# Edge Couplers in Silicon Photonic Integrated Circuits: A Review
## 3 Structural Transformations of Edge Couplers in Horizontal Direction
### Edge Couplers Based on a Multi-Tip Taper  
The coupling efficiency of a fiber-to-chip edge coupler can be simplified as the product of two factors: that is, the modal overlap between the fiber mode and the superimposed mode at edge coupler facet (\(\eta_{1}\)) and the mode conversion efficiency of the edge coupler (\(\eta_{2}\)). The definition of modal overlap efficiency \(\eta_{1}\) can be expressed as  
\[\eta_{1}=\frac{\left|\int E_{1}*E_{2}\,\mathrm{dA}\right|^{2}}{\int\left|E_{1} \right|^{2}\mathrm{dA}*\int\left|E_{2}\right|^{2}\mathrm{dA}} \tag{1}\]  
where A is the modal distribution area, while \(E_{1}\) and \(E_{2}\) stand for the electric fields of the fiber mode and the superimposed mode at the coupler facet separately [55]. Since the total coupling efficiency of an edge coupler is determined by both \(\eta_{1}\) and \(\eta_{2}\), it is intuitive to improve the overall coupling efficiency by increasing either \(\eta_{1}\), \(\eta_{2}\), or both of them.  
The multi-tip taper is widely applied to improve the modal overlap efficiency [56; 57; 58], and the schematic of a multi-tip taper is presented in Figure 5a. A superimposed mode can be formed at the taper facet of the single taper, which consists of multiple tips, and the modal field diameter (MFD) is thus increased to better match with the fiber mode regarding the aspect of modal size. The design parameters of a multi-tip taper include the number of tips, interval distance between adjacent tips, taper tip width, and taper length, which can be properly designed to achieve high modal overlap in different situations, and the shape of the multi-tip taper may resemble a fork or a comb. In the case of fiber-to-chip coupling, the edge coupler facet should be aligned with the fiber core accurately to ensure fine overlap; thus, coupling efficiency is highly sensitive to alignment condition. Since it is a set of multiple tips as a whole that interconnects with the fiber, the dependence on misalignment and fabrication deviation reduces, considering each single tip in a multi-tip taper coupler. A multi-tip taper with multiple tips can loosen the constraints of alignment and reach larger misalignment tolerance as well as fabrication deviation tolerance.  
Figure 4: The relationship between inverse taper length and (**a**) spot size and (**b**) coupling loss. Reproduced with permission from [54], Copyright Elsevier, 2011.  
The utilization of multiple tips can introduce a high degree of design freedom to realize the expected coupling efficiency within a notably small footprint. The total length of a multi-tip taper is usually below 100 \(\upmu\)m [59; 60]. As depicted in [60], an edge coupler on the basis of double-tip inverse tapers is designed, fabricated, and characterized. It reaches a low coupling efficiency of 1.1 dB for the TE mode and 1.52 dB for the TM mode respectively at 1550 nm within an ultra-compact size of 40 \(\upmu\)m. A narrow slot is formed between two adjacent tapers, and the tip width of the slot is typically the feature size of the entire structure, at a value of several tens of nanometers. On this basis, a multi-tip taper is commonly prepared via electron beam lithography (EBL) patterning, which is highly precise and convenient with a single-step process at the expense of being time-consuming and expensive. In addition to general applications in fiber-to-chip optical coupling, multi-tip tapers with flexible tip width and tip numbers can also work under chip-to-chip communication [56] or laser-to-chip optical interconnect [59].