## 2 Methods
### Interference Theory of Orthogonal Two-Axis Lloyd's Mirror Interferometer  
According to the analysis of the interferometer, the collimated beam was divided into five sections, as shown in the Figure 3a, called beam 1, 2, 3, 4, and 5. The beam directly projecting onto the substrate is beam 1; the beam propagating onto the substrate after being reflected by the X-mirror is beam 2; the beam propagating onto the substrate after being reflected by the Y-mirror is beam 3; the beams propagating onto the substrate after being reflected by the Y-mirror firstly and reflected by the X-mirrors secondly is beam 4; and the beams propagating onto the substrate after being reflected by the X-mirror firstly and reflected by the Y-mirrors secondly is beam 5.  
First of all, however, it should be noted that only beams 1, 2, and 3 are required for fabrication of the desired 2D gratings. Beams 4 and 5 will introduce unexpected grating patterns [14]. Thus, a physical filter is stuck in the optical path as shown in Figure 3b, a similar manner as that illustrated in present methods [16; 17; 18]. It should be noted that the incident plane wavefront beam still follows a Gaussian distribution, which will influence the grating shape uniformity among the whole grating surface.  
Theoretically, assuming that only three--beams 1, 2, and 3--have uniform intensity distribution at the wavefront, the electric field of a laser beam can be expressed as follows.  
\[\mathbf{E}_{i}=E_{i}\cdot\mathbf{e}_{i}\text{exp}(j\mathbf{k}_{i}\cdot\mathbf{r}+\gamma_{i}) \ (i=1,\ 2,\ 3,\ 4,\ 5), \tag{1}\]  
where \(\mathbf{r}\) is the position vector, \(\mathbf{E}_{i}\) is the real electric field amplitude, \(\mathbf{e}_{i}\) is a unit vector in the polarization direction of the laser beam, \(\gamma_{i}\) is the initial phase, and \(\mathbf{k}_{i}\) is the wave vector. The total electric field of needed incident beams can be expressed by  
\[\mathbf{\mathrm{e}}(\mathbf{\mathrm{r}})=\sum_{i=1}^{3}E_{i}\text{exp}(j\mathbf{k}_{i} \cdot\mathbf{r}+\gamma_{i})\ (i=1,\ 2,\ 3). \tag{2}\]  
The intensity distribution of interference field upon the substrate can be written as follows,  
\[\text{I}(\mathbf{\mathrm{r}})=\sum_{i=1}^{3}\left|E_{i}\right|^{2}+2\sum_{m=2}^{3 }\sum_{n<m}E_{m}E_{n}\mathbf{e}_{m}\mathbf{e}_{n}cos\{(\mathbf{k}_{n}-\mathbf{k}_{m})\cdot\bm {r}+\varphi_{n-}\varphi_{m}\}. \tag{3}\]  
Figure 3: Optical configurations of orthogonal two-axis [PERSON]’s mirror interferometer. (**a**) The sections of collimated beam (from the direction of the incident beam). (**b**) Improved physical filter.  
According to Equation (3), the periods of the interference fringes to be generated by the interference between the two of those beams can be expressed as follows,  
\[g_{nm}=\frac{2\pi}{|\mathbf{k}_{n}-\mathbf{k}_{n}|}\ (n<m\leq 3). \tag{4}\]  
Assume \(\varphi\) = 45\({}^{\circ}\) and the mirrors and substrate are orthogonal with each other, the X and Y-directional fringe periods and, respectively, can be expressed as follows,  
\[g_{x}=g_{y}=\frac{\lambda}{\sqrt{2}cos\theta}. \tag{5}\]  
Figure 4 summarizes the line interference fringes to be generated by the interference between the two of those beams. As we can be seen in Figure 4, the interference fringes generated by beams 1 and 2, and beams 1 and 3 are parallel to Y-axis and X-axis, respectively, as shown in Figure 4a,b. Obviously, this two interference fringes are perpendicular to each other. The interference fringe generated by beam 2 and beam 3 are 45\({}^{\circ}\) with respect to X-axis as shown in Figure 4c. The interference fringe is enough for fabricating a 2D-orthogonal grating with the interference fringes generated by beams 1 and 2 and beams 1 and 3. Consequently, it is necessary to eliminate the interference of beam 2 and beam 3.