1. Introduction

Efficient and coherent frequency conversions, including intra-band and distinct-band conversion, have several classical and quantum applications, such as, communication, sensing, imaging, and detection [1–3]; for example, casting signals to specific channels in all-optical processing, connecting the telecommunications band with the visible/short near-infrared wavelength of atomic systems and quantum memory and for time/frequency metrology [4], quantum interface [5], and quantum internet [6,7]. Compared to bulk optical materials [8] and meter-long optical fibers [9], the integrated photonic chip provides an attractive method for accomplishing frequency conversion in a compact, reconfigurable, high-efficiency, and low-power version [10]. Various χ⁽²⁾- and χ⁽³⁾- materials can be adopted for fabricating photonic chips, including lithium niobite [11], AlN [12], SOI [13], and silicon nitride [14], which support various parametric processes for frequency conversion.

Owing to its compatibility with CMOS technology, the silicon platform, including silicon nitride and silicon dioxide, has become a prominent candidate for frequency conversion, wherein χ⁽³⁾-based four-wave mixing (FWM) is the dominant process. A distinctive example is the four-wave-mixing Bragg scattering (FWM-BS) [15–18] process wherein a signal beam or single-photon state can be coherently transferred to a new frequency and the spectral translation is set by the difference between the two pump frequencies. The FWM-BS process can be understood by considering that the interference of the two pumps effectively creates a grating in the nonlinear medium that scatters the signal to two idlers. Microring resonators are the typical choice for FWM-BS owing to their large field enhancement and thus more efficient frequency conversion within a small mode volume. However, unlike in a waveguide where the frequency can be continuously tuned while the phase-mismatch usually lead to poor χ⁽³⁾ efficiency, in a resonator the phase matching is automatically conserved through the mode matching. Indeed, for CW wave on resonance, an increase of the azimuthal mode number by a single bin corresponds to a 2π phase shift. Yet the dispersion of the resonator leads to an uneven distribution of the frequencies of resonance, which leads to the failure to satisfy the energy conservation conditions. Thus, various frequency matching strategies have been reported, such as designing the waveguide geometry to engineer dispersion [19], a dual-cavity resonant structure to compensate for cavity dispersion [20] and suitable control of the pump power and offset frequency [21]. However, these frequency matching methods are not tunable and are typically valid for a specific wavelength. To match the light-matter interaction system, such as the atom and quantum memory system, or for matching the DWDM channel well, a tunable phase-matching scheme, that is, fine tuning of the converted wavelength, is highly desired.

Herein, we propose a tunable frequency matching strategy for tunable wavelength generation via the FWM-BS process in microring resonators. Frequency matching is achieved in a manner such that the frequency mismatching is compensated by the mode splitting resulting from the strong coupling between the clockwise (CW) and counterclockwise (CCW) propagating modes. Strong CW/CCW coupling can be realized by adding reflector into the ring [22–25] and periodically modulating the side wall of the waveguide [26,27], which has been applied in eliminating the effects of backscattering [25], realizing a wide free spectral range [24], frequency engineering [27], and topological physics [28]. However, most of the designs [22–25,26,28] aim for linear optics and no nonlinear optical processes are concerned. One of them adopts [27] the mode-splitting from CW/CCW coupling for frequency matching in BS-FWM process, but the amount of mode-splitting cannot be tuned. In our work, the amount of mode splitting is designed to be dynamically adjustable; thus, frequency matching is effective over a large bandwidth. In the present work, we proposed and designed tunable mode-splitting frequency matching in silicon nitride microrings. The idler can be tuned from one FSR to tens of FSR away from the signal. Further, we also showed that the frequency matching strategy is valid under different frequency combinations and is robust to waveguide geometry and fabrication variations. The proposed strategy can be widely adopted for most optical parametric processes, including but not limited to FWM-BS, and can establish a reliable connection not only in the intra-band but also in the far separated frequency bands. This will stimulate wide applications in integrated nonlinear photonics and quantum optics.

2. Tunable mode-splitting phase-matching scheme

In microring resonators, there exist two circulating modes (CW and CCW) that are originally degenerate and resonate at the same frequency. A reflection in the ring causes coupling between the CW and CCW propagating modes, and thus the degeneracy is broken. Consequently, resonance splitting occurs, and the distance of the splitting modes is determined by the reflection strength. A loop Mach–Zehnder interferometer (loop-MZI) has been proposed its use as a tunable reflector in the ring has been demonstrated [22–25].

In this work, microring with loop-MZI was adopted for nonlinear frequency conversion for the first time. The microring resonators with a loop-MZI reflector are shown in Fig. 1(a). The loop-MZI can be regarded as equivalent to a reflector with a tunable coupling coefficient between the CW and CCW propagating modes, as shown in Fig. 1(b). Here, coupling-induced mode splitting was proposed for the compensation of frequency mismatching in the FWM-BS, achieving a widely tunable frequency conversion that is typically not accessible in common microrings.

 

Fig. 1. (a) Schematic diagram of the microring resonators with a loop-MZI reflector inside. (b) Schematic diagram of loop-MZI reflector. (c) Schematic diagram of FWM-BS and mode splitting.

Download Full Size | PDF

Further, a loop-MZI reflector inside the ring was adopted. The loop-MZI reflector introduced the reflectivity and it could be designed such that the reflectivity in four-interacting beams varied with the phase applied differently to the two arms of the MZI. We considered the four-interacting beams in FWM-BS could be grouped into two bands that were far separated, such as the reflectivity of the longer wavelength varying from 0 to 1, while the reflectivity of the shorter wavelength was nearly 0 by the reasonable design of the coupling coefficient in the waveguide coupling region, as shown in Fig. 1(b). Thus, the frequency mismatching of the frequency conversion can be compensated for by mode splitting at longer wavelength bands. The tunable reflectivity results in tunable mode splitting; hence, it can fulfill the phase-match with a certain bandwidth, that is, transferring the signal wavelength into a variety of neighboring wavelengths.

Figure 1(c) shows a schematic diagram of the FWM-BS. The two idlers are labeled i+ and i- (i+ denotes the idler with a higher frequency). The energy conservation and phase matching of the FWM-BS can be expressed as

As seen in Eq. (5), mode splitting provides phase compensation for either −2|µ_r (θ_d)| or 2|µ_r (θ_d)|. To achieve phase compensation, the amount between |ω_i± - ω_s| and |ω_p1 - ω_p2| needs to be compared, which is determined by the dispersion of waveguides. In the case that |ω_i± - ω_s| > |ω_p1 - ω_p2|, 2|µ_r (θ_d)| is required to increase |ω_p1 - ω_p2| for frequency matching, while for |ω_i± - ω_s| < |ω_p1 - ω_p2|, −2|µ_r (θ_d)| is required to decrease |ω_p1 - ω_p2| for frequency matching. Thus, the two cases can be described as

3. Device analysis

For the microring resonator with a loop-MZI reflector, the Mach–Zehnder interferometer acts as a coupler that can provide precise and independent tuning of the coupling coefficient and transmission phase shift. We used the transmission matrix method to analyze the reflectivity tuning process [29]. Further, the two directional couplers were of the same size. The length of the waveguide-coupling region was L, while $\kappa$ and t denote the waveguide amplitude cross-coupling and transmission coefficients, respectively, satisfying ${\kappa ^2} + {t^2} = 1$. θ₁ and θ₂ are used to represent the propagation phase shifts of the MZI arms denoted by Arm 1 and Arm 2. The transmission matrix can be written as

By simplifying the reflection matrix in Eq. (8), and obtaining the reflection matrix

Temporal coupled-mode theory (t-CMT) is a very useful model for analyzing a single resonance of a ring resonator. t-CMT analyzed the mode splitting in the microring resonators with a loop-MZI reflector inside. S_t refers to the wave amplitude at the output port, and S_i refers to the wave amplitude at the input port. The relationship between S_t and S_i is as follows [17]:

It is evident that the resonance mode of ω₀ is split into ω₁= ω₀ + |µ_r(θ_d)| and ω₂= ω₀ - |µ_r(θ_d)|. The distance of the mode splitting can be adjusted by adjusting the reflectivity. Consequently, frequency matching can be achieved by using tunable mode splitting.

We designed a silicon nitride microring to numerically demonstrate the value of mode splitting. The silicon nitride (Si₃N₄) photonic platform can satisfy many required conditions for efficient and low-power frequency conversion, including high quality factor (Q) modes, wide wavelength span, and negligible nonlinear absorption. The straight silicon nitride waveguide was configured to have a section dimension of 1000 nm (width) × 800 nm (height). Further, the cladding and buried oxide layers were silica. Figure 2(a) shows the n_g from 1550 nm to 700 nm bands, while Fig. 2(b) and Fig. 2(c) show the fundamental mode’s mode field distribution at 1550 nm and 810 nm.The gap between the waveguide of the directional coupler was 300 nm, and the length was 18.9 µm, while the total length of the microring was 2800 µm. The two arms of the loop-MZI reflector are both 1000 µm which including a 800 µm length integrated heater as the phase modulator. The integrated heater on the two arms of the MZI is designed to be 100 µm away from the coupling zone, so as to ensure that the phase modulation will not cause crosstalk to the waveguide coupling. Figure 3 shows a comparison of mode splitting at R = 0 and R = 1. The pumping wavelength was set at 1550 nm, assuming that t = 0.9798 and a_l = 0.2.

 

Fig. 2. (a) Simulated group index from 1550 nm to 700 nm bands for waveguide cross-sections with 1000 nm × 800 nm. (b) Fundamental mode’s distribution at 1550 nm. (c) Fundamental mode’s field distribution at 810 nm.

Download Full Size | PDF

 

Fig. 3. Comparison of mode splits between R=0 and R=1

Download Full Size | PDF

4. Tunable FWM-BS

The FWM-BS was designed in a configuration with two pumps at approximately 1550 nm, and the signal is at approximately 810 nm. By using Lumerical Mode Solution, we can see that the directional coupler of the 1550 nm band is a 3 dB coupler with $\kappa$²_1550nm= 0.5, while the cross-coupling coefficient of the 810 nm band is $\kappa$²_810nm= 0.002. A large difference in the cross-coupling coefficients between these two bands was observed, and the cross-coupling coefficient in the 810 nm band was almost zero, implying that the tuning range was narrow and contribution to mode splitting was negligible. Thus, only tuning of the mode splitting in the 1550 band was needed to achieve frequency matching, which reduced the complexity of the device. It is evident from Eq. (17) that a large tuning range for reflectivity implies a large tuning range for the degree of mode splitting. Figure 4 shows the tuning range of |µ_r(θ_d)| in the 1550 nm band.

Next, we considered the degree of frequency mismatching in the FWM-BS process. The FWM-BS process itself has a strict frequency mismatching, which leads to extremely low conversion efficiency without satisfying it. The frequency mismatching term for this process is represented by Eq. (3) and Eq. (4). We considered m = m_p2 - m_p1, m_p1 as a fixed value indicating the resonant mode nearest to 1550 nm, while m_p2 indicates the gradually increasing number of modes starting with m_p1. m_s was also a fixed value, indicating the nearest resonant mode at 810 nm. As m was tuned, m_i+ and m_i- varied according to Eq. (2), and different frequency mismatching were observed as m varied.

Figure 5 represents the gradual increase in frequency mismatching as the number of modes between m_p1 and m_p2 increased, and it demonstrates the difference in phase mismatch between ω_i+ and ω_i- in the FWM-BS process, where the phase mismatch of ω_i- was larger than that of ω_i+ because of dispersion in waveguides. This implies that if the waveguide structure is not optimized for frequency matching in a normal waveguide, idlers attains a value close to ω_s in the FWM-BS process. Thus, it was difficult to tune idlers far from signal, which was a limitation of this process. Our scheme can compensate for this partial frequency mismatching. It was found that |ω_p1 - ω_p2| < |ω_i+ - ω_s| and the difference between the two terms becomes larger as m increases; therefore 2|µ_r(θ_d)| is required to increase |ω_p1 - ω_p2| to compensate for the mismatch. It is evident from Fig. 3 that |µ_r(θ_d)| is continuously tuned by θ_d, and the tuning range is 0–8 (2π GHz). For 2|µ_r(θ_d)|, it was 0–16(2π GHz), and the maximum tuning range is shown in Fig. 4. In Fig. 4, the tuning range for the ω_i+ generation process is 1–37 number of FSRs which is approximately 14 nm in 1550 nm band. Further, the results show that the idler can be tuned from one FSR to tens of FSR away from the signal. Based on the above analysis, our strategy could perfectly achieve a tunable FWM-BS process. We provided the experimental processing to explain tunable phase-matching more intuitively. In practical experiments, we first fixed the signal and thereafter selected the pump separation, with the phase of MZI set as required for obtaining the idler wavelength. As the frequency distance between the two pump lights increased, the generated idler departed from the signal with increasing separation.

 

Fig. 4. The tuning range of µ_r in 1550 nm band as a function of differential phase shifts.

Download Full Size | PDF

 

Fig. 5. Degree of frequency mismatching as the number of modes between m_p1 and m_p2 increases. i+ and i- represent the frequency mismatching of the two idlers. The blue line represents the maximum tuning range of 2|µ_r (θ_d)|.

Download Full Size | PDF

Our approach can also be applied to the waveguides of other cross-sections. We fixed the height of the waveguide to 800 nm and simulated the waveguide with a width ranging from 850 nm to 1050 nm (other parameters remained unchanged). Changing the waveguide structure affects |µ_r (θ_d); however, this change was found to be negligible, and the effect can therefore be neglected. The maximum tuning range of the FSR calculated for different structures (Fig. 6(a)) shows that different structures exert different effects on the tuning range. Even when the width was 850 nm, the tuning range could cover at least 30 number of FSRs, which verifies that our strategy is applicable to different structures.

 

Fig. 6. (a) The maximum tuning FSR ranging with the waveguide width. (b) The maximum tuning FSR ranging with the wavelength of signal light.

Download Full Size | PDF

In the above analysis, we assumed that the signal light was fixed at 810 nm. However, there are still several relevant wavelengths in the visible band, and the signal light needs to be tuned to meet the different wavelengths generated. As shown in Fig. 6(b), we calculated the FSR tuning range of FWM-BS for different wavelengths of signal light with the other conditions held constant. Changing the wavelength of the signal light caused the tunable range of the FWM-BS process to change, but a tunable range of at least 30 number of FSRs was still guaranteed at the calculated wavelengths, which demonstrates the tunability of our strategy at different signal wavelengths.

5. Discussion and conclusion

In our study, despite the efficiency from either port of the bus waveguide coupled to the ring will be max out at 50% due to the concurrent CW and CCW nonlinear interactions, without proper frequency matching, nonlinear process will not occur at all leading to an efficiency close to zero. Mode splitting for frequency matching possesses a large tuning range, which implies a broad tuning bandwidth of FWM-BS, in contrast to the traditional wavelength-fixed phase-matching methods. Furthermore, our phase-matching strategy is compatible with a variety of wavelengths and robust to fabrication variations, which greatly improves the success rate of nanofabrication.

In addition to the demonstrated intra-band conversion, that is, neighborhood frequency transition within the telecom band or the visible or near infrared (NIF), coherent large-separation inter-band spectral translation between telecom and visible/NIF was also accomplished using this phase-matching strategy, which promises wide applications in both classical and quantum optics. From the perspective of classical optics, the intra-band and inter-band conversion via FWM-BS can be engineered for tunable light sources, all-optical signal processing, and switching. In contrast, for quantum applications, the quantum state can be transferred to its close sideband or far distinct band, which can be quite useful for quantum computation in frequency space [31] and quantum interference in quantum internet [19].