# Prism-Based Spatial Heterodyne Spectrometer with a Fixed Fringe Localization Plane
## 2 Principles and Designs
### Spectral Inversion  
The parameters of the key components in the Zemax OpticStudio v.19.4 simulation are provided here first. The glass of each prism used is N-LASF9 (SCHOTT Group Corporation, Mainz, Germany) and the apex angle of each prism is 30\({}^{\circ}\). The half-width of the incident light is 20 mm. The imaging system has a magnification of 0.5. The parameters of the  
Figure 2: **(a)** Schematic diagram of the prism–bimirror–mirror structure. The angle between the two mirrors of the bimirror is 90\({}^{\circ}\). **(b)** Layout of the tunable spatial heterodyne spectrometer (TSHS) using a prism–bimirror–mirror structure.  
detector refer to the SONY IMX 174 (Sony Group Corporation, Tokyo, Japan), which has a pixel pitch of 5.86 \(\upmu\)m and 1936 \(\times\) 1216 pixels.  
Assuming that the wavelength of the incident light is not equal to the tuned Littrow wavelength, the equivalent fully transmissive optical system is shown in Figure 3a and the wave fronts intersecting the centers of the refracting surfaces of the prisms of the two arms are shown in Figure 3b. Assuming points A and B on the two wave fronts are from any point on the light source, the two beams propagate in the directions AC and CB, respectively. They are equivalent to intersecting at point C. Thus, the FLP intersects the center of the refracting surface of each prism and is perpendicular to the outgoing optical axis of each prism. The center of interference is always located at the center of the FLP and the values of the optical path difference (OPD) at both ends of the FLP are the largest.  
If the angle between each wave front and the FLP is \(\beta\), then the \(OPD\) is  
\[OPD=AC+CB=2x\text{sin}\;\beta \tag{1}\]  
Then, the spatial frequency \(f\) of the interferogram is  
\[f=2\sigma\text{sin}\;\beta \tag{2}\]  
where \(\sigma\) is the wavenumber of the incident light, which is the reciprocal of the wavelength \(\lambda\). Setting the Littrow wavelength as \(\lambda_{0}\), the angle \(\beta\) between the wave front and the FLP is  
\[\beta=\text{sin}^{-1}(n(\lambda_{0})\text{sin}(30^{\circ}))-\text{sin}^{-1}( n(\lambda)\text{sin}(30^{\circ})) \tag{3}\]  
Figure 3: **(a)** Schematic diagram of the equivalent fully transmissive model of the two arms of the interferometer using a prism–bimirror–mirror structure. The red arrows and optics represent the ray in one arm and the optics it passes through, while the green arrows and optics represent the ray in the other arm and the optics it passes through. The blue optics represent the optics that the ray in each of the two arms passes through. **(b)** Schematic diagram of the wave fronts intersecting the centers of the refracting surfaces of the prisms of the two arms.  
The refractive index \(n\) versus wavelength \(\lambda\) equation for the N-LASF9 provided in the Glass Catalog of ZEMAX is  
\[n(\lambda)^{2}-1=\frac{K_{1}\lambda^{2}}{\lambda^{2}-L_{1}}+\frac{K_{2}\lambda^{ 2}}{\lambda^{2}-L_{2}}+\frac{K_{3}\lambda^{2}}{\lambda^{2}-L_{3}} \tag{4}\]  
The coefficients of the formula are \(K_{1}\) = 2.00029547, \(L_{1}\) = 0.0121426017 \(\upmu\)m\({}^{2}\), \(K_{2}\) = 0.298926886, \(L_{2}\) = 0.0538736236 \(\upmu\)m\({}^{2}\), \(K_{3}\) = 1.80691843, and \(L_{3}\) = 156.530829 \(\upmu\)m\({}^{2}\). The correspondence between the spatial frequency \(f\) of the interferogram on the FLP and the wavelength \(\lambda\) of the incident light has been established up to this point. Compared to the SHS with a prism-mirror structure, it has a much simpler inversion formula.