## 3 Optical Error Analysis
### Optical Errors of FTIS
#### 3.2.1 Collimation Error  
In practical implementations, the beam exiting the collimation system from an object point exhibits non-ideal collimation with a finite divergence angle. As illustrated in Figure 9, rays with different divergence angles \(\theta\) generate distinct OPDs in both the FTIS and FPI systems. Assuming uniform intensity distribution across the divergence angle range \(2\theta_{0}\), the interference intensity of monochromatic light entering the FTIS can be expressed as follows:  
\[ \begin{array}{ll}I_{1-dire}(\
u,\Delta)&=\int_{0}^{2\pi}d\psi\int_{0}^{ \theta_{0}}B(\
u)\cos(2\pi\
u\Delta\cos\theta)\sin\theta d\theta\Big{/}\int_{ 0}^{2\pi}d\psi\int_{0}^{\theta_{0}}\sin\theta d\theta\\ &=B(\
u)\cos\bigg{[}2\pi\Delta\
u\bigg{(}1-\frac{1-\cos\theta_{0}}{2}\bigg{)} \bigg{]}\sin c[\Delta\
u(1-\cos\theta_{0})]\end{array}  \tag{20}\]  
Under ideal conditions, the interferometric spectrum produced using FTIS is a standard cosine function. Under the influence of the divergence angle, the FTIS two-beam interference introduces a modulation function \(\text{sinc}[\
u\Delta(1-\cos\theta_{0})]\) and a wavenumber shift factor \((1-\cos\theta_{0})/2\). The effect on the double-beam interferometric spectrum is shown in Figure 10. In Figure 10, with a divergence angle of \(1^{\circ}\) and OPD values of \(\Delta=1\), \(2\), and \(3\) mm, the interferometric spectrum under this error is compared with the ideal case. It can be seen that the divergence angle causes both a reduction in interference intensity and a drift in the wavenumber of the interferometric spectrum. As the OPD increases, the interference intensity at the same wavenumber position decreases, while the wavenumberdrift remains unchanged. For the same OPD, as the wavenumber increases, the interference intensity decreases according to the modulation function \(\text{sinc}[v\Delta(1-\text{cos}\theta_{0})]\), and the periodic frequency of the double-beam interferogram is reduced due to the presence of the wavenumber drift factor. The maximum incident light divergence angle corresponding to the spectral resolution limit of a conventional FTIS is given by the following:  
\[\theta_{0}=\sqrt{\lambda_{\text{min}}/L}. \tag{21}\]  
The total interference intensity \(I_{1}\) is  
\[ \begin{array}{ll}I_{1-drive}(\Delta)&=\int_{0}^{2\pi}d\psi\int_{0}^{\theta_{ 0}}I_{0}(\Delta\cos\theta)\sin\theta d\theta\Big{/}\int_{0}^{2\pi}d\psi\int_{ 0}^{\theta_{0}}\sin\theta d\theta\\ &=\int_{\text{cos}\theta_{0}}^{1}I_{0}(\Delta u)du/(1-\cos\theta_{0})\\ &=I_{0-drive}(\Delta)\end{array} . \tag{22}\]  
The FPI is also affected by the divergence angle, and the incident angle of the same light beam is identical for both FTIS and FPI. The interferogram \(I_{2}\) is given by the following:  
Figure 10: Assuming a divergence angle of \(1^{\circ}\), (**a**) FTIS with an OPD of 1 mm, (**b**) FTIS with an OPD of 2 mm, and (**c**) FTIS with an OPD of 3 mm: the overall comparison and local magnification of the FTIS double-beam interferometric spectrum. Figure 9: Beams with different divergence angles produce different OPDs in the FTIS and FPI. Here, \(\theta_{0}\) is the maximum divergence half-angle, and \(\psi\) is the azimuth angle. The red arrows indicate the phase differences caused by the central aperture light in the FTIS and FPI, respectively, and the black arrows indicate the phase differences caused by the edge aperture light in the FTIS and FPI, respectively.