1. Introduction

Integrated photonic resonators can greatly enhance the efficiency of nonlinear processes, with applications in frequency conversion [1–9] and photon-pair production [10–12]. In such devices, optimal performance occurs when the resonator has a set of high-quality-factor modes (collectively satisfying phase matching conditions) which are each centered at one of the participant frequencies of the targeted process. However, achieving this multiple-resonance condition for any set of frequencies — let alone for a specific set — requires precision beyond what is attainable by current fabrication processes. Even if such precision were attainable, transient fluctuations in temperature or humidity, or gradual material relaxation, will inevitably disrupt the multi-resonance structure. Consequently, practical implementations require a means of dynamically tuning the resonance frequencies in order to compensate for both fabricated and dynamic variations.

Photonic resonators can be tuned by a variety of mechanisms, including static methods such as post-fabrication modification of the resonator [13–16] or cladding [17–22], or active methods via temperature, electro-optic effect, or mechanical strain [1,2,5–8,12,23–25]. Such methods have been shown to enable double or even triple resonance provided the tuning mechanism has a different effect on each mode [1,8,23,25]. However, these techniques employ blanket alterations of the multi-mode structure which necessarily affect all resonances simultaneously [26]. Consequently, the particular wavelengths for which the multi-resonance condition can be achieved are still determined by the static device dimensions, including fabrication imperfections. Alternative, mode-targeted wavelength trimming methods [26–28] have also been demonstrated and could be used to achieve multi-resonance at specific frequencies, though these corrections are static and cannot be used to compensate for dynamic variations. An ideal tuning mechanism should combine the features of these paradigms, enabling dynamic tuning of each mode individually. Such control would not only increase tolerance to environmental and fabrication variances, but would also find use in quantum networks to correct the spectral inhomogeneity of optically active defects [29,30].

In this paper, we develop a complete description of an active tuning scheme for multi-resonant nonlinear resonators which uses coupled auxiliary resonators to independently tune the participant modes (Fig. 1). Through selective coupling in the frequency domain, a targeted mode can be isolated within the auxiliary resonator and tuned independently. While similar schemes have been explored experimentally [31–34], we derive, in full generality, an analytical model for the auxiliary-resonator-based tuning mechanism based on a temporal coupled-mode theory description. It is shown that an individual mode in an overcoupled nonlinear resonator can be tuned on a scale exceeding the inter-cavity coupling rate $|g|$ — which may already be significantly larger than the resonance linewidth — without significant degradation of the conversion efficiency. The formalism is directly applied to triply-resonant difference frequency conversion, in which we show there is negligible degradation of the conversion efficiency over several linewidths of detuning. Practical considerations, such as the range of physically attainable inter-cavity coupling rates and quality factors are studied in the case of small-diameter gallium phosphide ring resonators. For a main resonator with telecom-band intrinsic and coupling quality factors of $Q_i = 10^6$ and $Q_c = 10^5$ respectively, simulations find inter-cavity coupling rates as high as $|g|/2\pi =136$ GHz are attainable without inducing significant excess losses, yielding a tuning bandwidth of approximately $64$ linewidths. These calculations indicate that auxiliary resonator tuning may be the most promising tuning technology to realize high-yield multi-resonant devices at targeted operating frequencies.

 

Fig. 1. (a) Diagram of a main ring resonator with frequency $\omega _a$ and loss $\Gamma _a$ is coupled to an auxiliary cavity of frequency $\omega _b$ and loss $\Gamma _b$ at a rate of $|g|$. This system is modeled in Section 2. (b) A potential layout for sum/difference frequency generation using two auxiliary resonators. For simplicity, input/output coupling for all three main resonator modes is shown with a single waveguide. Each auxiliary resonator can only couple to one of the frequencies $\omega _j$ in the main resonator. This system is modeled in Section 3

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2. Orthogonal tuning by selective mode coupling

Auxiliary resonator tuning uses modifications of an auxiliary resonator structure to indirectly influence specific coupled modes in a main resonator while leaving other main resonator modes unperturbed. In principle, the tuning scheme could be implemented using any resonator tuning mechanism to control the auxiliary resonator, including thermal or electro-optic tuning. Many performance metrics, such as tuning speed and reversibility, are inherited from the chosen physical tuning mechanism. In this section, a model of the auxiliary resonator tuning system is developed in order to describe behaviors that are inherent to the tuning scheme.

A single-mode auxiliary resonator system is shown in Fig. 1(a). The main resonator supports a mode with complex field amplitude $a$ at an unperturbed frequency $\omega _a$, which is coupled to mode $b$ at frequency $\omega _b$ in the auxiliary resonator with an inter-cavity coupling rate $g$ as well as an input/output waveguide. The field evolution in this is system is described by the coupled-mode theory (CMT) model

The steady-state response to monotone driving at frequency $\omega$ is obtained by a Fourier transform: $a(t) \to A(\omega )$, $b(t) \to B(\omega )$, and $s_\pm (t)\to S_\pm (\omega )$. The auxiliary resonator field $B$ can be eliminated from the resulting algebraic system of equations to yield a single resonator equation

This can be compared to the maximum steady-state field in the absence of the auxiliary resonator $A_0/S_+ = 2\sqrt {\kappa }/\Gamma _a$ as obtained by setting $|g|=0$ and $\omega = \omega _a$. The resulting normalized steady-state field $|A|/|A_0|$ is shown in Fig. 2(a) as a function of the drive and resonator detunings ($\delta,\Delta _b$). We observe characteristic anti-crossing behavior with field maxima closely tracking the eigenvalues of an equivalent lossless system for $|g|\gtrsim \Gamma _a,\Gamma _b$. For large resonator detunings $|\Delta |\gg |g|$, the normalized steady-state field at $\delta = 0$ approaches unity and is largely insensitive to changes in $\Delta$ indicating decoupling of the auxiliary and main resonator modes. This demonstrates that the auxiliary resonator can affect a targeted mode in the main resonator without significantly perturbing other modes, provided significant detuning is maintained. While the main resonator may be affected by cross-talk from the physical tuning mechanism used to control the auxiliary resonator and vice versa, mode-specific tuning can still be achieved if each control signal has a distinct effect. Mechanism-specific design modifications, such as deeply etched trenches for thermal isolation [35], can mitigate cross-talk to improve the distinguishability of the control signals.

 

Fig. 2. (a) Main cavity field amplitude as a function of drive and resonator detunings normalized to the resonantly driven cavity in the absence of an auxiliary ring. Dashed lines track the resonances given by $\Delta _b^{\text {opt}}(\delta )$ (Eq. (6)). Here $\Gamma _a/|g| = 0.5$ and $\Gamma _b/|g| = 0.1$. (b) Normalized field amplitudes along the resonance peaks as a function of drive detunings for different values of $\Gamma _a/\Gamma _b$ with $\Gamma _a/|g|=0.1$. (c) The 3-dB tuning bandwidth $B_\delta$ (Eq. (7)) as a function of $\Gamma _1/\Gamma _2$ for different $|\Gamma _a|$. Points are bandwidths obtained from fits the corresponding curves in (b).

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This effective-resonator picture shows explicitly that the auxiliary frequency $\omega _b$ (and consequently resonator detuning $\Delta _b$) can be tuned to reduce the effective detuning $\delta _{\text {eff}}$ at the expense of introducing additional effective loss $\Gamma _{\text {eff}}\geq \Gamma _a$. These competing effects must be balanced to optimize the particular performance metric under consideration. In many cases, maximization of the steady-state resonator field Eq. (5) is desired which depends on minimizing $|i\delta _{\text {eff}}+\Gamma _{\text {eff}}/2|$. We determine the optimal resonator detuning in this case to be given by

This relation can be used to find the required tuning range for the auxiliary resonator in order to implement a desired tuning range for the main resonator. In the limit of $|g|\gg \Gamma _a,\Gamma _b$, the optimal detuning can be simplified to $\Delta _b^{\text {opt}} \approx -\delta + |g|^2/4\delta$.

The normalized cavity field $|A/A_0|$ along the $\Delta _b^\mathrm {opt}$ contour is plotted as a function of the drive detuning $\delta$ in Fig. 2(b). The field roughly follows a Lorentzian lineshape with full-width-at-half-max $\approx \sqrt {\Gamma _a/\Gamma _b}$. Explicit calculation of the normalized steady-state energy $|A/A_0|^2$ reveals a 3-dB tuning ($\delta$) bandwidth of

3. Nonlinear processes in auxiliary-resonator-tuned systems

To generalize the auxiliary resonator model for applications involving nonlinear mode interactions, a resonator configuration similar to Fig. 1(b) is considered. A single main resonator supports multiple modes $a_j$ at frequencies $\omega _{a_j}$, each of which may be coupled to an associated auxiliary resonator mode $b_j$ and an input/output mode $s^\pm _{j}$. The main resonator modes can interact through nonlinear functions $N_j(\textbf{a})$ corresponding to the phase-matched resonant process specifically targeted by the design. In contrast, the auxiliary resonator design(s) will generally not support phase-matched multi-resonant nonlinear processes. Thus, the system can be described with a series of equation pairs:

For a given nonlinear process the various modes will be driven, either externally or by the nonlinear interaction, at a single frequency $\omega _{d,j}$. The steady-state response is taken to be $a_j(t) = A_j\exp (-i\omega _{d,j} t)$ and $b_j(t) = B_j\exp (-i\omega _{d,j} t)$. This enables an effective-resonator description for each pair of modes $j$,

3.1 Difference-frequency generation

This section illustrates the effect of auxiliary resonator tuning on a nonlinear process using the specific case of resonantly-enhanced $\chi ^{(2)}$ difference frequency generation (DFG), which is of particular interest for quantum information applications such as long-distance entanglement distribution over silica optical fiber. In the multi-resonant DFG process ($\omega _i \leftrightarrow \omega _p + \omega _o$) shown in Fig. 1(b), the main resonator supports input $a_1$, pump $a_2$, and output $a_3$ modes at frequencies $\omega _1 \approx \omega _i$, $\omega _2 \approx \omega _p$, and $\omega _3 \approx \omega _o$. The input and pump resonant modes are driven by incoming waveguide modes $s^+_{1}$ and $s^+_{2}$, while all three resonator modes are coupled to outgoing waveguide modes $s^-_j$. Two separate auxiliary rings $b_2$ and $c_3$ have resonances $\omega _b \approx \omega _2$ and $\omega _c \approx \omega _3$ and are coupled to $a_2$ and $a_3$ with rates $g_b$ and $g_c$ respectively. The system dynamics are described by the CMT equations

The steady-state fields in the effective-resonator picture satisfy

In the small-signal limit of the DFG process, the converted light in the output-frequency resonator mode is negligible compared to either of the externally-driven modes. In this case, the small-signal conversion efficiency $\eta _{ss}$ can be derived by solving the steady-state equations while neglecting the nonlinear ($\beta$) terms in Eqs. (11a) and (11b:

The small signal efficiency is maximized when the steady-state field of each participant mode is independently maximized, so the same choice of resonator detuning in Eq. (6) remains optimal. Consequently, the modification of the small-signal efficiency and corresponding bandwidth match that of the single-resonator model Eq. (7). Similar to a triple-resonant process in a single resonator, $\eta _{ss}$ is maximized when all three main resonator modes are effectively critically coupled to the waveguide: $\kappa _j = \Gamma _{j,\mathrm {eff}}/2$. However, the optimal waveguide coupling for auxiliary-tuned modes can vary depending on $\Gamma _{\mathrm {eff}}$, which increases with the magnitude of the corrected detuning.

The quantum-limited maximum conversion efficiency can be investigated using the undepleted-pump approximation in which $|S^+_{1}|\ll |S^+_{2}|$. In this regime, only the nonlinear term in the pump mode Eq. (11b) can be neglected, resulting in an effective coupling between the input and output modes

Under the assumption that $\delta _1 = 0$ by global tuning of the main resonator, the conversion efficiency is given by

We find that $\eta _{\mathrm {opt}}$ is approximately maximized for resonator detunings which maximize the cavity fields Eq. (6). The resulting $\eta _{\mathrm {opt}}$ can be compared to the ideal system quantum limit [36]

4. Practical tuning range and bandwidth

The above analysis demonstrates the potential of the auxiliary resonator tuning mechanism in enabling efficient and targeted multi-resonant nonlinear processes. The bandwidth over which such processes can be achieved, however, is largely determined by the inter-cavity coupling rate $|g|$ (Eq. (7)). Consequently, the practicality of the tuning scheme depends on the range of experimentally obtainable coupling rates in realistic device designs. In this section, we simulate both the coupling rates and added scattering losses resulting from an auxiliary ring resonator coupled to a triply-resonant ring resonator design [8]. Based on these simulations, we then examine a resonant DFG process at targeted frequencies in the combined system in order to characterize the tuning bandwidth and performance.

4.1 Coupling rates and losses

To determine $|g|$ for a ring or disk resonator, the propagation of a broadband pulse through the coupling region, depicted in Fig. 3(a), is simulated with a variational finite-difference time-domain method (varFDTD, Lumerical) to determine the power scattering matrix. varFDTD performs an approximately equivalent two-dimensional simulation using effective indices derived from a three-dimensional structure. The normalized single-pass power coupling spectrum $|k|^2$ can then be related to the inter-cavity coupling rate $|g|$ as

 

Fig. 3. (a) Simulation diagram of the coupling region between two rings used to simulate single-pass transmission and inter-ring coupling. (b) Dependence of inter-cavity coupling rate on inter-cavity distance, derived from simulated single-pass coupling, for an auxiliary resonator design with $R_{\text {aux}}=3$ µm and $w_{\text {aux}}=400$ nm. The coupling $k$, transmission $t$, and scattering $s$ fractions refer to the field amplitude. (c) Excess scattering loss rates in the main cavity telecom and pump band modes induced by the coupled auxiliary resonators. For the telecom mode, equivalent scattering quality factor is shown on the right-side axis. Loss rates below $\mathrm {\sim 1\,GHz}$ at 1550 nm cannot be accurately estimated by varFDTD due to single-pass power loss dropping below the simulation noise floor. Exponential fits to the remaining points indicate a reduction in loss of about 2 and 3.5 orders of magnitude every 100 nm for 1550 nm and 1080 nm respectively.

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The main resonator geometry is based on the design from Ref. [8], which utilizes a hybrid-integrated gallium phosphide(GaP)-on-oxide(SiO$_2$) ridge-waveguide ring of inner radius of 5.3 µm, width of 690 nm, and height of 430 nm to target phase-matched resonances at 637 nm, 1080 nm, and 1550 nm for DFG. A range of auxiliary resonator radii $R_{\text {aux}}$, widths $w_{\text {aux}}$, and separations $d_{\text {gap}}$ were simulated ranging from 3–8 µm, 200–500 nm, and 0–300 nm respectively. Frequency-domain simulations (Lumerical MODE) are utilized to determine the group indices and bending losses for propagating modes near 1080 nm and 1550 nm. This enables the determination of the inter-cavity coupling rates via Eq. (18).

The coupling rate $|g|$ most strongly depends on the inter-cavity separation distance $d_{\text {gap}}$, as shown in Fig. 3(b). The dependence of $|g|$ on the auxiliary ring geometry is comparatively weak (Supplement 1 S2). In general, the coupling rate increases for smaller ring diameters with shorter round-trip propagation times and smaller ring widths with stronger evanescent fields, so long as the auxiliary resonator mode is not approaching cutoff. As expected, we observe a roughly exponential decay of the coupling rate $|g|$ with increasing inter-cavity separation. For small separations ($d_{\text {gap}} < 50$ nm), the auxiliary resonator begins to strongly perturb the main resonator modes causing increased coupling rates but also increased scattering losses $\Gamma _{cr}$ (Fig. 3(c)). Consequently, the optimal coupling distance corresponds to the smallest distance for which the scattering losses do not significantly contribute to the total quality factor. For a design targeting loaded quality factors of $Q=10^5$, a coupling distance of $d_{\text {gap}} = 125$ nm would provide a sufficiently high telecom coupling rate ($|g|\approx 90$ GHz) while introducing relatively minor losses ($Q_{cr} \approx 10^6$).

4.2 Difference-frequency generation

We now model the performance of a triply resonant DFG process (based on [8]) with the addition of auxiliary ring resonators. We consider an auxiliary ring design with nominal dimensions $R_{\text {aux}}=3$ µm, $w_{\text {aux}}=400$ nm, and $d_{\text {gap}}=150$ nm. The resulting pump (1080 nm) and output (1550 nm) inter-cavity coupling rates are $|g_b|/2\pi \approx 15$ GHz and $|g_c|/2\pi \approx 64$ GHz respectively. We assume all three modes have fabrication-limited intrinsic quality factors of $Q_i = 10^6$ (quality factors in GaP photonics at visible/telecom wavelengths have been observed to be well above $10^5$ [8,9,38]). Coupling-region loss at the chosen $d_{\text {gap}} = 150$ nm corresponds to scattering quality factors far exceeding $10^6$ (Fig. 3(c)) and may consequently be neglected.

Coupling to the waveguide introduces an additional loss channel within the main resonator but not the auxiliary resonators. We assume that the main resonator is significantly over-coupled, corresponding to a coupling quality factor of $Q_c=10^5$ for all three modes. The resulting total quality factors for the three main resonator modes are then $Q_j = 9.1\times 10^4$ ($j=1,2,3$). The auxiliary rings do not couple to the waveguide and so $Q_b = Q_c = 10^6$. We then compute the 3-dB tuning bandwidth Eq. (7) for the pump and output modes to be

The assumed resonator parameters and the corresponding bandwidths are summarized in Table 1. As described in Eq. (6), the accessible portion of this tuning curve is limited by the tuning range of the auxiliary resonator. Also, if a resonance is tuned by more than approximately 20% of a free spectral range, the conversion efficiency may be reduced due to phase matching effects (Supplement 1 S3), though this is not likely to become significant in high-finesse systems.

Table 1. Parameters for DFG performance estimation. All rates are in GHz.

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Finally, we compute the projected DFG performance metrics as shown in Fig. 4 assuming the input mode has been tuned globally onto resonance (i.e., $\delta _1=0$). We observe that the pump and output modes could be tuned over a range of 4 and 23 main-resonator linewidths, respectively, while maintaining over 90% of the maximum small signal conversion efficiency. The small signal efficiency can be increased at the expense of the pump mode tuning bandwidth by reducing the coupling rate $\kappa _2$ to the near-critically coupled regime ($\kappa _2 \approx \gamma _2$). Although the tuning bandwidth of the pump mode is diminished compared to the output mode, the critical power is relatively insensitive to modulation, remaining near 10 mW over the full range (Fig. 4(d)). Operation at the quantum efficiency limit can be achieved over the much larger telecom tuning bandwidth (Fig. 4(c)) even in spite of the comparably worse pump bandwidth. These results indicate that the auxiliary resonator tuning method could enable high-efficiency, frequency-targeted DFG over large bandwidths with minimal impact in performance.

 

Fig. 4. Small-signal photon conversion efficiency with and without an auxiliary resonator as a function of (a) output and (b) pump mode detuning. All detunings are measured relative to the unperturbed (sans auxiliary resonator) main resonator frequencies ($\omega _2$, $\omega _3$). Insets show the indicated regions with re-scaled detuning axis. (c) Quantum-efficiency-limited (QEL) photon conversion efficiency as a function of the output drive detuning $\delta _3$. Pump mode detunings do not affect the value of $\eta _{\text {QEL}}$ but are shown to modify the critical power in (d). Critical values are shown to remain in the few mW range over the tuning bandwidths of both pump $\delta _2$ and output $\delta _3$ modes.

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5. Conclusion and outlook

Coupled auxiliary resonators offer a straightforward method to actively and independently tune specific resonances in a multi-resonant device without altering the structure of the primary resonator. The tuning scheme can be implemented and used in conjunction with any resonator-specific tuning mechanism, allowing for compatibility with most material platforms while also limiting the required fabrication complexity. This technique enables the tuning of individual resonances over tens of linewidths and can be accurately modeled by a small number of fast and computationally inexpensive simulations.

The flexibility and control uniquely afforded by the auxiliary-resonator tuning scheme is particularly relevant to quantum information applications which impose strict restrictions on process wavelengths. For example, difference-frequency conversion of photons emitted from solid-state qubits—such as the nitrogen-vacancy [39] and silicon-vacancy [40] centers in diamond—to a specific telecom wavelength could be used to not only minimize losses on a fiber-based quantum network [30], but also correct for spectral inhomogeneity of the emitter nodes. In this way, the high conversion efficiency afforded by the multi-resonant nonlinear process, combined with the flexibility of the auxiliary resonator tuning mechanism, may serve an indispensable role in the development of large-scale quantum networks.