**1. Introduction**

The rotating element ellipsometer, after continuous development in many different configurations and applications, is widely employed as a primary technique in scientific research and industry [1–9]. The typical rotating element ellipsometer system comprises various types: the rotating polarizer ellipsometer (RPE) [10,11], the rotating analyzer ellipsometer (RAE) [12–14], and the rotating polarizer analyzer ellipsometer (RPAE).

The RPAE, which allows the polarizer and the analyzer to rotate in different angular velocities simultaneously, was firstly proposed by Azzam [15]. Intensive efforts have been devoted to RPAE from different aspects in past decades [16–20]. A self-established RPAE was constructed and presented in 1987 [16], with the polarizer and the analyzer rotating synchronously at an angular velocity ratio of 1:2. The system is superior in the elimination of DC signal error and phase-shift correction. Additionally, it provided two methods for the determination of ellipsometric parameters to realize the self-consistency of the data. Subsequently, the instrument was improved in 1994 [18], which enabled a fully variable incident angle by micro-stepping techniques, and employed a fixed polarizer to eliminate the effect of residual polarization from a light source.

The system and random errors have been extensively studied as an important topic in the development of ellipsometry. The accuracy and precision of ellipsometry can be

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effectively improved by performing error analysis and reduction. The analyses and corrections of errors, caused by the imperfect compensator and birefringence in window, were performed by McCrackin in 1970 [21]. Aspnes systematically presented the measurement and the correction of the first-order errors [22] and the uncertainties of ellipsometric parameters [23,24]. Azzam and Bashara investigated the errors from imperfect components, cell-window birefringence, and incorrect azimuth angles [25], and performed systematic error analysis on the RAE [26]. The errors in ellipsometry have been extensively analyzed in various aspects, such as the beam deviation [27], birefringence of window [28,29], incident angle [30], azimuthal errors, and residual ellipticity [31–33]. Moreover, the systematic error analyses on different configurations have been reported, including the RAE [34], PRPSE [35], multichannel ellipsometer [36], and the Mueller matrix ellipsometer [37,38].

Although the aforementioned error analyses are conditionally applicable to the RPAE, the error investigations specifically for this type of ellipsometer are still limited. The noise effect of Fourier coefficients on the RPAE with the same configuration as in [18] was analyzed by simulation [39]. In our previous work, the systematic error reduction, induced by the analytical discrete Fourier transform, was proposed theoretically and tested experimentally [40]. Apart from the effect from the Fourier transformation, the experimental performance affected by the systematic error is worth studying further.

In this work, a method to study the error analysis on the incident angle and azimuth was presented experimentally for the self-established RPAE with bulk Au. Both spectroscopic and monochromatic repeated measurements were carried out at various incident angles. The dielectric constants were determined from the measured ellipsometric parameters to study the accuracy and stability, which were evaluated according to the differences and dispersion degrees of experimental data compared with the reference values, respectively. The initial azimuths of polarizing elements were adjusted rotationally by groups of certain angles to study the effect of azimuthal errors in three modes. The performance of RPAE was evaluated quantitatively with the mean absolute relative error (MARE). The error analysis method proposed in this work is also useful for spectroscopic ellipsometry, including temperature-dependent properties of thin polymer films and metal nanoparticles [41–44].
