## 1 Introduction  
Thin films play an essential role in manufacturing semiconductors and display electronics. The thicknesses and refractive indices of thin films should be well controlled and measured for production efficiency improvement and yield control [1; 2]. Among the various thin-film sensing technologies [3; 4; 5; 6], ellipsometry is widely used due to its ability to carry out rapid, nondestructive, and high-precision measurement [7; 8; 9; 10; 11]. In recent times, ellipsometry has also been widely used in surface and interface measurement and characterization of nanostructures and optoelectronic materials [12; 13; 14]. Ellipsometric measurements depend on a mechanical system called a goniometer. Any vibration results in signal distortion problems in the ellipsometric measurements and consequent variation in the optical parameters of the investigated film [15; 16; 17]. The signal distortion problems resulting from the vibration are especially significant in on-line inspection cases using rotating compensator ellipsometry. Specifically, the signals of an ellipsometer captured at different times may have different errors or distortions due to the varying defocus distances caused by vibrations, making it difficult to characterize errors of subsequent thin-film parameter calculation. Therefore, a method to overcome vibration-induced measurement distortion in on-line inspection of thin films is urgently needed.  
Back focal plane ellipsometry (also known as angle-resolved ellipsometry or microellipometry) [18; 19; 20; 21] has advantages in overcoming the challenges of conventional ellipsometry in defocused measurements under vertical vibrations (vertical vibration defocus is simply referred to as defocus from hereon) due to its fast measurement characteristics based on a single-frame BFP imaging and equivalent sensitivity to conventional ellipsometry. This method can obtain single-frame BFP reflectance and ellipsometric information of different incident and azimuth angles through a high-_NA_ (numerical aperture) objective lens without using mechanical moving parts to realize angular or spectral scanning. Moreover, single-frame BFP images can also easily process defocus effects in real time. Compared to the inclined structure of conventional ellipsometry, the coaxial structure of BFP ellipsometry is spatially compact for on-line measurements and suffers from a smaller influence of defocused structures [22]. Different methods of improvement have been developed for single-frame BFP ellipsometry, e.g., modeling using the Mueller matrix [23], introducing compensator modulations [24; 25], capturing through a color camera [20], and applying on-line measurement of oil film gaps [26; 27]. However, in BFP ellipsometric measurement, when vibrational defocus occurs, the captured BFP images may suffer from blur and distortion, resulting in the distortion of incident angles and incorrect ellipsometric responses. This issue has not been analyzed and solved. Suppose an incident angle distortion and the corresponding ellipsometric error are calibrated at each defocus position. In that case, the BFP accuracy could be improved by compensating the errors related to the defocus effect in the single-frame measurement model [28; 29; 30].  
This work proposes an effective calibration and compensation method for sample vibration-induced defocus effect in BFP ellipsometry. The method begins by calibrating defocus-induced incident angle distortion using the Brewster angle of a standard material. Then, the ellipsometric errors are corrected by inserting a calibrated compensation matrix into a system model using a Jones matrix. During actual measurement, the defocus position is determined by analyzing the frequency domain and energy characteristics of a BFP image. Then, the previous calibration results compensate for incident angle distortion and ellipsometric errors. Several thin-film validation experiments were carried out, which verified that the method significantly improved the film thickness measurement accuracy of BFP ellipsometry.