Advanced optical filters with coupled Sagnac loop waveguide reflectors

We present theoretical designs of high performance optical filters in integrated silicon photonic nanowire resonators. We use mode interference in formed by zig-zag waveguide coupled Sagnac loop reflectors (ZWC-SLRs), tailored to achieve diverse filtering functions with good performance. These include compact bandpass filters with improved roll-off, optical analogues of Fano resonances with ultrahigh spectral extinction ratios (ERs) and slope rates, and resonance mode splitting with high ERs and low free spectral ranges. The analysis verifies the feasibility of multi-functional integrated photonic filters based on ZWC-SLR resonators for flexible spectral engineering in diverse applications. different filtering functions including compact bandpass filters with improved roll-off, optical analogues of Fano resonances with ultrahigh ERs and SRs, resonance mode splitting with high ERs and low FSRs. This work highlights the ZWC-SLR resonators as a robust and adaptable approach to flexible spectral engineering for a diverse range of applications .


COMPACT BANDPASS FILTERS WITH IMPROVED ROLL-OFF
In this section, we tailor the mode interference in the two ZWC-SLR resonator to realize compact BPFs with improved roll-off. Figures 2(a) and (b) show the power transmission spectrum and corresponding group delay response of the two ZWC -SLR resonator from Port 1 to Port 2 in the wavelength range of 1548.9 nm -1551.2 nm, respectively. There are wide-flat stopbands and a passband with improved roll-off, arising from coherent mode interference within the two ZWC -SLR resonator. The structural parameters are LSLR = L= 100 µm, ts = tb = 0.78.
To quantitatively analyze the improvement in the filtering roll-off, we further compare the 3-dB BW of the BPF based on two ZWC -SLRs (2-ZWC-SLRs) with BPFs considering other types of integrated photonic resonators, including a single add-drop MRR (1-MRR) [27,28], two cascaded SLRs (2-C-SLRs) [29], three cascaded SLRs (3-C-SLRs) [22], and two parallel coupled MRRs (2-MRRs) [27,28]. In comparison, the above filters were designed based on the same SOI wire waveguide (i.e., with the same ng = 4.3350 and α = 55 m -1 ). Figure 3(a) shows the normalized power transmission spectra of the BPFs considering the various types of integrated resonators mentioned above. The filtering spectra of all the devices were normalized to have the same ER (~10.36 dB) and full width at minimum (~230.6 GHz) as those of the BPF in Fig. 2(a). The corresponding 3-dB BWs are given in Fig. 3(b). It is clear that the BPF based on the two ZWC-SLRs resonator has the largest 3-dB BW and the best roll-off, reflecting enhanced mode interference in this compact device consisting of only two SLRs.

ULTRA-SHARP FANO RESONANCES
In this section, we tailor the spectral response of the three ZWC-SLR resonator structure to realize optical analogues of Fano resonances with high ERs and SRs. The power transmission spectrum from Port 2 to Port 4 of the three ZWC-SLR resonator is depicted in Fig. 4 Figure 4(b) shows a zoom-in view of Fig. 3(a) in the wavelength range of 1549.8 nm -1550.65 nm, which shows a Fano resonance with an ultra-high ER of 76.32 dB and an ultra-high SR of 997.66 dB/nm. The ER is defined as the difference between the maximum and the minimum transmission, and the SR is defined as the ratio of the ER to the wavelength difference between the resonance peak and notch (i.e., ∆λ in Fig. 4(b)). The high ER and SR reflect the high performance of the Fano resonances resulting from strong coherent optical mode interference in the compact resonator with only three SLRs. Further, the periodical filter shape of the zig-zag 3WC-SLR resonator is also useful for applications in WDM systems.   Fig. 4. The corresponding IL and SR are depicted in Fig. 5(b). The IL increases with ts, while the SR first increases and then decreases with ts, achieving a maximum value of 997.66 dB/nm at ts = 0.743. The non-monotonic relationship between the SR and ts is a combined result of both a decrease in ∆λ and a non-monotonic variation in ER. The latter mainly arises from the difference between the internal (transmission) and external (coupling) cavity loss, which is similar to that for different coupling regimes in microring resonators (MRRs) [30].

RESONANCE MODE SPLITTING
In this section, we tailor the mode interference in the three ZWC-SLR resonator to achieve resonance mode splitting with high ERs and low FSRs. The resonance mode splitting with multiple densely spaced resonances can break the dependence between the Q factor, FSR, and physical cavity length, thus allowing low FSRs and high Q factors in resonators with a compact footprint. Figure 6(a) shows the power transmission spectrum from Port 2 to Port 4 of the three ZWC-SLR resonator. The structural parameters are LSLR = L1,2,3,4 = 115 µm, ts = 0.72, and tb = 0.99, which are designed in order to achieve a WS of about 100 GHz between adjacent split resonances. In Fig. 6(a), WS1 = 98.33 GHz and WS2 = 102.26 GHz. There are two split resonances within a FSR of ~ 200.59 GHz. Figure 6(b) shows a zoom-in view of Fig. 6(a) in the wavelength range of 1549 nm -1550.7 nm. The IL, Q factor, ER1, and ER2 of the two split resonances in Fig. 4(b) are ~2.02 dB, ~6.03 × 10 4 , ~24.65 dB, and ~27.55 dB, respectively.   Fig. 6 (a). The Q factor and ERs (ER1 and ER2) as functions of ts are depicted in Fig. 7(b). As ts increases, the Q factor slightly decreases while the ER1 and ER2 change more dramatically, resulting in a change in the spectral response towards that of the Fano resonances in Fig. 4(a). The non-monotonic change in ER2 with ts follows the trend of the SR in Fig. 5(b) for similar reasons. In particular, ER1 equals to ER2 when ts = 0.7177. Under this condition, the Q factor and effective FSR are ~6.06 × 10 4 and ~100.30 GHz (i.e., half of the FSR in Fig. 6(a)), respectively. To achieve the same FSR, the circumference of a comparable MRR (with the same waveguide geometry and loss) is 690 µm, which is 6 times the length of the SLRs. This highlights the reduced cavity length enabled by the mode splitting in the 3WC-SLR resonator. On the other hand, the Q factor of a comparable MRR with the same FSR and ER is ~6.08 × 10 4almost the same as that of the zig-zag 3WC-SLR resonator. This indicates that the reduced cavity length did not come at the expense of a significant decrease in Q factor.
The number of split resonances can be changed by varying the length of the connecting bus waveguides. Figure 8(a) shows the power transmission spectrum from Port 1 to Port 3 of the three ZWC-SLR resonator. Clearly, there are four split resonances in each FSR. The structural parameters are LSLR = 115 µm, L1,3 = 115 µm, L2,4 = 230 µm, and ts = tb = 0.88. The WSs between the split resonances are WS1 = WS3 = 100.46 GHz and WS2 = 90.37 GHz. Figure 8(b) shows a zoom-in view of Fig. 8(a) in the wavelength range of 1548.7 nm -1550.7 nm. The power transmission spectra for different ts is shown in Fig. 9(a). The corresponding Q factors (Q1 and Q2) and ERs (ER1 and ER2) for the first two resonances from the left side are shown in Fig. 9(b). In Figs. 9(a) and (b), all the Q factors and ERs decrease with ts, along with slightly decreased ILs.

CONCLUSIONS
We theoretically investigate advanced multi-functional integrated photonic filters based on ZWC-SLR resonators. Mode interference in the ZWC-SLR resonators is tailored to achieve different filtering functions including compact bandpass filters with improved roll-off, optical analogues of Fano resonances with ultrahigh ERs and SRs, resonance mode splitting with high ERs and low FSRs. This work highlights the ZWC-SLR resonators as a robust and adaptable approach to flexible spectral engineering for a diverse range of applications.