# Edge Couplers in Silicon Photonic Integrated Circuits: A Review
## 3 Structural Transformations of Edge Couplers in Horizontal Direction  
### Edge Couplers Based on Inverse Taper with Nonlinear Profile  
An edge coupler based on one standard single inverse taper is the most basic and straightforward formation for all various types of edge couplers. When the thickness of the Si waveguide is constant while the width decreases gradually toward the fiber, the Si strip waveguide becomes a taper. Light cannot be confined properly within the narrow taper end. Thus, the mode is distributed in a larger area around the taper, which contributes to an enlarged modal size. As light emitting from the fiber is coupled into the Si taper and continues to travel along the widening taper, the mode tends to be better confined by the larger cross-section and can propagate with ultra-low loss into successive devices. As presented in Figure 3a, the linear profile is the most common form considering the contour of the inverse taper, but it may be not the most appropriate form to achieve the best performance. The main advantage of a linear-taper coupler is its simple structure and ease of fabrication, while this structural simplicity also leads to an extremely large size and limited coupling efficiency, especially for optical coupling to fibers with a large spot size. Research studies on different profiles of tapers have been conducted to realize better performance compared with linear-shape tapers in the aspects of smaller footprint, lower propagation loss, and broader bandwidth. Common transformations of linear tapers include multi-sectional tapers [49; 51; 52], parabolic tapers or quadratic tapers [53], and exponential tapers [54], as shown in Figure 3b-d separately. The mode propagation loss is dependent on multiple factors including taper length, taper tip width, and the slope of the contour profile. Generally, the mode spot size becomes smaller as the taper length increases when the widths at each facet are fixed, as shown in Figure 4a. This is attributed to the stronger mode confinement along with the larger cross-section of the inverse taper when light propagates. The modal distribution can be characterized by the spot size and narrow taper tip, while inadequate mode confinement contributes to a large spot size, since a major percentage of power is distributed in the surrounding lower cladding and upper cladding. When the inverse taper length increases and the cross-section is large enough to support the entire mode, most of the power is confined inside the silicon waveguide, and the spot size decreases. From Figure 4b, we can find that when the taper length is long enough over 200 \(\upmu\)m, the coupling efficiency tends to slowly decrease at a constant rate, and length is not the critical factor that influences the performance of the taper. In this circumstance, tapers with different profiles have almost the same coupling efficiency, and a linear-shape inverse taper will be the best choice with the easiest fabrication. Otherwise, parabolic tapers and exponential tapers with proper design can achieve a lower coupling loss and better misalignment tolerance within a smaller length [49; 54]. In addition, the mode can propagate adiabatically without excitation of higher-order modes when the slope of a taper profile is gentle enough. In the case of multi-sectional tapers [51], the enveloping contour is divided into multiple discrete steps. Scattering loss decreases as the step size is reduced, and a multi-sectional contour line with a small enough step size can be regarded as equivalent to a continuous straight contour line. Based on the inspiration of a multi-sectional taper, it has the potential to divide the contour line into multiple sections and optimize the mathematical expression of a taper profile in each section. Such division brings about a high degree of design freedom to the basic inverse taper, which can contribute to high performance for an edge coupler based on one single inverse taper.