# Grating-Assisted Fiber to Chip Coupling for SOI Photonic Circuits
## 3 Factors Affecting Grating Loss
### Mode Overlap  
The other principal source of loss in a generic uniform surface grating is that of mode mismatch between the fundamental fiber mode and that of diffracted grating field. The fiber mode is Gaussian while a uniform grating has an exponentially decaying power profile. It can be seen that coupled power \(P_{CE}\) depends on \(\eta\)[16] which is expressed as \(\int_{S}|E_{gr}\times H_{fib}^{*}dS|^{2}\), where \(E_{gr}\) is the grating field and \(H_{fib}\) is the fiber mode field and \(S\) corresponds to the surface of the fiber facet. The field overlap can be improved through implementing non-uniform grating couplers by way of apodization of individual grating scatterers. The power leakage profile of a uniform grating is expressed as:  
\[P(x)=P_{0}e^{-2\alpha x} \tag{5}\] \[\frac{dP}{dx}=-2\alpha P(x) \tag{6}\]  
where \(\alpha\) is the scattered field leakage coefficient described previously and is constant for a uniform grating and \(x\) is the propagation direction. For an apodized grating, the radiated field resembles a Gaussian, so the power leakage can be expressed as:  
\[\frac{dP}{dx}=-G^{2}(x) \tag{7}\]  
where \(G(x)\) corresponds to the Gaussian distribution of the emitted field. Integrating this equation we obtain:  
\[P(x)=-\int_{x_{0}}^{x}G^{2}(x)+C_{0}^{2} \tag{8}\]  
Figure 4: Two prominent means of suppressing substrate leakage (**a**) surface gratings with a distributed Bragg reflector (DBR) composed of alternating layers of \(Si/SiO_{2}\) and (**b**) with overlay gratings. In (**c**), the diffracted electric field intensity profile in log scale is plotted from 2-D FDTD simulations, in which the grating coupler is excited from fundamental mode in slab waveguide, in the three cases of \(Si\)-substrate, DBR substrate and \(Si\) overlays.  
Now substituting the variable power leakage \(dP(x)/dx\) in Equation (7) with the one in Equation (6) and assuming the field leakage parameter \(\alpha\) as a function of \(x\), we obtain:  
\[2\alpha(x)P(x)=G^{2}(x) \tag{9}\]  
and now substituting value of \(P(x)\) in Equation (9) with the one derived in Equation (8), we obtain:  
\[\alpha(x)=\frac{G^{2}(x)}{2(C_{0}^{2}-\int_{0}^{x}G^{2}(s)ds)} \tag{10}\]  
where \(C_{0}\) is a constant related to the electric field strength at the start of grating. This variation of power leakage of individual grating elements can be accomplished by modulation of fill factor or periodicity along propagating direction (as illustrated in Figure 5). On shallow etched platform, non-uniform grating couplers with coupling loss of -0.62 dB [13] and -1.9 dB [17] have been shown with and without bottom mirrors respectively. The choice of fiber tilt angle of illumination \(\theta\), is highly design and application specific. It is possible to have very high efficiency non-uniform couplers with both shallow [18] as well as deep angles of incidence [19]. It may be pointed out that, for non-uniform designs, the modulation of individual grating strength results in a higher effective grating index than uniform couplers. This results in non-uniform couplers having a reduced optical bandwidth of operation when compared to the latter. Furthermore, an optimal apodized non-uniform grating design may require feature sizes ranging from a few tens to a few hundreds of nm in dimension. The coupler design thus may be limited by the patterning processes such as electron beam or deep UV lithography.