## 3 Optical Error Analysis
### Optical Errors of FPI
#### 3.1.2 Non-Parallelism Error of FPI  
During the manufacturing and alignment processes of the FPI, a small angular deviation \(\beta_{FPI}\) between the two partially reflective surfaces introduces position-dependent OPD variations across the aperture. As shown in Figure 4, it should be assumed that one of the semi-reflective surfaces of the FPI tilts along the Y-axis direction. The aperture is circular with a radius of \(r_{0}\), and the cavity length error introduced by the non-parallelism is \(\sigma\Delta/2\), resulting in an optical path error of \(\sigma\Delta\). The normalized distribution function of the OPD error is as follows:  
\[g_{FPI}(\sigma\Delta)=\frac{1}{\pi\beta_{FPI}r_{0}}\sqrt{1-(\sigma\Delta/2 \beta_{FPI}r_{0})^{2}}. \tag{11}\]  
In the equation, \(r\) denotes the radius of the clear aperture. From Equation (11), we further derive the full width at the half maximum of g\({}_{FPI}(\sigma\Delta)\) as \(FWHM_{para}=2\sqrt{3}\beta_{FPI}r_{0}\). A point on the aperture at a height along the Y-axis can be expressed as \(h=\sigma\Delta/2\beta_{FPI}\), where \(\beta_{\rm x}\) is the tilt angle. Let \(\sigma\Delta=2\beta_{FPI}r_{0}\cos\varphi\), as shown in Figure 4, where \(\varphi\) is the azimuthal angle. Integrating over the entire aperture, the FPI's transmission spectrum is as follows:  
\[ \begin{array}{rl}T_{FPI-para}(\
u)&=\int_{-2\beta_{FPI}r_{0}}^{2\beta_{FPI}r _{0}}T_{FPI}(\
u,\Delta_{FPI}+\sigma\Delta)g(\sigma\Delta)d(\sigma\Delta)\\ &=a_{0}+\int_{0}^{\pi}\sum\limits_{m=1}^{\infty}\frac{2a_{m}}{\pi}\cos[2\pi \
u m(\Delta_{FPI}+2\beta_{FPI}r_{0}\cos\varphi)]\sin^{2}\varphi d\varphi\\ &=a_{0}+\sum\limits_{m=1}^{\infty}a_{m}\frac{2J_{1}(2\pi\
u m2r_{0}\beta_{FPI })}{2\pi\
u m2r_{0}\beta_{FPI}}\cos(2\pi\
u m\Delta_{FPI})\end{array} , \tag{12}\]  
where \(J_{1}(\rm z)\) denotes the first-order Bessel function of the first kind. The impact of non-parallelism errors on the transmission spectrum of the Fabry-Perot interferometer (FPI) is illustrated in Figure 5. Overall, such errors cause a certain degree of reduction in the interference intensity of \(T_{FPI-para}\). By decomposing the transmission spectrum into DC components, fundamental frequency components, and higher-order harmonic components, as shown in Figure 5b-e, it is observed that under ideal conditions, the fundamental and higher-order harmonic components exhibit standard cosine functions. However, non-parallelism errors lead to a decrease in the interference intensity of each component. Moreover, as the order \(m\) and wavenumber \(\
u\) increase, the attenuation of interference intensity becomes more pronounced. Additionally, non-parallelism errors do not cause changes in the periodic frequency of the harmonic components. Figure 4: (**a**) The schematic diagram of the FPI non-parallelism error in the direction perpendicular to the optical axis and a locally magnified schematic diagram of the flatness error. (**b**) The schematic diagram of the distribution of the non-parallelism error along the optical axis. According to Equation (6), the influence on the interferogram is as follows:  
\[ \begin{array}{ll}I_{2-para}(\Delta)&=\int_{-2\beta_{FP1}\tau_{0}}^{2\beta_{FP1} \tau_{0}}I_{0}(\Delta)*FT[T_{FPI}(\
u,\Delta_{FPI}+\sigma\Delta)]g(\sigma\Delta) d\sigma\Delta\\ &=I_{0}(\Delta)*\left\{a_{0}\delta(\Delta)+\sum\limits_{m=1}^{\infty}a_{m}[ \delta(\Delta+m\Delta_{FPI})+\delta(\Delta-m\Delta_{FPI})]*g_{m}(\Delta) \right\}\\ &=a_{0}I_{0}(\Delta)+\sum\limits_{m=1}^{\infty}a_{m}[I_{0}(\Delta+m\Delta_{ FPI})+I_{0}(\Delta-m\Delta_{FPI})]*g_{m}(\Delta)\end{array} . \tag{13}\]  
In the equation, \(g_{m}(\Delta)=g(\Delta/m)/m\). Consequently, compared to Equation (6), FPI non-parallelism introduces a convolution of frequency-shifted components response function \(a_{m}\delta(\Delta\pm m\Delta_{FPI})\) with \(g_{m}(\Delta)\), resulting in response function broadening where the broadening width increases by a factor of \(m\) and the amplitude reduces to \(1/m\) as the harmonic order \(m\) increases.