# Contrast Analysis of Polarization in Three-Beam Interference Lithography
## 3 Contrast in Three-Beam Interference Patterns
### Co-Directional Component Polarization  
When the polarization common component is the largest, the relationship between the azimuth and the polarization angle of N-beam interference is as follows:  
\[\alpha_{i}=2\pi\frac{i-1}{N},\phi_{i}=\phi-\alpha_{i} \tag{16}\]  
where \(\phi\) is the polarization angle of the beam whose azimuth is zero. The polarization vector can be written as follows:  
\[\mathbf{e}_{i}=\left[ \begin{array}{c}i\Bigl{(}\cos\theta\cos\Bigl{(}2\pi\frac{i -1}{N}\Bigr{)}\cos\Bigl{(}\phi-2\pi\frac{i-1}{N}\Bigr{)}-\sin\Bigl{(}2\pi\frac {i-1}{N}\Bigr{)}\sin\Bigl{(}\phi-2\pi\frac{i-1}{N}\Bigr{)}\\ j\Bigl{(}\cos\theta\sin\Bigl{(}2\pi\frac{i-1}{N}\Bigr{)}\cos\Bigl{(}\phi-2\pi \frac{i-1}{N}\Bigr{)}+\cos\Bigl{(}2\pi\frac{i-1}{N}\Bigr{)}\sin\Bigl{(}\phi-2 \pi\frac{i-1}{N}\Bigr{)}\Bigr{)}\\ -k\sin\theta\cos\Bigl{(}\phi-2\pi\frac{i-1}{N}\Bigr{)}\end{array} \right]^{T} \tag{17}\]  
As shown in Figure 6, the three incident beams follow a symmetrical configuration with the azimuth of \(\alpha_{1}=0^{\circ},\alpha_{2}=120^{\circ},\) and \(\alpha_{3}=240^{\circ}\). The incident angles of the three beams are set as \(\theta_{1}=\theta_{2}=\theta_{3}=\theta\). The polarization vectors of the three beams are  
\[\mathbf{e}_{1}=\left[ \begin{array}{cc}i\bigl{(}\frac{1}{4}\cos\theta\cos\phi- \frac{\sqrt{3}}{4}\cos\theta\sin\phi+\frac{\sqrt{3}}{4}\sin\phi+\frac{3}{4} \cos\phi\Bigr{)}\\ \mathbf{j}\Bigl{(}-\frac{\sqrt{3}}{4}\cos\theta\cos\phi+\frac{3}{4}\cos\theta\sin \phi+\frac{1}{4}\sin\phi+\frac{\sqrt{3}}{4}\cos\phi\Bigr{)}\\ \mathbf{k}\Bigl{(}\frac{1}{2}\sin\theta\cos\phi-\frac{\sqrt{3}}{2}\sin\theta\sin \phi\Bigr{)}\end{array} \right]^{T} \tag{18}\]  
The three beams are symmetrically incident, and the vibration components of the x-y plane are along the same angle \(\phi\). Substituting Equation (18) into Equation (15), considering the polarization and incident angle one at a time, the contrast as a function of two factors is plot in Figure 6. Similar to the two-beam non-coplanar interference, when interfering at a small angle, the polarization angle will not change the contrast. The contrast loss will only increase as the incident angle increases. As the incident angle continues to increase, the contrast is more sensitive to the changes in the polarization angle and is extremely susceptible to polarization angle deviations. In this case, taking the TE wave of one of the three beams as the common vibration direction can ensure that the contrast is sufficiently high even at a large incident angle. The polarization angle of beam 1 is \(\phi_{1}=\phi=90^{\circ}\). From Equation (16), the polarization vectors can be expressed as  
\[\begin{array}{c}\mathbf{e}_{1}=\left[ \begin{array}{cc}0&\mathbf{j}&0\end{array} \right] \\ \mathbf{e}_{2}=\left[ \begin{array}{cc}i\Big{(}-\frac{\sqrt{3}}{4}\cos\theta+\frac{ \sqrt{3}}{4}\Big{)}&\mathbf{j}\Big{(}\frac{3}{4}\cos\theta+\frac{1}{4}\Big{)}&\mathbf{k} \Big{(}-\frac{\sqrt{3}}{2}\sin\theta\Big{)}\\ \mathbf{e}_{3}=\left[\begin{array}{cc}i\Big{(}\frac{\sqrt{3}}{4}\cos\theta-\frac {\sqrt{3}}{4}\Big{)}&\mathbf{j}\Big{(}\frac{3}{4}\cos\theta+\frac{1}{4}\Big{)}&\bm {k}\Big{(}\frac{\sqrt{3}}{2}\sin\theta\Big{)}\end{array}\right]\end{array}  \tag{19}\]  
From Equation (14), the intensity distribution of three-beam interference can be calculated by  
\[ \begin{array}{c}I(x,y,z)=3+(3\cos\theta+1)\cos\big{(}\frac{3}{2}k\sin \theta\cdot x\big{)}\cos\big{(}\frac{\sqrt{3}}{2}k\sin\theta\cdot y\big{)}\\ +\left[\frac{9}{4}\Big{(}\cos\theta+\frac{1}{3}\Big{)}^{2}-2\right]\cos\Big{(} \sqrt{3}k\sin\theta\cdot y\Big{)}\end{array}  \tag{20}\]  
According to the above equation, the maximum intensity of the interference field is related to the value of the incident angle. As shown in Figure 7, after the incident angle is increased to \(32^{\circ}\), the contrast no longer decreases as the incident angle increases. At the same time, with the increase in incident angle, the pattern transitions from the hexagonal center maximum lattice to the hexagonal minimum lattice.