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

doi:10.1364/OE.19.026783

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

A.H. Quarterman, S. Carswell, G.J. Daniell, Z. Mihoubi, K.G. Wilcox, A.L. Chung, V Apostolopoulos, and A.C. Tropper »View Author Affiliations

Optics Express, Vol. 19, Issue 27, pp. 26783-26795 (2011)
http://dx.doi.org/10.1364/OE.19.026783

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▸ Article Outline
– Abstract
1. Introduction
2. Model
3. Numerical results and discussion
4. Conclusions
– References and links

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Abstract

The interaction of an optical pulse with a quantum well saturable absorber is simulated using a semi-classical two-level-atom model which has been modified to approximate spectral hole burning in the carrier distribution. Saturable absorption behaviour is examined 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. Examination of the spectral and temporal absorption profiles suggests that intense pulses experience additional pulse shortening from the optical Stark effect.

1. Introduction

Optically pumped Vertical-External-Cavity Surface-Emitting Lasers (VECSELs) have produced pulses with durations ranging from 60 fs [1

A. H. Quarterman, K. G. Wilcox, V. Apostolopoulos, Z. Mihoubi, S. P. Elsmere, I. Farrer, D. A. Ritchie, and A. Tropper, “A passively mode-locked external-cavity semiconductor laser emitting 60-fs pulses,” Nat. Photonics 3(12), 729–731 (2009). [CrossRef]

], through the femtosecond range [2

P. Klopp, U. Griebner, M. Zorn, and M. Weyers, “Pulse repetition rate up to 92 GHz or pulse duration shorter than 110 fs from a mode-locked semiconductor disk laser,” Appl. Phys. Lett. 98(7), 071103 (2011). [CrossRef]

–6

K. G. Wilcox, A. H. Quarterman, H. Beere, D. A. Ritchie, and A. C. Tropper, “High peak power femtosecond pulse passively mode-locked vertical-external-cavity surface-emitting laser,” IEEE Photon. Technol. Lett. 22(14), 1021–1023 (2010). [CrossRef]

] and up to several picoseconds [7

O. D. Sieber, M. Hoffmann, D. J. Maas, V. J. Wittwer, M. Golling, T. Sudmeyer, and U. Keller, “Experimental confirmation of quasi-soliton pulse formation in ultrafast VECSELs,” in Advanced Solid State Photonics, OSA Technical Digest Series (CD) (Optical Society of America, 2010), paper AMB26.

–9

M. Hoffmann, O. D. Sieber, D. J. H. C. Maas, V. J. Wittwer, M. Golling, T. Südmeyer, and U. Keller, “Experimental verification of soliton-like pulse-shaping mechanisms in passively mode-locked VECSELs,” Opt. Express 18(10), 10143–10153 (2010). [CrossRef] [PubMed]

] when mode-locked using semiconductor saturable absorber mirrors (SESAMs), but the pulse shaping mechanisms at femtosecond pulse durations 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 SESAMs have recovery times fast enough to produce femtosecond pulses, 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. [5

A. Garnache, S. Hoogland, A. C. Tropper, I. Sagnes, G. Saint-Girons, and J. S. Roberts, “Sub-500-fs soliton-like pulse in a passively mode-locked broadband surface-emitting laser with 100 mW average power,” Appl. Phys. Lett. 80(21), 3892–3894 (2002). [CrossRef]

] with recovery times of order 10 ps. Quasi-soliton modelocking has been shown to be the dominant mechanism in picosecond VECSELs [7

, 9

], 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 [10

H. A. Haus and M. N. Islam, “Theory of the soliton laser,” IEEE J. Quantum Electron. 21(8), 1172–1188 (1985). [CrossRef]

]. Femtosecond VECSELs do not show this property [3

P. Klopp, U. Griebner, M. Zorn, A. Klehr, A. Liero, M. Weyers, and G. Erbert, “Mode-locked InGaAs-AlGaAs disk laser generating sub-200-fs pulses, pulse picking and amplification by a tapered diode amplifier,” Opt. Express 17(13), 10820–10834 (2009). [CrossRef] [PubMed]

], implying that another pulse shortening mechanism is dominant at these pulse durations.

The optical Stark effect [11

S. Tsuda, W. H. Knox, S. T. Cundiff, W. Y. Jan, and J. E. Cunningham, “Mode-locking ultrafast solid-state lasers with saturable Bragg reflectors,” IEEE J. Sel. Top. Quantum Electron. 2(3), 454–464 (1996). [CrossRef]

] 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. Since the optical Stark effect is an intensity-dependent phenomenon the absorber response is quasi-instantaneous, so that this effect can provide an ideal fast saturable absorption mechanism [12

H. A. Haus, “Theory of mode-locking with a fast saturable absorber,” J. Appl. Phys. 46(7), 3049–3058 (1975). [CrossRef]

]. Previous simulations by Daniell [13

G. J. Daniell, Z. Mihoubi, K. G. Wilcox, and A. C. Tropper, “Numerical model of the optical Stark effect as a mode-locking mechanism for femtosecond vertical-external-cavity surface-emitting semiconductor lasers,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonics Applications Systems Technologies. OSA Technical Digest (CD)(Optical Society of America, 2008), paper CThF3.

] and Mihoubi [14

Z. Mihoubi, G. J. Daniell, K. G. Wilcox, and A. C. Tropper, “Numerical model of a vertical-external-cavity surface-emitting semiconductor lasers mode-locked by the optical Stark effect,” in Proceedings of the 8th International Conference on Numerical Simulation of Optoelectronic Devices. (IEEE, 2008), pp. 91–92.

] 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 at a similar rate to that observed in femtosecond mode-locked VECSELs. 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.

Various approaches have been used to model the action of a SESAM in a passively mode-locked laser. Haiml et al use population rate equations, neglecting relaxation processes, to represent absorption saturation in a slow saturable absorber [15

M. Haiml, R. Grange, and U. Keller, “Optical characterization of semiconductor saturable absorbers,” Appl. Phys. B 79(3), 331–339 (2004). [CrossRef]

]. A more rigorous approach would in principle be based on microscopic semiconductor Bloch equations, correctly describing both the effect of an intense coherent optical field, and many-body interactions [16

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 4th edition (World Scientific, 2004).

]. It has been shown that rigorous microscopic models can predict the lasing behaviour of optically pumped semiconductor quantum well gain media to a high degree of accuracy [17

J. V. Moloney, J. Hader, and S. W. Koch, “Quantum design of semiconductor active materials: laser and amplifier applications,” Laser Photonics Rev. 1(1), 24–43 (2007). [CrossRef]

]; however this work has not to date been applied to saturable absorbers, in part due to the computational scale of the task. Balle has shown that, for interaction with monochromatic fields, there are analytical approximations that considerably reduce the computational cost of microscopic models [18

S. Balle, “Simple analytical approximations for the gain and refractive index spectra in quantum well lasers,” Phys. Rev. A 57(2), 1304–1312 (1998). [CrossRef]

, 19

S. Balle, “Analytical description of spectral hole-burning effects in active semiconductors,” Opt. Lett. 27(21), 1923–1925 (2002). [CrossRef] [PubMed]

]. This approach has further been developed into a quasi-equilibrium time-domain model, simulating frequency-selective saturable absorption and pulse distortion in a quantum well [20

J. Javaloyes and S. Balle, “Quasi-equilibrium time-domain susceptibility of semiconductor quantum wells,” Phys. Rev. A 81(6), 062505 (2010). [CrossRef]

]. These promising approaches have not to date been applied to a study of the mechanisms of SESAM mode-locking. Inhomogeneous Bloch Eq. (-)based models have also been used to describe SESAM behaviour [21

V. P. Kalosha, M. Muller, and J. Herrmann, “Theory of solid-state laser mode locking by coherent semiconductor quantum-well absorbers,” J. Opt. Soc. Am. B 16(2), 323 (1999). [CrossRef]

], but only in the regime where the pulse duration is shorter than the carrier scattering time and coherent effects dominate.

Our aim in this paper is to introduce the simplest possible SESAM model that combines a two-level atom description of the light-matter interaction with the existence of a reservoir of dark states, coupled to the optically interactive states by phonon-carrier and carrier-carrier scattering. Of key relevance is the demonstration by Fan et al. of a dual wavelength quantum well VECSEL [22

L. Fan, M. Fallahi, J. Hader, A. R. Zakharian, J. V. Moloney, W. Stolz, S. W. Koch, R. Bedford, and J. T. Murray, “Linearly polarised dual-wavelength vertical-external-cavity surface-emitting laser,” Appl. Phys. Lett. 90(18), 181124 (2007). [CrossRef]

], and the analysis of its behaviour using a full microscopic model [23

A. Bäumner, S. W. Koch, and J. V. Moloney, “Non-equilibrium analysis of the two-color operation in semiconductor quantum-well lasers,” Phys. Status Solidi, B Basic Res. 248(4), 843–846 (2011). [CrossRef]

]. The authors of [23

] conclude that lasing at a particular wavelength depletes the inverted carrier distribution only at wavevectors which correspond to the laser transition energy. The inversion at other wavevectors is not directly depleted by lasing: carriers in these states populate a reservoir from which the lasing states can be refilled via scattering processes. We propose here that the interaction of a quantum well SESAM with intense femtosecond optical pulses similarly bleaches the carrier distribution in a region of wavevector space determined by the optical wavelength. This local bleaching, or wavevector holeburning, is counteracted by scattering processes, allowing dark states elsewhere in k-space to modify the optical response of the SESAM.

To investigate the behaviour of a SESAM with wavevector holeburning we represent the electron and hole states of the quantum well using two distinct populations of two-level atoms, labelled as ‘live’ and ‘dead’. A ‘live’ atom represents a coherent superposition of electron and hole states at the same wavevector, interacting resonantly with the optical field of the laser. A ‘dead’ atom represents a dark state, in which electron and hole are not optically coupled because they do not satisfy the wavevector selection rule. Atoms are transferred between these populations by carrier-carrier and carrier-phonon scattering processes. The model thus takes into account the presence of large numbers of electron and hole states which are not available for optical interactions, without incurring the complexity of a full band structure calculation. With this computationally efficient model we are able to simulate the optical response of the SESAM, and its effectiveness as a mode-locking element, over a wide range of intracavity pulse duration and intensity. The predictions of the model are found to be quantitatively consistent with the pulse shortening observed experimentally in particular quantum-well SESAM-mode-locked VECSELs.

2. Model

We begin by reproducing for completeness the derivation of polarisation and level populations of an ensemble of two-level atoms [24

W. W. Chow, S. W. Koch, and M. Sargent, Semiconductor-Laser Physics (Springer-Verlag, 1994).

]. The general state of a single live two-level atom can be described by the wavefunction

Φ (r, t) = c_{1} (t) ϕ_{1} (r) \exp (- i ω_{1} t) + c_{2} (t) ϕ_{2} (r) \exp (- i ω_{2} t),

(1)

where

{| c_{1, 2} |}^{2}

describe the probability of the atom occupying the ground and excited live levels and

ϕ_{1, 2}

are the spatial eigenfunctions of the two levels. In the case where the population interacts with an electric field, E(t), the time-dependent Schrödinger equation can be used to show that the evolution of

c_{1, 2}

are governed by

{\dot{c}}_{1} = i \frac{μ}{ℏ} E (t) \exp (- i ω t) c_{2},

(2)

{\dot{c}}_{2} = i \frac{μ}{ℏ} E (t) \exp (i ω t) c_{1},

(3)

where

ω = ω_{2} - ω_{1}

. The expectation value of the dipole moment operator, (q_ez) where q_e is the electronic charge, for an atom in the superposition state described by Eq. (1) is given by

\int Φ^{*} (q_{e} z) Φ d^{3} r = μ [c_{1}^{*} c_{2} \exp (- i ω t) + c_{1} c_{2}^{*} \exp (i ω t)] = p_{1} + p_{2},

(4)

where

p_{1} = μ c_{1}^{*} c_{2} \exp (- i ω t)

corresponds to a complex representation of the polarisation of the atom, and

p_{2} = p^{*}_{1}

. The dipole moment, μ, is given by

μ = \int ϕ_{1}^{*} (q_{e} z) ϕ_{2} d^{3} r .

(5)

where

ϕ_{1}

and

ϕ_{2}

are chosen such that

μ

is real. Differentiating Eq. (4) and substituting Eqs. (2) and 3 we find

{\dot{p}}_{1} = - i ω p_{1} - i \frac{μ^{2} E (t)}{ℏ} (N_{2} - N_{1}),

(6)

{\dot{p}}_{2} = i ω p_{2} + i \frac{μ^{2} E (t)}{ℏ} (N_{2} - N_{1}),

(7)

where the probabilities of the atom occupying the ground or excited live levels, N₁ and N₂, are defined by

N_{1} = c_{1}^{*} c_{1},

(8)

N_{2} = c_{2}^{*} c_{2} .

(9)

The polarisation of a medium with

ρ

identical two-level atoms is

P = ρ (p_{2} + p_{1})

, and it is convenient to introduce the quadrature component of the polarisation, defined by

i Q = ρ (p_{2} - p_{1})

. Equations for the evolution of

P

and

Q

can be derived using Eqs. (6) and 7:

\dot{P} = - ω Q,

(10)

\dot{Q} = ω P + \frac{2 ρ μ^{2}}{ℏ} E (t) (N_{2} - N_{1}),

(11)

where

N_{1}

and

N_{2}

now represent the fractions of the population in the ground and excited levels. Similarly, by differentiating Eqs. (8) and 9 and substituting Eqs. (2) and 3 expressions for the evolutions of

N_{1}

and

N_{2}

can be shown to be

{\dot{N}}_{1} = \frac{E Q}{ℏ ρ},

(12)

{\dot{N}}_{2} = - \frac{E Q}{ℏ ρ} .

(13)

These equations must be modified to include recombination and carrier scattering effects. Interband decays have a characteristic timescale

τ_{1}

. Both

P

and

Q

decay at the polarisation decay time

τ_{P}

[24

W. W. Chow, S. W. Koch, and M. Sargent, Semiconductor-Laser Physics (Springer-Verlag, 1994).

] which is given by the expression

\frac{1}{τ_{p}} = \frac{1}{2} (\frac{1}{τ_{2}} + \frac{1}{τ_{3}}),

(14)

where

τ_{2}

and

τ_{3}

are the scattering times of electrons and holes respectively. Transfer of atoms between live and dead levels also occurs with a timescale

τ_{p}

, as the same scattering processes are responsible for the transfer of carriers to different momentum states in the quantum well as for the decay of the polarisation. In a semiconductor, electrons and holes will have different scattering times due to their different effective masses. This detail is ignored in this paper, an assumption that is justified by noting that there will only be a significant difference for pulse durations below the polarisation decay time, a regime which is not examined in this paper.

The rate of scattering from live to dead states, or vice-versa, is proportional to the number of final states available in the process. It is therefore necessary to introduce a further parameter into the model, the ratio R of the number of ‘dead’ states to the number of ‘live’ states. This clearly represents a drastic and somewhat unphysical simplification of the system. Fortunately, however, we find that the simulation is not sensitive to the magnitude of this ratio, with results that vary little for values of R in the range 3 - 10. We therefore adopt the following approach, remembering that in a 2-D quantum well the density of electronic states is energy independent, and the total number of available states is proportional to the range of energy that they span. We take the overall bandwidth of the absorbing resonance (7.5 THz, representative of the SESAMs used in [1

K. G. Wilcox, Z. Mihoubi, G. J. Daniell, S. Elsmere, A. Quarterman, I. Farrer, D. A. Ritchie, and A. Tropper, “Ultrafast optical Stark mode-locked semiconductor laser,” Opt. Lett. 33(23), 2797–2799 (2008). [CrossRef] [PubMed]

–6

]) as a measure of the total number of available electronic states, and the FWHM bandwidth of the optical pulse as a measure of the live states. Our estimate for R is then the ratio of these two bandwidths. R is therefore not a constant parameter of the model, but varies with pulse duration.

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 populations in the dead and live levels remain in equilibrium in the absence of an optical field, and the factors of

R

in the interband decay rates between levels reflect the number of states making up both the initial and final levels of the transition. Including non-radiative transitions the equations for the evolution of

P

Q

N_{1}

and

N_{2}

become

\dot{P} = - ω Q - P / τ_{p},

(15)

\dot{Q} = ω P + \frac{2 ρ μ^{2}}{ℏ} E (t) (N_{2} - N_{1}) - Q / τ_{p},

(16)

{\dot{N}}_{1} = \frac{E Q}{ℏ ρ} - \frac{R N_{1}}{τ_{p}} + \frac{N_{2}}{τ_{1}} + \frac{N_{3}}{τ_{p}} + \frac{R N_{4}}{τ_{1}},

(17)

{\dot{N}}_{2} = - \frac{E Q}{ℏ ρ} - (\frac{1}{τ_{1}} + \frac{R}{τ_{1}} + \frac{R}{τ_{p}}) N_{2} + \frac{N_{4}}{τ_{p}},

(18)

and we must add two new equations for the evolution of the populations in the dead ground and excited levels,

N_{3}

and

N_{4}

;

Fig. 1 Energy level diagram of the live-dead two-level atom system. Populations of the levels are indicated by

N_{1 - 4}

. Non-radiative transition rates between levels are also shown.

{\dot{N}}_{3} = \frac{R N_{1}}{τ_{p}} + \frac{R N_{2}}{τ_{1}} - \frac{N_{3}}{τ_{p}} + \frac{R^{2} N_{4}}{τ_{1}},

(19)

{\dot{N}}_{4} = - (\frac{1}{τ_{p}} + \frac{R}{τ_{1}} + \frac{R^{2}}{τ_{1}}) N_{4} + \frac{R N_{2}}{τ_{p}} .

(20)

These are general equations which are valid for any electric field. An optical pulse can be written as

E = \frac{1}{2} [e \exp (i Ω t) + e^{*} \exp (- i Ω t)],

(21)

where

e

is a slowly varying envelope and

Ω

the carrier frequency.

P

and

Q

are driven to oscillate at the carrier frequency but the populations are driven to oscillate at twice the carrier frequency as a result of the EQ terms in Eqs. (17) and 18, and may also include a non-oscillatory component. We therefore have

P = \frac{1}{2} [p \exp (i Ω t) + p^{*} \exp (- i Ω t)],

(22)

Q = \frac{1}{2} [q \exp (i Ω t) + q^{*} \exp (i Ω t)],

(23)

and for any of the four populations,

N = \bar{n} + \frac{1}{2} [n \exp (2 i Ω t) + n^{*} \exp (- 2 i Ω t)],

(24)

where

\bar{n}

represents the non-oscillatory component and

n

the oscillatory component. Substituting Eqs. (21) to 24 into Eqs. (15) to 20 and equating oscillatory and non-oscillatory terms we can derive equations for the evolution of the envelopes:

\dot{p} + i Ω p = - ω q - p / τ_{p},

(25)

\dot{q} + i Ω q = ω p + \frac{ω_{p}^{2}}{ω} [e ({\bar{n}}_{2} - {\bar{n}}_{1}) + \frac{e^{*}}{2} (n_{2} - n_{1})] - q / τ_{p},

(26)

{\dot{\bar{n}}}_{1} = ω (e q^{*} + e^{*} q) - \frac{R {\bar{n}}_{1}}{τ_{p}} + \frac{{\bar{n}}_{2}}{τ_{1}} + \frac{{\bar{n}}_{3}}{τ_{p}} + \frac{R {\bar{n}}_{4}}{τ_{1}},

(27)

{\dot{n}}_{1} + 2 i Ω n_{1} = 2 ω e q - \frac{R n_{1}}{τ_{p}} + \frac{n_{2}}{τ_{1}} + \frac{n_{3}}{τ_{p}} + \frac{R n_{4}}{τ_{1}},

(28)

{\dot{\bar{n}}}_{2} = - ω (e q^{*} + e^{*} q) - (\frac{1}{τ_{1}} + \frac{R}{τ_{1}} + \frac{R}{τ_{p}}) {\bar{n}}_{2} + \frac{{\bar{n}}_{4}}{τ_{p}},

(29)

{\dot{n}}_{2} + 2 i Ω n_{2} = - 2 ω e q - (\frac{1}{τ_{1}} + \frac{R}{τ_{1}} + \frac{R}{τ_{p}}) n_{2} + \frac{n_{4}}{τ_{p}},

(30)

{\dot{\bar{n}}}_{3} = \frac{R {\bar{n}}_{1}}{τ_{p}} + \frac{R {\bar{n}}_{2}}{τ_{1}} - \frac{{\bar{n}}_{3}}{τ_{p}} + \frac{R^{2} {\bar{n}}_{4}}{τ_{1}},

(31)

{\dot{n}}_{3} + 2 i Ω n_{3} = \frac{R n_{1}}{τ_{p}} + \frac{R n_{2}}{τ_{1}} - \frac{n_{3}}{τ_{p}} + \frac{R^{2} n_{4}}{τ_{1}},

(32)

{\dot{\bar{n}}}_{4} = - (\frac{1}{τ_{p}} + \frac{R}{τ_{1}} + \frac{R^{2}}{τ_{1}}) {\bar{n}}_{4} + \frac{R {\bar{n}}_{2}}{τ_{p}},

(33)

{\dot{n}}_{4} + 2 i Ω n_{4} = - (\frac{1}{τ_{p}} + \frac{R}{τ_{1}} + \frac{R^{2}}{τ_{1}}) n_{4} + \frac{R n_{2}}{τ_{p}},

(34)

where electric fields are measured in units of

E_{n} = {(2 ρ ℏ ω / ϵ_{0})}^{1 / 2}

, defined by equating the energy stored in the electric field,

\frac{1}{2} ϵ_{0} E_{n}^{2}

, to the energy required to completely bleach the two-level atom ensemble,

ρ ℏ ω

p

and

q

are then measured in units of

P_{n} = ϵ_{0} E_{n}

. We also introduce the plasma frequency of the medium defined by

ω_{p}^{2} = 2 ρ ω μ^{2} / ℏ ϵ_{0}

and a characteristic electric field,

E_{S t a r k} = ℏ ω / 2 μ

, 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. These values are chosen to represent the 8 nm thick In_0.25Ga_0.75As quantum well absorbers used in [1

, 4

–6

]. The polarisation decay time is chosen to be 150 fs, which is consistent with spectroscopic measurements of similar quantum wells [25

W. H. Knox, C. Hirlimann, D. A. B. Miller, J. Shah, D. S. Chemla, and C. V. Shank, “Femtosecond excitation of nonthermal carrier populations in GaAs quantum wells,” Phys. Rev. Lett. 56(11), 1191–1193 (1986). [CrossRef] [PubMed]

]. The quantum well recovery time is taken to be 21 ps, as in [5

Table 1 Parameters of the two level atom distribution used to simulate an 8 nm thick In_0.25Ga_0.75As quantum well.

Polarisation decay time	$τ_{p}$	150 fs
Plasma frequency	_{$ω_{p}$}	2.099 $τ_{p}^{- 1}$
Interband recovery time	$τ_{1}$	140 $τ_{p}^{}$

The plasma frequency is difficult to determine directly and is therefore chosen to give the appropriate 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 the change in output pulse energy. Figure 2 shows the transmission of the absorber, calculated from the change in pulse energy, as a function of input pulse energy for sech squared profile pulses of duration 1.5 ps (

10 τ_{p}

) and whose centre wavelength is equal to the resonance centre wavelength. The curve is fit using the method in [15

M. Haiml, R. Grange, and U. Keller, “Optical characterization of semiconductor saturable absorbers,” Appl. Phys. B 79(3), 331–339 (2004). [CrossRef]

], giving a saturation pulse fluence of 0.447 E²_Starkτ_pand a modulation depth of 0.75% for the value of the plasma frequency stated in Table 1. The top x-axis in Fig. 2 is calculated by equating the saturation fluence of the two-level atom ensemble to the estimated saturation fluence of the SESAM used in [5

], and provides an approximate reference point for comparisons between the model and experiment.

Fig. 2 Transmission through the population of two level atoms as a function of input pulse fluence for 1.5 ps sech squared profile pulses. The curve is fit using the method in [15

M. Haiml, R. Grange, and U. Keller, “Optical characterization of semiconductor saturable absorbers,” Appl. Phys. B 79(3), 331–339 (2004). [CrossRef]

], giving a saturation fluence of 0.447 E²_Starkτ_p and a modulation depth of 0.75%. The parameters used are shown in Table 1. The top x-axis is based on the value of saturation fluence in [5

3. Numerical results and discussion

Figure 3 shows the time varying small-signal absorption of the two-level atom ensemble as a pulse passes through the medium. This is the absorption experienced by a small test field propagating through the two level atom ensemble in conjunction with the main optical pulse, and is a numerical analogue of a pump-probe measurement. In Fig. 3a the incident pulse has a duration of 10 times the polarisation decay time, and in Fig. 3b the pulse duration is equal to the polarisation decay time. The VECSELs in [4

] typically operate with close to 15 times the saturation fluence on the SESAM. We therefore use a fluence of

6.66 E_{S t a r k}^{2} τ_{p}

in both cases. The response in Fig. 3a is consistent with that of a slow saturable absorber, where the recovery time of the absorber is long compared to the pulse duration. Figure 3b shows the corresponding curve for a shorter pulse (duration

τ_{p}

) with the same fluence and centre wavelength. The total pulse energy absorption has decreased from 0.051% to 0.030%, and while there is still a slow component to the absorption recovery there is now also a fast component. This fast component does not follow the pulse profile exactly, as would be expected for a fast saturable absorber, but responds on the timescale of the polarisation decay time.

Fig. 3 Time-resolved absorption for pulses with durations of

10 τ_{p}

(Fig. 3a) and

τ_{p}

(Fig. 3b) and fluence

6.66 E_{S t a r k}^{2} τ_{p}

. Figure 3a shows a slow saturable absorber type absorption profile where the absorption recovery is long compared to the pulse duration. In Fig. 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 Fig. 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 relative to the absorber resonance centre wavelength, λ_c. Figure 4 shows the change in the absorption spectrum induced by

τ_{p}

duration sech² pulses of different fluences. 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,

τ_{p}

, and with centre wavelength

λ_{c}

. As the pulse fluence, and therefore peak intensity, increases the absorption experiences a broadening relative to the low fluence absorption resonance. The red curve in Fig. 4 shows the absorption spectrum at a pulse fluence of 6.66 E²_Starkτ_p. The intensity-dependent broadening and decrease in peak intensity is recognisable as the optical Stark effect [26

W. H. Knox, D. S. Chemla, D. A. B. Miller, J. B. Stark, and S. Schmitt-Rink, “Femtosecond ac Stark effect in semiconductor quantum wells: Extreme low- and high-intensity limits,” Phys. Rev. Lett. 62(10), 1189–1192 (1989). [CrossRef] [PubMed]

]. The high-intensity absorption spectrum peak is slightly blueshifted relative to the low intensity peak, but this effect is small compared to the broadening.

Fig. 4 Calculated absorption spectra of the two-level atom medium when interacting with sech² pulses of duration

τ_{p}

and with different peak intensities. As the peak intensity increases the optical Stark effect alters the resonance profile, broadening it and reducing its peak amplitude.

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,

τ_{p}

, which is approximately 150 fs in the InGaAs quantum wells commonly used in ultrafast VECSELs [25

]. 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. A fluence of

6.66 E_{S t a r k}^{2} τ_{p}

is close to the experimental values in the 260-fs and 500-fs pulse VECSELs in [4

] and [5

], but is significantly higher than the fluence used in the demonstration 60 fs pulses in [1

]. It is therefore likely that the optical Stark effect is not the dominant mechanism in determining the steady state behaviour of this shorter pulse VECSEL, though it may still affect the evolution to steady state.

Figure 5 shows the effect of a single pass through the absorbing medium on sech² profile pulses with fluence

6.66 E_{S t a r k}^{2} τ_{p}

and pulse durations from

τ_{p}

20 τ_{p}

. 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 of the absorber 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 [27

H. A. Haus, “Theory of mode-locking with a slow saturable absorber,” IEEE J. Quantum Electron. 11(9), 736–746 (1975). [CrossRef]

Fig. 5 Pulse shortening and change in pulse energy per single pass of the absorber as a function of pulse duration for a sech² pulse with fluence

6.66 E_{S t a r k}^{2} τ_{p}

. Spectral hole burning causes a significant drop in absorption for pulses shorter than

5 τ_{p}

. An increase in pulse shortening can be seen over the same range of pulse durations.

The x-intercept of a linear fit to this region of the graph gives the pulse duration at which bleaching ceases to shorten pulses, and therefore the shortest possible pulse that can be achieved by bleaching alone. For the parameters used in this model this pulse duration is 450 fs. 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 of a real VECSEL may therefore be significantly longer than this 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 Fig. 5 drops below

5 τ_{p}

the change in pulse energy 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 effect 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 is found to match that of the fast absorption component in Fig. 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 Fig. 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 450 fs limit from bleaching.

This point is illustrated in Fig. 6 . Here, the simulated pulse shortening per round trip due to the saturable absorber, and the calculated pulse lengthening per round trip due to gain filtering are shown for the experimentally measured parameters of the lasers described in [1

]. Pulse shortening is calculated numerically, using the model above, for absorbers with two different modulation depths. Pulse lengthening due to gain filtering is calculated for two different gain filter bandwidths. The bandwidths used were those of the gain samples used in [1

] and [4

], and measured using a spectrotemporal method, as described in [28

M. E. Barnes, Z. Mihoubi, K. G. Wilcox, A. H. Quarterman, I. Farrer, D. A. Ritchie, A. Garnache, S. Hoogland, V. Apostolopoulos, and A. C. Tropper, “Gain bandwidth characterisation of surface-emitting quantum well laser gain structures for femtosecond operation,” Opt. Express 18(20), 21330–21341 (2010). [CrossRef]

]. As gain filtering is the dominant pulse lengthening mechanism in femtosecond VECSELs, the steady-state pulse duration is set by the crossing point of the curves.

Fig. 6 Pulse shortening per round trip due to simulated saturable absorbers with 0.75% and 0.3% modulation depths, and pulse lengthening per round trip due to gain filters with 37 nm and 51 nm effective bandwidths. Intersections between the curves represent steady state pulse durations.

The gain sample and SESAM used to generate 500 fs pulses in [5

] are represented by the black and green curves. Their intersection occurs at a pulse duration of 550 fs, indicating good agreement between simulation and experiment. Likewise, for the samples used in [4

] (black and red curves) a pulse duration of 320 fs is predicted, while the experimentally measured pulse duration was 260 fs. This provides strong evidence that the optical Stark effect and the effect of gain filtering are the dominant pulse shaping mechanisms in these lasers. The experimentally measured 60 fs pulses in [1

], do not agree with the prediction of 200 fs (blue and red curves), indicating that additional pulse shaping mechanisms must be present. In this laser, gain saturation was observed to become stronger as a result of spectral hole burning as the pulse duration decreased. This effect reduces the gain available to the trailing edge of a pulse, and can therefore produce additional pulse shortening.

4. 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 that an additional pulse shortening mechanism has become active. An intensity dependent distortion of the absorption spectrum identifies this nonlinearity as the optical Stark effect. The performance and operating range of the modelled effects are consistent with experimental results from femtosecond VECSELs [4

], implying that the optical Stark effect is responsible for the shaping of these pulses.

While quasi-soliton modelocking has been identified as the mechanism responsible for the generation of picosecond pulses in VECSELs [7

], a dominant mechanism in sub-picosecond VECSELs has not been conclusively identified. 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 polarisation decay time. In VECSELs with large pulse energies, 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.

The intensity-dependent nature of this effect means that only VECSELs with high fluences on the saturable absorber will experience optical Stark modelocking. Certain femtosecond VECSELs, such as those described in [4

–6

], operate with these high SESAM fluences, and are therefore likely to be dominated by the optical Stark effect. Femtosecond pulse generation has also been observed in VECSELs with much lower SESAM fluences [1

–3

]. While the optical Stark effect may play some role in these lasers the low SESAM fluences mean that it is unlikely to be the dominant pulse shaping mechanism. It has been shown that spectral hole burning can play a major role in pulse shaping in these lasers [1

], but other effects, such as rapid carrier thermalisation in the SESAM, have also been suggested [3

In order to examine these modelocking mechanisms further, 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.

Acknowledgments

The authors acknowledge funding from the Engineering and Physical Sciences Research Council of the UK (EPSRC). Adrian Quarterman also wishes to acknowledge funding provided by the EPSRC’s PhD Plus Fellowship programme.

References and links

1.	A. H. Quarterman, K. G. Wilcox, V. Apostolopoulos, Z. Mihoubi, S. P. Elsmere, I. Farrer, D. A. Ritchie, and A. Tropper, “A passively mode-locked external-cavity semiconductor laser emitting 60-fs pulses,” Nat. Photonics 3(12), 729–731 (2009). [CrossRef]
2.	P. Klopp, U. Griebner, M. Zorn, and M. Weyers, “Pulse repetition rate up to 92 GHz or pulse duration shorter than 110 fs from a mode-locked semiconductor disk laser,” Appl. Phys. Lett. 98(7), 071103 (2011). [CrossRef]
3.	P. Klopp, U. Griebner, M. Zorn, A. Klehr, A. Liero, M. Weyers, and G. Erbert, “Mode-locked InGaAs-AlGaAs disk laser generating sub-200-fs pulses, pulse picking and amplification by a tapered diode amplifier,” Opt. Express 17(13), 10820–10834 (2009). [CrossRef] [PubMed]
4.	K. G. Wilcox, Z. Mihoubi, G. J. Daniell, S. Elsmere, A. Quarterman, I. Farrer, D. A. Ritchie, and A. Tropper, “Ultrafast optical Stark mode-locked semiconductor laser,” Opt. Lett. 33(23), 2797–2799 (2008). [CrossRef] [PubMed]
5.	A. Garnache, S. Hoogland, A. C. Tropper, I. Sagnes, G. Saint-Girons, and J. S. Roberts, “Sub-500-fs soliton-like pulse in a passively mode-locked broadband surface-emitting laser with 100 mW average power,” Appl. Phys. Lett. 80(21), 3892–3894 (2002). [CrossRef]
6.	K. G. Wilcox, A. H. Quarterman, H. Beere, D. A. Ritchie, and A. C. Tropper, “High peak power femtosecond pulse passively mode-locked vertical-external-cavity surface-emitting laser,” IEEE Photon. Technol. Lett. 22(14), 1021–1023 (2010). [CrossRef]
7.	O. D. Sieber, M. Hoffmann, D. J. Maas, V. J. Wittwer, M. Golling, T. Sudmeyer, and U. Keller, “Experimental confirmation of quasi-soliton pulse formation in ultrafast VECSELs,” in Advanced Solid State Photonics, OSA Technical Digest Series (CD) (Optical Society of America, 2010), paper AMB26.
8.	A. Aschwanden, D. Lorenser, H. J. Unold, R. Paschotta, E. Gini, and U. Keller, “2.1-W picosecond passively mode-locked external-cavity semiconductor laser,” Opt. Lett. 30(3), 272–274 (2005). [CrossRef] [PubMed]
9.	M. Hoffmann, O. D. Sieber, D. J. H. C. Maas, V. J. Wittwer, M. Golling, T. Südmeyer, and U. Keller, “Experimental verification of soliton-like pulse-shaping mechanisms in passively mode-locked VECSELs,” Opt. Express 18(10), 10143–10153 (2010). [CrossRef] [PubMed]
10.	H. A. Haus and M. N. Islam, “Theory of the soliton laser,” IEEE J. Quantum Electron. 21(8), 1172–1188 (1985). [CrossRef]
11.	S. Tsuda, W. H. Knox, S. T. Cundiff, W. Y. Jan, and J. E. Cunningham, “Mode-locking ultrafast solid-state lasers with saturable Bragg reflectors,” IEEE J. Sel. Top. Quantum Electron. 2(3), 454–464 (1996). [CrossRef]
12.	H. A. Haus, “Theory of mode-locking with a fast saturable absorber,” J. Appl. Phys. 46(7), 3049–3058 (1975). [CrossRef]
13.	G. J. Daniell, Z. Mihoubi, K. G. Wilcox, and A. C. Tropper, “Numerical model of the optical Stark effect as a mode-locking mechanism for femtosecond vertical-external-cavity surface-emitting semiconductor lasers,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonics Applications Systems Technologies. OSA Technical Digest (CD)(Optical Society of America, 2008), paper CThF3.
14.	Z. Mihoubi, G. J. Daniell, K. G. Wilcox, and A. C. Tropper, “Numerical model of a vertical-external-cavity surface-emitting semiconductor lasers mode-locked by the optical Stark effect,” in Proceedings of the 8th International Conference on Numerical Simulation of Optoelectronic Devices. (IEEE, 2008), pp. 91–92.
15.	M. Haiml, R. Grange, and U. Keller, “Optical characterization of semiconductor saturable absorbers,” Appl. Phys. B 79(3), 331–339 (2004). [CrossRef]
16.	H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 4th edition (World Scientific, 2004).
17.	J. V. Moloney, J. Hader, and S. W. Koch, “Quantum design of semiconductor active materials: laser and amplifier applications,” Laser Photonics Rev. 1(1), 24–43 (2007). [CrossRef]
18.	S. Balle, “Simple analytical approximations for the gain and refractive index spectra in quantum well lasers,” Phys. Rev. A 57(2), 1304–1312 (1998). [CrossRef]
19.	S. Balle, “Analytical description of spectral hole-burning effects in active semiconductors,” Opt. Lett. 27(21), 1923–1925 (2002). [CrossRef] [PubMed]
20.	J. Javaloyes and S. Balle, “Quasi-equilibrium time-domain susceptibility of semiconductor quantum wells,” Phys. Rev. A 81(6), 062505 (2010). [CrossRef]
21.	V. P. Kalosha, M. Muller, and J. Herrmann, “Theory of solid-state laser mode locking by coherent semiconductor quantum-well absorbers,” J. Opt. Soc. Am. B 16(2), 323 (1999). [CrossRef]
22.	L. Fan, M. Fallahi, J. Hader, A. R. Zakharian, J. V. Moloney, W. Stolz, S. W. Koch, R. Bedford, and J. T. Murray, “Linearly polarised dual-wavelength vertical-external-cavity surface-emitting laser,” Appl. Phys. Lett. 90(18), 181124 (2007). [CrossRef]
23.	A. Bäumner, S. W. Koch, and J. V. Moloney, “Non-equilibrium analysis of the two-color operation in semiconductor quantum-well lasers,” Phys. Status Solidi, B Basic Res. 248(4), 843–846 (2011). [CrossRef]
24.	W. W. Chow, S. W. Koch, and M. Sargent, Semiconductor-Laser Physics (Springer-Verlag, 1994).
25.	W. H. Knox, C. Hirlimann, D. A. B. Miller, J. Shah, D. S. Chemla, and C. V. Shank, “Femtosecond excitation of nonthermal carrier populations in GaAs quantum wells,” Phys. Rev. Lett. 56(11), 1191–1193 (1986). [CrossRef] [PubMed]
26.	W. H. Knox, D. S. Chemla, D. A. B. Miller, J. B. Stark, and S. Schmitt-Rink, “Femtosecond ac Stark effect in semiconductor quantum wells: Extreme low- and high-intensity limits,” Phys. Rev. Lett. 62(10), 1189–1192 (1989). [CrossRef] [PubMed]
27.	H. A. Haus, “Theory of mode-locking with a slow saturable absorber,” IEEE J. Quantum Electron. 11(9), 736–746 (1975). [CrossRef]
28.	M. E. Barnes, Z. Mihoubi, K. G. Wilcox, A. H. Quarterman, I. Farrer, D. A. Ritchie, A. Garnache, S. Hoogland, V. Apostolopoulos, and A. C. Tropper, “Gain bandwidth characterisation of surface-emitting quantum well laser gain structures for femtosecond operation,” Opt. Express 18(20), 21330–21341 (2010). [CrossRef]

OCIS Codes
(020.6580) Atomic and molecular physics : Stark effect
(140.4050) Lasers and laser optics : Mode-locked lasers
(140.5960) Lasers and laser optics : Semiconductor lasers

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: August 15, 2011
Revised Manuscript: October 25, 2011
Manuscript Accepted: October 25, 2011
Published: December 14, 2011

Citation
A.H. Quarterman, S. Carswell, G.J. Daniell, Z. Mihoubi, K.G. Wilcox, A.L. Chung, V Apostolopoulos, and A.C. Tropper, "Numerical simulation of optical Stark effect saturable absorbers in mode-locked femtosecond VECSELs using a modified two-level atom model.," Opt. Express 19, 26783-26795 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-27-26783

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References

A. H. Quarterman, K. G. Wilcox, V. Apostolopoulos, Z. Mihoubi, S. P. Elsmere, I. Farrer, D. A. Ritchie, and A. Tropper, “A passively mode-locked external-cavity semiconductor laser emitting 60-fs pulses,” Nat. Photonics3(12), 729–731 (2009). [CrossRef]
P. Klopp, U. Griebner, M. Zorn, and M. Weyers, “Pulse repetition rate up to 92 GHz or pulse duration shorter than 110 fs from a mode-locked semiconductor disk laser,” Appl. Phys. Lett.98(7), 071103 (2011). [CrossRef]
P. Klopp, U. Griebner, M. Zorn, A. Klehr, A. Liero, M. Weyers, and G. Erbert, “Mode-locked InGaAs-AlGaAs disk laser generating sub-200-fs pulses, pulse picking and amplification by a tapered diode amplifier,” Opt. Express17(13), 10820–10834 (2009). [CrossRef] [PubMed]
K. G. Wilcox, Z. Mihoubi, G. J. Daniell, S. Elsmere, A. Quarterman, I. Farrer, D. A. Ritchie, and A. Tropper, “Ultrafast optical Stark mode-locked semiconductor laser,” Opt. Lett.33(23), 2797–2799 (2008). [CrossRef] [PubMed]
A. Garnache, S. Hoogland, A. C. Tropper, I. Sagnes, G. Saint-Girons, and J. S. Roberts, “Sub-500-fs soliton-like pulse in a passively mode-locked broadband surface-emitting laser with 100 mW average power,” Appl. Phys. Lett.80(21), 3892–3894 (2002). [CrossRef]
K. G. Wilcox, A. H. Quarterman, H. Beere, D. A. Ritchie, and A. C. Tropper, “High peak power femtosecond pulse passively mode-locked vertical-external-cavity surface-emitting laser,” IEEE Photon. Technol. Lett.22(14), 1021–1023 (2010). [CrossRef]
O. D. Sieber, M. Hoffmann, D. J. Maas, V. J. Wittwer, M. Golling, T. Sudmeyer, and U. Keller, “Experimental confirmation of quasi-soliton pulse formation in ultrafast VECSELs,” in Advanced Solid State Photonics, OSA Technical Digest Series (CD) (Optical Society of America, 2010), paper AMB26.
A. Aschwanden, D. Lorenser, H. J. Unold, R. Paschotta, E. Gini, and U. Keller, “2.1-W picosecond passively mode-locked external-cavity semiconductor laser,” Opt. Lett.30(3), 272–274 (2005). [CrossRef] [PubMed]
M. Hoffmann, O. D. Sieber, D. J. H. C. Maas, V. J. Wittwer, M. Golling, T. Südmeyer, and U. Keller, “Experimental verification of soliton-like pulse-shaping mechanisms in passively mode-locked VECSELs,” Opt. Express18(10), 10143–10153 (2010). [CrossRef] [PubMed]
H. A. Haus and M. N. Islam, “Theory of the soliton laser,” IEEE J. Quantum Electron.21(8), 1172–1188 (1985). [CrossRef]
S. Tsuda, W. H. Knox, S. T. Cundiff, W. Y. Jan, and J. E. Cunningham, “Mode-locking ultrafast solid-state lasers with saturable Bragg reflectors,” IEEE J. Sel. Top. Quantum Electron.2(3), 454–464 (1996). [CrossRef]
H. A. Haus, “Theory of mode-locking with a fast saturable absorber,” J. Appl. Phys.46(7), 3049–3058 (1975). [CrossRef]
G. J. Daniell, Z. Mihoubi, K. G. Wilcox, and A. C. Tropper, “Numerical model of the optical Stark effect as a mode-locking mechanism for femtosecond vertical-external-cavity surface-emitting semiconductor lasers,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonics Applications Systems Technologies. OSA Technical Digest (CD)(Optical Society of America, 2008), paper CThF3.
Z. Mihoubi, G. J. Daniell, K. G. Wilcox, and A. C. Tropper, “Numerical model of a vertical-external-cavity surface-emitting semiconductor lasers mode-locked by the optical Stark effect,” in Proceedings of the 8th International Conference on Numerical Simulation of Optoelectronic Devices. (IEEE, 2008), pp. 91–92.
M. Haiml, R. Grange, and U. Keller, “Optical characterization of semiconductor saturable absorbers,” Appl. Phys. B79(3), 331–339 (2004). [CrossRef]
H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 4th edition (World Scientific, 2004).
J. V. Moloney, J. Hader, and S. W. Koch, “Quantum design of semiconductor active materials: laser and amplifier applications,” Laser Photonics Rev.1(1), 24–43 (2007). [CrossRef]
S. Balle, “Simple analytical approximations for the gain and refractive index spectra in quantum well lasers,” Phys. Rev. A57(2), 1304–1312 (1998). [CrossRef]
S. Balle, “Analytical description of spectral hole-burning effects in active semiconductors,” Opt. Lett.27(21), 1923–1925 (2002). [CrossRef] [PubMed]
J. Javaloyes and S. Balle, “Quasi-equilibrium time-domain susceptibility of semiconductor quantum wells,” Phys. Rev. A81(6), 062505 (2010). [CrossRef]
V. P. Kalosha, M. Muller, and J. Herrmann, “Theory of solid-state laser mode locking by coherent semiconductor quantum-well absorbers,” J. Opt. Soc. Am. B16(2), 323 (1999). [CrossRef]
L. Fan, M. Fallahi, J. Hader, A. R. Zakharian, J. V. Moloney, W. Stolz, S. W. Koch, R. Bedford, and J. T. Murray, “Linearly polarised dual-wavelength vertical-external-cavity surface-emitting laser,” Appl. Phys. Lett.90(18), 181124 (2007). [CrossRef]
A. Bäumner, S. W. Koch, and J. V. Moloney, “Non-equilibrium analysis of the two-color operation in semiconductor quantum-well lasers,” Phys. Status Solidi, B Basic Res.248(4), 843–846 (2011). [CrossRef]
W. W. Chow, S. W. Koch, and M. Sargent, Semiconductor-Laser Physics (Springer-Verlag, 1994).
W. H. Knox, C. Hirlimann, D. A. B. Miller, J. Shah, D. S. Chemla, and C. V. Shank, “Femtosecond excitation of nonthermal carrier populations in GaAs quantum wells,” Phys. Rev. Lett.56(11), 1191–1193 (1986). [CrossRef] [PubMed]
W. H. Knox, D. S. Chemla, D. A. B. Miller, J. B. Stark, and S. Schmitt-Rink, “Femtosecond ac Stark effect in semiconductor quantum wells: Extreme low- and high-intensity limits,” Phys. Rev. Lett.62(10), 1189–1192 (1989). [CrossRef] [PubMed]
H. A. Haus, “Theory of mode-locking with a slow saturable absorber,” IEEE J. Quantum Electron.11(9), 736–746 (1975). [CrossRef]
M. E. Barnes, Z. Mihoubi, K. G. Wilcox, A. H. Quarterman, I. Farrer, D. A. Ritchie, A. Garnache, S. Hoogland, V. Apostolopoulos, and A. C. Tropper, “Gain bandwidth characterisation of surface-emitting quantum well laser gain structures for femtosecond operation,” Opt. Express18(20), 21330–21341 (2010). [CrossRef]

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