# Method to Remove Tilt-to-Length Coupling Caused by Interference of Flat-Top Beam and Gaussian Beam
## 2 Optical Model  
A simplified schematic of an interferometer is shown in the upper part of Figure 1; this interferometer includes a flat-top beam, a Gaussian beam, an aperture stop and a quadrantphoto-diode (QPD). In the face of aperture, the flat-top beam can be regarded as a fraction of a plane wave. The electric field of an infinite plane wave can be given by:  
\[E_{plane}=A\exp[-i(\vec{k}\cdot\vec{r}-\omega_{1}t)] \tag{1}\]  
Please note that \(A\) is the amplitude and is irrelevant to the pathlength signal, so it is set to unity here; \(\omega_{1}\) is the angular frequency; \(\vec{k}\) is the wave vector and its direction is ordinarily the direction in which the plane wave is traveling, so \(|\vec{k}|=k=2\pi/\lambda\), where \(k\) is the wavenumber. \(\vec{r}\) is the position vector that defines a point in three-dimensional space. Figure 1 shows that the direction cosine of \(\vec{k}\) is \((\sin\alpha,0,\cos\alpha)\). The non-diffractive form of the flat-top beam can be derived simply. In this situation, R can be regarded as the radial dimension of the flat-top beam, and a suitable expression for the flat-top beam is derived as shown in Equation (2). It is then very simple to calculate the complex amplitude of the flat-top beam on the detector by replacing \"z\" with \"\(s\)\" in Equation (2), \(s\) is the position of the detector.  
\[E(x,y,z)_{flat}=\left\{\begin{array}{ll}\exp[-ik(x\sin\alpha+z\cos\alpha)+i \omega_{1}t]&x^{2}+y^{2}\leq R\\ 0&\text{else}\end{array}\right. \tag{2}\]  
The fundamental Gaussian beam serves as the most appropriate description for the output of most lasers. The amplitude of the electric field is also set to unity here and Gouy phase is assumed to be static phase and can thus be ignored in the situation shown in Figure 1, the electric field can then be written as:  
\[E(x,y,z)_{gauss}=\exp[-ik(\frac{x^{2}+y^{2}}{2q}+z-z_{0})+i\omega_{2}t] \tag{3}\]  
where the complex parameter \(q=(z-z_{0})+iz_{r}\), the Rayleigh range \(z_{r}=\pi\omega_{0}^{2}/\lambda\), \(\omega_{0}\) is the waist radius, \(z_{0}\) is the z-axis value of the waist position, and \(\omega_{2}\) is the frequency.  
Next, we analyze the diffraction form. When a beam travels through free space, [PERSON]'s diffraction formula can be used to calculate the beam model of diffracted propagation. As shown in Figure 1, the coordinates on the aperture are represented by \(x_{1}y_{1}\), while the coordinates on the detector are represented by \(x.y\). Therefore, the field distribution \(E(x,y)_{diff}\) of the diffracted beam on the detector can be calculated as:  
\[E(x,y)_{diff}=\frac{1}{t\lambda}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \frac{\exp(ikr)}{r}\frac{1+\cos\theta}{2}E_{aper}(x_{1},y_{1})dx_{1}dy_{1}\]  
\[r=\sqrt{z^{2}+(x-x_{1})^{2}+(y-y_{1})^{2}},\cos\theta=\frac{z}{r} \tag{4}\]  
where \(r\) is the distance between point \((x,y)\) and point \((x_{1},y_{1})\), and \(E_{aper}(x_{1},y_{1})\) is the pre-diffracted complex amplitude of beam on the aperture.  
In current high-precision laser interferometers such as LISA or Taiji, heterodyne interference is used as the method of measurement, the QPD is usually used as the detector. And phasemeter is then used to analyze the photocurrents generated on the QPD to obtain the pathlength signal [7]. The optical simulation software \"ASAP\", known throughout the optics industry for its accuracy and efficiency on optical modelling, is used as the modelling tool to simulate heterodyne interference and obtain beat signal on detector. Then MATLAB is used as signal-processing tool to imitate the quadrature-phase(IQ) demodulation technique, which is used on the phasemeter of LISA pathfinder to compute interference signals [7]. An example is shown in the lower part of Figure 1, where the sampling number of the beat period is 16. The following simulations in Section 3.1 adopted the above way to perform the numerical analyses.