## Abstract

Accurate angular phase data are extracted from angle-resolved scattering measurements made with polarized light using a technique developed in the laboratory. This Ellipsometry of Angle-Resolved Scattering (E.A.R.S.) technique makes it possible to distinguish surface scattering from bulk scattering independent of the scattering levels for different types of samples. Phase data are also investigated in the speckle pattern.

© 2005 Optical Society of America

## 1. Introduction

Far field light scattering from rough surfaces and inhomogeneous bulks has extensively been studied, with a major application in random media characterization [1–10]. For example polished surfaces and slightly inhomogeneous bulks create low level angle-resolved scattering (ARS) that is proportional to roughness or permittivity spectrum, whose integration provides root mean squares of roughness and heterogeneity in the optical bandwidth.

However most of these studies do not allow reconstruction of topography or surface profile, because they are limited to intensity data that only give access to the statistical moments of microstructure. Therefore the motivation of this work is to extend scattering data to angular phase measurements [11, 12], in order to address this problem. One basic idea is to create interferences between scattering from the sample and another reference source [13], but this technique leads to difficulties connected with the rotation of receiver, whose distance from the sample should not vary within an accuracy less than one wavelength… One way to solve this point consists in using interferences between the two polarizations of the scattered waves, that is, Ellipsometry of Angle-Resolved Scattering (E.A.R.S.).

In a recent work [14, 15] limited to slightly inhomogeneous samples, we demonstrated the validity of the E.A.R.S. technique and recorded accurate measurements of polarimetric phase difference in the scattering pattern, thanks to rotating analyzer and polarizer introduced in a well known scatterometer [16]. The results were applied to the separation of surface and bulk effects in low-loss samples, due to the fact that first-order scattering only depends on the origin of scattering, not on the specific topography or microstructure. Therefore the separation method for low-scattering samples is valid whatever their structural parameters, which provides an efficient characterization tool.

Now the major point that is addressed in this paper is the generalization of the separation technique (surface or bulk) to arbitrary heterogeneous samples with high level diffuse reflectance. The problem is very different since phase data from these samples depend on microstructure (topography or heterogeneity), not only on scattering origins. Moreover, cross-polarization effects play a key role and must be carefully considered.

## 2. E.A.R.S. principles

We consider (Fig. 1.) a monochromatic incident field E which is linearly polarized:

with **E**
_{S} and **E**
_{P} the proper states of polarization (transverse electric and magnetic). The incident beam passes through a polarization modulator before interaction with the sample under study, with the result:

where **E**
^{’} is the field after the modulator, and Δη=η_{S}-η_{P} the phase difference between the proper states:

In relation (3), Ω is the modulation frequency and α_{0} a residual term. The scattered field **A**(θ) is then recorded through an analyzer at each direction θ in the incident plane.

## 2.1 Case of slightly heterogeneous samples

As discussed in [14–15], the case of slightly heterogeneous samples is the simplest one, due to the absence of cross-polarized scattering in the incidence plane. Each incident polarization (${\mathbf{E}}_{\mathrm{S}}^{\u2019}$ or ${\mathbf{E}}_{\mathrm{P}}^{\u2019}$) creates its own scattering (**A**
_{S} or **A**
_{P}) that we write as:

with ν_{S} and ν_{P} scattering coefficients given by first-order vector theories of surface or bulk [17]. After projection by the analyzer these fields interfere and the resulting intensity I(θ) is recorded as:

with:

and:

These equations are similar to those of specular ellipsometry, due to the absence of cross-polarization. The output signal I involves a continuous component I_{C} and two harmonics I_{Ω} and I_{2Ω}. J_{1} and J_{2} are Bessel functions resulting from the first terms of series expansion of the signal. N_{S or P} and δ are connected with the modulus and argument of the scattered waves:

with:

Therefore calibration of the set-up should give us access to the polarimetric phase term δ(θ) that controls the elliptic polarization of the scattered wave. This term can be analyzed with electromagnetic theories under the Born approximation [17].

## 2.2 Case of arbitrary heterogeneous samples

Cross-polarization effects have now to be taken into account in the incidence plane, due to the high scattering levels. Each incident polarization (${\mathbf{E}}_{\mathrm{S}}^{\u2019}$ or ${\mathbf{E}}_{\mathrm{P}}^{\u2019}$) is both responsible for S and P scattered fields, so that the resulting proper states of scattering are written as:

where ν_{XY} are scattering coefficients given by rigorous theories of surface or bulk. After projection by the analyzer, all fields interfere and the two resulting harmonics are again given as:

Relation (8) shows that the equations are similar to those of the previous section, except that the equivalent amplitude and phase terms are now given as:

In the absence of cross polarization, the equivalent phase term δ^{’} is reduced to δ. In this work special attention will be devoted to this equivalent phase term δ^{′}.

## 3. Scattering origins

#### 3.1 Case of low level scattering

For slightly inhomogeneous substrates, first-order electromagnetic theory predicts all scattering coefficients to be proportional to the Fourier Transform of heterogeneities, that is, for each polarization:

where the subscripts (1) and (2) are respectively for surface and bulk scattering. *h*(σ) is the Fourier Transform of the surface profile h(**r**), while *p*(σ) is the Fourier Transform of the relative transverse variations p(**r**) of the refractive index inhomogeneities. The spatial pulsation σ is connected with scattering angle θ and wavelength λ: σ=(2*π*n_{0}/λ) sinθ (11) with n_{0} the refractive index of the external medium.

In relation (10) the optical factor C_{i} only depends on the origin of scattering (surface or bulk), refractive index of sample, and illumination and observation conditions (wavelength, direction and polarization). All information about microstructure is therefore included in the *h*(σ) and *p*(σ) Fourier transforms.

From relation (10) the polarimetric phase term δ_{i} can be calculated for surface and bulk scattering as:

with *γ*
_{i} the roughness or permittivity spectrum:

and ∑ the illuminated area on the sample. Therefore relation (13) shows that the phase term only depends on the origin of scattering, not on the specific microstructure. This result is valid whatever the structural parameters and hence allows discrimination of surface and bulk effects, provided that the two effects (surface and bulk) are not simultaneously present. Indeed when a competition exists between surface and bulk scattering, the micro-structural terms still remain included in the phase term, and relation (10) leads to:

The result can be a ripple in the phase pattern which reveals interferences between surface and bulk scattering. Such ripple will have a contrast connected with the relative contribution of surface and bulk effects.

We conclude in this section that for low-level scattering, the EARS technique allows one to detect the scattering origins, or to emphasize interferences between surface and bulk scattering. Experimental results were previously given [15] and have confirmed this prediction for fused silica and magnesium fluoride polished substrates.

## 3.2 Case of high level scattering

The results of the preceding section are not valid for arbitrary heterogeneous samples. Rigorous theories must now be used to emphasize the relationship between the equivalent phase term δ^{’} and the microstructure. A priori the results would be strongly dependent on the specific microstructure, even in the absence of surface and bulk interferences. Moreover, few numerical codes are available to predict 3-dimensional scattering from arbitrary surfaces and bulks, and these codes are time consuming. We will see in the last sections how a phenomenological approach may offer an investigation of experimental data. At this step, experiment will lead our investigation and help in the discrimination between surface and bulk effects.

## 4. Experimental setup

The experimental setup (Fig. 2.) is based on a ten-axis computer-controlled scatterometer that was previously described [16]. It allows recording of intensity patterns in the incidence plane at laser wavelengths ranging from 325nm to 10.6µm. The sample can be rotated or translated, and both incidence and scattering angles are variable. The dynamic of measurements is better than 7 decades in the visible, with respect to a lambertian etalon sample that we use for calibration. This dynamic is mainly limited at low angles by parasitic light originating from beam dumps.

In order to record angle-resolved ellipsometric data at each scattering direction, analyzers and polarizers were first introduced with step motors in the scatterometer [14]. More recently a piezo-optic polarization modulator was added to increase the acquisition data speed. Such device introduces a phase difference Δη(t) between the two proper polarization states of any incident polarized light, given by relation (3). The modulation frequency Ω is about 51 kHz. The amplitude Δη_{0} of the modulation can be modified with an electric gain or voltage. Calibration of the modified system was performed with a reference plate and additional data in the absence of sample. Repeatability of the phase term is better than 5%.

## 5. Application to substrates characterization

All measurements are here performed with a 4 mW He-Ne laser at wavelength 633nm. The angular step of scattering measurements is Δθ=1°. We considered 4 samples (Fig. 3.) with different properties:

• The first sample #1 is an opaque polished glass (RG1000 Schott) with low scattering levels. The total integrated scattering measured in the reflecting hemisphere is close to 3. 10^{-4}, which gives an approximate roughness value of 4nm. This kind of glass was previously used in the laboratory [1].

• The second sample #2 is also black glass but with coarse ground finish, so that all the incident light is scattered. The order of magnitude of roughness is 0.5µm, a value extracted from AFM measurements. The diffuse reflectance measured in the reflecting hemisphere is close to 4.4%. The average slope is less than 12°, so that multiple reflections would be minimized.

• The third sample #3 is a metallic (Au) “Infragold” diffuse reflectance material that we currently use for calibration in the mid-infrared. The diffuse reflectance measured in reflecting hemisphere is close to 98%. The roughness and slope are similar to those of sample #2.

• The last sample #4 is another dielectric “Spectralon” diffuse reflectance material (Polytetrafluoroethylene and Barium Sulfate) that we use for calibration in the visible. Its diffuse reflectance measured in the reflecting hemisphere is close to 100%. Roughness and slopes are similar to samples #2 and #3.

The scattering patterns of all samples were measured and are plotted in Fig. 3. Notice that all 3 samples with diffuse reflectance reveal a quasi-lambertian pattern.

## 5.1 Low loss sample

In order to eliminate bulk scattering, the polished opaque sample was first considered. Its real refractive index at 633 nm is given by n^{’}=1.548. The measured reflection from this glass is close to 4.6%, so that the imaginary index n” can be neglected in this first approach. Notice that the glass is opaque because of its thickness (5mm).

Figure 4. is given for theoretical predictions. The angular behavior of the phase term δ is plotted for normal (i=0°) and oblique (i=50°) illumination. At normal illumination, no discrimination is allowed to separate surface and bulk effects, whatever the angular range of scattering data. At the inverse, the phase step for surface scattering at oblique incidence can be directly used for discrimination of surface and bulk effects. This phase step is the consequence of a sign change of the scattered wave at angles close to the Brewster angle [5].

Figure 5. is given for experimental results. Phase data at normal illumination are similar to the predictions given in Fig. 4. Oblique illumination reveals the phase step characteristic of surface scattering, as predicted by theory. However this phase function is not purely a step function. This effect can be first attributed to the imaginary index n” of the opaque glass (Fig. 5.), but the resulting value would be too high (n”=0.15). Further calculation with first-order theory involving a single layer (Fig. 5.) shows that the departure from a step function can be explained by the presence of a transition layer with optical thickness (ne=55nm) at the substrate surface of the opaque glass. This result is similar to those found with classical ellipsometry on the specular beams (contamination layers). In summary, we here conclude that the phase term behaviour of this first polished sample reveals the scattering origins (roughness), and that a transition layer is detected.

Notice in this subsection that a specific bulk sample would be required to complete the first-order study and emphasize a phase term with specific bulk behavior. However this sample should scatter low bulk scattering predominant over surface scattering, and we did not have such adequate etalon. Indeed all samples with predominant bulk scattering exhibited high-level losses, which cannot be predicted with first-order theory (see next section).

## 5.2 High loss samples

Here we consider the other three samples with total diffuse reflectance, which are glass and the two Spectralon and Infragold samples. The equivalent phase data are measured and plotted in Fig. 6. and 7., for normal (i=0°) and oblique (i=50°) illumination.

• We first observe that the glass and gold samples emphasize an angular behavior similar to that predicted for first-order surface scattering, with a slight ripple around the average curve. This ripple does not exceed some 10° in the angular range, and will be attributed to cross-polarization effects. Notice the average slope of the curves at oblique illumination. In the case of the glass sample, this curve gets closer to the step function of Fig. 4.

• At the inverse, the phase term of the dielectric Spectralon sample exhibits a large ripple uniformly distributed within the [-π, π] interval, whatever the illumination incidence. This effect is specific of bulk scattering.

Therefore the results presented in Fig. 6. and 7. allow a direct discrimination of surface and bulk effects. Indeed scattering from the dielectric Spectralon sample is expected to originate from bulk, since its diffuse reflectance (100%) is greater than reflection (some %). This result is not true for the Au Infragold sample, whose reflection is close to 98%. Moreover, scattering from the Au sample is expected to originate from roughness, due to its opacity. At last, scattering from the glass sample is expected to originate from roughness, due to the well known glass homogeneity and to the fact that its diffuse reflectance is close to reflection (4.6%).

## 5.3 Phase data in the speckle pattern

To go further in the investigation, the two Spectralon and Infragold samples were measured with a higher angular resolution given by Δθ=0.05°. The incident spot area of the illumination beam is ∑=4mm^{2}, which gives the angular size (δθ=0.02°) of the speckle pattern. The optical fiber collects the scattered light at a distance 80cm from the sample, so that the integration angle due to the receiver is close to 0.07° (the fiber diameter is 1 mm). Hence an angular step of Δθ=0.05° is rather enough to resolve and measure the polarimetric phase term in the speckle pattern.

Results are given in Fig. 8. in the angular range (30°–31°), and reveal properties similar to those given at low angular resolution in Fig. 6. and 7. For the dielectric Spectralon sample, the equivalent phase term is still uniformly distributed in the [-π, π] interval, while variations are much less for the metallic Infragold sample. Repeatability of measurements was carefully checked for all these measurements, with accuracy better than 5%.

## 6. Theoretical investigation

As said in section 3, strictly speaking all results for arbitrary surface and bulk scattering could be predicted with rigorous electromagnetic theories [18, 19] involving integral, modal, differential methods and others. However few calculation codes are at disposal when two-variable surfaces and bulks are considered and these codes are time-consuming. To overcome this difficulty, we use two approaches.

## 6.1 Basic analysis

For surface scattering we have observed a reduced ripple in the equivalent phase term, whatever the scattering levels. An approximate analysis can be provided under the assumption that scattering can be predicted with local reflection of rays at tangential planes. According to this basic theory, the phase values can be deduced from those calculated for specular reflection, that are plotted in Fig. 9. In this figure we notice that the phase step occurs at a starting angle i_{s} which is material dependent. Now when the surface is illuminated at incidence i, scattering originates from local reflection at tangential planes with local incidence il given by:

with β the normal angle of tangential plane with respect to Oz. Therefore the phase step can only be reached for incidence values i_{1} greater than i_{s}, with the result:

At normal illumination (β> i_{s}), it should be necessary for the surface to have slopes greater than i_{s}, which is hardly probable because of its high value (i_{s} >40° for both glass and gold). Such prediction is in agreement with the low phase value measured for all rough surfaces in this paper. Now at oblique illumination close to the starting angle (i ≈ i_{s}), relation (17) is reduced to β>0, so that the phase step may occur even for low slope values, which is in agreement with experimental data. Such interpretation will be confirmed in section 7.

## 6.2 Phenomenological investigation

In relation (9) all scattering coefficients ν_{XY} are complex numbers with modulus ${\mathrm{N}}_{\text{XY}}^{0.5}$ and argument δ_{XY}:

We now introduce an amplitude cross-polarization ratio τ as:

With the result:

Where T is the energy cross-polarization ratio. These relations give the equivalent phase term as:

In most situations experiment reveals the same order of magnitude for T_{SP} and T_{PS}, so that we here assume for the sake of simplicity:

At this step the phase ripple may be both explained by one (δ) or the other (arg(z_{1})) argument, if we assume all phase terms δ_{XX} and κ_{XY} to be random variables. To go further in the investigation and emphasize modifications in regard to first-order results, we introduce additional parameters:

In this relation the subscript (1) is for first-order scattering (ν_{1,XX}), while subscript (2) indicates the difference (ν_{2,XX}) between total scattering ν_{XX} and first-order scattering (ν_{1,XX}). Therefore α_{XX} expresses the relative weight of first-order scattering in the resulting SS or PP scattering process. It is again assumed that α_{SS} and α_{PP} have identical modulus, as often confirmed by experiment. The final result is at last given as:

Where δ_{1} is the first-order phase term of Fig. 4., and:

With:

Numerical calculation now allows us to study any departure of δ′ from the first-order phase δ_{1}, due to increasing heterogeneities. We only plot the z_{1} ripple from relation (22) because z_{1} and z_{2} are equivalent. Illumination is assumed to be oblique (i=50°). In Fig. 10. we observe an increase of ripple with values of cross-polarization ratio T. Low T values create a slight ripple analogous to surface scattering of Fig. 5. and 7. At the inverse, a maximum ripple is reached for a unity value of T, as for the case of bulk scattering in Fig. 6. and 7. Notice here that though T is limited to 1, higher values of ρ can be considered and create a ripple analogous to the case T=1.

If we compare these results with experimental data of Fig. 5. and 7., some conclusions can be drawn. If we first consider surface scattering where the ripple is slight whatever the scattering levels, both phase terms arg(z_{1}) and arg(z_{2}) must be reduced in relation (22). There are 2 ways to reduce these arguments, which are low modulus or low deterministic phase. Concerning the cross-polarization ratio, the result is a low T value (T≈0.01) for the glass and Au samples, associated with random phase behaviour of κ. Concerning parameter ρ, and because it cannot be much less than unity in the presence of diffuse scattering, the result is a low deterministic phase for ξ. At last to explain bulk scattering (dielectric Spectralon sample), one or the other phase (κ or ξ) must be random, together with a high modulus value (T or ρ). These results will be confirmed in the next section.

## 7. Further confirmation

In order to check the values predicted for cross-polarization ratio, we have measured the angular scattering for all SS, SP, PP and PS polarizations. The results are given in Fig. 11. and confirm a unity T value for the bulk sample (dielectric Spectralon), and a low value for the metallic Infragold (T≈0.02) and glass (T≈0.07) samples. The lowest T values are in agreement with the lowest phase ripples. Notice also that our assumption T_{SP}≈T_{PS} is confirmed by experiment. Fig. 12. is an additional confirmation, since it shows the absence of modulation for the bulk sample whatever the analyzer position.

The last confirmation is given by atomic force measurements of roughness. The AFM sampling interval was first chosen (Δx=160 nm) to reach a band-pass or resolution similar to the optical one, which guarantees that the measured roughness is that responsible for far field scattering [20, 21]. As can be checked for the glass sample in Fig. 13., the roughness is close to 0.5 µm but the average slope magnitude remains less than 12°, which explains the slight values of cross-polarization ratio and associated ripple. Therefore care must be taken before general conclusions, because surfaces with high slopes responsible for multiple reflections could behave as heterogeneous bulks. The same remark is valid for surfaces that are not functions but parametric curves.

## 8. Conclusion

Angle-resolved variations of the polarimetric phase term can be recorded in the scattering pattern of any sample, thanks to the E.A.R.S technique. High resolution data can easily be reached, which allows to investigate the resolved speckle. For polished samples that can be described with first-order electromagnetic theories, the phase variations offer a direct discrimination of surface and bulk effects. Interferences between surface and bulk patterns could also be revealed when a ripple is observed on the curve.

In the case of non polished samples with diffuse reflectance, we have shown how these techniques can be extended. In particular the phase variations of an inhomogeneous dielectric Spectralon sample were shown to exhibit strong variations randomly distributed in the (-π; π) interval, which is specific of bulk scattering. At the inverse, the same curve measured for a rough Infragold or glass surface emphasized variations strongly limited within ±10°, which is specific of surface scattering. These results were shown to be in agreement with predictions issued from a basic theory of local reflection at tangential planes. In addition and thanks to a phenomenological approach, the ripple contrast in the phase term was shown to increase with cross-polarization ratios. These ratios were extracted from measurements and confirmed by additional experimental data.

In summary all results prove that EARS constitutes a promising tool for probing and discrimination of arbitrary heterogeneities in surfaces and bulks, whatever the scattering levels. However all surfaces in this paper have slopes lower than 12°, so that additional work remains to be done on surfaces with higher slopes.

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