Advanced Integrated Photonic Filters Designed with Coupled Sagnac Loop Reflectors

We theoretically investigate integrated photonic resonators formed by two mutually coupled Sagnac loop reflectors (MC-SLRs). Mode interference in the MC-SLR resonators is tailored to achieve versatile filter shapes with high performance, which enable flexible spectral engineering for diverse applications. By adjusting the reflectivity of the Sagnac loop reflectors (SLRs) as well as the coupling strength between different SLRs, we achieve optical analogues of Fano resonance with ultrahigh spectral slope rates, wavelength interleaving / non-blocking switching functions with significantly enhanced filtering flatness, and compact bandpass filters with improved roll-off. In our designs the requirements for practical applications are considered, together with detailed analyses of the impact of structural parameters and fabrication tolerances. These results highlight the strong potential of MC-SLR resonators as advanced multi-functional integrated photonic filters for flexible spectral engineering in optical communications systems.

Fano resonances, that feature asymmetric resonant lineshape profiles, are a fundamental physical phenomenon induced by interference between a discrete localized state and a continuum state [11][12][13]. It was first reported early in the 20th century [14,15], and has been widely used in atom spectroscopy since then [12]. Recent advances in photonics and nanotechnology has led to new ways of realizing optical analogues of Fano resonances with broad applications in light focusing beyond the diffraction limit, optical switching, sensing, data storage, topological optics, and many others [16][17][18][19][20]. These optical phenomena have been demonstrated in many types of resonant cavities such as dielectric rods, disordered structures, lattices of nanospheres, metasurfaces, and integrated photonic resonators [16,17,19,[21][22][23][24].
Optical interleavers, switching nodes and bandpass filters (BPFs) are core components for signal multiplexing/demultiplexing, routing and monitoring in wavelength division multiplexing (WDM) optical communication systems [25][26][27]. To realize these devices, optical filters with a flat-top spectral response that can minimize filtering distortion, are highly desirable. To date, various schemes have been proposed to improve the roll-off of optical filters for achieving quasi flat-top spectral responses [27,[28][29][30][31][32][33][34]. However, these schemes, based on either finite-impulse-response (FIR) filters such as Mach-Zehnder interferometers (MZIs) or infinite-impulse-response (IIR) filters such as Fabry-Perot (FP) cavities and microring resonators (MRRs), usually achieve flat-top spectral responses via cascading many subunits [35]. This not only results in a bulky device footprint but also imposes stringent requirements on the alignment of resonant wavelengths from separate sub-components. Moreover, it is challenging to maintain the desired spectral response given the unequal wavelength drifts for different sub-components induced by the thermo-optic effect [36,37].
To realize Fano-resonance based devices, optical interleavers as well as switching nodes and BPFs in the form of photonic integrated circuits could reap the greatest dividends in terms of compact footprint, high stability, high scalability and mass-producibility for practical applications. Recently we demonstrated multi-functional photonic filters based on cascaded Sagnac loop reflectors (CSLR) in silicon-on-insulator (SOI) nanowires [33]. Here, we theoretically investigate more advanced filter structuresnamely, mutually coupled Sagnac loop reflectors (MC-SLR) − using similar principles. As compared with the CSLR resonators that include only IIR filter elements, the MC-SLR resonators that consist of both FIR and IIR filter elements provide more versatile mode interference and greatly improved flexibility for spectral engineering. We tailor the mode interference in MC-SLR resonators to achieve optical analogues of Fano resonances, or alternatively EIT or Autler-Towns splitting, that yield BPFs with ultrahigh slope rates, wavelength interleaving and non-blocking switching functions with significantly enhanced filtering flatness with improved roll-off, all in a compact footprint.
Detailed analyses of the impact of varying the structural parameters, including fabrication tolerances, are provided to facilitate device design and optimization. For practical applications, the key requirements including a high extinction ratio, low insertion loss, low crosstalk and meeting the ITU-T spectral grid [38] are also considered. These results verify the effectiveness of using MC-SLR resonators as advanced multi-functional integrated photonic filters for flexible spectral engineering in optical communication systems. Figure 1(a) illustrates the schematic configuration of the MC-SLR resonators. We investigate two types MC-SLR resonators: the first, consisting of two parallel Sagnac loop reflectors (SLRs) coupled to a top bus waveguide, is termed a parallel MC-SLR resonator while the second, consisting of two inversely coupled SLRs, is termed a zig-zag MC-SLR resonator. In both resonators the bus waveguides introduce additional feedback paths for coherent optical mode interference, which provide greatly improved flexibility for engineering the spectral response. We model the MC-SLR resonators using the scattering matrix method [33,39], where the waveguide and coupler parameters are defined in Table I. To simplify the comparison, we assume that the two SLRs are identical for each individual MC-SLR resonator, i.e., LSLR1 = LSLR2 = LSLR, L1 = L2 = L, ts1 = ts2 = ts, tb1 = tb2 = tb.  Table I. When ts=1, the parallel MC-SLR is equivalent to an MZI (i.e., FIR filter), and it is equivalent to a FP cavity (i.e., IIR filter) when ts = 1 and tb = 1, respectively. On the other hand, the zigzag MC-SLR resonator is equivalent to a MZI combined with a SLR when ts = 1 and tb = 1, respectively. When ts ≠ 1 and tb ≠ 1, both can be regarded as a hybrid filter consisting of both FIR and IIR filter elements. The mutual interaction between the FIR and IIR filter elements yields a very versatile coherent optical mode interference. The freedom in designing the reflectivity of the SLRs (i.e., ts), the coupling strength between the SLRs and bus waveguides (i.e., tb), and the lengths of the SLRs (i.e., LSLR) as well as the connecting bus waveguides (i.e., L) forms the basis for engineering the spectral response of the MC-SLR resonators, which leads to diverse applications.
In the following sections, we tailor the spectral response of MC-SLR resonators to achieve various filtering functions with high performance, including optical analogues of Fano resonances, wavelength interleaving and non-blocking switching, and BPFs. The devices are designed based on, but not limited to, the SOI integrated platform. In our design, we use values obtained from our previously fabricated SOI devices [33,40] Fig. 2 respectively. In particular, the resonance spectrum shows a high slope rate (SR, defined as the ratio of the ER to the corresponding wavelength difference at the Fano resonance) of 389 dB/nm, indicating strong coherent optical mode interference in the parallel MC-SLR resonator.
Compared to the FP cavity based on two cascaded SLRs [41], the top bus waveguide in the parallel MC-SLR resonator forms an additional feedback path that allows more versatile coherent optical mode interference between the two SLRs. As compared with previous work in achieving Fano resonances based on integrated MRRs [42,43], the mutual interference between the FIR and IIR filter elements yields Fano resonances with a high SR in a compact device, requiring only two SLRs. parameter, keeping the others the same as in Fig. 2(a). Figure 3(a-i) shows the power transmission spectra for various values of ts, with the calculated SR and IL as functions of ts depicted in Fig. 3(a-ii). Clearly both the SR and the IL decrease with ts, reflecting the trade-off between them. Figure 3(b-i) shows the power transmission spectra for different values of tb while the corresponding values of IL and SR are depicted in Fig. 3(b-ii). The change in SR and IL with tb shows the opposite trend to their dependence on ts while still maintaining the tradeoff between them. Figure 3(c-i) shows the power transmission spectra for various L. Figure   3(c-ii) depicts the calculated SR and IL as functions of L. One can see that the resonant wavelength redshifts as L increases, indicating that it can be tuned by adjusting the phase shift with thermo-optic micro-heaters [37,39] or carrier-injection electrodes [44,45] along the connecting bus waveguides. In Fig. 3(c-ii), both SR and IL increase with L, while the change in SR is more dramatic than that for IL. This indicates that the SR of the Fano resonance can be significantly improved at the expense of a slightly increased IL within a reasonable range.   normalized root-mean-square deviation (NRMSD) within the 1-dB BW range as a function of ts are depicted in Fig. 5(b). The 1-dB BW decreases with ts, and the corresponding NRMSD increases with ts. This reflects the deterioration of the filtering flatness for an increased ts. The spectral responses of the parallel MC-SLR resonator for various tb are shown in Fig. 6(a). The 1-dB BW and the corresponding NRMSD within 1-dB BW range versus tb are plotted in Fig.   6(b). Their changes with tb shows an opposite trend to their change with ts, indicating improved filtering flatness for an increased tb.  the MC-SLR resonator shows an increased 1-dB BW and improved filtering flatness, at the expense of reduced ERs and increased ILs within reasonable ranges. Note that we used a moderately low waveguide propagation loss (α = 55 m -1 , i.e., 2.4 dB/cm) in our design, but well within experimental capability for SOI nanowires. For waveguides with lower propagation loss, such as is achievable silicon nitride or doped silica waveguides , for example, a more significant improvement in the 1-dB BWs and filtering flatness can be achieved.    Table III, together with those of the device with a CS of 100 GHz. The high 1-dB BW to CS ratios highlight the filtering flatness. The almost equal 3-dB BW to the CS ratios and the ERs for the complementary output ports also reflect very symmetric wavelength interleaving / de-interleaving for these devices.  Given the characteristics of the parallel MC-SLR resonator as a four-port device, a 2 × 2 nonblocking switching unit was further designed based on it. We chose the Benes switching architecture since it exhibits minimum complexity among various non-blocking switching architectures [76]. Figure 8(a) shows the (i) cross and (ii) bar states of the non-blocking switching unit and the corresponding spectral responses between the different ports, shown in Fig. 8(b). The structural parameters of the parallel MC-SLR resonator were the same as those in Fig. 4(b). Two resonance channels centered at wavelengths of λ1 = 1549.4938 nm and λ2 = 1550.2945 nm were selected for the operation of the cross and bar states, respectively. When the resonance channel at λ1 is red shifted to λ2, the switching unit changes from the cross state to the bar state. Practically, the red shift can be realized by slightly increasing the chip temperature via temperature controllers or injecting a high-power pump at other resonance wavelengths [26,32,77,78]. Figure 8(c) shows the shift of the center wavelengths of the resonance channels at (i) λ1 and (ii) λ2 as a function of chip temperature variation ΔT. The thermo-optic coefficient (dn / dT =1.8 × 10 -4 / °C) of silicon used in our calculation was the same as that used elsewhere [76]. It can be seen that the resonance channel red shifts when increasing ΔT. Table IV shows the ERs, ILs and crosstalk for the 2 × 2 non-blocking switching unit based on the parallel MC-SLR resonator. As can be seen, flat-top spectral response with high ERs, low ILs and low crosstalk is achieved. When the input port is changed to Port 2, the wavelength channels for the cross and bar states remain unchanged, i.e., λ1 = λ1 ′ and λ2 = λ2 ′ .   We also investigate the impact of varying ts and tb on the IL and ER, which are important parameters for the non-blocking switching unit. Figure 9 (a) plots the IL and ER of the transmission spectra from Port 1 to (i) Port 3 and (ii) Port 4 of the parallel MC-SLR resonator versus ts. The other structural parameters were the same as those in Fig. 4(b). Clearly the IL decreases with the ts while the ER shows the opposite trend, reflecting a trade-off between them. The IL and ER as functions of tb are plotted in Fig. 9(b). As shown in Fig. 9  V. COMPACT BPFS WITH IMPROVED ROLL-OFF In this section, we tailor the mode interference in the zig-zag MC-SLR resonator to realize compact BPFs with improved roll-off. Figure 10 Figure 10(b) shows the corresponding group delay response of the BPF in Fig. 10(a). To quantitatively analyze the improvement in the filtering roll-off, we further compare the 3-dB BW of the BPF based on two zig-zag MC-SLRs (2-Z-SLRs) with BPFs based on other types of integrated photonic resonators, including a single add-drop MRR (1-MRR) [77,78], two cascaded SLRs (2-C-SLRs) [41], three cascaded SLRs (3-C-SLRs) [33], and two parallel coupled MRRs (2-MRRs) [77,78]. For 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 ) and had the same ER and free spectral range (FSR) as those of the BPF in Fig. 10(a).  Fig. 10(d). It is clear that the BPF based on the two zig-zag MC-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.
We further investigate the impact of ts, tb, and L on the performance of the BPF based on the zig-zag MC-SLR resonator. We only changed one structural parameter, keeping the others the same as those in Fig. 10 increasing ts and keeping constant tb or vice versa, both the ER and 3-dB BW increase, together with a slightly increased IL. This indicates that the ER and 3-dB BW can be further improved by sacrificing IL within a reasonable range. In Fig. 11(c), the ER, IL and 3-dB BW remain unchanged for different L when LSLR is constant. This not only verifies the feasibility to implement tunable BPFs by introducing thermo-optic micro-heaters [37,39] or carrierinjection electrodes [41,45], but also highlights the high fabrication tolerance of the BPF.
Finally, this approach towards integrated optical filters is also applicable for phase-filters such as tunable dispersion compensators and delays [79 -83].  VI. CONCLUSION We theoretically investigate advanced filter structures based on MC-SLR resonators consisting of both FIR and IIR filter elements. Mode interference in the MC-SLR resonators is tailored to provide optical analogues of Fano resonances with ultrahigh SRs, a flat-top spectral response for wavelength interleaving/non-blocking switching functions, and compact bandpass filters with improved roll-off. A detailed analysis of the impact of varying the structural parameters is presented, with a particular focus on the requirements for practical applications. This work highlights the strong potential of MC-SLR resonators as multi-functional integrated photonic filters for flexible spectral engineering in optical communication systems.