Integrated waveguides and nanowires for optical spectral shaping by using Sagnac loop reectors

We propose and theoretically investigate integrated photonic lters based on two coupled Sagnac loop reectors (SLRs) formed by a self-coupled optical waveguide. Recently we investigated integrated photonic lters based on cascaded SLRs and coupled SLRs. Here, we advance this eld by presenting a unique approach of using coupled SLRs formed by a self-coupled optical waveguide. This enables us to achieve high performance lter functions including Fano-like resonances and wavelength interleaving with a simpler design and a higher fabrication tolerance by tailoring coherent mode interference in the device. Our design takes into account the device fabrication issues as well as the requirements for practical applications. As a guide for practical device fabrication, an analysis of the impact of the structural parameters and fabrication tolerance on each lter function is also provided. The Fano-like resonances show a low insertion loss (IL) of 1.1 dB, a high extinction ratio of 30.2 dB, and a high slope rate (SR) of 747.64 dB/nm. The combination of low IL and high SR promises this device for Fano resonance applications. Our device also can achieve wavelength de-interleaving function with high fabrication tolerance which is attractive for optical interleavers that need a at-top symmetric lter shape. Optical interleavers and de-interleavers are core elements for signal multiplexing and demultiplexing in wavelength division multiplexing optical communication systems. Versatile spectral responses with a simple design, compact device footprint, and high fabrication tolerance make this approach highly promising for exible response shaping in a wide variety of applications.


Introduction
Integrated photonic resonators (IPRs) have allowed a variety of functional optical devices, including lters, modulators, sensors, switches, and logic gates, thanks to their compact footprint, exible topology, and great scalability [1][2][3][4]. Compared to the IPRS formed by photonic crystal structures [5] and gratings [6] that have sub-wavelength cavity lengths, the IPRs formed by directional-coupled wire waveguides with longer cavity lengths have smaller free spectral ranges (FSRs) that match the spectral grids of state-ofthe-art wavelength division multiplexing (WDM) optical communication systems, thus making them more widely applicable to these systems. In addition, sub-wavelength dimensions of photonic crystal cavity and Bragg grating structures are more prone to fabrication tolerances as compared with directionalcoupled wire waveguides. Ring resonators (RRs), and Sagnac loop re ectors (SLRs), which are essential building blocks for IPRs, are made up of directional couplers. Unlike RRs, which only allow for unidirectional light propagation, SLRs allow for bidirectional light propagation as well as mutual coupling between light travelling in opposing directions, resulting in a more versatile coherent mode interference and spectral response. Furthermore, a standing-wave (SW) resonator made up of cascaded SLRs has a cavity length about half that of a traveling wave (TW) resonator made up of a ring resonator with the same FSR, allowing for a more compact device footprint.
We investigated integrated photonic lters based on cascaded SLRs [7,8] and coupled SLRs [9,10] in our previous work. Here, we advance this eld by presenting the novel approach of using two coupled SLRs with a feedback loop formed by a self-coupled wire waveguide that yield different response shapes including Fano-like resonances and wavelength de-interleaving [11].
In our design, we take into account the device fabrication issues experienced in Refs. [7,8] as well as the needs for practical applications. As a guide for practical device fabrication, an analysis of the in uence of structural parameters and fabrication tolerance is also provided.

Device Con guration
The proposed structure is illustrated schematically in Fig. 1, which consists of two inverse-coupled SLRs with a feedback loop formed by a single self-coupled wire waveguide. Table 1 details the device's structural parameters. To simplify the discussion, we assume that L SLR1 = L SLR2 = L SLR . The spectral response of the device is calculated using the scattering matrix method [7,9]. In the device model, we use waveguide group index of n g = 4.3350 (transverse electric (TE) mode) and propagation loss of α = 55 m −1 (i.e., 2.4 dB/cm), which are in line with our previously fabricated silicon-on-insulator (SOI) devices [7,8,12]. The device is designed based on, but not limited to, the SOI platform. Coupler between SLR s t 2 κ 2 a a f = exp(-αL FL / 2), a si = exp(-αL SLRi / 2), α is the power propagation loss factor. b φ f = 2πn g L FL / λ, φ si = 2πn g L SLRi / λ, n g is the group index and λ is the wavelength. c t i 2 + κ i 2 = 1 for lossless coupling are assumed for all the directional couplers In the following sections, mode interference in the device is tailored to achieve high-performance ltering functions, including Fano-like resonances and wavelength de-interleaving.

Fano-like Resonances
Fano resonances are a fundamental physical phenomenon demonstrating an asymmetric spectral lineshape arising from quantum interference between discrete and continuum states [13,14]. They have underpinned many applications such as optical switching, data storage, sensing, and topological optics, due to their unique physics and capability of providing ultra-narrow spectral linewidths [13][14][15]. In this section, the spectral response of the device in Fig. 1 is tailored to realize Fano-like resonances with high slope rates (SRs) and low insertion loss (IL). The power transmission and re ection spectra is depicted in Fig. 2 Table 2. For comparison, the device structural parameters (L SLRi , n g , and α) of all the three structures were kept the same except for the transmission coe cients (t i ) that were tuned to obtain the highest SR for each structure. As compared with previous devices, the device presented here has a much lower IL of 1.1 dB, along with a slightly improved SR. The combination of high SR and low IL promises this device for Fano resonance applications. We note that a low IL of 1.1 dB is outstanding among the reported Fano-resonance devices on the SOI platform [17, 18], which makes the device here more attractive for practical applications. In Figs. 3(a)-(c), we further investigate the impact of the device structural parameters including t i (i = 1-3) and length variations of feedback loop (∆L FL ) on the performance of the Fano resonance. In each gure, we changed only one structural parameter, keeping the others the same as those in Fig. 2 (a). In Figs. 3(a)-(c), (i) shows power transmission spectra and (ii) shows the corresponding IL and SR for different t i (i = 1-3), and ∆L FL , respectively. The SR decreases with t i (i = 1, 3), while the IL rst decreases with t i (i = 1, 3) and then remains almost unchanged. The SR decreases with t 2 , while the IL shows an opposite trend, re ecting that both of the two parameters can be improved by enhancing the coupling strength between SLR 1 and SLR 2 . As shown in Fig. 3(c)

Wavelength De-interleaving Function
Optical interleavers and de-interleavers are core elements for signal multiplexing/demultiplexing in wavelength division multiplexing (WDM) optical communication systems [20,21]. In this section, we engineer the spectral response of the device in Fig. 1 to achieve wavelength de-interleaving function. Flattop spectral response of de-interleavers minimize the ltering distortions and group delay variation and high ER minimize signal crosstalk between adjacent channels [22]. Fig. 4(a) shows the power transmission and re ection spectra when the device structural parameters are L SLR = 100 µm, L FL = 300 µm, t 1 = 0.992, and t 2 = t 3 = 0.95. The IL, ER, and 3-dB bandwidth for the passband at output port are 0.36 dB, 12.7 dB, and 83.65 GHz, respectively. The IL, ER, and 3-dB bandwidth for the re ection spectrum at input port are 0.33 dB, 12 dB, and 91.9 GHz, respectively. As compared with at-top lters based on cascaded ring resonators [23], ring-assisted Mach-Zehnder interferometers [24], and cascaded SLRs [7], our device can achieve the same level of ltering atness with fewer subunits.
We further investigate the impact of varied t i (i = 1-3) in Figs. 4(b)-(d), respectively. For simpli cation, we only show the spectral response at output port. In Fig. 4(b), as t 1 increases, the ER of the passband decreases while the top atness improves, re ecting the trade-off between them. In Figs. 4(c)-(d), the bandwidth of the passband increases with t 2 , t 3 , respectively, while the ER shows an opposite trend.
We also investigate the impact of varied ∆L SLRi (i = 1, 2) and ΔL FL in Figs. 5(a)-(c), respectively. In Figs. 5(a)-(c), as ∆L SLRi (i = 1, 2) or ΔL FL increases, the lter shape remains unchanged while the resonance redshifts. Since the resonant cavity of the device is formed by a single self-coupled wire waveguide, random length fabrication errors in different parts (i.e., SLR 1 in Fig. 5(a), SLR 2 in Fig. 5(b), and feedback loop in Fig. 5(c),) will not induce any asymmetry in the lter shape. This yields a higher fabrication tolerance as compared with the coupled SLRs in Refs. [9,10], which is particularly attractive for optical interleavers that require a at-top symmetric lter shape.From Figs. 4(b)-(c) and Fig. 5, it can be seen that the slight changes in the structural parameters induced by fabrication disorders have no major impact on device performance.

Conclusions
We have theoretically investigated integrated photonic lters based on two coupled SLRs with a feedback loop formed by a self-coupled optical waveguide. High performance lter functions including Fano-like resonances and wavelength de-interleaving are achieved by tailoring coherent mode interference in the device. Our design takes into account the device fabrication experience as well as the requirements for practical applications. The impact of device structural parameters on each lter function is analyzed to facilitate optimized performance. Versatile spectral responses, compact device footprint, and high fabrication tolerance make this approach highly promising for exible response shaping in a wide variety of applications including potentially optical microcombs for advanced dispersion design for many applications.  Declarations Competing interests: The authors declare no competing interests.   Figure 1 Schematic con guration of device. The de nitions of t i (i = 1, 2, 3), L SLRi (i = 1, 2), and L FL are given in Table 1.