# Prism-Based Spatial Heterodyne Spectrometer with a Fixed Fringe Localization Plane
## 2 Principles and Designs  
### Movement of the FLP in Prism-Based SHSs  
The principle of prism-based SHSs and the phenomenon of the movement of the FLP are briefly described using a typical prism-based SHS [19]. As shown in Figure 1, it is based on a Michelson interferometer modified by replacing the mirror in each arm with a prism-mirror assembly. Both mirrors are tilted at the same angle with respect to the prisms in their respective arms. The collimated light enters the beam splitter and is divided into two beams. The beam in each arm passes sequentially through the prism, mirror, prism again, and then returns to the beam splitter. The beam with a certain wavelength, called the Littrow wavelength, will incident perpendicular to the mirror and return parallel to the optical axis. The wave fronts with the Littrow wavelength exiting from the interferometer are perpendicular to the optical axis and produce an interferogram with zero frequency. For a wavelength different from the Littrow wavelength, since the prism has different refractive indices at different wavelengths, the wave fronts of the two outgoing beams are no longer parallel, but tilt at the same angle relative to the optical axis in opposite directions, equivalently producing fringes on the FLP. The FLP can be visualized as a plane of recombination of the two backpropagating beams exiting from the interferometer and should be imaged by the imaging system onto the detector plane.  
For traditional grating-based SHSs, the location of the FLP is very intuitive as the beam in each arm is diffracted by the grating only once, so the FLP intersects the center of each grating. However, for the SHSs with a prism-mirror structure, the beam in each arm is refracted more than once by the prism, so there is uncertainty in the location of the recombination plane obtained by extending the outgoing rays from the two arms in the reverse direction. For example, if the apex angle of the prism is set to 30\({}^{\circ}\), the height of the prism is set to 25 mm, the material is set to N-LASF9, the length of the optical axis between the prism and the mirror is set to 16.2 mm, and the Littrow wavelength is 650 nm, the distances from the FLP to the entering surface of the prism if the wavelengths of the incident ray are 400 nm, 600 nm, and 1000 nm are, respectively, about 73.6 mm, 83.5 mm, and 88.6 mm. It can be seen that the location of the FLP is different for different wavelengths of the incident ray, which is consistent with the phenomenon in the experiments done by [PERSON] et al. [18]: when focusing the imaging optics on a particular FLP distance, a spectral response peaking was obtained at that corresponding wavelength. In fact, the FLP of a point source (ideally) is unlocalized, except that the size of the overlap region of the beams from the two arms varies at different locations. However, even a laser is not an ideal point source and the FLP is localized [5]. Maintaining a fixed FLP can eliminate the need to calculate the FLP's location and move the imaging device based on the predicted wavelength of the incident light during use.