## 2 Sampled Apodized Grating Waveguide Structure and Design
### Optical Modes Analysis
\[\
abla^{2}E_{t}+\
abla\Big{(}\frac{1}{n^{2}}\
abla(n^{2})\cdot E_{t}\Big{)}+ n^{2}k^{2}E_{t}\ =\ \beta^{2}E_{t} \tag{2}\]  
where \(k\) is the wave vector and \(\beta\) is the propagation constant. By using the transverse components of \(E_{x}\) and \(E_{y}\), Equation (2) can be expressed as the matrix form of Equation (3). \[\left[ \begin{array}{cc}P_{xx}&P_{xy}\\ P_{yx}&P_{yy}\end{array} \right]\left[ \begin{array}{c}E_{x}\\ E_{y}\end{array} \right]\ =\ \beta^{2}\left[ \begin{array}{c}E_{x}\\ E_{y}\end{array} \right] \tag{3}\]  
Figure 2: **(a)** Overall diagrammatic sketch of the sampled apodized waveguide grating; **(b)** Sketch of metal-printing sampled apodized waveguide grating in one sampling period; **(c)** Diagrammatic sketch view of the grating section with some important parameters; **(d)** Cross-section profile of this grating. It is usual for many optical waveguide devices that the coupling between the two polarizations is weak and negligible. At this time, the semivectorial treatment which refers to TE and TM modes is sufficient. By neglecting the crosscoupling terms \(P_{xy}\) and \(P_{yx}\), the fullvectorial reduces to Equation (4). The two decoupled semivectorial equations of Equations (5) and (6) are applied to calculate the effective refractive indices of TE and TM modes, respectively [30; 31]. \[\left[ \begin{array}{cc}P_{xx}&0\\ 0&P_{yy}\\ \end{array} \right]\left[ \begin{array}{c}E_{x}\\ E_{y}\\ \end{array} \right]=\beta^{2}\left[ \begin{array}{c}E_{x}\\ E_{y}\\ \end{array} \right] \tag{4}\]  
\[P_{xx}E_{x} = \frac{\partial}{\partial x}\Bigg{[}\frac{1}{n^{2}}\frac{\partial \left(n^{2}E_{x}\right)}{\partial x}\Bigg{]}+\frac{\partial^{2}E_{x}}{\partial y ^{2}}+n^{2}k^{2}E_{x}=\beta_{\text{TE}}{}^{2}E_{x} \tag{5}\] \[P_{yy}E_{y} = \frac{\partial^{2}E_{y}}{\partial x^{2}}+\frac{\partial}{\partial y }\Bigg{[}\frac{1}{n^{2}}\frac{\partial\left(n^{2}E_{y}\right)}{\partial y} \Bigg{]}+n^{2}k^{2}E_{y}=\beta_{\text{TM}}{}^{2}E_{y} \tag{6}\]  
In order to solve this Eigen Equation, the waveguide cross section in Figure 2d is discretized into small squares of size \(\Delta\text{x}\times\Delta\text{y}\) (In our calculation, \(\Delta\text{x}=\Delta\text{y}=10\) nm). The grid being computed is defined as \(P\) and the adjoining neighbor points of upper, lower, left and right directions of computed grid \(P\) are marked with subscripts \(S\), \(N\), \(E\) and \(W\). The refractive indices of these points are defined as \(n_{S}\), \(n_{N}\), \(n_{E}\), \(n_{W}\) and \(n_{P}\), respectively. The expressions of \(P_{xx}\) and \(P_{yy}\) are listed as Equations (7) and (8). \[P_{xx}:\left[ \begin{array}{cc}0&\frac{1}{\left(\Delta\text{y} \right)^{2}}&0\\ \frac{\alpha_{W}}{\left(\Delta\text{x}\right)^{2}}&n_{P}^{2}k^{2}-\frac{2}{ \left(\Delta\text{y}\right)^{2}}-\frac{2\alpha_{P}}{\left(\Delta\text{x} \right)^{2}}&\frac{\alpha_{E}}{\left(\Delta\text{x}\right)^{2}}\\ 0&\frac{1}{\left(\Delta\text{y}\right)^{2}}&0\\ \end{array} \right] \tag{7}\] \[\alpha_{W} = \frac{4(n_{P}^{2}+n_{W}^{2}n_{P}^{2})}{n_{P}^{2}+2n_{W}^{2}n_{P}^{ 2}+2n_{W}^{2}n_{W}^{2}+3n_{W}^{2}n_{W}^{2}}\] \[\alpha_{P} = \frac{2(2n_{P}^{2}+n_{W}^{2}n_{P}^{2}+n_{W}^{2}n_{P}^{2})}{n_{P}^{ 2}+2n_{W}^{2}n_{W}^{2}+2n_{W}^{2}n_{W}^{2}}\] \[\alpha_{E} = \frac{4(n_{P}^{2}+n_{W}^{2}n_{E}^{2})}{n_{P}^{2}+2n_{W}^{2}n_{P}^{ 2}+3n_{W}^{2}n_{W}^{2}}\]  
\[P_{yy}:\left[ \begin{array}{cc}0&\frac{y_{N}}{\left(\Delta\text{y} \right)^{2}}&0\\ \frac{1}{\left(\Delta\text{x}\right)^{2}}&n_{P}^{2}k^{2}-\frac{2\alpha_{P}}{ \left(\Delta\text{y}\right)^{2}}-\frac{2}{\left(\Delta\text{x}\right)^{2}}& \frac{1}{\left(\Delta\text{x}\right)^{2}}\\ 0&\frac{8}{\left(\Delta\text{y}\right)^{2}}&0\\ \end{array}