Polarization characteristics of scattered light from macroscopically rough surfaces

Abstract

The polarization status of in- and out-of-plane scattered light from macroscopically rough surfaces was measured and expressed in a scattering coordinate system. In this coordinate system, the direction of the scattered light was decided by the transverse and longitudinal scattering angles, both of which are defined by using the incident ray and its section plane as the fundamental axis and plane. The polarization characteristics of scattering, which consist of patterns of polarization degrees and the Stokes of the scattered light in terms of transverse and longitudinal scattering angles, was presented and investigated. The Stokes patterns of scattered light exhibited independence from the transverse scattering angle. Furthermore, the ellipsometric parameters of samples, which were deduced from the polarization status of full scattering, showed the validity as well as limitations of the facet model for macroscopically rough surface.

Appendix

The Stokes of the reflected lights are related to the polarization state of the incident light (right circularly polarized in this work), and given by [19],

$$S_{0} = r_{p}^{2} + r_{s}^{2} ,$$

(A-1)

$$S_{1} = r_{p}^{2} - r_{s}^{2} = - \cos 2\varPsi ,$$

(A-2)

$$S_{2} = 2r_{p} r_{s} \cos (\Delta + 90^\circ ) = - \sin 2\varPsi \sin \Delta ,$$

(A-3)

$$S_{3} = 2r_{p} r_{s} \sin (\Delta + 90^\circ ) = \sin 2\varPsi \cos \Delta ,$$

(A-4)

where Δ is the phase difference of p and s polarized light by reflection, and Ψ is the ratio of p and s polarized light. The phase 90° in Eqs. (A-3) and (A-4) is invoked due to the right circularly polarized incident light.

Fresnel reflection coefficients r _p and r _s are reflection layer related factors. For the reflection at a single interface between medium 1 (refractive index N ₁) and medium 3 (refractive index N ₃), the coefficients become,

$$r_{p}^{13} = \frac{{N_{1} \sin \frac{{\theta_{\text{ls}} }}{2} - N_{3} \left[ {1 - (N_{1} /N_{3} )^{2} \cos^{2} \frac{{\theta_{\text{ls}} }}{2}} \right]^{1/2} }}{{N_{1} \sin \frac{{\theta_{\text{ls}} }}{2} + N_{3} \left[ {1 - (N_{1} /N_{3} )^{2} \cos^{2} \frac{{\theta_{\text{ls}} }}{2}} \right]^{1/2} }} ,$$

(A-5)

$$r_{s}^{13} = \frac{{N_{3} \sin \frac{{\theta_{\text{ls}} }}{2} - N_{1} \left[ {1 - (N_{1} /N_{3} )^{2} \cos^{2} \frac{{\theta_{\text{ls}} }}{2}} \right]^{1/2} }}{{N_{3} \sin \frac{{\theta_{\text{ls}} }}{2} + N_{1} \left[ {1 - (N_{1} /N_{3} )^{2} \cos^{2} \frac{{\theta_{\text{ls}} }}{2}} \right]^{1/2} }} .$$

(A-6)

For the successive reflection from two interfaces between medium 1, medium 2 (refractive index N ₂), and medium 3, the total reflection coefficients will be [20],

$$r_{p}^{123} = \frac{{r_{p}^{12} + r_{p}^{23} \exp ( - j2\beta )}}{{1 + r_{p}^{12} r_{p}^{23} \exp ( - j2\beta )}} ,$$

(A-7)

$$r_{s}^{123} = \frac{{r_{s}^{12} + r_{s}^{23} \exp ( - j2\beta )}}{{1 + r_{s}^{12} r_{s}^{23} \exp ( - j2\beta )}} ,$$

(A-8)

where the subscript “12” and “23” denote that the coefficients are for interfaces between medium 1 and medium 2, and medium 2 and medium 3, respectively. β is the phase thickness of the medium 2 and is given by [20]

$$\beta = \frac{2\pi d}{\lambda }N_{2} \left[ {1 - (N_{1} /N_{2} )^{2} \cos^{2} \frac{{\theta_{\text{ls}} }}{2}} \right]^{1/2} ,$$

(A-9)

where d is the thickness of medium 2.