Monolithic nonplanar ring oscillators (NPROs) are widely used in high-precision laser interferometry owing to their compact structure, low susceptibility to external perturbations, and flexible frequency-tuning capabilities. Such optical resonators are typically fabricated from single crystals. To ensure the lasing performance while maintaining a cost-effective production, the geometric tolerance of the resonator must be determined prior to its optical processing. Central to this task is identifying a self-producing eigen-optical path inside the resonator based on predefined positions and orientations of the multiple reflection planes forming the resonator.

To analyze the effects of the geometric variations of a monolithic NPRO on the intracavity eigen-optical path, we developed a technique to solve the eigen-optical-path problem. Beginning from the basic law of light reflection by flat mirrors, we showed the conditions that must be satisfied such that the path traversed by an optical beam reflecting from multiple planes is closed and self-reproducing. These conditions allow the construction of a set of matrix equations with which each reflection point on the optical path is located, thus yielding an eigen-optical path for a specific resonator geometry. Subsequently, the general procedure for analyzing the geometric tolerances of a resonator was summarized by converting the various limiting factors into constraints on the resonator geometry. These factors include but are not limited to the intracavity optical path, diffraction loss, and coupling with external optics. Following this procedure, two NPROs with different geometries (Fig. 6 and Table 1) were analyzed via numerical simulations to obtain the tolerances (Fig. 8) of the four reflecting surfaces of each resonant in terms of a set of predetermined constraints (Table 2). To demonstrate the feasibility of the proposed method, we conducted a case study in which a large translational deviation of one surface of an NPRO (Fig. 10) was observed after an initial round of optical processing, which complicated the coupling of the output laser to the external optics. The deviations and possible corrections were analyzed using the proposed method, and the effectiveness of the correction was examined.

Changes in the location of the on-plane laser spot, round-trip diffraction loss, and output-axis offset with translations of the four NPRO surfaces (h₀?h₃) along their normal directions were first calculated (Fig. 7) in the numerical simulation. This information, along with the corresponding constraints (Table 2), provides itemized tolerances for the four surfaces of the NPRO. Figure 8 shows a comparison of the tolerances required for two NPROs with the same geometric configuration but different sizes. For the larger resonators, the assembly restrictions impose stricter tolerances on the deviations of all optical surfaces, thus indicating that fabrication errors have a high probability of inducing alignment issues in the assembly stage. However, for the small resonators, the diffraction loss can potentially increase, which may impose a stricter tolerance. The constraints originating from the optical path and the diffraction loss result in an almost identical tolerance. This similarity is primarily because the beam spot size is smaller than the dimensions of the reflecting surface, as well as because the beam can be approximated as a geometric ray. However, in extreme cases where the dimensions of the optical surface are comparable to or smaller than the beam spot size, the diffraction loss becomes pronounced and hence must be included in the analysis of geometric tolerance. For an NPRO with a fabrication error, as shown in Fig. 10, the resultant intracavity eigen-optical path was analyzed. This analysis shows that a viable solution is to translate both the h₁ and h₃ surfaces inward along their normal directions by 0.23 mm, thereby restoring the original intracavity optical path and the positions of the pump and output laser beams. After the correction is implemented, the pump and output beams are within the adjustment constraints when the laser is assembled, whereas the lasing of the NPRO is unaffected.

Geometric tolerances of monolithic NPROs were analyzed by solving the eigen-optical-path problem in systems with multiple reflecting planes. By analyzing the effect of changes in the resonator geometry on the eigen-optical path, combined with constraints on the intracavity optical path, the diffraction loss, optical alignment, and geometric tolerance of a monolithic NPRO can be analyzed and discussed. If a large geometrical deviation occurs during optical fabrication, then its effect on the optical properties of the resonator can be evaluated based on changes in the eigen-optical path, thereby providing guidance for implementing a cost-effective correction. The method introduced herein is applicable to the design and optical fabrication of laser gyroscopes and optical resonators with complex geometries.