Nonequilibrium and thermal effects in mode-locked VECSELs

J.V Moloney; I. Kilen; A. Bäumner; M. Scheller; S.W. Koch

doi:10.1364/OE.22.006422

1. Introduction

Optically-pumped vertical external cavity surface emitting lasers (VECSELs) or disk lasers have been shown to be ideal as wavelength agile, high brightness sources for numerous applications including raw power [1], multi-Watt single frequency [2] and high average power under various mode-locking scenarios. Specifically, mode-locking has been observed with external semiconductor saturable absorber mirrors (SESAM) [3–5], with integrated quantum well and quantum dot SESAMS (MIXSEL) [6], with graphene [6,7] and carbon nanotube [8] saturable absorbers. To our knowledge, the shortest fundamental mode-locked pulse duration to date has been 107 fs albeit at very low average power [9]. While shorter 60fs pulses have been reported [10], these have been harmonically mode-locked transients within longer few picosecond “pulse molecules”. Rate equation level models using parameters extracted from experiment have proved successful in capturing the mode-locking behavior for longer duration pulses [11]. As typical intraband carrier scattering times are on the order of 100 fs, it is expected that dynamically changing nonequilibrium distributions will not have a chance to relax to quasi-equilibrium Fermi-Dirac distributions during the pulse itself. The complexity of the nonequilibrium many-body dynamics has limited studies to date to a relatively simple one of multiple QW single pass geometries.

In this paper, we present what to our knowledge is the first attempt to simulate microscopic many body effects in a simple mode-locked geometry by solving the Maxwell - Semiconductor Bloch equations (MSBE) in the Hartree Fock limit using a 2-band model. Our preliminary finding is that sub 100fs mode-locking is feasible in the low gain limit where all carriers are utilized in the pulse forming stage. In higher gain situations unused carriers can destabilize the pulse by providing unused gain in spectral regions that exist external to the interacting nonequilibrium system.

In the following section we then present a brief assessment of the intriguing preliminary observation of self-Kerr Lens mode-locked pulse trains in VECSELs [12] and extend the discussion to include augmenting the ultrafast Kerr effect with a nonlinear crystal in the VECSEL cavity.

2. Microscopic Theory and Simulation of short-cavity Mode-locking

The theoretical analysis of the VECSEL modelocking dynamics requires the simulation of the light-field $E (z, t)$ propagation inside the laser cavity as well as the nonlinear interaction of this cavity field with the material polarizations $P (z, t)$ of the optically active QW gain region and the QW saturable absorber (SESAM). Assuming that the cavity light-field circulates in $z$ -direction perpendicular to the QW planes, Maxwell's wave equation can be written in the simple form

\begin{array}{l} [\frac{\partial^{2}}{\partial z^{2}} - \frac{n^{2}}{c_{0}^{2}} \frac{\partial^{2}}{\partial t^{2}}] E (z, t) = μ_{0} \frac{\partial^{2}}{\partial t^{2}} P (z, t) \end{array}

where the constants

n

and

c_{0}

are the background refractive index and the vacuum velocity of light and

μ_{0}

denotes the vacuum permeability, respectively. The macroscopic polarization

P (z, t)

is obtained by the summation of the microscopic polarizations

p_{λ, ν, k}

over the crystal momentum

k

in the plane of the QWs. The dynamics of the microscopic polarization

p_{λ, ν, k}

follows from the multiband semiconductor Bloch equations [13] (SBE)

\begin{array}{l} \frac{\partial}{\partial t} p_{λ, ν, k} = - \frac{i}{ℏ} \sum_{λ_{1}, ν_{1}} (e_{λ, λ_{1}, k}^{e} δ_{ν, ν_{1}} + e_{ν, ν_{1}, k}^{h} δ_{λ, λ_{1}}) p_{λ_{1}, ν_{1}, k} - i (n_{λ, k}^{e} + n_{ν, k}^{h} - 1) Ω_{λ, ν, k} + Γ_{λ, ν; deph} \\ \frac{\partial}{\partial t} n_{λ (ν), k}^{e (h)} = - 2 I m {(Ω_{λ, ν, k} (p_{λ, ν, k})}^{⋆}) + Γ_{λ (ν); scatt}^{e (h)} . \end{array}

These equations couple the dynamics of the microscopic polarization

p_{λ, ν, k}

and the occupation functions

n_{λ (ν), k}^{e (h)}

determining the probability that a state

k

in the conduction band

λ

(valence band

ν

) is occupied by an electron (hole). The renormalized single-particle energies in the Hartree-Fock approximation are

e_{λ, λ_{1}, k}^{e} = ε_{λ, k}^{e} δ_{λ, λ_{1}} - \sum_{λ_{2}, q} V_{∣ k - q ∣}^{λ, λ_{2}, λ_{1}, λ_{2}} n_{λ_{2}, q}^{e} e_{ν, ν_{1}, k}^{h} = ε_{ν, k}^{h} δ_{ν, ν_{1}} - \sum_{ν_{2}, q} V_{∣ k - q ∣}^{ν, ν_{2}, ν_{1}, ν_{2}} n_{ν_{2}, q}^{h}

and the renormalized Rabi frequency is

Ω_{λ, ν, k} = ω_{R} + \frac{1}{ℏ} \sum_{λ_{1}, ν_{1}, q \neq k} V_{∣ k - q ∣}^{λ, ν_{1}, ν, λ_{1}} p_{λ_{1}, ν_{1}, q} .

Here,

V (∣ k - q ∣)

is the Coulomb potential and

ω_{R} = d_{k}^{λ, ν} E (z, t) / ℏ

where

d_{k}^{λ, ν}

is the dipole-matrix element.

Contributions to the SBE that go beyond the Hartree-Fock approximation describe correlation effects such as dephasing of the polarization ( $Γ_{deph}$ ), carrier scattering ( $Γ_{scatt}$ ), and Coulomb screening. In the equation of motion for the carrier distributions Eq. (2) the correlation effects in second-Born Markov approximation lead to Boltzmann type scattering terms.

Together with the wave equation, the SBE establish the Maxwell-semiconductor Bloch equations (MSBE). Since the numerical evaluation of the microscopic scattering terms in each time step of a numerical laser-dynamics simulation is excessively CPU-time demanding, it is often necessary to include the correlation effects on the level of effective rates expressing the net effect of the underlying microscopic processes. In this limit, the dephasing of the polarization is described by a simple decay contribution with the characteristic dephasing time $τ_{d e p}$ ; $\begin{array}{l} Γ_{λ, ν; deph} = - p_{λ, ν, k} / τ_{d e p h} \end{array}$ and, similarly, the carrier-scattering contribution modeling the equilibration of the carrier system follows from $\begin{array}{l} Γ_{λ (ν); scatt} = - (n_{λ (ν), k}^{e (h)} - f_{λ (ν), k}^{e (h)}) \end{array} / τ_{s c a t t}$ where $f_{λ (υ), k}^{e (h)}$ is the background quasi-equilibrium distribution of the optically pumped QW and $τ_{scatt}$ governs the characteristic time scales of scattering events. To test our model [14] and to study the relevance of nonequilibrium effects, we restrict ourselves to a two-band model, postulating strong confinement of electrons and holes such that only the lowest subbands need to be taken into account. In addition, we limit the consideration to a parabolic band structure where the transition energy, i.e. the difference between the conduction and valence band energy, $ℏ ω_{k} = \frac{ℏ^{2} k^{2}}{2 m_{e}} + \frac{ℏ^{2} k^{2}}{2 m_{h}} + E_{g}$ is expressible in terms of the effective masses of the electrons, $m_{e}$ , holes, $m_{h}$ , and the band-gap energy $E_{g}$ . As a sanity check we employed a full multi-band microscopic simulation in a single pass setting that goes beyond the Hartree-Fock limit to assess the role of ultrafast correlations in the hot carriers (electron/hole) and carrier capture from the pumped barrier to inverted well states. In the low gain limit studied here, these correlations do not affect the results.

As a first effort we employ a simple linear one-dimensional VECSEL cavity with a 2 ps round-trip time, with an active mirror consisting of a DBR and a resonant-periodic gain medium on the one end and the SESAM attached to a 1% outcoupling mirror on the other end. We assume a 3ps (0.5 ps) relaxation time for the QW (SESAM) populations back to a reference 300 K quasi-equilibrium distribution of density 2.0e16 [1/m²] (5.0e14 [1/m²]) and InGaAs/AlGaAs bandstructure parameters. The gain medium consists of 10 effective 8nm QWs and the bandgap of the single SESAM QW is energetically 10 meV below that of the gain QWs. To simulate the transverse focusing onto the SESAM used in mode-locking experiments, we use a tenfold field enhancement in the SESAM. With this input, we numerically solved the MSBE starting with a very weak initial pulse of 200 fs duration (FWHM). By changing the initial pulse duration and amplitude, we verified that the results are independent of the initialization details.

An example of the numerical results is presented in Fig. 1. The upper left panel shows the reference carrier distribution in the gain QWs together with the nonequilibrium distribution induced by the intra-cavity circulating pulse. We clearly notice the pronounced kinetic hole-burning in the distributions. The corresponding nonequilibrium gain spectrum is shown in the upper right panel of Fig. 1. We clearly see a significant gain reduction and flattening in comparison to the quasi-equilibrium gain spectrum of the background distributions. The lower left panel depicts the SESAM carrier distribution induced by the pulse and the stable converged pulse shape is plotted in the lower right panel, respectively. From the insets, we can see that pulse amplitude and FWHM stabilize approximately 2ns after the initialization. The resulting 75 fs FWHM mode-locked pulse has a somewhat complex shape and is clearly not Fourier-transform limited since we did not implement any attempts of round-trip dispersion management.

Fig. 1 Example for the results obtains by numerical solution of the MSBE for mode-locking operation. The upper left panel shows the momentum resolved reference carrier distributions f_k^e(h) in the gain QWs together with the nonequilibrium distribution n_k^e(h) induced by the intra-cavity circulating pulse. The corresponding nonequilibrium gain spectrum is shown in the upper right panel (dotted line) together with the reference gain spectrum of the background distributions (solid line). The SESAM carrier distribution and the converged stable pulse are plotted in the lower left and right panels, respectively. The insets show the temporal stabilization of the pulse amplitude and the FWHM.

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Whereas the chosen example shows a successfully mode-locked configuration, we have encountered many situations with unstable results where the pulse develops complicated multi-peaked shapes or does not reach a stable configuration. We are in the process of mapping the parameter space to identify the best mode-locking conditions and to explore the limitations on achievable pulse durations and intensities.

3. Assessment of Kerr Lens mode-locking

As pointed out in the introduction, the intrinsic Kerr effect within the VECSEL itself might be employed to achieve mode-locking as discussed in [12]. We conducted similar experiments to study this “self-mode-locking” phenomenon and indeed observe the occurrence of a pulse train by measuring the RF signal with a fast photodiode. However, in our study we noticed that the train fluctuates over a time scale of some microseconds as also observed in [12] which might suggest slow thermal effects or unavoidable fluctuations in the optical pump intensity. In our experimental setup we were unable to detect stable mode-locking operation in this configuration. Furthermore, it is expected that despite the high Kerr coefficient of GaAs at wavelengths around 1 micron, the tiny interaction length within the VECSEL chip will result in a Kerr-lens extremely weak compared to the thermal lens formed by the profile of the pump spot.

We then attempted traditional Kerr-lens mode locking by adding a nonlinear crystal to the cavity – the latter should boost the Kerr nonlinearity inside the cavity to a level comparable with other Kerr-lens mode locked lasers such as titanium sapphire (TiSa) lasers. For this, we utilize an undoped yttrium orthvanadate (YVO₄) crystal which exhibits a nonlinear refractive index three times higher than that of TiSa. We built a cavity as illustrated in Fig. 2 with a VECSEL chip emitting around 1micron acting as gain element and we place the Kerr crystal close to one end mirror. The intensity induced Kerr effect will form a lens within the crystal. An aperture is inserted in the cavity such that diffraction losses occur when the laser operates in pure cw mode. Once pulses build up, the intensity increases and thus, the Kerr lens reduces the mode size at the aperture positions and saturates the losses. The AR coated 10 mm long YVO4 crystal was cut at 45° with respect to the optical axis to additionally act as beam displacer for the extraordinary beam. After passing through the crystal, both polarization components are displaced by one millimeter and the aperture can be used to select the lasing polarization. The YVO₄ crystal induces positive dispersion within the cavity. As compensation we employed two dielectric Gires-Tournois-Interferometer (GTI) mirrors with a combined negative dispersion of 5200 fs² per cavity round-trip. The other cavity elements including the AR-coated VECSEL chip have a relatively flat dispersion at the emission wavelength of 1035nm. Thus, the linear dispersion of the YVO₄ crystal (about 4000 fs²) is slightly overcompensated as is favorable for Kerr-lens mode-locking. In the experiments, we pump the VECSEL with a fiber coupled diode laser emitting around 808nm with up to 20W of pump power. We optimize the cavity for the most stable pulse train and autocorrelation trace. Under optimal conditions, the average output power is 1.5 W and we measure a RF signal as shown in Fig. 2. While the signal shows a pulse train with stability over a time frame of few µs, over longer times (tens of µs) there are significant fluctuations in the trains as also observed in attempts to achieve mode-locking employing the intrinsic Kerr effect in VECSEL [11] and thus, the system is not stably mode locked. This instability is more apparently in the measured the auto-correlation trace as shown in Fig. 3.A short pulse (with FWHM of about 850fs) sits on a high background pedestal indicating a strong quasi-CW component in the emission and thus, no stable mode-locking.

Fig. 2 (a) Schematics of the laser cavity which is formed by a highly reflective (HR) curved mirror, a flat output coupler (OC) with 1% transmission, a flat HR end mirror, the VECSEL chip as well as two dispersion compensation mirrors (DM). As Kerr-medium (KM) a YVO₄ crystal is used. (b) The YVO₄ crystal also provides polarization control: The ordinary laser beam passes the crystal directly while the orthogonal polarization is transversal displaced and is blocked by the aperture (A).

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Fig. 3 RF signal for the emitted laser light for different time windows (10ns, 2µs, 10µs). Measured autocorrelation signal recorded with a background free SHG autocorrelator. A short spike is observed towering above a significant background pedestal.

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4. Summary

The research area of femtosecond pulsed mode-locking in VECSELs is a relatively young field despite the many successes demonstrated to date. Stable pulse mode-locking has been observed using saturable absorbers of various types and in varying VECSEL chip geometries. Most common is an external SESAM consisting of either a single QW or QD layer. The MIXSEL configuration provides the possibility of compact integrated high repetition rate pulsed lasers. The limitation of SESAMS is relatively slow recombination times which makes the recent demonstrations with grapheme and carbon nanotubes particularly interesting. The latter exhibits recombination times on the order of 100fs which is coincidentally the time scale on which microscopic many-body effects are expected to be influential. The challenge in modeling the latter is the extreme computational complexity of the many-body physics including carrier capture between bands including barrier states and intraband correlations involving carrier-carrier and carrier phonon scattering. The first simulation presented in this paper, while modest, is a key first step to assessing the role of nonequilibrium processes in limiting pulse duration or maximal achievable intensity or repetition rate. The next step is to move beyond the Hartree Fock limit and study the influence of ultrafast carrier scattering within individual carrier plasmas.

Kerr lens mode-locking would essentially provide an instantaneous response supplanting any need for saturable absorber solutions. However, KLM in VECSELs presents its own set of challenges due to the small loss modulation and strong sensitivity to the laser peak intensity. While our results can only be indicative of longer term instability, this behavior is likely to persist unless ways can be found to deal with noise sources such as thermal fluctuations within the VECSEL chip itself or possibly in the pump optics.

5. Conclusion

In conclusion, we have carried out the first microscopic simulation of mode-locking in a simple VECSEL cavity. Our results indicate that clean sub 100fs pulses can be generated in the low gain limit. At higher gain and consequently higher peak intensities, we have preliminary evidence that unused carriers can destabilize the pulse waveform. In parallel, we revisited the possibility of achieving KLM in such VECSEL cavities. Our experience is that the intrinsic and probably weak Kerr lens induced in the VECSEL chip causes long time instabilities of the pulse wavetrains even in the presence of a passive nonlinear crystal in the cavity.

Acknowledgment

We gratefully acknowledge funding support from the U.S Air Force Office for Scientific Research under BRI grant FA9550-14-1-0062 and from the German Science Foundation (DFG: GRK 1782). We also acknowledge fruitful discussions with Alexander Albrecht and Mansoor Sheik-Bahae at the University of New Mexico.

References and links

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Nonequilibrium and thermal effects in mode-locked VECSELs

Abstract

1. Introduction

2. Microscopic Theory and Simulation of short-cavity Mode-locking

3. Assessment of Kerr Lens mode-locking

4. Summary

5. Conclusion

Acknowledgment

References and links

Cited By

Figures (3)

Equations (4)

Optics Express