# High-Tolerance Grating Couplers for Vertical Backside Coupling
## 3 Results and Discussion
### Optimization of Grating Coupler
The schematic representation of the focusing 3D GC coupled from the backside is shown in Figure 5a, where L\({}_{t}\) represents the length of the so-called taper part (unetched region of GC), L\({}_{d}\) denotes the total length of the grating region (etched region), while the dimensions of the focusing GC are dictated by the taper angle \(\theta_{t}\). In 2D-FDTD simulations, the beam propagation is not considered along the z-direction. Consequently, there is a discrepancy between the simulated results from 2D and 3D GCs, especially in asymmetric structures [25]. In order to obtain a higher CE at  
Figure 4: (a) A 2D simulated electric field profile within the GC, for a fiber-to-chip coupling from the backside. (b) Transmission spectra for each direction of the diffracted light from the GC, (which can be interpreted as diffraction efficiency). 1550 nm, we thus ran PSO algorithms on the initial 3D-focused GC structure. Moreover, the 3D GC was also optimized in a way to minimize the mode mismatch between the light's mode inside GC and the fundamental TE mode of the output waveguide. After optimization, the parameters were as follows: P1 = 369 nm, P2 = 200 nm, P3 = 333 nm, P4 = 200 nm, T1 = 600 nm, T2 = 354 nm, T3 = 200 nm, T4 = 385 nm, Lt = 20 \(\upmu\)m, Ld = 22.52 \(\upmu\)m, Lf = 26.4 \(\upmu\)m, and \(\theta_{\text{t}}=30^{\circ}\). The parameter N represents the number of grating periods, set to nine in the 3D simulation to ensure that the final GC footprint (GC total footprint = 48.92 \(\upmu\)m) fitted well on the fiber's facet (diameter = 120 \(\upmu\)m). This allowed the input light to efficiently transmit between two opposite cores of a four-core MCF (core spacing = 70.71 \(\upmu\)m) while also providing enough space for additional necessary PIC components, such as bent waveguides and ring resonators for sensing. The incident light entered vertically from the bottom MCF into the binary blazed grating area and was diffracted into the left output waveguide when the Bragg condition was met. Figure 5b shows the top view of the electric field profile at 1550 nm, while using the fundamental z-polarized mode from fiber's core (SM--4C1550) as the incident light from the backside of the GC. The figure illustrates the effective coupling of light into the output left waveguide via the compact 3D GC. As the output, a rib waveguide with an etching depth of 70 nm was used, while the width of the waveguide was set to 600 nm to minimize transmission losses [26]. Figure 5c shows the directionality of the system. The CE at the output port shows a slight decrease compared to the 2D case, yet it still reached 40% at the wavelength of 1550 nm, while transmission and reflection stood at approximately 38% and 17%, respectively. This GC exhibited sufficient performance for future applications in fiber-tip sensing, if combined, for example, with ring resonators within a single transferred PIC. Indeed, we optimized in parallel a ring resonator to be coupled with the rib waveguide in between two GCs. Its design and therefore performance rely mostly on it having the same etching depth as the GCs, and its footprint (must fit on non-patterned area on fiber's facet). After  
Figure 5: (**a**) Sketch of the 3D-focusing GC structure. (**b**) The top-view E-field profile of a 3D-focusing GC from the 3D-FDTD simulation. (**c**) Directionality of transmission in the 3D-focusing GC. optimization, a Q-factor value between 2000 and 3000, and a bandwidth at 1550 nm of 20 nm were achieved, matching the values of one of the GCs. The use of higher etching depth as well as more exotic ring shapes (adiabatic widening, Bezier curves, etc.) could drastically improve ring resonator's performances, however at a cost of complexity of fabrication, thus making our fiber-tip sensors less mass-producible.