progressive recovering of polarization. These dynamic data could be more accurate with a specific zooming around the minima positions, but the procedure is time consuming. The maximum dynamic is reached for the single pixel (zone 3) for which the angular aperture is lower than the speckle size. Notice that in the absence of the procedure here discussed, an apparent phase term δ* or partial polarization would be measured for each angular aperture.

In Fig. 9 (a, b, c) we give the pictures recorded for each zone at the minima and the maxima average signals. The left, middle and right figures are respectively for zones 1, 2 and 3. For each zone the top picture is for the maximum signal, and the bottom picture is for the minimum signal. We observe as predicted that the pictures obtained for the minima are different for each zone.

## 6- Conclusion

Recent results [5, 6] allowed us to address the field of imaging in random media via a selective cancellation procedure of scattering sources. The procedure was derived from what could be called an advanced ellipsometric imaging in the scattering pattern. However though the technique revealed successful performances for slightly inhomogeneous media, key limitations were emphasized in the case of arbitrary scattering samples, due to partial or total depolarization.

In this paper we showed that scattering processes of a polarized incident beam do not create partial polarization in the classical sense. What we observe is an average process of polarization states over the receiver solid angle, that can be quantified versus solid angle and correlation length of scatterers. An analytical formula was given to investigate these effects that were shown not to lead to an equivalent polarization, but that can be interpreted as partial polarization.

Experimental data were related to a multiscale study of light scattering polarization versus solid angle, thanks to a CCD sensor. The polarization behaviour of each pixel was analyzed, as well as the behaviour of pixel collections. The results clearly showed that the dynamic of the ellipsometric curve increased when the solid angle is reduced, which confirms that partial polarization is the result of an average process. Therefore it is possible to recover full polarization when the measurement distance is increased. The distance at which polarization is recovered depends on the nature of the scatterers, whose correlation length is responsible for the derivative of polarization parameters versus scattering angle. Hence the selective cancellation technique presented in [5,6] should remain valid in all situations provided that the measurement procedure is adapted. The procedure is turned more complex in consequence since each pixel must be separately cancelled. Liquid crystals matrices may replace the retardation or the quarterwave plate in order to reduce the acquisition time.

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