## 2 Sampled Apodized Grating Waveguide Structure and Design
### Lateral Shift Apodized Waveguide Grating Design  
In order to realize a sampled apodized grating with the design aim described in Section 2.1, one should first consider a single traditional Bragg grating unit with TM\({}_{0}\) and TM\({}_{1}\) mode conversion. The value of \(\Delta\lambda_{b}\) is approximate 5.0 nm and SLSR should exceed 20 dB in the single Bragg grating unit. In the traditional mode conversion Bragg grating design, there should be the same value of ridge dislocations \(\Delta S_{0}\) at every grating period to ensure the coupling between TM\({}_{0}\) and TM\({}_{1}\) optical modes. In general, the value \(\Delta S_{0}\) is generally designed as \(\Lambda\)/2 to satisfy maximum coupling of TM\({}_{0}\) and TM\({}_{1}\) optical modes. \(\Lambda\) should be satisfied with Bragg phase matching condition, \(\lambda_{0}=2\) [\(n_{\mathit{eff}0}\times(1-Duty)+n_{\mathit{eff}1}\times Duty\)] \(\Lambda\)/\(m\), where \(m\) is the grating order designed as 9, and the fabrication tolerance is larger than that of \(m=1\). For general Bragg grating, the reflectivity is zero when \(m\) is even. So odd number of grating order is selected. Further, when \(m\) is 9, the minimum fabrication linewidth is obviously larger than when \(m\) is 1. In addition, according to the grating equation mentioned above, due to the fabrication error, the offset of resonance wavelength is one ninth of that when \(m=1\). So, the grating order \(m\) is selected as 9. In this expression, \(n_{\mathit{eff}0}\) and \(n_{\mathit{eff}1}\) are the effective refractive indices of TM\({}_{0}\) and TM\({}_{1}\) modes, respectively. Figure 3: (**a**) Relationship between \(N_{\mathit{clud}}\) and \(b\); (**b**) Relationship between \(N_{\mathit{core}}\) and \(b\); (**c**) Relationship between \(n_{\mathit{eff}}\) and \(c\); (**d**) Cross-section optical field of metal-defined waveguide structure. The transfer matrix method is adopted to solve the single Bragg grating reflection spectrum according to the couple mode equation written in the matrix form. The grating is divided into \(M\) uniform sections which are used as 100 in the transfer matrix method calculation. The amplitudes of forward and backward optical mode are expressed as \(C_{i}\) and \(D_{i}\), respectively, where \(i\) is between 1 and \(M\). In this simulation, \(C_{0}=C(P)=1\) and \(D_{0}=D(P)=0\). The propagation through each uniform section is described by a matrix defined as \(F_{i}\) in Equations (9) and (10), which is utilized to obtain the reflection spectrum of Bragg grating [32]. \[\left[ \begin{array}{c}C_{i}\\ D_{i}\end{array} \right]\ =\ F_{i}\!\!\left[ \begin{array}{c}C_{i-1}\\ D_{i-1}\end{array} \right] \tag{9}\]  
\[ \begin{array}{c}F_{grat}\ =\ F_{M}F_{M-1}\cdots F_{i}\ =\ \left[\begin{array}{ cc}S_{11}&S_{12}\\ S_{21}&S_{22}\end{array}\right](i\ =\ 1,2,\cdots M)\\ R\ =\ \left|\frac{5_{21}}{5_{11}}\right|^{2}F_{i}\ =\ \left[\begin{array}{ cc}s_{11}&s_{12}\\ s_{21}&s_{22}\end{array}\right]\\ s_{11}\ =\ \cosh[s(z_{i+1}-z_{i})]-j\frac{\xi}{5}\!\sinh[s(z_{i+1}-z_{i})]\\ s_{12}\ =\ -j\frac{k_{0}}{5}\!\sinh[s(z_{i+1}-z_{i})]\\ s_{21}\ =\ \cosh[s(z_{i+1}-z_{i})]+j\frac{\xi}{5}\!\sinh[s(z_{i+1}-z_{i})]\\ k_{0}\ =\ \iint E_{0}(x,y)\Delta E_{1}^{*}(x,y)\,dxdy\ =\ \frac{(N_{0}^{2} \alpha r_{0}^{-N_{0}^{2}})}{n_{eff}0}\frac{\sin(m\chi Duty)}{m}\Gamma_{grating}\\ \xi\ =\ \delta+\sigma_{0},s\ =\ \sqrt{k_{0}^{2}-\xi^{2}}\\ \delta\ =\ 2n\pi_{eff}(\frac{1}{\lambda}-\frac{1}{\lambda_{0}}),\sigma_{0}\ =\ \frac{2\pi}{3}\bar{\kappa}_{neff}\end{array}  \tag{10}\]  
In this expression, \(z\), \(k_{0}\), \(\sigma_{0}\), and \(\Gamma_{grating}\) are the light propagation length, coupling coefficient, refractive index modulation, and power ratio restricted in the region between two metal strips for the traditional mode conversion Bragg grating, respectively. The distance between \(z_{i-1}\) and \(z_{i}\) is \(P\!/M\) which indicates every section length is the same. \(\lambda_{0}\) is the resonance wavelength mentioned in the Bragg phase matching condition.