Quartz crystal is an important birefringence material, which is widely used in optical related fields. The two main optical parameters of quartz crystal wave plate are phase retardation and fast axis azimuth. Due to the influence of manufacturing process, these two actual parameters will deviate from the theoretical value, so it is usually necessary to accurately measure the optical parameters before use. Ellipsometry is usually used to measure the parameters of quartz crystal in a wide spectrum, but the existing ellipsometric measuring instruments often assume that the optical axis of the crystal is aligned with the measuring optical path, which introduces measurement error, especially in the ultraviolet band. Therefore, it is necessary to propose a fitting model for accurate measurement of quartz crystal parameters. The model contains rich information and the fitting results are accurate. This model has important reference value for measuring the accurate parameters of anisotropic materials by ellipsometry.

A Mueller matrix model for accurate measurement of quartz crystal parameters by ellipsometry is proposed. Firstly, the crystal coordinate system (a, b, c) is transformed into the measurement coordinate system (x, y, z) by coordinate transformation, which involves three Euler angles ?_E, θ_E, and ψ_E (Fig. 1). After coordinate transformation, the expression of dielectric tensor of quartz crystal in the measurement coordinate system can be obtained. Then, the Berreman 4×4 matrix theory is used to establish the correlation between quartz crystal parameters and Mueller matrix. The Mueller matrix measurement value of the sample is obtained by the Mueller matrix ellipsometer, and then the Mueller matrix model is used for iterative fitting. The Levenberg-Marquardt algorithm is used for fitting, and the evaluation function is defined as root mean square error (RMSE). The fitting parameters are adjusted by nonlinear iterative regression to minimize the evaluation function, that is, when the evaluation function converges to the global minimum, the actual parameters of the sample are obtained. Finally, the thickness of the crystal, Euler angles and the phase retardation can be obtained by fitting calculation.

In order to fully demonstrate the effect of the fitting model, we measured two samples with different thicknesses. Both samples were placed in different directions in turn, and different incident angles were selected for measurement at each placement azimuth (Fig. 2). Firstly, the built-in model of ellipsometer is used to fit the phase retardation. In the ultraviolet band, the fitting results of the phase retardation at different azimuth angles show significant non-zero values, and the maximum value is close to 4° (Fig. 3). Obviously, the built-in model has defects in fitting the phase retardation of quartz crystal. The Mueller matrix model described in this paper is then used for experiments. Taking the thick sample as an example, the actual measured Mueller matrix dispersion curve [Fig. 4(a)], the Mueller matrix dispersion curve fitted by the Mueller matrix model [Fig. 4(b)], and the difference between the two curves [Fig. 4(c)] can be obtained. The average value of the difference is -0.0007, and the maximum value is 0.231. At different sample azimuth angles and incident angles, the maximum and average RMSE of thick samples are 4.182 and 4.127, respectively, and the maximum and average RMSE of thin samples are 3.906 and 3.770, respectively (Fig. 5). The fitting thicknesses of thick and thin samples are 0.832 mm and 0.691 mm, respectively. In comparison, the measured thicknesses using micrometers are 0.834 mm and 0.695 mm, respectively. The relative errors are 0.24% and 0.57%, respectively (Table 1). The Euler angles θ_E of thick and thin samples are 1.902' and 1.932', respectively.

This paper proposes a Mueller matrix model for accurately measuring quartz crystal parameters. Based on the coordinate transformation and Berreman 4×4 matrix theory, the correlation model between the measured values of Mueller matrix and crystal thickness, Euler angle and dielectric tensor is established, and the fitting effect of the model is evaluated by using the RMSE as the evaluation function. The experimental results show that the fitted Mueller matrix dispersion curves are highly consistent with the measured dispersion curves. The RMSE of the model can be stabilized in a small range (<5) under different sample azimuth angles and incident angles. The thickness obtained by fitting is similar to that measured by micrometer (relative error <1%), and the fitted Euler angle is consistent with the measurement results. The experimental results fully show the accuracy of the fitting results of the established Mueller matrix model, and the thickness, Euler angle and phase retardation of the quartz crystal are successfully obtained. Using this model combined with the dual-rotating compensator Mueller matrix ellipsometer, rich information of the sample can be accurately obtained through simple measurement steps and model fitting, which provides an important reference for accurately measuring the parameters of anisotropic materials.