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Abstract
Tunable ring resonators are essential devices in integrated circuits. Compared to conventional ring resonators, valley photonic crystal (VPC) ring resonators have a compact design and high quality factor (Q-factor), attracting broad attention. However, tunable VPC ring resonators haven't been demonstrated. Here we theoretically demonstrate the first tunable VPC ring resonator in the telecommunication wavelength region, the resonance peaks of which are tuned by controlling the temperature based on the thermal-optic effect of silicon. The design is ultracompact (12.05 µm by 10.44 µm), with a high Q-factor of 1281.00. By tuning the temperature from 100 K to 750 K, the phase modulation can reach 7.70 π, and the adjustment efficiency is 0.062 nm/K. Since thermal tuning has been broadly applied in silicon photonics, our design can be readily applied in integrated photonic circuits and will find broad applications. Furthermore, our work opens new possibilities and deepens the understanding of designing novel tunable VPC photonic devices.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Currently, silicon photonics has been regarded as one of the most promising photonic integration platforms due to the combination of a very high index contrast and the availability of complementary metal-oxide semiconductor (CMOS) fabrication technology [1–5], which allows the fabrication of photonic circuitry chips. Silicon photonic devices have shown unprecedented compactness, especially wavelength-selective devices, among which ring resonators play an essential role in the success of silicon photonics because silicon enables ring resonators of unprecedented small size.
In general, an optical ring resonator is a set of optical waveguides in which at least one is a closed loop coupled to input and output waveguides. The working principle of optical ring resonators is based on the constructive interference of the light at resonant wavelength, which builds up in intensity over multiple roundtrips. Since resonances only occur when the resonator's optical path length is exactly a whole number of wavelengths [6], the optical ring resonator can work as a filter. In addition, two or more ring resonators and waveguides can be coupled to form an add/drop optical filter [7] or an all-pass notch filter [6]. Furthermore, by different combinations, ring resonators can be applied in wave-division-multiplexing (WDM) devices [8], logic gates [9], optical modulators [10], tunable lasers [11], etc.
The spacing between these resonances is defined as the free spectral range (FSR). For many applications, a relatively large FSR (several nm) is preferred, which requires the size of rings to be small (a few microns). Although the high refractive contrast between silicon and air can facilitate strong confinement for waveguides, the imperfections of the structure during the manufacturing process still introduce substantial optical loss, degrading the micro ring resonators’ (MRRs) performance. These effects happen to conventional silicon waveguides and waveguides based on photonic crystal (PC) structures [12–21].
In comparison, topological photonic crystals (TPCs) [22–30] have unique anti-scattering unidirectional transmission properties. The recent development of valley photonic crystal (VPC) structures allows the design and fabrication of VPC devices at the telecommunication wavelength region [22,24], enabling highly efficient optical MRRs with the desired features of small size, low losses, and integrability into (existing) optical networks. Additionally, since the resonance wavelengths can be changed by manipulating the effective optical path length of MRRs, the structures can be tuned by controlling the effective refractive index of the structure without changing the geometric structure [31–33], which is vital in the applications of silicon photonic devices [3,4]. However, so far, tunable VPC ring resonators haven't been demonstrated.
Here we theoretically demonstrate the first tunable topological all-pass notch filter (APNF) based on silicon material in the telecommunication wavelength region. The APNF comprises a topological waveguide and a topological MRR, which can achieve a high Q factor of 1314.45 at the working wavelength of 1577.34 nm and a large FSR of 10.54 nm. The tuning is conducted based on the thermo-optic effect of silicon. Within the tunning temperature range from 100 K to 750 K, the overall phase shift is 7.80π. It requires only 83.33 K tuning to realize a phase shift of π. The tuning efficiency is calculated to be 0.063 nm/K. These designs confirm the feasibility of designing thermo-optic modulators based on VPC structures, which can be fabricated using CMOS nanofabrication technology. Thus, our designs open new possibilities in designing nanophotonic devices based on VPCs and will find broad applications in optical communications and quantum computing [34,35].
2. Structural design of thermo-optic tunable VPC structures
The schematic of the thermo-optic tunable APNF is shown in Fig. 1, which is embedded in a free-standing silicon substrate with a thickness of h = 220 nm. The APNF comprises a topological MRR and a straight topological waveguide. The MRR and the waveguide are constructed using two VPC structures, namely VPC1 and VPC2. Those VPC structures are the honeycomb lattice with circular air holes. Here the quantum emitters are placed on the straight waveguide. The arrows indicate the light propagation direction of right-handed circularly polarized (RCP) light in the APNF. The blue and red represent the working states at low and high temperatures. The tuning can be implemented experimentally by heating the entire structure within a closed environment, for example, a heating chamber. By tuning the temperature from T1 and T2, the resonance peak shifts from λ1 and λ2. The entire tuning process is reversible as long as the maximum temperature is below silicon's thermal damaging threshold temperature (∼973 K). Other tuning mechanisms can be applied to achieve tunings, such as the nonlinear optical effect [36] and electro-optic effect [37], which will require highly doped silicon and sophisticated designs [38–40] that are beyond the scope of this work. Here we demonstrate the tuning based on the thermo-optic effect [41] in silicon without doping and consider the case that the temperature distribution in the silicon substrate is uniform.
Fig. 1. Schematic of the thermos-optic tunable topological APNF working at low temperature (a) and high temperature (b). The yellow arrows mark the light path of right-handed circularly polarized (RCP) light in the APNF. h is the thickness of the silicon plate. (c) Schematic of tuning the filtering peak position via temperature. The λ1 and λ2 correspond to the resonance peak positions at the temperatures of T1 and T2, respectively. The tuning process is reversible.
To realize the spin-valley locking effect, we first build a honeycomb PC structure using air hole arrays (lattice constant a = 482 nm) with C6v symmetry (R1 = R2 = 86 nm) to achieve a Dirac point in the photonic band structure (Fig. 2(b)). Previously, we demonstrated high-performance VPC waveguides based on triangular-shaped air holes [42] working in the telecommunication wavelength region. Despite the broad working bandwidth, the triangular shape lattice suffers relatively high loss at the sharp corners. Therefore, it is not suitable for applications in designing MRRs. Thus, in this work, circular air holes are used. By enlarging the radius of a set of air holes and reducing the other (R1 = 129 nm and R2 = 43 nm for VPC1 and R1 = 43 nm and R2 = 129 nm for VPC2), the C6v rotation symmetry can be reduced to C3v, which opens a topological photonic bandgap at the K point to form VPC1 and VPC2 (Fig. 2(b)). The bandgap is in the wavelength range of 0.27-0.33 a/λ (1455.31 nm-1776.72 nm), indicated by the shadow area in Fig. 2(b). The values of the Berry curvature and topologically invariant valley Chern number CV corresponding to the two VPCs were calculated, which are CV = -1 for VPC1 and CV = 1 for VPC2 [42,43], respectively.
Fig. 2. (a) Schematics of the unit cells of the initial honeycomb PC and the VPC1 structures. a represents the lattice constant, and R1 and R2 represent the radius of the air circular hole. (b) Photonic band diagrams of the PC and VPC1 structures. The black and red dotted lines represent the photonic band structure of PC and VPC1, respectively. The light blue rectangular region represents the photonic bandgap range (1455.31nm-1776.72 nm), and the gray part represents the light cone. (c) The tuning curve of the photonic band gap range with temperature. (d) Topological edge state curves at 100 K, 450 K, and 750 K.
Here we first study the effect of temperature tuning on the position and width of the bandgap, which is a result of different dispersive refractive indices at different working temperatures [44] (The detail is shown in Supplementary Section 1). The temperature range used in this study is from 100 K to 750 K, which is well below the thermal damage threshold of silicon at 973 K. Therefore, the entire tuning process can be dynamic and reversible. The step is 50 K, except from 250 K to 293 K, in which the temperature is increased by 43 K. Since 293 K is a frequently used room temperature, we would like to include this particular point in this study. The central wavelength of the bandgap redshifts along with the increase of working temperature due to the positive thermo-optic coefficient of silicon (the refractive index increases along with the increase of temperature), as shown in Fig. 2(c). The working bandwidth varies from 1450.00 nm-1759.10 nm at 100 K to 1481.11 nm-1801.99 nm at 750 K. Meanwhile, the central wavelength redshifts from 1604.50 nm to 1641.00 nm at a tuning rate of 0.056 nm/K. The gap width (marked by the blue in Fig. 2(c)) becomes larger (from 309.1 nm at 100 K to 320.88 nm at 750 K).
There are two possible combinations to construct a waveguide using VPC1 and VPC2, namely the zigzag and beard geometries, which support different topological edge states, as shown in Figs. S2(a) and (b) (The detail is shown in Supplementary Section 2). Compared to the beard geometry, the zigzag geometry (Fig.S2(a)) can have a strong spin-valley locking effect (Fig. S2(c)), thus having higher transmittance and broad bandwidth occupying the entire topological bandgap (Figs. S2(d) and (f)). Due to the spin-valley locking effect, different polarizations are locked to different valleys (K or K’ valley) to achieve a high transmittance near unity (Fig. S2(f)). Thus, the zigzag geometry is chosen in this study. We further study the effects of temperature tuning on the topological edge states. The resulting edge state curves are shown in Fig. 2(d) and Fig. S3 (The detail is shown in Supplementary Section 3), which show a redshift along with the temperature increase. Since all the edge states can cross the entire bandgap, the working bandwidth still depends on the bandgap, confirmed by the transmittance spectra shown in Fig. S2(f).
3. Realization of the thermo-optic tunable topological ring resonator
Due to the spin-valley locking effect, circularly polarized light (CPL) can propagate along the Γ-K or Γ-K’ direction with a minimal loss depending on the handiness of the circular polarization, even at a very sharp angle (for example, 60°). We construct a triangular MRR using the straight topological waveguides based on zigzag geometry, schematically shown in Fig. 3(a). 3D finite difference time domain (FDTD) method (commercial Lumerical FDTD software) is used to study the optical properties. Here the emitter is embedded in the MRR to explore the coupling between the emitter and topological edge states. The total length of the ring (3 L, L is the side length of the triangle) can be used to control the number and the positions of the resonance peaks. Here we focus on the case L = 25a (12.05 μm) to study the thermo-optic tuning, which is an optimized design according to our calculation. And the tuning method can generally be applied to topological MRRs of any length. The transmittance spectrum at 300 K is shown in Fig. 3(b), where one can see several peaks within the bandwidth (marked by the grey area). All these peaks belong to the topological edge states. The FSR is 10.54 nm, as marked in the figure. The highest peak is at the wavelength of λ=1577.34 nm, and the full width at half maximum (FWHM) is Δλ=1.20 nm, which corresponds to a quality factor of Q = λ/Δλ = 1314.45, which is comparable to the state-of-the-art Q factor of topological MRR [45]. According to the electric field density distribution shown in Fig. 3(c), all the optical energy is well confined in the MRR without strong scattering, even at the sharp corners, due to the spin-valley locking effect. In addition, from the plot of Poynting vector distribution inside the MRR shown in Fig. 3(d), one can see the optical wave circulate the triangular MRR. We further applied different temperatures to study the tuning range of the peaks, shown in Fig. S4 (The detail is shown in Supplementary Section 4) and Fig. 3(e). As expected, the peak position redshifts along with the temperature increase, providing a phase modulation of 7.80π (41.12 nm wavelength shift) within the temperature range from 100 K to 750 K. The curve of peak position versus temperature is plotted in Fig. 3(f), which shows a linear relationship. The tuning efficiency can be calculated by fitting the curve according to the following equation [31]
Fig. 3. (a) Schematic of a topological MRR structure. The side length of the triangular MRR is L = 25a, and RCP is the light source on the MRR. The yellow arrows indicate the direction in which light waves are traveling. (b) The transmittance spectrum of the MRR shown in (a); the gray shaded area marks the working bandwidth. (c) Electric field intensity distribution at 1577.34 nm in the MRR. (d) Poynting vector distribution at 1577.34 nm in the MRR, the red arrows indicate the vectorial direction. (e) The transmittance spectra of a specific resonant peak (at 1577.34 nm when the temperature is 300 K)in the MRR at different temperatures. (f) The curves of the relationship between the wavelength (at 1577.34 nm when the temperature is 300 K) and Q factor of the resonant peak versus temperature (dots) and linear fitting (solid lines).
4. Realization of thermo-optic tunable APNF
It is ideal to use the same topological edge states in the straight waveguide and the MRR to achieve critical coupling between the two elements to form a high-performance APNF. Therefore, the zigzag geometry is also applied to design the straight waveguide. The distance between the straight waveguide and the MRR controls the coupling condition at different wavelengths. The distance in an APNF based on conventional silicon rigid waveguides and MRRs can be freely tuned to achieve critical coupling at a specific wavelength. In comparison, the distance in any APNF based on PC structures can only move units of the lattice constant. Therefore, the tuning capability of PC-based APNFs is relatively limited. In our design, the straight waveguide touches a corner of the MRR. The RCP light source is placed on the straight waveguide. As expected, the APNF shows a high transmittance (>0.8) within the working bandwidth, except for the dips coupled to the MRR (Fig. 4(b)). The minimum of the dips depends on the coupling condition between the straight waveguide and the MRR. A value of 0 can be achieved under the critical coupling condition [46]. Here the distance between the straight waveguide and the MRR, defined as the distance from the central line of the straight waveguide to the center of the MRR, can be tuned at a step of Δd = Na. N is an integer, and a is the lattice constant. Therefore, it is challenging to fine-tune the coupling condition for each peak, which will require nanometer tuning accuracy. A minimum of 0.043 is achieved at the wavelength of 1537.21 nm, which is very close to the critical coupling condition. The FWHM of the dip at 1537.21 nm is 1.20 nm, which results in a Q factor of 1281.00. Meanwhile, the FSR of 10.54 nm is maintained. The electric field distributions at different wavelengths are plotted in Figs. 4(c) and (e), showing the dips come from the efficient coupling to the resonant modes of the MRR. One can see that the resonant mode at 1537.21 nm is circulating within the MRR (Fig. 4(c)). The black arrows mark the propagation direction. In comparison, the mode at 1546.72 nm cannot be coupled to the MRR (Fig. 4(d)), thus resulting in a high transmittance.The notch filtering process can be further understood by considering the following physical scheme. Assuming a lossless balanced, i.e., a 50:50 beam splitting process at the branch, which is the touching point between the straight waveguide and the MRR, the transmittance can be expressed as [6]
where ${a_l}$ and $\delta = 6\pi {n_{eff}}L\textrm{ / }\lambda $ are the roundtrip loss coefficient and phase shift, respectively. ${n_{eff}}$ is the effective refractive index of the propagation mode, L is the length of the MRR, and $\lambda $ is the wavelength. When the phase shift δ is a multiple of 2π, the light is on resonance and shows a notch in the transmittance spectrum.Fig. 4. (a) Schematic of topological APNF comprising an MRR and a straight waveguide. (b) The transmittance spectrum of the APNF. (c) and (d) The electric field intensity distributions in the APNF at 1537.21 nm and 1546.72 nm. (e) Transmittance spectrum of a resonant peak (at 1537.21 nm when the temperature is 300 K) of the APNF at different temperatures. (f) The curves of the relationship between the wavelength (at 1537.21 nm when the temperature is 300 K) and Q factor of the resonant peak versus temperature (dots) and linear fitting (solid lines).
We further apply thermo-optic tuning of the APNF, expecting a similar trend to the ring-resonator, which is shown in Fig. 4(e) and Fig. S5 (The detail is shown in Supplementary Section5). The redshift for the dip at 1537.21 nm (at 300 K) is 40.59 nm in the temperature range, corresponding to 3.85 FSR, translating into a phase shift of 7.70 π, which is relatively large compared with the conventional MRRs. The tuning efficiency is 0.062 nm/K, comparable to a typical conventional MRR [47](0.052 nm/K) and a PC MRR [48] (0.08 nm/K). A π phase shift can be achieved with a temperature change of 84.42 K, which is relatively small compared to conventional MRR [47] (430.77 K for π phase shift) and a PC MRR [48] (281.25 K for π phase shift) calculated according to parameters shown in the papers. The lower temperature promises lower power consumption in the optical modulation process, which meets the future trend of highly integrated photonic chips. Meanwhile, the Q-factor and FSR have a similar tendency to the MRR, as shown in Table 2.
In order to demonstrate the unique defect immune transmission property of the topological APNF, we study the performance of the APNFs with different types of defects (the detail is shown in Supplementary Section 6), which shows neglectable shift (<0.1 nm) in the position of resonance dips and the same Q-factors. On the other hand, we also considered ±10% random errors in the hole size introduced by the fabrication process (the detail is shown in Supplementary Section 7). The transmittance spectrum of the APNF with defects is identical to the one without errors. Thus, neither defects nor fabrication errors degrade the performance of the APNFs, confirming the high feasibility in the experimental realization of the designed structures.
5. Conclusion
In conclusion, we have theoretically demonstrated a thermo-optic tunable topological APNF based on VPC structures, achieving a high Q factor, a large FSR, and a large tuning range. We have presented a comprehensive and systematic study on the thermo-optic effect of silicon on the photonic bandgap (as well as the working bandwidth), the topological edge states, and the peak/dip positions of the MRR/APNF. The design and tuning principle can be broadly applied to other thermo-optic tunable topological photonic devices, such as optical modulators, Mach-Zehnder interferometers, and directional couplers. The design is suitable for experimental fabrication using the CMOS nanofabrication technique. In addition, compared with conventional and photonic crystal MRRs, the performance of topological MRRs is not affected by defects and fabrication errors. Thus, it will find broad applications in optical communications and quantum computing.
Funding
Key R&D Program of Shanxi Province (International Cooperation) (201903D421052); National Natural Science Foundation of China (62175178); National Natural Science Foundation of China (U22A20258); Young Scientists Fund of the National Natural Science Foundation of China (11904255).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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