Figure 1 illustrates the schematic configuration of the integrated CSLR resonator. It consists of N SLRs (SLR1, SLR2, …, SLRN) formed by a selfcoupled 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.17 The two adjacent SLRs together with the connecting waveguide form a FP cavity, thereby N cascaded SLRs can also be regarded as N1 cascaded FP cavities (FPC1, FPC2, …, FPCN−1), similar to Bragg gratings.30, 31 To study the CSLR resonator based on the scattering matrix method,32–33 we define the waveguide and coupler parameters of the CSLR resonator in Table I, and so the field transmission function from port IN to port OUT can be written as:
TABLE I. Definitions of waveguide and coupler parameters of the CSLR resonator
Waveguide

Length

Transmission factora

Phase shiftb

waveguide connecting
SLRi to SLRi+1 (i = 1, 2, .., N1)

Li

ai

φi

Sagnac loops in
SLRi (i = 1, 2, .., N)

Lsi

asi

φsi

Coupler

Coupling lengthc

Field transmission coefficientd

Field crosscoupling
coefficientd

couplers in SLRi (i = 1, 2, .., N)

Lci

ti

κi

a ai = exp(−αLi /2), asi = exp(−αLsi /2), α is the power propagation loss factor.
b φi = 2πngLi /λ, φsi = 2πngLsi /λ, ng is the group index and λ is the wavelength.
c Lci (i = 1, 2, .., N) are the straight coupling lengths shown in Fig.1. They are included in Li.
d In our calculation, we assume ti2 + κi2 =1 for lossless coupling in all the directional couplers.
In Eq. (1), Tsiand Rsi(i = 1, 2, .., N) denote the field transmission and reflection functions of SLRi given by:
T i (i = 1, 2, .., N1) represent the field transmission functions of the waveguide connecting SLRi to SLRi+1, which can be expressed as:
For the CSLR resonators implemented by SLR1, SLR2, …, and SLRi (i =1, 2, .., N), TCSLR (i) are the field transmission functions, RCSLR+(i) and RCSLR−(i) are the field reflection functions for light input from left and right sides, respectively, which can be given by:
In Eqs. (2) and (3), it can be seen that the transmittance and reflectivity of the SLRi depend on the ti, In terms of practical fabrication, the ti can be engineered by changing the coupling length Lci. The large dynamic range in the transmittance and reflectivity of individual SLRs that can be engineered by changing ti or κi makes the CSLR resonator more flexible for spectral engineering as compared with Bragg gratings. On the other hand, according to Eq. (4), the transmission spectra of the CSLR resonators can also be tailored by changing φi (i = 1, 2, .., N1)  i.e., the phase shifts along the connecting waveguides. The freedom in designing ti (i =1, 2, .., N) and φi (i = 1, 2, .., N1) is the basis for flexible spectral engineering based on the CSLR resonators, which can lead to versatile applications. In Eqs. (5) and (6), RCSLR+(i) equals to RCSLR−(i) only when SLR1, SLR2, …, and SLRi are identical, i.e., when SLR1, SLR2, …, and SLRi are not identical, there are nonreciprocal reflections from the CSLR resonators for light input from different directions. These nonreciprocal reflections are induced by different losses within the resonant cavity  the CSLR resonators themselves will still have reciprocal transmission for light input from different directions.
CSLR resonators with two SLRs (N = 2) can be regarded as single FP cavities without mode splitting.23,29 Here, we start from the CSLR resonators with three SLRs (N = 3). Based on Eqs. (1) – (6), 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 singlemode silicon photonic nanowire waveguides with a crosssection of 500 nm × 260 nm, we use values based on our previously fabricated devices for the waveguide group index of the transverse electric (TE) mode (ng = 4.3350) and the propagation loss (α = 55 m1 (2.4 dB/cm)). The same values of ng and α are also used for the calculations of other transmission and group delay spectra in this section. The calculated power transmission spectra of the CSLR resonator (N = 3) for various t2 when t1 = t3 = 0.87 are shown in Fig. 2(a). The corresponding group delay spectra are shown in Fig. 2(b). 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 bandpass Butterworth filter34 with a flattop filter shape can be realized, which is desirable for signal filtering in optical communications systems.35,36 On the other hand, when t2 = 0.742, the CSLR resonator exhibits a flattop group delay spectrum, which can be used as a Bessel filter for optical buffering.37,38 When t2 =\(\sqrt{\text{1/2}}\), SLR2 works as a total reflector, and so there is null transmission for the CSLR resonator. The same goes for t1 =\(\sqrt{\text{1/2}}\) or t3 =\(\sqrt{\text{1/2}}\). Figure 2(c) shows the calculated power transmission spectra of the CSLR resonator (N = 3) for various t1 = t3 when t2 = 0.97. The group delay spectra are depicted in Fig. 2(d) accordingly. One can see that decreasing t1 and t3 (i.e., enhancing the coupling strengths) results in increased Q factor, extinction ratio, and group delay, at the expense of an increase in the insertion loss. The sharpening of the filter shape can be attributed to coherent interference within the coupled resonant cavities, which could be useful for implementation of highQ filters.7,32.
Figure 3 shows the calculated power transmission spectra of asymmetric CSLR resonators (N = 3) when L1 ≠ L2. For comparison, we use the same field transmission coefficients [t1, t2, t3] = [0.87, 0.77, 0,87] in the calculation. In Fig. 3(a), we plot the calculated power transmission spectra around one resonance when there are relatively small differences between L1 and L2. It can be seen that the differences between L1 and L2 lead to different filter shapes of the CSLR resonator. In particular, when L2 = 100.18 µm, the transmission spectrum of the CSLR resonator is almost the same as when L2 = 100.00 µm. This is because in such a condition the difference between the phase along L1 and L2 is approximately π. Considering that the physical cavity length is half of the effective cavity length for a SW resonator,23 the effective phase difference is about 2π, and so there are almost the same transmission spectra resulting from coherent interference within in the resonant cavity. The calculated power transmission spectra in Fig. 3(a) also indicate that the filter shape of the CSLR resonator can be tuned or optimized by introducing thermooptic microheaters19,33 or carrierinjection electrodes39,40 along L1,2 to tune the phase shift. Figure 3(b) presents the calculated power transmission spectra when there are relatively large differences between L1 and L2. Due to the Vernierlike effect between the FPC1 and FPC2, diverse mode splitting filter shapes are achieved at different resonances of the transmission spectra, which can be utilized to select resonances with desired filter shapes for passive photonic devices.41 Such differences in the filter shapes become more obvious for an increased difference between L1 and L2. In Fig. 3(c), we plot the calculated power transmission spectra when L1= 0 and L2 = mLs1 (m = 1, 2, 3, 4). Since the effective cavity length of FPCi equals to Lsi + 2Li + Lsi+1 (i = 1, 2),23 the various sets of L1 and L2 in Fig. 3(c) correspond to the conditions that the effective cavity length of FPC2 is integer multiples of that of FPC1. We can see that there are split resonances with different numbers of transmission peaks in the spectra. Unlike the split resonances in Fig. 3(b), there are identical filter shapes in each period.
Figure 4 shows the calculated power transmission spectra of the CSLR resonators with four SLRs (N = 4). In the calculation, we assume Ls1 = Ls2 = Ls3 = Ls4 =129.66 µm, and L1 = L2 = L3 =100 µm. To simplify the comparison, we only show plots for the conditions that t1 = t4, t2 = t3 (Figs. 4(a) and (b)) and t1 = t3, t2 = t4 (Figs. 4(c) and (d)). The calculated power transmission spectra for various t2 = t3 when t1 = t4 = 0.87 are shown in Fig. 4(a). The corresponding group delay spectra are provided in Fig. 4(b). It can be seen that the three split resonant peaks gradually merge to a single one as t2 and t3 decrease (i.e., the coupling strengths increase). After that, by further decreasing t2 and t3, the Q factor, extinction ratio, and group delay of the combined single resonance increase, together with an increase in the insertion loss. This trend is similar to that in Figs. 3(a) and (b) for N = 3. Figure 4(c) depicts the calculated power transmission spectra for various t2 = t4 when t1 = t3 = 0.85. The calculated group delay spectra are shown in Fig. 4(d) accordingly. For this condition, the CSLR resonator is no longer axisymmetric, and so the trend in Figs. 4(c) and (d) is different from that in Figs. 4(a) and (b). As t2 and t4 decrease (i.e., the coupling strengths increase), the transmission peak in the centre starts to appear and then becomes more pronounced, together with a decreased spectral range between the resonant peaks on both sides.
Figure 5(a) shows the calculated power transmission spectra of the CSLR resonators with different numbers of SLRs (N). In the calculation, we use SLRs and connecting waveguides with the same lengths as those in Figs. 2 and 4. We also assume that t1 = t2 =…= tN = 0.85. 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 N1. The differences between the maximum transmission of different resonant peaks are determined by the waveguide propagation loss, and these can be mitigated by decreasing the waveguide propagation loss (α) to the point where, in the limit of zero loss, they would no longer exist. In Fig. 5(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. Figures 5(c) − (f) show three specific optical filters designed based on the CSLR resonators with eight SLRs (N = 8). The filter in Figs. 5(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.17 In the calculation, we assume that t1 = t2 =…= t8 = 0.97. With enhanced light trapping, there are increased time delays and enhanced lightmatter interactions, which are useful in nonlinear optics and laser excitation.10,11,27,28 In Fig. 5(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. 5(d). This group delay can be increased further by using more cascaded SLRs. The filter in Fig. 5(e) is an 8thorder Butterworth filter with a flattop filter shape. The field transmission coefficients of SLRi (i = 1, 2, …, 8) are [t1, t2, t3, t4, t5, t6, t7, t8] = [0.98, 0.94, 0.91, 0.90, 0.90, 0.91, 0.94, 0.98], respectively. Figure 5(f) shows the designed optical filter with multiple transmission peaks in the spectrum. Each transmission peak has a high extinction ratio of over 10 dB. The field transmission coefficients of SLRi (i = 1, 2, …, 8) in this filter are [t1, t2, t3, t4, t5, t6, t7, t8] = [0.84, 0.935, 0.945, 0.955, 0.955, 0.945, 0.935, 0.84], respectively. By tailoring the transmission of individual resonant peaks via changing ti and φi, these CSLR resonators could potentially find applications in RF spectral shaping and broadband arbitrary RF waveform generation42,43. Based on twophoton absorption (TPA)induced free carrier dispersion (FCD) 44, CSLR resonators with multiple transmission peaks could also be used for wavelength multicasting in wavelength division multiplexing (WDM) systems.9,45 It is also worth mentioning that the narrow bandwidth between the split resonances arises from coherent interference within the CSLR resonators. For ring resonators, such a narrow bandwidth can only be achieved by using much larger loop circumferences, thus leading to much larger device footprints. In addition, the CSLR resonator is a SW resonator, and so the cavity length is nearly half that of a TW resonator (e.g., ring resonator) with the same FSR, which enables even more compact device footprints.