Numerical simulation of optical Stark effect saturable absorbers in mode-locked femtosecond VECSELs using a modified two-level atom model

The interaction of an optical pulse with a quantum well saturable absorber is simulated using a two level atom model in the limit where pulse duration approaches the carrier-carrier scattering time. For long pulses bleaching dominates the absorber response but as the pulse duration approaches the carrier-carrier scattering timescale an additional pulse shaping mechanism becomes active, allowing the absorber to continue to shorten pulses beyond the limit set by bleaching. Examinations of the spectral and temporal absorption profiles identify the additional pulse shortening mechanism as the optical Stark effect.


Introduction
Optically pumped Vertical-External-Cavity Surface-Emitting Lasers (VECSELs) have produced pulses with durations ranging from 60 fs [1], through the femtosecond range [2][3][4] and up to several picoseconds [5,6] when mode-locked using semiconductor saturable absorber mirrors (SESAMs), but the pulse shaping mechanisms in these lasers are not well understood.The most common techniques for producing sub-picosecond pulses in solid state and fibre lasers cannot be used in VECSELs for various reasons.Low temperature grown or ion implanted semiconductor saturable absorbers (SESAMs) have recovery times fast enough to produce femtosecond pulse durations, but their high insertion losses limit their use in low gain lasers such as VECSELs.The low pulse energies and short material interaction lengths in VECSELs prevent the Kerr effect from producing significant self lensing or self phase shifts.
The SESAMs used in ultrashort pulse VECSELs are typically of the surface recombination type first introduced by Garnache et al. [4] with recovery times of order 10 ps.Quasi-soliton modelocking has been shown to be the dominant mechanism in picosecond VECSELs [5], where the necessary self phase modulation results from the rapid changes in carrier density in the quantum well gain medium rather than the Kerr effect.Soliton modelocking predicts an inverse relationship between pulse energy and pulse duration [8].Femtosecond VECSELs do not show this property [2], implying that another pulse shortening mechanism is dominant at these pulse durations.
The optical Stark effect [10] has been proposed as a candidate for the shaping of these pulses.Here the quantum well absorption spectrum is distorted by the optical pulse to produce self-absorption modulation.As the optical Stark effect is an intensity-dependent phenomenon the absorber response is instantaneous, meaning that this effect can provide an ideal fast saturable absorption mechanism [9].Previous simulations by Daniell [11] and Mihoubi [12] have modelled a quantum well as an ensemble of two-level atoms and have shown that in the absence of bleaching the optical Stark effect can shorten pulses.In this paper we extend this two-level atom model to include bleaching and carrier-carrier scattering effects, and show that the optical Stark effect can indeed provide an intensity-dependent change in absorption which is capable of pulse shaping under realistic VECSEL operating conditions.
The interaction between a semiconductor quantum well and an optical pulse can be described by propagating the pulse through an ensemble of two-level atoms.The populations of the levels can be calculated numerically allowing the polarisation of the medium, and therefore the effect of the medium on the pulse to be found.Models of this type have been used to describe the saturation properties of quantum well based saturable absorbers [13] but, to the best of our knowledge, all such models assume that all the two-level atoms in the ensemble interact equally with the optical field.
This assumption of homogeneity is valid for pulse propagation calculations where the pulse duration is long compared to the carrier-carrier scattering times in the semiconductor.In this case the distortion of the carrier distribution caused by the narrowband optical pulse is compensated for by fast carrier scattering.A Fermi-Dirac distribution is maintained and the medium behaves homogeneously.At shorter pulse durations carrier scattering can no longer act quickly enough to maintain a Fermi-Dirac distribution.In an inverted medium this will result in spectral hole burning in the gain distribution and thus reduced gain for shorter pulses as seen in [1].In a saturable absorber it will result in spectral hole burning in the absorption distribution as the number of carriers available to provide absorption over the laser bandwidth is reduced.In order to model the effect of a saturable absorber on short pulses we must use a model that includes carrier scattering effects.
In this paper we use a modified two-level atom model to calculate the interaction between a quantum well and an optical pulse as the pulse duration approaches the carrier-carrier scattering timescale.The ensemble of two-level atoms is divided into two separate populations; 'live' atoms interact with the optical field whereas 'dead' atoms do not.Atoms are transferred between these populations by carrier-carrier scattering.All the two-level atoms are assumed to have the same transition energy.In this respect the model is incomplete.A Bloch equation based model would reproduce the band structure more realistically, but if we assume that the laser bandwidth is small compared to the spread of carrier energies within the band the two-level atom model is valid.We also ignore any carrier-phonon scattering effects.

Model
We begin by deriving equations for the polarisation and level populations of a standard two-level atom system.The population of live two-level atoms can be described by the wavefunction where describe the occupancy of the ground and excited live levels and are the spatial wavefunctions of the two levels.From the time-dependent Schrödinger equation we can find the evolution of to be governed by The dipole moment of a single two-level atom is defined as Scattering rates between states depend on the ratio of the number of dead states to the number of live states, , as this determines the available number of states into which the carriers can scatter.represents the ratio of the spread of energies in the semiconductor Fermi-Dirac distribution to the bandwidth of the optical pulse.In this paper we assume that this ratio remains constant, which is equivalent to assuming that all pulses have the same optical bandwidth.This assumption is justified on the basis that carrier scattering only affects pulses with shorter durations, meaning that the value of R used will have very limited effects on pulses longer than ~10 .The value of R used throughout this paper is therefore chosen to be representative of typical bandwidths observed in VECSELs producing pulses approaching carrier scattering times [2][3][4].
Figure 1 shows a schematic of the 4 energy levels in the live-dead two-level atom scheme.Scattering rates between levels are such that the ratio, , between populations in the dead and live levels is maintained in the absence of an optical field.Including non-radiative transitions the equations for the evolution of , , and become where electric fields are measured in units of , the electric field required to completely bleach the live population.and are then measured in units of .We also introduce the plasma frequency of the medium defined by and a characteristic electric field, , the saturation field strength of the Stark effect.
Equations 25-34 are solved using a Runge Kutta algorithm to find the polarisation of the medium and therefore the effect on a pulse passing through the medium.Table 1 shows the parameters used for the simulations below.values are chosen to represent the 8 nm thick In 0.25 Ga 0.75 As quantum well absorbers used in [1,3,4].The electron scattering time is chosen to be 100 fs, which is consistent with spectroscopic measurements [14], and the hole scattering time 600 fs, in keeping with the ratio of the electron and hole effective masses.These values are chosen so as to give a polarisation decay time of 170 fs, which sets the small-signal absorption bandwidth to be X nm, as measured in .The quantum well recovery time is taken to be 21 ps and the modulation depth 0.75 %, as in [4].The plasma frequency is difficult to determine directly and is therefore chosen to match the experimentally measured modulation depth.The modulation depth of the simulated absorber can be found by running the model for different input pulse energies at a fixed pulse duration and finding change in pulse energy.Figure 2 shows the transmission of the absorber as a function of input pulse energy for sech squared profile pulses of duration 2 ps and whose centre wavelength is equal to the resonance centre wavelength.The curve is fit using the method in [13], giving a saturation pulse energy of 3.03 E 2  Stark τ 2 and a modulation depth of 0.75 %.The curve is fit using the method in [13], giving a saturation pulse energy of 3.03 E 2 Starkτ2 and a modulation depth of 0.75 %.The parameters used are shown in table 1.

Numerical results and discussion
Figure 3a shows the absorption as a function of time for a pulse with energy whose duration ( ) is significantly longer than the carrier-carrier scattering time.This absorption response is consistent with slow saturable absorber modelocking, where the recovery time of the absorber is long compared to the pulse duration.Figure 2b shows the corresponding curve for a shorter pulse ( ) with the same energy and centre wavelength.The total pulse absorption has decreased from 0.097 % to 0.069 %, and while there is still a slow component to the absorption recovery there is now also a fast component.This fast component is not intensity dependent but responds on the timescale of the polarisation decay time.Rabi oscillations are visible close to zero delay in figure 3b, but they are small and strongly damped and therefore do not have a significant effect on the output pulse envelope.A similar but even weaker feature is visible in figure 3a.
(a) (b) Figure 3. Time-resolved absorption for pulses with energy and durations of (figure 3a) and (figure 3b). Figure 3a shows a slow saturable absorber type absorption profile where the absorption recovery is long compared to the pulse duration.In figure 3b there is still a slow component to the absorption recovery but there is also a fast, nearly-intensity-dependent component.
The mechanism responsible for the fast component in figure 3b can be identified as the optical Stark effect by examining the effect of pulse peak intensity on the absorption spectrum.Absorption spectra are calculated by finding the change in pulse energy as a function of the pulse centre wavelength, λ, relative to the absorber resonance centre wavelength, λ c . Figure 4 shows the change in the absorption spectrum induced by duration sech 2 pulses of different pulse energies.At low excitation the absorption spectrum of the two-level atoms has a Lorentzian shape whose width is set by the inverse of the polarisation decay time, , and with centre wavelength .As the pulse energy, and therefore peak intensity increases the absorption experiences a broadening.This broadening is associated with a drop in the peak amplitude of the resonance as the area under the resonance remains constant.The intensitydependent broadening and decrease in peak intensity is recognisable as the optical Stark effect.The highintensity absorption spectrum peak is slightly blueshifted relative to the low intensity peak, but this effect is small compared to the broadening and is therefore not visible in figure 4.This intensity dependent decrease in absorption is clearly capable of acting as a saturable absorption mechanism.Crucially, the response time of this mechanism is of the order of the polarisation decay time, , which is approximately 150 fs in the InGaAs quantum wells commonly used in ultrafast VECSELs [14].This timescale is short compared to most sub-picosecond VECSEL pulses meaning that the optical Stark effect can act as a close-to-ideal fast saturable absorber.
Figure 5 shows the effect of a single pass through the absorbing medium on sech 2 profile pulses with energy and pulse durations from to .As the pulse energy is constant a decrease in the pulse duration corresponds to an increase in the peak intensity.At long pulse durations the fractional change in pulse energy decreases slowly towards shorter pulse durations due to the reduced recovery as the ratio of pulse duration to recovery time decreases.The change in pulse duration is approximately proportional to the pulse duration over this range and can therefore be attributed to bleaching.) and fractional change in pulse energy ( ) as a function of pulse duration for a sech 2 pulse of energy .Carrier-carrier scattering causes a significant drop in absorption for pulses shorter than .An increase in pulse shortening can be seen over the same range of pulse durations.
Extrapolating a linear fit to this region of the graph gives a minimum pulse duration of that can be achieved by slow saturable absorber bleaching alone.In practice, the steady state pulse duration of the laser will be the pulse duration at which the pulse shortening per round trip is equal to the pulse lengthening due to gain dispersion and group delay dispersion.The steady state pulse duration may therefore be significantly longer than the limit.This highlights the importance of designing VECSEL samples so as to minimise the effects of gain dispersion and group delay dispersion.
As the pulse duration in figure 5 drops below the pulse absorption decreases rapidly due to spectral hole burning.In the same range of pulse durations the pulse shortening can be seen to increase above that expected for slow saturable absorber bleaching.This indicates that an additional pulse shortening mechanism becomes active as the pulse duration decreases.This mechanism becomes stronger as the absorption decreases, showing that the mechanism does not result from absorption.The pulse duration range over which this mechanism is active matches that of the fast absorption component in figure 3, demonstrating that the additional pulse shortening is due to the optical Stark effect.This additional pulse shortening effect will allow the saturable absorber to compete more effectively with the pulse lengthening at short pulse durations, meaning that shorter steady state pulse durations will be achieved.Crucially, as shown in figure 4, the additional pulse shortening results from a fast effect.Given small enough pulse lengthening effects, this allows the SESAM to continue to shorten pulses below the limit from bleaching.

Conclusions
We examine the interaction between an optical pulse and a quantum well saturable absorber by simulating the absorber using a two-level atom model which has been modified to take carrier-carrier scattering into account.In addition to the change in absorption due to bleaching this model predicts a significant nonlinear change in absorption with a response time similar to the polarisation decay time.At pulse durations approaching carrier scattering timescales the pulse shortening increases beyond that due to bleaching, indicating an additional pulse shortening mechanism.An intensity dependent distortion of the absorption spectrum identifies this nonlinearity as the optical Stark effect.
While quasi-soliton modelocking has been identified as the mechanism responsible for the generation of picosecond pulses in VECSELs [5], the dominant mechanism in sub-picosecond VECSELs has not been conclusively identified until mow.Numerical modelling shows that the optical Stark effect provides an additional pulse shortening mechanism which, while negligible at longer pulse durations, becomes dominant as pulse durations approach the carrier-carrier scattering time.The additional pulse shortening allows shorter steady state pulse durations to be reached, while the near-instantaneous response of the effect allows it to shorten pulses beyond the limit due to bleaching alone.
In order to examine this modelocking mechanism, experimental studies of the pulse shortening dynamics in sub-picosecond VECSELs are already underway.A semiconductor Bloch equation based numerical model of a quantum well saturable absorber is also under development.When completed this model will allow a more detailed examination of the optical Stark effect pulse shortening mechanism to be undertaken.

Figure 1 .
Figure 1.Energy level diagram of the live-dead two-level atom system.Populations of the states are indicated by .Nonradiative transition rates between levels are also shown.

Figure 2 .
Figure 2. Transmission of the population of two level atoms as a function of input pulse energy for 2 ps sech squared profile pulses.The curve is fit using the method in[13], giving a saturation pulse energy of 3.03 E 2Starkτ2 and a modulation depth of 0.75 %.The parameters used are shown in table 1.

Figure 4 .
Figure 4. Change in the absorption spectrum of the two-level atom medium caused by sech 2 pulses of duration

Table 1 .
Parameters of the two level atom distribution used to simulate an 8 nm thick In0.25Ga0.75Asquantum well.