Elsevier

Thin Solid Films

Volumes 455–456, 1 May 2004, Pages 50-53
Thin Solid Films

Optical anisotropy relevant to rotating-compensator polarimeters: application to the monoplate retarder

https://doi.org/10.1016/j.tsf.2003.12.044Get rights and content

Abstract

We analyze the optical properties of the monoplate compensator under misalignment conditions. The coupling between the boundary conditions and the secular equation in the crystal frame leads to a quartic secular equation, and certain representations of the mode vectors vanish for particular combinations of parameters. Misalignment affects mainly odd harmonics in the transmitted intensity. The even harmonics, which carry sample information, are affected only to second order. Results are in good agreement with experiment.

Introduction

Monoplate compensators [1] are a practical alternative to biplate and Berek [2] compensators, not generating the multiple-internal-reflectance artifacts of the biplate and substantially reducing the beam displacement characteristic of the Berek plate. Monoplates are fabricated from optically uniaxial material such as MgF2 with the c-axis oriented internally at an angle, typically near 7°, relative to the surface normal. They thus differ from the Berek plate, where the c-axis is normal to the surface and the entire plate is inclined.

Here, we analyze the monoplate where both the unit vector normal to the surface and the entering wave vector are misaligned with respect to the rotation axis ω. In a rotating-compensator ellipsometer such misalignment generates 1ωt, 3ωt and 5ωt coefficients in the transmitted intensity in addition to the d.c., 2ωt and 4ωt coefficients that carry the information about the sample, where ω is the (mechanical) angular velocity of the plate. The extent to which the even coefficients are affected has not yet been established. While the analysis of the aligned monoplate is straightforward, in the misaligned case coupling between the boundary conditions and the dielectric tensor in the crystal frame yields a quartic dispersion equation. In addition, the lengths of some representations of the normal-mode vectors vanish at certain azimuth angles of the plate, preventing a direct determination of their relative projections on the coordinate axes. If this difficulty is not properly taken into account the inability to calculate these projections lead to large inaccuracies in subsequent numerical calculations. These observations motivated a second purpose of this work, to discuss practical aspects of the solution of the dispersion equation for propagation in anisotropic materials, and thus supplement the original work of Yeh [3].

We have investigated the problem both numerically and by a first-order analytic expansion, although only the former results will be discussed here. The odd-ωt terms are found to arise from a modulation of the relative retardance δ of the principal axes of the plate as well as an effective modulation of its azimuth C about its expected (mechanical) value. The even-ωt coefficients are affected only to second order, so some level of misalignment can be tolerated without adversely affecting the information about the sample. Both numerical and analytic results show that all odd coefficients should vanish simultaneously, so the observation that the 1ωt and 3ωt coefficients vanish under different conditions indicates the existence of another mechanism affecting the retardation properties of the plate, probably stress.

Section snippets

Theory

We consider specifically a rotating-compensator ellipsometer of the PCSA type, with which our comparison data were obtained. Because, our interest is centered on the compensator, all other components are considered ideal. If the compensator is also ideal the transmitted intensity is given by the absolute square of the transmitted field, which is given by the Jones-matrix productE′xE′y=1000cosAsinAsinAcosArp00rscosCsinCsinCcosC×t1100t22cos(C−P)sin(C−P)sin(C−P)cos(C−P)1000ExEywhere (Ex,Ey) is

Application

We apply this analysis to our PCSA ellipsometer operating with a Si wafer and A=P=0°. To the extent that optical activity in the quartz Rochon prisms can be ignored, this is equivalent to straight-through operation. The system uses a MgF2 monoplate with an internal angle of 7.660°, which gives it a retardation of 90° at a photon energy of 2.915 eV. The monoplate is carried in a hollow-shaft motor that is mounted on a base plate that functions as a geometric clamp whose polar and azimuthal

Acknowledgements

We are pleased to acknowledge support by the US Office of Naval Research.

References (3)

  • P. Yeh

    Surf. Sci.

    (1980)
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