## Abstract

An ellipsometric technique based on angle-resolved light scattering is addressed to open applications in the field of imaging in random media. The first experimental demonstration is given to prove the selective extinction of different scattering sources such as surface roughness and bulk heterogeneity in optical components and liquids. The results are compared with theory.

© 2007 Optical Society of America

## 1. Introduction

Numerous studies have been performed to increase the sensitivity of light scattering to specific sources, with the objective to localize, identify and characterize particular micro-defects or objects. Most studies involve angular [1–3], wavelength [4, 5] and polarization [6, 7] behaviors of scattering, as well as dependence versus overcoatings [8, 9]…, and have produced successful results. More recently, polarization techniques were improved and have led to the development of an ellipsometric procedure [10, 11] for the determination of polarimetric phase terms in each scattering direction. These last techniques emphasized the possibility to improve the sensitivity of scattering to surface roughness, bulk heterogeneity or localized defects. Moreover, they were shown to offer a direct discrimination of the origins of light scattering [12, 13], whatever the scattering levels. One key result was given by the angular variations of the phase term in the scattering pattern, which emphasized random variations for bulk scattering, and smooth variations for surface scattering [12, 13].

Because the scattering speckle can be completely polarized in each scattering direction, these polarization techniques were again extended to show [14], from a theoretical point of view, that scattering or specular sources could be selectively eliminated. In this paper, we describe the first experimental results that concern this extinction prediction and its comparison with theory. Either surface or bulk scattering can be eliminated in optical components or scattering liquids which allows the procedure to be extended to imaging in random media.

## 2. Principles

The theory was presented in two recent papers [14, 15] that deal with scattered and specular beams. It is based on the simple idea that allows elimination of polarized light at each direction of space, provided that a rotating analyzer and a tunable retardation plate are used (Fig. 1). The analyzer angle *Ψ(θ)* and the tunable phase term Δ*η*(θ)* are matched at each direction *θ* of space in order to fulfill the extinction conditions based on intensity and phase equations [14, 15]. Moreover, the illumination conditions must be chosen for these extinction conditions to exist and to differ with the origins of light scattering. Lastly, the technique works for polarized light. For this reason, it was shown in reference [12] that the scattering speckle resulting from monochromatic and polarized illumination was fully polarized in the far field pattern for most samples.

#### 2.1. Elimination of polarized scattering

These principles can be summarized in a more formal way as follows. Consider two micro-objects that are illuminated with a monochromatic, plane polarized beam (Fig. 2). For simplicity, we assume that each incident polarization has the same energy, and that the time difference between the two polarization modes is given [14, 15] by Δ*η*/*ω*, with *ω* the temporal frequency. With these parameters, linear polarization is described with Δ*η*=0.

The electric field direction is chosen to be 45°, as is usually done for specular ellipsometry to reach a maximum contrast. The resulting diffraction or scattering pattern at direction *θ* can be written as:

with *E*̄ the polarized complex electric field that describes diffraction or scattering from the two objects, *E*̄_{1} the diffracted field from the first object that would exist in the absence of the second, and *E*̄_{2} the diffracted field of the second object in the absence of the first one. The last term *E*
_{12}̄ characterizes interaction between the two objects and can be defined from Eq. (1). Notice here that an intensity calculation would lead to interference described by the square of the total field given in Eq. (1). This interference occurs in the entire space where the fields exist.

After passing through an analyzer and a retardation plate in direction *θ*, the resulting field is transformed (proportional to) into an algebric sum in the complex plane [14, 15]:

with *E _{s}* and

*E*the polarization modes of the field that interfere after projection on the analyzer axis, and

_{p}*α*is a complex number that can be adjusted via rotation of the analyzer and adjustment of retardation plate:

Eqs. (2–3) show that the *f* transformation can be reduced to zero at each direction, provided that the *α* number is adequately chosen:

Since the *α* number can be arbitrarily chosen in the complex plane, there always exists a solution (*Ψ*,Δ*η**) for this extinction condition through the analyzer and retardation plate parameters:

$$\Delta {\eta}^{*}\left(\theta \right)=\pi -\left[\Delta \delta \left(\theta \right)+\Delta \eta \right]$$

with Δ*δ*(*θ*) the phase difference created between the polarization modes, that results from the diffraction or scattering process:

#### 2.2. Selective elimination

However the *α* number can also be chosen to eliminate the influence of a particular term in the sum given in Eq. (1). Indeed Eq. (2) can be split to emphasize the contribution of each object:

with:

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}f\left({\overrightarrow{E}}_{12}\right)={E}_{12s}\left(\theta \right)+\alpha \left(\theta \right){E}_{12p}\left(\theta \right)$$

At this step, it is clear that the *α* number can be matched to achieve different extinction conditions, such as the elimination of the diffraction pattern from one object:

or the elimination of the diffraction pattern arising from the interaction between the two objects:

In a more general way, any combination can be found among different combinations, since *f*(*E*⃗+*E*⃗) and *f*(*E*⃗* _{i}*+

*E*⃗

_{12}) can also be eliminated, etc… In other words, it is possible to observe the objects together or separately, or to limit the observation to their interaction (Fig. 2). This technique can be generalized to an arbitrary number N of objects:

which may lead to complex series caused by multiple interactions between K objects, with K<N. In Eq. (10)
*E*̄ designates the diffraction pattern of the object (i) that would exist in the absence of the others, while *E*̄_{1→N} describes the interaction between all objects. However for the method to work, the extinction conditions must be different for each term in the series of Eq. (10), that is, for all objects and their interaction, which requires adjusting the illumination conditions and noting specific polarization behaviors that offer the possibility of discrimination.

#### 2.3. Relation to microstructure

One additional question concerns the speed variations of parameters *Ψ(θ)* and Δ*η**(*θ*) that must be adjusted for scattering elimination at each direction of space. In the case of low level scattering, first-order electromagnetic theories [1, 5, 16] predict smooth variations of these parameters in the whole angular *θ* range, which is practical from the point of view of experiment. Moreover within the framework of these perturbation theories that predict a scattered or diffraction pattern proportional to the Fourier Transform of the objects (i.e., of the surface profile or of the refractive index variations), these variations only depend [11] on the origins of scattering (surface or bulk) and on the opto-geometrical parameters (dimensions and refractive indices…) of the objects under study, but not on shape, microstructure or topography. Under these conditions the extinction conditions can be predicted without any knowledge of microstructure, which constitutes a key advantage [11, 12].

The inverse case of arbitrary scattering levels is more complex and rigorous electromagnetic theories must be used to predict the extinction conditions that depend on microstructure [12, 14]. When the microstructure is not known, prediction cannot be performed and the analyzer and retardation plate parameters must be scanned to pass through different zero conditions among eight combinations (case of 2 objects). Moreover for these high scattering levels, large variations of the derivatives *∂Ψ/∂θ* and *∂(Δη*)∂θ* require the analyser and retardation parameters to be retuned within a narrow angular step Δ*θ* in the scattering pattern, which leads to experimental difficulties. Extreme cases may occur when these variations are too large within the receiver solid angle, in which case polarization fails (becomes partial) and the method is no longer valid. The same result can be obtained whatever the receiver solid angle, when parameters *Ψ(θ)* and Δ*η*(θ)* rapidly vary within the intrinsic angular size of the speckle, of the order of Δ*θ _{s}=λ/a*, with

*λ*the wavelength and a the radius of the spot area on the sample.

Notice also that the procedure is valid for deterministic or random objects. In what follows, we focus on the study of surface roughness and bulk heterogeneities within optical substrates, coatings and liquids. The components under study exhibit low level scattering, so that the extinction conditions can be predicted with perturbation theories [11, 12].

## 3. The case of random roughness and bulk heterogeneity

In this section the two objects under study are the random roughness of a polished surface, and the random bulk heterogeneity of an optical glass substrate of real refractive index 1.5. The surface topography is described by a profile function *h(x, y)*, while the transverse random variations in the bulk refractive index are described by a function *p(x, y)*=Δ*ε/ε*, with *ε* the average bulk permittivity. As recalled in the first section and shown in references [11, 12], the extinction conditions do not depend on microstructure (on functions *h* and *p*) within the framework of perturbation theories [1, 5, 16]. Moreover, no crossed polarization is predicted in the incidence plane.

Numerical calculations were performed to predict the angular variations of both the phase term Δ*η*(θ)* and the analyzer angle *Ψ(θ)* that allow the extinction conditions to be matched in each direction. The calculation is based on first-order electromagnetic theories of surface and bulk scattering [1, 16], a comparison of which can be found in detail in Ref. [5]. The analytical formulas of the scattered fields are too complex to be recalled in this paper but can be found in references [1, 5, and 16].

These intensity and phase variations of scattering are calculated in the incidence plane, both for surface and bulk scattering (Figs. 3 and 4). The incident polarization is assumed to be linear (Δ*η*=0) with 45° direction. In Fig. 3 the analyzer angle *Ψ* is given versus direction *θ* for both surface and bulk scattering, in the case of a slightly rough and heterogeneous glass substrate. Calculation is plotted for two illumination incidences (i=0° and i=50°). Similar data are given in Fig. 4 but concern the angular variations of the phase term.

We first observe that at normal illumination (i=0°), the angular behavior of the analyzer angle is identical for surface and bulk scattering. Similarly, no difference is observed for the phase term at normal illumination, so that discrimination between surface and bulk scattering is not possible. In other words, surface and bulk scattering are simultaneously eliminated when the *Ψ* and Δ*η** parameters are matched for normal illumination. Notice that this result is not general since it depends on refractive indices, in particular for thin metallic films and multilayers where all fields are complex. Indeed the extinction conditions are calculated from the ratio of the field values [Eq. (4)] that both vary with the scattering origins and the refractive indices. However, in our case of dielectric substrates, this result does not vary with refractive index.

The discrimination is more successful at oblique illumination (i=50°), since the departure between surface and bulk curves is clearly emphasized for both *Ψ* and Δ*η**, that describe intensity and phase conditions. Notice the phase step at large angles for surface scattering, which does not occur for bulk scattering. This step is the result of a sign change of the p-polarized wave scattered from a surface, analogous to the classical Brewster angle given for specular waves [17]. It constitutes a signature of surface scattering in comparison to bulk scattering. This scattering angle *θ _{b}* at which the sign change occurs for surface scattering is given for a single surface as [17]:

with *n* the refractive index of the sample and *i* the angle of incidence.

These theoretical results first show the importance of the illumination conditions that are a key for discrimination. When these conditions are chosen, the curves in Figs. 3 and 4 give the intensity and phase parameters that allow elimination of surface scattering, or elimination of bulk scattering. Following Eqs. (1) and (7), the resulting signal that can be recorded after elimination of bulk scattering is reduced to:

and after elimination of surface scattering:

Therefore the choice of our *Ψ* and Δ*η** parameters does not allow elimination of the interaction pattern *f*(*E*̄_{12}) between surface and bulk scattering. However this is not a difficulty with these low level scattering samples, since the interaction between surface and bulk is of the second order according to first-order theories. Indeed within the framework of perturbation theories [5, 16], the surface roughness is assumed to be much less than the incident wavelength, and the relative bulk heterogeneity is assumed to be much less than unity. Under these conditions the field scattered from the surface is a first-order field proportional to the Fourier transform of the surface profile, and the field scattered from the bulk is a first-order field proportional to the Fourier transform of the random variations of the refractive index. As a consequence, an interaction between surface and bulk would involve the product of these two Fourier transforms, resulting in a second-order field. difficulty with these low level scattering samples, since the interaction between surface and bulk is of the second order according to first-order theories. Indeed within the framework of perturbation theories [5, 16], the surface roughness is assumed to be much less than the incident wavelength, and the relative bulk heterogeneity is assumed to be much less than unity. Under these conditions the field scattered from the surface is a first-order field proportional to the Fourier transform of the surface profile, and the field scattered from the bulk is a first-order field proportional to the Fourier transform of the random variations of the refractive index. As a consequence, an interaction between surface and bulk would involve the product of these two Fourier transforms, resulting in a second-order field. difficulty with these low level scattering samples, since the interaction between surface and bulk is of the second order according to first-order theories. Indeed within the framework of perturbation theories [5, 16], the surface roughness is assumed to be much less than the incident wavelength, and the relative bulk heterogeneity is assumed to be much less than unity. Under these conditions the field scattered from the surface is a first-order field proportional to the Fourier transform of the surface profile, and the field scattered from the bulk is a first-order field proportional to the Fourier transform of the random variations of the refractive index. As a consequence, an interaction between surface and bulk would involve the product of these two Fourier transforms, resulting in a second-order field.

In the general case of arbitrary scattering levels, one may keep in mind that the interaction must be simultaneously eliminated to reduce the observation to a single object as if it were alone under the illumination beam, that is:

with i≠j. To predict this last extinction condition, it would be necessary to mix surface and bulk effects within rigorous computer codes, which we do not address here since the condition would depend on microstructure. Moreover, we may notice that the two surface and bulk scattering sources are most often independent, which may reduce the interaction to zero.

## 4. Application to experiment

The experimental apparatus is shown in Fig. 1. The incident beam comes from a 4 mW, 40 cm long, 633 nm wavelength He-Ne laser that is linearly polarized at 45° to the plane of incidence. Although this laser may produce different output modes, it can be considered to be a monochromatic source for our application because of the achromatic behavior of scattering for single substrates. Moreover, the polarizer guarantees that the incident beam is linearly polarized.

The other optical devices are an achromatic half-wave plate and a sheet polarizer. For each scattering angle, the phase difference and analyzer rotation angle are adjusted to maximize and minimize scattering. Light scattering is recorded with a CCD camera placed at the appropriate scattering angle. In this section, we consider two angles of incidence: normal incidence (i=0°) and oblique incidence (i=56°) close to Brewster angle. This second angle of incidence is chosen to optimize the contrast between surface and bulk scattering, but other angles of incidence could also be used to achieve similar results, following Eq. (11).

The exposure time was 2.0 sec for all CCD recordings, so that all pairs of images could be directly compared in Figs. 5 to 11. However, the incident power could be different for each figure. The images were scaled to 256 gray levels in all figures.

#### 4.1. Rough surface in the absence of bulk scattering

Opaque polished black glass was first chosen so that only surface scattering needed to be eliminated. The retardation plate and analyzer parameters were set to maximize and minimize the light scattering, as shown in Figs. 5 and 6 (the left photo is the maximum scattering and the right photo is the minimum scattering in all figures). The calculated values of these parameters are given in Table 1 and are compared to the measured values of the extinction parameters. There is good agreement, since slight departures between theory and experiment (2.8° at normal illumination and 0.4° at oblique illumination for the analyzer angle) could have resulted from the measurement accuracy of the angles and detectivity levels, or from additional scattering from dust and pits on the surface that do not follow first-order predictions.

In Figs. 5 and 6, the differences between maximum and minimum scattering levels are more pronounced at oblique illumination (i=56°), because of the lowest (i-*θ*) difference. This is a classical result because first-order scattering is larger at angles close to the specular direction (θ=i). A faint signal can still be recorded at the minimum scattering level, but may be the result of several effects such as the analyzer efficiency or the accuracy of the retardation plate, as well as additional scattering.

#### 4.2. Heterogeneous bulk scattering in the absence of surface scattering

For experiments on bulk scattering, materials with controllable scattering levels are required. Although some substrate materials such as Zerodur and MgF_{2} have low bulk scattering, these levels are not controllable. Moreover, the polished front and back surfaces of most samples are rough and surface scattering generally predominates over bulk scattering. In order to avoid such difficulties and only deal with bulk scattering, we used a turbid liquid (dilute solution of alcohol in water) with controllable micro-impurities. The liquid was 45% ethanol, 55% water and 2 g/L of anethol molecules [18]. These molecules have a very low solubility in water. If water is added to this solution, an emulsion forms spontaneously. Droplets are created and scatter the light. Here the concentration of the liquid is 12.5% (and so 5.6% of ethanol). The flask had a rectangular cross section and the sides were superpolished (Figs. 7 and 8) so that their scattering was much less than the bulk scattering from the liquid. The latter was adjusted for the experiment.

As shown in Figs. 7 and 8, bulk scattering could be easily recorded. Results comparable to those in the previous section are given here. The calculated values of the extinction parameters are given in Table 2 along with the measured values. For normal incidence illumination, the 2° agreement for the analyzer angle is good. At 56° illumination angle, the 7.4° error for the analyzer angle is considerably larger, possibly because the first-order theory was no longer valid, or because larger bubbles remained in the solution. The cause of the discrepancy should be further investigated. Also, the size of the particles in the emulsion that forms spontaneously when the anethol molecules are added should be measured and other properties of the emulsion determined [18].

The remaining air bubbles, that had a large difference in refractive index from the liquid, were not taken into account in the first-order theory. Note that we used an achromatic waveplate and a sheet polarizer. A Glan-Thompson prism polarizer was also tried, but it did not improve the results.

So far we have shown that it is possible to match the parameters to eliminate surface scattering (section 4.1) or bulk scattering (section 4.2). Now we consider the case where both scattering sources are present.

#### 4.3. Contrast in a scattering medium: selective discrimination

In this section we address the case where both surface and bulk scattering are present. The substrate was a sample of Zerodur that had bulk scattering. The bulk scattering was relatively low and did not mask the scattering from the two surfaces, as shown in Fig. 9. In the left photo in Fig. 9, the analyser and retardation plate were adjusted to maximize surface scattering. The bulk scattering can be clearly seen although the surface scattering dominates. In the right photo, the parameters were adjusted to extinguish surface scattering. There is a noticeable reduction of the surface scattering from the two substrate faces while there is little change in the bulk scattering. The average gray level for the surface scattering in the left photo was N=254 (overloaded) and N=63 for the bulk scattering, while the values in the right photo were reduced to N=94 for the surface scattering and N=50 for the bulk scattering. Quantitative information about the relative reduction of surface and bulk scattering would require a larger dynamic range in the CCD camera. Although much work remains to be done with a better camera having 65,000 gray levels, these results show that it is possible to distinguish between surface and bulk scattering.

The same procedure can be used for applications in the field of imaging in random media as shown in the following experiment. Turbid liquid was placed in a flask with letters written on a piece of adhesive tape attached to the back surface, as shown in Fig. 10. Normal incidence illumination was used, and the parameters were adjusted to eliminate surface scattering from both sides of the flask. These conditions also allowed bulk scattering to be eliminated, as shown in the previous section for normal incidence illumination and scattering viewed at 30°. The two photos were then recorded to show maximum and minimum scattering levels. When the parameters were matched to eliminate scattering, the letters clearly appeared on the back surface because of their specific polarization behavior. Note that scattering from the letters was not eliminated because it did not follow first-order theory, so that the extinction conditions were different.

#### 4.4. Additional experiment

We now return to surface scattering and consider two kinds of roughness scattering sources. First, we consider a multilayer coating deposited on the central region of a bare glass substrate (Fig. 11). The central overcoated circle was the first object while the bare surrounding region was the second object. The normally incident beam was focused on the boundary between the two regions so that scattering from both areas could be viewed simultaneously (the camera viewing angle was 30°).

As before, photos were taken for maximum and minimum scattering levels. The extinction conditions were only matched for the bare glass substrate, so that scattering from the multilayer coating, which was elliptically polarized, remained. The difference in the polarization behaviors of the glass and multilayer coating allowed the discrimination technique to work.

Scattering could also be eliminated from the overcoated region, but that would require knowledge of the coating design in order to predict the extinction parameters. Also, the interaction between the two regions should not be neglected.

## 5. Conclusion

Three recent theorical papers [12, 14, 15] have suggested the possibility of studying further effects of light scattering with potential applications in the field of imaging in random media. In the first paper [12] it was shown that angular scattering from random roughness or bulk heterogeneity was most often fully polarized in the far field speckle pattern, with polarimetric phase terms uniformly distributed or not within the interval [0,2*π*], depending on the scattering origins. In the second paper [14] this polarization behavior was used to suggest a procedure for selectively eliminating specific scattering sources. In the third paper [15], the same procedure was applied to specular beams to probe the thickness of multilayers with adequate z-resolution.

In this paper, we have experimentally tested the validity of these theories by using a selective extinction procedure and applied it to surface scattering, bulk scattering, and imaging through a turbid liquid. It was found that surface scattering can be reduced or eliminated while bulk scattering is still accurately recorded and vice versa. The procedure can be applied to random or deterministic objects and can be generalized to discriminate between several objects. However, this work has been limited to low level scattering that can be predicted using first-order perturbation theories. These theories have the advantage that the extinction conditions do not depend on microstructure. The next step would be to study arbitrary scattering levels, but that requires knowledge of microstructure and is also complicated by the large values of derivatives *∂Ψ/∂θ* and ∂(Δ*η**)*∂θ*. As shown in Ref. [14], the technique should still work for rough surfaces and bulk materials that scatter all the incident light, provided that the correlation length is not too short. For shorter correlation lengths, large variations of *∂Ψ/∂θ* and create partial polarization or total depolarization that cause the method to fail.

Finally, the procedure described in this paper may open new fields of imaging in random media since it can be applied to specular beams or scattering including elastic scattering, diffraction, luminescence, and other effects.

## 6. Acknowledgments

This project was supported by DGA (Direction Générale de l’Armement) and the Provence Alpes Cote d’Azur region. The authors would like to thank Dr Jean Bennett and the reviewers for helpful suggestions.

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