Silicon photonic filters based on cascaded Sagnac loop resonators

We demonstrate advanced integrated photonic filters in silicon-on-insulator (SOI) nanowires implemented by cascaded Sagnac loop reflector (CSLR) resonators. We investigate mode splitting in these standing-wave (SW) resonators and demonstrate its use for engineering the spectral profile of on-chip photonic filters. By changing the reflectivity of the Sagnac loop reflectors (SLRs) and the phase shifts along the connecting waveguides, we tailor mode splitting in the CSLR resonators to achieve a wide range of filter shapes for diverse applications including enhanced light trapping, flat-top filtering, Q factor enhancement, and signal reshaping. We present the theoretical designs and compare the CSLR resonators with three, four, and eight SLRs fabricated in SOI. We achieve versatile filter shapes in the measured transmission spectra via diverse mode splitting that agree well with theory. This work confirms the effectiveness of using CSLR resonators as integrated multi-functional SW filters for flexible spectral engineering.

By changing the reflectivity of the SLRs and the phase shifts along the connecting waveguides, mode splitting in the CSLR resonators can be tailored for diverse applications such as enhanced light trapping, flat-top filtering, Q factor enhancement, and signal reshaping. We achieve high performance and versatile filter shapes that correspond to diverse mode splitting conditions. The experimental results confirm the effectiveness of our approach towards realizing integrated multi-functional SW filters for flexible spectral engineering. 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. 17 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. 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:

II. DEVICE CONFIGURATION AND OPERATION PRINCIPLE
R si = 2jt i κ i a si e −jφ si , i = 1, 2, .., N Ti (i = 1, 2, .., N-1) represent the field transmission functions of the waveguide connecting SLRi to SLRi+1, which can be expressed as: (4) 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, .., 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. In Eqs. (5) and (6) 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.  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 band-pass Butterworth filter 34 with a flat-top 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 flat-top group delay spectrum, which can be used as a Bessel filter for optical buffering. 37,38 When t2 =√1/2, SLR2 works as a total reflector, and so there is null transmission for the CSLR resonator. The same goes for t1 =√1/2 or t3 =√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 high-Q filters. 7,32 .   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 thermo-optic micro-heaters 19,33 or carrierinjection electrodes 39,40 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 Vernier-like 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 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.  In the calculation, we assume that t1 = t2 =…= t8 = 0.97. With enhanced light trapping, there are increased time delays and enhanced light-matter 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 42,43 . Based on two-photon 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.

III. DEVICE FABRICATION
We fabricated a series of CSLR filters based on the designs in Section II, on an SOI wafer with a 260-nm- where Li′ (i = 1, 2, …, N-1) are the lengths of the connecting waveguides excluding the straight coupling lengths. For practical fabrication, we used the same Sagnac loop structure for each SLR, with the differences in Lci (i = 1, 2, …, N) compensated by slightly changing Li′ (i = 1, 2, …, N-1) according to Eq. (7).  Table II, which are close to our expectations from the design. For ng, α, and ti, the difference between the fit and designed values are smaller than 0.01, 20 m -1 (0.6 dB/cm), and 0.05, respectively. The residual differences between the fit ng and α can be attributed to slight variations between the fabricated samples. In Fig. 7(a), various mode splitting spectra of the fabricated devices with different Lc2 are obtained, which are consistent with the theory in Fig. 2(a). The measured spectra of the fabricated devices with different Lc1 = Lc3 in Fig. 7(b) also agree well with the theory in Fig. 2(c).
These experimental results verify that the transmission spectra of the CSLR resonators can be tailored by changing the coupling strengths of the directional couplers in the SLRs. Since we have demonstrated in Ref.
29 that dynamic tuning of the coupling strengths can be realized by using interferometric couplers to replace the directional couplers and tuning them in a differential mode, tuning of the transmission spectra of the CSLR resonators can also be achieved in the same way.   μm. In the device pattern, the loop region of SLR1 was rotated to the bottom side of SLR2. It can be seen that the measured spectrum shows good agreement with the theory in Fig. 3(c), with the discrepancies mainly arising from the grating coupler spectral response as well as slight variations in coupling coefficients with wavelength. 46 The dispersion of the SOI nanowire waveguides is another factor that could account for the discrepancies, since we used ng rather than the wavelength-dependent effective index to match the FSR in our theoretical calculations. are shown in Fig. 9(a). The fitting parameters are listed in Table II Table II. The device in Fig. 8(b) was designed for enhanced light trapping, and the measured transmission spectrum is similar to the calculated spectrum in Fig. 5(c). The measured filter shape in Fig. 9(b) exhibits a slight asymmetry and this is because the additional phase shift along L4 is not exactly π/2. By introducing thermo-optic microheaters or carrier-injection electrodes along L4 to tune the phase shift, the symmetry of the filter shape can be improved further. The device in Fig. 9(c) was designed to perform as an 8th-order Butterworth filter with a flat-top filter shape. As can be seen, the passband is almost flat, which is close to the calculated spectrum in Fig. 5(e). The 3-dB bandwidth is ~0.7 nm, which is significantly narrower than what can typically be achieved by silicon waveguide Bragg gratings 30 . By either increasing the relevant ti or by increasing the number of SLRs, the 3-dB bandwidth can be further improved. The slight unevenness of the top of the transmission band can be attributed to discrepancies between the designed and practical coupling coefficients. Figure 9(d) shows the measured transmission spectrum with multiple resonant peaks. The minimum extinction ratio of the transmission peaks is ~7.8 dB, which is slightly lower than that in Fig. 5(f). This is mainly because the waveguide propagation loss of the fabricated devices (α = 64 m -1 ) is slightly higher than we assumed in the calculation (α = 55 m -1 ). By further reducing the propagation loss, higher extinction ratios of the split resonances can be obtained.

V. CONCLUSION
In summary, we design and fabricate sophisticated and high performance optical filters in SOI nanowires through the use of mode splitting in integrated CSLR resonators, by designing the reflectivity of the SLRs and the phase shifts along the connecting waveguides. These filters are extremely useful for a wide range of applications including enhanced light trapping, flat-top filtering, Q factor enhancement, and signal reshaping.
We theoretically analyse and practically fabricate devices with up to eight SLRs. We measure the transmission spectra of the fabricated devices and obtain versatile filter shapes corresponding to diverse mode splitting conditions. The experimental results are consistent with theory and validate the CSLR resonator as a powerful and versatile approach to realise multi-functional SW filters for flexible spectral engineering in photonic integrated circuits.

ACKNOWLEDGMENTS
This work was supported by the Australian Research Council Discovery Projects Program (DP150104327).
We also acknowledge the Melbourne Centre for Nanofabrication (MCN) for the support in device fabrication and the Swinburne Nano Lab for the support in device characterisation.