The Herriott cell is an optical system composed of two concave mirrors, and features stable light paths and simple structures. However, as industrial demands for design specifications become increasingly stringent, the power efficiency loss in the cell assembly is more pronounced. Currently, there is a lack of a systematic theoretical model as academic support for this problem. Thus, our study is based on the fundamental principles of the Herriott cell and the extended matrix of the ABCD matrix, analyzing the types of tolerances that affect the cell parameters. Mathematical models for the tolerances of two mirror distances, tilt, and eccentricity are built. Finally, a simulation analysis is conducted on the influence of tolerances on the exit light power utilization under different equivalent optical paths. The proposed mathematical models and analysis methods can be applied to any single-ring Herriott cell and provide reasonable allowable variations for two mirror distances, tilt, and eccentricity tolerances in the design of practical cells. Meanwhile, significant theoretical implications are provided for the engineering implementation of cells.

First, the Herriott theory is employed to deduce the sizes of all light spots in the cell, laying the groundwork for subsequent theory and analysis. Second, based on the fundamental principles of the Herriott cell and the extended matrix of the ABCD matrix, we build mathematical models for the most critical two mirror distances, tilt, and eccentricity in the assembly tolerances of the Herriott cell. The calculation formulas for the above assembly tolerances can be adopted to compute the position information of all light spots in the Herriott cell under error occurrence. Third, since the light source in our study is assumed to be uniform, the utilization rate of exit light power is calculated using the ratio of residual light spot area to original light spot area. Fourth, the effects of assembly tolerances on the change in light spot positions in the Herriott cell are analyzed in various scenarios, and formulas for calculating the utilization rate of exit light power for each scenario are created based on geometric relationships. Finally, a set of basic parameters for the Herriott cell are adopted for simulation, with the allowable ranges of the three assembly tolerances analyzed. The downward trend of residual light power is examined. Additionally, by adjusting the mirror spacing, we determine the change in the allowable range of tolerances for different optical path lengths in the cell and summarize the patterns.

We list a total of five different mirror distances to alter the equivalent optical path length in the Herriott cell. Firstly, an analysis of distance tolerance is conducted to provide a relationship curve between distance variation and exit light power utilization (Fig. 10). As the optical path gradually increases, the distance tolerance becomes more lenient. Under 99% power utilization, the allowable tolerance range for a distance of 450 mm is approximately ±1.05 mm, and for a distance of 250 mm, the range is approximately ±0.58 mm. Subsequently, the relationship between relative distance tolerance and power utilization is explored (Fig. 11), showing an identical relationship curve for relative distance tolerance and power utilization under different optical paths. At 95% power utilization, the relative distance tolerance is approximately 1.2%. Next, an analysis of tilt tolerance is conducted. Under a constant optical path, the tilt tolerance range of the mirror around the x-axis in the positive direction is smaller than in the negative direction, and the decrease in power utilization rate is faster in the positive direction (Fig. 12). Additionally, with the increasing optical path, the tilt tolerance range will significantly decrease, and the decreasing trend will gradually slow down. At 95% power utilization, the allowable tilt tolerance range for a distance of 450 mm is approximately -0.97°-0.28°, and for a distance of 250 mm, it is approximately -2.19°-0.52°. The tilt tolerance around the y-axis is essentially symmetrical (Fig. 13), and at 95% power utilization, the allowable tilt tolerance range for a distance of 450 mm is approximately ±0.15°. Meanwhile, for a distance of 250 mm, it is approximately ±0.68°. Finally, for eccentricity tolerance, the curve of the mirror’s eccentricity tolerance around the y-axis is essentially symmetrical (Fig. 14) and gradually decreases with the rising optical path. At 95% power utilization, the allowable eccentricity tolerance range for a distance of 450 mm is approximately ±3.96 mm, and for a distance of 250 mm, it is approximately ±10.06 mm. The positive tolerance range of eccentricity in the x-axis direction is larger than the negative direction, and the tolerance range for a large optical path is slightly smaller than that for a small optical path (Fig. 15). At 95% power utilization, the allowable eccentricity tolerance range for a distance of 450 mm is approximately -7.31-14.63 mm, and for a distance of 250 mm, it is approximately -7.72-16 mm.

We analyze the fundamental changes in three types of tolerances based on the relationship curves between assembly tolerances and exit light power utilization. In different optical paths, the slackness of distance tolerance is precisely opposite to the ranges of tilt and eccentricity tolerances. A comparison between tilt and eccentricity tolerances reveals their similarities in tolerance curve trends. However, tilt tolerance is more sensitive than eccentricity tolerance, and the relationship between relative distance tolerance and power utilization is a constant curve, with the lowest sensitivity among the three tolerances being distance tolerance. Finally, the mathematical models and analysis methods in our study are adopted to conduct a comprehensive selection of various parameters such as total optical path and number of reflections during the design of Herriott cells. Finally, low-sensitivity optimization design is achieved for long optical path cells, with assembly costs compressed during production.