Figure 1 illustrates the schematic configuration of the integrated CSLR resonator. It consists of N SLRs (SLR1, SLR2, …, SLRN) formed by a self-coupled nanowire waveguide loop. In the CSLR resonator, each SLR performs as a reflection/transmission element and contributes to the overall transmission spectra from port IN to port OUT in Fig. 1. Therefore, the cascaded SLRs with a periodic loop lattice show similar transmission characteristics to that of photonic crystals [25, 27]. The two adjacent SLRs together with the connecting waveguide form a FP cavity, thereby N cascaded SLRs can also be regarded as N-1 cascaded FP cavities (FPC1, FPC2, …, FPCN−1), similar to Bragg gratings [16, 28, 29]. To study the CSLR resonator based on the scattering matrix method (SMM) [30–32], we define the waveguide and coupler parameters of the CSLR resonator in Table I. The large dynamic range in engineering the transmittance and reflectivity of individual SLRs via changing ti or κi makes the CSLR resonator more flexible for spectral engineering as compared with Bragg gratings. On the other hand, the transmission spectra of the CSLR resonators can also be tailored by changing φi (i = 1, 2, .., N-1), i.e., the phase shifts along the connecting waveguides. The freedom in designing ti (i = 1, 2, .., N) and φi (i = 1, 2, .., N-1) is the basis for flexible spectral engineering based on the CSLR resonators, which can lead to versatile applications.
CSLR resonators with two SLRs (N = 2) can be regarded as single FP cavities without mode splitting [23, 26]. Here, we start from the CSLR resonators with three SLRs (N = 3). The calculated power transmission spectra and group delay spectra of the CSLR resonators with three SLRs (N = 3) are depicted in Fig. 2. The structural parameters are chosen as follows: Ls1 = Ls2 = Ls3 = 129.66 μm, and L1 = L2 = 100 μm. For single-mode silicon photonic nanowire waveguides with a cross-section of 500 nm × 260 nm, we also assume that the waveguide group index of the transverse electric (TE) mode is ng = 4.3350 and the power propagation loss factor is α = 55 m-1 (2.4 dB/cm) based on our previously fabricated devices. The same ng and α are also employed for the calculations of other transmission and group delay spectra in this section. It is clear that different degrees of mode splitting can be achieved by varying t2. As t2 decreases (i.e., the coupling strength increases), the spectral range between the two adjacent resonant peaks decreases until the split peaks finally merge into one. By further decreasing t2, the Q factor, extinction ratio, and group delay of the combined single resonance increases, together with an increase in the insertion loss. In particular, when t2 = 0.77, a band-pass Butterworth filter with a flat-top filter shape can be realized, which is desirable for signal filtering in optical communications systems. On the other hand, when t2 = 0.742, the CSLR resonator exhibits a flat-top group delay spectrum, which can be used as a Bessel filter for optical buffering.
Figure 3(a) shows the calculated power transmission spectra of the CSLR resonators with different numbers of SLRs (N). It can be seen that as N increases, the number of split resonances within one FSR also increases. For a CSLR resonator consisting of N SLRs, the maximum number of split resonances within one FSR is N-1. In Fig. 3(b), we plot the calculated power transmission spectra of the CSLR resonator (N = 8) for different t1 = t2 =…= t8. As ti (i = 1, 2, …, 8) increases (i.e., the coupling strengths decrease), the bandwidth of the passband also increases, together with a decrease in insertion loss. In principle, the bandwidth of the passband is limited by the FSR of the CSLR resonator. The filter in Figs. 3(c) and (d) is designed for enhanced light trapping by introducing an additional π/2 phase shift along the centre FPC (i.e., L4 for N = 8), which is similar to enhancing light trapping in photonic crystals by introducing defects [25]. With enhanced light trapping, there are increased time delays and enhanced light-matter interactions, which are useful in nonlinear optics and laser excitation [27]. In Fig. 3(c), one can see that there are central transmission peaks induced by an additional phase shift along L4, which correspond to a group delay 2.1 times higher than that of the CSLR resonator without the additional phase shift in Fig. 3(d). This group delay can be increased further by using more cascaded SLRs. The filter in Fig. 3(e) is an 8th-order Butterworth filter with a flat-top filter shape. Figure 3(f) shows the designed optical filter with multiple transmission peaks in the spectrum. Each transmission peak has a high extinction ratio over 10 dB.