## 2 Sampled Apodized Grating Waveguide Structure and Design
### Optical Modes Analysis  
Our designed device mainly realizes wideband reflection spectrum and periodic filtering at 1550 nm wavelength, with mode conversion of TM\({}_{0}\) and TM\({}_{1}\). The mode conversion is convenient for signal extraction, larger fabrication tolerance and cascading with other photon devices. This paper mainly discusses the function realization of the device, rather than the application of mode conversion. The overall diagrammatic sketch of the sampled apodized waveguide grating is shown in Figure 2a. This sampled apodized waveguide grating is composed of input waveguide, several same sampling periods and output waveguide. Each sampled period consists of an apodized Bragg grating and straight waveguide. The length of sampling period, the grating and duty cycle are defined as \(d\), \(P\) and \(t=P/d\), respectively. The sampled grating contains several parts of the same sampling period, whose number is defined as \(N\). The direction of the power transmitting in the waveguide is defined as z axis. The sketch of metal-printing sampled apodized waveguide grating in one sampling period is \begin{table}
\begin{tabular}{c c c c c} \hline
**Mixture** & **FSU-8 Composition (wt. \%)** & **Tg (\({}^{\circ}\)C)** & **Td\({}^{\text{C}}\) (\({}^{\circ}\)C)** & **Td\({}^{\text{D}}\) (\({}^{\circ}\)C)** \\ \hline \multirow{3}{*}{FSU-8} & 0 & 155.7 & 270.7 & 294.9 \\  & 25 & 158.4 & 277.7 & 302.9 \\  & 50 & 161.9 & 261.4 & 300.3 \\  & 75 & 156.0 & 271.8 & 300.0 \\ \hline \end{tabular}
\end{table} 
Table 1: Thermal properties of materials with different content of FSU-8. Figure 1: (a) Molecular structure of fluorinated photopolymer (FBPA-PC EP and FSU-8); (b) Near-infrared absorption spectrum of FSU-8 (0%, 25%, 50%, and 75%)/FBPA-PC EP and Commercial SU-8. shown in Figure 2b. The electrode heater is set to be upon the core FBPA-PC EP. The width of the electrode is designed to be 25 \(\upmu\)m and the length of each electrode has the same length of each apodized Bragg grating. Figure 2c shows the diagrammatic sketch view of the grating section. The duty cycle (\(Duty=W/\Lambda\)) of a Bragg grating is defined as the ratio of length without metal ridge (\(W\)) to Bragg grating period (\(\Lambda\)). The lateral shifts (\(\Delta S\)) of ridges on both sides are zero at the beginning and end of apodized Bragg grating. The value of \(\Delta S\) is defined as \(\Delta S_{0}\) at the center of the apodized Bragg grating. The ridge corrugation width (\(\Delta h\)) is also shown in Figure 2c. \(\Delta h\) has important effect on the coupling coefficient and the reflection spectrum. The cross-section profile of this grating is shown in Figure 2d. The 5 \(\upmu\)m thickness SiO\({}_{2}\) lower cladding lays on the Si bottom substrate, whose thickness is 500 \(\upmu\)m. The 2.5 \(\upmu\)m thickness fluorinated photopolymer film is applied as the core layer. The sampled grating and straight waveguides are defined by two Al electrodes with 40 nm thickness and most optical power is limited between metals. The effective refractive index of this structure can be calculated by finite difference method (FDM). The refractive indices of the SiO\({}_{2}\) bottom cladding layer, fluorinated photopolymer, Al layer, and air are measured as 1.444, 1.526, 1.579 + 11.568 i, and 1.000 at 1550 nm wavelength, respectively. The algorithm of FDM is shown as below. From [PERSON]'s Equations, the full vector wave equation for the electric field can be obtained as:  
\[\
abla_{t}\Big{(}n^{2}E_{t}\Big{)}+\frac{\partial n^{2}}{\partial z}E_{z}+n^{2 }\frac{\partial E_{z}}{\partial z}\ =\ 0 \tag{1}\]  
where \(E_{t}\) is the electric field intensity along the x and y directions, \(E_{z}\) is the electric field along \(z\) direction, and \(n\) is used as the refractive index distribution shown in Figure 2d. Because \(n\) is \(z\)-invariant, \((\partial n^{2})/\partial z=0\), and \(E(x,y,z)=E_{t}\) e\({}^{-\beta z}\), the vectorial wave Equation (2) can be derived [30].