1. Introduction

Nonlinear optical phenomena have been widely investigated in high-Q microcavities, where light-matter interaction is enhanced due to confinement of the light field [1–3]. Various dielectric materials with high refractive indices including silica [4] and silicon [5], polydimethylsiloxane (PDMS) [6], silicon nitride (SiN) [7,8], and gallium arsenide (GaAs) [9] have been explored. Numerous nonlinear optical applications have been realized in microcavity platforms, such as second-harmonic (SH) generation [9–13], four-wave mixing [14,15], optical parametric oscillation [16–18], Raman laser [16,19,20] and Kerr frequency combs [21–24].

Due to the relatively weak nonlinear optical coefficients, a high intracavity circulating power of light is required to stimulate the nonlinear frequency conversion processes. The inevitable material absorption of the dielectric leads to considerable heating of microcavities. The thermal effect induced by the pump laser is much more severe in integrated microcavities due to their small volume, leading to large thermal shifts of the resonant frequency that usually exceeds 100 times the resonance linewidth [16,25,26] and instability of the system [27–30]. Therefore, thermo-optic effects impose great experimental challenges to highly efficient nonlinear optical applications based on microcavities. For example, phase-matching or resonance matching conditions [7,11–13,31,32] vary with the environment temperature, pump power and frequency detuning. Thus without carefully considering all the effects of temperature, the performances of microcavity-enhanced nonlinear optical effects and other practical applications of microcavities will be drastically limited. Yet, the interplay between thermo-optic and coherent nonlinear optical effects and their control is rarely studied.

There are some proposals and experiments to suppress or compensate for the thermal effect in microcavities. One approach utilizes different thermo-optic coefficients of different materials, where the thermal drift can be compensated by intricate designs or with the use of extra materials [26,33,34], but it is only applicable to certain materials. Another approach is directly adjusting the ambient temperature of the photonic chip which tunes the resonant frequency by thermal expansion and thermo-refractive effects [7,8,11–13,31]. This method is limited by the slow thermal response, high power consumption, and its inability to selectively tune a targeted device on the chip. Furthermore, one problem of such passive control is that red or blue detuning, which is vital for certain nonlinear optical processes [21,22,35–37], is not experimentally accessible because of thermal bistability or multi-stability.

In this paper, we demonstrate the all-optical control of the thermal behavior associated with second-harmonic generation (SHG) in a selected microring resonator on an aluminum nitride (AlN) photonic chip. Inspired by recent experiments on the thermal control in dissipative Kerr soliton generation [38–40], we introduce a strong telecom auxiliary laser to couple with the selected microring and realize both the precise tuning of the doubly-resonant condition and the elimination of thermo-optic bistability of the SHG. The experimental results are consistent with our triple-mode model for the interaction between thermo-optic and nonlinear optical effects. Our scheme holds many advantages over previous works, including the fast modulation rate at about $256\,\mathrm {kHz}$ due to the all-optical control, device selectivity on a photonic chip, and the universality of the mechanism that is free from the limitation on fabrication or materials.

2. Experimental system for SHG

The experimental setup for the on-chip controllable SHG is illustrated in Fig. 1(a), where telecom pump and auxiliary lasers are combined by a 50:50 fiber beamsplitter (BS) and coupled into the input port of the on-chip AlN frequency doubler by a lensed fiber. The device is placed on a feedback-controlled heater to adjust the chip temperature. Figure 1(b) shows the SEM image of the device, in which the array of microrings are connected by the same bus waveguide, and a detailed schematic illustration of a single frequency doubler extracted from the device which consists of a microring resonator and two waveguides with different widths. The free spectral range (FSR) of the microring at the telecom band is about $11\,\mathrm {nm}$ (or about $1.35\,\mathrm {THz}$), while its radius and width are optimized to achieve the highest SHG efficiency. Due to the different microring-to-waveguide mode matching conditions, the width of the straight waveguide ($\mathrm {w_{1}}$) is designed for telecom lasers while the wrap-around waveguide ($\mathrm {w_{2}}$) is for visible signals. By utilizing an on-chip wavelength division multiplexer (WDM), signal and pump lasers at different wavelengths can be combined into the same waveguide at the width of $\mathrm {w_{1}}$, so that all the output lasers can be collected by the same lensed fiber. Other detailed information about the device can be found in Refs. [11,13].

 

Fig. 1. (a) Experimental setup of the system for the on-chip doubly-resonant SHG, including telecom pump (Pump) and auxiliary (Aux) lasers; beamsplitter (50:50 BS); electro-optic modulator (EOM); wavelength division multiplexer ($\mathrm {WDM_{1(2)}}$); and photodetectors ($D_{1\left (2\right )}$). The photonic chip is placed on a heater with a temperature stabilizer. The transmission of the SH signal power ($P_{\mathrm {SHG}}$) and the pump laser ($\mathcal {T}_{\mathrm {pump}}$) are detected by $D_{1}$ and $D_{2}$ respectively after $\mathrm {WDM_{1}}$ and $\mathrm {WDM_{2}}$. The components in the green dash square are only used for measuring the temperature control response time (Fig. 4). (b) The SEM image of the device and the schematic illustration of a single AlN microring structure. The telecom pump laser is coupled into the microring by the straight upper waveguide and the visible SH signal is extracted through the lower wrap-around waveguide in the opposite direction. Due to the different microring-to-waveguide mode matching conditions, the width of the straight waveguide ($\mathrm {w_{1}}$) and the wrap-around waveguide ($\mathrm {w_{2}}$) are different. The telecom auxiliary laser is injected into the system from the same side as the pump laser. (c) and (d) show the detailed spectra of the pump lasers (red lines) and SH signals (blue lines) when the on-chip pump power $P_{\mathrm {p}}$ is fixed at $0.2\,\mathrm {mW}$ and $3.6\,\mathrm {mW}$, respectively. The black dash lines are the theoretical results fitted by our model.

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When the phase-matching conditions are satisfied for visible and telecom modes in a microring, the visible SH signal can be efficiently generated by the pump laser, and all the three lasers including pump laser, auxiliary laser and SH signal can be collected by the same lensed fiber at the output port of the device. The SH signal is detected by a visible photodetector ($D_{1}$) after $\mathrm {WDM_{1}},$ in which the visible SH signal and telecom pump and auxiliary lasers can be separated into two arms. Transmitted pump laser is separated from the auxiliary laser by $\mathrm {WDM_{2}}$ and detected by a photodetector ($D_{2}$), as shown in Fig. 1(a). To begin our model for this system, we introduce $a$ and $b$ as dimensionless bosonic operators for two modes with resonant frequencies at the fundamental frequency ($\omega _{a}$) and the second-harmonic frequency ($\omega _{b}$). The Hamiltonian of the system can be written as ($\hbar =1$)

In the rotating frame of the pump laser $H_{0}=\omega _{p}a^{\dagger }a+2\omega _{p}b^{\dagger }b$, the dynamics of the system can be described by the Heisenberg equations

In practice, we have $K_{b}\neq 2K_{a}$ due to materials and modal dispersion properties, thus controlling $T_{\mathrm {chip}}$ could effectively tune the frequency matching and achieve the doubly-resonant condition for SHG, i.e. $\omega _{a}\left (T_{\mathrm {ring}}\right )=2\omega _{b}\left (T_{\mathrm {ring}}\right )$, if $T_{\mathrm {ring}}=T_{\mathrm {chip}}$. However, this equality is not valid for typical nonlinear optical experiments due to heating effects induced by the pump laser in the microring ($T_{\mathrm {ring}}>T_{\mathrm {chip}}$). Here, we first study the thermal behavior caused by the pump laser without an auxiliary laser.

Since the device is optimized for SHG phase-matching conditions, we can observe an SH signal in a weak pump laser. Figure 1(c) shows the transmitted pump laser (red line) and the generated SH signal (blue line) when the pump power ($P_{\mathrm {p}}$) is $0.2\,\mathrm {mW}$. For such a weak pump laser power, the temperature variation of the microring induced by the pump laser $T_{\mathrm {ring}}-T_{\mathrm {chip}}$ is negligible. As a result, both the transmitted and SH spectra exhibit Lorentz shapes without obvious cavity resonance asymmetry, as shown in Fig. 1(c). However, when the pump laser is strong and induces significant heating of the microring, the cavity resonance frequency shifts as the pump laser frequency approaches the resonance. As shown in Fig. 1(d), both the transmitted (red line) and SH (blue line) spectra are triangular-shaped when $P_{\mathrm {p}}=3.6\,\mathrm {mW}.$ Such linewidth broadening and asymmetric lineshape are attributed to the thermal bistability, which was first revealed in Ref. [25] .

This thermal bistability can be incorporated into the SHG model by including the cavity absorption model which describes the cavity temperature changing as a function of the intracavity power. In the current experimental setup, external laser parameters vary slowly with respect to the thermal response of the microring, thus we study the steady-state behavior of the system, which gives

To test the SHG thermal bistability model, we numerically solve the above equations and fit the spectra in both Figs. 1(c) and (d). The agreement between the theoretical and experimental results validates our model. Here, we adopt the approximation $\Delta T\approx \alpha _{a}\left |a\right |^{2}$ under the low conversion efficiency approximation $\left |b\right |^{2}\ll \left |a\right |^{2}$ with parameters in Fig. 1. By changing the chip temperature ($T_{\mathrm {chip}}$), $K_{a}$ and $K_{b}$ can be measured and the results are $K_{a}/2\pi =-1.8\times 10^{3}\,\mathrm {MHz/K}$ and $K_{b}/2\pi =-3.47\times 10^{3}\,\mathrm {MHz/K}$. Other fitted parameters are $\kappa _{a,\mathrm {ex}}/2\pi =512\,\mathrm {MHz}$, $\kappa _{a,\mathrm {in}}/2\pi =352\,\mathrm {MHz}$, $\kappa _{b,\mathrm {ex}}/2\pi =1160\,\mathrm {MHz}$, $\kappa _{b,\mathrm {in}}/2\pi =874\,\mathrm {MHz}$, $\alpha _{a}=\alpha _{b}=3.5\times 10^{-7}\,\mathrm {K}$, and $g/2\pi =38.5\,\mathrm {kHz}$.

3. Tuning of the doubly-resonant condition

The thermal bistability accompanying with the SHG revealed in the last section implies several issues in practical high efficient SHG experiments: $T_{\mathrm {ring}}$ is pump power dependent, thus the optimal SHG may not be always achievable. The triangle-shape spectra also mean limited control of the SHG. Although the chip temperature $T_{\mathrm {chip}}$ provides a knob for compensating the thermal effect, the temperature stabilization of a whole chip is more challenging and very slow, and also all the optical devices on the same chip will be unavoidably affected by $T_{\mathrm {chip}}$. Therefore, we introduce an auxiliary laser to couple with the selected microcavity, whose temperature is controlled optically.

The model is shown in Fig. 1(b), in which the auxiliary laser with the frequency $\omega _{\mathrm {Aux}}$ significantly different from the pump laser is injected into the microring. By scanning the auxiliary laser wavelength, the target ring could be identified by observing the change of the SH signal when the auxiliary laser is on-resonance with a cavity mode. In the selected microring, the auxiliary laser is associated with an auxiliary mode $c$, which satisfies the condition that $\omega _{a}-\omega _{c}\thickapprox n\times FSR$ (n is an integer) to prevent the auxiliary laser from producing the SH signal. Note that it is not necessary to choose an auxiliary mode in a similar waveband as the pump laser, and the mode family (or polarization) can be different from the pump mode.

The flow chart of the proposed triple-mode model for all-optical thermal control is shown in Fig. 2(a), where both pump and auxiliary lasers could affect $T_{\mathrm {ring}}$. Similar to Eq. (1), the auxiliary laser induces $H_{\mathrm {Aux}}=\omega _{c}c^{\dagger }c+\varepsilon _{\mathrm {Aux}}(ce^{i\omega _{\mathrm {Aux}}t}+c^{\dagger }e^{-i\omega _{\mathrm {Aux}}t})$, where $\varepsilon _{\mathrm {Aux}}=\sqrt {2\kappa _{c,\mathrm {ex}}P_{\mathrm {Aux}}/\hbar \omega _{\mathrm {Aux}}}$. $P_{\mathrm {Aux}}$ and $\omega _{\mathrm {Aux}}$ are the power and frequency of the auxiliary laser, and $\kappa _{c,\mathrm {ex(in)}}$ is the external (intrinsic) loss rate of $c$, while $\kappa _{c,\mathrm {load}}=\kappa _{c,\mathrm {in}}+\kappa _{c,\mathrm {ex}}$. Then we can solve the steady state cavity amplitude similarly as

 

Fig. 2. (a) The schematic illustration of the triple-mode model for all-optical thermal control. An auxiliary mode $c$ is added into the model, where $\omega _{a}-\omega _{c}\thickapprox n\times FSR$. The auxiliary laser resonated with $c$ alters $\Delta T$, which controls the frequencies of modes $a$, $b$, and $c$. (b) The transmission of the auxiliary laser by upward and backward laser wavelength scanning directions. Due to the thermal bistability of the microring cavity, the FWHMs of the two spectra are distinct. The green dots represent the theoretical fitting result of the thermal bistability induced by the auxiliary laser. (c) The estimated temperature variation of microring (${\textstyle \Delta T}$) when scanning the auxiliary laser wavelength. The upward scan reaches a maximum $\Delta T$ at $3.86\,\mathrm {K}$ while the backward scan corresponds to $2.17\,\mathrm {K}$.

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The effect of the auxiliary laser is characterized by scanning the laser wavelength and measuring the spectra, as shown in Fig. 2(b), with the on-chip auxiliary laser power $P_{\mathrm {Aux}}=6.5\,\mathrm {mW}$. The red line represents the upward wavelength scan while the purple line represents the backward wavelength scan. To ensure that the system is always at a steady-state, the scan speed of the auxiliary laser reduces to $0.2\,\mathrm {nm/s}$. Note that the FWHM of the red line ($4\,\mathrm {GHz}$) and purple line ($1.4\,\mathrm {GHz}$) are distinct due to the bistability of the microring cavity. The green hollow dots in Fig. 2(b) are theoretical results fitted by Eq. 11), based on which we calculated $\Delta T$ controlled by the auxiliary laser, as plotted in Fig. 2(c). We find that when we scan the auxiliary laser in an upward wavelength direction $\Delta T$ reaches $3.86\,\mathrm {K}$, while $\Delta T=2.17\,\mathrm {K}$ along the opposite direction. Here $\Delta T$ is a theoretical result according to the experimentally determined parameters $K_{a}$ and $K_{b}$, which just provides a rough estimation of the intracavity temperature changed by the auxiliary laser and could not be directly experimentally verified.

Figures 3(a)-(e) present the experimental demonstration on the control of the SHG wavelength by the auxiliary laser. During this process, a scanning weak pump laser with $P_{p}=1.6\,\mathrm {mW}$ generates SH signals while a strong auxiliary laser with $P_{\mathrm {Aux}}=9\,\mathrm {mW}$ is added to the system in a fixed frequency to control the thermal effect intracavity. Note that in every single graph of Figs. 3(a)-(e), the auxiliary frequencies are different, while the detailed frequencies are denoted by the color stars in Fig. 3(f). The stars in Fig. 3(f) indicate the discrete scan of the auxiliary laser, while the red line in the same graph shows the continuous scan. The doubly-resonant wavelength shifts when the auxiliary laser approaches the resonance of the auxiliary mode, consistent with the prediction of auxiliary-controlled temperature in Fig. 2(c). Particularly, the purple spectrum in Fig. 3(e) can be regarded as the reference SH signal without an auxiliary laser. It is because the spectrum is measured after the occurring of a sudden jump of the transmitted auxiliary laser power (shown as the purple star in Fig. 3(f)), where the sudden jump indicates a large jump of cavity temperature and then the detuning of mode $c$ $\delta _{c}\gg \kappa _{c,\mathrm {load}}$ (also shown in Fig. 2(c)). Therefore, for Fig. 3(e) the auxiliary laser is far off-resonant with the mode c and the intracavity auxiliary power is negligible. Furthermore, we measure the maximum doubly-resonant SHG signal frequency shift ($\delta _{\mathrm {SHG}}$) induced by different auxiliary laser power with $P_{\mathrm {Aux}}=1.6\,\mathrm {mW},4.8\,\mathrm {mW},7.36\,\mathrm {mW}$ and $9\,\mathrm {mW}$, respectively. $\delta _{\mathrm {SHG}}$ is defined as the difference between the two resonances ($\omega _{\mathrm {start}}$ and $\omega _{\mathrm {end}}$) under or out of the impact of auxiliary laser. For example, $\delta _{\mathrm {SHG}}=\left |\omega _{\mathrm {start}}-\omega _{\mathrm {end}}\right |=19.4\,\mathrm {GHz}$ when $P_{\mathrm {Aux}}=9\,\mathrm {mW}$ as shown in Figs. 3(a)-(e). The results are summarized as the hollow green squares in Fig. 3(h), which show a linear dependence. According to the Eqs. (3), (4) and (10), we numerically solve the $\delta _{\mathrm {SHG}}$ when $P_{\mathrm {Aux}}$ varies from $0\,\mathrm {mW}$ to $11\,\mathrm {mW}$ by steps, shown as the blue dots in Fig. 3(h). The results show a broken line when $P_{\mathrm {Aux}}<4\,\mathrm {mW}$, but it becomes linear-dependent at higher $P_{\mathrm {Aux}}$. It is because the auxiliary laser cannot dominate the thermal effect of the system at low power, hence the interplay between pump and auxiliary lasers impacts the thermal response of the system, leading to the fluctuation of $\delta _{\mathrm {SHG}}$. Furthermore, since the coupling efficiency between the lensed fiber and the waveguide will be impacted by the fluctuation of the input polarization and $T_{\mathrm {chip}}$, the on-chip auxiliary laser power is difficult to calibrate, hence there are deviations between the experimental and theoretical results.

 

Fig. 3. (a)-(e) The tuning of SHG by the auxiliary laser. The SH signals (color lines) are generated by the telecom pump laser at $1.6\,\mathrm {mW}$, while $P_{\mathrm {Aux}}$ is $9\,\mathrm {mW}$. The resonant frequencies and lineshapes of signals change by the variation of $\omega _{\mathrm {Aux}}$. The black dash lines are the theoretical result fitted by the triple-mode thermal-control model. The resonance of SH signals could be tuned over $19.4\,\mathrm {GHz}$ by the all-optical thermal control. $\lambda _{c}/\lambda _{1}/\lambda _{2}$ is the wavelength corresponding to the maximum/half maximum of the measured spectrum. (f) An upward scan of the auxiliary laser with $P_{\mathrm {Aux}}=9\,\mathrm {mW}$ (red solid line), where the color-coded stars correspond to the wavelength of the auxiliary laser at which each spectrum in (a)-(e) is obtained. (g) The symmetry parameter of the lineshapes against the wavelength of the auxiliary laser when $P_{\mathrm {Aux}}=9\,\mathrm {mW}$. (h) The variation of $\delta _{\mathrm {SHG}}$ against different $P_{\mathrm {Aux}}$. The hollow green squares represent the measured $\delta _{\mathrm {SHG}}$ while the blue dots are calculated by the theoretical model.

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The tuning range ($\delta _{\mathrm {SHG}}$) of our system is actually limited by the linewidths of the microcavity modes ($\kappa _{a,\mathrm {load}}$, $\kappa _{b,\mathrm {load}}$). Here we assume the system should satisfy the condition $P_{\textrm {SHG}}\geqslant P_{\textrm {SHG, {\textrm {max}}}}/2$. According to Eq. (6), we find that when $-\kappa _{b,\mathrm {load}}\leqslant \left |\delta _{a}-\delta _{b}\right |\leqslant \kappa _{b,\mathrm {load}}$ is satisfied, we can always achieve $P_{\mathrm {SHG}}>\frac {1}{2}P_{\mathrm {SHG},\mathrm {max}}$ with $P_{\mathrm {SHG},\mathrm {max}}$ denotes the maximally achievable $P_{\mathrm {SHG}}$. Such as relation implies that the system high efficient SHG ($P_{\mathrm {SHG}}>\frac {1}{2}P_{\mathrm {SHG},\mathrm {max}}$) could always be achieved with certain temperature tuning range that satisfies $-\kappa _{b,\mathrm {load}}\leqslant \left |2K_{a}-K_{b}\right |\Delta T\leqslant \kappa _{b,\mathrm {load}}$, i.e. $-15.4\leqslant \Delta T\leqslant 15.4\,\mathrm {K}$. From Eq. (2), the corresponding tuning range of signal frequency can be written as $\delta _{\mathrm {SHG}}=\left |K_{a}\right |\Delta T$, which gives $-27.7\leqslant \delta _{\mathrm {SHG}}\leqslant 27.7\,\mathrm {GHz}$. It is obvious that the cavity with high quality factor or high thermal drift coefficient $K_{a(b)}$ can have a larger tuning range ($\frac {2K_{a}\kappa _{b,\mathrm {load}}}{\left |2K_{a}-K_{b}\right |}$), but the variation of temperature ($\Delta T$) should be adjusted precisely to maintain the phase-matching conditions. Furthermore, the damaged power threshold of the material is about $300\,\mathrm {mW}$ (Refs. [11,13]) limits the achievable tuning range as well. To dominate the temperature tuning by auxiliary laser, the power of auxiliary laser should also be much stronger than the pump laser. Therefore, we can find the practical power limitation in our system is $P_{\textrm {Aux}}\leq 300\,\mathrm {mW}$ and $P_{p}\leq 30\,\mathrm {mW}$, respectively.

The demonstration of temperature controlled by the auxiliary laser also indicates that the SHG for a fixed pump wavelength could be switched optically. Hence, the temporal response of the all-optical control of SHG is experimentally investigated. Previously, we made the approximation of steady state as the wavelength scan speed $d\omega _{\mathrm {p}}/dt$ satisfies $d\omega _{\mathrm {p}}/dt\times \frac {1}{\gamma _{\mathrm {th}}}\ll \kappa _{a,\mathrm {load}},\kappa _{b,\mathrm {load}}$ and also the power changing fulfills $\frac {dP_{\mathrm {p,Aux}}}{dt}\times \frac {1}{\gamma _{\mathrm {th}}}\ll P_{\mathrm {p,Aux}}$, in which $\gamma _{\mathrm {th}}$ is the cavity thermal relaxation rate, thus the transient response of the cavity is negligible. For the purpose of controlling temperature, we can quickly modulate the $P_{\mathrm {Aux}}$ to tune SHG doubly-resonant conditions, hence the steady-state approximation does not apply and the system will show a fast switching of SHG signal. Shown by the green dash region in Fig. 1(a), a telecom electro-optic intensity modulator (EOM) driven by a modulated RF signal is applied to generate the intensity-modulated auxiliary laser. Here, the frequency of one output sideband of EOM, which is controlled by a 20 GHz microwave generator, is tuned to be resonant with the auxiliary mode, and the intensity of the sideband is switched by a low-frequency RF source. The pump laser at $P_{\mathrm {p}}=4\,\mathrm {mW}$ is off-resonance with $\omega _{a}$ to prevent the generation of the SH signal when the auxiliary laser is off. When the auxiliary laser ($P_{\mathrm {Aux}}=9\,\mathrm {mW}$) is on, $\omega _{a}$ will be tuned to the near-resonant frequency of the pump laser which leads to the generation of the SH signal. Figures 4(a) and (b) show the modulated SH signals in different frequencies of sine waves corresponding to $32\,\mathrm {kHz}$ and $1024\,\mathrm {kHz},$ respectively. When the modulation rate reaches $32\,\mathrm {kHz}$, the SH signal follows the modulation very well, showing the largest modulation extinction. But when the modulation rate approaches $1024\,\mathrm {kHz}$, the modulation extinction decreases obviously, shown as Fig. 4(b). Figure 4(c) depicts the power extinction of the generated SH signal against the modulation rate, and shows an all-optical modulation bandwidth of about $256\,\mathrm {kHz}$.

 

Fig. 4. Modulation of the SHG by the auxiliary laser. (a) and (b) The temporal response of the dynamic control with the modulation rate of $32\,\mathrm {kHz}$ and $1024\,\mathrm {kHz}$, respectively. (c) The extinction of SH power against the modulation rates.

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4. Elimination of thermo-optic bistability by the auxiliary laser

Under the auxiliary control laser, not only can the doubly-resonant wavelength be controlled, the thermal response of SHG is significantly changed. As shown in Figs. 3(c) and (d), some spectra of SH signals show Lorentz-like peaks (green and blue lines) in contrast to the triangle-shaped peaks shown in Figs. 3(a) and (e). These results prove that the auxiliary laser could also control the thermal response behavior of the SHG process. In the previous section, we intend to control $T_{\mathrm {ring}}$ by the auxiliary laser but neglect the thermal effect of the pump laser. In fact, both auxiliary and pump lasers coupled into the microring cavity impact $\Delta T$ simultaneously (Eq. (12)), and meanwhile the temperature variation ($\Delta T$), in turn, affects the intracavity power. Therefore, the triple-mode system also functions as a feedback control model of the thermal effect. According to Eq. (12), we find that the effect of the feedback control is mainly depending on the power of the two lasers. Under the approximation that $P_{\mathrm {Aux}}$ is much stronger than $P_{\mathrm {p}}$, i.e. $\left |a\right |^{2}\ll \left |c\right |^{2}$, the pump laser can be regarded as the perturbation of the triple-mode system and we can get the first order Taylor expansion of Eq. (10) as

According to our experiments, the effective detuning between $\omega _{\mathrm {Aux}}$ and $\omega _{c}$ defined as $\delta _{c}=\delta _{c,0}+\alpha _{c}K_{c}\left |c_{0}\right |^{2}$ is comparable with $\kappa _{c,\mathrm {load}}$ when the auxiliary laser could efficiently excite the auxiliary mode. As a consequence, the corresponding thermo-optic mode frequency shift $\alpha _{c}K_{c}\left |c_{0}\right |^{2}$ due to the auxiliary laser is much larger than $\kappa _{c}$ (Fig. 2). Thus, in practice we get

5. Conclusion

In this paper, we studied the control of thermal effect associated with the second-harmonic generation by an auxiliary laser in an AlN integrated microcavity system for a targeted microring resonator. We demonstrated a precise and fast control of the microring temperature with a relatively large dynamic range, which provided an efficient and convenient approach to tune the doubly-resonant condition of nonlinear optical frequency conversion, and can be utilized for the precise frequency alignment between telecom photons and atomic transitions [12]. Additionally, our approach suppressed the thermal bistability induced by the pump laser, which is equivalent to canceling the thermo-optic coefficient of the microcavity that is widely studied in previous research [26,33,34]. Comparing to the previous works, our demonstration will not be limited by fabrication techniques, design and materials, therefore it can be extended to various microcavity platforms and also other nonlinear optical effects.