## 3 Optical Error Analysis
### Optical Errors of FPI
#### 3.1.4 Reflectivity Changes in FPI  
In ideal FPI configurations, the reflectivity \(R\) of semi-reflective surfaces is typically treated as a constant value. However, practical implementations exhibit wavelength-dependent reflectivity variations expressed as \(R(\
u)=R_{0}+\delta R(\
u)\), where \(R_{0}\) represents the mean reflectivity, and \(\delta R(\
u)\) denotes the spectral deviation term. This variation primarily affects the harmonic coefficients \(a_{m}\), which can be reformulated through Taylor expansion about \(R_{0}\) as follows:  
\[\left\{ \begin{aligned} & a_{m}(\
u)=a_{m}+a^{\prime}m\delta R(\
u)+o \big{[}\delta R(\
u)^{n}\big{]}\\ &\left\{\begin{array}{l}a_{m}=R_{0}^{m}\frac{1-R_{0}}{1+R_{0}} \\ a^{\prime}m=\frac{R_{0}^{m-1}\left[m\left(1-R_{0}^{2}\right)-2R_{0}\right]}{ \left(1+R_{0}\right)^{2}}\end{array}\right..\end{aligned} \right. \tag{18}\]  
In the equation, the influence of reflectivity variations on the coefficients is denoted by \(a^{\prime}m\delta R(\
u)\). The \(a^{\prime}m\) is the first derivative of \(a_{m}\) with respect to \(R_{0}\), and it quantitatively represents the sensitivity of \(a_{m}\) to changes in reflectivity. From the analytical expression of \(a^{\prime}m\), the functional dependencies of \(a^{\prime}0\), \(a^{\prime}1\), \(a^{\prime}2\), and \(a^{\prime}3\) on \(R_{0}\) can be derived and plotted as characteristic curves, as shown in Figure 7. For the MJI-HI system achieving twofold spectral super-resolution, the dominant contributions originate from the \(m=0\) and \(1\) frequency-shifted components, whereas multifold super-resolution necessitates the inclusion of higher-order terms (\(m\geq 2\)), each successively introducing finer spectral modulation through harmonic interference coupling proportional to its respective sensitivity coefficients \(a^{\prime}m\). For the \(m=0\) order, the sensitivity coefficient is given by \(a^{\prime}0=-2/(1+R_{0})^{2}\), with its value ranging from \(-2\) to \(-0.5\). As \(R_{0}\) increases, the influence of reflectivity variations on \(a_{0}(\
u)\) diminishes. For the \(m=1\) order, \(a^{\prime}1=0\) when \(R_{0}=-1=41.4\%\), rendering the coefficient minimally sensitive to small reflectivity fluctuations. Similarly, for the \(m=2\) order, \(a^{\prime}2=0\) at \(R_{0}=(\sqrt{5}-1)/2=61.8\%\). However, it is fundamentally impossible to simultaneously nullify the sensitivity of all \(m\)-order frequency-shifted components to \(\delta R(\
u)\). For twofold spectral super-resolution, selecting an FPI with \(R_{0}\approx 41.4\%\) effectively reduces the sensitivity of \(a_{1}\). For threefold super-resolution, FPIs with \(R_{0}\in[41.4\%,61.8\%]\) achieve concurrently low sensitivity for both \(a_{1}\) and \(a_{2}\), thereby optimizing performance across multiple harmonic orders while maintaining robustness against reflectivity variations. Taking \(R_{0}=40\%\) as an example, a randomly generated reflectance variation curve \(\delta R(\
u)\) is introduced. Under this error condition, the variations in the FPI transmission spectrum \(T_{FPI-R}\) and its constituent components are illustrated in Figure 8. Overall, the reflectance variation induces changes in the finesse across different wavelengths. By decomposing the spectrum into the DC component and harmonic components, it is observed that \(a_{0}\), which is also determined by the reflectance, varies with wavenumber. Under this error, the interference intensities of other harmonic components shift at different wavenumber positions. Notably, since the selected \(R_{0}\) is close to the zero point of \(a^{\prime}_{1}\) (\(R_{0}=41.4\%\)), the fundamental harmonic component exhibits minimal deviation under this error in Figure 8c, remaining largely consistent with the ideal case.