# Grating-Assisted Fiber to Chip Coupling for SOI Photonic Circuits
## 5 Lowering Footprint
### Focusing Gratings  
One way to reduce the tapering footprint is by incorporating focusing gratings [32]. This approach involves curving the individual grating lines in order to achieve in-plane focusing of incident plain wavefront to a single spot. Each of these grating lines behaves as a series of confocal ellipses where one of the two focal points varies with the grating number and the other point is fixed. The fixed point is typically at the starting of a wire waveguide. Focusing grating lines can be defined by modifying the phase matching condition in Equation (2) as:  
\[p\lambda=n_{g}\sqrt{x^{2}+y^{2}}-xn_{c}sin\theta \tag{16}\] \begin{table}
\begin{tabular}{l c c c c c c c} \hline
**SOI** & **Max** & **Bandwidth** & \(\lambda_{max}\) & **Uniform (U)/** & **Fiber Tilt** & **Etch** & **Bottom** \\
**Coupler** & **CE (-dB)** & **3 dB (nm)** & **(\upmu)** & **Apodized (A)** & **Angle (\theta)** & **Type** & **Reflector** \\ \hline Ref [26] & 3 & 58 & 1.31 & U & 12 & Shallow & N \\ Ref [27] & 2 & 50 & 1.302 & A & 17 & Double-etch & N \\ Ref [28] & 1.9 & 23 (1 dB) & 1.31 & A & 10 & Double-etch & N \\ Ref [29] & 1.05 & 50 & 1.551 & A & 8 & Shallow & N \\ Ref [11] & 1.6 & 63 & 1.515 & U & 10 & Shallow & Y \\ Ref [10] & 1.6 & 60 & 1.54 & U & - & Shallow & Y \\ Ref [15] & 1.6 & 80 & 1.53 & U & 13 & Shallow & N \\ Ref [17] & 1.9 & 43 (1 dB) & 1.524 & A & 15 & Shallow & N \\ Ref [18] & 1.2 & 45 & 1.533 & A & 10 & Shallow & N \\ Ref [13] & 1.08 & 42 (1 dB) & 1.551 & U & 9 & Shallow & Y \\ Ref [13] & 0.62 & 40 (1 dB) & 1.531 & A & 11 & Shallow & Y \\ Ref [30] & 2.5 & - & 1.325 & A & 33 & Full & N \\ Ref [31] & 3.2 & 36 (1 dB) & 1.59 & A & -31 & Full & N \\ Ref [19] & 2.16 & 64 & 1.543 & A & 27 & Full & N \\ Ref [25] & 0.58 & 71 & 1.56 & A & 13 & Full & Y \\ \hline \end{tabular}
\end{table} 
Table 1: Summary of various Silicon-on-Insulator (SOI) based fiber to chip grating couplers that include uniform as well as non-uniform (apodized) designs. Full etch signifies \(SWG\) couplers. CE refers to the coupling efficiency. Figure 8: Highly efficient grating with an aluminium bottom reflector with a coupling efficiency of -0.58 (Image reprinted with permission from [25]). where \(x\) is the propagation and \(y\), is the lateral axis (see Figure 9a) and \(n_{c}\) is the cladding index. \(p\) corresponds to the number of grating line. The above expression is valid when \(n_{g}\) is approximated to the fundamental effective slab mode index which may not be feasible for a deep etch. For that reason a further optimization using 3-D simulations or a design sweep of gratings with different focal lengths and periods is required. Several photonic design kits (PDKs) are now available for making design layouts for fabrication of integrated photonic components such as IPKISS and OptoDesigner (from PhoeniX). We have designed the focusing gratings using IPKISS which is a python based design tool from Luceda Photonics Inc. Figure 9 shows some of the fabricated focusing gratings. The simulated and measured coupling efficiencies for TE and TM gratings are plotted in Figure 10. For TE gratings, peak simulated fiber to chip coupling efficiency is -3.12 dB at 1.562 \(\upmu\)m with a 3 dB bandwidth of 75 nm at 640 nm period, 50% fill factor and 15\({}^{\circ}\) inclination angle. The experimentally observed peak efficiency for the same parameters with focusing gratings is -3.6 dB with a 57 nm 3 dB bandwidth at a focal length of 24 \(\upmu\)m. For TM gratings with 980 nm period, and similar fill factor and inclination, the corresponding simulated and experimentally observed figures are -3.84 dB at 1.571 \(\upmu\)m and -5.25 dB at 1.583 \(\upmu\)m respectively for a focal length of 20 \(\upmu\)m. 3 dB bandwidth for simulation and experiments are nearly similar at 72 and 65 nm respectively. TM gratings in general exhibit lower coupling efficiency compared to TE ones. This can be attributed to the lower effective index of the TM fundamental mode which in turn results in a low average grating index and thus low optical confinement of the guided mode.