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Abstract
Microresonator-based optical frequency combs have been greatly developed in the last decade and have shown great potential for many applications. A dual-comb scheme is usually required for lidar ranging, spectroscopy, spectrometer and microwave photonic channelizer. However, dual-comb generation with microresonators would require doubled hardware resources and more complex feedback control. Here we propose a novel scheme for dual-comb generation with a single laser diode self-injection locked to a single microresonator. The output of the laser diode is split and pumps the microresonator in clockwise and counter-clockwise directions. The scheme is investigated intensely through numerical simulations based on a set of coupled Lugiato-Lefever equations. Turnkey counter-propagating single soliton generation and repetition rate tuning are demonstrated.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Microresonator-based optical frequency combs (microcombs) have attracted wide attention in recent years for their advantages of compact volume, low power consumption and chip-level integration [1–3]. Many applications have been demonstrated with microcombs, such as optical atomic clock [4], optical frequency synthesizer [5], spectroscopy [6,7], microwave generation [8], radio-frequency filters [9,10], optical data transmission [11,12], astrocomb [13], lidar ranging [14,15], and spectrometer [16], etc. For spectroscopy, lidar ranging, and spectrometer, a well-known dual-comb configuration is employed which combines two frequency combs with slightly different repetition rates to retrieve the amplitude and phase of each comb mode. Directly using two individual comb generation devices is a viable solution, but at the cost of doubled system volume. Dual-comb generation with a single microresonator can not only reduce the complexity but also improve the mutual coherence between the frequency combs. Several methods have been reported previously for dual-comb or multi-comb generation with a single microresonator. These include bidirectional pumping with different detunings or powers [17,18], pumping two different polarization modes [19], as well as pumping different spatial modes [20].
In experiments, the generation of optical frequency combs is generally accomplished by sweeping the laser frequency across the resonance from the blue detuned side to the red detuned side. The sweeping speed and stop point need to be carefully controlled to overcome the thermal instability and stabilize the soliton state [21–23]. Recently, a scheme of self-injection-locked soliton generation has been reported which greatly simplifies the strategy of microcomb generation. An ordinary distributed feedback (DFB) laser or Fabry–Pérot laser diode is self-injection locked to a high-Q microresonator to simultaneously achieve linewidth narrowing and turn-key comb generation [24–27]. Self-injection-locked soliton microcomb generation has been demonstrated with various microresonator platforms. Recently, an integrated dual-microcomb source that is electrically driven and allows turnkey operation has been demonstrated [28]. However, this scheme requires doubled hardware complexity including two laser diodes and two microresonators. Furthermore, dual-comb generation with a single laser diode self-injection locked to a single microresonator has not been reported, to the best of our knowledge. Exploring such a possibility would further reduce the size, power consumption and cost of a dual-comb source for a wide variety of applications.
In this work, we propose a novel scheme of dual-comb generation based on self-injection-locked counter-propagating solitons. A single microresonator is pumped by a DFB laser diode in the clockwise (CW) and counter-clockwise (CCW) directions. The optical field from the microresonator is feedback injected to the pump laser through a symmetric through-drop port design. We perform detailed numerical simulations of this structure based on a set of coupled Lugiato-Lefever equations. Turnkey counter-propagating soliton generation is demonstrated in a nearly symmetric coupling condition. Once generated, the repetition rates of counter-propagating solitons can then be tuned by varying their pump power levels. Our work provides a promising path to on-chip integration of compact dual-comb sources.
2. Theoretical model of self-injection locked counter-propagating solitons
The scheme for self-injection locked dual-comb generation is illustrated in Fig. 1. The output of a DFB laser diode is split into two branches through a 1x2 Mach-Zehnder interferometer (MZI), and pumps a microresonator in the CW and CCW directions. With a symmetric through-drop port design, the optical field from the microresonator is naturally reflected back to the laser to ensure self-injection locking. The output intensities of the MZI, i.e., the transmission from the laser to the CW and CCW pumping directions, can be adjusted by changing the relative phase of the MZI arms. As will be seen in the next, the ability of manipulating the pump splitting ratio is essential to enable bidirectional turnkey soliton generation and repetition rate tuning.
Fig. 1. The scheme for self-injection-locked dual-comb generation. The arrows indicating the CW and CCW fields are marked in red and blue, respectively. The complex transmission coefficients from the laser diode to the microresonator coupling points are represented by $\sqrt {\alpha _{1}}e^{i\varphi _{1}}$ and $\sqrt {\alpha _{2}}e^{i\varphi _{2}}$, respectively.
The evolution of the intracavity fields can be characterized by the following Lugiato–Lefever equations [17]:
Here, $E_a$ is the intracavity CW field, $E_b$ is the CCW field and $E_L$ is the field in the laser diode; the square of the field amplitude $|E|^2$ represents the optical power, and $\overline {|E|^2}$ represents the average power over the microcavity (i.e., $\overline {|E|^2}=\int |E|^2d\theta /2\pi$); $\theta$ is the microcavity angular coordinate; $t$ is the evolution time; $\kappa$ is the total loss rate of the cavity including the intrinsic loss $\kappa _0$ and the external coupling loss $\kappa _{ex1,2}$; $\delta \omega$ is the detuning of the cold-cavity resonance relative to the injection-locked laser; $D_1$ is related to the free spectral range (FSR) and $D_2$ is the second-order dispersion coefficient according to the Taylor expansion of the resonant modes around the pumped mode (i.e., $\omega _\mu = \omega _0 + D_1 \mu + D_2\mu ^2/2+\cdots$); $g$ is the Kerr nonlinear coefficient; $\kappa _L$ is the coupling rate of the laser cavity. The terms containing $f_R$ and $\tau _R$ on the right-hand side of the equations represent the Raman effect, where $f_R$ is the Raman fraction, $\tau _R$ is the Raman shock time [29,30].
The pump laser self-injection locking process is dominantly governed by the reflected pumped mode. Therefore, the pumped mode which can be represented by the average field $\overline {E}=\int {Ed\theta /2\pi }$, is assumed to be injected into the laser from the microresonator. The evolution of the laser field can be described by a differential equation as [24]
Different from the self-injection-locked microcomb generation scheme reported previously [24] where the reflected light is due to weak Rayleigh backscattering in the cavity, in our configuration, the strong reflected light comes directly from drop ports. For simplicity, the weak Rayleigh backscattering in the cavity is neglected in our current model.
The following normalized Eqs. (4) and (5) are obtained by substituting $E_a=\sqrt {\kappa /(2g)}A$, $E_b=\sqrt {\kappa /(2g)}B$ and $E_L=\sqrt {\kappa /(2g)}F_L$ into Eqs. (1) and (2), and dividing both sides by $\kappa /2$. Equation (3) can be normalized in a similar way. To yield Eq. (6), it is further assumed that the dynamics of laser injection locking is much faster compared to the field evolution in the microresonator. The laser power thus adiabatically tracks the feedback (i.e., $d{E_L}/dt=0$) [24].
Here, the normalized intensities of the pumped mode are represented by $\bar A = \int Ad\theta /2\pi$, $\bar B = \int Bd\theta /2\pi$; $t' = \kappa t/2$ is the normalized evolution time; $d_2 = D_2/\kappa$ is the normalized dispersion coefficient; $d_{1\tau } = \tau _R D_1$ is the normalized Raman coefficient; $\Delta \omega =2\delta \omega /\kappa$ is the normalized detuning of the cold-cavity resonance relative to the injection-locked laser; $\Delta \omega _L=(2\delta \omega _L+\alpha _g\gamma )/\kappa$ is the normalized detuning of the cold-cavity resonance relative to the free-running laser; $C_{1,2} =2\sqrt {\kappa _{ex1,2}\kappa _{L}}/\kappa$ are the normalized coupling strengths.
3. Numerical simulation results
3.1 Turnkey counter-propagating soliton generation
It has been reported that turnkey operation can be achieved for conventional unidirectional self-injection-locked microcomb generation [24]. To verify that turnkey generation can also be achieved for counter-propagating solitons, we perform some numerical simulations and the results are presented in Fig. 2. The simulation parameters are as follows: $d_2 = 0.02, d_{1\tau } = 0.0013, f_R = 0.2, C_1 = C_2 = 8.92, \varphi _1 = \varphi _2 = 0.16\pi, F_L = 0.32, \alpha _g = 0$. We first simulate the case with perfectly symmetric pump strengths in the CW and CCW directions, i.e., $\alpha _1=\alpha _2=0.5$. Figures 2(a) and 2(b) show the evolution of the intracavity power with time. One single soliton per direction is spontaneously generated from the continuous-wave background after the laser is turned on. Turnkey generation of counter-propagating self-injection-locked solitons has been evidenced. Drifts of the solitons in Figs. 2(a) and 2(b), are attributed to the Raman induced soliton self-frequency shift. Figure 2(i) illustrates the hysteresis curve of the intracavity pump power versus the detuning. The black line indicates the trajectory upon turning on the laser. The intracavity pump transitions to a new operating point at which solitons form. Both CW and CCW directions show identical behavior similar to that of unidirectional self-injection-locked soliton generation [24]. Next, we try the case with asymmetric pump strengths, by setting $\alpha _1=0.48$ and $\alpha _2=0.52$. The results of one simulation run are shown in Figs. 2(d)-(f), (h) and (j). One single soliton is generated in the CCW direction, while no soliton is generated in the CW direction. Figure 2(j) shows the intracavity pump power versus the detuning. The CW and CCW curves are no longer degenerate and look rather different compared to the case with symmetric pumping. Under this condition, the intracavity pump power is higher in the CCW direction so that a CCW soliton first emerges. The CCW intracavity power then decreases after the formation of the soliton pulse. As a consequence of cross phase modulation (XPM) effect, the power in the CW direction also drops simultaneously before soliton formation. No soliton is thus generated in the CW direction. These observations imply that the simultaneous formation of self-injection-locked counter-propagating solitons critically depends on the pumping conditions in the CW and CCW directions. To verify the repeatability of turnkey operation, a large number of simulations with different pump splitting ratios are performed and the statistical results are presented in Fig. 2(k). In the case of perfectly symmetric pumping, bidirectional single soliton generation can be achieved with a high reliability of $\sim 70{\% }$. The probability decreases with the increase of pump asymmetry, and is only $\sim 4{\% }$ when $\alpha _1=0.48$ and $\alpha _2=0.52$. In practical experimental implementations, the pumping conditions in the CW and CCW directions should be as symmetric as possible.
Fig. 2. Simulation results of turnkey counter-propagating soliton generation. (a),(b) Evolution of the intracavity field in the CW and CCW directions when the pump strengths are symmetric ($\alpha _1 = \alpha _2 = 0.5$). (c) Snapshots of the intracavity field at the end of simulation. One single soliton in each direction can be observed. (g) Average intracavity power as a function of evolution time. (i) Intracavity power with respect to the pump detuning. The black line indicates the transition trajectory during the simulation, and the red star indicates the operating point when the steady state is reached. The curves in the CW and CCW directions are identical and completely overlaid with each other. (d)-(f),(h),(j) Similar to panels (a)-(c),(g),(i), but with asymmetric pump strengths ($\alpha _1 = 0.48,\alpha _2 = 0.52$). One single soliton is generated in the CCW direction, while no soliton is generated in the CW direction. The average intracavity power in both directions is remarkably close during this process, leading to the nearly overlapping transition trajectory curves (i.e., black lines) in panel (j). (k) Histogram of the simulation results with different pump splitting ratios. Deep purple: one single soliton is generated in each direction; Light purple: solitons can be generated in both directions, but there are multiple solitons in one direction; Gray: no soliton is generated in one direction.
3.2 Dual-comb repetition rate tuning
When the bidirectional pump strengths are perfectly symmetric, the counter-propagating solitons will have identical repetition rates. However, for most dual-comb-based applications, the repetition rates of the two frequency combs should be slightly different. To tune the repetition rates of counter-propagating solitons, we can gradually increase the imbalance of the bidirectional pump power by tuning the MZI, after the solitons are generated. A series of simulations are performed to verify the repetition rate tuning process, with the same parameters used in the previous section. Figure 3 shows the results when the CW and CCW pump power is first set symmetric with $\alpha _1 = \alpha _2 = 0.5$ for turnkey soliton generation and then slowly tuned to $\alpha _1 = 0.77$, $\alpha _2 = 0.23$. The single soliton state in both directions can be maintained in this process. As the pump strength increases in the CW direction, the CW intracavity power and soliton peak power increases. The comb spectrum is slightly broadened. An opposite trend is observed for the CCW direction with decreasing pump strength. The 10-dB bandwidth of the dual frequency comb is calculated and plotted in Fig. 3(h). The results show that the comb bandwidth is positively correlated with the pumping strength. The ratio of the bandwidth in the CW direction before and after the tuning is 1.05, while in the CCW direction it is 0.86. In other words, the ratio of the bandwidth between the CW and CCW directions is 1.23 after gradually tuning the pump splitting ratio. Moreover, the soliton group velocities change with different rates in the CW and CCW directions, as can be deduced from the soliton drifting traces shown in Figs. 3(c) and 3(d). The different soliton group velocities correspond to different comb repetition rates. This repetition rate tuning ability is attributed to the Raman effect induced soliton self-frequency shift [17,29,30], as will be investigated in detail in the following paragraphs. The imbalance of the splitting ratio cannot be too large in the tuning process. Otherwise, the lower pump power may be too weak to maintain the soliton in its direction. In our simulations, we find that the soliton in the CCW direction will cease to exist when the pump splitting ratio is tuned to $\alpha _1 = 0.8,\alpha _2 = 0.2$.
Fig. 3. Simulation results of dual-comb repetition rate tuning. Turnkey counter-propagating soliton generation is first achieved with $\alpha _1 = \alpha _2 = 0.5$. From the normalized time of $3\times 10^4$ to $6\times 10^4$ (marked with white dash lines), the pump splitting ratio is gradually tuned to $\alpha _1 = 0.77,\alpha _2 = 0.23$. (a),(b) Evolution of the comb spectrum. (c),(d) Evolution of the intracavity field. (e),(f) Snapshots of spectra before and after tuning the pump splitting ratio. (g) Pump splitting ratios in the tuning process. (h) Evolution of the 10-dB comb bandwidth. (i),(j) Evolution of average intracavity power and soliton peak power respectively.
To gain further insights into the CW and CCW fields, we define modified detuning as the frequency difference between the pump laser and the XPM shifted resonance, given by
where $P_\text {CCW} = \overline {|B|^2}$ and $P_{\text {CW}} = \overline {|A|^2}$ are the average power in the corresponding directions. By substituting the modified detunings into Eqs. (4) and (5), the equations become a form similar to the unidirectionally driven Lugiato–Lefever equation [31].For a general model with unidirectional pump, the soliton peak power is mainly determined by the cold cavity detuning [31,32]. A larger detuning corresponds to a higher soliton peak power. As illustrated in Fig. 4, if the CW direction has a larger pump power than the CCW direction, the CW intracavity power will be higher, i.e., $P_{\text {CW}}>P_{\text {CCW}}$. Thus the modified detuning $\Delta \omega _{\text {mod,CW}}>\Delta \omega _{\text {mod,CCW}}$. As a consequence, the soliton peak power in the CW direction becomes larger than that in the CCW direction. Therefore, by tuning the MZI to change the pump splitting ratio, different modified detunings are introduced in the CW and CCW directions through the XPM effect, leading to different soliton peak power in the two directions. The evolution of the average intracavity power and soliton peak power shown in Figs. 3(i) and 3(j) is consistent with the above analysis. To further validate the conclusion, we run simulations with different pump splitting ratios and calculate the modified detunings. Figure 5(a) illustrates the relationship between the pump splitting ratio and the modified detuning. Figure 5(b) illustrates the relationship between the soliton peak power and the modified detuning. Due to symmetry, the curves are degenerated in the CW and CCW directions.
Fig. 4. Illustration showing the different resonance shifts and modified detunings induced by XPM in the CW and CCW directions.
Fig. 5. (a) The modified detuning $\Delta \omega _{\text {mod}}$ as a function of the pump splitting ratio. (b) Soliton peak power as a function of the modified detuning $\Delta \omega _{\text {mod}}$. (c) Differences of the modified detunings (left) and the repetition rates (right) as a function of the logarithmic pump power ratio.
It has been known that the soliton group velocity and repetition rate can be changed due to Raman induced soliton self-frequency shift [17]. Therefore, the different soliton peak power in the CW and CCW directions will cause different Raman self-frequency shifts and different soliton repetition rates. Theoretically, the normalized repetition rate difference is related to the detuning by [17]
The simulated $|\Delta \omega ^2_{\text {mod,CW}}-\Delta \omega ^2_{\text {mod,CCW}}|$ as a function of the logarithmic pump splitting ratio is plotted in Fig. 5(c), which shows a linear relationship. This implies that the repetition rate difference should also be linearly related to the logarithmic pump power ratio. The simulated plot of repetition rate difference is shown in Fig. 5(c), which is indeed linear. The tuning rate is $3.91\times 10^{-4}/\text {dB}$. In comparison, the tuning rate inferred from Eq. (9) is about $2.54\times 10^{-4}/\text {dB}$. The discrepancy is attributed to the approximations in obtaining Eq. (9).
The results above are obtained with normalized equations and parameters. To provide a more practical evaluation with real device parameters, we take silicon nitride microresonator as an example. The parameters are assumed to be $Q\approx 1\times 10^7, D_1 = 2\pi \times 10\text {GHz}, \tau _R = 20\text {fs}$. The estimated maximum repetition rate difference is up to 128.8kHz, comparable to the value reported before for a silica microresonator when the detunings in the CW and CCW directions are separately tuned without self-injection locking [17]. It should be noted that it is generally more challenging to generate microwave-rate frequency combs with silicon nitride microresonators. For microresonators with FSRs of 50 GHz and 100 GHz, the simulated maximum repetition rate differences are 557.6 kHz and 800.7 kHz, respectively.
4. Summary and discussion
In summary, we have proposed a novel scheme for dual-comb generation by self-injection locking. The output of a laser diode is split by a MZI and pumps a microresonator in the CW and CCW directions. Turnkey counter-propagating soliton generation can be achieved with symmetric pump strengths. The repetition rates of the dual soliton combs can then be tuned by changing the pump splitting ratio. The repetition rate difference is proportional to the logarithmic pump power ratio. The repetition rate tuning mechanism is explained by Raman induced soliton self-frequency shift. The maximum achievable repetition rate difference is comparable to that of previously reported scheme with bidirectional pumping [17].
In our scheme, the repetition rates of the dual combs are tuned by tuning the bidirectional pump strengths with an MZI structure. In principle, it may also be achieved by tuning the external coupling coefficients (i.e., $C_1$, $C_2$). But it is more challenging to dynamically tune the coupling condition in practice. When tuning the pump splitting ratio, the MZI transmission phase should ideally be kept constant because the self-injection locking state is sensitive to feedback phase [24]. Therefore, the MZI should be tuned in a push-and-pull mode. Our simulations also show that the self-injection-locked dual-comb generation protocol is not sensitive to the asymmetry of the MZI. Even when the splitting ratios of the 1x2 and 2x2 couplers inside the MZI are 47:53 and 54:46, bidirectional single soliton generation can also be secured. Such an accuracy can be well guaranteed by the state-of-the-art photonic integration technique.
Funding
National Key Research and Development Program of China (2018YFB2201702); National Natural Science Foundation of China (62127805); Fundamental Research Funds for the Central Universities (2242022k60006).
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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