Rotatable broadband retarders for far-infrared spectroscopic ellipsometry

Abstract

Rotatable retarders have been developed for applications in spectroscopic, full Mueller Matrix ellipsometry in the far-IR spectral range. Several materials, such as silicon, KRS-5, and a commercial polymer plastic (TOPAS) have been utilized to achieve a fully adjustable retardation between 0° and 90°. Experimental characteristics of the rotatable retarders that utilize three- and four-bounce designs are compared with calculations. We discuss the effect of light focusing on the performance of these rotatable retarders.

Introduction

Broadband optical retarders are required for spectroscopic ellipsometry in its full Mueller matrix (MM) realization. Performance of the MM ellipsometer depends on the capability to produce substantially linearly-independent Stokes vectors for the light incident onto the sample. As has been shown [1], the errors in the measured MM of the sample are proportional to the condition number of the 4 × 4 matrix composed of the Stokes vectors of four polarization states incident at the sample. It can be proven that it is impossible to cover the Poincare sphere with linearly-independent Stokes vectors by only changing the linear polarization at the input surface of a stationary retarder. As we will illustrate further in this paper, total coverage of the Poincare sphere is possible by rotating a tandem of a linear polarizer and a retarder with a retardation of 90°. It is this goal that we are trying to achieve in the retarder designs described in this paper.

Traditionally, broadband retarders for the far-IR ellipsometry are made from a single triangular prism where the phase retardation between the s and p polarizations is achieved by total internal reflection inside the prism. The prism is cut for normal incidence at both the entrance and exit surfaces, thus eliminating any polarization effects at these surfaces. For the internal reflection, the incident angle θ should be greater than the critical angle to avoid intensity losses. By using Fresnel reflection coefficients, the one-bounce retardation δ that occurs from total internal reflection at incident angle θ in a prism material is obtained [2], [3]:

$$tan(\delta /2)={({sin}^2\theta -n^{-2})}^{1/2}/(sin\theta tan\theta )$$

where n is the ratio of refractive indices of the incidence and refraction media, respectively. Calculations for the maximum retardation provided by a single prism made of different materials are shown in Fig. 1(a). Transparent and isotropic materials are typically used, such as Si, KRS-5, polycrystalline ZnSe, or transparent polymer plastics. One can see that a 90° retardation is not always possible with a single reflection, especially for low-index materials, such as TOPAS plastic. However, a 90° retardation can be obtained with two or more internal reflections, such as found in a Fresnel rhomb. The choice of the retarder material is determined by the spectral range of the measurements. For example, the free-carrier absorption at low frequencies and the weak phonon absorption band at 520 cm⁻¹ can affect the performance of a Si retarder. KRS-5 can be used only in the frequency range above 400 cm⁻¹ due to phonon absorption.

The single-prism (= one internal reflection) retarder design has an obvious disadvantage for the broadband spectroscopic MM ellipsometry. To create at least four linearly-independent Stokes vectors on the sample surface, the retarder should be rotatable, which will ultimately steer the light beam off the optical axis of the ellipsometer. To return the beam back to the optical axis of the ellipsometer, one needs at least two more reflections. However, metal mirrors positioned at a high incident angle are not always suitable for this purpose due to dispersion that would result in a strong spectral dependence of retardation even in the far-IR spectral range. Another problem is related to metal mirror contamination that will also change the retardation over time and will require a frequent recalibration. To avoid this problem, various designs for rotating retarders relying on the double-Fresnel-rhomb approach have been proposed [4], [5]. However, no commercial solution is known for the far-IR spectral range.

A far-IR spectroscopic ellipsometer with the full MM capability has been recently developed at the U4IR beamline of the National Synchrotron Light Source (NSLS) at Brookhaven Nat'l Lab (BNL). We investigated the design and performance of retarders made from Si, KRS-5, and TOPAS materials for use in the spectral range between 10 and several thousands of cm⁻¹. Note that TOPAS plastic is transparent in both, visible and far-IR spectral ranges, so this greatly facilitates the procedure for accurately aligning the optical components inside the ellipsometric optical system. This is another reason why TOPAS plastic was chosen, over other common polymer materials for the far-infrared. In this paper, we focus on their performance in the most challenging far-IR spectral range between 10 and 100 cm⁻¹.

Alternatively, birefringent materials could be used as retarders. However, most of the birefringent materials have IR-active optical phonons, so the absorption due to optical phonons will affect their performance in a broad spectral range including the far infrared. Materials, like Si, do not have IR active phonons, but they are not birefringent. Application of external electric field to Si, which can cause birefringence, is not trivial, since it is hard to obtain a uniform field distribution across the large acceptance area of a retarder. Another alternative approach could be based on the application of thin films. However, in this case one can expect a strong dependence of retardation across the optical spectrum. This approach works much better for single wavelength ellipsometry, but in the far-IR the thin-film approach does not seem to be easy.