# Prism-Based Spatial Heterodyne Spectrometer with a Fixed Fringe Localization Plane
## 3 Simulation Experiments and Results  
The system was simulated using ZEMAX. We used a structure that has a simplified motion control system, as shown in Figure 5. The parameters of some components are given in Section 2.3 and the dimensions of the optics of the interferometer are shown in Table 1. A 40 mm \(\times\) 10 mm rectangular light source with collimation characteristics is selected, eliminating the setting of entrance optics. The imaging lenses are designed as object-space telecentric lenses. The aperture stop is set at the focal plane of the imaging system to act as a band-pass filter. First, the wavelength of the incident light is set to 390 nm and the Littrow wavelength is set to 398 nm (the angle of mirror 2 tilt is set to 20.99\({}^{\circ}\) according to Section 2.4 and the bimirror assembly is tilted at twice the angle of mirror 2). The interferogram received by the detector, the 2D power spectrum obtained by 2D Fourier transform of the 2D interferogram, and the 1D spectrum obtained by 1D Fourier transform of the 1D interferogram are shown in Figure 6; the upper half of the interferogram is the 2D interferogram and the lower half is the 1D interferogram. Since the impact spectrum of the 2D power spectrum is located in the second and fourth quadrants, only the portion of the 1D spectrum that is smaller than the Littrow wavelength is intercepted. From the 1D spectrum, the full width at half maximum (FWHM) is about 0.019 nm, which is close to the theoretical maximum spectral resolution of 0.014 nm calculated from Equation (12). Similarly, set the incident light wavelengths to 530 nm and 997 nm and repeat the work above. The FWHM values are about 0.071 nm and 1.07 \begin{table}
\begin{tabular}{c c} \hline
**Components** & **Dimensions** \\ \hline Cube beam splitter (50:50) & 60 mm \(\times\) 60 mm \(\times\) 60 mm \\ Mirror 1 & 50 mm \(\times\) 60 mm \\ Bimirror * & 35 mm \(\times\) 60 mm \\ Right-angle prism * & 70 mm \(\times\) 70 mm \\ Prisms 1 and 2 & 50 mm \(\times\) 25 mm \(\times\) 28.87 mm \\ Bimirror assembly * & 40 mm \(\times\) 40 mm \\ Mirror 2 & 44 mm \(\times\) 74 mm \\ \hline \end{tabular}
\end{table} 
Table 1: The geometrical dimensions of the optics of the interferometer in the simulation. Figure 6: Results of simulation at different wavelengths. the corresponding theoretical maximum spectral resolutions are 0.061 nm and 0.72 nm, respectively. The spectral resolving power obtained from the simulation is smaller than the theoretical spectral resolving power, which is a normal phenomenon. This discrepancy is mainly due to asynchronous sampling. To verify the constancy of the FLP location, we can observe the patterns on the detector. If the FLP is not optically conjugated to the detector plane, the beams from the two arms do not completely overlap at the detector plane and there should be regions around the interferogram where the irradiance is not zero that have no fringes. Therefore, the interferograms in Figure 6 verify that the FLPs of different wavelengths are all precisely located at the optical conjugate plane of the detector plane. To verify the FOV, a 1D interferogram of the limiting off-axis rays is traced in ZEMAX. The wavelength is set to 397 nm because the usable FOV at this wavelength is close to that of the instrument and the interference curve is easy to observe. The angles of the off-axis rays are calculated according to Equation (14); for example, \(\phi\) = 0.052\({}^{\circ}\) and \(\varphi\) = 0.1\({}^{\circ}\). Figure 7a shows the interference curves for the on-axis rays and the limiting off-axis rays. The 0 th pixel is the center of the detector. It shows that the OPD changes by about 1/2 of the wavelength at the edge of the fringe pattern, which is consistent with the theoretical result. To verify that the usable FOV of the instrument is applicable across the spectral range, another wavelength of 825 nm is set, as shown in Figure 7b. The OPD changes less than 1/2 of the wavelength.