# Method to Remove Tilt-to-Length Coupling Caused by Interference of Flat-Top Beam and Gaussian Beam
## 3 Theoretical Analysis
### Non-Diffractive Form  
The aperture stop in Figure 1 is moved to the front of the detector along the z-axis, but the rotation point of the flat-top beam is fixed and its size is assumed to be larger than the aperture stop in this analysis. So, beams can propagate freely without being diffracted. Then we derive the analytical expressions for the complex amplitude of each quadrant on the detector without diffraction. To research how the different factors are coupled into the measured path length, the final computation results will be visually decomposed into the sum of the different corresponding formulas in the  
Figure 1: The upper part of the figure is a simplified schematic of an interferometer. Here, we set a Gaussian beam as the reference beam that propagates along the \(z\) axis and strikes the center of the detector; We also set a flat-top beam as the measurement beam that is tilted by an angle \(\alpha\) around the \(y\) axis, and to simplify the calculations, we set its pivot as the coordinate origin \(O\). The aperture stop is placed at the coordinate origin and is perpendicular to the z-axis and \(R\) is its radial dimension. \(s\) is the position of the QPD. \(d_{offset}\) is the distance between the centers of the two beams. \(z_{0}\) is the waist position. \(D\) and \(L\) are the diameter size and the gap size of the QPD, respectively. The interference pattern per sampling on the detector shown as the left side of the lower part of the figure, is respectively summed over the area of detector to acquire a beat signal shown as the right side. The signal-processing tool based on MATLAB then can process the beat signal to obtain the phase information. simplification process. In Figure 1, the flat-top beam is rotated along the y-axis, so quadrants 1, and 2 have the same field distribution, and quadrants 3, and 4 have the same field distribution:  
\[ \begin{split} C_{1}=C_{2}&=\frac{1}{2}\int_{-R}^{R} \bigg{(}\int_{L/2}^{R}E_{flat}E_{gauss}^{*}dx\bigg{)}dy\\ &\approx\frac{1}{2}\epsilon\exp\big{(}-\frac{1}{2}ikz\sin^{2}( \alpha)+ikz(1-\cos(\alpha))+\frac{1}{2}ikz_{0}\sin^{2}(\alpha)\big{)}(F(L/2)+F( R))\end{split}  \tag{7}\]  
\[ \begin{split} C_{3}=C_{4}&=\frac{1}{2}\int_{-R}^{R} \bigg{(}\int_{-R}^{-L/2}E_{flat}E_{gauss}^{*}dx\bigg{)}dy\\ &\approx\frac{1}{2}\epsilon\exp\big{(}-\frac{1}{2}ikz\sin^{2}( \alpha)+ikz(1-\cos(\alpha))+\frac{1}{2}ikz_{0}\sin^{2}(\alpha)\big{)}(F(-L/2)+ F(-R))\end{split}  \tag{8}\]  
with:  
\[F(r)=\text{Erfi}\bigg{[}\frac{\left(\frac{1}{2}+\frac{i}{2}\right)\sqrt{k}(r-q \sin(\alpha))}{\sqrt{q}}\bigg{]} \tag{9}\]  
where \(\epsilon\) represents the collection of terms that are unrelated to \(\alpha\), and will be ignored in the following calculation. The longitudinal pathlength signal (LPS) that results from LPF signal can then be derived as:  
\[ \begin{split} LPS_{LPF}&=\frac{\arg(\sum_{n=1}^{4} C_{n})}{k}\\ &\approx-\frac{1}{2}sa^{2}+\frac{1}{2}(s-z_{0})a^{2}\\ &+\frac{\arg\bigg{[}2\Big{(}F(L/2)+F\big{(}R\big{)}\Big{)}+2 \Big{(}F(-L/2)+F\big{(}-R\big{)}\Big{)}\bigg{]}}{k}\end{split}  \tag{10}\]  
We then set \(L=0\) and \(D/2=R=\infty\) to give the idealized detector shape and integral area. Formulas are presented using Taylor expansions based on the premise of preserving the precision of the calculations to obtain more intuitive analytical expressions. The \(LPS_{LPF}\) is then changed to:  
\[LPS_{LPF}\approx-\frac{1}{2}sa^{2}+\frac{1}{2}(s-z_{0})a^{2}+O(\alpha^{4}) \tag{11}\]  
In the case shown in Figure 1, the geometrical optical path difference (\(LPS_{OPD}\)) between the two beams can be derived easily as:  
\[LPS_{OPD}=\Big{(}\frac{1}{\cos\alpha}-1\Big{)}\cdot s\approx-\frac{1}{2}sa^{2 }+O(\alpha^{4}) \tag{12}\]  
The above obviously shows that when compared with \(LPS_{OPD}\), \(LPS_{LPF}\) contains extra terms, including the factor \"\(s-z_{0}\)\". Figure 1 shows that \"\(s-z_{0}\)\" is the distance between the waist position and the detector, which determines the wavefront curvature of the Gaussian beam on the detector.