# Prism-Based Spatial Heterodyne Spectrometer with a Fixed Fringe Localization Plane
## 1 Introduction  
In 1971, spatial heterodyne spectroscopy (SHS) was pioneered at Osaka University in Japan [1]. SHS offers superior robustness and higher spectral resolving power compared to traditional Fourier-transform spectroscopy, making it a focal point in spectroscopy research [2]. Typically, SHS employs gratings as dispersion elements [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], but these gratings can introduce several challenges. Firstly, multiple diffraction orders disperse energy across different orders. To prevent inter-order disturbances, a baffling system needs to be set and in some cases this baffling system can be only partially effective [16]. Secondly, [PERSON] et al. point out that an interferogram is formed by wave packets with a titled energy front when diffracted from gratings, and this fact limits spectral resolving power [17]. Thirdly, in tunable spatial heterodyne spectrometers (TSHSs) using blazed gratings, adjusting the Littrow angle by tilting the gratings widens the spectral range but reduces energy in interfering beams by deviating from the gratings' blaze angle. Fourthly, compared to prisms, gratings generally generate more diffuse light [18].  
Due to the above limitations of grating-based SHSs, prism-based SHSs have increasingly captured researchers' interest in recent years. In 2019, [PERSON] et al. proposed a prism-based SHS, replacing the grating used in traditional SHSs with a prism-mirror structure [19]. In 2021, [PERSON] et al. experimentally demonstrated the beneficial effects on the signal-to-noise ratio that the substitution of gratings with prisms produces in a SHS [16]. In 2022, [PERSON] et al. proposed an orthogonal TSHS, utilizing a Prism-Mirror instead of a grating and designing an orthogonal configuration to simplify the motion control system [20]. In 2024, [PERSON] et al. designed a prism-based monolithicSHS by coating prisms with mirrors [18]. These prism-based SHS designs all leverage prism-mirror structures.  
Although prism-based SHS has a higher signal-to-noise ratio under certain conditions, this technology is currently immature. The fact that prism dispersion is based on Snell's law and the nonlinear relationship between wavelength and refractive index limits the instrument's usable Field-of-View (FOV), especially in the direction of dispersion, and spectral resolving power, especially at long wavelengths. There is still a lack of research on the FOV and methods to improve the spectral resolving power of prism-based SHS. In addition, the location of the fringe localization plane (FLP) varies with incident light wavelength. The FLP is where the interference fringes can be viewed with maximum clarity, so the fringes can be obtained at maximum contrast at this plane [5]. To observe fringes, the FLP must be optically conjugated to the detector plane using an imaging system. Therefore, if the incident light lacks spatially full coherence, the fringe contrast on the detector degrades when the FLP is not precisely conjugated with the detector plane. [PERSON] et al. [18] also noted that the instrumental response changes as the FLP location varies with wavelength. This paper focuses on the problem of the movement of the FLP and proposes a design to address it.