# Grating-Assisted Fiber to Chip Coupling for SOI Photonic Circuits
## 4 Subwavelength Gratings  
In recent years, a new class of grating couplers based on subwavelength structures have emerged. Basically, a subwavelength grating (\(SWG\)) is a periodically interleaved structure composed of low and high index materials which due to its subwavelength dimension and depending on incident polarization, behaves like a homogeneous medium. Although the concept of subwavelength structures behaving as homogeneous media was known for some time, it has only recently found wide ranging applications in silicon photonics. This is primarily due to the technological advances in various lithographic processes that allow patterning of subwavelength geometries with a relatively high level of accuracy and repeatability [21]. Use of \(SWG\) structures allows for artificial engineering of a wide range of optical indices from high contrasting materials. SWG based structures have also been demonstrated for various other integrated applications like grating polarizers, multimode interference splitters and ring resonators [22; 23]. When it comes to surface grating couplers, a major drawback of shallow etched designs is the difficulty in tuning grating effective index (as observed from Equation (11)) which demands critical control over etch depth. SWG based couplers, however, allow for greater flexibility in design, since the structures are fully etched and waveguide index depends only on the subwavelength period and fill factor. As per the zeroth order approximation [24], the refractive index of the composite medium consisting of \(Si\) and \(air\) for transverse electric (TE) and transverse magnetic (TM) polarizations is expressed as:  
\[\frac{1}{n_{0}^{TE}} = \Big{(}\frac{f_{y}}{n_{air}^{2}}+\frac{(1-f_{y})}{n_{Si}^{2}} \Big{)}^{\frac{1}{2}} \tag{12}\] \[n_{0}^{TM} = \big{(}f_{y}n_{air}^{2}+(1-f_{y})n_{Si}^{2}\big{)}^{\frac{1}{2}} \tag{13}\]  
where \(\Lambda_{y}\) and \(f_{y}\) are the lateral subwavelength periods and fill factors respectively. When the ratio \(P=n_{eff}^{TE/TM}\Lambda_{y}/\lambda_{0}\) (\(n_{eff}^{TE/TM}\) being the effective TE or TM slab mode index) approaches 1, the second order approximation of effective medium theory is considered. \[n_{2}^{TE} =n_{0}^{TE}\Big{(}1+\frac{\pi^{2}}{3}P^{2}f_{y}^{2}(1-f_{y})^{2}(n_ {Si}^{2}-n_{air}^{2})^{2}(\frac{n_{0}^{TM}}{n_{eff}^{TE}})^{2}\big{(}\frac{n_{ 0}^{TE}}{n_{air}n_{Si}}\big{)}^{4}\Big{)}^{1/2} \tag{14}\] \[n_{2}^{TM} =n_{0}^{TM}\Big{(}1+\frac{\pi^{2}}{3}P^{2}f_{y}^{2}(1-f_{y})^{2} \big{(}\frac{n_{Si}^{2}-n_{air}^{2}}{n_{eff}^{TM}n_{0}^{TM}}\big{)}^{2}\Big{)} ^{1/2} \tag{15}\]  
Figure 6a,b, illustrate the cross sections of a typical \(SWG\) composed of \(Si\) and \(air\), where the olive green regions signify the subwavelength trenches. Using the above equations, the polarization dependent subwavelength indices for \(Si\) and \(air\) are plotted in Figure 6c. Depending on the requirement of enhanced coupling or bandwidth, different kinds of SWG based couplers (either uniform or non-uniform) can be fabricated (as shown in Figure 7). As mentioned earlier in the case of shallow etched couplers, ideal coupling performance is achieved with a combination of high directionality as well as mode overlap. This was demonstrated in [25] where researchers achieved a coupling loss of only -0.58 dB using \(SWG\) with an aluminium mirror and measured a 3 dB bandwidth of 71 nm (see Figure 8). Table 1 summarizes the state of art of different kinds of experimentally demonstrated SOI based fiber to chip grating couplers. Figure 6: **(a)** A typical subwavelength grating (\(SWG\)) layout. The subwavelength index is decided by the period width \(\Lambda_{y}\) and the dark blue trench width \(f_{y}\Lambda_{y}\) (**b**) the side view cross-section of an _air-Si SWG_ grating on SOI. **(c)** Variation of \(SWG\) index with fill factor of air on a 220 nm thick SOI. Figure 7: Scanning electron microscope (SEM) image of four different kinds of \(SWG\) couplers fabricated on SOI (inset shows magnified image).