1. Introduction

The generation of dissipative Kerr solitons in optical microresonators has emerged as a crucial technique for generating microcombs which are chip-scale miniaturizations of optical frequency combs playing an essential role in various fields of science and technology [1]. The soliton microcombs have been successfully applied to diverse applications, including spectroscopy [2,3], precise clocks [4], low-phase-noise microwave generation [5–8], optical frequency synthesis [9], and optical communications [10,11]. Despite the notable advancements made in recent years, the intricate procedures needed to achieve the soliton state that exist in an unstable region governed by thermo-optical nonlinearity [12,13] still have significant challenges. Up to now, the previously suggested approaches such as power kicking [14], high-speed pump frequency sweeping [15], high-speed thermal tuning [16], pulse driving [17], low-speed pump modulation [18], and optically-assisted thermal counterbalancing [19,20], have depended on the utilization of feedback loops, external control systems, or additional optical components. These additional requirements for soliton locking significantly contribute to the complexity of the overall system and restrict the extent of integration and miniaturization.

Recently, there have been attempts to address this complexity by implementing a self-locking mechanism via combining effects from multiple nonlinearities which reside in the microcavitiy. Particularly, stimulated Brillouin scattering has been used as a secondary pump source for thermal counterbalancing, which enables a self-stabilized soliton state without the need for external feedback mechanisms or additional auxiliary lasers [21–24].

Another representative nonlinear phenomenon, stimulated Raman scattering, has been studied in the context of the transition between Kerr and Raman-Kerr comb states [25–27], along with spectral broadening of the Kerr comb [28,29] through control of the coupling and dispersion conditions. However, the complex interactions between two distinct nonlinear phenomena, optical parametric oscillation (OPO) and stimulated Raman scattering, makes it difficult to reliably observe Raman solitons. To overcome this difficulty, recent studies have focused on suppressing Kerr nonlinearities near the pump wavelength by pumping in the normal dispersion regime while generating Raman-scattered photons in the anomalous dispersion regime, thereby generating a phase-locked Raman comb [30–32] or a Raman comb with a sech² envelope in optical domain [33–36]. In these studies, the two distinct nonlinearities are kept independent rather than utilizing joint effects.

One successful method for generating Raman solitons which use the two distinct nonlinearities together relies on the Stokes process to transfer a Kerr soliton near the pump wavelength to a longer wavelength soliton in a mode of another mode family, where both mode families are in the anomalous dispersion region [37]. Nevertheless, this method still requires a complex Kerr soliton mode-locking scheme, in addition to difficulties regarding dispersion matching between different modes at relatively distant wavelengths.

In this study, we successfully controlled the dynamics of the interaction between two distinct nonlinear phenomena, OPO based Four-wave mixing (FWM) and stimulated Raman scattering, within the microresonator. This was achieved by manipulating the coupling between the resonator and the waveguide, thereby controlling the threshold powers for each nonlinear phenomenon. Consequently, we classify the resulting frequency comb dynamics into three regimes based on their spectral responses. For the regime where the threshold powers for the pump-induced OPO and stimulated Raman scattering are comparable, the two nonlinear phenomena cooperate to generate Raman single-solitons that are spontaneous while concurrently deterministic. Notably, the generated Raman single-soliton demonstrates self-locking behavior, remaining stable over more than two hours without external feedback controls. Additionally, in the regime where the threshold power for Raman scattering is lower than that of the OPO at the pump wavelength, we cannot achieve a single-soliton state. Instead, a Raman comb with a sech² envelope, albeit exhibiting a considerably broad RF linewidth compared to that of a single-soliton state, is generated. This study implies that the interplay between Raman scattering and OPO is necessary for achieving the single-soliton state. Also, in this regime, a frequency comb with an atypical envelope (Lorentzian ${\times} $ sech² function) can be observed around the pump wavelength that is generated via the anti-Stokes process from the Raman comb.

2. Adjusting nonlinear threshold powers to control the interaction between OPO and Raman processes

In order to utilize both frequency combs at the pump and its Raman-scattered wavelengths and their interaction via Raman scattering process, it is imperative to satisfy an anomalous dispersion at both wavelengths. In this dispersion regime where these two nonlinear phenomena can take place, their order of occurrence can be controlled by adjusting their threshold powers. Since the distance between the pump and the Raman-scattered wavelengths is sufficiently large (about 100 nm in our experiments), we controlled their threshold powers by adjusting the total quality factor (Q-factor) at each wavelength. This relation is evident in the following equations describing the nonlinear threshold powers of interest in this study, namely for the OPO in the pump wavelength ($P_{OPO,P}^{th}$) and the Raman lasing induced by the pump wavelength photons ($P_{Raman}^{th}$) [38,39].

Here, ${Q_t}$ is the total Q-factor, which consists of an intrinsic Q-factor, ${Q_i}$ and an extrinsic Q-factor, ${Q_e}$ ($\frac{1}{{{Q_t}}} = \frac{1}{{{Q_i}}} + \frac{1}{{{Q_e}}}$). ${V_{eff}}$ and ${A_{eff}}$ are the effective mode volume and area [40,41], FSR is the Free Spectral Range of the comb, and $\lambda $ is the wavelength. The subscripts P and R denote the respective properties at the pump and Raman-scattered wavelengths. $\varGamma $ is the spatial mode overlap factor between the pump and the Raman modes [42], ${g_R}$ is the nonlinear bulk Raman gain coefficient, and B is a correction factor of the circulating power due to internal backscattering ($0.5 \le B \le 1$). n is the refractive index of silica, ${n_2}$ is the nonlinear Kerr coefficient, c is the velocity of light.

The ratio between two nonlinear threshold powers ($P_{Raman}^{th}/P_{OPO}^{th}$) can be examined using the expressions above to determine the amount of the total Q-factor adjustment needed to practically manipulate the nonlinear threshold powers. For example, the threshold powers of the Raman lasing and the OPO at ${\lambda _P}$ can be compared by dividing them as follows.

We assume a large mode overlap between the pump and Raman modes and almost no backscattering (i.e., $\varGamma ,B = 1$). Once the material and structure of the microresonator are determined, most variables in the equation above can be estimated through numerical calculation via COMSOL Multiphysics. For the particular silica wedge microresonator employed in this study $n = 1.442,\; {n_2} = 2.8 \times {10^{ - 20}}{m^2}/W,$ ${\lambda _R} = 1670nm,\; {g_R} = 6.2 \times {10^{ - 14}}m/W,$ ${V_{eff}} = 9.78 \times {10^{ - 13}}{m^3},\; {A_{eff}} = 5.34 \times {10^{ - 11}}{m^2}$, so we can estimate $P_{Raman}^{th}/P_{OPO,P}^{th} \approx 3.3 \times {Q_{t,P}}/{Q_{t,R}}$. This implies that, under the coupling regime where ${Q_{t,P}} \approx {Q_{t,R}}$, which is generally used in various nonlinear studies, pump-induced OPO emerges, followed by a frequency comb generation. However, if ${Q_{t,R}}\; $ is increased by approximately three-fold in comparison to ${Q_{t,P}}$, the two thresholds become similar ($P_{Raman}^{th} \approx P_{OPO,P}^{th}$). Therefore, stimulated Raman scattering and pump-induced OPO can occur concurrently. If ${Q_{t,R}}$ increased even further such that $P_{Raman}^{th}$ is now lower than $P_{OPO,P}^{th}$, we can observe a situation where the comb at Raman-scattered wavelength (Raman comb) is generated without the comb at pump wavelength (pump comb). Note that the thresholds of the OPO and a cascade Raman process (specifically, the second Stokes of the original pump) at ${\lambda _R}\; $ can be perceived in a similar manner as previously described for the case at ${\lambda _P}$. Specifically, if the total Q-factor at the second cascade Raman-scattered wavelength is not significantly different to that at ${\lambda _R}$, the threshold for the OPO at ${\lambda _R}\; $ will be lower than that for the cascade Raman process. This was experimentally confirmed in all the aforementioned regimes where a Raman comb is generated before the second Stokes process is observed.

The wavelength-dependent total Q-factor can be effectively controlled by adjusting its coupling contribution, ${Q_e}$ through the implementation of a coupling scheme that can adjust the wavelength-dependent interference over the coupling region. Previous studies, such as pulley coupling [43] and multiple point coupling [44–46], have successfully employed this approach. In this study, a tapered-fiber was used for coupling control, where the fiber was affixed to the resonator to induce the coupling over an extended length long enough for wavelength-dependent interference control (inset image in Fig. 1(a)). By manipulating the coupling length, the interference was precisely controlled, thereby enabling sufficient control of ${Q_e}$ at different wavelengths independently.

 

Fig. 1. (a) Experimental setup for analyzing frequency comb dynamics and generating Raman soliton. an erbium-doped fiber amplifier (EDFA, Amonics AEDFA-33-B) was used to amplify the light coupled out from the external cavity diode laser (ECDL, Toptica CTL 1550), which was evanescently coupled to the resonator via tapered-fiber. Magnified inset shows a microscope image of the tapered-fiber coupling apparatus for a given coupling length. An optical spectrum analyzer (Yokogawa AQ6370D) and a highspeed detector (Thorlabs DXM30AF) connected to an RF spectrum analyzer (Rohde & Schwarz FSW-26) were used to measure its spectral shape and beat note signal. A side view SEM image of the microresonator along with two top view microscope images representing two different coupling regimes are shown. Case1 shows the coupling state where a gap is present between the tapered-fiber and resonator. Case2 shows the situation where the tapered-fiber and resonator are affixed with a certain coupling length to control wavelength-dependent interference. The depicted contact region is approximately estimated based on the abrupt change in fringe pattern that occurs when the tapered-fiber is attached to the microresonator. (b) Measured frequency dispersion of the pump mode (green line) with two examined wavelength regions indicated with different colors (blue represents data near 1515 nm and red represents data near 1625 nm). The distribution of total Q-factors (c), (d) and their representative transmission spectra (e), (f) in different coupling regimes were also illustrated. The relative mode numbers for the long (red) and short (blue) wavelength data are labeled on the top and bottom axes, respectively. Note that the yellow star in (d) represents the results obtained when the total Q-factor exhibits the most significant difference, and the results in (e), (f) were measured at the relative mode numbers indicated by the dashed lines in (c), (d).

Download Full Size | PDF

The efficacy of this coupling scheme was experimentally validated by assessing the ${Q_t}$ values at different wavelengths. Due to the lack of a suitable light source at the Raman-scattered wavelength (∼1670 nm) in this study, measurements were conducted near 1515 nm and 1625 nm instead, since their distance corresponds to a Raman shift frequency (∼12 THz). The measured wavelength regions are marked by shaded ovals in Fig. 1(b), with the shorter wavelength data shown in blue and the longer wavelength data shown in red.

Figure 1(c)-(d) shows the distributions of ${Q_t}$ for the pump mode families near the 1515 nm and 1625 nm, respectively. Also, their representative transmission spectra (represented by the dotted line in Fig. 1(c)-(d)) is depicted in Fig. 1(e)-(f). Measurements for two different coupling regimes were carried out: the coupling state commonly used in various microcavity studies where the tapered-fiber and resonator are in close proximity but separated by a certain gap [Fig. 1(c), (e)] and where they are affixed with certain coupling length to control wavelength-dependent interference [Fig. 1(d), (f)], respectively. The experimental results indicated that the ${Q_t}$ in the former coupling states were comparable at both wavelengths, as shown in Fig. 1(c). Specifically, ${Q_{t,\; 1515}} = 1.49\; \times \; {10^8}$ (${Q_{e,1515}} = 4.91 \times \; {10^8}$, ${Q_{i,1515}} = 2.14 \times \; {10^8}$) and ${Q_{t,1625}} = 1.26\; \times \; {10^8}$ (${Q_{e,1625}} = 3.11 \times \; {10^8}$, ${Q_{i,1625}} = 2.12\; \times \; {10^8}$), respectively [Fig. 1(e)]. However, a significant difference in the total Q-factor could be observed for the later coupling state, as shown in Fig. 1(d). In particular, ${Q_{t,\; 1515}} = 2.87\; \times \; {10^7}$ (${Q_{e,1515}} = 3.32 \times \; {10^7}$, ${Q_{i,1515}} = 2.11 \times \; {10^8}$) and ${Q_{t,1625}} = 1.26\; \times \; {10^8}$ (${Q_{e,1625}} = 3.08 \times \; {10^8}$, ${Q_{i,1625}} = 2.13\; \times \; {10^8}$), respectively [Fig. 1(f)]. Notably, the average values of ${Q_t}$ at longer wavelengths are $1.28\; \times \; {10^8}$, whereas at shorter wavelengths, they are $3.83\; \times \; {10^7}$. Additionally, Fig. 1(d) illustrates that the ratio of ${Q_t}$ between the longer and shorter wavelengths exhibits a remarkable increase of over one order of magnitude (marked with a yellow star). These findings validate the coupling scheme's effectiveness in modifying the threshold power ratio between Raman lasing and pump-induced OPO, as mentioned earlier.

In this experiment, we employed a tapered fiber that is close to an ideal coupler, leading us to believe that serious backscattering did not occur. Additionally, despite the physical contact between the tapered fiber and the microresonator, unlike conventional coupling methods, we did not observe any significant backscattering enhancement experimentally. In particular, the most dominant influence was attributed to the coupling Q-factor, which can vary by one order of magnitude. This variation of the Q-factor is thought to overshadow other effects, making them negligible in comparison.

3. Three distinct optical spectral regimes of frequency combs and the spontaneous soliton mode-locking

In this study, a silica wedge resonator with a thickness of 8µm, a diameter of 5.9 mm, and a wedge angle of 20◦ was used, suitable for the production of Kerr frequency combs. More details regarding the fabrication process can be found in previous research [47]. The higher order mode is used as a pump mode as this phenomenon is not limited to specific modes, and its ${Q_i}$ was measured to be $2.13\; \times \; {10^8}$ at a pump wavelength of 1563 nm, the free spectral range (FSR) is 11.1045 GHz, and second-order dispersion for the pump mode exhibits anomalous behavior with a value of 4.86 kHz at 1563 nm, as shown in Fig. 1(b). All of the obtained results were calibrated using a Mach–Zehnder interferometer (MZI) with an FSR of 100.6305 MHz.

The interaction between the pump and Raman frequency combs that can be achieved in our scheme by tuning the nonlinear threshold powers via adjusting ${Q_e}$, was experimentally investigated by observing the resulting optical spectra while adiabatically tuning the pump frequency, i.e., ensuring thermal equilibrium is always achieved during the soliton generating process [24]. We characterized the optical spectra of frequency combs into three regimes. Frequency comb measurements in Fig. 2(a), (d), (g) were generated under the coupling regime where $P_{OPO,P}^{th} < P_{Raman}^{th}$, in which case, only the pump-induced OPO is observed. Figure 2(b), (e), (h) correspond to the coupling regime where $P_{OPO,P}^{th} \approx P_{Raman}^{th}$, so that the pump-induced OPO and stimulated Raman scattering appear concurrently. Figure 2(c), (f), (i) represent measurements in the coupling regime where $P_{Raman}^{th} < P_{OPO,P}^{th}$, hence the Raman comb occurs before the pump comb. Note that Fig. 2(d), (e), (f) represent the change of optical spectra (from top to bottom) by red-detuning of the pump frequency, resulting in the increase of the coupled pump power.

 

Fig. 2. Three distinct spectral regimes of frequency combs depending on the relative threshold powers between OPO and Raman processes. The conceptual schemes when (a) only pump-induced OPO appear, (b) pump-induced OPO and stimulated Raman scattering appear concurrently, and (c) Raman comb and its anti-Stokes comb appear are illustrated. This is followed by the variation of optical spectra via pump frequency detuning (d)-(f) and RF spectra of the resulting frequency comb states (g)-(i). Note that blue data represents optical spectra generated near the pump wavelength (1563 nm) and red data represents optical spectra generated near the Raman-scattered wavelength (∼1670 nm). Also, gray line in (h) is firstly generated beat frequency which comes from Raman comb and black line is a beat frequency from Raman soliton which appears only at specific pump detuning condition.

Download Full Size | PDF

Under the first regime where $P_{OPO,P}^{th} < P_{Raman}^{th}$, the pump-induced OPO emerges as the coupled pump power increases. This is followed by the appearance of a frequency comb spectrum with modulation instability (MI) at the pump wavelength without any occurrence of Raman scattering [Fig. 2(d)]. In this configuration, it was not possible to achieve a Kerr soliton state due to the lack of external locking mechanisms. This corresponds to the usual observation of OPO and frequency comb state when pumped in the anomalous dispersion regime.

For the second regime where $P_{OPO,P}^{th} \approx P_{Raman}^{th}$, both the pump comb and Raman comb are generated concurrently, as shown in Fig. 2(e). As the coupled pump power increases, OPO at the pump wavelength is induced and followed by MI-comb generation. Simultaneously, stimulated Raman-scattered photons emerge near 1670 nm, around the maximum Raman gain. By increasing coupled pump power further, a Raman frequency comb with a sech² envelope profile, which is a representative characteristic of soliton, is generated. However, as shown with a gray line in Fig. 2(h), the beat note of the Raman comb has a linewidth of 320 kHz, which is noticeably broader than the linewidth of a mode-locked soliton. This is an indicator that the comb lines constituting this Raman comb may not be phase-locked to the same extent as a typical Kerr soliton. Note that the generation of this sech² envelope Raman comb does not require any external locking mechanisms, in contrast to the first coupling regime. This feature is attributed to the fact that the optical power present in the pump modes has a potential to counterbalance the thermal instability arising within the Raman comb modes [19–24].

When the coupled pump power is increased further after the Raman comb with a sech² envelope appears, the Raman single-soliton is spontaneously but deterministically generated under specific pump detuning [Fig. 3(a)]. The generated Raman single-soliton can be self-locked and stably maintained over more than two hours without any external feedback systems, and its generation can be confirmed by the narrow linewidth RF beat note (25 Hz), shown as a black line in Fig. 2(h), with a magnified inset. Note that a broad linewidth RF beat note co-exists with the RF beat note of the single-soliton, indicating that a single-soliton state co-exists with the sech² envelope Raman comb described above. Despite their optical spectra being indistinguishable, we can confirm that the two combs originate from distinct mode families from dispersion measurements.

 

Fig. 3. (a) Transmission spectra (blue line) and optical power of frequency comb at pump wavelength (yellow line) and Raman wavelength (red line) were measured by varying the pump frequency detuning when the Raman single-soliton was generated. Gray area represents the specific detuning where the Raman single-soliton is generated. (b) The RF beat spectrum of the single-soliton state was observed to be stably maintained for over more than two hours when pumped with an unlocked free-running laser. Here, ${{\boldsymbol \nu }_{\boldsymbol s}}$ denotes the repetition rate of the single-soliton.

Download Full Size | PDF

For the last regime where $P_{Raman}^{th} < P_{OPO,P}^{th}$, the Raman comb is generated before the pump-induced OPO is observed. The generated Raman comb exhibits a sech² envelope, while its RF beat note exhibits broad linewidth (320kHz), similar to the previous coupling regime. However, contrary to the previous case, we could not achieve the single-soliton state via pump frequency detuning, which implies that the previous single-soliton state is achieved through the interplay between the Raman and OPO process. Additionally, a frequency comb with an atypical envelope is generated at the pump wavelength alongside the Raman comb [Fig. 2(f)], which has a similar broad linewidth (320 kHz) as the Raman comb [Fig. 2(i)].

4. Anti-Stokes Raman comb with atypical envelope

Figure 4(a) provides a schematic illustration of the generation process for the atypical envelope of the pump comb observed in Fig. 2(f) through the anti-Stokes process. When the Raman comb with a sech² envelope builds up enough power to induce an anti-Stokes process, the Raman comb is transferred back to the pump wavelength region. In this case, the Raman frequency comb and its anti-Stokes comb have the same repetition rate, which is determined by the FSR of the original Raman comb modes at ${\lambda _R}$. However, due to the second-order dispersion of the microresonator, the FSR of the pump comb modes at ${\lambda _P}$ is different from the repetition rate of the anti-Stokes comb, resulting in a frequency mismatch ($\mathrm{\Delta }FSR$ ∼ 6 MHz estimated in this experiment). Note that the actual frequency difference between the comb line and corresponding resonant frequency, where the mode number deviates from the pump wavelength by m, will increase by $m \times \mathrm{\Delta }FSR$. As a result, the optical intensity of each comb line in the anti-Stokes comb diminishes in comparison to that of the original Raman comb. The extent of this reduction is determined by the line-shape (Lorentzian) and linewidth (7.1 MHz calculated from total Q-factor) of the pump mode.

 

Fig. 4. (a) Schematic illustration of atypical envelope frequency comb generation via anti-Stokes process from Raman comb. (b) Fit results of measured anti-Stokes comb envelope with Lorentzian ${\times} $ sech² (left) and Raman comb with sech² (right) function.

Download Full Size | PDF

Consequently, the sech² envelope of the Raman comb is modified by a Lorentzian ${\times} $ sech² function $(A \times sec{h^2}({x/B} )\times ({1/({{x^2} + {{({C/2} )}^2}} )+ D} )$, $A = 6.5 \times {10^{ - 3}},\; \; B = 190,\; \; C = 15,\; \; D = 2.3 \times {10^{ - 5}}$, when x has units of GHz, in linear-scale), which shows the good agreement between the measured and calculated envelopes, as depicted in Fig. 4(b). Additionally, the RF beat note at the pump wavelength observed under the second coupling regime in Fig. 2(h) also exhibits a distinct RF peak near 11.098 GHz within the broad RF beat note of MI-comb. This peak indicates the presence of an anti-Stokes Raman comb within this regime as well, albeit nearly obscured by the pump comb.

5. Conclusion

In summary, we have successfully generated a self-locked Raman single-soliton and characterized the different spectral dynamics of frequency combs by adjusting the relative threshold powers between the pump-induced OPO and stimulated Raman lasing. It was possible to generate a sech² envelope Raman comb with a broad RF linewidth (320kHz) via Raman-scattered photons, while under specific conditions, concurrently spontaneous and deterministic generation of a Raman single-soliton was achieved. The self-locked single-soliton can be generated and maintained without external locking mechanisms, which was confirmed by measuring the 25 Hz linewidth RF beat note for more than two hours. Note that the coupling mechanism needed for Raman soliton generation can also be fulfilled via monolithically integrated waveguide instead of tapered-fiber we used. In addition, in contrast to Brillouin solitons which require the suitable FSR for SBS generation in microcavities, Raman solitons do not have such constraints giving the potential to generate various repetition rate solitons. Lastly, frequency combs with unconventional envelopes generated through the anti-Stokes process from the Raman comb is observed.

To the best of our knowledge, this study represents the first experimental confirmation of successfully generating a self-locked Raman single-soliton, and it contributes to the classification of Raman soliton generating regimes. Further investigations are still needed to achieve a comprehensive understanding of the precise mechanisms underlying the spontaneous generation of single-soliton states and the governing mechanism of self-locking, including numerical calculations based on the LLE equation considering the Raman response and multiple mode interactions. Precise mode analysis for identifying and controlling the mode family used in Raman comb and soliton generation along with further reduction of the complexity of the system by suppressing non-phase-locked components of the Raman comb is also required for the next stage of advancements in this research.