Introduction. Ultrafast mode-locked laser oscillators, pervasive in modern-day applications, range from all-fiber-based MHz repetition rate oscillators using saturable absorbers (SAs) [1], and nonlinear phenomena, to hundreds of GHz repetition rate micro-ring resonators [2]. There are various applications for these ultrafast oscillators which require tunability of wavelength, multiple mode-locked oscillator sources, or benefit from multi-GHz repetition rates with large peak powers such as multi-photon microscopy [3], multi-comb spectroscopy [4], and pump–probe microscopy [5]. Optically pumped vertical external cavity semiconductor lasers (VECSELs), which fill a niche with repetition rates between fiber and micro-ring systems, utilize broadband semiconductor gain media to produce frequency tunable, extremely low-noise [6], robust, and low-cost mode-locked laser oscillator solutions.

VECSELs make ideal testbeds for studying nonlinear systems driven far from equilibrium. Earlier works have established that the microscopic physics is driven by photon extraction via kinetic hole burning and replenishment of carriers (electron/hole) as the mode-locked pulse traverses the resonant periodic gain quantum well (QW) stack. [7] Prior to and after the pulse, carriers are returned to lattice temperature Fermi distributions of the initial state. If pumped too hard by the external multimode fiber pump, mode-locking can be destabilized when the inversion can undergo rapid transient absorption.

One goal of this work is to explore the gain dynamics of semiconductor systems and exploit novel cavity geometries to generate independent pulse trains that still maintain a high degree of mutual coherence. In the previous work, we demonstrated the use of a single VECSEL-based frequency comb for spectroscopic studies in the mid-IR [8]. Due to the GHz level repetition rate that can be achieved with VECSEL laser systems, we subsequently demonstrated the use of a virtually imaged phase array (VIPA) to directly measure the individual comb “teeth” for spectroscopic absorption measurements of methane [9]. An alternative and more flexible approach for frequency comb spectroscopy that is often employed is based on a pair of mutually coherent frequency combs that have adjustable repetition rates [4]. The adjustable and high repetition rates provided by VECSELs make them a unique system for dual-comb spectroscopy enabling higher time resolution [10]. The ability to operate multiple laser systems from a single gain chip with flexibility in operational wavelength and repetition rates will benefit a wide range of applications such as dual-comb spectroscopy, as well as time-resolved pump–probe spectroscopy where low timing jitter between pulses is essential.

We explore a coupled cavity laser design in which two optical V-cavities spatially overlap on a common semiconductor gain medium as shown in Fig. 1. The laser cavities are individually mode-locked utilizing separate semiconductor saturable absorber mirrors (SESAMs). We explore numerically and experimentally the complex dynamics resulting from the gain competition between the two laser systems, looking for regimes of stable mode-locked operation. The coupled cavity design can provide a high level of coherence between the two pulse trains due to rejection of a pump-induced noise that is common to both cavities while still allowing for the ability to easily adjust the relative repetition rates. The Keller group [10] provides a solution where one cavity shares two polarization multiplexed channels, which lacks the ability to independently control oscillator channels and lacks wavelength tunability. In addition, the two pulse trains do not overlap on the VECSEL gain chip, and hence the system does not exhibit the interesting dynamics investigated in this work. Here, we utilize different angles of incidence on the semiconductor gain chip to minimize gain competition and enable operation at two different wavelengths. In this work, the use of two external cavities enables additional flexibility to control the relative repetition rate of the pulse trains.

 

Fig. 1. (a) Coupled co-mode-locked VECSEL cavities. The outer cavity is shown in purple, the inner cavity shown in red, and the pump shown in orange. SESAMs are mounted in the gray mounts. Output couplers are mounted in the onyx mounts, and the common VECSEL gain chip shown on the bottom. (b) Schematic representation, where the inner cavity (blue) has a 21° half-angle and the outer cavity (red) has a 44° half-angle. The SESAMs have 1 QW and the gain chip has 10 QWs each with a respective DBR.

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Microscopic modeling of this coupled cavity system requires that one move beyond the paraxial assumption and solve the Maxwell–semiconductor–Bloch equations (MSBEs) [7] taking account of higher-order transverse spatial gratings encoded in the gain medium. Depending on the angle of incidence of a particular branch of the cavity, the expansion of the SBE in higher-order Fourier modes exhibits a natural cutoff. Previous studies were limited to a Fourier mode expansion assuming paraxial small angles and suggested that these grating terms act to further stabilize the mode-locking.

Experimental setup. Figure 1 shows the layout of the coupled co-mode-locked cavities. The cavity geometry chosen is the V-cavity configuration, as it is the simplest and is relatively robust. Each V-cavity consists of a SESAM, a 0.6${\% }$ curved output coupler, and a shared VECSEL resonant periodic gain (RPG) chip. The VECSEL gain chip selected was designed to operate at around 1028 nm at normal incidence. The lasing wavelength increases as a function of the angle, as a consequence of the effective quantum well spacing increasing at larger angles. In developing a co-mode-locked dual cavity VECSEL, selection of a proper SESAM is crucial to operation. This SESAM must operate across a relatively large continuous spectral bandwidth and have a relatively short carrier lifetime. Both of the cavities have an adjustable length of $\approx 6.2$ cm, which is half of the pulse round trip optical path length.

Experimental results. Dual cavity mode-locking was successfully demonstrated for different cavity angles. Autocorrelator traces for stably mode-locked operation are shown in Fig. 2(a). The autocorrelator uses sum-frequency $\chi _2$ nonlinear process to measure the pulse envelope as shown in Fig. 2. The average powers of the inner and outer cavities are 28 and 32 mW, respectively, with pulse widths of 610 and 595 fs, respectively. The repetition rates were measured for the inner and outer cavities to be 2.11 and 2.17 GHz, respectively.

 

Fig. 2. (a) Autocorrelation was taken on an FR-103 autocorrelator. One can see the pulses from both the inner and outer cavities. (b) Spectral shift due to co-mode-locking frequency pulling. Thicker lines denote simultaneous operation, whereas thinner lines are independent cavity operation.

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This co-mode-locked VECSEL system is stably mode-locked on two well-separated central wavelengths due to the low mutual pulse interaction. The low pulse cross talk was verified by simply blocking the respective V-cavity arms and measuring the shift in the spectral response as shown in Fig. 2. This is mainly attributed to extraction of light from within two separate spectral windows lying within the full gain bandwidth as discussed later. In principle, more parallel channels can be accommodated as long as they are contained within the gain bandwidth.

Simulation of dual V-cavity. The Maxwell–semiconductor–Bloch equations (MSBEs) are a microscopic first-principles many-body approach essential to uncover ultrafast pulse dynamics on time scales comparable to typical dephasing times, thereby supplanting traditional “gain”-based methods. In these ultrafast systems, carrier distributions are driven into extreme nonequilibrium states, a situation not observable in the gain which integrates over such distributions. Moreover, they incorporate key many-body Coulomb interactions that influence dynamic bandgap reduction through energy and also field renormalization (Hartree–Fock level) and carrier–carrier and carrier–phonon scattering. The latter two have been shown to be adequately captured in terms of a microscopically derived dual rate approximation. These damping rates described below [11] are summarized with the following notation: $\Gamma _\text {deph}$ (polarization dephasing), $\Gamma _\text {scatt}$(slow charge carrier recovery time), $\Gamma _\text {fill}$ (fast charge carrier recovery time), and $\Lambda _\text {spont}$ (spontaneous emission noise). The renormalized Rabi energy appearing in the $\Omega _k$ terms below is defined by $d\cdot E + \sum _{k'} V_{k-k'} p_k'$, where $d$ is the dipole matrix element between the conduction and valence band and $V$ is the screened Coulomb matrix element. The renormalized carrier frequency appears as the $\omega _k$, which is defined as $\frac {1}{\hbar } [ \epsilon _k - \sum _{k'} V_{k-k'} (n_{k'}^e + n_{k'}^h) ]$:

 

Fig. 3. (a) Example of 2 QW stack, where basic unit cells are shown for simulating an RPG QW stack, where QWs shown represented in purple solve SBE which couple to the material gap regions, shown in light red and light blue by sourcing Maxwell’s equations. (b) Fields in grating basis are depicted, where one field $q_j$ interacts with a grating formed by two other fields (in-plane, not depicted) along $\hat {x}_\text {scat}$ within the QW.

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The RPG structure consists of a periodic stack of QWs with spacer layers and the SESAMs, a single QW with spacer layer. The stencil shown in Fig. 3 is the elemental building block implemented at each QW, adjoining the spacer layer and replicated throughout the dual cavity geometry to build our complete simulation framework. Details of this simulation will be published elsewhere. Finally, we introduce group delay dispersion (GDD) into our simulation framework to account for intrinsic dispersion arising from gain chip, SESAM, and other cavity components. This data is available to us experimentally and is responsible for the relatively long pulse durations observed in Fig. 2. The GDD’s effect is a quadratic dependence of phase with frequency in the spectral domain:

Simulation results. Prior to showing our simulation results, we can infer the angular dependences of the lasing emission from a consideration of the partial coherence of a periodic array of emitters corresponding to our 10 QW RPG stack as shown in Fig. 4(a). The solid line is the analytic prediction for a set of periodic emitters, the red dots show results from numerical simulation data from single V-cavity Maxwell–SBE models, the blue dots are from our actual experimental results shown in Fig. 4(b), and the three green dots are from independent single V-cavity measurements with the same gain chip and SESAM. We display the final filtered spectral bandwidths resulting from single V-cavity simulations lasing at different incident angles.

 

Fig. 4. (a) Solid line comes from maximum coherence amplitude, where simulation is the result from microscopic simulations of single VECSEL V-cavities. Dual V (Exp) is the experimental results from the experimental body of this work. Single V (Exp) are results for independent V-cavities in setting up the dual-V experiment. (b) Optical spectra from the microscopic simulations’ pulse spectra are shown.

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Next, we run the full dual cavity simulation for the inner cavity and outer cavity angles, corresponding to those in the above experiment. Just as in the experiment, we need to adjust the cavity parameters such as outcoupling loss to balance dual-wavelength channel mode-locked pulse relative intensities. We impose a GDD of +100 fs$^2$ which lengthens the pulse durations to about 300 fs. The simulation is run for 1000 cavity round trips at which point stable mode-locking is well established. We remark here that we accelerate convergence by using weak seed pulses rather than allow buildup from noise. Despite deliberately offsetting the initial seed pulse relative spacing in each cavity, we observe that pulses tend to bunch up, hitting the gain chip with a small relative delay as shown in Fig. 5(a) in our limiting case where the two cavity lengths are very nearly matched ($\Delta L = 1\mu$m). Figure 5(b) shows the nonequilibrium evolution of the carrier inversion, starting from the external pumped Fermi distribution shown in red. After hundreds of round trips, the pulses self-replicate indicating stable mode-locking. The supplement shows transient kinetic hole burning (Visualization 1) and the final repetitive hole burning in the stable mode-locked state of Fig. 5(b) (Visualization 2). Dual kinetic holes (black curve) with the same delay as the pulses on the left are burned in the carrier inversion. Stable mode-locking is ensured as the inversion always remains positive. After each pulse, the carrier distribution recovers to a hot near-Fermi distribution. It is important to stress that the kinetic holes are not uniformly burned within each of the 10 QWs in the RPG stack.

 

Fig. 5. (a) Simulated dual cavity pulse waveforms, where $\Delta \phi$ is depicted. (b) Corresponding dual kinetic hole burning in carrier distribution, $\alpha$ is redshifted from kinetic hole $\beta$.

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The reason for the bunching of the mode-locked pulses within each V-cavity arm on the gain chip is not self-evident. One might argue that, because the individual pulses are extracting carriers from separate reservoirs, the net gain (integral over the distributions) reduction is larger leading to a larger transient cooling of the system. An alternative argument, that both pulses are attracted through mutual phase locking, is supported by the relative phase jump between both pulses plotted in red in Fig. 5(a). There is a significant body of evidence in the literature for soliton attraction through phase locking and that this locking is evident from an approximate $\Delta \phi = \frac {\pi }{2}$ jump in the phase between both pulses. Gagnon et al. [13] argue that mutual attraction will occur if the phase deviates from $\Delta \phi = \frac {\pi }{2}$. Even more compelling experimental evidence [14] on mode-locked, phase-locked systems in ring fiber cavities support that this phase should slightly deviate from this $\frac {\pi }{2}$ especially as pulses maintain closer distances to each other. They demonstrate two pulses within 2.739 ps of each other and measured the phase difference of the pulses to be $\simeq 0.483\pi$, with 1e-3 radian phase measurement accuracy. In this work, the simulated pulses are within 36 fs of each other with a max phase shift measured to be $\simeq 0.462\pi$.

Conclusion. In this work, we first identify a truly novel and highly nontrivial example of a nonlinear dynamical system driven far out of equilibrium illustrated by experimentally demonstrating a two color co-mode-locked VECSEL source. Inverted carriers are almost simultaneously extracted for separate reservoirs. Many-body microscopic Maxwell–semiconductor–Bloch simulations strongly support our experimental results, allowing us to further elaborate on a mechanism for stable dual-wavelength channel mode-locking. It should be evident that this co-mode-locked setting can be further extended to support multiple wavelength channels as long as some uninverted carrier reservoirs remain undepleted.