1. Introduction

Elimination of light scattering [1,2] constitutes a challenge for most applications involving imaging systems, optical communication and characterization techniques. Numerous efforts have been performed in the thin film community to minimize scattering levels in optical coatings, thanks to polishing and cleaning techniques, as well as new deposition processes … However a relevant question lies in the possibility to reduce scattering whatever the materials, polishing processes or deposition technologies. As an example, and concerning optical multilayers whose surfaces are known to scatter most of the incident light, special designs have been emphasized to shift the stationary electric field [3,4] within the bulk of materials, with limited success. Another way to reduce scattering has consisted in the phase opposition of waves scattered by internal surfaces of the stack, with large success confirmed by experiment [5,6,7]. However this technique can only be applied to single layers and narrowband filters, not to mirrors where the scattered waves are in phase. Moreover, it only works for correlated stacks [8]. Lastly, it cannot work for bare substrates where only one surface is responsible for scattering.

In this work we present a procedure to eliminate angle-resolved polarized light scattering from optical substrates and coatings under monochromatic polarized illumination. The method is based on the angular behavior of the polarization of light scattering, a topic that was addressed by several authors [9–15] and that has led to the introduction of ellipsometric techniques [16–20] in the scattering pattern. These polarization techniques have shown that the sensitivity of scattering to specific sources or scatterers could be largely improved to provide a discrimination of surface and bulk effects [16–18], localized defects and others … Moreover, they were used to extract roughness and cross-correlation parameters in single layers, as shown by numerous results given by Germer [19–20]. In a general way, all these methods are valid for low scattering levels that can be calculated with perturbation theories that predict a polarization ratio not dependent on micro-structural parameters such as roughness and heterogeneity.

However one can go further in this investigation and search for the conditions of zero scattering at each direction of space, thanks to interferences between waves scattered from each polarization of the scattered light. A former result can be found in [11] where light scattering from a single surface was theoretically shown to be zero in the azimuthal plane π/4, under the assumption of a circularly polarized illumination. In a more general way and due to the fact that angle resolved scattering can be fully polarized, one can theoretically search for amplitude and phase distributions that allow to reach an annulment condition at each direction of space. When these conditions cannot be directly satisfied, they can be fulfilled by the introduction of single optical devices such as a rotating analyzer and a tunable retardation plate. Because the annulment conditions differ for surface roughness and bulk heterogeneity, the procedure allows a complete separation of surface and bulk effects. In other words, it allows direct probing of bulks in the absence of roughness effect, or direct probing of roughness in the absence of bulk effect. At low scattering levels, the efficiency of the method is connected to the fact that the polarization ratio does not depend on structural parameters. Within the framework of perturbation theories, the annulment procedure can be directly generalized to multilayers.

The last section of the paper addresses the case of high scattering levels that originate from arbitrary surface roughness and bulk heterogeneity. Because an equivalent polarization can still be defined and measured [17] in the scattering pattern of samples with diffuse reflectance, the procedure can be directly again generalized. However in this case the annulment conditions depend on the structural parameters (topography and heterogeneity …), which rises additional difficulties that we emphasize.

It should be noticed that our procedure can be seen as the continuation of previous works from the authors [15–17] and Germer [18–20], that describe enhanced sensitivity of polarized scattering to specific defects. The key points of this paper lie in the annulment conditions and equations that allow direct imaging of particular defects, and in the generalization of the procedure to arbitrary scattering levels.

2. Principles

2.1 Incident illumination

As usual in the ellipsometric techniques, the method requires a polarized illumination of the sample, so that interferences may occur between waves scattered by each incident polarization. We consider (Fig. 1) an incident plane wave given as:

where ρ = (x,y,z) is the spatial coordinates, and k ⁺ the incident wave vector:

with λ the wavelength, n₀ and n₁ the refractive index of superstrate and substrate, i the incidence angle and k = 2πn₀/λ.

 

Fig. 1. Incident polarized plane wave on a sample with normal z

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In relation (1) ${{\mathbf{A}}_{\mathrm{S}}}^{+}$ and ${{\mathbf{A}}_{\mathrm{P}}}^{+}$ respectively designate the proper polarization states of the electric field, whose complex amplitudes are given as:

with η_S and η_P the polarization phase terms. Relations (1–3) are valid for elliptical (η_S ≠ η_P) or linear (η_S = η_P) polarizations, depending on the relative values of the phase terms.

2.2 Polarized scattered waves

Whatever the origin of scattering (surface or bulk), the polarized fields scattered in the far field (Fig. 2) by each polarization of the incident beam can be written at direction (θ,ϕ) of space as:

where the superscript (±) is for reflection (-) or transmission (+), and k ^± the scattered wave vector:

with k = 2πn₀/λ (reflection hemisphere) or k = 2πn₁/λ (transmission hemisphere). The subscripts UV designate the V polarization of the wave scattered by the U polarization of the incident beam.

 

Fig. 2: Scattered wave vectors k^± at directions (θ₀,ϕ) by reflection, and (θ₁,ϕ) by transmission. The two wave vectors k^- and k⁺ have the same tangential component which is the spatial pulsation σ = 2π ν, with ν the spatial frequency.

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Let us now be interested in optical samples with low-level scattering, so that first-order electromagnetic theory [2, 21–24] can be used. The complex amplitudes of the scattered waves can furthermore be detailed as:

where the normal angle θ is given for the half hemispheres of reflection (θ = θ₀) and transmission (θ = θ₁). In relation (5) all information about microstructure is included in the f(θ,ϕ) term that describes [23] the Fourier Transform of the surface profile (case of surface scattering) or the Fourier Transform of the relative variations of refractive index (case of bulk scattering). This result is characteristic of first-order theories that predict the scattered field as the product of two terms, the first f(θ,ϕ) being related to micro-structural parameters, and the second ${{\mathrm{C}}_{\text{UV}}}^{\pm }$(θ,ϕ) being related to optical properties of the ideal sample (perfectly smooth and homogeneous). For more information, the optical factor ${{\mathrm{C}}_{\text{UV}}}^{\pm }$(θ,ϕ) is connected with illumination and observation conditions such as illumination incidence, polarization and wavelength, scattering angles. Moreover, it depends on the refractive indices of materials. Lastly, this factor depends on the origin of scattering (surface or bulk), and can be found in numerous papers [2, 21–24]. In particular in reference [23] a detailed comparison of surface and bulk scattering can be found, with analytical formulas for these coefficients. All scattering coefficients are derived from first-order development of the Fourier Transform of the field relationships at interfaces. However, in the case of bulk scattering, relation (5) is only valid in the case where the random index variations are assumed to be transverse [21]. In other situations, the bulk optical factor would depend on structural parameters in a complex way [21]. Notice also in the general case that first-order theories do not predict any cross-polarized scattering in the incidence plane (ϕ= 0), that is: ${{\mathbf{A}}_{\text{SP}}}^{\pm }$ = ${{\mathbf{A}}_{\text{PS}}}^{\pm }$ = 0.

2.3 The interferential zero condition

In order to project and align all polarizations of the scattered wave, we introduce an analyzer on the scattered beam at direction (θ,ϕ). The resulting wave is the interferential sum:

with:

and:

with ψ the position angle of the analyzer with respect to the tangential S (or TE) direction. Therefore a condition of zero scattering (A = 0) can be written at direction (θ,ϕ) of space as:

This relation is given for annulment of surface scattering, or for annulment of bulk scattering. We notice that the condition of zero scattering does not depend on the micro-structural term f(θ,ϕ), a result which is specific of first-order theory. Therefore the annulment condition is only connected with the scattering origins via the C_UV coefficients, which provides a major result. Now in the next section we investigate solutions for equation (8).

3. First-order solutions

Under the assumption of equal energy in each incident polarization (|${{\mathrm{A}}_{\mathrm{S}}}^{+}$| = |${{\mathrm{A}}_{\mathrm{P}}}^{+}$|), relation (8) can be rewritten as:

with Δη the polarization parameter of the incident beam:

Solutions of (9) require 2 conditions on modulus and argument at each scattering direction, whereas the only free parameter is the analyzer angle ψ. Though ψ(θ,ϕ) can be adjusted versus (θ,ϕ), an additional and controllable phase term Δη^*(θ,ϕ) must be introduced to satisfy the argument condition. This objective may be reached thanks to a tunable phase retardation device. The fast axis of this device should be parallel to the S direction of the scattered wave, which is parallel to the average sample surface and perpendicular to the spatial pulsation σ (see fig. 2). With such a device introduced on the scattered beam, relation (9) is turned into:

with Δη^* (θ,ϕ) the retardation phase term that describes an additional delay between the two polarization states of the scattered wave. Therefore at each direction of space, a condition for zero scattering can be reached by simultaneous matching of the analyzer angle ψ(θ,ϕ) and the phase term Δη^*(θ,ϕ). The analyzer angle (ψ>0) is given by the modulus condition:

and the phase term is given as:

Notice from Eqs. (12) and (13) that ψ(θ,ϕ) and Δη^*(θ,ϕ) can be calculated independently at each scattering direction, which is practical from the point of view of experiment. It should not be the case if the retardation plate element were introduced on the incident beam rather than on the scattered beam. Indeed one can check that in this situation the phase term Δη of relation (9) would be replaced by a new Δδ term:

with:

and Δη^* the retardation phase term that describes an additional delay between the two polarization states of the incident wave. We obtain:

with the argument and modulus conditions:

We notice from relation (17) that the modulus condition cannot be satisfied in the general case, for which reason the retardation device should not be introduced on the incident beam.

4. Numerical calculation

First of all we notice that the procedure is also valid for optical coatings, provided that all surfaces or bulks are identical within the multilayers. Under this assumption all cross-correlation coefficients are unity [8,11,23] and relation (5) is not modified for substrates or multilayers, with equivalent optical factors C_UV [8]. In this section we limit ourselves to angle-resolved scattering by reflection in the incidence plane (ϕ=0), which is easier to investigate because of the absence of cross-polarized light. Moreover this configuration may easily fit experiment since most scatterometers work in the incidence plane [25]. With a retardation plate on the scattered beam at direction θ, relation (11) is therefore reduced to:

so that ψ and Δη⁸ are given by:

In what follows we assume the incident polarization to be linear (Δη=0). Notice that in many situations the optical factors C_UU are slightly different, in particular at low scattering angles and for single designs under normal illumination. For this reason optimal values of ψ and Δη^* will often be close to π/4 and π, respectively. However these optimal values should be different for surface and bulk scattering, in order to eliminate one (surface roughness) or the other (bulk heterogeneity) scattering component. Therefore specific designs or illumination angles must be used to allow discrimination of effects.

Numerous results can be found in the literature [1–9, 23] concerning the angular behaviour of the modulus | C_SS(θ)/C_PP(θ) |. At the inverse, few studies [15–20] concern the Δη^*(θ) variations that must be controlled to eliminate scattering. These variations are calculated in Fig. 3–5 for substrates and correlated designs of single layers, mirrors and filters. They are given for surface and bulk scattering, respectively. The wavelength under study is 633nm. Figure 3 is calculated for a fused silica substrate illuminated at 56° incidence. At angles lower than 56°, the phase term is identical for surface and bulk origins, so that both components of scattering should be eliminated at the same time. At larger angles, the phase terms are different and differ from a π value, which allows to eliminate one (surface) or the other (bulk) scattering component. Notice that these differences in the angular behaviors of the bulk and surface phase factors are similar to what is observed for specular reflection after the Brewster angle. In other words, there is a sign change for the field scattered from surface roughness, but this is not the case for the field scattered from bulk heterogeneity.

Figure 4 is given for a single half-wave thin film SiO₂ layer of refractive index 1.49, with similar conclusions. Fig. 5 is calculated for a narrow-band filter of design HLHLH(6L)HLHLH, where H and L designate high (TiO₂) and low (SiO₂) index thin film materials with quarter-wave optical thicknesses. The refractive index of TiO₂ is 2.15 at wavelength 633nm. For this coating the separation of surface and bulk effects is effective in practically the whole angular range.

In most situations illumination and observation conditions will be found to enhance discrimination of phase terms characteristic of surface and bulk effects. The reason is connected with the fact that surface scattering originates from electric and magnetic currents located at surfaces, while bulk scattering originates from electric currents within volumes [23].

 

Fig. 3. Angular variations of the phase term Δθ^*(θ) (see text) calculated for bulk and surface scattering at wavelength 633nm. The illumination incidence is 56°. The sample is a fused silica substrate (n₁ = 1.50).

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Fig. 4. Angular variations of the phase term Δη^*(θ) (see text) calculated for bulk and surface scattering at wavelength 633nm. The illumination incidence is 56°. The sample is a single thin film half-wave SiO₂ layer.

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Fig. 5. Angular variations of the phase term Δη^*(θ) (see text) calculated for bulk and surface scattering at wavelength 633nm. The illumination incidence is 56°. The sample is a narrowband filter of design HLHLH (6L) HLHLH (see text).

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5. Case of arbitrary roughness and heterogeneity

Until now we discussed the case of slightly heterogeneous samples whose scattering can be predicted with first-order theory. Strictly speaking the previous procedure for scattering elimination should also be valid for high scattering levels resulting from arbitrary defects. The zero condition is still given by:

A first difference lies in the presence of cross-polarization terms (A_UV) that do not vanish in the incidence plane. But the key difference results from the fact that the scattered fields A_UV are no longer proportional to the Fourier Transform of defects (surface profile or random permittivity), so that relation (5) cannot be used any more. Indeed all fields are now connected with microstructure (roughness or heterogeneity) via complex relationships such as integral equations [26,27] or others. The result is that the zero condition will also be dependent on microstructure at each scattering direction, which is less practical. With a phase retardation device on the scattered beam, we obtain:

where ν _UV are scattering coefficients that are micro-structural dependent:

The analyzer angle and additional phase term are again given by the modulus and argument conditions at each scattering direction (θ,ϕ) as:

Relations (24–25) offer ψ and Δη^* values for elimination of surface or bulk scattering (or both) at each direction. However these values can be calculated only if the specific sample microstructure is known, so that the scattering coefficients ν _UV may be predicted with a rigorous theory [26,27] involving structural data.

Notice also that experiment could be directly used to search the zero conditions by scanning all ψ and Δη^* values at each scattering direction. However with this procedure we would only be able to eliminate the sum of surface and bulk scattering, not to discriminate between these effects. Indeed though the scanning procedure would involve analyzer and retardation values (ψ, Δη^*) that eliminate specific effects such as roughness or bulk scattering, there is no way to detect or track these values in the absence of microstructure knowledge. In other words, the experimental scanning procedure would only provide detection of the annulment conditions for the total scattering.

At last we notice in this section that the phase term is expected to show rapid and large angular variations in the scattering pattern. In a recent work [17] we showed that the polarization state could still be perfectly predicted and measured in the far field of the speckle pattern, so that all phase terms can still be measured with high-angle resolution, and then used for the annulment procedure.

6. Conclusion

We have first shown how surface or bulk scattering can be eliminated in low-loss optical coatings and substrates. The method is based on polarized interferences and requires both a controllable analyzer and a retardation phase device. Under these conditions, angle-resolved scattering can be eliminated at each particular scattering direction. The annulment condition does not depend on the specific sample microstructure, but only on the scattering origins. This result allows a complete discrimination of surface and bulk effects. It is therefore possible to probe bulks after elimination of surface scattering, or to probe surfaces after elimination of bulk scattering.

The procedure can be directly extended to samples with arbitrary surface roughness or bulk heterogeneity. The annulment conditions are given in a similar way, but the major difference lies in the fact that these conditions depend on the specific sample microstructure, which should be preliminary measured. In all cases the annulment of scattering can be reached by scanning all analyzer and retardation plate positions.