## 2 Sampled Apodized Grating Waveguide Structure and Design
### Lateral Shift Apodized Waveguide Grating Design
Further, \(R\) is the calculated reflectivity. In order to reduce the Bragg grating length, \(k_{0}\) should be designed as a large value and it can be adjusted by \(\Delta t\). Figure 4a shows the relationship between \(\Delta h\) and \(k_{0}\). It can be found that \(k_{0}\) increases faster when \(\Delta h\) is smaller than 4.0 \(\upmu\)m, and it begins stabilize when the value is larger than 5.5 \(\upmu\)m. The selected \(\Delta h\) is 5.5 \(\upmu\)m in our design and the optimized \(k_{0}\) can reach 200.5 mm\({}^{-1}\). Figure 4b shows the relationship between the coupling coefficient \(k_{0}\) and \(Duty\) when \(\Delta h\) is optimized as 4.0 \(\upmu\)m. In the traditional design of grating, \(Duty\) is often set as 0.5. As \(Duty\) is greater than 0.5 and increases, the sine term reduces in the expression of \(k_{0}\). However, the weighted refractive index increases, and a higher power portion is confined in the region between two metal strips, that makes \(\Gamma_{grating}\) increase. Thus, the value of \(k_{0}\) depends on the product of sine term and \(\Gamma_{grating}\)[25]. The best choice of \(Duty\) is 0.8 according to Figure 4b. The optimized \(\Lambda\) is 4.8 \(\upmu\)m according to the referred Bragg phase matching condition. Figure 4c gives the reflection spectrum at different grating length when \(Duty\) and \(\Delta h\) are chosen as 0.8 \(\upmu\)m and 4.0 \(\upmu\)m at the same time. As \(P\) increases, the peak reflectivity and \(\Delta\lambda_{b}\) increases. When \(P\) is 400.0 \(\upmu\)m, \(\Delta\lambda_{b}\) achieves 5.3 nm and it meets our requirements. However, the side-lobe suppression ratio (SLSR) can be calculated as 5.6 dB, which means the side-lobes are large. There is a great need to reduce the side-lobes. In order to suppress side-lobes and enlarge the SLSR, traditional apodized grating is used to improve the reflection spectrum. In the design of traditional apodized grating, the apodization function is often realized by the height modulation of ridges. The height of ridge is almost zero at the input and output of grating, meanwhile the height has a maximum value at the center of grating. The height of ridge is usually adopted as the Gaussian apodized function, and then the refractive index modulation \(\sigma_{0}\) is realized. The expression of \(\sigma_{0}\) in Equation (10) is related to the location of the grating, which means that \(\sigma_{0}\) should be written as \(\sigma(z)\). The expression of \(\sigma(z)\) is listed as Equation (11). \[\sigma(z)\ =\ \sigma_{0}\ \exp[-B(z-P/2)^{2}/P^{2}] \tag{11}\]In this expression, \(B\) is usually adopted as 10. For calculating the apodized grating, \(\sigma_{0}\) is replaced by \(\sigma(z)\) in Equation (10). With the calculation of Equations (10) and (11), the reflection spectrum of the traditional apodized grating is displayed as Figure 5a. The traditional apodized Bragg grating can improve the SLSR to 22.1 dB on one side of the spectrum and the other side of SLSR hardly decreases (4.1 dB). In addition, fabricating the apodized grating by conventional apodized phase mask and electric arc scanning apodization approach should ensure a precise control of experimental instrument and the fabrication process is complex. Based on the condition, the method of changing \(\Delta S\) of ridges is proposed. The \(\Delta S\) and coupling coefficient \(\kappa(z)\) of the grating are listed as Equation (12). \[ \begin{array}{c}\Delta S(z)\ =\ \Delta S_{0}\ \exp\left[-B(z-\frac{p}{z})^{2}/P^{ 2}\right]\\ \kappa(z)\ =\ \frac{1}{2}k_{0}\big{|}1-\exp(i\pi\cdot(1-2\Delta S/\Lambda))\big{|}\ =\ k_{0}\sin(\frac{\pi\Delta S}{\Lambda})\end{array}  \tag{12}\]  
Figure 5b shows the relationship of \(\kappa\) and the location \(z\) of the grating by calculating with Equation (12). It can be found that \(\kappa\) has a larger value of 200.5 mm\({}^{-1}\) at the center of the grating and exhibits an obvious decrease at both edge sides of the grating. As the apodized coefficient \(B\) increases, \(\kappa\) also decreases at both sides of the grating.