# Prism-Based Spatial Heterodyne Spectrometer with a Fixed Fringe Localization Plane
## 2 Principles and Designs
### Preliminary Optical Design  
To keep the location of the FLP constant, we can make the optical axes of the two arms be turned in opposite directions with respect to each other only once due to refraction. This requires the beam in each arm to pass through the prism only once. After the beam passes through the prism, it needs to return to the beam splitter without passing through the prism again. Therefore, some form of optics is needed to reflect the beam and provide a shift. To realize this idea, we propose to use a prism-bimirror-mirror assembly instead of the prism-mirror structure in each arm, as shown in Figure 2a, with the angle made by the two planes of the bimirror being a right angle. The collimated light is split by a beam splitter and is divided into two paths. Now we will describe only one of the paths. The beam is incident vertically into a prism and refracted at its exiting surface. Then, the beam is incident to one of the planes of the bimirror. The intersection line of the two planes of the bimirror is perpendicular to the incoming axis of the bimirror and parallel to the side surfaces of the prism. In terms of the direction of the beam, a right-angle bimirror is equivalent to a mirror, except that the beam is shifted by an extra segment of the optical axis between the incoming and outgoing axes, which staggers the incoming and outgoing beams. To allow the beam to return to the beam splitter for recombination with another one, a mirror is added behind the bimirror to change the direction of the optical axis. This mirror is tilted at a proper angle so that its outgoing optical axis is perpendicular to the surface of the cube beam splitter. The system is a TSHS in which the Littrow wavelength can be tuned by tilting the bimirrors and the mirrors. The rotating spindle is located at the refracting  
Figure 1: Layout of the interferometer of a typical prism-based spatial heterodyne spectrometer (SHS) [19]. A point source containing three wavelengths is set. Arrows of different colors represent rays with different wavelengths. surface of the prism, intersecting the center of the refracting surface, as well as at the mirror, intersecting the center of the mirror. The bimirror and mirror share this spindle, where the bimirror needs to be tilted twice as much as the mirror during the tuning process. The prism-bimirror-mirror structures of the two arms are upside down with respect to each other in order to ensure that the outgoing beams from the two arms are tilted in opposite directions, as shown in Figure 2b. If the wavelength of the incident light is not equal to the tuned Littrow wavelength, the wave fronts of the beams of the two arms are tilted by the same angle relative to the optical axis in opposite directions, equivalently producing the fringes on the FLP. The center of the FLP overlaps with the center of the refracting surface of each prism. The distance from the FLP to the imaging optics is equal to the length of the optical axis between the refracting surface of each prism and the imaging optics. This length does not change, regardless of the tuned Littrow wavelength and the wavelength of the incident light, solving the problem of the FLP location varying with the wavelength that exists in current prism-based SHS designs. Besides diffused light, stray light can be generated by specular reflection on the surfaces of the cube beam splitter and prisms. Anti-reflection (AR) coatings are needed to improve transmittance. In the arrangement shown in Figure 2, the splitting and recombination of the beams occur at different locations in the beam splitter, preventing the beams from reaching the detector without passing through any arm. In addition, the prisms are single- rather than double-passed, which may result in less chance for the beams to be reflected by the prisms. Stray light analysis requires the consideration of many other factors such as mechanical structure and still requires further research.