Power splitting is a one of the most fundamental application for DCs and has been playing an important role for various photonic integrated devices like MRRs. For MRRs, the coupling ratio is usually chosen according to the requirement for the 3 dB bandwidth. However, the regular DCs might not be a good option for some specific cases, even though it is easy to realize an arbitrary power splitting ratio (or coupling ratio) by choosing the length of the coupling region. For example, when MRRs are used as an optical filter, a box-like filtering response is often desired so that it can tolerate a wavelength shift due to any environmental change. This response for optical filters based on MMRs can be synthesized with multiple rings [
52–
55]. When using multiple microrings to achieve a box-like filtering response, all the couplers involved should be designed optimally to achieve the coupling coefficients as desired [
56]. For example, for the ultra-compact 5th order MRR optical filters demonstrated in Ref. [
57], the power coupling ratios
k2 for all the couplers are chosen as 0.45, 0.09, 0.05, 0.05, 0.09, and 0.45, respectively. In order to achieve the sufficient power coupling coefficient as high as 0.45, one usually has to use the design of race-track resonators with long coupling region or narrow gap
wgap in the coupling region. However, a longer coupling region causes a smaller free-spectral range (FSR), which limits the channel number available in wavelength-division-multiplexing (WDM) systems. When choosing a narrow gap (e.g., ~20 nm), the beating length becomes small. However, the fabrication (e.g., the lithography and etching processes) becomes difficult and the coupling ratio is very sensitive to the beating length variation [
57]. In order to overcome these issues, MMI couplers were used as the coupler with a power ratio of 45%:55% in Ref. [
57]. One should realize that the design with MMI couplers has some disadvantages. First, an MMI coupler usually has a few percent of excess loss due to the non-perfect self-imaging in the multimode section [
57], and there is also mode conversion loss between the straight section and the bending section of a microring (particularly when the bending radius is ultrasmall). As a result, some notable excess loss will be introduced for an MRR filter. Second, the power splitting ratio of an MMI coupler is fixed [
58] and thus it is not flexible to choose any other power splitting ratios in order to achieve a desired 3 dB bandwidth. Third, the length
LMMI of the MMI section is usually several microns (e.g.,
LMMI=~3.5 mm [
57]), which makes the cavity length increased by 2
LMMI and thus reduces the FSR. These problems can be solved by using bent DCs, as shown in Fig. 1(a) [
59].