## 3 Optical Error Analysis
### Optical Errors of FTIS
#### 3.2.4 The Combined Error of FPI Non-Parallelism and FTIS Mirror Tilt Error  
Non-parallelism errors in the FPI and tilt errors in the FTIS both induce wavefront angles between the coherent beams, where tilt errors in the FTIS manifest as a convolution of \(I_{0}(\Delta)\) with \(g_{FTIR}(\sigma\Delta)\), while non-parallelism errors in the FPI result in the convolution of each frequency-shifted component \(a_{m}I_{0}(\Delta\pm\Delta_{FPI})\) with \(g_{m}(\sigma\Delta)\). Since both the OPD errors in the FTIS and cavity length errors in the FPI are spatially dependent on the beam's position within the aperture, the combined effects of these errors cannot be linearly superimposed when analyzing the system. Assuming a circular clear aperture, a polar coordinate system \((r,\varphi)\) can be established to represent points within the aperture. As shown in Figure 11, let the FTIS be tilted along the \(\varphi_{FTIR}\) direction with a wavefront tilt angle \(2\beta_{FTIR}\) and the FPI be tilted along the \(\varphi_{\text{FPI}}\) direction with a wavefront tilt angle \(2\beta_{FPI}\). For a point \((r,\varphi)\), the OPD error introduced by the FTIS is \(\sigma\Delta=2r\beta_{FTIR}\text{cos}(\varphi-\varphi_{FTIR})\), while the cavity length error from the FPI is \(\sigma\Delta_{FPI}=2r\beta_{FPI}\text{cos}(\varphi-\varphi_{FPI})\). Consequently, the monochromatic interference intensity of the interferogram \(I_{2}\) can be expressed as follows:  
\[\left\{ \begin{array}{ll}I_{2-comb}(v,\Delta)&=\int_{0}^{r_{0}}\int_{0}^{2\pi }rB(v)\cos\delta_{FTIS}\Big{\{}a_{0}+\sum\limits_{m=1}^{\infty}2a_{m}\cos m \delta_{FPI}\Big{\}}d\varphi dr/\left(\pi r_{0}^{2}\right)\\ &=\int_{0}^{r_{0}}\int_{0}^{2\pi}a_{0}rB(v)\cos\delta_{FTIS}+rB(v)\sum\limits_{ m=1}^{\infty}a_{m}[\cos(\delta_{FTIS}+m\delta_{FPI})+\cos(\delta_{FTIS}-m \delta_{FPI})]d\varphi dr/\left(\pi r_{0}^{2}\right)\\ \delta_{FTIS}&=2\pi v[\Delta+2r\beta_{FTIR}\cos(\varphi-\varphi_{FTIS})\\ \delta_{FPI}&=2\pi v[\Delta_{FPI}+2r\beta_{FPI}\cos(\varphi-\varphi_{FPI})] \end{array} \right.. \tag{31}\]  
In the equation, \(\delta_{FTIS}\) represents the interference phase of the FTIS, and \(\delta_{FPI}\) denotes the interference phase of the FPI. This expression comprises three components: the DC component of \(I_{2}\) is given by \(\delta_{0}rB(v)\text{cos}\delta_{FTIS}\), while the \(m\)-th and \(-m\)-th order frequency-shifted components correspond to \(\beta_{m}rB(v)\text{cos}(\delta_{FTIS}--m\delta_{FPI})\) and \(\beta_{m}rB(v)\text{cos}(\delta_{FTIS}+m\delta_{FPI})\), respectively. Substituting the phase terms and simplifying yields the final form of the interferogram:  
\[\left\{ \begin{array}{ll}\delta_{FTIS}-m\delta_{FPI}=2\pi v[(\Delta-m \Delta_{FPI})+2r\beta_{m}\cos(\varphi-\varphi_{m})]\\ \delta_{FTIS}+m\delta_{FPI}=2\pi v[(\Delta+m\Delta_{FPI})+2r\beta_{-m}\cos( \varphi-\varphi_{-m})]\\ \beta_{m}=\sqrt{\beta_{FTIS}^{2}+m^{2}\beta_{FPI}^{2}-2m\beta_{FTIS}\beta_{ FPI}\cos(\varphi_{FTIS}-\varphi_{FPI})}\\ \beta_{-m}=\sqrt{\beta_{FTIS}^{2}+m^{2}\beta_{FPI}^{2}+2m\beta_{FTIS}\beta_{ FPI}\cos(\varphi_{FTIS}-\varphi_{FPI})}\\ \varphi_{m}=\arctan[(\beta_{FTIS}\sin\varphi_{FTIS}-m\beta_{FPI}\sin\varphi_{ FPI})/(\beta_{FTIS}\cos\varphi_{FTIS}-m\beta_{FPI}\cos\varphi_{FPI})]\\ \varphi_{-m}=\arctan[(\beta_{FTIS}\sin\varphi_{FTIS}+m\beta_{FPI}\sin\varphi_{ FPI})/(\beta_{FTIS}\cos\varphi_{FTIS}+m\beta_{FPI}\cos\varphi_{FPI})]\end{array} \right.. \tag{32}\]  
Therefore, for the \(m\)-th order frequency-shifted component, the combined errors from the FTIS and FPI can be equivalently modeled by generating a new effective tilt angle \(\beta_{m}\) oriented at \(\varphi_{m}\). The same is true for the \(-m\)-th order frequency-shifted component.