Since the first demonstration in the 1960s [1,2], laser diodes have been the subject of active research, thanks to which many areas of modern science and technology have been developed, including high-speed telecommunications, spectroscopy, microwave photonics, sensing, metrology, etc. Nevertheless, due to the rich nonlinear dynamics and complex behavior within a gain medium, laser diodes still attract considerable attention from the scientific community. A vivid example of such a nonlinear phenomenon is the four-wave mixing process in a laser diode gain medium, stimulated by external emission at a frequency shifted from the operational one [3].

Some particular cases of this phenomenon, arising due to the injection of an external signal into the laser gain medium from another laser, are known as “period-one” oscillations [4]. This effect has been extensively investigated since its first theoretical description was presented in Ref. [5]. These “period-one” oscillations turned out to be useful in diverse applications. For instance, they are used for the generation of microwave signals [6–9], with frequencies reaching up to 60 GHz [10–13]. Additionally, these oscillations have been associated with the generation of terahertz (THz) signals [14]. Moreover, this effect plays a key role in photonic microwave amplification [15], enabling the transmission of radio signals through optical fibers using radio-over-fiber (RoF) technology [16–18]. Furthermore, “period-one” oscillations have contributed to the development of LIDAR and speed measurement systems [19]. Different injection locking regimes have been investigated, focusing on “period-one” oscillations [20,21] and the resulting microwave signal linewidth narrowing [22–24].

At the same time, it is of great interest to investigate configurations that allow for the exclusion of the second laser source. Recent research has investigated setups that involve external injection into a single laser diode through passive optical feedback systems, such as mirrors [25] and Bragg gratings [26]. In this paper, we experimentally and numerically investigate the phenomenon of four-wave mixing in a laser diode gain medium or laser four-wave mixing (LFWM) induced by the Rayleigh-backscattering feedback [27] from a high-quality-factor (high-Q) integrated microring resonator. To the best of our knowledge such a phenomenon was not previously investigated in the proposed configuration.

Generally, interaction between a laser diode without an optical isolator and a high-Q microring resonator, under certain conditions, can lead to a self-injection locking regime [28]. Self-injection locking (SIL) is a technique that involves locking the frequency of a diode laser to an eigenfrequency of a high-Q microresonator. This approach relies on Rayleigh backscattering from inhomogeneities in the microresonator’s volume and surface, which reflects a portion of the incident light towards the laser diode. The backscattered light provides fast optical feedback that locks and stabilizes the laser diode frequency. The SIL approach is a straightforward and highly efficient laser diodes stabilization technique that provides significant spectral characteristics improvement [28] of both single-frequency [29] and multi-frequency sources [30,31], across a wide range of wavelengths, from ultraviolet to mid-infrared [32–36]. Extensive investigations of the SIL effect in the “laser diode-microresonator” system [37–41] made it possible to demonstrate on-chip sub-Hz linewidth lasers [42–45] and achieved impressive results in the mutual stabilization of multiple sources [46–48]. Furthermore, SIL phenomena have been efficiently utilized for the formation of both bright [49–55] and dark solitons [45,56], as well as in the development of ultra-low-noise photonic microwave oscillators [57] and frequency-modulated continuous wave LIDARs [58].

It has been demonstrated that the SIL effect strongly depends on the phase ψ0 of the wave backscattered from the microresonator (locking phase), defined by the distance between the laser and the microresonator [28].

As shown in Ref. [39], under non-optimal locking phase values, the frequency locking range of the laser can split into two regions. However, due to the strong nonlinearity in the gain medium of the laser source, at certain phase values, the presence of a backward wave can lead to nonlinear interaction between the wave at the laser operational frequency and the backward wave frequency within the laser active medium, resulting in four-wave mixing. As a result of this process, the laser can experience transition from a single-frequency regime to a multi-frequency one.

Such self-oscillation process in the considered system is shown schematically in Fig. 1. This process occurs before the transition of the laser diode from the free-running to the self-injection locking regime.

In this study, we reveal the possibility of observing LFWM in a controllable manner through diode current tuning for a certain level of detuning between the laser diode frequency and the microresonator eigenfrequency prior to the SIL regime. Having investigated the phenomenon numerically, we defined the boundaries for LFWM and SIL in terms of the “laser diode-microresonator” detuning. It was also demonstrated that LFWM strongly depends on the phase of the wave backscattered from the microresonator. The obtained experimental results are in good agreement with the numerical simulations and fully confirm the proposed model. Besides the controllable excitation of LFWM oscillations, the experimentally observed resulting LFWM-induced microwave signal could be continuously tuned in frequency by adjusting the “laser-microresonator” detuning, defined by the diode current, and demonstrated high stability, featuring a 3-kHz linewidth at 139-ms measurement time. Our findings uncover exciting nonlinear dynamics of the “laser diode-microresonator” systems and provide the opportunity to develop a wide range of chip-scale radiophotonic devices. Moreover, such a phenomenon should be taken into account in the development of SIL-based devices.

A schematic configuration of the system under study is shown in Fig. 1(a). The setup consists of a single-mode semiconductor laser, which experiences optical injection from the light backreflected from a high-Q optical microresonator. This system can be described by a set of differential equations, including those for the complex amplitude of the laser field A and the charge carrier density above the transparency threshold of the laser N [59,60], accompanied by the equations for forward and backward waves’ complex amplitudes A+ and A− [27,61].

We introduce additional normalization N˜=NNth, A˜=ASsat, where Nth is the carrier density at the lasing threshold, and Ssat is the saturating photon concentration. The system is written in slow time notation, where τ=t·κm/2 is the normalized time. The equations under study can be written in the following form, where we will use normalized variables everywhere else, but omit the primes (see Appendix A for details): $$\begin{gathered} dA˜dτ=γ˜p2(N˜−1)(1−iαH)A˜+i(ξ0+νωτ)A˜−A˜ext, \\ dN˜dτ=−γ˜e(−μ+N˜+N˜|A˜|2), \\ dA˜+dτ=−A˜++iβA˜−−2ηToTmeiωmτs/2A˜, \\ dA˜−dτ=−A˜−+iβA˜+, \end{gathered}$$(1)where A˜ext=2κdoκmTmT0eiωmτs/2A˜−. In this system, τp is the photon lifetime inside the laser resonator (γp˜=2kmτp), τe is the relaxation time of carriers (γe˜=2kmτe), κm is the microresonator decay rate, and ξ0=2(ωm−ωd0)/κm represents the initial normalized detuning between free-running laser frequency ωd0=ωd(τ=0) and the microresonator eigenfrequency ωm (laser-microresonator detuning). The normalized laser frequency sweep rate is νω (here and below we use normalization to κm/2), αH is the Henry factor, κdo is the laser output coupling rate, and Tm=ηκmτm, where η is the coupling efficiency to the microresonator and τm is the microresonator round-trip time. T0 is the amplitude transmission coefficient of the laser output mirror, τs is the time of propagation of light from the laser to the resonator and back, ψ0=ωmτs is backscattered wave phase (locking phase), and β is the normalized dimensionless coupling rate between counter-propagating modes inside the microresonator (equal to the ratio of the linear mode splitting value and loaded microresonator linewidth). Here, we also introduce diode current J normalized to the threshold current μ=JJth.

The resulting system of Eq. (1) is used to simulate the processes arising in the coupled “laser diode-microresonator” system upon sweeping the laser diode frequency using the Runge-Kutta method of order 5(4) with maximum step size of 10−4. The values used for numerical analysis are typical for such systems: τp=1.87×10−12s, τe=1.67×10−10s, αH=1.5, β=2.91, Tm/To=0.04, μ=1.2, κdo/2π=11GHz. The parameters of the microresonator mode used for modeling are κm/2π=κc/2π+κo/2π=209MHz, and η=0.35. Locking phase ψ0=0.2π. κm, β, and η are consistent with the experimental values.

We conducted a comprehensive analysis of the considered processes over a wide range of parameters, such as the locking phase ψ0, the laser output coupling rate kdo, and the coupling rate between counter-propagating modes inside the microresonator β. The main focus was on the dynamics of the laser field as the frequency of the laser diode approaches the microresonator eigenfrequency. For the numerical data analysis, the spectrogram method was used [53]. For this purpose, the spectrum of the laser radiation was calculated for various detuning values ξ.

The presence of a backscattered wave is of crucial importance. Initially, the model does not account for the laser linewidth or noise-like input needed for four-wave mixing initiation. However, as the system approaches the microresonator eigenfrequency, a noticeable backscattered wave emerges and changes the laser carrier density. This leads to the modulation instability with complex dynamics of the transient process, causing the appearance of additional laser frequency components. It is important to note that two types of oscillations are detected as a result of simulations. The first one is with a dramatic change in the intensity of laser radiation due to a change in the operating regime of the laser from free-running to locked, namely, relaxation oscillations, while the second one is a result of the four-wave mixing process in the laser diode gain medium. We carefully checked the stability of the regimes stopping the frequency scan (νω=0) at the point where the regimes occur. We found that the first regime does not correspond to the steady-state solution, and the relaxation oscillations rapidly disappear once the laser frequency scan is stopped. On the other hand, the second one, which is the result of the four-wave mixing process, remains stable, when the frequency tuning is stopped. Figure 2 shows a spectrogram of the laser emission for one of the regimes corresponding to LFWM oscillations. The laser diode operation frequency ξ is tuned from 70 to −20, with the laser frequency sweep rate νω=π/800. On the abscissa axis, there is the detuning ξ=ξ0+νωτ of the hot resonance in the laser cavity ωd from the microresonator eigenfrequency ωm. Along the ordinate axis, we have the generated frequency detuning from the same resonance in the microresonator ζ=2(ω−ωm)/κm. The generation frequency can be obtained as ω=ωm+ddτargA˜. The spectra of laser emission at different points of the laser frequency detuning ξ are shown in Figs. 2(b)–2(e). Figure 2(b) corresponds to the free-running laser far from the SIL regime. As it approaches locking, oscillations begin to occur [see Figs. 2(c) and 2(d)]. In the radiation spectrum, additional equidistant spectral components with frequency spacing Δν appear around the initial laser frequency with intervals in the radio frequency range. Depending on the detuning ξ, the oscillation frequency changes. Here, we can see that the oscillation frequency decreases with the detuning, but the behavior of the oscillation frequencies depends on the locking phase under consideration (for details see Fig. 3). Then, the laser is locked to the eigenfrequency of the microresonator [Fig. 2(e)].

LFWM oscillations are found to be significantly influenced by the locking phase ψ0 (Fig. 3) and are observed for phases that are tuned from non-optimal values, in which there is a slight asymmetry of the tuning curve ζ(ξ) [27]. For the fixed κdo/κm=24.39 we found the existence of microwave oscillation in the range of locking phase ω0τs∈(−0.0114π,0.046π),(0.086π,0.326π) (Fig. 3). Additionally, with an increase in κdo the frequency range at which LFWM oscillations can be observed expands. It follows from this that with increasing κdo, we observe oscillations at a larger set of initial phase values. In Fig. 3(b) the locking phase value corresponds to the initiation of LFWM. The involved frequency components are the laser frequency (ζ≈10) and stable branch of the transmission curve (ζ≈−5). In Figs. 3(c), 3(d), and 3(f) LFWM takes place involving both stable branches of the transmission curve. These reveal the origin of the discussed oscillations: one might propose that four-wave mixing in the laser diode gain medium arises from two initial signals with different frequencies. The spectrum of an unstabilized laser outside the locking regime is sufficiently broad. Thus, by tuning the laser frequency near the microresonator eigenfrequency at a specific locking phase value—thereby providing a smooth transition to the locking regime—one can shift to a regime in which a part of the laser emission, closest in frequency to the microresonator mode, is reflected back into the laser gain medium and amplified there [see Fig. 1(b)]. Consequently, there are two main frequencies in laser field spectrum: one frequency corresponds to the stable branch of SIL, and the second is the laser frequency itself. By adjusting parameters to achieve stronger LFWM, we drive more and more power into the resonator, and the backscattering wave turns out to be more and more powerful. This backscattering wave changes the refractive index of the laser diode, which, according to the Kramers-Kronig relation, changes the frequency of the laser. It is important that in describing the LFWM effect we do not introduce nonlinearity in the microresonator. This does not reduce the generality of the consideration of the effect since only a minor part of the laser output excites the microresonator mode. Thus, it turns out that the effect is linear from the microresonator point of view even for high pump powers.

Additional frequency oscillations inside the SIL region were observed and presented in Figs. 3(g) and 3(h) for ζ∈(22.5,32.5) and ζ∈(20,32.5), correspondingly. We related such type of oscillations with the microring resonator mode splitting due to nonzero coupling rate β. Due to the splitting of the resonator mode, energy exchange occurs between frequencies, leading to frequency modulation. This, in turn, results in nonlinear interaction within the laser’s gain medium and the generation of sidebands.

The experimental investigation of the described LFWM phenomena is conducted using the setup presented in Fig. 4(a). The laser diode and lensed fiber are butt-coupled to the photonic chip containing silicon nitride high-Q microring resonators [Fig. 4(b)]. The collected output signal is divided and sent to an oscilloscope (OSC) through a 400-kHz bandwidth photodetector (PD), to an electrical spectrum analyzer (ESA) through a fast 45-GHz bandwidth photodetector (FPD), and to an optical spectrum analyzer (OSA). The proposed experimental setup allows for a detailed investigation of the system’s dynamics by simultaneously monitoring all necessary aspects of the output signal.

The photonic chip contains structures consisting of a bus waveguide and a 25-μm-radius microring resonator. All structures have been initially characterized using an optically isolated narrow-linewidth laser Toptica CTL-1550. The microresonators feature ∼998GHz free spectral range (FSR) and anomalous group velocity dispersion. The parameters of the mode pumped by the laser diode are determined by fitting with a Lorentzian function, taking backscattering into account [61]: the intrinsic loss coefficient κ0/2π=135MHz, the coupling loss coefficient κc/2π=74MHz, and the backward-wave coupling rate (mode splitting) γ/2π=304MHz [Fig. 4(c)]. As a pump source, we use a laser diode with distributed feedback (DFB) featuring single-frequency emission at ∼1564nW and ∼100mW maximal output power.

In the experiment, we slowly scan the laser diode’s frequency across the vicinity of the microresonator’s eigenfrequency by monotonously tuning the diode current. During the scan, we synchronously record the following parameters of the output signal: power (LI-curve) with an OSC, microwave spectrum with an ESA, and optical spectrum with an OSA [Figs. 4(d)–4(f)]. The scanning speed in the experiment is ∼1GHz/s. At the beginning of the scan, the laser diode emits in a free-running regime, as its frequency is far from the microresonator eigenfrequency, indicating no backscattering from the microresonator (“Free Run” area). As the diode’s frequency approaches the resonance, weak backscattering from the microresonator appears. The intensity of this backscattered light is too weak to impose microresonator eigenfrequency and lock the laser diode frequency, as it happens in the SIL effect. However, it is sufficient to excite four-wave mixing oscillations in the gain medium (“LFWM” area). The transition to “LFWM state” is accompanied by a slight change of the LI-curve slope [Fig. 4(d)], the emergence of sidebands on the optical spectrum [Fig. 4(f)], and the corresponding beatnote signal [Fig. 4(e)]. By further changing the diode’s current, we decrease the frequency interval between the intrinsic laser diode frequency and the microresonator mode and increase the intensity of the feedback from the microresonator. Due to this fact, the intensity of the sidebands becomes higher (LFWM oscillations become more intense), and the spacing between them decreases (beatnote signal frequency decreases). At ∼10 and ∼12s in the scan, new optical sidebands appear in the center of the intervals between existing ones (new components of microwave beatnote signal), as had been predicted in numerical simulation for several phases of the backscattered wave; see Fig. 3(g) for ψ0=0.246π. Then, the laser diode transits to a self-injection locked state (“SIL” area) and its frequency locked with the microresonator mode, causing the optical sidebands and microwave beatnote components to collapse. Finally, after passing through the locking range, the laser diode returns to a free-running state (“Free Run” area).

During the experimental investigation, it was revealed that the possibility of observing LFWM oscillations and their stability strongly depends on the phase of the wave backscattered from the microresonator (locking phase). This phase is determined by the distance between the laser diode and the photonic chip facets, controlled by a piezo with ∼20-nm accuracy. By adjusting the locking phase, it is possible to switch between different “LFWM states” or even suppress this regime. This behavior is in complete agreement with data from numerical simulations (Fig. 3). Nevertheless, with the fixed backscattered phase, it is possible to achieve the “LFWM state” in a controllable manner and adjust the resulting beatnote microwave signal within this state by tuning the diode’s current. The observed smooth tuning range of the microwave signal for the optimal locking phase is about 2–3 GHz [Fig. 4(e)].

Typical spectral characteristics of the output signal in the “LFWM state” are presented in Fig. 5. The optical spectrum consists of the intrinsic laser diode line and a set of sidebands resulting from the LFWM oscillations [Fig. 5(a)]. The resulting microwave spectrum of the output signal is a set of equidistant narrow lines, which are the beatnotes between pairs of optical lines [Fig. 5(b)]. To characterize the beatnote signal between LFWM lines and estimate its stability, it is measured with high resolution and fitted with a Lorentzian profile [Fig. 5(c)]. It demonstrates a linewidth of the beatnote signal of 3 kHz over a measurement time of 139 ms at a resolution bandwidth of 1 kHz, which indicates a high signal stability, comparable to the beatnote signal of Kerr frequency microcombs [53,54].

As an additional validation of the proposed model and verification of the obtained simulation results, we repeat the experiment described above using different laser diodes and microresonators. In particular, we test photonic chips with 1-THz, 150-GHz, and 15-GHz FSR microresonators and alternately pump them with DFB diodes from three different manufacturers and multi-frequency Fabry–Perot diodes. For all setup configurations, we observed similar behavior patterns, implying the presence of LFWM oscillations prior to SIL states for a certain range of the backscattered wave phase.

The obtained experimental results are in good agreement with the numerical simulations and convincingly confirm them both qualitatively and quantitatively. Nevertheless, it should be mentioned that some parameters, such as the Henry factor of the laser diode and the absolute phase of the backscattered wave, are challenging to accurately determine in experiments, so there is a degree of freedom for numerical simulations.

The presented approach demonstrates linewidth and phase noise similar to those of a self-injection locked laser coupled to a microring resonator with specific parameters [46]. Over longer measurement times, LFWM appears more stable than SIL, as the beating optical components share the same origin. Compared to the period-one approach, LFWM demonstrates a narrower or comparable linewidth [62], whereas the “period-one” technique requires a master laser with low phase noise and certain fiber components. While dissipative Kerr solitons surpass LFWM in terms of stability [45], they demand a more complex and resource-intensive setup. LFWM, by contrast, does not require pump power exceeding the nonlinearity threshold or a specific dispersion law. A microresonator with a moderate Q-factor of around one million is sufficient to produce the microwave signal presented in our research. An important aspect of LFWM as a microwave generator is its capability for frequency tuning. In our experiment, we achieved a continuous frequency tuning range of up to 3 GHz, accomplished by adjusting the laser frequency. This sets LFWM apart from microwave signals generated via dissipative Kerr solitons. Another advantage of LFWM is that it is an entirely photonic setup with modest component requirements. Compared to opto-electronic oscillators (OEOs), which have significantly lower phase noise [63,64], our approach provides a simpler, more straightforward, and more versatile setup. Notably, SIL has been suggested as a promising way to improve OEO phase noise [65]. In Ref. [66], a self-modulated laser modulation approach, coupled with a spiral waveguide, uses laser diode current modulation, leading to additional frequency components. LFWM is a self-oscillating process that can be enhanced within a comparable scheme. Thus, in the context of SIL, LFWM could be applied to OEOs to improve their performance.

In summary, we have investigated theoretically and experimentally the effect of four-wave mixing in the laser diode gain medium, induced by the feedback from a high-Q microring resonator. We have developed a theoretical model describing the nonlinear interaction within the laser diode gain medium between two waves: intrinsic laser diode emission and the wave backscattered from the microresonator. We have demonstrated that this interaction leads to non-trivial dynamics accompanied by the four-wave mixing phenomenon. It is demonstrated that LFWM-induced microwave generation strongly depends on the locking phase and can be observed prior to the transition from a free-running regime to self-injection locking. We have defined the boundaries for LFWM and SIL in terms of “laser-microresonator” detuning and locking phase. The obtained numerical simulation results have been verified experimentally by conducting a series of experiments with different laser diodes and integrated microring resonators.

Also, it is experimentally demonstrated that the LFWM state of the laser diode can be reached in a controllable manner by smoothly tuning the diode current, and it provides the generation of a stable, narrow-linewidth, continuously tunable signal in the microwave range. The resulting microwave signal, observed for a 1-THz-FSR microring resonator with a Q-factor ∼0.9×106, pumped with a DFB laser diode, features 3kHz Lorentzian linewidth at 139-ms measurement time and a 3-GHz tunability range. Note that the spectral parameters of the resulting microwave signal, in addition to the phase of the backscattered wave, are strongly dependent on the microresonator’s parameters, such as the Q-factor and coupling rate. Optimization of these parameters may lead to further improvements in its stability and spectral purity.

The obtained results uncover the exciting nonlinear dynamics of the nonlinear interaction in the coupled system consisting of a laser diode and a high-Q microring resonator and describe the LFWM regime, which has not yet been investigated. Additionally, they may be used as a basis for further research aimed at determining the optimal parameters for LFWM-induced microwave generation. Our findings present an opportunity for GHz frequency synthesis using an all-optical setup and potentially at on-chip design. One of the key ingredients is the possibility of effective frequency tuning of the generated microwave signal. That may find applications as a microwave photonic generator, at precision spectroscopy, and for radio-over-fiber and other applications. Another important aspect of the study is the fact that the transition of the laser to a multi-frequency operation regime due to LFWM can significantly affect the dynamics of the entire system. Since many nanophotonic devices are currently being developed using self-injection locking phenomena [28], it is crucial to consider the possibility of the discussed nonlinear processes in such systems.

We believe that our findings significantly contribute to the understanding of the rich dynamics of the “laser diode-microresonator” system and pave the way for compact on-chip microwave photonic devices with outstanding performance in telecommunications, metrology, and sensing.

Acknowledgment. The authors express their gratitude to the State Corporation Rosatom for support within the framework of the Quantum Computing Roadmap (No. 868-1.3-15/15-2021) of the research presented in Section 2.

Author Contribution.N.Y.D., S.N.K., and A.E.S. conducted the experiment. D.M.S., D.A.C., and V.E.L. developed a theoretical model and performed numerical simulations. All authors analyzed the data. D.A.C., V.E.L., N.Y.D, D.M.S., and A.E.S. prepared the manuscript. A.E.S. conceived the project. I.A.B., D.A.C., and A.E.S. supervised the project. All authors reviewed and accepted the manuscript.