Broadband high-Q multimode silicon concentric racetrack resonators for widely tunable Raman lasers

Multimode silicon resonators with ultralow propagation losses for ultrahigh quality (Q) factors have been attracting attention recently. However, conventional multimode silicon resonators only have high Q factors at certain wavelengths because the Q factors are reduced at wavelengths where fundamental modes and higher-order modes are both near resonances. Here, by implementing a broadband pulley directional coupler and concentric racetracks, we present a broadband high-Q multimode silicon resonator with average loaded Q factors of 1.4 × 106 over a wavelength range of 440 nm (1240–1680 nm). The mutual coupling between the two multimode racetracks can lead to two supermodes that mitigate the reduction in Q factors caused by the mode coupling of the higher-order modes. Based on the broadband high-Q multimode resonator, we experimentally demonstrated a broadly tunable Raman silicon laser with over 516 nm wavelength tuning range (1325–1841 nm), a threshold power of (0.4 ± 0.1) mW and a slope efficiency of (8.5 ± 1.5) % at 25 V reverse bias.


S1. Mode coupling in a multimode single resonator
Fig. S1 | Schematic diagram of a multimode single racetrack resonator. The input pump is injected into the bus waveguide and couples to the racetrack. The mode coupling between the TE0 mode and higher-order mode happens inside the racetrack.
The loaded quality (Q) factor is a parameter that depends on the total loss of a resonant cavity, including coupling loss and propagation loss. A cavity with broadband high loaded Q factors is desired for nonlinear applications, potentially useful for enabling lower input powers by enabling high enhancement over a broadband range. Firstly, we would like to explain why mode coupling can be a problem on the loaded Q factors in a multimode single resonator.
For a multimode single resonator, the influence of mode coupling on loaded Q factors can be explained from equations (S1-S2) by modeling the multimode single resonator with only fundamental (TE0) mode and higher-order mode 1 . Parameters A0 and A1, ω0 and ω1, τ0 and τ1, κ0 and κ1 are the amplitudes, resonant frequencies, photon lifetimes, and coupling coefficients of the TE0 mode and higher-order mode, respectively. ut is the mutual coupling coefficient between the two modes. Sin and Sout are the amplitudes of the input and output pump waves.  In a multimode single resonator like Fig. S1, by assuming the TE0 mode and higher-order mode with parameters listed in Table S1, we solved equations (S1-S2) with different mutual coupling coefficients and obtained the transmission spectra in Fig. S2. Here, two adjacent wavelengths instead of equal resonant wavelengths are selected to display the independent loaded Q factors of the two modes by setting the mutual coupling coefficient ut to 0. In reality, ut is a fixed value for a given structure. When the two resonant frequencies approach each other, amplitudes of the two modes would increase, leading to the enhancement of the coupling term -jutA0 or -jutA1. In the simulation, we fixed the two resonant frequencies and gradually increased ut. With the obtained transmission spectra in Fig. S2, we calculated the loaded Q factors for different ut and summarized them in Table S2. With the increase of ut, the loaded Q factor for TE0 mode gradually reduces. Because the resonances of the TE0 mode and a higher-order mode get closer in the wavelength domain, the TE0 mode thus suffers from additional effective loss from the coupling to the higher-order mode and thus its loaded Q factor is reduced. The loaded Q factor of the TE0 mode can reduce by about 50% compared with that without the mode coupling as summarized in Table S2.
The performances of the two modes in the multimode single resonator can also be described by equations  coupling occurs at about 1556 nm. The loaded Q factor was calculated to be 4.9 × 10 5 compared to the loaded Q factor of 1.0 × 10 6 at 1553 nm which is away from the mode coupling. There is a 51% reduction in the loaded Q factor at the mode coupling region. The experimental transmission spectrum in Fig. S3b agreed well with the theoretical prediction in Fig. S3a.    Table S3, we solved the equations (2a-2c) 1 in the paper and obtained the transmission spectra of the multimode concentric resonator with different mutual coupling coefficients ut as shown in Fig. 1e of the paper. We calculated the loaded Q factors and summarized them in Table S4. We can see that the loaded Q factor of the OutTE0 slightly enhances as ut increases. It can be understood that the OutTE0 with higher propagation loss couples to the InTE0 with lower propagation loss. The InTE0 shares the propagation loss of the OutTE0. Thus, the loaded Q factor of the OutTE0 increases. In addition, also for the lower propagation loss of the inner resonator, when the mutual coupling happens, the InTE0 can present a higher loaded Q factor than that of the OutTE0 without mutual coupling. In other words, two resonances with higher loaded Q factors than that of OutTE0 without mutual coupling can appear in one FSR as shown in Table S4. We found that both the OutTE0 and InTE0 can keep loaded Q factors over 10 6 after the interaction. That is, even though the mode coupling from the higher-order mode can highly decrease the loaded Q factor at TE0 mode, as present in the multimode single resonator, the existence of the two high-Q resonances at TE0 mode in the multimode concentric resonator can keep at least one high-Q resonance at TE0 mode in each free spectral range (FSR).  We first experimentally characterized the transmission spectra of the two resonators. Even employing the broadband pulley directional coupler, we found that the multimode single resonator still suffers from the effects of the higher-order modes periodically in Fig. S7. Because the higher-order modes can become nearly coincident with the fundamental modes periodically. We magnify the affected resonances of the multimode single resonator at the mode coupling region in Fig. S8a. As the higher-order mode (triangle mark) gets close to the TE0 mode, more loss is coupled into the TE0 mode which results in a high decrease in the extinction ratio and the loaded Q factor. However, for the same wavelength range of the multimode concentric resonator in Fig. S8b, due to the split of two high-Q TE0 resonances at 1591 nm, even though the higher-order mode slightly catches the right TE0 resonance, the left TE0 resonance is barely affected. The principle of concentric resonators mitigating loaded Q factor reduction can be stated as follow. When a TE0 mode couples to a higher-order mode with a much larger propagation loss in a multimode single resonator, the loaded Q factor would be reduced seriously. However, in a multimode concentric resonator, when OutTE0 couples to InTE0 with slightly smaller propagation loss, both consequently split resonances present higher loaded Q factors comparable to that of the main resonant mode without mutual coupling. Therefore, by adding an inner resonator, we add an additional high-Q resonance within one FSR when independent resonant wavelengths of OutTE0 and InTE0 approach each other. If one mode encounters a higher-order mode and suffers from a serious reduction in loaded Q factor, there is still another mode propagating mainly in another resonator with displaying a high loaded Q factor.

S3. Broadband high-Q multimode concentric resonator
In detail, when a higher-order mode slightly couples to a TE0 mode, they share the coupling loss and propagation loss of each other. However, the propagation loss of the higher-order mode can be up to 8 times larger than that of the TE0 mode. Therefore, in the mode coupling region, the effective propagation loss of the TE0 mode would increase. Conventionally, in a multimode single resonator, the higher-order mode is unwanted and its coupling from the bus waveguide should be suppressed. The small coupling ratio of the higher-order mode can lower the effective coupling ratio of the TE0 mode when they couple to each other and share coupling ratios. Therefore, if the reduced coupling loss of the TE0 mode is smaller than its increased propagation loss, the total loss of the TE0 mode still increases, leading to the reduced loaded Q factor. TE0 mode always presents the reduced loaded Q factor in the mode coupling region for near-critical and under coupling cases. On the contrary, when the reduced coupling loss of the TE0 mode is larger than its increased propagation loss, the enhancement of the loaded Q factor occurs. We observed such a phenomenon in experiments only when TE0 mode is seriously over coupled with a low extinction ratio of only 6 dB. For nonlinear application, the cavity usually would not be so seriously over coupled where the loaded Q factor is quite small. Even though it can get a higher loaded Q factor from the mode coupling, most resonances are still with low Q factors. Therefore, we didn't consider it here. In our designs, the main resonant mode which is away from the mode coupling region presents slightly over coupling with a large extinction ratio of 15 dB under a low pump power from the laser. We classify the slightly over-coupling with extinction ratio over 10 dB into the near-critical coupling of which the loaded Q factors are still high at the order of 10 6 . As above mentioned, because of the mutual coupling from the inner resonator with lower propagation loss, two resonances with higher loaded Q factors than OutTE0 without mutual coupling can occur. If the higher-order mode affects one mode, there remains another mode presenting a high loaded Q factor. That is why QL_MC/QL_NMC can be larger than 1 in some mode coupling regions, as indicated in Fig. S9. We To further validate our results experimentally, we fabricated two additional multimode concentric resonators. We kept the gaps between the bus waveguide and the outer racetracks as 400 nm. But we varied the gaps between the inner and outer racetracks as 500 nm and 300 nm in the two devices.
Compared to the original device with a gap of 400 nm, the number of the mode coupling regions has increased by 45% and 19%. It may imply that the gap between the inner and outer racetracks is an important aspect to engineer for fewer mode coupling regions. Another two multimode concentric resonators with changing the width of the inner racetrack to 1.5 µm (the same width of the outer racetrack) and 3.25 µm were also fabricated in the same chip. The number of the mode coupling regions has increased by 15% and 6%. Very large reductions of the loaded Q factors in the multimode concentric resonator with a 1.5-µm -width inner racetrack were found in three of the mode coupling regions. A wide inner racetrack can reduce the loss, but it can also induce more higher-order modes. Thus, careful engineering of the widths of the inner racetrack is also critical. To reduce the higher-order modes in the multimode concentric resonator, other approaches, like single-mode bends, Euler bends, and Bezier bends are useful.

S4. Modes in the multimode concentric resonator
The simulated eigenstates of the supermodes in the concentric racetrack resonator are shown in Fig. S10.
For the wavelength range from 1240 nm to 1680 nm, the TE2 mode in the multimode outer resonator is strong at a short wavelength and gradually becomes less confined in the long wavelength. The inner resonator has a width of 3 µm which can support at least five modes as the below simulated mode profile.
That is, the multimode concentric resonator is composed of multimode waveguides for both the outer and inner resonators.

S5. Raman lasing characteristics
For resonators, maintaining a high Q factor is important in some nonlinear devices because the high Q can directly enable low-input power devices. The enhancement factor, denoting the largest build-up of light intensity in the cavity, is defined as the light intensity coupled into the ring over the light intensity in the bus waveguide 3 : ̂ is the transmission coefficient and a is the dimensionless loss coefficient. We calculated the ratios of the enhancement factors in the mode coupling region over the enhancement factors out of the mode coupling region in Fig. S11. Generally, the multimode concentric resonators exhibit smaller variation in enhancement factor than the multimode single resonators, since MMC/MNMC is closer to one in Fig. S11 for the multimode concentric resonators. For the values of the enhancement factors, the multimode concentric resonators also manifest larger values compared to the multimode single resonators at the mode coupling regions. As an example of the benefit of using the multimode concentric resonator, we demonstrated its use in an integrated widely tunable Raman laser, where the enhancement factor directly affects the required input pump power to achieve Raman lasing threshold Pth as follows.
s eff th r αs is the linear loss coefficient for the Stokes, Aeff is the effective area, gr is the Raman gain coefficient and T is the transmission factor (output power over input power). In the mode coupling region, we separately chose one resonance of the multimode single resonator and one resonance of the multimode concentric resonator to compare their Raman lasing threshold powers.
They have corresponding enhancement factors as 50.8 and 15.3. Using equation S11, we can calculate the Raman lasing threshold power as 0.3 mW and 1.3 mW. Fig. S12 shows the Raman lasing threshold powers were measured as 0.5 mW and 1.5 mW, in consonance with the theoretical predictions. Besides, due to the highly decreased Q factor in the mode coupling region of the multimode single resonator, the Stokes output power is largely reduced by 500 times smaller at the similar pump powers. If using the highly affected resonance with the largely decreased Q factor in Fig. S4, the theoretical threshold power increases to 94 mW, which is 200 times larger than that in the multimode concentric resonator.

S6. Raman lasing experimental setup
For the Raman lasing measurement, the pump light was injected from a tunable laser. A polarization controller was used to adjust the light into the quasi-transverse-electric polarization. Then the light was coupled into the device via the edge couplers and lensed fibers. A 1:99 fiber coupler divided the light into a power meter to measure the output power and an optical spectrum analyzer to record the output spectrum.

S7. Resonator characteristics
For an all-pass resonator, the loaded quality (QL) factor can be expressed as 4 : g L= , FWHM (1 ) where FWHM is the full width at half maximum of the resonance; ng is the group velocity; a = exp(-αL/2) is the amplitude transmission and α is the power attenuation coefficient; L is the roundtrip length of the resonator; λ is the resonant wavelength; ̂ is the self-coupling coefficient of the directional coupler. The intrinsic quality (Qi) factor and coupling Q (Qc) factor can be calculated by [5][6][7] : where ω is the resonant frequency; R is the radius of the resonator; T0 is the normalized transmission at the resonance. Equation S13 takes -sign for over-coupled regime while + sign for under-coupled regime.
To satisfy the energy conservation, ̂2 = 1 −̂2, where ̂ is the cross-coupling coefficient of the directional coupler. To calculate , a supermode approach is used 8 : where uL = πΔn/λ is coupling strength; Ldc is the length of the directional coupler; Δβ is the difference between the propagation constants of the bus waveguide and the outer racetrack. In the design, we only consider the eigenmode, OutTE0, of the concentric racetracks, since OutTE0 is directly coupled with the bus waveguide and hence can be seen in transmission even without mode coupling. If the power attenuation coefficient of OutTE0 is assumed to be wavelength-independent, the required power coupling fraction from the bus waveguide to the concentric racetracks for loaded Q factor around 10 6 can be taken as a constant value, denoted as ĉ 2 . In other words, for a broadband resonator 9 , d ĉ 2 /d ≈ 0 . If a symmetric directional coupler is applied, Δβ = 0 and ĉ 2 = sin 2 ( L • dc ), meaning that ĉ 2 depends totally on uL.
Because uL is positively related to wavelength, i.e., the longer wavelength yields the less confined modes and more coupling. Therefore, it is difficult to get broadband critical coupling for the symmetric directional coupler. We adopt an asymmetric directional coupler here. Thus, a phase-mismatched condition is introduced to compensate for the change of uL with wavelength.

S8. Comparison of silicon Raman lasers
In Table S5, we summarize the progress on integrated silicon Raman lasers from the pulsed Raman laser to continuous-wave Raman laser 10,11 . Attempts on developing a low-threshold Raman laser and extending the lasing wavelength band from C-band to the mid-infrared region have also been included 3,12 . A further approach to hybrid silicon with rare-earth can reduce the Raman lasing threshold power and extend the lasing wavelength to 1.9 μm, which is promising for mid-infrared applications 13 .  Choosing the pump power around the lasing threshold of 0.3 mW, we measure the photocurrents with reverse bias voltages from 0 V to 30 V in Fig. S14. The photocurrents increase rapidly from 0 V to 15 V and are prone to saturate after 25 V. Therefore, we adopt 25 V reverse bias voltages in the Raman lasing measurements.

S10. Development of low-loss silicon nano waveguide
The 450-nm SOI waveguides fabricated by a commercial multi-project wafer foundry have standard propagation losses of around 2 dB/cm. The wider multimode waveguides offer smaller propagation losses for the fundamental mode because of the reduced modal overlap with the sidewall roughness. Therefore, our fabricated multimode waveguide racetrack resonators can achieve loaded Q factors over 10 6 . Apart from the multimode structure, recent progress on nano waveguides can achieve propagation loss as low as 0.4 dB/cm by optimizing the fabrication such as the etchless thermal oxidation 14,15 , high-resolution ArF immersion lithography 16 , and post-etching roughness removal processes 17 . They are summarized in Table   S6. We used the commercial foundry for fabrication, so we used the multimode waveguide to obtain low loss and further achieved a high-Q resonator.