Quantum cascade laser combs: effects of modulation and dispersion

Gustavo Villares; Jérôme Faist

doi:10.1364/OE.23.001651

1. Introduction

In a mode-locked laser, a fixed phase relationship is established between the longitudinal modes of the laser cavity [1]. This phase relation is generally established by either a subtile compensation between a negative dispersion and the Kerr effect, giving rise to the propagation of a temporal soliton, or by a saturable absorber opening a time-window of low loss for a pulsed output. In both cases, the net result is the production of short pulses that will be produced at the roundtrip frequency of the cavity and, in well designed laser, an average power commensurate to the one of a continuous wave laser.

In fact, the terminology of mode-locked laser is now tacitly used to design lasers which do not only have a fixed relative phase between the modes, but one where this phase difference between adjacent modes is constant, such that the solution is a single pulse as described above. In such pulsed lasers, the upper state lifetime of the laser transition should be much longer than the round-trip time, so that the energy could be accumulated in the upper state in-between successive pulses.

Quantum cascade lasers (QCL) [2], being based on intersubband transitions in quantum wells, exhibit very short upper state lifetime τ₂ [3], in the sub picosecond range (τ₂ ≈ 0.6 ps) in high performance devices operating at room temperature. This time is much shorter than the typical cavity round trip time τ_rt, 64 ps for a 3 mm long device, such that the product ωτ₂ << 1, where ω = 2π/τ_rt is the angular frequency corresponding to the longitudinal mode spacing. As a result, passive mode-locking in the sense described above is not possible. Active mode-locking has been achieved in devices operating at cryogenic temperature in the THz [4, 5] or with photon-assisted tunneling transitions [6] such that the condition ωτ₂ ≈ 1 was satisfied. However, compared to passive mode-locking in which the pulse length can routinely reach the inverse of the gain bandwidth Ω_g, fundamental active mode-locking leads to much longer pulses since the pulse length τ is now a geometrical average of the gain bandwidth and the round-trip frequency ω as given by the Kuizenga-Siegman formula [1, 7]:

τ = \sqrt[4]{\frac{2 g}{M ω^{2} Ω_{g}^{2}}},

where M is the modulation depth and g the (single pass) gain. For a 3 mm long mid-infrared (MIR) device with a modulation depth M = 0.2, a total mirror and waveguide loss of 13 cm⁻¹, and with a gain bandwidth of 25meV, representative of the devices used in [6], the formula above predicts a pulse length of τ = 1.1 ps, still significantly shorter than the observed minimum pulse length of 3 ps as Eq. (1) assumes ωτ₂ >> 1. All in all, the reduced resulting mode-locking bandwidth therefore limits the usefulness of such devices for broadband spectroscopy.

In contrast, it was recently shown that broadband quantum cascade laser, engineered in low dispersion waveguides, can also operate as frequency combs such that the phase relation between the modes is also fixed [8], and preliminary results using these devices demonstrated the possibility of achieving dual-comb spectroscopy [9]. In the case of QCLs, however, locking between the modes is achieved by four-wave mixing (FWM) [10], and the phase relation between the modes is such that the optical intensity is approximately constant in the cavity. In this picture, the beating between two modes create sidebands [10] that injection-lock the adjacent modes. Such model of multimode laser and the resulting locking to modes equally spaced was already explained theoretically by Lamb in the limit of three modes [11]. Recently, such a Maxwell-Bloch model in frequency space was extended to the case of multimode operation, in order to describe the nature of the comb behavior of quantum cascade laser and contrast it to the case of semiconductor lasers with long recovery times [12]. In this study, some simplifications were assumed in the numerical implementation such as neglecting dispersion altogether, assuming that the gain was due to a single active region stack, and not allowing the possibility of modulating a section of the device as was done in [6, 5].

In this paper, we extend the same model to broadband gain to take into account the dispersion caused by both the gain medium as well as by the cavity. We also examine the influence of an amplitude modulation applied to a section of the device. Finally, we also look at the case of multiband stacks.

2. Theory

Simple time-dependent rate equations [13, 14] are sufficient to model the propagation of pulses in a quantum cascade laser cavity and agreed with ultrafast pump and probe measurements [14]. In particular, these rate equations predict, as shown in Fig. 1(a), a strong damping of pulses upon propagation in the active region of a QCL device illustrating the difficulty of sustaining mode-locked pulses. However, these models are not able to predict the dynamics of multimode operation since there is not coherent coupling between the populations and the electric field.

Fig. 1 Pulse propagation in QCL and theoretical framework used to study the laser dynamics. a) Simulation, based on rate equations, of a 2ps long pulse propagating in a quantum cascade laser. Because of the very fast gain relaxation time, the pulse is damped after only a few millimeter of propagation. b) Maxwell-Bloch equations for the evolution of the populations ρ₁₁ and ρ₂₂ as well as the coherences ρ₁₂ and ρ₂₁. A modal decomposition is used in order to study lasers exhibiting a frequency modulated output.

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The Maxwell-Bloch model followed here uses a modal decomposition of the cavity field [12] as shown schematically in Fig. 1(b), and not a propagation in the time domain, as was done in many early theoretical studies of multimode instabilities in quantum cascade lasers [15, 16]. While the latter technique is very well adapted to follow the propagation of a pulse inside the laser cavity, it is ill-conditioned to study the output of a laser that exhibits mostly a frequency modulated output.

2.1. Density matrix

The model is based on the evolution of a density matrix where the active region is approximated by a two-level system, following the evolution equation:

\dot{ρ} = {\frac{- i}{ℏ} [ℋ, ρ] + \frac{\partial ρ}{\partial t})}_{coll}

where ρ is the density matrix containing the populations ρ₁₁ and ρ₂₂ as well as the coherences ρ₁₂ and ρ₂₁, ℋ is the Hamiltonian of the two-level system in the presence of the optical (classical) field E(t). The Hamiltonian can be written, in a matrix format:

ℋ = (\begin{matrix} ℏ ω_{1} & ℏ {\tilde{Ω}}_{12} (t) \\ ℏ {\tilde{Ω}}_{21} (t) & ℏ ω_{2} \end{matrix})

where ħω₁ (respectively ħω₂) is the energy of level 1 (respectively 2), ħΩ̃₁₂(t) is the interaction energy between the electrons and the field, taken in the dipole approximation and the time-dependent Rabi frequency:

{\tilde{Ω}}_{12} (t) = {\tilde{Ω}}_{21} (t) = μ_{21} E (t) / ℏ

is assumed to be real as the dipole matrix element μ₂₁ = −ez₂₁ can be assumed to be so without loss of generality. The lifetimes and the loss of coherence are expressed through the collision term by the following matrix:

{\frac{\partial ρ}{\partial t})}_{coll} = - (\begin{matrix} γ_{11} (ρ_{11} - ρ_{11, p}) & γ_{12} ρ_{12} \\ γ_{12} ρ_{21} & γ_{22} (ρ_{22} - ρ_{22, p}) \end{matrix})

where

γ_{11} = τ_{1}^{- 1}

and

γ_{22} = τ_{2}^{- 1}

are the scattering rates out of the ground state and excited state, respectively, and

γ_{12} = τ_{coh}^{- 1}

is the loss of coherence of the transition. A constant pumping with a rate γ₂₂ρ_22,p is such that an equilibrium population ρ_22,p of the upper state can be reached. Similarly, the term ρ_11,p would express a thermal population of the lower state. Solving the density matrix evolution leads to the well-known Bloch equations for the individual components of the density matrix which are, in the limit of negligible lower state population (ρ₁₁ = 0):

\begin{array}{l} Δ \dot{ρ} = i {\tilde{Ω}}_{12} (ρ_{21} - ρ_{12}) - γ_{22} (Δ ρ - Δ ρ_{p}) \\ {\dot{ρ}}_{21} = - i ω_{21} ρ_{21} + i {\tilde{Ω}}_{12} Δ ρ - γ_{12} ρ_{21} \\ ρ_{21} = ρ_{12}^{*} \end{array}

where ω₂₁ = ω₂ − ω₁ is the center of the laser transition. All these quantities are time-dependent in a non-trivial manner. We do an expansion on the modes of an ideal cavity without dispersion. These modes are written as:

ω_{l} = ω_{21} + l ω l \in {- n_{m}, - n_{m} + 1, ., 0, ., n_{m}}

so we have a total of (2n_m + 1) modes spaced by the angular frequency ω, also called the comb frequency spacing. The electric field and Rabi frequencies are expanded in term of these modes as:

\begin{array}{r} E (t) = \frac{1}{2} e^{- i ω_{21} t} \sum_{l = - n_{m}}^{l = n_{m}} A_{l} e^{- i l ω t} + \frac{1}{2} e^{i ω_{21} t} \sum_{l = - n_{m}}^{l = n_{m}} A_{l}^{*} e^{i l ω t} \\ {\tilde{Ω}}_{12} (t) = \frac{1}{2} e^{- i ω_{21} t} \sum_{l = - n_{m}}^{l = n_{m}} Ω_{l} e^{- i l ω t} + \frac{1}{2} e^{i ω_{21} t} \sum_{l = - n_{m}}^{l = n_{m}} Ω_{l}^{*} e^{i l ω t} \end{array}

where the Rabi frequencies are related to the amplitudes through:

Ω_{l} = \frac{μ_{21} A_{l}}{ℏ}

We expand the density matrix in a Fourier series with the following time dependence:

\begin{array}{l} ρ_{21} = e^{- i ω_{21} t} \sum_{k = - n_{m}}^{k = n_{m}} σ_{21, k} e^{- i k ω t} \\ ρ_{12} = e^{i ω_{21} t} \sum_{k = - n_{m}}^{k = n_{m}} σ_{21, k}^{*} e^{i k ω t} \\ Δ ρ = \sum_{r = - n_{m}}^{r = n_{m}} Δ ρ_{r} e^{- i r ω t} \end{array}

In order to take into account the modulation of the gain at the round trip frequency ω, we also expand the equilibrium part of the density matrix and assume that the latter is the addition of a constant pumping and a sinusoidal modulation at the frequency ω:

Δ ρ_{p} = Δ ρ_{p, 0} + \frac{1}{2} (Δ ρ_{p, M} e^{- i ω t} + Δ ρ_{p, M}^{*} e^{i ω t})

We then substitute the Eqs. (8), (10) and (11) in Eq. (6). The rotating wave approximation consists, in this context, to neglect the terms that would lead to e^{±i2ω₂₁t} time dependence. We have therefore for the first equation:

\begin{array}{l} \sum_{r} (- i r ω + γ_{22}) Δ ρ_{r} e^{- i r ω t} = & γ_{22} (Δ ρ_{p, 0} + \frac{1}{2} (Δ ρ_{p, M} e^{- i ω t} + Δ ρ_{p, M}^{*} e^{i ω t})) \\ + \frac{i}{2} \sum_{k} \sum_{l} (Ω_{l}^{*} σ_{21, k} e^{- i (k - l) ω t} - Ω_{l} σ_{21, k}^{*} e^{i (k - l) ω t}) \end{array}

And then for the second:

\sum_{k} (γ_{12} - i k ω) σ_{21, k} e^{- i k ω t} = \frac{i}{2} \sum_{l} \sum_{r} Ω_{l} Δ ρ_{r} e^{- i (l + r) ω t}

Both equations are solved by matching the time dependences e^ikωt on both sides. For the first one, l = k + r and l = k − r in the sum, which yields:

Δ ρ_{r} = \frac{1}{- i r ω + γ_{22}} [γ_{22} Δ ρ_{p, 0} δ_{r, 0} + \frac{γ_{22} Δ ρ_{p, M}^{*}}{2} δ_{r, - 1} + \frac{γ_{22} Δ ρ_{p, M}}{2} δ_{r, 1} + \frac{i}{2} \sum_{k} (Ω_{k - r}^{*} σ_{21, k} - Ω_{k + r} σ_{21, k}^{*})]

The second one l + r = k and yields:

σ_{21, k} = \frac{i}{2 (γ_{12} - i k ω)} \sum_{l} Ω_{l} Δ ρ_{k - l}

We then solve these equations using a perturbative approach on Δρ_r and σ_21,k. The zeroth order for $Δ ρ_{r}^{(0)}$ yields from Eq. (14):

Δ ρ_{r = 0}^{(0)} = Δ ρ_{p, 0}

Δ ρ_{r = 1}^{(0)} = \frac{γ_{22}}{γ_{22} - i ω} \frac{Δ ρ_{p, M}}{2}

Δ ρ_{r = - 1}^{(0)} = \frac{γ_{22}}{γ_{22} + i ω} \frac{Δ ρ_{p, M}^{*}}{2}

We have taken

σ_{21, k}^{(0)} = 0

. Inserting then this values for Δρ⁽⁰⁾ in Eq. (15) then yields:

σ_{21, k}^{(1)} = \frac{- 1}{2 (i γ_{12} + k ω)} [Ω_{k} Δ ρ_{p, 0} + Ω_{k + 1} \frac{γ_{22}}{γ_{22} + i ω} \frac{Δ ρ_{p, M}^{*}}{2} + Ω_{k - 1} \frac{γ_{22}}{γ_{22} - i ω} \frac{Δ ρ_{p, M}}{2}]

Inserting now Eq. (16) in Eq. (12) yields (for values of r ≠ (−1, 0, −1)):

Δ ρ_{r}^{(2)} = \frac{i}{2 (γ_{22} - i r ω)} \sum_{k} (Ω_{k - r}^{*} σ_{21, k}^{(1)} - Ω_{k + r} σ_{21, k}^{(1) *})

Inserting now the value of

σ_{21, k}^{(1)}

yields:

\begin{array}{r} Δ ρ_{r}^{(2)} = \frac{i}{2 (γ_{22} - i r ω)} \sum_{k} (\frac{Ω_{k - r}^{*}}{2} \frac{- 1}{(i γ_{12} + k ω)} [Ω_{k} Δ ρ_{p, 0} + Ω_{k + 1} \frac{γ_{22}}{γ_{22} + i ω} \frac{Δ ρ_{p, M}^{*}}{2} + Ω_{k - 1} \frac{γ_{22}}{γ_{22} - i ω} \frac{Δ ρ_{p, M}}{2}] \\ - \frac{Ω_{k + r}}{2} \frac{- 1}{(- i γ_{12} + k ω)} [Ω_{k}^{*} Δ ρ_{p, 0} + Ω_{k + 1}^{*} \frac{γ_{22}}{γ_{22} - i ω} \frac{Δ ρ_{p, M}}{2} + Ω_{k - 1}^{*} \frac{γ_{22}}{γ_{22} + i ω} \frac{Δ ρ_{p, M}^{*}}{2}]) \end{array}

After an algebraic arrangement, it yields:

\begin{array}{r} Δ ρ_{r}^{(2)} = \frac{i}{2 (γ_{22} - i r ω)} \sum_{k} {(\frac{Ω_{k - r}^{*} Ω_{k} Δ ρ_{p, 0}}{2 (- i γ_{12} - k ω)} - \frac{Ω_{k + r} Ω_{k}^{*} Δ ρ_{p, 0}}{2 (i γ_{12} - k ω)}) \\ + (\frac{γ_{22}}{γ_{22} + i ω} \frac{Ω_{k - r}^{*} Ω_{k + 1} Δ ρ_{p, M}^{*} / 2}{2 (- i γ_{12} - k ω)} - \frac{γ_{22}}{γ_{22} - i ω} \frac{Ω_{k + r} Ω_{k + 1}^{*} Δ ρ_{p, M} / 2}{2 (i γ_{12} - k ω)}) \\ + (\frac{γ_{22}}{γ_{22} - i ω} \frac{Ω_{k - r}^{*} Ω_{k - 1} Δ ρ_{p, M} / 2}{2 (- i γ_{12} - k ω)} - \frac{γ_{22}}{γ_{22} + i ω} \frac{Ω_{k + r} Ω_{k - 1}^{*} Δ ρ_{p, M}^{*} / 2}{2 (i γ_{12} - k ω)})} \end{array}

We now apply the substitution k′ = k − r, k′ = k + 1 − r and k′ = k − 1 − r for the first terms of the first, second and third lines respectively, which yields:

\begin{array}{r} Δ ρ_{r}^{(2)} = \frac{1}{4} \frac{i}{(γ_{22} - i r ω)} \sum_{k} Ω_{k + r} {Ω_{k}^{*} Δ ρ_{p, 0} (\frac{1}{- i γ_{12} - (k + r) ω} - \frac{1}{i γ_{12} - k ω}) \\ + Ω_{k - 1}^{*} \frac{γ_{22}}{γ_{22} + i ω} \frac{Δ ρ_{p, M}^{*}}{2} (\frac{1}{- i γ_{12} - (k - 1 + r) ω} - \frac{1}{i γ_{12} - k ω}) \\ + Ω_{k + 1}^{*} \frac{γ_{22}}{γ_{22} - i ω} \frac{Δ ρ_{p, M}}{2} (\frac{1}{- i γ_{12} - (k + 1 + r) ω} - \frac{1}{i γ_{12} - k ω})} \end{array}

To find the third order term

σ_{21, k}^{(3)}

we write using Eq. (15):

σ_{21, n}^{(3)} = \frac{i}{2 (γ_{12} - i n ω)} \sum_{l} Ω_{l} Δ ρ_{n - l}^{(2)}

By inserting Eq. (23) into Eq. (24), we have:

\begin{array}{r} σ_{21, n}^{(3)} = \frac{i}{2 (γ_{12} - i n ω)} \frac{1}{4} \sum_{l} \sum_{k} Ω_{l} \frac{i}{(γ_{22} - i (n - l) ω)} Ω_{k + n - l} {Ω_{k}^{*} Δ ρ_{p, 0} (\frac{1}{- i γ_{12} - (k + n - l) ω} - \frac{1}{i γ_{12} - k ω}) \\ + Ω_{k - 1}^{*} \frac{γ_{22}}{γ_{22} + i ω} \frac{Δ ρ_{p, M}^{*}}{2} (\frac{1}{- i γ_{12} - (k - 1 + n - l) ω} - \frac{1}{i γ_{12} - k ω}) \\ + Ω_{k + 1}^{*} \frac{γ_{22}}{γ_{22} - i ω} \frac{Δ ρ_{p, M}}{2} (\frac{1}{- i γ_{12} - (k + 1 + n - l) ω)} - \frac{1}{i γ_{12} - k ω})} \end{array}

By doing the following substitution l′ = n + k − l, we get:

\begin{array}{r} σ_{21, n}^{(3)} = \frac{i}{2} \frac{1}{γ_{12} - i n ω} \frac{1}{4} \sum_{l} \sum_{k} Ω_{n + k - l} \frac{i}{(γ_{22} - i (l - k) ω)} Ω_{l} {Ω_{k}^{*} Δ ρ_{p, 0} (\frac{1}{- i γ_{12} - l ω} - \frac{1}{i γ_{12} - k ω}) \\ + Ω_{k - 1}^{*} \frac{γ_{22}}{γ_{22} + i ω} \frac{Δ ρ_{p, M}^{*}}{2} (\frac{1}{- i γ_{12} - (l - 1) ω} - \frac{1}{i γ_{12} - k ω}) \\ + Ω_{k + 1}^{*} \frac{γ_{22}}{γ_{22} - i ω} \frac{Δ ρ_{p, M}}{2} (\frac{1}{- i γ_{12} - (l + 1) ω)} - \frac{1}{i γ_{12} - k ω})} \end{array}

where l′ was then substituted by l for clarity. The first line corresponds to the result of [12].

To ease the implementation, a number of simplifications are introduced. In general, we have that γ₁₂ ≫ ω, as the gain is broad enough to sustain many modes. Therefore, we can assume that:

\frac{1}{- i γ_{12} - (l \pm 1) ω} \approx \frac{1}{- i γ_{12} - l ω}

As a result, the modulation term has the same overall form as the FWM term, to the exception of the pre factors:

M_{\pm} = \frac{γ_{22}}{γ_{22} \pm i ω}

substituting the direct injection Δρ_p,0 by Δρ_p,M/2 on both modes n ± 1. This enables an efficient numerical implementation since the kernel has to be computed only once. We also define the amplitude of the coherent population oscillations:

C_{k l} = \frac{γ_{22}}{γ_{22} - i (l - k) ω}

as well as the transition gain:

{\tilde{G}}_{n} = \frac{i γ_{12}}{n ω + i γ_{12}}

the (complex) gain coefficient for the mode n. We also write the term driving the width of the FWM gain:

B_{k l} = \frac{γ_{12}}{2 i} (\frac{1}{- i γ_{12} - l ω} - \frac{1}{i γ_{12} - k ω})

The same result may be rewritten using normalized coefficients (and with the approximation mentioned above):

σ_{21, n}^{(3)} = \frac{- i}{4 γ_{22} γ_{12}^{2}} {\tilde{G}}_{n} \sum_{l} \sum_{k} C_{k l} B_{k l} Ω_{n + k - l} Ω_{l} {Ω_{k}^{*} Δ ρ_{p, 0} + Ω_{k - 1}^{*} M_{+} \frac{Δ ρ_{p, M}^{*}}{2} + Ω_{k + 1}^{*} M_{-} \frac{Δ ρ_{p, M}}{2}}

σ_{21, n}^{(1)} = \frac{i}{2 γ_{12}} {\tilde{G}}_{n} [Ω_{n} Δ ρ_{p, 0} + Ω_{n + 1} M_{+} \frac{Δ ρ_{p, M}^{*}}{2} + Ω_{n - 1} M_{-} \frac{Δ ρ_{p, M}}{2}]

2.2. Cavity modes

In this section, we give the expression of the intracavity field E(t), which is a solution of Maxwell’s equation and has to obey:

\nabla^{2} E - μ_{0} σ \frac{\partial E}{\partial t} - \frac{ε_{r}}{c^{2}} \frac{\partial^{2} E}{\partial t^{2}} = μ_{0} \frac{\partial^{2} P}{\partial t^{2}}

where we have introduced the cavity losses by the conductivity σ. Both electrical fields E and polarisation P are real quantities and are also expanded as Eq. (8):

\begin{array}{l} E (t) = \frac{1}{2} \sum_{l = - n_{m}}^{l = n_{m}} A_{l} e^{- i ω_{l} t} + \frac{1}{2} \sum_{l = - n_{m}}^{l = n_{m}} A_{l}^{*} e^{i ω_{l} t} \\ P (t) = \frac{1}{2} \sum_{l = - n_{m}}^{l = n_{m}} P_{l} e^{- i ω_{l} t} + \frac{1}{2} \sum_{l = - n_{m}}^{l = n_{m}} P_{l}^{*} e^{i ω_{l} t} \end{array}

The modes have a spatial dependence given by an envelope A_n(z):

A_{n} (z) = A_{n} sin (k_{n} z)

The polarization for the mode n is written as:

P_{n} = - N 〈 μ 〉 = - N Tr [ρ μ] = - N μ_{21} σ_{21, n}

where the sum on N represents the sum over all states of the periods of the active region.

We prescribe that all the modes must be eigenmodes of the cavity, assumed to have no losses through the mirrors. This is a simplification in semiconductor lasers since usually at least one of the mirror has a relatively low reflectivity and therefore the field amplitude close to the facet is not a perfect standing wave. We define then the wavevector k_n for the n-th eigenmode by:

k_{n} l_{c} = (N_{0} + n) π

where N₀ is the index of the center frequency of our laser. The ”cold cavity” frequencies ω_nc are then defined by:

ω_{n c} = \frac{k_{n} c}{n_{0}}

where

n_{0} = \sqrt{ε_{r}}

is the refractive index. The frequencies ω_nc are usually different from the frequencies ω_n in a real cavity, as the cavity dispersion will shift the resonances ω_nc according to the ideal dispersionless cavity represented by the frequencies ω_n = (n + N₀)ω.

By inserting Eqs. (35)--(37) into (34), we obtain an equation for the evolution of the amplitudes, using the slowly varying envelope approximation (i.e. Ä ≈ 0):

\frac{1}{2} sin (k_{n} z) {- k_{n}^{2} A_{n} - μ_{0} σ ({\dot{A}}_{n} - i ω_{n} A_{n}) - \frac{ε_{r}}{c^{2}} (- ω_{n}^{2} A_{n} - 2 i ω_{n} {\dot{A}}_{n})} = ω_{n}^{2} μ_{0} N μ_{21} \frac{1}{2} σ_{21, n} (z)

that can be simplified (multipling both sides by

c^{2} / n_{0}^{2}

and using the definition of ω_nc) into:

{((ω_{n}^{2} - ω_{n c}^{2}) + \frac{i ω_{n} σ}{ε_{0} ε_{r}}) A_{n} + (2 i ω_{n} - \frac{σ}{ε_{0} ε_{r}}) {\dot{A}}_{n}} sin (k_{n} z) = \frac{1}{ε_{0} ε_{r}} ω_{n}^{2} N μ_{21} σ_{21, n} (z)

We assume that the cavity has a relatively large Q so that ω_n ≫ σ/(ε₀ε_r), which yields:

2 i ω_{n} {\dot{A}}_{n} = - ((ω_{n}^{2} - ω_{n c}^{2}) + \frac{i ω_{n} σ}{ε_{0} ε_{r}}) A_{n} + \frac{ω_{n}^{2} N μ_{21} σ_{21, n} (z)}{ε_{0} n_{0}^{2} sin (k_{n} z)}

To give an interpretation of our conductivity σ, we assume we have a resonant mode (ω_n = ω_nc) and no driving polarization (σ_21,n = 0). Then the amplitude satisfies:

\dot{A} = - \frac{σ}{2 ε_{0} ε_{r}} A

We have for the photon lifetime in the cavity τ_c the equation for the amplitude:

| \dot{A} | = - \frac{| A |}{2 τ_{c}}

justifying that we interpret σ/(ε₀ε_r) as the cavity photon energy decay rate

τ_{c}^{- 1}

. And finally we obtain:

{\dot{A}}_{n} - i (\frac{ω_{n}^{2} - ω_{n c}^{2}}{2 ω_{n}}) A_{n} = - \frac{i ω_{n} N μ_{21} σ_{21, n} (z)}{2 ε_{0} n_{0}^{2} sin (k_{n} z)} - \frac{1}{2 τ_{c}} A_{n}

To go further, we take Eq. (45), multiply both sides by sin(k_nz)², divide by the cavity length l_c, and integrate over the cavity length:

{\dot{A}}_{n} - i (\frac{ω_{n}^{2} - ω_{n c}^{2}}{2 ω_{n}}) A_{n} = - \frac{i ω_{n} N μ_{21}}{l_{c} ε_{0} n_{0}^{2}} \int_{0}^{l_{c}} σ_{21, n} (z) sin (k_{n} z) d z - \frac{1}{2 τ_{c}} A_{n}

Now we substitute σ_21,n(z) by its expansion:

σ_{21, n} (z) = σ_{21, n}^{(1)} (z) + σ_{21, n}^{(3)} (z)

given by Eq. (33) and (32).

Focusing first on the integral on the spatial coordinate, and recalling the relationship between Ω_l and A_l, we observe that the term $σ_{21, n}^{(1)} (z)$ will lead to a integral with the form:

\frac{1}{l_{c}} \int_{0}^{l_{c}} A_{n} sin {(k_{n} z)}^{2} d z = A_{n} κ_{n, n} = A_{n} \frac{1}{2}

corresponding to the gain, while the terms due to the amplitude modulation will yield to values of κ_n,n±1 = 0. For the terms

σ_{21, n}^{(3)} (z)

, we have then integral of the form:

\frac{1}{l_{c}} \int_{0}^{l_{c}} A_{n + k - l} sin (k_{n + k - l} z) A_{l} sin (k_{l} z) A_{k}^{*} sin (k_{k} z) sin (k_{n} z) d z = A_{n + k - l} A_{l} A_{k}^{*} κ_{n, l, k, n + k - l}

where we have defined the coefficient κ_n,l,k,m as:

κ_{n, l, k, m} = \frac{1}{l_{c}} \int_{0}^{l_{c}} sin (k_{n} z) sin (k_{l} z) sin (k_{k} z) sin (k_{m} z) d z

The coefficient κ_{n,l,k,n+k−l} has the following values:

\begin{array}{r} κ_{n, n, n, n} = \frac{3}{8} & (k = l = n) \\ κ_{n, n, m, n} = \frac{1}{4} & (k = l \neq n) or (l = n) \\ κ_{n, l, k, n + k - l} = \frac{1}{8} & otherwise \end{array}

It is well known that active mode-locking is achieved in semiconductor lasers by modulating only a section of the active region. For this reason, we assume now that the modulation term Δρ_p,M has a spatial dependence so that a fraction 0 < l_m < 1 of the total length l_c is modulated:

Δ ρ_{p, M} (z) = Θ (l_{m} l_{c} - z) Δ ρ_{p, M}

where Θ(z) is the Heaviside step function. We define now the coefficient κ′_n,l,k,m(l_m) as:

κ_{n, l, k, m}^{'} (l_{m}) = \frac{1}{l_{c}} \int_{0}^{l_{m} l_{c}} sin (k_{n} z) sin (k_{l} z) sin (k_{k} z) sin (k_{m} z) d z

As a result, the factor κ′_{n,l,k,n+k−l}(l_m) depends explicitly on l_m. An important consequence is that κ′_n,n±1 is non-zero and has the value:

\begin{array}{l} κ_{n, n + 1}^{'} & = \frac{1}{l_{c}} \int_{0}^{l_{m} l_{c}} sin (k_{n} z) sin (k_{n + 1} z) d z \\ = \frac{1}{2 π} {sin (π l_{m}) - \frac{sin (π (2 (n + N_{0}) + 1) l_{m})}{(2 (n + N_{0}) + 1)}} \end{array}

where N₀ is the index of the center frequency. Moreover, the value of κ′_n,n−1 is obtained by substituting n → n − 1 in the above formula. Similarly:

κ_{n, l, k, n + k - l}^{'} = κ_{n, l, k, n + k - l}^{'} (l_{m})

is computed algebraically.

2.3. Equation for the mode amplitudes

We obtain finally:

\begin{array}{r} {\dot{A}}_{n} - i (\frac{ω_{n}^{2} - ω_{n c}^{2}}{2 ω_{n}}) A_{n} = - \frac{1}{2 τ_{c}} A_{n} \\ - \frac{i ω_{n} N μ_{21}}{ε_{0} n_{0}^{2}} \frac{i}{2 γ_{12}} {\tilde{G}}_{n} [Ω_{n} Δ ρ_{p, 0} κ_{n, n} + Ω_{n + 1} M_{+} κ_{n, n + 1}^{'} \frac{Δ ρ_{p, M}^{*}}{2} + Ω_{n - 1} M_{-} κ_{n, n - 1}^{'} \frac{Δ ρ_{p, M}}{2}] \\ - \frac{i ω_{n} N μ_{21}}{ε_{0} n_{0}^{2}} \frac{- i}{4 γ_{22} γ_{12}^{2}} {\tilde{G}}_{n} \sum_{l} \sum_{k} C_{k l} B_{k l} Ω_{n + k - l} Ω_{l} {Ω_{k}^{*} κ_{n, k, l, n + k - l} Δ ρ_{p, 0} \\ + Ω_{k - 1}^{*} M_{+} κ_{n, k - 1, l, n + k - 1 - l}^{'} \frac{Δ ρ_{p, M}^{*}}{2} + Ω_{k + 1}^{*} M_{-} κ_{n, k + 1, l, n + k + 1 - l}^{'} \frac{Δ ρ_{p, M}}{2}} \end{array}

We rewrite the above equation using the threshold gain g₀:

g_{0} = \frac{ω_{n} τ_{c} N μ_{21}^{2} Δ ρ_{p, 0}}{2 ε_{0} n_{0}^{2} ℏ γ_{12}}

as well as the modulation fraction δ :

δ = \frac{Δ ρ_{p, m}}{2 Δ ρ_{p, 0}}

and the fact that

κ_{n, n} = \frac{1}{2}

. We have then:

\begin{array}{r} 2 τ_{c} {\dot{A}}_{n} - 2 i τ_{c} (\frac{ω_{n}^{2} - ω_{n c}^{2}}{2 ω_{n}}) A_{n} = - A_{n} + g_{0} {\tilde{G}}_{n} A_{n} + 2 g_{0} {\tilde{G}}_{n} [A_{n + 1} M_{+} δ κ_{n, n + 1}^{'} + A_{n - 1} M_{-} δ^{*} κ_{n, n - 1}^{'}] \\ - \frac{μ_{21}^{2}}{ℏ^{2} γ_{22} γ_{12}} g_{0} {\tilde{G}}_{n} \sum_{k, l} C_{k l} B_{k l} A_{m} A_{l} {A_{k}^{*} κ_{n, k, l, m} + A_{k - 1}^{*} M_{+} δ^{*} κ_{n, k - 1, l, m_{+}}^{'} + A_{k + 1}^{*} M_{-} δ κ_{n, k + 1, l, m_{-}}^{'}} . \end{array}

with

m = n + k - l

m_{+} = n + k - 1 - l

m_{-} = n + k + 1 - l

As the next step, we normalize the fields to the saturation field:

A_{sat} = \frac{ℏ \sqrt{γ_{12} γ_{22}}}{μ_{21}}

and rewrite the gain G_n = g₀G̃_n to obtain:

\begin{array}{r} 2 τ_{c} {\dot{A}}_{n} = {G_{n} - 1 + i 2 τ_{c} (\frac{ω_{n}^{2} - ω_{n c}^{2}}{2 ω_{n}})} A_{n} + 2 G_{n} [A_{n + 1} M_{+} δ^{*} κ_{n, n + 1}^{'} + A_{n - 1} M_{-} δ κ_{n, n - 1}^{'}] \\ - G_{n} \sum_{k, l} C_{k l} B_{k l} A_{m} A_{l} {A_{k}^{*} κ_{n, k, l, m} + A_{k - 1}^{*} M_{+} δ^{*} κ_{n, k - 1, l, m_{+}}^{'} + A_{k + 1}^{*} M_{-} δ κ_{n, k + 1, l, m_{-}}^{'}} . \end{array}

As the last step, we normalize the time to 2τ_c to obtain our final result:

\begin{array}{r} {\dot{A}}_{n} = {\underset{Net gain}{\underset{︸}{G_{n} - 1}} + i \underset{Cavity dispersion}{\underset{︸}{(\frac{ω_{n}^{2} - ω_{n c}^{2}}{2 ω_{n}})}}} A_{n} + 2 G_{n} \underset{Modulation}{\underset{︸}{[A_{n + 1} M_{+} δ^{*} κ_{n, n + 1}^{'} + A_{n - 1} M_{-} δ κ_{n, n - 1}^{'}]}} \\ - G_{n} \sum_{k, l} \underset{FWM term}{\underset{︸}{C_{k l} B_{k l} A_{m} A_{l} {A_{k}^{*} κ_{n, k, l, m} + A_{k - 1}^{*} M_{+} δ^{*} κ_{n, k - 1, l, m_{+}}^{'} + A_{k + 1}^{*} M_{-} δ κ_{n, k + 1, l, m_{-}}^{'}}}} . \end{array}

This equation is identical to the result presented in [12] to the exception of the addition of the dispersion term and of the modulation.

2.4. Specific cases and simplifications

The expression is simplified when the whole device is uniformly modulated, i.e. l_m = 1. In that case, κ′_n,n±1 = 0 and we have:

\begin{array}{r} {\dot{A}}_{n} = {G_{n} - 1 + i (\frac{ω_{n}^{2} - ω_{n c}^{2}}{2 ω_{n}})} A_{n} \\ - G_{n} \sum_{k, l} C_{k l} B_{k l} A_{m} A_{l} {A_{k}^{*} κ_{n, k, l, m} + A_{k - 1}^{*} M_{+} δ^{*} κ_{n, k - 1, l, m_{+}} + A_{k + 1}^{*} M_{-} δ κ_{n, k + 1, l, m_{-}}} \end{array}

meaning that the modulation enters only through the FWM term. Denoting the FWM sum:

S_{n} = \sum_{k, l} C_{k l} B_{k l} A_{m} A_{l} A_{k}^{*} κ_{n, k, l, m}

and assuming that:

\begin{array}{l} C_{k \pm 1, l} \approx C_{k l} \\ B_{k \pm 1, l} \approx B_{k l} \end{array}

(since we assume ω ≪ γ₁₂, γ₂₂) we note that the modulation terms are the FWM terms, evaluated for the modes n ± 1. As a result, we have then a form that is numerically more efficient to evaluate:

{\dot{A}}_{n} = {G_{n} - 1 + i (\frac{ω_{n}^{2} - ω_{n c}^{2}}{2 ω_{n}})} A_{n} - G_{n} (S_{n} + δ^{*} M_{+} S_{n + 1} + δ M_{-} S_{n - 1}) .

The same procedure can be done in the general case (l_m ≠ 1), by using:

S_{n}^{'} = \sum_{k, l} C_{k l} B_{k l} A_{m} A_{l} A_{k}^{*} κ_{n, k, l, m}^{'},

Eq. (65) is now

\begin{array}{r} {\dot{A}}_{n} = {G_{n} - 1 + i (\frac{ω_{n}^{2} - ω_{n c}^{2}}{2 ω_{n}})} A_{n} + 2 G_{n} [A_{n + 1} M_{+} δ^{*} κ_{n, n + 1}^{'} + A_{n - 1} M_{-} δ κ_{n, n - 1}^{'}] \\ - G_{n} (S_{n} + δ^{*} M_{+} S_{n + 1}^{'} + δ M_{-} S_{n - 1}^{'}) \end{array}

2.5. Dispersion

Althought it was not discussed in [12], the formalism developed allows to take into account dispersion. The latter will arise from both the gain G_n as well as from the four-wave mixing and modulation terms M_±, C_kl, B_kl that are all complex quantities. Because of the dispersion, the frequencies are pulled from their equidistant values, and as a result the phase of the mode amplitudes A_n will be rotating with time. Actually, after solving for the time-dependent amplitude A_n, the frequency offset δω_n of a specific mode n from its reference value is obtained directly using:

δ ω_{n} = \frac{d (arg (A_{n}))}{d t}

The dispersion in the gain will be compensated, at least partially, by the dispersion of the cavity given by the term:

D_{n} = (\frac{ω_{n}^{2} - ω_{n c}^{2}}{2 ω_{n}})

where the empty cavity frequency ω_nc takes into account all the remaining dispersion due to the material and the waveguide and could have a priori any form. To enable a parametrization of that dispersion, we assume the cavity is characterized by a group index n_g and a constant group velocity dispersion (GVD).

The resonance condition is that, for the mode number n, we have:

k (ω_{n}) = \frac{(N_{0} + n) π}{l_{c}} .

We use a second order expansion for k(ω_n) to parametrize the dispersion:

k (ω_{n c}) = k (ω_{0}) + \frac{\partial k}{\partial ω} (ω_{n c} - ω_{0}) + \frac{1}{2} \frac{\partial^{2} k}{\partial ω^{2}} {(ω_{n c} - ω_{0})}^{2}

The group index is by definition:

n_{g} = c \frac{\partial k}{\partial ω}

while the GVD is:

GVD = \frac{\partial^{2} k}{\partial ω^{2}}

Furthermore, we assume the length of the cavity is matched in the center of the gain, so ω₀ = ω₂₁ and thus

k (ω_{0}) = \frac{N_{0} π}{l_{c}}

. We obtain therefore a second order algebraic equation for δω_nc = ω_nc − ω₂₁:

δ ω_{n c}^{2} + 2 \frac{n_{g}}{c GVD} δ ω_{n c} - \frac{2 n π}{l_{c} GVD} = 0 .

We introduce the cavity mode spacing ω_c given by:

ω_{c} = \frac{π c}{n_{g} l_{c}}

as well as a normalized (unitless) GVD parameter given by:

β_{GVD} = \frac{ω_{c} c GVD}{n_{g}}

The solution of Eq. (78) is then:

δ ω_{n c} = \frac{ω_{c}}{β_{GVD}} (- 1 + \sqrt{1 + 2 n β_{GVD}})

which gives, after expanding the square root to the second order:

δ ω_{n c} = n ω_{c} (1 - \frac{n}{2} β_{GVD})

The dispersion term can then be computed:

D_{n} = (\frac{ω_{n}^{2} - ω_{n c}^{2}}{2 ω_{n}}) = (\frac{{(ω_{21} + n ω)}^{2} - {(ω_{21} + δ ω_{n c})}^{2}}{2 (ω_{21} + n ω)})

After some algebra and expanding the square roots to the second order we find:

D_{n} = ω {\frac{n N_{0}}{N_{0} + n} [1 - {\tilde{ω}}_{c} + \frac{1}{2} n {\tilde{ω}}_{c} β_{GVD}] + \frac{n^{2}}{2 (N_{0} + n)} (1 - {\tilde{ω}}_{c}^{2})}

where:

{\tilde{ω}}_{c} = \frac{ω_{c}}{ω}

is the normalized round trip frequency. In practice the parameter ω̃_c is adjusted manually to minimize the offset frequencies, as in real device ω is not imposed externally (except for radio frequency injection locking) but is obtained from the device itself.

2.6. Broad gain devices

MIR comb operation was actually demonstrated in broad-gain devices. To simulate such a structure, one should think of a number of two-level systems coupled to the same cavity modes. Each two-level system will create its own polarization that has to be summed afterwards. We concentrate here on the simplest structure comprising two active regions with equal gain, each offset from ω₀ by a quantity ±n_offω. As such, the total gain $G_{n}^{(1 + 2)}$ of such a stack would be:

G_{n}^{(1 + 2)} = \frac{g_{0}}{2} (\frac{1}{1 - i (n - n_{off}) ω τ_{coh}} + \frac{1}{1 - i (n + n_{off}) ω τ_{coh}})

Similarly, the coefficient B_kl for the active region (1) tuned to (n − n_off)ω will yield:

B_{k l}^{(1)} = \frac{1}{2} (\frac{1}{1 - i (l - n_{off}) ω τ_{coh}} + \frac{1}{1 + i (k - n_{off}) ω τ_{coh}}),

while the term C_kl is unaffected since it depends only on the difference of the index k and l.

A possibility to reduce the computation load is to make the following approximation when computing the FWM term:

G_{n}^{(1)} \sum_{k, l} C_{k l} B_{k l}^{(1)} A_{m} A_{l} A_{k}^{*} + G_{n}^{(2)} \sum_{k, l} C_{k l} B_{k l}^{(2)} A_{m} A_{l} A_{k}^{*} \approx G_{n}^{(1 + 2)} \sum_{k, l} C_{k l} B_{k l}^{(1 + 2)} A_{m} A_{l} A_{k}^{*}

as the resonant nature of G_n makes it a reasonable approximation. In this way, the general form of the equation 65 is unchanged, only the parameters G_n and B_kl must be recomputed.

3. Results

3.1. Dispersion in MIR QCL frequency combs

As already mentioned, the influence of dispersion on the behaviour of the modes was not considered in the first attempt to simulate QCL frequency combs [12]. Nevertheless, dispersion plays a crucial role on mode-locking of lasers [1]. Dispersion in QCL may arise from the dispersion of the different materials used in active region and in the cladding layers, from the dispersion created by the cavity (also called modal dispersion). Ultimately, an effective dispersion is induced by the laser gain as well as by the FWM gain, as the terms G_n, δ, M_±, C_kl and B_kl can assume complex values and their imaginary parts can therefore induce dispersion. In this section we use our model to investigate the impact of all these sources of dispersion on the modal behaviour of a MIR QCL. For this purpose, we simulate a typical MIR QCL by setting τ₂ = 0.5 ps, τ_coh = 0.1 ps. The device studied is 6 mm long (τ_rt = 128 ps) and the photon lifetime in the cavity τ_c = 26 ps, corresponding to total losses of 4.1 cm ⁻¹. The device is assumed to be free-running and no active modulation or saturable absorber is considered. However, all the dispersion terms are taken into account in the simulation. Therefore, Eq. (65) reduces now to:

{\dot{A}}_{n} = {G_{n} - 1 + i D_{n}} A_{n} - G_{n} \sum_{k, l} C_{k l} B_{k l} A_{m} A_{l} A_{k}^{*} κ_{n, k, l, m}

The normalised GVD parameter β_GVD is used in order to account for material and modal dispersion. Laser gain and FWM gain induced dispersion are also considered by letting the terms G_n, C_kl and B_kl assume complex values.

The results of a typical MIR QCL frequency comb with GVD = −500 fs²mm⁻¹ are shown in Fig. 2. The optical amplitude spectrum as well as the phase spectrum are represented in Fig. 2(a). The amplitudes of the modes are distributed in a somehow random way, characteristic of MIR QCL frequency combs. In addition, the phase difference between adjacent modes is not the same for all pairs of modes, as it is the case for lasers emitting transform limited pulses. Instead, the amplitudes and the phases are distributed in way such that the total power is nearly constant, as shown in Fig. 2(b), where the output power is represented as a function of time. This behaviour is characteristic of a frequency-modulated laser and is also illustrated in Fig. 2(c), where the oscillations of the instantaneous frequency with time are represented. Finally, Fig. 2(d) shows the amplitude spectrum of the detected power (corresponding to the spectrum of the photocurrent when measured by a spectrum analyser). We observe that all the harmonics of the detected power are around 3 orders of magnitudes lower than the direct current (DC) value, corresponding to a constant detected power. The phase-locking mechanism of such a free-running, continuous-wave QCL is four-wave mixing which, combined with the short gain recovery time of a QCL, leads to a phase signature comparable to a frequency modulated laser.

Fig. 2 Free-running MIR QCL (see Media 1). a) Optical amplitude and phase spectrum. b) Instantaneous power as function of time. c) Instantaneous frequency as function of time. d) Radio frequency amplitude spectrum of the detected power, equivalent to the photocurrent spectrum measured with a spectrum analyzer.

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We characterise the impact of the dispersion by computing the value $\bar{ν_{n}}$ corresponding to the average of the frequency of the mode n, expressed by:

\bar{ν_{n}} = \frac{τ_{r t}}{δ t} \frac{1}{N_{steps}} \sum_{i} \frac{φ_{n}^{i + 1} - φ_{n}^{i}}{δ t}

where

φ_{n}^{i}

represents the phase of the mode n at the computational step i, δt represents the simulation time step and N_steps is the total number of simulation steps. A linear variation of

\bar{ν_{n}}

with the mode number n is observed when the cavity mode spacing ω_c and the comb line spacing ω are not matched. As discussed before, the normalized round trip frequency ω̃_c is manually adjusted to minimize the offset between these two frequencies. If ω_c and ω are perfectly matched,

\bar{ν_{n}}

should be identically zero for all values of n. However, if dispersion is present, the comb line spacing will start to increase (or decrease) with the mode number n and a curvature of

\bar{ν_{n}}

shall be observed. This is schematically explained in Fig. 3(a).

Fig. 3 Impact of the cavity dispersion on a free-running MIR QCL. a) Schematic description of the impact of dispersion on the laser modes. b) Average of the frequency of the mode n $\bar{ν_{n}}$ for three values of simulated dispersion (GVD = −500 fs²mm⁻¹ and GVD = ±30000 fs²mm⁻¹).

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Figure 3(b) shows the frequency offset $\bar{ν_{n}}$ of the MIR QCL previously described. We observe that $\bar{ν_{n}}$ is identically zero over the entire laser spectrum when GVD = −500 fs².mm⁻¹, showing that this value of GVD is relatively low and does not disturb the comb formation. In addition, we study the same devices with a stronger value of dispersion (GVD = ±30000 fs²mm⁻¹). These two results are also shown in Fig. 3(b). In agreement with our predictions, we observe a curvature of $\bar{ν_{n}}$ , demonstrating that these values of dispersion are enough to disturb the frequency comb operation of the QCL. Moreover, we see that changing the sign of the GVD modifies the sign of the curvature of $\bar{ν_{n}}$ .

Finally, as the first QCL frequency comb was demonstrated using a broad gain device [8], we use our model to simulate a MIR QCL containing two active regions with different central frequencies. We assume the same values for τ₂, τ_coh and τ_rt as the device previously studied. The total losses are assumed to be slightly higher than the previous case, corresponding to a cavity lifetime of τ_c = 26 ps, as broad gain devices usually experience higher total losses than single active region devices. The two active regions have equal gain and are centered at 1375 cm⁻¹ and 1436 cm⁻¹, corresponding to an offset of 61 cm⁻¹. Figure 4(a) shows the gain profile as well as the optical amplitude spectrum of such a broad gain device. The flat broad gain profile allows multimode emission, as more than 150 modes are present in the optical amplitude spectrum. Figure 4(b) shows the amplitude spectrum of the detected power, which indicates that no amplitude modulation is observed in this broad gain device. The results of our model agree qualitatively with the measurements reported in [8], which are represented in Fig. 4(a) and (b).

Fig. 4 Broad gain MIR QCL. a) Gain profile as well as optical amplitude spectrum of a MIR QCL containing two active regions centered at 1375 cm⁻¹ and 1436 cm⁻¹, corresponding to an offset of 61 cm⁻¹. b) Radio frequency amplitude spectrum of the detected power, equivalent to the photocurrent spectrum measured with a spectrum analyzer. c) Optical spectrum of a MIR QCL frequency comb, measured with a Fourier transform infrared spectrometer (0.12 cm⁻¹ resolution), centered at 1430 cm⁻¹ and covering 60 cm⁻¹. d) Radio frequency amplitude spectrum of the detected power, measured with fast quantum well infrared detector (QWIP) and a spectrum analyzer (RBW = 10 Hz), showing an narrow beat note at the round trip frequency with an extremely low amplitude.

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3.2. THz QCL active mode-locking

In the mid-infrared, active mode-locking of ”conventional” QCL devices, i.e. devices with sub picosecond upper state lifetimes, remains extremely unfavorable because of the short upper state lifetime. In recent experiments, the beatnote of a mid-infrared QCL device was locked by electrical injection up to a frequency of 14GHz [17] but later measurements of interferometric autocorrelation did not show evidence for pulses, much like earlier studies [15]. Actually, RF injection at the round-trip frequency in a mid-IR QCL comb was found to destabilize the comb formation above a certain injection power level [8].

In contrast, active mode-locking of a THz QCL has been experimentally demonstrated by modulating the QCL bias current at the round-trip frequency on a small section of the device [5]. A coherent sampling technique - which can be seen as an optical sampling oscilloscope acquisition scheme - was used to measure transform-limited pulses with a duration of ∼10 ps. In this section we show that our model can also predict active mode-locking of THZ QCLs. As laser action in THz QCLs occurs between two states with energy spacing below the optical phonon energy, the typical lifetimes in THz QCLs are significantly higher than of MIR QCLs. This has a significant impact on the mode-locking mechanism. For a typical THz QCL, we assume τ₂ = 12 ps, τ_coh = 0.7 ps. The device studied is 3 mm long (τ_rt = 63 ps) and the photon lifetime is τ_c = 21 ps, corresponding to a total losses of 5.1 cm ⁻¹. The device is actively modulated at the round trip frequency ω. The device is DC-biased (1.2G₀) over 90% of its length and a modulation (3.5G₀) at the round trip frequency ω is applied on the other section of the device. Moreover, all the dispersion terms are taken into account in the simulation. In this case, all the terms of Eq. (65) are taken into account in this study.

Figure 5(a) shows the optical amplitude spectrum as well as the phase spectrum of an active mode-locked THz QCL. We observe a somehow gaussian distribution for the amplitude of the different modes and the phases show a linear variation across the spectrum, in agreement with the behaviour of an active mode-locked laser. The instantaneous power is plotted in Fig. 5(b). Narrow pulses with a FWHM of 7.3 ps are predicted in this case, in agreement with the value measured in [5]. The amplitude spectrum of the detected power, shown in Fig. 5(c), also is characteristic of a pulsed output with high power harmonics. Figure 5(d) shows the time domain evolution of the amplitude of each mode. We observe a transient regime (< 5ns) where the laser turns on and therefore the amplitudes are not stable. After this transient regime, we observe a steady-state regime where all the mode amplitudes are extremely stable with time.

Fig. 5 Active mode-locked THz QCL (see Media 2). a Optical amplitude and phase spectrum. b Instantaneous power as function of time. c Radio frequency amplitude spectrum of the detected power, equivalent to the photocurrent spectrum measured with a spectrum analyzer. d Time domain evolution of the modes amplitude.

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3.3. Transport

One very specific feature of quantum cascade lasers, due to the very fast scattering time, is the fact that the transport is driven by the photon field up to very high frequencies. As a result, any amplitude modulation of the optical field will be reflected on the driving current of the device. This fact was observed in many experiments in both the mid- and far-infrared frequencies where the same beatnote was measured either using an external detector or a probe on the current of the device. For this reason, we do have a positive feedback loop, where the (self) generation of an amplitude modulation does create a current modulation of the device.

Because our model is both able to predict the amount of amplitude modulation (by beating of the mode amplitude) and the effect of a current, the effect of such a feedback can be computed. In our model, the driving current is proportional to g₀, and an uniteless photon flux can be written as:

S = \frac{3}{8} \sum_{n} {| A_{n} |}^{2}

so that the slope efficiency at threshold, in the perfect case:

\frac{\partial S}{\partial g_{0}} = 1 .

Assuming the device would have a zero differential resistance above threshold and, because of the capacitance of the active region, its voltage would remain constant above threshold, any modulation in the photon flux ΔS would translate in an identical modulation of the gain Δg₀ and therefore in Δρ_m. In contrast, if the device had a very large differential resistance, the same ΔS would induce no current modulation at all (in case the current is completely controlled by the tunneling through the injection barrier). For this reason, we introduce a dimensionless feedback parameter 0 < |μ_m| < 1 that describes the amount of current feedback arising from the modulation, such that we use, at each self-consistent step:

g_{m} = μ_{m} \sum_{n} A_{n} A_{n + 1}^{*}

and we neglect the higher frequency modulation terms arising from

A_{n} A_{n + 2, 3 . .}^{*}

. In principle the value of μ_m could be deduced from a microscopic model of the active region, more specifically using models able to solve the voltage-current characteristics in presence of a light field [18]. For this purpose, we assumed the same THz QCL investigated before and we replace the active modulation at the round trip frequency by the self-generated current modulation expressed by Eq. (93). The results show an optical amplitude spectrum consisting of 2 modes, similar to the amplitude spectrum of the same device when no modulation (external or self-generated) is applied. Also, no sign of pulse generation is observed.

4. Conclusion

We have theoretically studied the frequency comb operation of quantum cascade lasers using a coupled-mode formalism based on Maxwell-Bloch equations. Cavity dispersion and external modulation were taken into account in the model. The model agreed with the experimental results both in the mid- and far-infrared. In particular, it confirmed that active mode-locking could be induced by modulating a short section of the device, when the lifetime of the upper state is long enough. The model, however, confirmed the experimental facts that such active mode-locking occurs in a rather short range of operating conditions and lead to a relatively limited bandwidth. Even more difficult seems the achievement of mode-locking using self-induced transparency as gain and loss have to be carefully calibrated in the same cavity [19, 20]. For this reason, it appears that mode-locking based on four-wave mixing in broadband gain, low dispersion cavities is the most promising way of achieving broadband quantum cascade laser optical frequency combs. A very interesting extension of that model would be to include the Langevin noise terms in order to predict the frequency and amplitude noise, for which data are becoming available [9].

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Quantum cascade laser combs: effects of modulation and dispersion

Abstract

1. Introduction

2. Theory

2.1. Density matrix

2.2. Cavity modes

2.3. Equation for the mode amplitudes

2.4. Specific cases and simplifications

2.5. Dispersion

2.6. Broad gain devices

3. Results

3.1. Dispersion in MIR QCL frequency combs

3.2. THz QCL active mode-locking

3.3. Transport

4. Conclusion

References and links

Supplementary Material (6)

Cited By

Figures (5)

Equations (93)

Optics Express