Spontaneous mode locking of a multimode semiconductor laser under continuous wave operation

Self-starting mode-locking is observed in a laser based on a compact III-V diode-pumped quantum-well surface-emitting semiconductor laser technology with a saturable-absorber-free but dispersive cavity. Continuous wave generation of picosecond pulses at a rate of 100 GHz is demonstrated by recording microwave intensity noises, beat frequency, time-resolved optical spectra, and intensity autocorrelation. Coherence of the pulse train is obtained through the frequency noise measurement of the demodulated beat note, demonstrating a timing jitter as low as 110 fs, near the quantum limit. Using a theoretical model based on a generalized Haus master equation, we demonstrate the existence of this mode locked state without the need for saturable absorption. The fundamental physical mechanism is the interplay between self-phase modulation and anomalous dispersion like in cavity soliton together with light–matter interaction-induced time symmetry breaking.


Introduction
Semiconductor mode-locked diode lasers (MLLDs) generating short pulses show potential in many applications, such as high-speed optical interconnects  or optical time domain multiplexing (Jiang et al., 2007;Mollenauer et al., 2000). In contrast with active MLLDs (Bowers et al., 1989), passive MLLDs do not require any external electrical oscillator. Passive mode-locking has been achieved in semiconductor lasers by the use of strongly intensity-dependent elements in the laser cavity where saturable absorption provides a coupling mechanism between laser modes, which typically results in short pulses in the time domain with repetition rates ranging from a few to hundreds of GHz (Marsh & Portnoi, 2000;Vladimirov & Turaev, 2004). However, owing to the spontaneous emission noise and their low-finesse cavity, MLLDs' optical pulse train exhibit significant phase noise and timing jitter (Hou et al., 2011). In contrast to edgeemitting devices, passively mode-locked vertical-external-cavity surface-emitting lasers (VECSELs) are sources of sub-ps pulse trains with excellent low jitter in a high beam quality (Garnache et al., 2002). In particular, the so-called mode-locked integrated externalcavity surface-emitting laser (MIXSEL), in which both the gain and the saturable absorber share the same microcavity, represents a compact source with a repetition rate up to 100 GHz (Maas et al., 2007). However, the gain and absorber layers, which require different quantum confinements, have to be optimized separately by using different growth parameters.
On the other hand, mode-locking can also occur in semiconductor lasers in the absence of a saturable absorber (Day et al., 2020;Sato, 2003;Rosales et al., 2012). In this case, it is understood that the beating of different laser modes leads to oscillations of the carrier density in the gain medium at multiples of the repetition frequency (corresponding to longitudinal mode spacing), which, in turn, again couple with the laser modes. In the presence of spatial inhomogeneous gain saturation (spatial hole burning), this locking phenomenon leads to frequency modulation rather than amplitude modulation, even though selfgeneration of optical pulses has been reported in single-section quantum-dash lasers (Gosset et al., 2006).
In this article, we report a self-starting mode-locked system generating picosecond pulses at a rate of 100 GHz based on an absorber free short cavity VECSEL. Stable mode-locked operation is triggered by an instability known in the framework of the complex Ginzburg-Landau equation (CGLE) as phase turbulence (Gil & Lippi, 2014). It predicts that a semiconductor laser can undergo a transition to a multimode regime due to non-linearity and phase dispersion. In the absence of spatial hole burning (SHB), this coherent non-linear process gives rise to phase-locked frequency combs first observed in quantum cascade lasers (Piccardo et al., 2020). In a VECSEL, as the quantum well (QW) thickness is very small compared to the wavelength and QWs are usually located at the antinodes of the laser electric field, all the modes are in phase on the gain medium. This prevents longitudinal SHB, and thanks to the QWs homogeneous broadened gain, the multi-longitudinal dynamics is only driven by coherent non-linear mode-mode coupling (Garnache et al., 2007;Chomet et al., 2023). Here, we take advantage of the VECSEL technology and physical concept advantages to precisely engineer the amount of phase dispersion in FIGURE 1 (A) Experimental setup for the characterization of the multimode VECSEL dynamics. (B) Top plot, calculated second-order dispersion of the microcavity VECSEL. Bottom plot, optical spectrum of the laser for an optical output power of 42 mW and the fit of the state with a spectral sech2 envelope. Inset: Intensity autocorrelation signal obtained through second harmonic generation (blue) and calculated from the sech2 envelope (red).

Frontiers in Photonics
frontiersin.org 02 the laser cavity. The signature of a stable comb regime is demonstrated through time resolved optical spectra and a narrow beating at the mode spacing frequency by excitation of a unitraveling-carrier photodiode. The latter is shown to have a phase noise near the quantum limit. Using a generalized Haus master equation (HME), we reproduce this stable locked state and discuss the impact of gain recovery on dynamics.

Methods
The laser cavity shown in Figure 1A includes a GaA-based gain mirror and an output coupler (99%) separated by an air gap of L c = 1.5 mm , leading to a free spectral range (FSR) of 100 GHz. The gain structure (GaAs 229) is temperature-stabilized thanks to a Peltier cooler and optically pumped by a 808-nm 400-mW low-noise single-mode diode on a radius of 30 μm (at~70°Brewster angle). A 1.1λ/4 SiN thick antireflection coating is deposited on the top surface to reduce the microcavity resonance filtering effect on gain bandwidth and to introduce the anomalous group velocity dispersion essential for triggering comb formation. The calculated spectral dependence of the dispersion is shown in Figure 1B. More details about the gain structure and dispersion calculation can be found in Supplementary Material S1.
The laser operates at a Peltier temperature T p = 15°C in continuous wave (cw) at a wavelength of~1.01 μm. Laser emission is reached for a threshold density of 1.8 kW/cm 2 , and a maximum output power of 60 mW is achieved. The laser beam is collimated using a 75-mm-focal achromatic lens, followed by the application of an optical isolator to prevent parasitic feedback. Just above the laser threshold, the device emits in a single longitudinal mode. However, for an output power of 42 mW, the spectrum ( Figure 1B) consists of a few strong modes superimposed on a sech 2 -shaped lobe. The peak of the spectrum arises close to the zero-crossing of the dispersion in the anomalous dispersion regime. The generation of a pulse train was indicated by using a background free auto-correlator (FR-103XL) with a delay of 150 ps (inset of Figure 1B). Pulse to pulse duration was 10 ps, which corresponds to a round-trip time of the cavity. The second harmonic generated (SHG) signal reached 60% of modulation, meaning that continuum is still present. The dotted line is the SHG profile calculated from the Fourier transform of the optical spectrum, which is close to the measured one at the vicinity of the peak maxima, demonstrating the stabilization close to the Fourier transform limited pulses. The calculated pulse width was 1.4 ps (FWHM).
Although the SHG signal strongly indicates the presence of a pulse train, stability of the comb lines has to be evaluated to confirm stable mode locking. Here, we employed two techniques, timeresolved spectra and RF spectra of the pulse train. The experimental setup is shown in Figure 1A.

Observation of a highly coherent selfmode-locked light state
The first confirmation of mode locking was obtained by recording the cw optical spectrum at the output of a high-resolution grating monochromator (JY HR1000), using a streak camera (Hamamatsu C10910-02) with a time resolution of 25 ns, as shown in Figure 2A.
The single-shot trace shows a cw operation for the comb lines, with a frequency difference of approximately 100 GHz (resolution of 25 GHz). Detuning the peak of the spectrum with respect to the zero-crossing of the dispersion leads to an unstable behavior of the comb where fast-intensity oscillations of each mode are observable ( Figure 2B). Signature of these oscillations at~4.5 MHz (and higher harmonics) was visible in the total laser output intensity fluctuations recorded with a 2-GHz bandwidth low-noise photodiode for 1.2 mA of photocurrent ( Figure 2C). In the mode-locked regime, intensity noise is pump noise limited up to the laser cut-off frequency (~120 MHz) and reaches above the shot noise level. The pulse train is then very stable with relative rms intensity fluctuations below 0.1% in the 100 kHz−1 GHz frequency range.
Second confirmation of mode locking was obtained by recording the RF spectra of the pulse train by excitation of a commercial unitraveling-carrier photodiode (IOD-PMAN-13001-1 from NTT) (Blin et al., 2017). The microwave signal that comes from the photodiode is collimated using a 10-cm focal-length Teflon lens, then focused using an identical lens into a calibrated heterodyne head receiver that allows observation of the beating at the round-trip frequency on a RF analyzer, as shown in the inset of Figure 2D. The center frequency was 111.24 GHz, and the linewidth was < 200 kHz for a 10-ms measuring time, which tends to be limited by the 100-kHz resolution bandwidth of the RF analyzer. This beat note was then downshifted below 1 GHz and amplified using a 20-dB-gain 0.1-1 GHz bandwidth low-noise amplifier (Minicircuit ZFL-1000LN). More details about the setup can be found in Supplementary Material and Supplementary Section S2. The resulting signal was then analyzed thanks to a numerical Hilbert transform, in order to extract the single-sided spectral density of phase-noise fluctuations, as described by Bandel et al. (2016). The result is shown in Figure 2D together with its quantum limited case calculated from Paschotta et al. (2006). From this, we obtained an integrated timing jitter δT of 110 fs in a bandwidth of 1 kHz-1 MHz (Haus & Mecozzi, 1993;Scott et al., 2001). This low value has to be related to the high repetition rate of the pulse and gives an absolute jitter δT/T~1%, which is higher by two orders of magnitude than typical values obtained by SESAM mode-locked VECSELs with the GHz repetition rate in approximately the same integration bandwidth (δT/T~0.01%) (Wittwer et al., 2011;Quarterman et al., 2008). Nevertheless, in our case, the phase relationships between the modes are ultra-stable as the phase noise reaches its quantum limit above 10 kHz of RF frequency (see Figure 2D). The laser then emits a highly coherent train of picosecond pulses, the timing jitter of which is close to its quantum limited case (90 fs). This demonstrates that the non-linear process that triggers this locked state is highly coherent. However, we experimentally found that a stable comb formation occurs only for a short range of dispersion, as discussed previously ( Figures 2B, C).

Justification of the existence of selfstarting frequency combs
To confirm that the observed locked state is a result of the interplay between anomalous dispersion and non-linear phase Frontiers in Photonics frontiersin.org 03 changes, we simulate the laser dynamics using a generalized HME (Hausen et al., 2020). Such an equation can be reduced to the CGLE that has been shown to support comb formation in the case where ultrafast gain relaxation and near-threshold operation are considered. The latter considerations are not suitable with our device as comb formation occurs well above the laser threshold and the carrier lifetime is much longer than the cavity round-trip time. The generalized HME reads as follows: with ξ being an integer describing the evolution from one round-trip toward the next; it represents a slow time scale, while the second variable t refers to the time evolution normalized to one round-trip. Eq. 1  describes the electric field evolution from round-trip to round-trip z ξ A and the variation of the carrier G on the fast time scale t. Eq. 3 is used to maintain the long-term carrier dynamics. Here, γ e is the carrier relaxation rate in the gain, γ is the gain bandwidth, G 0 is the unsaturated gain (Loss), α is the amplitude phase coupling, η is the pump over the threshold ratio, and T is the cold cavity round-trip time. Φ n are the different orders of phase dispersion. Here, we consider the expansion until n = 3. Phase dispersion in VECSEL cavities is mainly introduced through the amplitude reflectivity of the gain mirror, which acts as a Gires-Tournois interferometer (GTI). Thus, phase dispersion can Frontiers in Photonics frontiersin.org be precisely engineered through the round-trip time of the microcavity and top VECSEL reflection coefficient (see Supplementary Material S1). For example, the impact of thirdorder dispersion (TOD) has been demonstrated in the multilongitudinal dynamics of a MIXSEL (Schelte et al., 2020). Evaluating the phase reflectivity of our gain structure at the vicinity of the lasing wavelength, we consider a time delay Φ 1 = 80 fs, group velocity dispersion (GDD) Φ 2 = −80 fs 2 , and a TOD Φ 3 = −53000 fs 3 . The resulting Eqs 1-3 were solved using standard split-step methods, such as the outlined method, for example, by Gurevich and Javaloyes (2017). Figure 3A shows the two time scale representations of the dynamics of the laser output intensity. After a relatively long time interval (about 200,000 round-trips) the laser reaches a frequency comb regime, where a single pulse is stabilized in the cavity. Strong gain depletion and time symmetry breaking terms (time delay and TOD) induce a drift of the pulse within the cavity (Javaloyes et al., 2016). The lack of time symmetry can also be seen in Figure 3B, where the pulse shows an asymmetric profile because of the long time response of the medium. The pulse FWHM is~2 ps and lies on a continuum background in good agreement with autocorrelation measurements ( Figure 1B). Experiments indicate that a precise control of the GDD is needed to generate this highly coherent state, and we investigate its stability in the parameter space GDD-η. Figure 3C displays a colormap of the calculated autocorrelation of the laser intensity. The autocorrelation in this context means that we compare the intensity over one round-trip with the intensity delayed by several round trips. In a comb, they match perfectly, and the autocorrelation is exactly unity. A part of the parameter space is a single frequency state (gray area), and instability of this solution (black dash line) occurs for an amount of anomalous dispersion given by GDD < − 1 α 4G 0 /γ 2~− 25 fs 2 (Chomet et al., 2023). This instability generates intensity fluctuations that grow with an increase of GDD in a strictly periodic manner, generating a comb. In the absence of fast positive self-amplitude modulation (SAM), thanks to a saturable absorber process, this stable pulsed state finds its origin in time symmetry breaking under weak gain-SAM-saturation/ modulation. Thus, a short pulse is allowed to propagate with causality principle breaking by drifting strongly backward to maximize the amplification gain ( > 0 SAM), as seen in Figures  3A, B. This pulse regime will thus be slightly red shifted. We believe that in this light state, GDD dispersion tends to balance self-phase modulation (SPM) associated with the amplitude phase coupling factor as in the soliton-like laser mode-locking mechanism (Zolotovskii et al., 2015). The typical value of pulse duration δτ in this mode-locked regime at the vicinity of the next unstable region is as follows: Equation 4 is analogous to that describing the width of a fundamental soliton, where αγ e TG 0 is the effective SPM coefficient. With the values of Figure 3B, δτ~2.1 ps is in good agreement with the width of the pulse shown. Equation 4 dictates an Frontiers in Photonics frontiersin.org 05 ever-decreasing pulse width for increasing pumping power or cavity round-trip time. This only works until the pulse experiences very strong losses due to the finite-gain bandwidth. The pulse will then break in a double-pulse solution such that each pulse is longer and experiences a reduced loss due to the filter. To show this, we investigate the mode-locking behavior of the laser with an increase in the cavity length. Increasing the latter will result in the gain responding faster to the electric field beating at the cavity round-trip frequency, and thus, a larger SPM coefficient. Figure 4A shows the optical spectrum of the laser operating in the anomalous dispersion regime close to zero GDD (in blue P out = 27 mW, in red P out = 42 mW) in a long cavity configuration L c = 8.5 mm (FSR = 18 GHz). The spectrum corresponding to P out = 27 mW (blue curve in Figure 4A) shows a mode spacing~54 GHz = 3 × FSR, which suggests a regime of harmonic mode locking (HML), where three pulses instead of one circulate in the cavity. Increasing the pump power leads to oscillation of more cavity modes (red curve in Figure 4A), but a streak camera photograph reveals an unstable behavior of the optical spectrum ( Figure 4B). Figure 4C presents the results of numerical simulations of Eqs 1-3 Jiang et al., 2007), which confirms the experimental observations in which the laser intensity is now seen to stabilize not a single but three pulses within the cavity. This stable HML regime is, however, easily destabilized with a change of the control parameters (pumping power, dispersion, cavity length. . .), which results in pulses drifting with different group velocities. We show in Supplementary Section S3 that a non-null third-order dispersion is necessary to stabilize this multiple pulse solution; this is interesting as TOD is, here again, a parameter that can be engineered.

Discussion
In summary, we have presented a regime of spontaneous mode locking in a short cavity VECSEL. A train of pulses with a low timing jitter was demonstrated to be generated in the cavity. Existence of this comb regime was demonstrated by a generalized HME. Its formation is due to the interplay with dispersion and a slow light-matter interaction that induces time symmetry breaking and positive SAM even in the absence of a saturable absorber. Increasing the cavity length was seen to result in a breakup into multiple pulses and a regime of HML. Stabilization of the latter could possibly be engineered with TOD. In contrast with MIXSEL, where the gain and absorber layers have to be carefully optimized, here, we have used a common GaA-based gain mirror, where phase dispersion can be tailored by dielectric coatings. The use of metamaterials could also lead to a fine tuning of the dispersive effects.
We believe this spontaneous mode-locking laser regime could be observed in other slow gain laser materials with γ e T (η − 1) < 1 at a repetition rate 1/T in the presence of phase amplitude coupling: a ns pulse train at 110 MHz has been observed in a cw traveling wave Ti: sapphire laser system including a spectral filter but was attributed to the Kerr lens effect, whereas below the Kerr non-linearity intensity threshold (Akachanov & Fstoeckel, 1995), an unstable mode locking at 100 MHz has been observed in a ring dye laser under strong dispersion (Vinogradov et al., 1992). A frequency comb at 11 GHz has been demonstrated based on a dispersive QCL ring microresonator (Piccardo et al., 2020), but in this Kerr-like fast media system γ e T > 1, an HML regime takes place.
An interesting application of such spontaneous mode-locking processes characterized by a low non-linear intensity threshold < 10 kW/cm 2 would be to seed the start of the passively mode-locked system exhibiting a high non-linear SAM threshold intensity above ≫ 10 kW/cm 2 (like in Kerr-based mode locking).

Data availability statement
The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.  Figure 3A but T = 60 ps, η = 4, and Φ 2 = −60 fs 2 . In this long cavity configuration, the laser stabilizes three pulses within a round-trip, resulting in a regime of harmonic mode locking.