# Contrast Analysis of Polarization in Three-Beam Interference Lithography
## 3 Contrast in Three-Beam Interference Patterns
### TE-TE-TE Polarization  
Both co-directional component TE polarization and co-directional component TM polarization were analyzed above. In both cases, neither the high contrast of the light field nor the stability of the interference system can be guaranteed under the conditions of large-angle interference. As shown in Figure 8d, another polarization mode is proposed for interference lithography under large-angle incidence. Figure 6: On the left is a top view of three beams incident, and the arrows represent the polarization direction of the beams on the x-y plane. The figure on the right shows that the maximum co-directional polarization component is still applicable to the small-angle incident situation of multi-beam interference. However, as the incident angle increases, the stability of the multi-beam interference system will be very poor. When the polarization of the three beams are in _TE-TE-TE_ mode, the polarization angle of three beams is \(\phi_{1}=\phi_{2}=\phi_{3}=90^{\circ}\). The polarization vectors can be expressed as  
\[ \begin{array}{c}\mathbf{e}_{1}=\left[\begin{array}{ccc}0&\mathbf{j}&0\\ \end{array}\right]\\ \mathbf{e}_{2}=\left[\begin{array}{ccc}-i\frac{\sqrt{3}}{2}&-j\frac{1}{2}&0\\ \end{array}\right]\\ \mathbf{e}_{3}=\left[\begin{array}{ccc}\mathbf{i}\frac{\sqrt{3}}{2}&-j\frac{1}{2}&0 \\ \end{array}\right]\end{array}  \tag{21}\]  
From Equation (14), the intensity distribution of three-beam interference can be calculated by  
\[I(x,y,z)=3-2\cos(\frac{3}{2}k\sin\theta\cdot x)\cos(\frac{\sqrt{3}}{2}k\sin \theta\cdot y)-\cos(\sqrt{3}k\sin\theta\cdot y) \tag{22}\]  
It can be observed from Equation (22) that the maximum intensity remains unchanged at 4.5 when the incident is in TE-TE-TE mode, which is only half of the theoretical value. The increase in the incident angle will not cause additional contrast loss and it makes it  
Figure 8: Schematic diagram of polarization mode of three-beam interference. (**a**) The incident azimuth of the interference of the three beams. (**b–d**) Three different types of polarization mode combinations, the arrows represent the polarization direction of the beams on the x-y plane. Figure 7: The interference of the three beams incident with the maximum co-directional TE polarization. The light field intensity distribution at different incident angles: the maximum value of the light field intensity changes with the incident angle and the relationship between the contrast and the incident angle. (**a**) The intensity distribution of the light field with an incident angle equal to \(15^{\circ}\). (**b**) The intensity distribution of the light field with an incident angle equal to \(60^{\circ}\). (**c**,**d**) The maximum value of the light field intensity and the contrast change due to the incident angle. easier for us to control the exposure dose at different incident angles. Although the increase in exposure time may introduce additional environmental errors to the interference light field, it is clearly easier to obtain a pattern with a smaller feature size when the light field has a high contrast. It can be seen from Figure 6 that the contrast loss caused by the polarization angle deviation must be considered, especially under the condition of large-angle interference. Under _TE-TE_ incident condition, the polarization angle deviations are \(\delta_{1},\delta_{2},\delta_{3}\), and the polarization angle of three beams are \(\phi_{1}=90^{\circ}+\delta_{1},\phi_{2}=90^{\circ}+\delta_{2},\phi_{3}=90^{ \circ}+\delta_{3}\). The polarization vector can be expressed as  
\[\begin{array}{c}\mathbf{e}_{1}=\left[ \begin{array}{cc}-i\cos\theta\sin\delta _{1}&j\cos\delta_{1}&\mathbf{k}\sin\theta\sin\delta_{1}\end{array} \right]\\ \mathbf{e}_{2}=\left[