Polarization Characteristics of Scattered Radiation from a Diffraction Grating by Ellipsometry with Application to Surface Roughness

The polarization properties of the various diffracted orders from a grating are determined ellipsometrically. The ratio ${\rho }_m$ of the far-field complex reflection coefficients for the $p$ and $s$ polarizations for any order $m$ is measured. This information is complementary to the data available from the partition of the scattered energy over the radiating orders. By examining the polarization of the specularly reflected zeroth-order beam at large angles of incidence the optical effect of surface roughness is established. The roughened surface layer is shown to be equivalent to a film whose index of refraction is the average of the indices of the material of the surface and the immersion medium, in accordance with the Maxwell Garnett theory. The film thickness is a measure of the surface roughness. To explain the ellipsometric results on the various orders we derived an expression for ${\rho }_m$ starting from the Stratton-Silver-Chu integral using the physical-optics approximation. There is agreement between the gross features of the theory and experiment, but the exact magnitudes could not be compared. This points to the need for an exact solution of the grating-diffraction problem for both polarizations, including the effect of a finite conductivity at optical frequencies.