# Contrast Analysis of Polarization in Three-Beam Interference Lithography
## 2 Contrast in Double-Beam Non-Coplanar Interference Patterns  
When two laser beams from the same light source coincide, an interference pattern is produced, which contains all of the information including the intensity and phase of the two interfering beams. The interference of two beams is the superposition of electric field vectors. The electric field vector of \(i\)th beam vector can be written as  
\[E_{i}=A_{i}\mathbf{e}_{i}\exp[\mathbf{i}(\mathbf{k}_{i}\cdot\mathbf{r}+\varphi_{i})] \tag{1}\]  
where \(A_{i}\) is the amplitude, \(\mathbf{e}_{i}\) is the polarization vector, \(\mathbf{k}_{i}\) is the wave propagation vector, \(\mathbf{r}\) is the position vector at interference area, and \(\varphi_{i}\) is the initial phase of the light wave. The intensity of two-beam interference can be calculated as follows:  
\[I_{\mathrm{2-beam}}=(E_{1}+E_{2})(E_{1}+E_{2})^{*}=|A_{1}|^{2}+|A_{2}|^{2}+2A_{ 1}A_{2}|\mathbf{e}_{1}\mathbf{e}_{2}^{T}\big{|}\cos[(k_{1}-k_{2})\cdot\mathbf{r}+(\varphi_{ 1}-\varphi_{2})] \tag{2}\]  
Interference contrast is defined as  
\[K=\frac{I_{\mathrm{max}}-I_{\mathrm{min}}}{I_{\mathrm{max}}+I_{\mathrm{min}}} \tag{3}\]  
\(I_{\mathrm{max}}\) and \(I_{\mathrm{min}}\) represent the minimum and the maximum intensity of the interference field. Substituting Equation (2) into Equation (3) yields  
\[K\propto\frac{2A_{1}A_{2}|\mathbf{e}_{1}\mathbf{e}_{2}^{T}\big{|}}{|A_{1}|^{2}+|A_{2}| ^{2}} \tag{4}\]  
Equation (4) states that contrast is affected by the intensity mismatch and the polarization states for each beam. We already know that the contrast loss caused by the unequal amplitude of the two beams is very low. Even if the amplitude ratio of the two beams reaches 2:3, the contrast loss is only 0.076. Therefore, the main source of contrast loss for non-coplanar interference is the polarization state of the two beams. In Equation (1), according to the coordinate rotation transformation matrix \(\mathbf{e}_{i}\), \(\mathbf{k}_{i}\) and \(\mathbf{r}\) can be expressed as  
\[\mathbf{e}_{i}=\left[ \begin{array}{c}\mathbf{e}_{ix}\\ \mathbf{e}_{iy}\\ \mathbf{e}_{iz}\end{array} \right]^{T}=\left[ \begin{array}{c}\mathbf{i}(\cos\alpha_ {i}\cos\theta_{i}\cos\phi_{i}-\sin\alpha_{i}\sin\phi_{i})\\ \mathbf{j}(\sin\alpha_{i}\cos\theta_{i}\cos\phi_{i}+\cos\alpha_{i}\sin\phi_{i}) \end{array} \right]^{T} \tag{5}\]  
\[\mathbf{k}_{i}=\left[ \begin{array}{cc}\mathbf{k}_{ix}&\mathbf{k}_{iy}&\mathbf{k}_{iz}\\ \end{array} \right]=k\big{[} \begin{array}{cc}\mathbf{i}\cos\alpha_{i}\sin\theta_{ i}&j\sin\alpha_{i}\sin\theta_{i}&-k\cos\theta_{i}\end{array} \right] \tag{6}\]\[\mathbf{r}=\left[ \begin{array}{cc}ix&jy&\mathbf{kz}\end{array} \right]^{T} \tag{7}\]  
Here, \(\alpha_{i}\) and \(\theta_{i}\) are the azimuthal angle and the incident angle, respectively; \(\phi_{i}\) is the polarization angle; \(k=2\pi/\lambda\) is the wavenumber; and \(\lambda\) is the wavelength of the laser. As shown in Figure 1, the azimuth angles of the two non-coplanar coherent beams are \(\alpha_{1}=0\) and \(\alpha_{2}=\alpha(0<\alpha<\pi)\), and they incident at the identical angle \(\theta\). When \(\alpha\) is equal to \(\pi\), it turns into coplanar interference. Using Equations (5)-(7), the contrast for two-beam interference can be further reduced to  
\[K\propto\left| \begin{array}{cc}\sin\alpha\cos\theta\sin(\phi_{1}-\phi_{2})+ \cos\alpha\cos(\phi_{1}-\phi_{2})\\ +(1-\cos\alpha)\sin^{2}\theta\cos\phi_{1}\cos\phi_{2}\end{array} \right| \tag{8}\]  
Equation (8) shows that for different azimuth and incident angles, we need to choose different beam polarization modes to maximize the contrast of the interference field. Owing to the incident surfaces of the two light waves not being coplanar, there is no case where the horizontal polarization component is totally in the same direction. We can only reduce the contrast loss by ensuring that their co-directional TE vibration components achieve the maximum value.