# High-Tolerance Grating Couplers for Vertical Backside Coupling
## 2 Materials and Methods  
### Grating Coupler Design  
To efficiently couple light from fiber to chip, it is necessary to minimize the mode mismatch between the optical fiber mode (Gaussian) and the diffracted light from the grating. Achieving efficient coupling heavily relies on meeting the Bragg condition, also known as the phase matching condition (PMC). To efficiently couple light of a specific wavelength into the GC, its period \(\Delta\) must be set at a specific value to achieve Bragg diffraction. This grating period \(\Delta\) can be defined as:  
\[\Delta=\frac{m\lambda}{\left(N_{eff}-n_{u}\times\sin\theta\right)} \tag{1}\]  
where \(m\) is the diffraction order of the GC, \(n_{u}\) represents the refractive index of the surrounding medium from which light is incident (here silica), \(N_{eff}\) represents the average effective refractive index of the waveguide in the grating region [22], and \(\lambda\) represents the wavelength of the incident light. In this work, since light is incident perfectly vertically from the backside of the GC, the angle of incidence \(\theta\) is equal to zero. Therefore, Equation (1) can be simplified to:  
\[\Delta=\frac{m\lambda}{N_{eff}} \tag{2}\]  
For the design, we consider a 220 nm thick silicon membrane; however, a similar approach and geometry can also be applied to InP membranes [3]. Figure 2 shows the 2D schematic diagram of the backside vertical incidence GC design based on the binary blazed GC geometry. The GC is placed on top of a multi-core optical fiber (MCF: SM--4C1550; NA = 0.14-0.17; n\({}_{\text{core(1550 nm)}}\) = 1.4508-1.4540; n\({}_{\text{cladding(1550 nm)}}\) = 1.444; MED = 7.5 um-8.4 um; Fibercore company, University Parkway, Southampton, UK), with air above it (see Figure 3). In order to ensure the mechanical stability of the GC while it is suspended during the transfer [3], the designed GC features a shallow etch of 70 nm, retaining Si with a thickness of e = 150 nm. The binary blazed grating type of GC includes asymmetric sub-grating structures. In a grating period \(\Lambda\), the designed GC has four sub-gratings with different fill factors (simulations showed that three or five sub-gratings are less efficient), and the widths of the pitches and trenches are expressed as P1-P4 and T1-T4, respectively. The distance from the center of the fiber's core to the starting point of the grating region is defined as L\({}_{\text{f}}\). When the GC is symmetric, light is evenly diffracted into both sides of the waveguide [13]. Therefore, by implementing an asymmetric structure for the GC and optimizing it through PSO, it is possible to increase the output light intensity of the left waveguide while preventing light from undergoing negative-order diffraction (into the right side waveguide on Figure 2). A high fabrication tolerance can be achieved by using a larger minimum feature size. However, to satisfy first-order Bragg diffraction, the grating period \(\Lambda\) of the binary blazed GC is typically set to 500 nm [15]. This also means that for a minimum feature size of the four sub-gratings greater than 200 nm, second-order diffraction in the GC must be utilized, resulting in a total grating period exceeding 1.6 \(\upmu\)m.