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Abstract
The nonlinear dynamical behaviors of a semiconductor microcavity laser with frequency comb injection have been experimentally and numerically investigated. The microcavity laser is harmonically locked to a unit fraction of the comb spacing due to the undamped relaxation oscillation at certain conditions, creating additional comb lines with reduced frequency spacing. The stability maps indicating various locking states are obtained based on rate equations, which demonstrates that the locking regions are closely related to the relaxation oscillation. Moreover, the microcavity laser with comb injection leads to spectral broadening of the original comb and the number of comb lines raises from 3 to 13. Owing to the large modulation bandwidth of the microcavity laser, the comb lines and the frequency spacing can be tailored over a wide range by varying the injection parameters.
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1. Introduction
Nonlinear dynamics of semiconductor lasers induced by optical injection have received increasing attention in the past few decades owing to their potential applications in various fields. For example, optical injection-locked lasers were extensively studied for improving the dynamical modulation characteristics and the intrinsic laser noise [1–5]. The semiconductor laser with optical injection could bifurcate to complex dynamics, including periodic oscillations and chaotic states, due to the amplitude-phase coupling in the electric field of laser modes [6–9]. The rich nonlinear dynamics open the way for novel applications in radio-over-fiber communication [10,11], microwave signal generation [12,13], optical sensing [14], and random bit generation [15–18].
In recent years, semiconductor lasers subject to optical frequency comb (OFC) injection have been proposed to bring additional nonlinear dynamical phenomena. Optical injection of frequency combs was demonstrated to selectively amplify individual comb lines as an effective filter [19–21], and has been widely applied in optical communication [22–25], millimeter wave generation [26,27], and high-resolution spectroscopy [28]. The side-mode suppression ratio (SMSR) could be improved by optimizing the Q value or the comb spacing [29,30], and the noise behaviors including residual side modes were studied numerically [31,32] and experimentally [33]. The frequency comb injection exhibited a different injection locking solution compared to the case of single frequency laser injection, which can be applied to generate mode-locked OFCs. The desired properties of OFCs, such as comb bandwidth, comb spacing, and signal-to-noise ratio, would vary according to different applications. Therefore, the lasers with a comb injection have been experimentally and numerically studied to tailor the properties of the output combs by varying the injection parameters [34,35]. Harmonic locking of the injected laser was investigated to decrease the output comb spacing [36–38]. In addition, the generation of harmonics can be applied in narrow-linewidth microwave generation [39] and may be beneficial for the spectroscopy measurement requiring comb spacing below 1 GHz [40]. Furthermore, single-polarization or dual-polarization frequency combs were generated in the vertical-cavity surface-emitting laser through the comb injection, which can be used for polarization division multiplexing [41,42].
Whispering-gallery-mode microcavity lasers, with the merits of high-quality factor, small mode volume, and capability of planar integration, have attracted great attention as potential light sources for on-chip photonic integration. Microdisk lasers have been proven to exhibit various nonlinear dynamics under different injection conditions [43,44]. In [44], external single optical injection is applied to the microdisk laser to improve the modulation bandwidth from 8.7 to 38 GHz under injection locking state. Besides, the integrated coupled microcavity laser, such as twin-microdisk [45] and twin-microring lasers [46], enables the application of optical injection technology in monolithic integration. The coupled twin-microdisk and twin-microring lasers were investigated for presenting rich dynamical states, such as four-wave mixing, injection locking, and periodic oscillation states. At present, the nonlinear dynamics produced by injecting optical frequency combs into semiconductor microcavity lasers have not yet been studied. In this paper, we investigate the nonlinear dynamics of a square semiconductor microcavity laser with frequency comb injection. The microcavity laser can be harmonically locked to a unit fraction of injected comb line spacing at certain conditions. The optical and electrical spectra of the microcavity laser with comb injection are explored by varying the detuning frequency between the laser mode and comb lines. The harmonic locking occurs when the frequency difference between the microcavity laser and the adjacent comb line or the relaxation oscillation frequency is a unit fraction of the spacing of the injected comb. To further study the nonlinear dynamics of the comb injection, the stability maps indicating nonlinear dynamics of the microcavity laser are obtained based on rate equations, which illustrate the frequency spacing of the laser lines in different locked states. The simulation results reveal that the locking regions are related to the relaxation oscillation of the microcavity laser. Finally, the microcavity laser with the comb injection also performs spectral broadening. The number of comb teeth is more than 13 with the injected frequency comb spacing varying from 6 GHz to 14 GHz.
2. Lasing characteristics of a square microcavity laser
Microcavity lasers have large modulation bandwidths due to their small active region volume. Among them, the square microcavity has higher output power and better single-mode characteristics owing to uniform mode field distribution and carrier injection efficiency [47]. A square semiconductor microcavity laser with a side length of 16 µm and a 2µm-wide output waveguide directly connecting to one vertex of the square is used in the comb injection experiment. The square microcavity laser is fabricated using an AlGaInAs/InP multiple quantum well epitaxial wafer similar to that in [48]. In the following experiment, the device is fixed on a thermoelectric cooler maintaining a substrate temperature of 288 K. Figure 1(a) depicts the optical power collected by the integrated sphere and the applied voltage versus the injection current. The threshold current is 6 mA through the clear inflection point of the power curve and the output power is 0.6 mW at 30 mA. Figure 1(b) shows the lasing spectra of the microcavity laser measured at injection currents of 16 and 30 mA. The longitudinal mode interval is 15.6 nm. At the injection current of 30 mA, single-mode operation with an SMSR of 50.2 dB is realized. The small-signal modulation responses are measured through a 20GHz vector network analyzer combined with a 50GHz high-speed photodetector. The relaxation resonance frequencies of 8.0 GHz and 9.3 GHz, and the 3-dB modulation bandwidths of 12.1 GHz and 15.5 GHz are obtained for the microcavity laser at 16 mA and 30 mA, respectively. The injection current of the microcavity laser is set to 30 mA in the following experiment due to the larger modulation bandwidth.
Fig. 1. (a) The output power collected by an integrated sphere and applied voltage versus the injection current, and the inset is the optical microscope image of the microcavity laser, (b) lasing spectra under injection currents of 16 mA and 30 mA, and (c) small-signal modulation responses at 16 and 30 mA for a square microcavity laser with a side length of 16 µm.
3. Experimental setup
Figure 2 shows the experimental setup for investigating the nonlinear dynamics of the microcavity laser subject to frequency comb injection. A commercial tunable laser with a linewidth of 100 kHz is polarization aligned by a polarization controller (PC1) and modulated by a Mach-Zehnder modulator (MZM) to generate three flat comb lines. The modulation bandwidth and the half-wave voltage of the MZM are 25 GHz and 3.5 V, respectively. The average optical power after the MZM is amplified to 15 dBm by an erbium-doped fiber amplifier (EDFA1), and the noise introduced by the EDFA is removed by an optical band-pass filter (OBPF1). Then the light is injected into the microcavity laser through a fiber circulator. The injection power is adjusted by a tunable optical attenuator, and the polarization of the injected light is aligned with the microcavity laser mode by tuning PC2 to maximize the injection efficiency. The output of the microcavity laser is amplified by EDFA2, filtered out by an OBPF2, and divided into two parts through a 1:1 optical coupler. One part of the light is detected by a photodetector with a bandwidth of 40 GHz and measured by an electrical spectrum analyzer (ESA). The other part of the light is used to measure the optical spectra by an optical spectrum analyzer (OSA) with a resolution of 0.02 nm. The injection efficiency is estimated by measuring the photocurrent of the microcavity laser at 0 V bias voltage. The photocurrent is 304 µA with an injection power of 3 mW, and then the injection efficiency is about 0.20 by assuming a photoelectric responsivity of 0.5 A/W. In this paper, the injected optical power is defined as the optical power feeding into the circulator, and the detuning frequency is defined as the center frequency of the injected comb subtracting the free-running lasing frequency of the microcavity laser.
Fig. 2. Experimental setup for investigating nonlinear dynamics of the microcavity laser subject to frequency comb injection. TL, tunable laser; PC, polarization controller; SG, signal generator; MZM, Mach-Zehnder modulator; EDFA, erbium-doped fiber amplifier; OBPF, optical bandpass filter; VOA, variable optical attenuator; OC, optical coupler; PD, photodetector; OSA, optical spectrum analyzer; ESA, electrical spectrum analyzer.
4. Nonlinear dynamics under frequency comb injection
4.1 Harmonic frequency locking
Firstly, we study the harmonic locking effect caused by the comb injection into the microcavity laser. Figure 3 shows the output of the microcavity laser subject to the optical injection of a frequency comb with three 16GHz-spaced comb lines. When the detuning frequency is -5 GHz and the injection power is 0.9 mW, the original microcavity laser mode is injection locked by the center comb line, and additional lasing peaks are generated at the bisector points of the original adjacent comb lines, and the spacing of the peaks is reduced from 16 GHz to 8 GHz, as shown in Fig. 3(a). The reason is that when the relaxation oscillation frequency of the microcavity laser is close to one-half of the comb spacing, the relaxation oscillation becomes undamped and harmonic locking occurs. The electrical spectrum measured by a high-speed photodetector and an ESA is further used to verify the establishment of the harmonic locking. The microwave signal at 8 GHz is observed as shown in Fig. 3(b). The microwave signal is further characterized by the ESA with a resolution of 10 kHz, as shown in Fig. 3(c). The linewidth of the 8GHz signal is 36.3 kHz by fitting the peak with a Lorentzian function, which indicates the harmonic locking of the microcavity laser. In addition, Fig. 3(d) depicts the microwave signal at 16 GHz with a linewidth smaller than 510 Hz limited by the spectral resolution. It is noted that since the mode-locking mechanism is not introduced by the nonlinear phenomenon caused by the optical injection, the newly generated harmonics are not completely phase-dependent. Therefore, the optical beatnote of the 8 GHz signal generated from the harmonics has a much wider linewidth compared to that of the injected comb.
Fig. 3. (a) Optical spectrum, and (b) electrical spectrum of the microcavity laser with the injection of 16GHz-spaced comb. (c) Electrical spectra of the 8 GHz, and (d) 16 GHz microwave signals. RBW: resolution bandwidth; VBW: video bandwidth.
The output spectra of the microcavity laser with an 18GHz-spaced comb injection are measured at the different detuning frequencies. The detuning frequency between the center comb line and the microcavity laser mode is tuned from -36 GHz to 36 GHz. Figure 4(a) shows the optical spectra of the microcavity laser with a fixed injection power of 0.24 mW. When the detuning frequency is around -18 GHz and 0 GHz, the microcavity laser is injection locked to the adjacent comb line. The relaxation oscillation of the microcavity laser becomes undamped and is harmonically locked to one-half of the comb spacing. The harmonic locking is achieved owing to the intensity modulation induced by undamped relaxation oscillation. To further show the detail of the output light, the electrical spectra are recorded by the high-speed photodetector and ESA and shown in Fig. 4(b). A microwave signal of 18 GHz is observed during the whole tuning process, due to the beating frequency of the injected comb lines. The microwave signals at 9 GHz are generated when the detuning frequency closes to -18 GHz and 0 GHz consistent with the optical spectra. One can find that the harmonic locking range is large when the microcavity laser is locked to the middle comb line. In addition, the microwave signal at 9 GHz appears when the detuning frequency is -9 GHz and 9 GHz, corresponding to the bisecting point between two adjacent lines. This coincides with the 9 GHz comb spacing of the optical spectra. The reason is that the resonance occurs when the frequency difference between the microcavity laser and the adjacent comb line closes to one-half of the comb spacing, which is commonly considered as the Arnol’d tongue [49,50]. Similarly, the 6GHz microwave signal and its harmonics, corresponding to one-third of the comb spacing, are also observed at the detuning frequency of -6 GHz and 12.5 GHz in the electrical spectra.
Fig. 4. (a) Experimental optical spectra, and (b) corresponding electrical spectra of the microcavity laser subject to frequency comb injection with different detuning frequencies. The enlarged views show the electrical spectra with detuning frequencies of -6 and 12.5 GHz. (c) Theoretical optical spectra, and (d) corresponding electrical spectra obtained from rate equation simulations using the same parameters in the experiment. The enlarged views show the electrical spectra with detuning frequencies of -6.75 and 11.25 GHz.
To numerically simulate the comb-injection experiment described above, the rate equations for a microcavity laser with comb injection are used. The complex electric field of the injected comb is written as:
where Ej, ωj, and φj are the amplitude, the angular frequency, and the initial phase of the jth comb line, respectively. The injected optical comb has three comb lines, and j is equal to 1, 2, and 3, respectively. Since the injected comb is originated from the electro-optics modulation, the relative phase of the optical tones is constant. The simulation results are independent of the relative phase values of the three comb teeth. Therefore, we set the initial phase to be 0 for our simulation for convenience. Besides, each comb line has the same amplitude. The rate equations for the microcavity laser with comb injection are written as:The rate equations Eqs. (3)–(5) for the photon density, carrier density, and phase are derived from Eq. (2). In these equations, ${E_s}(t )= E\textrm{exp}[{j\varphi (t )} ]\; $ is the complex amplitude of the electric field of the microcavity laser. The frequency detuning between the injected optical frequency ωj and the free-running frequency ω0 is defined as Δωj = ωj - ω0. n, s, and φ are the carrier density, the photon density and the phase of the lasing mode, respectively. sj is represented as the photon density for the jth optical tone. I is the injection current, η is the current injection efficiency, q is the electron charge, and Vα = α2dα is the active region volume. A, B, and C are the defect, bimolecular, and Auger recombination coefficients, respectively. υg = c / ng is the group velocity, where c is the speed of light in vacuum, and ng is the group refractive index. Γ is the optical confinement factor, αi is the material internal absorption loss, and τpc = Q / ω0 is the passive mode lifetime, where Q is the passive mode quality factor. β is the spontaneous emission factor, α is the linewidth enhancement factor, and ${\kappa _c} = \sqrt {{\eta _{\textrm{inj}}}} \; /\; {\tau _{\textrm{pc}}}\; $ is the coupling rate, where ηinj represents the injection efficiency from the fiber to the laser and is estimated to be 0.20. Considering the gain suppression effect, the gain coefficient is assumed to be a logarithmic function:
Table 1. Parameters used in the rate equations
By solving the rate equations Eqs. (3)–(5), the optical and electrical spectra with varied detuning frequency Δf are obtained with a fixed injection strength Rinj. Here the injection strength is defined as Rinj = sinj/s0, and s0 is the free-running steady-state lasing mode photon density. A fourth-order Runge-Kutta method with a time step of 200 fs and a time span of 20 ns is utilized to calculate the time series output. The optical and electrical spectra are obtained from the Fourier transform of the complex field amplitude E and |E|2. The relationship between the photon density s and the laser field amplitude E is:
In the simulation, the injection current of the microcavity is maintained as 30 mA with a relaxation resonance frequency of 9 GHz. The optical and electrical spectra is calculated at Rinj = -20.3 dB with the same detuning frequencies used in the experiment. In this case, the relative intensity of the optical and electrical spectra for detuning frequencies ranged from -36 to 36 GHz are recorded and plotted as maps, as shown in Fig. 4(c) and Fig. 4(d). The relaxation oscillation and Arnol’d harmonic locking between comb lines can be observed. The trend of the simulated results is consistent with the experimental results, indicating that the theoretical model can carry out further research on harmonic locking.
Based on the numerical simulation model, the theoretical stability maps for the microcavity laser subject to frequency comb injection are systematically studied over the injection parameter space (Rinj, Δf). Figure 5(a) shows the nonlinear map of the microcavity laser with 18GHz-spaced optical combs injection. Since the positive detuning and the negative detuning have similar locking phenomena, only the results for positive detuning are shown. In the simulation, the frequency spacing of the generated optical comb is identified by the interval of the photon density time series and represented by different colors. The interval is calculated via the time difference between the adjacent maximum value of the photon density. When the calculated time intervals are different, it is defined as an unlocked state and represented by white regions. At the detuning frequency of 0 GHz, the microcavity laser is injection locked to one of the injected comb lines at low injection intensity. Increasing the injection intensity gives rise to a large region where the frequency spacing is reduced to one-half of that of the injected comb, which means that 1/2 harmonic locking is achieved. At even higher injection intensity, a small region indicating the 1/4 harmonic locking appears before entering the unlocked state. It is worth noting that, harmonic locking is hardly observed at the detuning frequency of 18 GHz, consistent with the experimental spectra shown in Fig. 4. In addition, Arnol’d tongue locking is also clearly seen when the frequency difference between the microcavity laser and the adjacent comb line is a unit fraction of the comb spacing. The 1:2 resonance tongues to 1:9 resonance tongues are shown. The range of 1/2 harmonic locking at 9 GHz detuning frequency is the largest of these tongues, and the locking regions gradually shrink as the denominator of the fraction increases. Moreover, when the injection strength is lower than -19 dBm, the locking region expands with the increase of the injection strength. As in the case of relaxation oscillation locking, period-doubling locking occurs at higher injection intensity within this locking tongue. The results indicate that the frequency spacing of the generated comb can be tuned by changing the injection parameters.
Fig. 5. Calculated nonlinear stability maps over the injection parameter space (Rinj, Δf) for the microcavity laser with the relaxation oscillation frequency of (a) 9 GHz, and (b) 6 GHz.
Furthermore, the nonlinear dynamic stability map of the microcavity laser with a relaxation oscillation frequency of 6 GHz is simulated, which corresponds to one-third of the comb spacing. As shown in Fig. 5(b), the relaxation oscillation locking region is significantly reduced due to the unstable harmonic locking for the bigger denominator of the rational fraction. In addition, the Arnol’d locking region also shrinks compared to that with a relaxation oscillation frequency of 9 GHz, which may result from the carrier density fluctuations that cannot follow the higher speed modulation caused by the periodic forcing of the neighboring comb lines. This verifies that the microcavity laser is beneficial to generate harmonics with a wider range of repetition frequencies due to its large modulation bandwidth.
4.2 Spectral broadening
In addition, the effects of injection locking and spectral broadening for the microcavity laser with comb injection are studied. Figure 6 shows the optical spectra and corresponding electrical spectra of the microcavity laser with 8GHz-spaced comb injection. With a small injection power of 0.1 mW and a detuning frequency of 0 GHz, the microcavity laser achieves mode selection and amplification, as shown in Figs. 6(a) and 6(b). The SMSR is 13.2 dB. Extra comb lines appear due to the nonlinear dynamics in the microcavity laser. Figures 6(c) and 6(d) show the case of spectral broadening when the injection power and the detuning frequency are set as 5 mW and -11 GHz, respectively. The number of optical comb lines raises from 3 to 14. When the injection power is 8.5 mW and the detuning frequency is 87 GHz, which is more than 10 times the comb spacing, an optical comb with 21 comb teeth is obtained, as shown in Fig. 6(e). The spectrum in the gray region is filtered out by an OBPF and detected by a photodetector. The obtained beat frequency is shown in Fig. 6(f). The linewidth of the 8 GHz signal is estimated to be less than 510 Hz from the zoom-in spectrum in the inset, which is consistent with the repetition frequency signal of the injected comb, and indicates that the optical comb generated by the microcavity laser is stable. Compared to the optical injection with a single frequency source, the stable dynamic range is enlarged for the microcavity laser subject to comb injection. The experimental results show that the output optical comb from the microcavity laser can be broadened effectively.
Fig. 6. The output optical spectra and corresponding electrical spectra of the microcavity laser subject to 8GHz-spaced comb injection with the detuning frequencies of (a)(b) 0 GHz, (c)(d) -11 GHz, and (e)(f) 87 GHz.
Finally, the spectral broadening of the microcavity laser subject to comb injection with different frequency spacing are studied. Figure 7(a) shows the output optical spectra subject to the comb injection with the frequency spacing of 10 GHz, 12 GHz, and 14 GHz, where the detuning frequencies in the three cases are maintained at -16 GHz, and the injection powers are 7.1 mW, 8.9 mW, and 10.4 mW, respectively. Figure 7(b) shows the number of comb teeth versus the frequency spacing of the optical comb. The number of comb teeth is more than 13 when the spacing of the injected combs varies from 6 GHz to 14 GHz. It can be seen that the number of comb teeth increases as the frequency spacing of the injected comb is close to the resonance frequency of the microcavity laser. The number of the comb lines reaches 16 at the 10 GHz repetition frequency. When the frequency spacing continues to increase, the number of comb teeth gradually decreases, because the carrier density fluctuation cannot follow the photon beating frequency due to the limited modulation bandwidth as shown in Fig. 2(b).
Fig. 7. (a) The output optical spectra of the microcavity laser subject to the optical comb with 10 GHz, 12 GHz, and 14 GHz spacing injection. (b) The number of comb lines generated versus injected comb spacing.
5. Conclusion
In conclusion, we have experimentally and numerically investigated the nonlinear dynamics for a square semiconductor microcavity with frequency comb injection. By sweeping the detuning frequency between the injected comb and the microcavity laser mode, the microcavity laser emits combs with reduced frequency spacing, which is harmonically locked to unit fractions of the comb spacing owing to the undamped relaxation oscillation or Arnol’d-type oscillation. The nonlinear stability map is simulated based on the rate equations, and the frequency spacing under different lock states is explored. The numerical simulation reveals that the locking region shrinks as the relaxation oscillation frequency of the microcavity laser decreases. Moreover, the comb-injected microcavity laser is verified to induce spectral broadening with undeteriorated coherence. When the frequency spacing varies in the range from 6 GHz to 14 GHz, the number of comb teeth increases to more than 13. Due to the large modulation bandwidth of the microcavity laser, it is beneficial to generate a broadened optical comb with a wide range of frequency spacing. The results show that the frequency spacing and the number of comb teeth can be adjusted for the microcavity laser with frequency comb injection.
Funding
National Natural Science Foundation of China (61875188, 61935018, 62122073).
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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