# Method to Remove Tilt-to-Length Coupling Caused by Interference of Flat-Top Beam and Gaussian Beam
## 3 Theoretical Analysis
### Diffraction  
The basis of the situation shown in Figure 2a is that no diffraction occurs during the beam propagation process. However, the amplitude and the phase of flat-top beam wave on the detector could not be flat if diffraction occurred during propagation, but diffraction is very common in practical situations. Therefore, it is necessary to research whether the TTL coupling can also be eliminated as shown in Figure 2a when diffraction occurs. The aperture stop is located on the coordinate origin \(O\) as shown in Figure 1 in this case. For consistency with Figure 2a, we also set \(s=500\) mm, \(\omega_{0}=0.5\) mm, \(L=0\), and \(z_{0}=0\) in the analysis. The diffraction effects regime could be determined by using the Fresnel number as follows:  
\[N_{F}=\frac{R^{2}}{s\lambda} \tag{13}\]  
Fresnel numbers of around 1 or higher characterize a case of Fresnel diffraction (near-field diffraction). From Equation (13) we know that when \(N_{F}\) is approximately 1, then \(R\) is approximately 0.7 mm. We thus set \(R=0.7\) mm to study the effect of the Fresnel diffraction and set \(D=30\) mm to ensure that almost all the energy is detected. Equation (4) is used to calculate the field distribution of the diffracted beam. The field distribution of the diffracted flat-top beam on detector in polar coordinates(i.e., \(x_{1}=\rho_{1}\cos\psi_{1},y_{1}=\rho_{1}\sin\psi_{1},x=\rho\cos\psi,y=\rho\sin\psi\)) can then be written as:  
\[E(\rho,\psi)_{flat-diff}=\frac{1}{i\lambda}\int_{0}^{2\pi}\int_{0}^{R}\frac{ \exp(ikr)}{r}\frac{1+\cos\theta}{2}\exp[-ik(\rho_{1}\cos\psi_{1}\sin\alpha)] \rho_{1}d\rho_{1}d\psi_{1} \tag{14}\]  
The field distribution of the diffracted Gaussian beam on detector can then be written as  
\[E(\rho,\psi)_{Gauss-diff}=\frac{1}{i\lambda}\int_{0}^{2\pi}\int_{0}^{R}\frac {\exp(ikr)}{r}\frac{1+\cos\theta}{2}\exp[-ik(\frac{\rho_{1}^{2}}{2(iz_{r}-z_{0 })}-z_{0})]\rho_{1}d\rho_{1}d\psi_{1} \tag{15}\]  
where \(r=\sqrt{s^{2}+(\rho\cos\psi-\rho_{1}\cos\psi_{1})^{2}+(\rho\sin\psi-\rho_{1} \sin\psi_{1})^{2}}\), \(\cos\theta=\frac{\kappa}{r}\). There we analyze two situations. The first is the case where the aperture stop only works for the flat-top beam and not for  
Figure 2: (**a**) When the waist position of the Gaussian beam coincides with the rotation point of the flat-top beam and the detector is an large enough plane with no gap, \(LPS_{LPF}\) is shown to be significantly below the picometer scale. (the differences are numerical errors related to the ray tracing algorithm). (**b**) When the gap size of detector is 20 \(\mu\)m, the balance of (**a**) is disturbed and \(LPS_{LPF}\) is shown to be increased to 10 pm order. the Gaussian beam (sometimes no stop is present in the transmission path of the Gaussian beam). This leads to the path length change:  
\[LPS_{LPF}=\frac{\arg(\int_{0}^{2\pi}\int_{0}^{D/2}E_{flat-diff}E_{gauss}^{*}pd \rho d\psi)}{k} \tag{16}\]  
The second case is where both interfering beams are affected by the aperture diffraction. This leads to the path length change:  
\[LPS_{LPF}=\frac{\arg(\int_{0}^{2\pi}\int_{0}^{D/2}E_{flat-diff}E_{gauss-diff}^{ *}pd\rho d\psi)}{k} \tag{17}\]  
It is difficult to obtain analytic solutions for Equations (16) and (17), so numerical integrals are used to perform the analysis by using MATLAB (ASAP software was not selected due to data accuracy). The results for the two cases are shown in Figure 3a,b, respectively. Simultaneously, the analytical solution Equation (10) from the previous section is used to calculate the result for the case without diffraction for the comparison(the aperture stop is located on the detector). The result is shown in Figure 3c  
Comparison of Figure 3a-c shows that in the case of interference between the non-diffracted flat-top beam and the Gaussian beam, the value of \(LPS_{LPF}\) is close to that of \(LPS_{OPD}\). The reason for this is that the small size of the flat-top beam reduces the degree of wavefront mismatch between the two beams and the geometric noise then becomes the main component of the TTL noise.