## 3 Optical Error Analysis
### Optical Errors of FPI
#### 3.1.3 Non-Flatness Error of FPI  
Surface imperfections inherent in fabricating FPI partially reflective surfaces induce position-dependent OPD across the clear aperture, as indicated by \(\sigma\Delta\) in Figure 4. The normalized distribution of OPD errors resulting from these non-flatness error can be expressed as follows:  
\[p(\sigma\Delta)=\frac{1}{\sqrt{\pi}\sigma\Delta_{D}}e^{-\frac{\sigma\Delta^{2 }}{\sigma\Delta_{D}^{2}}}, \tag{14}\]  
Figure 5: (**a**) The comparison and local magnification of the FPI transmittance spectra under ideal conditions (\(T_{FPI}\)) and with parallelism errors (\(T_{FPI-para}\)); \(T_{FPI}\) and \(T_{FPI-para}\) are decomposed into (**b**) the DC component, (**c**) fundamental frequency component, (**d**) second harmonic component, and (**e**) third harmonic component, with corresponding comparisons and local magnifications. Here, \(z\) is defined as \(2\pi\
u 2\tau_{0}\beta_{FP1}\) in Equation (12), and the curves \(\pm 2a_{m}I_{1}(mz)/(mz)\) can be regarded as the envelopes of the fundamental frequency and higher harmonic components of \(T_{FPI-para}\). where \(\sigma\Delta_{D}\) denotes the surface defect factor. Following Equation (14), the FWHM of \(p(\sigma\Delta)\) is expressed as \(FWHM_{flat}=2\sqrt{ln2}\sigma\Delta_{D}\). The transmission function of the FPI can then be formulated as follows:  
\[ \begin{array}{rl}T_{FPI-flat}(\
u)&=\int_{-\infty}^{\infty}T_{FPI}(\
u, \Delta_{FPI}+\sigma\Delta)p(\sigma\Delta)d(\sigma\Delta)\\ &=a_{0}+\sum\limits_{m=1}^{\infty}2a_{m}\cos(2\pi m\Delta_{FPI})*p(\Delta_{ FPI})\\ &=a_{0}+\sum\limits_{m=1}^{\infty}2a_{m}e^{-(\pi m\sigma\Delta_{D}\
u)}\cos(2 \pi vm\Delta_{FPI})\end{array} . \tag{15}\]  
Consequently, the non-flatness error induces Gaussian smoothing of the transmission spectrum \(T_{FPI}\) (\(\
u\)) through convolution with a Gaussian kernel \(p(\sigma\Delta)\), as shown in Figure 6. When \(T_{FPI-flat}\) is decomposed into its DC component and various harmonic components, its effect is similar to that of non-parallelism error. The interference intensity of each harmonic component decreases, which can be regarded as producing an interference intensity factor of \(\exp[-(m\pi\sigma\Delta_{D}\
u)^{2}]\). Therefore, as the order \(m\) and the wavenumber \(\
u\) increase, the interference intensity decreases more dramatically. Accounting for this imperfection, the joint interferogram can be expressed as follows:  
\[ \begin{array}{rl}I_{2-flat}(\Delta)&=\int_{-\infty}^{\infty}I_{0}(\Delta)*FT [T_{FPI}(\
u,\Delta_{FPI}+\sigma\Delta)]p(\sigma\Delta)d\sigma\Delta\\ &=I_{0}(\Delta)*\left[a_{0}\delta(\Delta)+\sum\limits_{m=1}^{\infty}a_{m}[ \delta(\Delta+m\Delta_{FPI})+\delta(\Delta-m\Delta_{FPI})]*p_{m}(\Delta) \right]\\ &=a_{0}I_{0}(\Delta)+\sum\limits_{m=1}^{\infty}a_{m}[I_{0}(\Delta+m\Delta_{ FPI})+I_{0}(\Delta-m\Delta_{FPI})]*p_{m}(\Delta)\end{array} \, \tag{16}\]  
where \(p_{m}(\Delta)=p\ (\Delta/m)/m\). Consequently, analogous to the effects of non-parallelism, FPI non-flatness error causes the frequency-shifted component response function to convolve  
Figure 6: (**a**) The comparison and local magnification of the FPI transmittance spectra under ideal conditions (\(T_{FPI}\)) and with parallelism errors (\(T_{FPI-flat}\)); \(T_{FPI}\) and \(T_{FPI-flat}\) are decomposed into (**b**) the DC component, (**c**) fundamental frequency component, (**d**) second harmonic component, and (**e**) third harmonic component, with corresponding comparisons and local magnifications. Here, the curves \(\pm 2a_{m}\exp[-(m\pi\sigma\Delta_{D}\
u)^{2}]\) can be regarded as the envelopes of the fundamental frequency and higher harmonic components of \(T_{FPI-flat}\). and broaden, resulting in \(a_{m}\delta(\Delta\pm m\Delta_{FPI})*p_{m}(\Delta)\). The width and height of the broadening are both related to the order \(m\). Based on the above analysis, the effects of FPI non-parallelism and non-flatness on the interferogram can be regarded as two equivalent errors.