## 3 Optical Error Analysis
### Optical Errors of FTIS
#### 3.2.3 Mirror Tilt Error  
In FTIS, optical components such as the fixed mirror, moving mirror, and beam splitter may deviate from their ideal positions due to alignment errors or external vibrations, leading to tilt and other issues. The two coherent parallel beams exiting the interferometer exhibit a wavefront tilt with an angle of \(2\beta_{FTIS}\). This tilt causes variations in the OPDat different positions within the aperture. When the aperture is circular, the normalized distribution of the OPD error is given by the following:  
\[g_{FTIS}(\sigma\Delta)=\frac{1}{\pi\beta_{FTIS}r_{0}}\sqrt{1-\left(\sigma\Delta /2\beta_{FTIS}r_{0}\right)^{2}}. \tag{26}\]  
According to Equation (26), the FWHM of \(g_{FTIS}(\sigma\Delta)\) is given by \(FWHM_{tilt}=2\sqrt{3}\beta_{FTIS}r_{0}\). \(I_{1}\) can be expressed as follows:  
\[\begin{array}{ll}I_{1-tilt}(\Delta)&=\int dv\int_{-\beta_{FTIS}r_{0}}^{ \beta_{FTIS}r_{0}}B(v)\cos[2\pi v(\Delta+\sigma\Delta)]g_{FTIS}(\sigma\Delta)d (\sigma\Delta)\\ &=\int\left[B(v)\frac{2J_{1}(2\pi v2r_{0}\beta_{FTIS})}{2\pi v2r_{0}\beta_{ FTIS}}\right]\cos(2\pi v\Delta)d\
u\end{array}. \tag{27}\]  
The Bessel function forms the modulation function, and the higher the wavenumber, the faster the modulation depth decreases. This error does not affect the subsequent FPI, meaning \(T_{FPI}(\
u)\) remains unchanged. Similarly, \(I_{2}\) can be expressed as follows:  
\[\begin{array}{ll}I_{2-tilt}(\
u)&=\int dv\int_{-\beta_{FTIS}r_{0}}^{\beta_{ FTIS}r_{0}}B(v)T_{FPI}(v)\cos[2\pi v(\Delta+\sigma\Delta)]g_{FTIS}(\sigma \Delta)d(\sigma\Delta)\\ &=\int\left[B(v)T_{FPI}(v)\frac{2J_{1}(2\pi v_{0}\beta_{FTIS})}{2\pi v\pi v_{ 0}\beta_{FTIS}}\right]\cos(2\pi v\Delta)d\
u\end{array}. \tag{28}\]  
The two interferograms can also be expressed in an integral form as follows:  
\[\left\{\begin{array}{ll}I_{1-tilt}(\Delta)=\int_{-2\beta_{FTIS}r_{0}}^{2 \beta_{FTIS}r_{0}}I_{0}(\Delta+\sigma\Delta)g_{FTIS}(\sigma\Delta)d\sigma \Delta\\ I_{2-tilt}(\Delta)=\int_{-2\beta_{FTIS}r_{0}}^{2\beta_{FTIS}r_{0}}I_{0}( \Delta+\sigma\Delta)*FT[T_{FPI}(v)]g_{FTIS}(\sigma\Delta)d\sigma\Delta\end{array} \.\right. \tag{29}\]  
In FTIS, the moving mirror's positional stability during scanning is critical for maintaining interferometric accuracy. As the moving mirror translates, random tilt angles \(\beta_{FTIS}\) may arise due to mechanical imperfections, vibrations, or control system limitations. These tilts introduce spatially varying OPDs across the beam aperture. The \(g_{FTIS}(\sigma\Delta)\) changes with \(\Delta\). Consequently, the acquired interferograms \(I_{1}\) and \(I_{2}\) deviate from their idealized forms, manifesting as position-dependent phase noise that scales with \(\beta_{FTIS}\).  
For the fixed optical components, such as mirrors in SMFTIS and TSMFTIS, or stationary components, like the fixed mirror or beam splitter in FTIS, the tilt angles they introduce are fixed. Consequently, the OPD error distribution \(g_{FTIS}(\sigma\Delta)\) becomes a stationary function, allowing the interferogram to be expressed in a convolution form:  
\[\left\{\begin{array}{ll}I_{1-tilt}(\Delta)&=I_{0}(\Delta)*g_{FTIS}(\Delta) \\ &=I_{0-tilt}(\Delta)\\ I_{2-tilt}(\Delta)&=I_{0}(\Delta)*g_{FTIS}(\Delta)*FT[T_{FPI}(v)]\\ &=I_{0-tilt}(\Delta)*\left[a_{0}\delta(\Delta)+\sum_{m=1}^{\infty}a_{m}\delta (\Delta\pm m\cdot\Delta_{FPI})\right]\end{array}\.\right. \tag{30}\]  
Both \(I_{1-tilt}\) and \(I_{2-tilt}\) maintain linear relationships with \(I_{0-tilt}\). The response function \(a_{m}\delta(\Delta\pm m\Delta_{FPI})\) remains unchanged, and the fixed-tilt-induced errors do not affect the core interferogram inversion algorithm. The reconstructed interferogram \(I_{0}\) converges to \(I_{0-tilt}\), demonstrating equivalence to interferograms acquired by an FTIS system with identical tilt errors.