Effect of Thermal Tuning and Mode Coupling on Soliton Microcombs in ALN Microresonators

Optical resonator-based Kerr frequency combs allow an ultra-precise time and frequency measurements for a wide range of applications including ranging, biosensing, high-capacity communications, and quantum information technology. Kerr solitons are coherent frequency combs that are crucial for frequency synthesizer and chronometric instruments. The behavior of the soliton is determined by multiple factors while the key challenge for stable operation is to maintain thermal equilibrium in the resonator. In this work, owing to the stabilizing thermal effect caused by a neighboring auxiliary mode (TE10), we demonstrate the generation of octave-spanning (1100-2380 nm) single-soliton within a wide laser detuning range (up to 16 GHz) by pumping the TE00 mode. The influence of temperature on the microcombs behavior is also explored. Firstly, diverse two-solitons are achieved by adjusting the temperature. Secondly, the soliton access possibility is reduced when further increasing the temperature due to the avoided mode crossing between the pump mode (TE00) and another adjacent fundamental transverse-magnetic mode. At the same time, the spectral bandwidth of the microcomb that originates from TM00 modes is significantly enhanced. Finally, we show that the carrier offset frequency, which is evaluated to be ∼30-40 GHz, increases as the temperature is increased.

last for many hours. These unique characteristics of the soliton microcombs make it a potential candidate in many commercial applications such as optical frequency synthesizer [2], opticalclock [3], communications [4], [5] and LiDAR [6]. Particularly, for timing and precision measurements, the chip-scale soliton microcomb with an octave-span and integrated with a pump laser is required for stable measurement of the comb via the self-referencing technique [7]. Some remarkable results related to the octave-spanning soliton microcombs have been reported recently. By directly butt coupling the high-Q microresonators with a semiconductor laser pump without an optical isolator [8], [9], self-injection-locking was demonstrated to be a very promising integration strategy to stably produce an octavespanning soliton [10]. In addition, the presence of a neighboring auxiliary mode near the pump mode has been demonstrated to be a key approach to overcome the thermal instability in the microresonators allowing access the octave-soliton in AlN and Si 3 N 4 platforms [11]- [13]. Owing to the wide soliton existence step based on this strategy, we have reported direct and deterministic creation of octave-spanning Kerr solitons in an AlN microresonator by switching the laser wavelength on and off resonance [11]. The soliton existence step is the laser detuning range where the soliton state stably exists.
Mode coupling is a very common phenomenon in resonators that can occur between two close modes that have different transverse orders or polarizations [14]. In terms of the microcomb, such mode interaction has been exploited to generate dark-pulse [15], [16], dispersive wave (DW) [17] and soliton crystals [18]. Temperature plays an important role in microcomb formation. Firstly, the thermal instability when the pump enters the reddetuned side of the resonance can hinder the soliton stabilization. Second, it can affect the resonance characteristics including the repetition rate and carrier-envelope offset frequency (f ceo ) of the soliton combs [19].
In this work, using an AlN microresonator, we investigate the effect of temperature control on mode coupling. Firstly, the stable access to octave-spanning single-soliton is achieved by pumping the fundamental transverse-electric (TE 00 ) mode, which has an auxiliary first-order transverse-electric (TE 10 , on the red-side of the TE 00 mode) for thermal compensation. A record soliton existence step of ∼16 GHz is demonstrated. Versatile two-soliton states with almost an octave-spanning can be accessed at slightly high temperature. In addition, another adjacent mode but with cross-polarization (fundamental transverse-magnetic TM 00 , on the blue-side of TE 00 mode) appears at higher temperature and leads to a strengthened mode coupling with the TE 00 mode, which then hinders the soliton access. At a relatively high temperature due to strong mode coupling, significant enhancement of the microcomb bandwidth of the TM 00 mode with the assistance of the dispersive wave (DW) of the TE 00 is obtained. The f ceo of the single-soliton and its dependence on temperature was also investigated experimentally.

II. RESONANCE CHARACTERISTICS OF THE MICRORESONATOR
Control of dispersion and Q factor are crucial to determine the microcomb bandwidth and the soliton access. The integrated dispersion (D int ) curves of the TE 00 (upper panel), TE 10 (middle panel) and TM 00 (lower panel) mode families, in the 60-μmradius AlN microresonator, are calculated via commercially available software COMSOL and summarized in Fig. 1. We evaluate the effect of changing the ring waveguide width and keeping a fixed AlN thickness of 1.2 μm. As shown in Fig. 1, changing the width results in an obvious shift in position of the DW, i.e., D int = 0. Near zero anomalous dispersion with access to the DW is vital for efficient four wave mixing (FWM) and broadband combs generation [20]. The TE 00 mode shows anomalous dispersion (D>0) for waveguide widths smaller than 2.5 μm. Whereas, the TE 10 mode has large anomalous dispersion for the narrower rings. The TM 00 mode has large anomalous dispersion profiles at ∼2.3 μm. The desired dispersion of the TE 10 and TM 00 modes with a short-wavelength DW is achieved for relatively wider resonators, which have been experimentally used for an octave-spanning comb generations [21], [22].
In this paper, we employ a resonator with a width of 2.29 μm and height of 1.2 μm that was fabricated by standard photolithography and dry etching processes, fabrication details can be found in [21]. The TE 00 mode at ∼193 THz has a simulated FSR, D int /2π and DW of 374 GHz, ∼4.8 MHz and ∼255 THz, respectively. Fig. 2(a) shows the measured transmission spectra for the TE polarization, where the TE 00 and TE 10 modes are denoted by blue and green dash lines. Clearly, all the resonance wavelengths red shift with increase of TEC temperature due to the thermo-optic effect. The insertion loss is around 6.4 dB, and the transmission curves are shifted by 10 dB for clarity. Possessing a larger FSR, the TE 00 modes are located on the blue side of the TE 10 modes in wavelength regions (i) and (ii), while in (iii) and (iv) they move to the red side. Fig. 2(b) is a zoomed-in image of the two adjacent modes in region (ii) near 1550 nm. It is noticeable that in region (ii) another mode, the TM 00 (marked with pink coloured square) appears at high temperature, due to the mode coupling with the TE 00 mode. Therefore, for region (ii), we mark the two fundamental modes to red-and blue-sides as mode A and B, respectively, corresponding to TE 00 and TM 00 modes when temperature is below ∼35°C . Fig. 2(c)(upper panel) shows the wavelength separations between the TE 10 modes and adjacent fundamental modes, i.e., all TE 00 modes and the TM 00 mode in region (ii), which is defined as The extinction ratios (ERs) of fundamental modes in region (ii). At 14°C, the resonance wavelengths of mode A (TE 00 ) and B (TM 00 ) are ∼41 and ∼79 pm less than that of the TE 10 modes, while with an increase of the temperature, the two branches have an avoided crossing accompanied by the significant change in ERs, indicating the mode coupling between them. It is anticipated that such mode coupling would affect the soliton dynamics. The separations between two adjacent TE modes in region (i) decrease linearly from 102.3 to 91.7 pm when the temperature increases from 20°C to 55°C. Similarly, for regions (iii) and (iv), small separation increases of 9.4 and 10.6 pm are observed. Without mode coupling, the separation between the two TE modes in region (ii) is expected to be ∼31 pm at 60°C, which would be sufficient to stably access the soliton as demonstrated at room-temperature [11].

III. EXPERIMENTAL RESULTS
In this section, we present the main experimental soliton comb results when pumping the TE 00 mode near 1550 nm under different TEC temperatures.

A. Octave-Spanning Single-Solitons Under Various TEC Temperatures
The experimental setup is plotted in the Fig. 3(a). Firstly, we set the substrate TEC temperature and couple the fibers with the waveguide. The resonator is pumped by a continuous wave laser (CW) laser which is amplified by the erbium-doped fiber amplifier (EDFA). A fiber polarization controller (FPC) is utilized to select the laser polarization mode (TE mode in this case). We demonstrated the single-soliton generation by sweeping the pump laser with a relatively slow speed 1 nm/s. The top stack of Fig. 3(b) shows the simultaneously measured pump-and all-transmission, at 16°C and 320 mW on-chip power, with the help of a band-pass filter. The difference of the transmissions was calculated and plotted in the bottom stack of Fig. 3(b), showing an apparent decrease around 1550.575 nm when the soliton was produced. It should be noted that this difference curve is different from the comb power which is achieved by suppressing the pump directly during sweeping. Despite, the soliton state can be found from the step-like features and the monitored spectrum in OSA. However, the soliton existence window is not clear due to the lack of a soliton annihilation position, where the power generally equals to 0. This unusual phenomenon was observed due to the presence of an adjacent TE 10 mode. Based on the measurement on other devices without this adjacent mode, the nonlinear red shift of the TE 00 mode is around ∼150 pm [23], much greater than the mode separation between TE 00 and TE 10 modes. Therefore, the transmissions of the two modes merge in a wide wavelength region. More specifically, a pump power with the same level as the soliton comb power was absorbed and coupled into the TE 10 mode directly once the soliton is annihilated. When further increasing the laser detuning, a breather soliton can be generated from the TE 10 mode [see discussion in section D]. By repeatedly (50 times) sweeping the laser wavelengths to different positions and recording the comb state, we plot the soliton access possibility with brown colored circle symbols. A thick double arrow line is depicted to mark the soliton access window (126 pm, ∼16 GHz) where the possibility is higher than 80%. Similar results were achieved at 28°C [see Fig. 3(c)], while one can see a complete and clear soliton step with a range of 80 pm (∼10 GHz), consistent with the discrete soliton possibility measurement. The wide and stable soliton access step is significant and crucial for many practical applications without the requirement of power kicking techniques [20], [24]. Fig. 3(d) plots the typical single-soliton spectra obtained at 16°C and 28°C, with the pump laser wavelength of 1550.6 and 1550.8 nm, respectively. The 3-dB bandwidth of the solitons is fitted to be ∼15 THz. The DW waves that occur around 270 THz agree well with the simulated dispersion profile. Some comb lines at the long-wavelength edge are found due to the 2 nd -order diffraction of the optical spectrum analyzer (OSA). As depicted in the inset of Fig. 3(d), the 1 st -order comb lines marked with circle symbols have the same separation as the cavity FSR. However, the intensity of 2 nd -order lines (square symbol) increases versus the wavelength increase and shows a separation of half the cavity FSR. The single-soliton spectrum has a sech 2 spectral envelope fitting, shown by the red dashed line in Fig. 3(d). Hence, the octave-spanning soliton microcomb ranging from 126 to 272 THz is deterministically obtained. We also pumped the mode A beyond 50°C where the mode coupling is mitigated, while only MI combs are generated. This can be attributed to the change of the extinction ratio, Q-factor and the separation between the pump and auxiliary resonance modes. A relatively poor stability of the device holder at high temperature is another limitation. Fig. 4(a) is a histogram illustrating the soliton access steps under various temperatures and powers. When sweeping the laser with a speed of 1 nm/s, the single-solitons can be easily accessed between 290 and 480 mW on-chip power depending on the temperature. The soliton steps show an overall decreasing trend along with an increase of the on-chip power due to thermal effects in the resonator. The difference curves, between all-and pump-transmission, with the widest soliton access window at each discrete temperature are plotted and compared in Fig. 4(b). As marked by the blue rectangles, obvious soliton steps between 22°C and 28°C were observed, before the power drops. Without a clear soliton step, the soliton available window at 14°C, 16°C, 18°C, and 20°C are found to be ∼16, ∼14, ∼12 and ∼10 GHz, respectively, with the similar repeated scanning method illustrated in Fig. 3(a). The single-soliton cannot be accessed beyond 28°C. Considering that the spacing of the two TE transverse modes has no significant change (only 3 pm) from 16°C to 28°C, we would say the mode coupling between the two fundamental modes affects the extinction ratio and thus limits the soliton generation. This can also explain that the soliton access step is much wider at relatively low temperature, where it is far from the strong mode coupling region.

B. Two-Solitons Generation
In addition to the single-soliton, two-solitons are also demonstrated by increasing the temperature slightly to 29°C. By repeatedly sweeping the laser wavelength at 1nm/s, we plot the recorded comb power profile in Fig. 5(a), where the discrete pronounced steps indicate the generation of solitons with a number (N) of 2 or 1. At 29°C, the nonlinear shift is widened due to the mode coupling between two fundamental modes. The two-solitons can be repeatedly accessed but are more sensitive to the polarization compared with the pure single-soliton. We record and plot the spectra of various two-solitons in Fig. 5(b), which reach almost an octave-spanning from 130 to 260 THz. The two-solitons are accessed by stopping the pump laser at where μ is relative mode index to the pump mode, S (1) (μ) is spectral envelope of the single-soliton and ϕ is an azimuthal angle in between 0 and 2π. No solitons are observed beyond 29°C due to the strong mode coupling between the two fundamental polarization modes.

C. Microcombs From Other Mode Families
To better understand the nonlinear dynamics for the adjacent modes, we also pump the TM 00 and TE 10 modes at different temperatures. As shown in Fig. 6(a), the spectral bandwidth of the MI comb originating from the TM 00 mode is enhanced from ∼82 THz at 16°C to ∼130 THz, i.e., an octave-span, with 28°C. Coincidentally, like the TE 00 soliton microcomb [ Fig. 3(d)], here the DW hump appears at 250 THz, which might be caused by the mode-crossing induced dispersion perturbation [17], [25]. Such spectrum broadening may also result from a thermal-optical change of global dispersion (not only local dispersion) for TM 00 mode at higher temperature, which we will explore more in future. For comparison, we also pumped other TM 00 modes that are far from the TE 00 modes with the same power, while various   around 45 GHz at 26°C is observed. This adjustment is attributed to the modified FSR, and the thermo-optical effect induced red shift of resonance wavelength, which was investigated in [30]. Different from the typical f-2f self-referencing schemes, this is a pure mathematical calculation with a relatively poor accuracy of ∼1 GHz. Nevertheless, this helped us to estimate the level of the offset frequency and will guide the future design. Also, a further decrease on the f ceo is possible by decreasing the temperature. temperatures were set. We found that the comb spectra always have a bandwidth of ∼80 THz, which is narrower than the comb shown in the Fig. 6(a) because of the lack of an accessible DW. For the auxiliary TE 10 mode, the opposite phenomenon was found and depicted in Fig. 6(b). At relatively high temperatures (higher than 22°C), it is hard to get a stable comb from the TE 10 mode. However, a breather soliton can be accessed at 16°C with a pump wavelength of 1550.62 nm. The breather soliton was verified from the electrical spectrum analyzer (ESA) noise, which changes from a broad-band beat note to several beating lines [see Fig. 6(c)] while the comb profile changes slightly [26], [27]. The frequency comb was suppressed at high temperature because of the enhanced non-linear resonance shift and the competition from the pump mode.

D. F ceo Frequency Characterization
The characterization and stabilization of the microcombs is also essential for practical applications. f ceo is an important parameter for the microcomb and can be detected through the self-referencing f -2f [6] and 2f -3f [28] techniques. Tuning the f ceo close to zero with good stability is of practical significance in frequency synthesis applications [2].
In this section, we show the relation between temperature and f ceo of the soliton microcombs. As depicted in Fig. 3(d) inset, the 1 st -order and 2 nd -order induced comb lines will coincide around 2340 nm, with a frequency of f n = n×f rep +f ceo and f 2n /2 = n×f rep +f ceo /2, where n is the comb number. By setting the OSA resolution and sampling interval at 0.01 and 0.025 nm, respectively, we measured the soliton comb spectra in a narrow wavelength region. The f ceo can be calculated through the mathematical calculations as follows Fig. 7(a) shows the measured spectra at 16°C by increasing the laser wavelength over a 0.15 nm range. As the square marked symbols show in Fig. 7(b), the f ceo is estimated to be ∼ 38 GHz, while there is sudden decrease to 33 GHz due to the change of the single-soliton states [11]. The obtained value is almost equal to one reported in [29], which utilized an auxiliary 2-μm laser and an on-chip integrated AlN waveguide to realize self-referencing. By adjusting the TEC temperature, f ceo has a minimum value of around 33 GHz at 14°C while an increase to

IV. CONCLUSION
In summary, we investigate the effect of temperature control on the soliton microcomb behavior. With an auxiliary resonance near the pump, an octave-spanning dissipative Kerr soliton is deterministically accessed in an AlN microresonator with an ultra-wide soliton step of ∼16 GHz at 16°C. This was achieved with a slow pump tuning speed of 1nm/s. Using a rather simple experimental setup with a thermoelectric controller, we explore the soliton existence step versus temperature. The breather soliton was generated from the auxiliary high-order mode at low temperature due to a large mode separation. We also investigate avoided mode crossings between TE 00 and TM 00 at higher TEC temperature, which can hinder the soliton access from the pump mode but enhance the bandwidth of the microcomb from the TM 00 mode. Near octave-spanning two-solitons are also demonstrated in the resonator with a relatively high temperature. The mode coupling and thermal tuning was found to alter the local dispersion profile of the TM 00 mode and significantly enhance the comb bandwidth to an octave-spanning range at a relatively high temperature. Hence, mode coupling seems is a very useful tool for soliton generation and can be further explored by designing microcavities with engineered precise positions for such mode coupling. Our work also shows that AlN is another platform for soliton generation and control. Further studies of temperature control are of interest since this material and its sapphire substrate are of high thermal conductivity.