Based on the theory of the Abeles matrix, the reflections from the multilayer dielectric mirrors in a non-planar ring resonator are discussed in detail considering the imperfect layer thickness and the incident angle errors. The polarization of Gaussian beams in nonplanar ring resonators are analyzed by using a statistical method and the way to analyze the stability of a nonplanar ring resonator in the perspective of polarization is also presented. The results show that a nonlanar ring resonator with a finite mode size may not have the reasonable polarization performances sometimes; the eigenmodes might be right- and left-handed elliptically polarized instead of circularly polarized, and they also have different ellipticity and round trip losses, which are critical for a nonplanar ring resonator. These interesting findings are important to the cavity designs of ring resonators with a nonplanar structure.
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