Advanced Classical Optical Filters with Three Waveguide Coupled Sagnac Loop Reectors

We theoretically investigate advanced multi-functional integrated photonic lters formed by three waveguide coupled Sagnac loop reectors (3WC-SLRs). By tailoring the coherent mode interference, the spectral response of the 3WC-SLR resonators is engineered to achieve diverse ltering functions with high performance. These include optical analogues of Fano resonances that yield ultrahigh spectral extinction ratios (ERs) and slope rates, resonance mode splitting with high ERs and low free spectral ranges, and classical Butterworth, Bessel, Chebyshev, and elliptic lters. A detailed analysis of the impact of the structural parameters and fabrication tolerances is provided to facilitate device design and optimization. The requirements for practical applications are also considered. These results theoretically verify the effectiveness of using 3WC-SLR resonators as multi-functional integrated photonic lters for exible spectral engineering in diverse applications. We have theoretically investigated advanced multi-functional integrated photonic lters based on 3WC-SLR resonators. Mode interference in the 3WC-SLR resonators is tailored to achieve high performance ltering functions including optical analogues of Fano resonances with ultrahigh ERs and SRs, resonance mode splitting with high ERs and low FSRs, and classical Butterworth, Bessel, Chebyshev, and elliptic lters. The requirements for practical applications are considered in our designs, together with detailed analysis of the impact of structural parameters and fabrication tolerances. This work highlights the 3WC-SLR resonators as a powerful and versatile approach to exible spectral engineering for a diverse range of applications.

The zig-zag 3WC-SLR resonator is equivalent to two cascaded Mach-Zehnder interferometers (MZIs, which is a finite-impulse-response (FIR) filter) and a Fabry-Perot cavity (which is an infinite-impulse-response (IIR) filter) when t s = 1 and t b = 1, respectively. On the other hand, the parallel 3WC-SLR resonator is equivalent to two cascaded MZIs and three cascaded SLRs (which is an IIR filter) when t s = 1 and t b = 1, respectively. When t s ≠ 1 and t b ≠ 1, both types of filters can be considered to be hybrid, consisting of both FIR and IIR filter elements that allows more versatile coherent mode interference induced by mutual interaction. As compared with the 2WC-SLR resonators [164], the 3WC-SLR resonators have an extra SLR and additional feedback paths that introduce more complex coherent mode interference, which can lead to enhanced filter performance and versatility. The freedom in designing the reflectivity of the SLRs (i.e., t s ), the coupling strength between the SLRs and connecting bus waveguides (i.e., t b ), and the waveguide lengths (i.e., L SLR and L i ) enables flexible spectral engineering based on the 3WC-SLR resonators, which can lead to diverse filtering functions.
In the following sections, mode interference in the 3WC-SLR resonators is tailored to achieve highperformance filtering functions, including optical analogues of Fano resonances (Section III), resonance mode splitting (Section IV), and classical Butterworth, Chebyshev, Bessel, and elliptic filters (Section V). In our design, we use values obtained from our previously fabricated SOI devices [163,166] for the waveguide group index (n g = 4.3350, transverse electric (TE) mode) and the propagation loss (α = 55 m -1 , i.e., 2.4 dB/cm). The devices are designed based on but not limited to the SOI integrated platform. t bi k bi a a i = exp (-αL i / 2), a si = exp (-αL SLRi / 2), α is the power propagation loss factor. b φ i = 2πn g L i / λ, φ si = 2πn g L SLRi / λ, n g is the group index and is the wavelength.
c t si 2 + κ si 2 = 1 and t bi 2 + κ bi 2 = 1 for lossless coupling are assumed for all the directional couplers.

Ultra-sharp Fano Resonances
In this section, we tailor the spectral response of the zig-zag 3WC-SLR resonator to realize optical analogues of Fano resonances with high ERs and SRs. The power transmission spectrum from Port 2 to Port 4 of the zig-zag 3WC-SLR resonator is depicted in Fig. 2 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. 2(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. Table II compares the performance of the Fano resonances generated by the parallel 2WC-SLR resonator [164] and the zig-zag 3WC-SLR resonator. As compared with the parallel 2WC-SLR resonator, the zig-zag 3WC-SLR resonator can generate Fano resonances with increased ER and SR as well as decreased insertion loss (IL), all of which are highly desirable in practical applications.
Moreover, the periodical filter shape of the zig-zag 3WC-SLR resonator is also useful for applications in wavelength division multiplexed (WDM) systems.  between the SR and t s 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) [167]. Figure 3(b-i) compares the power transmission spectra for various t b . The IL and SR functions of t b are depicted in Fig. 3(b-ii). Both the IL and SR increase with t b , re ecting a trade-off between them. Note that although the ER for t b = 0.994 is higher than for t b = 0.998, the SR for t b = 0.994 is higher than that for t b = 0.998 due to a more signi cantly decreased ∆λ. In Figs. 3(c-i) and (c-ii), we compare the corresponding results for various ΔL, which is the length variation of the connecting bus waveguides. To simplify the comparison, we assume the same ΔL for each connecting bus waveguides L 1,2,3,4 and keep L SLR constant. As ΔL increases, the IL and SR remain unchanged while the resonance redshifts. This highlights the high fabrication tolerance and also indicates that the resonance wavelength can be tuned by introducing thermo-optic micro-heaters [149,168,169] or carrierinjection electrodes [142] along the connecting bus waveguides to tune the phase shift.

Resonance Mode Splitting
In this section, we tailor the mode interference in the zig-zag 3WC-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 intrinsic 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 4(b) shows a zoom-in view of Fig. 4(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.
We further investigate the impact of varying t s , t b , and L = L 1,2,3,4 on the Q factor, ER1, and ER2 of the split resonances, all of which are important parameters reflecting the degree of mode splitting. In Figs. 5(a) -(c), we only changed one structural parameter in each figure, keeping the others the same as those in Fig. 4. Figure 5(ai) shows the spectral response for various t s . The Q factor and ERs (ER1 and ER2) as functions of t s are depicted in Fig. 5(a-ii). As t s 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. 2(a).
The non-monotonic change in ER2 with t s follows the trend of the SR in Fig. 3(a-ii) for similar reasons. In particular, ER1 equals to ER2 when t s = 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. 4(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 4 -almost 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 spectral response for different t b are shown in Fig. 5(b-i). The corresponding Q factor and ERs are depicted in Fig. 5(b-ii). The ER1 remains almost unchanged while both the ER2 and the Q factor increase with t b , at the expense of a slightly increased IL. The corresponding results for different ∆L are shown in Fig. 5(c-i) and (c-ii). Following the trend in Fig. 2(c), the filter shape remains unchanged while the resonance redshifts as ΔL increases.
The number of split resonances can be changed by varying the length of the connecting bus waveguides.  Table III we further compare the resonance mode splitting in the zig-zag 3WC-SLR resonator with that of ve cascaded SLRs [163] which can also generate four split resonances. As compared with the ve cascaded SLRs that only include standing-wave (SW) lter elements, the mode interference between the SW and travelling-wave (TW) lter elements in the zigzag 3WC-SLR resonator yields higher ERs for the split resonances, a smaller difference between the ERs of the split resonances, and fewer required SLRs. Ref.  In Figs. 7(a) − (c), we investigate the impact of t s , t b , and ∆L on the performance of the resonance mode splitting based on the zig-zag 3WC-SLR resonator in Fig. 6. We only changed one structural parameter in each figure, keeping the others the same as those in Fig. 6 (a). The power transmission spectra for different t s and t b are shown in Figs. 7(a-i) and (b-i), respectively. The corresponding Q factors (Q1 and Q2) and ERs (ER1 and ER2) for the first two resonances from the left side are shown in Figs. 7(a-ii) and (b-ii) respectively. In Fig. 7(a), all the Q factors and ERs decrease with t s , along with slightly decreased ILs. In Fig. 7(b), as t b increases, the difference between the two Q factors as well as the difference between the two ERs gradually decrease, resulting in a more symmetric resonance line-shape, which is desirable for reducing filtering distortions. Figures  7(c-i) and (c-ii) compares the corresponding results for various ΔL. As ΔL increases, the ER1 slightly increase and ER2 slightly decrease while both Q factors slightly increase, which make the filter shapes more asymmetric.

High Performance Classical Filters
In this section, we tailor the mode interference in the parallel 3WC-SLR resonator to realize classical filters including Butterworth, Bessel, Chebyshev, and elliptic filters, that all exhibit broad filtering bandwidths and high ERs. The spectral response of these practical filters (solid lines) together with the ideal passband filter (dashed line) are shown in Fig. 8. The Butterworth filter has a flat passband response, while the Bessel filter has a linear phase response over the passband. The Chebyshev filters have either passband ripples (Type I) or stopband ripples (Type II) together with a flat response in the opposite band, resulting in a steeper roll-off than the Butterworth filter. The elliptic filter has both passband and stopband ripples that yields the steepest roll-off among the four types of filters [153]. Figure 9(a) shows the power transmission spectrum and corresponding group delay response of the parallel 3WC-SLR resonator from Port 2 to Port 3. As can be seen, there is Butterworth filter shape with flat-top passbands arising from coherent mode interference within the parallel 3WC-SLR resonator, which can be used for low-distortion signal filtering in optical communication systems [171,172]. The structural parameters are provided in Table IV. The IL, ER, and 3-dB bandwidth (BW) of the Butterworth filter are ~1.71 dB, ~7.12 dB, and 28.08 GHz, respectively. We then investigate the impact of t s , t b , and ∆L on the performance of the Butterworth filter. The results are shown in Figs. 9(b) -(d), respectively. In Fig. 9(b), the bandwidth of the passband increases with t s , together with slightly degraded filtering flatness. In Fig. 9(c), the resonance is split, with the spectral range between the split resonances increasing with t b . This indicates that the Butterworth filter shape gradually transitions to a Chebyshev Type I filter shape with improved roll-off and degraded flatness. In Fig.  9(d), as ∆L increases, the filter shape remains unchanged while the resonance redshifts, showing similar trends to the Fano resonances in Fig. 3(c-i) and the resonance mode splitting in Fig. 5(c-i). This reflects the high fabrication tolerance and the feasibility to realize tunable Butterworth filters.
The spectral and group delay responses of the Bessel filter based on the parallel 3WC-SLR resonator are shown in Fig. 10(a). The input and output ports are Port 2 and Port 3, respectively, the same as those for the Butterworth filter in Fig. 9. The structural parameters are provided in Table IV. As can be seen, the Bessel filter with a flat-top group delay response is achieved, which is useful for applications such as optical buffering and delay lines [173,174]. The group delay response versus t s , t b , and ∆L are shown in Figs. 10(b) -(d), respectively. In Fig. 10(b), the bandwidth of the group delay response increases with t s , at the expense of a decreased maximum group delay value and degraded flatness. In Fig. 10(c), the maximum group delay on both sides increases with t b , while the group delay at the center wavelength shows the opposite trend, resulting in higher unevenness in the group delay response. In Fig. 10(d), the group delay response remains unchanged but redshifts as ∆L increases. Figure 11(a) shows the power transmission spectrum and group delay response of the Chebyshev Type II filter based on the parallel 3WC-SLR resonator. The input port is Port 2, which is the same as those for the Butterworth and Bessel filters, while the output port is changed from Port 3 to Port 4. The structural parameters are provided in Table IV. Clearly, there is a Chebyshev Type II filter shape with equal stopband ripples and a flat response in the passband. The IL, maximum stopband ripple, ER, and 3-dB BW are ~1.26 dB, ~0.037 dB, 27.33 dB, and ~18.77 GHz, respectively. This filter function with a very flat passband and strongly rejected band can be useful for cleaning and extracting channels from crosstalk in a WDM optical communications system, respectively [175]. The power transmission spectra versus t s , t b , and ∆L are shown in Figs. 11(b) -(d), respectively. As shown in Figs. 11(b) and (c), by increasing t s and keeping constant t b or vice versa, the notch depth of the single resonance first increases and then the single resonance is split with an increased spectral range between the split resonances. This is a typical phenomenon for resonance mode splitting, which has also been observed in Refs. [149,176]. In Fig. 11(d), the filter shape remains unchanged but redshifts as ∆L increases, following the trends for previous filters.
Finally, we tailor the mode interference in the parallel 3WC-SLR resonator to realize elliptic filters. As compared with the Butterworth and Chebyshev type I filters that have all-pole transfer functions, elliptic filters include both poles and zeros in the transfer function, which can provide higher stopband extinction levels and better rolloff [160]. Figure 12(a) shows the power transmission spectrum and group delay of the parallel 3WC-SLR resonator from Port 2 to Port 4. There is an elliptic filter shape with ripples in both the passband and stopband. The structural parameters are also provided in Table IV. The IL and notch depth of filter are ~0.59 dB and ~31.5 dB, respectively. The maximum passband and stopband ripples are ~1.56 dB and ~18.25 dB, respectively. The ripples in the passband and stopband make the filtering roll-off steeper. These ripples together with the steep roll-off result in a reasonable trade-off between the minimum signal degradation and the maximum noise/interference rejection [177]. The power transmission spectra versus t s , t b , and ∆L are shown in Figs. 12(b) -(d), respectively. In Figs. 12(b) and (c), the evolution of the split in Figs. 11 (b) and (c). In Fig. 12(d), unlike the trends for previous filters that exhibit an unchanged filter shape when ∆L is varied, the filter shape shows a slight asymmetry in the stop-band when ∆L is away from 0. This is mainly due to the asynchronous feedback in the elliptic filter, which results in asymmetrically located zeros around the center frequency [178].

Conclusions
We have theoretically investigated advanced multi-functional integrated photonic lters based on 3WC-SLR resonators. Mode interference in the 3WC-SLR resonators is tailored to achieve high performance ltering functions including optical analogues of Fano resonances with ultrahigh ERs and SRs, resonance mode splitting with high ERs and low FSRs, and classical Butterworth, Bessel, Chebyshev, and elliptic lters. The requirements for practical applications are considered in our designs, together with detailed analysis of the impact of structural parameters and fabrication tolerances. This work highlights the 3WC-SLR resonators as a powerful and versatile approach to exible spectral engineering for a diverse range of applications. Figure 1 Page 23/28