For gradient refractive index media with large refractive index gradients, traditional ray tracing methods based on refined elements or spatial geometric steps have problems such as low tracing accuracy and efficiency. The ray tracing method based on refractive index steps proposed in this paper can effectively solve this problem. This method uses the refractive index step to replace the spatial geometric step. The starting point and the end point of each ray tracing step are on the constant refractive-index surfaces. It avoids the problem that the traditional tracing method cannot adapt to the area of sudden change in the refractive index and the area where the refractive index changes sharply. Therefore, a suitable distance can be performed in the iterative process. It can achieve high-efficiency and precise ray tracing in areas whether the refractive index changes slowly or sharply. According to the comparison of calculation examples, this method can achieve a tracing accuracy of ${10}^{-5}$ mm. The speed and precision of ray tracing are better than traditional methods.

The main disadvantage of windowless beam expansion systems is that they cannot achieve a good sealing effect. Turbulence and impurities in the environment can easily affect the imaging and primary mirror. Thus, in this study, a matrix of small holes was introduced for inflation to form a stable and smooth flow inside the system to avoid these disadvantages. In order to study the layout of the matrix, the flow state of the model was analysed, and the Lorentz–Lorenz formula and Barron gradient operator were used for ray tracing. Simulation results show that when the matrix of small holes is arranged in 16 rows with 360 holes in each row, inflation has a lesser effect on the wavefront aberration of the system. Moreover, the root mean square (RMS) of wavefront aberration was only 0.077 μm, which was superior to the other layouts considered. Experimental results show that the RMS was 0.08 μm in this state, which is consistent with the analysis. This indicates that this analysis method can meet actual work needs. The calculation methods and calculation results have high reliability and, thus, can be also used in similar situations.