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Abstract
The hybrid microcavity composed of different materials shows unique thermal-optical properties such as resonance frequency shift and small thermal noise fluctuations with the temperature variation. Here, we have fabricated the hybrid Si3N4 − TiO2 microring, which decreases the effective thermo-optical coefficients (TOC) from 23.2pm/K to 11.05pm/K due to the opposite TOC of these two materials. In this hybrid microring, we experimentally study the thermal dynamic with different input powers and scanning speeds. The distorted transmission and thermal oscillation are observed, which results from the non-uniform scanning speed and the different thermal relaxation times of the Si3N4 and the TiO2. We calibrate the distorted transmission spectrum for the resonance measurement at the reverse scanning direction and explain the thermal oscillation with a thermal-optical coupled model. Finally, we analyse the thermal oscillation condition and give the diagram about the oscillation region, which has significant guidance for the occurrence and avoidance of the thermal oscillation in practical applications.
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1. Introduction
Whispering-gallery-mode (WGM) microresonators have the advantages of the ultra-high quality factor and small mode volume which maintains the high light intensity and enhances the light-matter interactions [1–7]. Due to the absorption of the material, the circulating light in the microresonator will inevitably heat the material, leading to the shift of the resonance frequency via the thermo-optic and thermal expansion effects [8–11]. These two effects convert temperature fluctuations into changes in the material refractive index and the cavity size, respectively, with different response times [9,12,13]. Thermal nonlinear effects like linewidth broadening, linewidth compression, and thermal oscillation have been widely studied and used as tools for thermal sensing, thermal imaging, and thermal locking [8,9,14]. However, in most applications, the thermal stability of the resonance frequency is of great importance, such as lasing, optical sensing, optical signal processing, and dispersion engineering [15–19]. Especially, for soliton microcombs, which have attracted increasing interest in the past decade, thermal instability can shorten the existence range of the soliton microcomb and introduce thermorefractive noise [5,10,20,21].
To achieve the temperature-independent resonance frequency, various approaches have been proposed, including designing special photonic geometries, optimizing the thermal dissipation structures, and hybridizing materials with opposite TOCs [22–30]. Among these approaches, the hybrid material approach has been widely implemented in silicon, silicon oxide, silicon nitride ($Si_{3}N_{4}$), and lithium niobate platforms, and the choice of negative TOC materials is also diverse, such as polymers PDMS and PMMA, and dielectric materials titanium oxide ($TiO_{2}$) [26–35]. Compared to polymers, $TiO_{2}$ is more suitable for integrated applications due to its chemical, mechanical stability and CMOS compatibility. However, in the hybrid microresonator, the competition between the opposite thermo-optic effects can lead to the thermal oscillation, as demonstrated in the PDMS-coated microresonators [34,36,37]. Thermal oscillation has the potential for the highly sensitive sensing and light modulation [37–39], but it also breaks the stability of resonance frequency. Therefore, it is meaningful and necessary to investigate whether thermal oscillation exists in the $TiO_{2}$-cladded hybrid microresonators and the mechanism and regime of the thermal oscillation.
In this paper, we have fabricated the hybrid $Si_{3}N_{4}-TiO_{2}$ microring resonator with 850-nm-thick $Si_{3}N_{4}$ core layer and 180-nm-thick $TiO_{2}$ cladding layer. The effective TOC of the microring drops from $23.2\:\mathrm{pm/K}$ to $11.05\:\mathrm{pm/K}$. Moreover, we observe and systematically study the distorted transmission spectrum and the thermal oscillation resulting from the competition between opposite thermo-optic effects in $Si_{3}N_{4}$ core layer and $TiO_{2}$ cladding layer under different pump power and scanning speed. Finally, a thermal-optical coupled model is established to explain the mechanism of thermal oscillation, and the conditions of thermal oscillation are analyzed to provide guidance for generating or avoiding thermal oscillation in the practical applications.
2. Device fabrication and experimental setup
The $Si_{3}N_{4}$ microring resonator is fabricated on a 850-nm-thick stoichiometric $Si_{3}N_{4}$ film with $3 \mu m$ buried silicon dioxide ($\mathrm{SiO}_{2}$) layer. Detailed fabrication process can be found in Ref. [40]. After finishing the microring preparation, the $TiO_{2}$ cladding layer is then deposited by sputtering. According to our previous theoretical paper [41], for the purpose of counteracting the thermo-optical effect while satisfying the anomalous dispersion, the cross section of the microring is $1800\:\mathrm{nm}\times 850\:\mathrm{nm}$, and the thickness of the $TiO_{2}$ cladding layer is about $180\:nm$, as shown in the inset of Fig. 1(d). The schematic of the experimental setup is shown in Fig. 1(a). A tunable laser (Toptica CTL) is amplified by an erbium-doped fiber amplifier (EDFA) and coupled into the chip by the lensed fiber. The fiber polarization controller (FPC) is used to control the polarization of the laser. The output light is collected by another lensed fiber and recorded by a photodetector. The fiber-to-chip coupling loss of each chip facet is about 5.2 dB.
Fig. 1. (a) Sketch of the experimental setup. AFG, arbitrary function generator; DSO, digital storage oscilloscope; EDFA, erbium-doped fibre amplifier; WM, wavelength meter; PD, photodetector; FPC, fibre polarization controller. Inset: optical microscope image of the $Si_{3}N_{4}-TiO_{2}$ microring with radius of $100\:\mu m$. (b) The typical transmission of the hybrid microring. (c) A detailed transmission around the wavelength of $1595.47\:\mathrm{nm}$ has a linewidth of $0.015\:nm$ according to the Lorentzian fitting (red line), corresponding to a loaded Q of $1.06\times 10^{5}$. (d) The frequency shift with the temperature in the microring without (blue square) and with (red triangle) the $TiO_{2}$ deposition. The inset shows the simulated cross section of the waveguide.
Figure 1(b) shows the typical transmission spectrum of the TE polarization of the hybrid microring. Detailed characterization of a cavity resonance around the wavelength of $1595.47\:\mathrm{nm}$ is shown in the Fig. 1(c), exhibiting the loaded Q factor of $1.06\times 10^{5}$. The average loaded Q factor of fundamental TE modes is around $1\times 10^{5}$. Because the mode profile of the TM modes in the air-cladding waveguide does not match that of the optical modes in the lensed fiber, the TM polarization is difficult to be launched into the chip through the lensed fiber. To character the effective TOC of the hybrid microring, we slowly increase the temperature of the chip from room temperature to $55\:\mathrm{{^\circ }\mathrm{C}}$ and record the wavelength shift of the resonance mode, as shown as the red triangles in Fig. 1(d). The fitted curve exhibits a linear slope of $11.05\:\mathrm{pm/K}$. As a comparison, we also shows the frequency shift of TE modes in the pure $Si_{3}N_{4}$ microring by the blue squares, which exhibits a linear slope of $23.2\:\mathrm{pm/K}$. It is clear shown that the effective TOC is reduced by the $TiO_{2}$ cladding layer. Due to the difference of the refractive index and TOC between the actual deposited $TiO_{2}$ film and the values adopted in the theoretical work, as well as the fabrication errors in the process, the effective TOC does not decrease as much as expected.
3. Distorted transmission and thermal oscillation
Next, we finely study the dynamical process of the optical resonance mode in the microring via a triangle signal, which is a general technique [8,42–49]. However, we find that the transmission spectrum can be distorted by the nonuniform scanning speed, as shown in the Fig. 2. The blue curve in Fig. 2(a) is wavelength variation we would expect to obtain from the triangle signal we applied, while the black curve is the actual wavelength of the tunable laser we calibrated with a wavelength meter with the resolution of $0.8\,pm$. The offset between them can be up to $0.12\,nm$, as shown by the red curve in Fig. 2(a). We believe that it is derived from the nonlinear response between the piezo signal on the laser and the applied triangle signal. We confirmed this phenomenon using different lasers at different scanning speeds. Therefore, since the actual scanning speed at the beginning (end) of linear part of the triangle signal is slower (faster) than the average scanning speed, the measured transmission spectrum will be compressed(stretched), resulting in distortion of the transmission spectrum. The blue curve in Fig. 2(b) obviously shows the asymmetric transmission when the optical mode is close to the beginning and end regime, where the on-chip power is about 7.5mW to avoid the obvious thermal broadening and compression. If the mode is shifted to the mid-regime, the transmission looks symmetric, as shown the black curve in Fig. 2(b). To eliminate this distortion, we measure the actual wavelength related to the real time by the wavelength meter, and calibrate the distorted transmission spectrum. Figure 2(c) shows the transmission spectra after the calibration. It can be seen that the same resonance mode is consistent at different positions during the up and down scanning, which ensures the accuracy of the spectral information we measured.
Fig. 2. The calibration for the laser’s wavelength scanning by the triangle wave from AFG. (a) The black curve represents the actual wavelength of the tunable laser by the wavelength meter centered around $1566\:nm$. The blue curve represents the virtual linear wavelength scan which connect the ends. The red curve is the offset between the actual and desired wavelength. The transmission spectra before (b) and after (c) the calibration. The black and blue curves are the back and forth transmission spectra of the same resonance mode with different scanning center wavelengths. The red lines in (b-c) with the same scale clearly show the distorted transmission before the calibration.
Figure 3(a) illustrates the transmission spectra with different input powers and different scanning speeds. With higher input power, the thermal broadening becomes more obvious, and the resonance frequency shift also becomes larger, as shown in Fig. 3(b). However, the thermal broadening increases linearly with the input power only when the scanning speed is about 2000 GHz/s, and the thermal broadening is not regular when the scanning speed is relatively small because of the occurrence of the thermal oscillation under the appropriate power. The transmission spectra obviously shows that the thermal oscillation is related to the input power, the scanning speed and the laser-cavity detuning. Figure 3(c) shows a magnified view of the thermal oscillation when the on-chip power is 33mW and the scanning speed is about 20 GHz/s. It is noted that the thermal oscillation time in the transmission is asymmetric, indicated by the longer thermal oscillation time at the beginning of the triangle signal. It results from that the speed slows down at the beginning of down wavelength scan, leading to the longer retention time within the oscillation region.
Fig. 3. (a) The calibrated transmission of the resonance with different on-chip power 72mW, 58mW, 44mW, 33mW, 15mW from top to bottom, and scanning time of 2ms, 2s and 20s. (b) Relationship between the resonance frequency broadening and the on-chip power with different scanning speeds. (c) A magnified view of the thermal oscillation when the power is 33 mW and the scanning speed is 20 GHz/s.
This thermal oscillation origins from the different thermal relaxation time and opposite thermal-optical coefficient of the $Si_{3}N_{4}$ and $TiO_{2}$. The similar oscillation behavior have been observed in polymer-coated microresonators [34]. To further understand the thermal oscillation behavior in this hybrid $Si_{3}N_{4}-TiO_{2}$ microresonator, we simulate it with the thermal-optical coupled equation [34,50]:
Table 1. Parameters used in the numerical simulations.
Figure 4(a) illustrate the measured and simulated results of the optical transmission spectra. To understand the process of the thermal oscillation, the numerical simulations of temperature fluctuations and the resonance frequency shift are presented for the related $Si_{3}N_{4}$, $TiO_{2}$ and the hybrid microcavity, as shown in Fig. 4(b-c). One period of thermal oscillation could be divided into four stages to understand the whole process, as shown in the inset of Fig. 4.
Fig. 4. (a) Measured (black curve) and simulated (red curve) results of the optical transmission spectra. (b) Numerical simulations of temperature fluctuations in the $Si_{3}N_{4}$ and $TiO_{2}$ layers, respectively. (c) Numerical simulations of resonance frequency shift in two layers and the hybrid microresonator. Right: Magnified views of the area of the thermal oscillation in each figure. Four stages are shaded by blue (I), yellow (II), green (III) and purple (IV).
Stage I: In the first stage, the pump laser wavelength is scanned from blue to red, gradually approaching the the effective (hybrid) resonance wavelength, and the pump laser is gradually coupled into the microresonator. In this process, due to the temperature increase caused by material absorption, the refractive index of $Si_{3}N_{4}$ layer gradually increases, while that of $TiO_{2}$ layer gradually decreases. Because the positive thermo-optic effect is more significant in the $Si_{3}N_{4}$ layer, the effective TOC of the hybrid microring is positive, and the effective resonance wavelength is dominated by the thermo-optic effect in $Si_{3}N_{4}$ layer, exhibiting a red shift. The transmission spectrum at this region shows a typical thermal stability behavior.
Stage II: In the second stage, once the pump laser is scanned through the resonance mode, the pump laser enters the effective red detuning region. Since the thermal relaxation time of $Si_{3}N_{4}$ layer is relatively small, the heat dissipation reduces the temperature of the microresonator, which leads to the blue shift of the effective resonance wavelength and the greater effective detuning, thus the extinction ratio of the transmission spectrum decreases rapidly.
Stage III: In the third stage, as the heat dissipation also begins to occur in the $TiO_{2}$ layer, the decrease of the temperature in the $TiO_{2}$ layer leads to a red shift of the effective resonance wavelength. The effective detuning thus decreases, which heats the $Si_{3}N_{4}$ layer again and further decreases the effective detuning. When the $Si_{3}N_{4}$ layer absorbs great heat from the pump laser, the effective resonance red shifts overwhelmingly and the pump wavelength is caught up subsequently. After arriving the effective blue detuning, the extinction ratio of the transmission spectrum increases naturally and the $Si_{3}N_{4}$ layer reaches the thermal balance of the absorption and dissipation.
Stage IV: In the fourth stage, as the temperature in the $TiO_{2}$ layer increases again, the effective resonance wavelength begins to blue shift and eventually crosses the pump laser wavelength, leading the system to enter the stage II again.
Stage I serves as the starting stage, and the cycle of stages II, III, and IV produces continuous thermal oscillations until the pump wavelength is scanned away from the region where the resonance wavelength can recoil.
4. Thermal oscillation regime
To further explore the thermal oscillation condition, we fix the pump wavelength at a certain detuning and record the transmission spectrum. As can be seen from Fig. 5(a), in addition to scanning the resonance mode, thermal oscillation can also occur and maintain when stopped at a specific detuning. If the input power is too low or too high, the thermal oscillation disappears, as shown in Fig. 3. It is better to understand the thermal oscillation regime and avoid it during the practical applications. The thermal oscillation condition could be analysed through the steady state condition of the Eq. (1–3) by solving the eigenvalue of the thermal equation around the stable solution based on the Lyapunov stability analysis [53,54]. Firstly, as we research the process in the time of the thermal respond level, the photon amplitude could be treated as the steady state because the intracavity photon amplitude can be stabilized in cavity decay time at nanoseconds level. Therefore, the Eq. (1) equals to zero and we derive the $\tilde {a}=\frac {\sqrt {\frac {\kappa _{ex}P_{in}}{\hbar \omega }}}{\frac {\kappa }{2}+i\left (\omega _{0}-\omega _{p}-\omega _{0}\left (\frac {1}{n_{eff}}\sum _{i}\eta _{i}\frac {dn_{i}}{dT}\Delta T_{i}\right )\right )}$. Then, the stable solution of the $\Delta T_{i}$ represented by $\Delta T_{i,0}$ could be solved from Eq. (1–3) by $\frac {d\Delta T_{i}}{dt}=0$ :
Fig. 5. Thermal oscillation condition. (a) A frequency fixed pump laser at the red side of the resonance which shows the aperiodic thermal oscillation due to the experiment condition of the laser and the surroundings. (b) The real part of the three solutions of the $\widetilde {T_{0}}$. (c-d) The solved thermal oscillation regime determined by the sign of the eigenvalue of the temperature perturbation equation. The thermal oscillation region is surrounded by the black line. The input power for (c) is 20mW and the $\eta _{Si_{3}N_{4}}$ for (d) is 0.85. Other parameters are used in the Table 1.
To analyse the steady state condition, we introduce the perturbation $\widetilde {\delta T_{i}}=\widetilde {T_{i}}-\widetilde {T_{0}}$ into the Eq. (4) and derive the temperature perturbation equation:
5. Conclusion
In conclusion, we characterize the thermal dynamics in the $Si_{3}N_{4}-TiO_{2}$ hybrid microring including the thermal broadening and oscillation. The non-uniform laser scanning speed should be considered to revise the thermal broadening and oscillation time. By simulating the thermal oscillation with the thermal-optical coupled equation, we give an insight into the principle of thermal oscillation and the oscillation condition with the temperature perturbation equation. The principle of the oscillation condition in $Si_{3}N_{4}-TiO_{2}$ microresonators is universal for the various hybrid system which is not limited to the microresonator and provide a powerful tool to explore hybrid system.
Funding
National Natural Science Foundation of China (11934012, 12104442, 12293052, 92050109, 92250302); CAS Project for Young Scientists in Basic Research (YSBR-069).
Acknowledgments
C.-H. Dong was supported by the State Key Laboratory of Advanced Optical Communication Systems and Networks, Shanghai Jiao Tong University, China. This work was partially carried out at the USTC Center for Micro and Nanoscale Research and Fabrication.
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.
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