Abstract

Frequency comb formation in quantum cascade lasers is studied theoretically using a Maxwell-Bloch formalism based on a modal decomposition, where dispersion is considered. In the mid-infrared, comb formation persists in the presence of weak cavity dispersion (500 fs2 mm−1) but disappears when much larger values are used (30’000 fs2 mm−1). Active modulation at the round-trip frequency is found to induce mode-locking in THz devices, where the upper state lifetime is in the tens of picoseconds. Our results show 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 frequency combs.

© 2015 Optical Society of America

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 τrt, 64 ps for a 3 mm long device, such that the product ωτ2 << 1, where ω = 2π/τrt 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 Ωg, 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=2gMω2Ωg24,
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.

 figure: Fig. 1

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:

ρ˙=i[,ρ]+ρt)coll
where ρ is the density matrix containing the populations ρ11 and ρ22 as well as the coherences ρ12 and ρ21, 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:
=(ω1Ω˜12(t)Ω˜21(t)ω2)
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:
Ω˜12(t)=Ω˜21(t)=μ21E(t)/
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:
ρt)coll=(γ11(ρ11ρ11,p)γ12ρ12γ12ρ21γ22(ρ22ρ22,p))
where γ11=τ11 and γ22=τ21 are the scattering rates out of the ground state and excited state, respectively, and γ12=τcoh1 is the loss of coherence of the transition. A constant pumping with a rate γ22ρ22,p is such that an equilibrium population ρ22,p of the upper state can be reached. Similarly, the term ρ11,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):
Δρ˙=iΩ˜12(ρ21ρ12)γ22(ΔρΔρp)ρ˙21=iω21ρ21+iΩ˜12Δργ12ρ21ρ21=ρ12*
where ω21 = ω2ω1 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:
ωl=ω21+lωl{nm,nm+1,.,0,.,nm}
so we have a total of (2nm + 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:
E(t)=12eiω21tl=nml=nmAleilωt+12eiω21tl=nml=nmAl*eilωtΩ˜12(t)=12eiω21tl=nml=nmΩleilωt+12eiω21tl=nml=nmΩl*eilωt
where the Rabi frequencies are related to the amplitudes through:
Ωl=μ21Al
We expand the density matrix in a Fourier series with the following time dependence:
ρ21=eiω21tk=nmk=nmσ21,keikωtρ12=eiω21tk=nmk=nmσ21,k*eikωtΔρ=r=nmr=nmΔρreirωt
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:
Δρp=Δρp,0+12(Δρp,Meiωt+Δρp,M*eiωt)
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:
r(irω+γ22)Δρreirωt=γ22(Δρp,0+12(Δρp,Meiωt+Δρp,M*eiωt))+i2kl(Ωl*σ21,kei(kl)ωtΩlσ21,k*ei(kl)ωt)
And then for the second:
k(γ12ikω)σ21,keikωt=i2lrΩlΔρrei(l+r)ωt
Both equations are solved by matching the time dependences eikωt on both sides. For the first one, l = k + r and l = kr in the sum, which yields:
Δρr=1irω+γ22[γ22Δρp,0δr,0+γ22Δρp,M*2δr,1+γ22Δρp,M2δr,1+i2k(Ωkr*σ21,kΩk+rσ21,k*)]
The second one l + r = k and yields:
σ21,k=i2(γ12ikω)lΩlΔρkl

We then solve these equations using a perturbative approach on Δρr and σ21,k. The zeroth order for Δρr(0) yields from Eq. (14):

Δρr=0(0)=Δρp,0
Δρr=1(0)=γ22γ22iωΔρp,M2
Δρr=1(0)=γ22γ22+iωΔρp,M*2
We have taken σ21,k(0)=0. Inserting then this values for Δρ(0) in Eq. (15) then yields:
σ21,k(1)=12(iγ12+kω)[ΩkΔρp,0+Ωk+1γ22γ22+iωΔρp,M*2+Ωk1γ22γ22iωΔρp,M2]
Inserting now Eq. (16) in Eq. (12) yields (for values of r ≠ (−1, 0, −1)):
Δρr(2)=i2(γ22irω)k(Ωkr*σ21,k(1)Ωk+rσ21,k(1)*)
Inserting now the value of σ21,k(1) yields:
Δρr(2)=i2(γ22irω)k(Ωkr*21(iγ12+kω)[ΩkΔρp,0+Ωk+1γ22γ22+iωΔρp,M*2+Ωk1γ22γ22iωΔρp,M2]Ωk+r21(iγ12+kω)[Ωk*Δρp,0+Ωk+1*γ22γ22iωΔρp,M2+Ωk1*γ22γ22+iωΔρp,M*2])
After an algebraic arrangement, it yields:
Δρr(2)=i2(γ22irω)k{(Ωkr*ΩkΔρp,02(iγ12kω)Ωk+rΩk*Δρp,02(iγ12kω))+(γ22γ22+iωΩkr*Ωk+1Δρp,M*/22(iγ12kω)γ22γ22iωΩk+rΩk+1*Δρp,M/22(iγ12kω))+(γ22γ22iωΩkr*Ωk1Δρp,M/22(iγ12kω)γ22γ22+iωΩk+rΩk1*Δρp,M*/22(iγ12kω))}

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

Δρr(2)=14i(γ22irω)kΩk+r{Ωk*Δρp,0(1iγ12(k+r)ω1iγ12kω)+Ωk1*γ22γ22+iωΔρp,M*2(1iγ12(k1+r)ω1iγ12kω)+Ωk+1*γ22γ22iωΔρp,M2(1iγ12(k+1+r)ω1iγ12kω)}
To find the third order term σ21,k(3) we write using Eq. (15):
σ21,n(3)=i2(γ12inω)lΩlΔρnl(2)
By inserting Eq. (23) into Eq. (24), we have:
σ21,n(3)=i2(γ12inω)14lkΩli(γ22i(nl)ω)Ωk+nl{Ωk*Δρp,0(1iγ12(k+nl)ω1iγ12kω)+Ωk1*γ22γ22+iωΔρp,M*2(1iγ12(k1+nl)ω1iγ12kω)+Ωk+1*γ22γ22iωΔρp,M2(1iγ12(k+1+nl)ω)1iγ12kω)}
By doing the following substitution l′ = n + kl, we get:
σ21,n(3)=i21γ12inω14lkΩn+kli(γ22i(lk)ω)Ωl{Ωk*Δρp,0(1iγ12lω1iγ12kω)+Ωk1*γ22γ22+iωΔρp,M*2(1iγ12(l1)ω1iγ12kω)+Ωk+1*γ22γ22iωΔρp,M2(1iγ12(l+1)ω)1iγ12kω)}
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 γ12ω, as the gain is broad enough to sustain many modes. Therefore, we can assume that:

1iγ12(l±1)ω1iγ12lω
As a result, the modulation term has the same overall form as the FWM term, to the exception of the pre factors:
M±=γ22γ22±iω
substituting the direct injection Δρp,0 by Δρp,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:
Ckl=γ22γ22i(lk)ω
as well as the transition gain:
G˜n=iγ12nω+iγ12
the (complex) gain coefficient for the mode n. We also write the term driving the width of the FWM gain:
Bkl=γ122i(1iγ12lω1iγ12kω)

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

σ21,n(3)=i4γ22γ122G˜nlkCklBklΩn+klΩl{Ωk*Δρp,0+Ωk1*M+Δρp,M*2+Ωk+1*MΔρp,M2}
σ21,n(1)=i2γ12G˜n[ΩnΔρp,0+Ωn+1M+Δρp,M*2+Ωn1MΔρp,M2]

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:

2Eμ0σEtεrc22Et2=μ02Pt2
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):
E(t)=12l=nml=nmAleiωlt+12l=nml=nmAl*eiωltP(t)=12l=nml=nmPleiωlt+12l=nml=nmPl*eiωlt
The modes have a spatial dependence given by an envelope An(z):
An(z)=Ansin(knz)
The polarization for the mode n is written as:
Pn=Nμ=NTr[ρμ]=Nμ21σ21,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 kn for the n-th eigenmode by:

knlc=(N0+n)π
where N0 is the index of the center frequency of our laser. The ”cold cavity” frequencies ωnc are then defined by:
ωnc=kncn0
where n0=εr is the refractive index. The frequencies ωnc are usually different from the frequencies ωn in a real cavity, as the cavity dispersion will shift the resonances ωnc according to the ideal dispersionless cavity represented by the frequencies ωn = (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):

12sin(knz){kn2Anμ0σ(A˙niωnAn)εrc2(ωn2An2iωnA˙n)}=ωn2μ0Nμ2112σ21,n(z)
that can be simplified (multipling both sides by c2/n02 and using the definition of ωnc) into:
{((ωn2ωnc2)+iωnσε0εr)An+(2iωnσε0εr)A˙n}sin(knz)=1ε0εrωn2Nμ21σ21,n(z)
We assume that the cavity has a relatively large Q so that ωnσ/(ε0εr), which yields:
2iωnA˙n=((ωn2ωnc2)+iωnσε0εr)An+ωn2Nμ21σ21,n(z)ε0n02sin(knz)
To give an interpretation of our conductivity σ, we assume we have a resonant mode (ωn = ωnc) and no driving polarization (σ21,n = 0). Then the amplitude satisfies:
A˙=σ2ε0εrA
We have for the photon lifetime in the cavity τc the equation for the amplitude:
|A˙|=|A|2τc
justifying that we interpret σ/(ε0εr) as the cavity photon energy decay rate τc1. And finally we obtain:
A˙ni(ωn2ωnc22ωn)An=iωnNμ21σ21,n(z)2ε0n02sin(knz)12τcAn

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

A˙ni(ωn2ωnc22ωn)An=iωnNμ21lcε0n020lcσ21,n(z)sin(knz)dz12τcAn
Now we substitute σ21,n(z) by its expansion:
σ21,n(z)=σ21,n(1)(z)+σ21,n(3)(z)
given by Eq. (33) and (32).

Focusing first on the integral on the spatial coordinate, and recalling the relationship between Ωl and Al, we observe that the term σ21,n(1)(z) will lead to a integral with the form:

1lc0lcAnsin(knz)2dz=Anκn,n=An12
corresponding to the gain, while the terms due to the amplitude modulation will yield to values of κn,n±1 = 0. For the terms σ21,n(3)(z), we have then integral of the form:
1lc0lcAn+klsin(kn+klz)Alsin(klz)Ak*sin(kkz)sin(knz)dz=An+klAlAk*κn,l,k,n+kl
where we have defined the coefficient κn,l,k,m as:
κn,l,k,m=1lc0lcsin(knz)sin(klz)sin(kkz)sin(kmz)dz
The coefficient κn,l,k,n+kl has the following values:
κn,n,n,n=38(k=l=n)κn,n,m,n=14(k=ln)or(l=n)κn,l,k,n+kl=18otherwise

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 Δρp,M has a spatial dependence so that a fraction 0 < lm < 1 of the total length lc is modulated:

Δρp,M(z)=Θ(lmlcz)Δρp,M
where Θ(z) is the Heaviside step function. We define now the coefficient κ′n,l,k,m(lm) as:
κn,l,k,m(lm)=1lc0lmlcsin(knz)sin(klz)sin(kkz)sin(kmz)dz

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

κn,n+1=1lc0lmlcsin(knz)sin(kn+1z)dz=12π{sin(πlm)sin(π(2(n+N0)+1)lm)(2(n+N0)+1)}
where N0 is the index of the center frequency. Moreover, the value of κ′n,n−1 is obtained by substituting nn − 1 in the above formula. Similarly:
κn,l,k,n+kl=κn,l,k,n+kl(lm)
is computed algebraically.

2.3. Equation for the mode amplitudes

We obtain finally:

A˙ni(ωn2ωnc22ωn)An=12τcAniωnNμ21ε0n02i2γ12G˜n[ΩnΔρp,0κn,n+Ωn+1M+κn,n+1Δρp,M*2+Ωn1Mκn,n1Δρp,M2]iωnNμ21ε0n02i4γ22γ122G˜nlkCklBklΩn+klΩl{Ωk*κn,k,l,n+klΔρp,0+Ωk1*M+κn,k1,l,n+k1lΔρp,M*2+Ωk+1*Mκn,k+1,l,n+k+1lΔρp,M2}
We rewrite the above equation using the threshold gain g0:
g0=ωnτcNμ212Δρp,02ε0n02γ12
as well as the modulation fraction δ :
δ=Δρp,m2Δρp,0
and the fact that κn,n=12. We have then:
2τcA˙n2iτc(ωn2ωnc22ωn)An=An+g0G˜nAn+2g0G˜n[An+1M+δκn,n+1+An1Mδ*κn,n1]μ2122γ22γ12g0G˜nk,lCklBklAmAl{Ak*κn,k,l,m+Ak1*M+δ*κn,k1,l,m++Ak+1*Mδκn,k+1,l,m}.
with
m=n+kl
m+=n+k1l
m=n+k+1l
As the next step, we normalize the fields to the saturation field:
Asat=γ12γ22μ21
and rewrite the gain Gn = g0n to obtain:
2τcA˙n={Gn1+i2τc(ωn2ωnc22ωn)}An+2Gn[An+1M+δ*κn,n+1+An1Mδκn,n1]Gnk,lCklBklAmAl{Ak*κn,k,l,m+Ak1*M+δ*κn,k1,l,m++Ak+1*Mδκn,k+1,l,m}.
As the last step, we normalize the time to 2τc to obtain our final result:
A˙n={Gn1Netgain+i(ωn2ωnc22ωn)Cavitydispersion}An+2Gn[An+1M+δ*κn,n+1+An1Mδκn,n1]ModulationGnk,lCklBklAmAl{Ak*κn,k,l,m+Ak1*M+δ*κn,k1,l,m++Ak+1*Mδκn,k+1,l,m}FWMterm.
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. lm = 1. In that case, κ′n,n±1 = 0 and we have:

A˙n={Gn1+i(ωn2ωnc22ωn)}AnGnk,lCklBklAmAl{Ak*κn,k,l,m+Ak1*M+δ*κn,k1,l,m++Ak+1*Mδκn,k+1,l,m}
meaning that the modulation enters only through the FWM term. Denoting the FWM sum:
Sn=k,lCklBklAmAlAk*κn,k,l,m
and assuming that:
Ck±1,lCklBk±1,lBkl
(since we assume ωγ12, 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:
A˙n={Gn1+i(ωn2ωnc22ωn)}AnGn(Sn+δ*M+Sn+1+δMSn1).
The same procedure can be done in the general case (lm ≠ 1), by using:
Sn=k,lCklBklAmAlAk*κn,k,l,m,
Eq. (65) is now
A˙n={Gn1+i(ωn2ωnc22ωn)}An+2Gn[An+1M+δ*κn,n+1+An1Mδκn,n1]Gn(Sn+δ*M+Sn+1+δMSn1)

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 Gn as well as from the four-wave mixing and modulation terms M±, Ckl, Bkl 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 An will be rotating with time. Actually, after solving for the time-dependent amplitude An, the frequency offset δωn of a specific mode n from its reference value is obtained directly using:

δωn=d(arg(An))dt
The dispersion in the gain will be compensated, at least partially, by the dispersion of the cavity given by the term:
Dn=(ωn2ωnc22ωn)
where the empty cavity frequency ωnc 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 ng and a constant group velocity dispersion (GVD).

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

k(ωn)=(N0+n)πlc.
We use a second order expansion for k(ωn) to parametrize the dispersion:
k(ωnc)=k(ω0)+kω(ωncω0)+122kω2(ωncω0)2
The group index is by definition:
ng=ckω
while the GVD is:
GVD=2kω2
Furthermore, we assume the length of the cavity is matched in the center of the gain, so ω0 = ω21 and thus k(ω0)=N0πlc. We obtain therefore a second order algebraic equation for δωnc = ωncω21:
δωnc2+2ngcGVDδωnc2nπlcGVD=0.
We introduce the cavity mode spacing ωc given by:
ωc=πcnglc
as well as a normalized (unitless) GVD parameter given by:
βGVD=ωccGVDng
The solution of Eq. (78) is then:
δωnc=ωcβGVD(1+1+2nβGVD)
which gives, after expanding the square root to the second order:
δωnc=nωc(1n2βGVD)
The dispersion term can then be computed:
Dn=(ωn2ωnc22ωn)=((ω21+nω)2(ω21+δωnc)22(ω21+nω))
After some algebra and expanding the square roots to the second order we find:
Dn=ω{nN0N0+n[1ω˜c+12nω˜cβGVD]+n22(N0+n)(1ω˜c2)}
where:
ω˜c=ωcω
is the normalized round trip frequency. In practice the parameter ω̃c 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 ±noffω. As such, the total gain Gn(1+2) of such a stack would be:

Gn(1+2)=g02(11i(nnoff)ωτcoh+11i(n+noff)ωτcoh)
Similarly, the coefficient Bkl for the active region (1) tuned to (nnoff)ω will yield:
Bkl(1)=12(11i(lnoff)ωτcoh+11+i(knoff)ωτcoh),
while the term Ckl 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:

Gn(1)k,lCklBkl(1)AmAlAk*+Gn(2)k,lCklBkl(2)AmAlAk*Gn(1+2)k,lCklBkl(1+2)AmAlAk*
as the resonant nature of Gn makes it a reasonable approximation. In this way, the general form of the equation 65 is unchanged, only the parameters Gn and Bkl 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 Gn, δ, M±, Ckl and Bkl 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, τcoh = 0.1 ps. The device studied is 6 mm long (τrt = 128 ps) and the photon lifetime in the cavity τc = 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:

A˙n={Gn1+iDn}AnGnk,lCklBklAmAlAk*κn,k,l,m
The normalised GVD parameter βGVD 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 Gn, Ckl and Bkl 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.

 figure: Fig. 2

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 νn¯ corresponding to the average of the frequency of the mode n, expressed by:

νn¯=τrtδt1Nstepsiφni+1φniδt
where φni represents the phase of the mode n at the computational step i, δt represents the simulation time step and Nsteps is the total number of simulation steps. A linear variation of νn¯ with the mode number n is observed when the cavity mode spacing ωc and the comb line spacing ω are not matched. As discussed before, the normalized round trip frequency ω̃c is manually adjusted to minimize the offset between these two frequencies. If ωc and ω are perfectly matched, νn¯ 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 νn¯ shall be observed. This is schematically explained in Fig. 3(a).

 figure: Fig. 3

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 νn¯ 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 νn¯ of the MIR QCL previously described. We observe that νn¯ 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 νn¯, 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 νn¯.

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, τcoh and τrt as the device previously studied. The total losses are assumed to be slightly higher than the previous case, corresponding to a cavity lifetime of τc = 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).

 figure: Fig. 4

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, τcoh = 0.7 ps. The device studied is 3 mm long (τrt = 63 ps) and the photon lifetime is τc = 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.

 figure: Fig. 5

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=38n|An|2
so that the slope efficiency at threshold, in the perfect case:
Sg0=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 Δρm. 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 < |μm| < 1 that describes the amount of current feedback arising from the modulation, such that we use, at each self-consistent step:
gm=μmnAnAn+1*
and we neglect the higher frequency modulation terms arising from AnAn+2,3..*. In principle the value of μm 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].

References and links

1. H. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1173–1185 (2000). [CrossRef]  

2. J. Faist, F. Capasso, D. Sivco, C. Sirtori, A. Hutchinson, and A. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994). [CrossRef]   [PubMed]  

3. M. C. Tatham, J. F. Ryan, and C. T. Foxon, “Time-resolved Raman measurements of intersubband relaxation in GaAs quantum qells,” Phys. Rev. Lett. 63(15), 1637–1640 (1989). [CrossRef]   [PubMed]  

4. S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nat. Photonics 4(9), 636–640 (2010). [CrossRef]  

5. S. Barbieri, M. Ravaro, P. Gellie, G. Santarelli, C. Manquest, C. Sirtori, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Coherent sampling of active mode-locked terahertz quantum cascade lasers and frequency synthesis (vol 5, pg 306, 2011),” Nat. Photonics 5(6), 378 (2011).

6. C. Y. Wang, L. Kuznetsova, V. M. Gkortsas, L. Diehl, F. X. Kaertner, M. A. Belkin, A. Belyanin, X. Li, D. Ham, H. Schneider, P. Grant, C. Y. Song, S. Haffouz, Z. R. Wasilewski, H. C. Liu, and F. Capasso, “Mode-locked pulses from mid-infrared quantum cascade lasers,” Opt. Express 17(15), 12929–12943 (2009). [CrossRef]   [PubMed]  

7. D. J. Kuizenga and A. Siegman, “FM and AM mode locking of homogeneous Laser-Part I: Theory,” IEEE J. Quantum Electron. 6(11), 694 (1970). [CrossRef]  

8. A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature (London) 492(7428), 229–233 (2012). [CrossRef]  

9. G. Villares, A. Hugi, S. Blaser, and J. Faist, “Dual-comb spectroscopy based on quantum-cascade-laser frequency combs,” Nat. Commun. 5, 1–9 (2014). [CrossRef]  

10. P. Friedli, H. Sigg, B. Hinkov, A. Hugi, S. Riedi, M. Beck, and J. Faist, “Four-wave mixing in a quantum cascade laser amplifier,” Appl. Phys. Lett. 102(22), 222104 (2013). [CrossRef]  

11. W. E. Lamb Jr, “Theory of an optical maser,” Phys. Rev. 134(6A), A1429–A1450 (1964). [CrossRef]  

12. J. B. Khurgin, Y. Dikmelik, A. Hugi, and J. Faist, “Coherent frequency combs produced by self frequency modulation in quantum cascade lasers,” Appl. Phys. Lett. 104(8), 081118 (2014). [CrossRef]  

13. J. Faist, Quantum Cascade Lasers, 1st ed. (Oxford University, 2013). [CrossRef]  

14. H. Choi, L. Diehl, Z.-K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. Norris, “Time-resolved investigations of electronic transport dynamics in quantum cascade lasers based on diagonal lasing transition,” IEEE J. Quantum Electron. 45(4), 307–321 (2009). [CrossRef]  

15. A. Gordon, C. Y. Wang, L. Diehl, F. X. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77(5), 053804 (2008). [CrossRef]  

16. V. M. Gkortsas, C. Wang, L. Kuznetsova, L. Diehl, A. Gordon, C. Jirauschek, M. A. Belkin, A. Belyanin, F. Capasso, and F. X. Kartner, “Dynamics of actively mode-locked quantum cascade lasers,” Opt. Express 18(13), 13616–13630 (2010). [CrossRef]   [PubMed]  

17. M. R. St-Jean, M. I. Amanti, A. Bernard, A. Calvar, A. Bismuto, E. Gini, M. Beck, J. Faist, H. C. Liu, and C. Sirtori, “Injection locking of mid-infrared quantum cascade laser at 14 GHz, by direct microwave modulation,” Laser Photon. Rev. 8(3), 443–449 (2014). [CrossRef]  

18. R. Terazzi and J. Faist, “A density matrix model of transport and radiation in quantum cascade lasers,” New Journal of Physics 12, 033045 (2010). [CrossRef]  

19. V V Kozlov, “Self-induced transparency soliton laser via coherent mode locking,” Phys. Rev. A 56(2), 1607– 1612 (1997). [CrossRef]  

20. C. Menyuk and M. Talukder, “Self-induced transparency modelocking of quantum cascade lasers,” Phys. Rev. Lett. 102(2), 023903 (2009). [CrossRef]   [PubMed]  

References

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  1. H. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1173–1185 (2000).
    [Crossref]
  2. J. Faist, F. Capasso, D. Sivco, C. Sirtori, A. Hutchinson, and A. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
    [Crossref] [PubMed]
  3. M. C. Tatham, J. F. Ryan, and C. T. Foxon, “Time-resolved Raman measurements of intersubband relaxation in GaAs quantum qells,” Phys. Rev. Lett. 63(15), 1637–1640 (1989).
    [Crossref] [PubMed]
  4. S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nat. Photonics 4(9), 636–640 (2010).
    [Crossref]
  5. S. Barbieri, M. Ravaro, P. Gellie, G. Santarelli, C. Manquest, C. Sirtori, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Coherent sampling of active mode-locked terahertz quantum cascade lasers and frequency synthesis (vol 5, pg 306, 2011),” Nat. Photonics 5(6), 378 (2011).
  6. C. Y. Wang, L. Kuznetsova, V. M. Gkortsas, L. Diehl, F. X. Kaertner, M. A. Belkin, A. Belyanin, X. Li, D. Ham, H. Schneider, P. Grant, C. Y. Song, S. Haffouz, Z. R. Wasilewski, H. C. Liu, and F. Capasso, “Mode-locked pulses from mid-infrared quantum cascade lasers,” Opt. Express 17(15), 12929–12943 (2009).
    [Crossref] [PubMed]
  7. D. J. Kuizenga and A. Siegman, “FM and AM mode locking of homogeneous Laser-Part I: Theory,” IEEE J. Quantum Electron. 6(11), 694 (1970).
    [Crossref]
  8. A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature (London) 492(7428), 229–233 (2012).
    [Crossref]
  9. G. Villares, A. Hugi, S. Blaser, and J. Faist, “Dual-comb spectroscopy based on quantum-cascade-laser frequency combs,” Nat. Commun. 5, 1–9 (2014).
    [Crossref]
  10. P. Friedli, H. Sigg, B. Hinkov, A. Hugi, S. Riedi, M. Beck, and J. Faist, “Four-wave mixing in a quantum cascade laser amplifier,” Appl. Phys. Lett. 102(22), 222104 (2013).
    [Crossref]
  11. W. E. Lamb, “Theory of an optical maser,” Phys. Rev. 134(6A), A1429–A1450 (1964).
    [Crossref]
  12. J. B. Khurgin, Y. Dikmelik, A. Hugi, and J. Faist, “Coherent frequency combs produced by self frequency modulation in quantum cascade lasers,” Appl. Phys. Lett. 104(8), 081118 (2014).
    [Crossref]
  13. J. Faist, Quantum Cascade Lasers, 1st ed. (Oxford University, 2013).
    [Crossref]
  14. H. Choi, L. Diehl, Z.-K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. Norris, “Time-resolved investigations of electronic transport dynamics in quantum cascade lasers based on diagonal lasing transition,” IEEE J. Quantum Electron. 45(4), 307–321 (2009).
    [Crossref]
  15. A. Gordon, C. Y. Wang, L. Diehl, F. X. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77(5), 053804 (2008).
    [Crossref]
  16. V. M. Gkortsas, C. Wang, L. Kuznetsova, L. Diehl, A. Gordon, C. Jirauschek, M. A. Belkin, A. Belyanin, F. Capasso, and F. X. Kartner, “Dynamics of actively mode-locked quantum cascade lasers,” Opt. Express 18(13), 13616–13630 (2010).
    [Crossref] [PubMed]
  17. M. R. St-Jean, M. I. Amanti, A. Bernard, A. Calvar, A. Bismuto, E. Gini, M. Beck, J. Faist, H. C. Liu, and C. Sirtori, “Injection locking of mid-infrared quantum cascade laser at 14 GHz, by direct microwave modulation,” Laser Photon. Rev. 8(3), 443–449 (2014).
    [Crossref]
  18. R. Terazzi and J. Faist, “A density matrix model of transport and radiation in quantum cascade lasers,” New Journal of Physics 12, 033045 (2010).
    [Crossref]
  19. V V Kozlov, “Self-induced transparency soliton laser via coherent mode locking,” Phys. Rev. A 56(2), 1607– 1612 (1997).
    [Crossref]
  20. C. Menyuk and M. Talukder, “Self-induced transparency modelocking of quantum cascade lasers,” Phys. Rev. Lett. 102(2), 023903 (2009).
    [Crossref] [PubMed]

2014 (3)

G. Villares, A. Hugi, S. Blaser, and J. Faist, “Dual-comb spectroscopy based on quantum-cascade-laser frequency combs,” Nat. Commun. 5, 1–9 (2014).
[Crossref]

J. B. Khurgin, Y. Dikmelik, A. Hugi, and J. Faist, “Coherent frequency combs produced by self frequency modulation in quantum cascade lasers,” Appl. Phys. Lett. 104(8), 081118 (2014).
[Crossref]

M. R. St-Jean, M. I. Amanti, A. Bernard, A. Calvar, A. Bismuto, E. Gini, M. Beck, J. Faist, H. C. Liu, and C. Sirtori, “Injection locking of mid-infrared quantum cascade laser at 14 GHz, by direct microwave modulation,” Laser Photon. Rev. 8(3), 443–449 (2014).
[Crossref]

2013 (1)

P. Friedli, H. Sigg, B. Hinkov, A. Hugi, S. Riedi, M. Beck, and J. Faist, “Four-wave mixing in a quantum cascade laser amplifier,” Appl. Phys. Lett. 102(22), 222104 (2013).
[Crossref]

2012 (1)

A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature (London) 492(7428), 229–233 (2012).
[Crossref]

2011 (1)

S. Barbieri, M. Ravaro, P. Gellie, G. Santarelli, C. Manquest, C. Sirtori, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Coherent sampling of active mode-locked terahertz quantum cascade lasers and frequency synthesis (vol 5, pg 306, 2011),” Nat. Photonics 5(6), 378 (2011).

2010 (3)

S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nat. Photonics 4(9), 636–640 (2010).
[Crossref]

R. Terazzi and J. Faist, “A density matrix model of transport and radiation in quantum cascade lasers,” New Journal of Physics 12, 033045 (2010).
[Crossref]

V. M. Gkortsas, C. Wang, L. Kuznetsova, L. Diehl, A. Gordon, C. Jirauschek, M. A. Belkin, A. Belyanin, F. Capasso, and F. X. Kartner, “Dynamics of actively mode-locked quantum cascade lasers,” Opt. Express 18(13), 13616–13630 (2010).
[Crossref] [PubMed]

2009 (3)

C. Menyuk and M. Talukder, “Self-induced transparency modelocking of quantum cascade lasers,” Phys. Rev. Lett. 102(2), 023903 (2009).
[Crossref] [PubMed]

H. Choi, L. Diehl, Z.-K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. Norris, “Time-resolved investigations of electronic transport dynamics in quantum cascade lasers based on diagonal lasing transition,” IEEE J. Quantum Electron. 45(4), 307–321 (2009).
[Crossref]

C. Y. Wang, L. Kuznetsova, V. M. Gkortsas, L. Diehl, F. X. Kaertner, M. A. Belkin, A. Belyanin, X. Li, D. Ham, H. Schneider, P. Grant, C. Y. Song, S. Haffouz, Z. R. Wasilewski, H. C. Liu, and F. Capasso, “Mode-locked pulses from mid-infrared quantum cascade lasers,” Opt. Express 17(15), 12929–12943 (2009).
[Crossref] [PubMed]

2008 (1)

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77(5), 053804 (2008).
[Crossref]

2000 (1)

H. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1173–1185 (2000).
[Crossref]

1997 (1)

V V Kozlov, “Self-induced transparency soliton laser via coherent mode locking,” Phys. Rev. A 56(2), 1607– 1612 (1997).
[Crossref]

1994 (1)

J. Faist, F. Capasso, D. Sivco, C. Sirtori, A. Hutchinson, and A. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[Crossref] [PubMed]

1989 (1)

M. C. Tatham, J. F. Ryan, and C. T. Foxon, “Time-resolved Raman measurements of intersubband relaxation in GaAs quantum qells,” Phys. Rev. Lett. 63(15), 1637–1640 (1989).
[Crossref] [PubMed]

1970 (1)

D. J. Kuizenga and A. Siegman, “FM and AM mode locking of homogeneous Laser-Part I: Theory,” IEEE J. Quantum Electron. 6(11), 694 (1970).
[Crossref]

1964 (1)

W. E. Lamb, “Theory of an optical maser,” Phys. Rev. 134(6A), A1429–A1450 (1964).
[Crossref]

Amanti, M. I.

M. R. St-Jean, M. I. Amanti, A. Bernard, A. Calvar, A. Bismuto, E. Gini, M. Beck, J. Faist, H. C. Liu, and C. Sirtori, “Injection locking of mid-infrared quantum cascade laser at 14 GHz, by direct microwave modulation,” Laser Photon. Rev. 8(3), 443–449 (2014).
[Crossref]

Barbieri, S.

S. Barbieri, M. Ravaro, P. Gellie, G. Santarelli, C. Manquest, C. Sirtori, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Coherent sampling of active mode-locked terahertz quantum cascade lasers and frequency synthesis (vol 5, pg 306, 2011),” Nat. Photonics 5(6), 378 (2011).

S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nat. Photonics 4(9), 636–640 (2010).
[Crossref]

Beck, M.

M. R. St-Jean, M. I. Amanti, A. Bernard, A. Calvar, A. Bismuto, E. Gini, M. Beck, J. Faist, H. C. Liu, and C. Sirtori, “Injection locking of mid-infrared quantum cascade laser at 14 GHz, by direct microwave modulation,” Laser Photon. Rev. 8(3), 443–449 (2014).
[Crossref]

P. Friedli, H. Sigg, B. Hinkov, A. Hugi, S. Riedi, M. Beck, and J. Faist, “Four-wave mixing in a quantum cascade laser amplifier,” Appl. Phys. Lett. 102(22), 222104 (2013).
[Crossref]

Beere, H.

S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nat. Photonics 4(9), 636–640 (2010).
[Crossref]

Belkin, M. A.

Belyanin, A.

Bernard, A.

M. R. St-Jean, M. I. Amanti, A. Bernard, A. Calvar, A. Bismuto, E. Gini, M. Beck, J. Faist, H. C. Liu, and C. Sirtori, “Injection locking of mid-infrared quantum cascade laser at 14 GHz, by direct microwave modulation,” Laser Photon. Rev. 8(3), 443–449 (2014).
[Crossref]

Bismuto, A.

M. R. St-Jean, M. I. Amanti, A. Bernard, A. Calvar, A. Bismuto, E. Gini, M. Beck, J. Faist, H. C. Liu, and C. Sirtori, “Injection locking of mid-infrared quantum cascade laser at 14 GHz, by direct microwave modulation,” Laser Photon. Rev. 8(3), 443–449 (2014).
[Crossref]

Blaser, S.

G. Villares, A. Hugi, S. Blaser, and J. Faist, “Dual-comb spectroscopy based on quantum-cascade-laser frequency combs,” Nat. Commun. 5, 1–9 (2014).
[Crossref]

A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature (London) 492(7428), 229–233 (2012).
[Crossref]

Bour, D.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77(5), 053804 (2008).
[Crossref]

Calvar, A.

M. R. St-Jean, M. I. Amanti, A. Bernard, A. Calvar, A. Bismuto, E. Gini, M. Beck, J. Faist, H. C. Liu, and C. Sirtori, “Injection locking of mid-infrared quantum cascade laser at 14 GHz, by direct microwave modulation,” Laser Photon. Rev. 8(3), 443–449 (2014).
[Crossref]

Capasso, F.

V. M. Gkortsas, C. Wang, L. Kuznetsova, L. Diehl, A. Gordon, C. Jirauschek, M. A. Belkin, A. Belyanin, F. Capasso, and F. X. Kartner, “Dynamics of actively mode-locked quantum cascade lasers,” Opt. Express 18(13), 13616–13630 (2010).
[Crossref] [PubMed]

H. Choi, L. Diehl, Z.-K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. Norris, “Time-resolved investigations of electronic transport dynamics in quantum cascade lasers based on diagonal lasing transition,” IEEE J. Quantum Electron. 45(4), 307–321 (2009).
[Crossref]

C. Y. Wang, L. Kuznetsova, V. M. Gkortsas, L. Diehl, F. X. Kaertner, M. A. Belkin, A. Belyanin, X. Li, D. Ham, H. Schneider, P. Grant, C. Y. Song, S. Haffouz, Z. R. Wasilewski, H. C. Liu, and F. Capasso, “Mode-locked pulses from mid-infrared quantum cascade lasers,” Opt. Express 17(15), 12929–12943 (2009).
[Crossref] [PubMed]

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77(5), 053804 (2008).
[Crossref]

J. Faist, F. Capasso, D. Sivco, C. Sirtori, A. Hutchinson, and A. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[Crossref] [PubMed]

Cho, A.

J. Faist, F. Capasso, D. Sivco, C. Sirtori, A. Hutchinson, and A. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[Crossref] [PubMed]

Choi, H.

H. Choi, L. Diehl, Z.-K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. Norris, “Time-resolved investigations of electronic transport dynamics in quantum cascade lasers based on diagonal lasing transition,” IEEE J. Quantum Electron. 45(4), 307–321 (2009).
[Crossref]

Colombelli, R.

S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nat. Photonics 4(9), 636–640 (2010).
[Crossref]

Corzine, S.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77(5), 053804 (2008).
[Crossref]

Davies, A. G.

S. Barbieri, M. Ravaro, P. Gellie, G. Santarelli, C. Manquest, C. Sirtori, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Coherent sampling of active mode-locked terahertz quantum cascade lasers and frequency synthesis (vol 5, pg 306, 2011),” Nat. Photonics 5(6), 378 (2011).

Diehl, L.

V. M. Gkortsas, C. Wang, L. Kuznetsova, L. Diehl, A. Gordon, C. Jirauschek, M. A. Belkin, A. Belyanin, F. Capasso, and F. X. Kartner, “Dynamics of actively mode-locked quantum cascade lasers,” Opt. Express 18(13), 13616–13630 (2010).
[Crossref] [PubMed]

C. Y. Wang, L. Kuznetsova, V. M. Gkortsas, L. Diehl, F. X. Kaertner, M. A. Belkin, A. Belyanin, X. Li, D. Ham, H. Schneider, P. Grant, C. Y. Song, S. Haffouz, Z. R. Wasilewski, H. C. Liu, and F. Capasso, “Mode-locked pulses from mid-infrared quantum cascade lasers,” Opt. Express 17(15), 12929–12943 (2009).
[Crossref] [PubMed]

H. Choi, L. Diehl, Z.-K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. Norris, “Time-resolved investigations of electronic transport dynamics in quantum cascade lasers based on diagonal lasing transition,” IEEE J. Quantum Electron. 45(4), 307–321 (2009).
[Crossref]

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77(5), 053804 (2008).
[Crossref]

Dikmelik, Y.

J. B. Khurgin, Y. Dikmelik, A. Hugi, and J. Faist, “Coherent frequency combs produced by self frequency modulation in quantum cascade lasers,” Appl. Phys. Lett. 104(8), 081118 (2014).
[Crossref]

Ding, L.

S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nat. Photonics 4(9), 636–640 (2010).
[Crossref]

Faist, J.

J. B. Khurgin, Y. Dikmelik, A. Hugi, and J. Faist, “Coherent frequency combs produced by self frequency modulation in quantum cascade lasers,” Appl. Phys. Lett. 104(8), 081118 (2014).
[Crossref]

G. Villares, A. Hugi, S. Blaser, and J. Faist, “Dual-comb spectroscopy based on quantum-cascade-laser frequency combs,” Nat. Commun. 5, 1–9 (2014).
[Crossref]

M. R. St-Jean, M. I. Amanti, A. Bernard, A. Calvar, A. Bismuto, E. Gini, M. Beck, J. Faist, H. C. Liu, and C. Sirtori, “Injection locking of mid-infrared quantum cascade laser at 14 GHz, by direct microwave modulation,” Laser Photon. Rev. 8(3), 443–449 (2014).
[Crossref]

P. Friedli, H. Sigg, B. Hinkov, A. Hugi, S. Riedi, M. Beck, and J. Faist, “Four-wave mixing in a quantum cascade laser amplifier,” Appl. Phys. Lett. 102(22), 222104 (2013).
[Crossref]

A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature (London) 492(7428), 229–233 (2012).
[Crossref]

R. Terazzi and J. Faist, “A density matrix model of transport and radiation in quantum cascade lasers,” New Journal of Physics 12, 033045 (2010).
[Crossref]

H. Choi, L. Diehl, Z.-K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. Norris, “Time-resolved investigations of electronic transport dynamics in quantum cascade lasers based on diagonal lasing transition,” IEEE J. Quantum Electron. 45(4), 307–321 (2009).
[Crossref]

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77(5), 053804 (2008).
[Crossref]

J. Faist, F. Capasso, D. Sivco, C. Sirtori, A. Hutchinson, and A. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[Crossref] [PubMed]

J. Faist, Quantum Cascade Lasers, 1st ed. (Oxford University, 2013).
[Crossref]

Foxon, C. T.

M. C. Tatham, J. F. Ryan, and C. T. Foxon, “Time-resolved Raman measurements of intersubband relaxation in GaAs quantum qells,” Phys. Rev. Lett. 63(15), 1637–1640 (1989).
[Crossref] [PubMed]

Friedli, P.

P. Friedli, H. Sigg, B. Hinkov, A. Hugi, S. Riedi, M. Beck, and J. Faist, “Four-wave mixing in a quantum cascade laser amplifier,” Appl. Phys. Lett. 102(22), 222104 (2013).
[Crossref]

Gellie, P.

S. Barbieri, M. Ravaro, P. Gellie, G. Santarelli, C. Manquest, C. Sirtori, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Coherent sampling of active mode-locked terahertz quantum cascade lasers and frequency synthesis (vol 5, pg 306, 2011),” Nat. Photonics 5(6), 378 (2011).

S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nat. Photonics 4(9), 636–640 (2010).
[Crossref]

Gini, E.

M. R. St-Jean, M. I. Amanti, A. Bernard, A. Calvar, A. Bismuto, E. Gini, M. Beck, J. Faist, H. C. Liu, and C. Sirtori, “Injection locking of mid-infrared quantum cascade laser at 14 GHz, by direct microwave modulation,” Laser Photon. Rev. 8(3), 443–449 (2014).
[Crossref]

Giovannini, M.

H. Choi, L. Diehl, Z.-K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. Norris, “Time-resolved investigations of electronic transport dynamics in quantum cascade lasers based on diagonal lasing transition,” IEEE J. Quantum Electron. 45(4), 307–321 (2009).
[Crossref]

Gkortsas, V. M.

Gordon, A.

V. M. Gkortsas, C. Wang, L. Kuznetsova, L. Diehl, A. Gordon, C. Jirauschek, M. A. Belkin, A. Belyanin, F. Capasso, and F. X. Kartner, “Dynamics of actively mode-locked quantum cascade lasers,” Opt. Express 18(13), 13616–13630 (2010).
[Crossref] [PubMed]

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77(5), 053804 (2008).
[Crossref]

Grant, P.

Haffouz, S.

Ham, D.

Haus, H.

H. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1173–1185 (2000).
[Crossref]

Hinkov, B.

P. Friedli, H. Sigg, B. Hinkov, A. Hugi, S. Riedi, M. Beck, and J. Faist, “Four-wave mixing in a quantum cascade laser amplifier,” Appl. Phys. Lett. 102(22), 222104 (2013).
[Crossref]

Hoefler, G.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77(5), 053804 (2008).
[Crossref]

Hugi, A.

J. B. Khurgin, Y. Dikmelik, A. Hugi, and J. Faist, “Coherent frequency combs produced by self frequency modulation in quantum cascade lasers,” Appl. Phys. Lett. 104(8), 081118 (2014).
[Crossref]

G. Villares, A. Hugi, S. Blaser, and J. Faist, “Dual-comb spectroscopy based on quantum-cascade-laser frequency combs,” Nat. Commun. 5, 1–9 (2014).
[Crossref]

P. Friedli, H. Sigg, B. Hinkov, A. Hugi, S. Riedi, M. Beck, and J. Faist, “Four-wave mixing in a quantum cascade laser amplifier,” Appl. Phys. Lett. 102(22), 222104 (2013).
[Crossref]

A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature (London) 492(7428), 229–233 (2012).
[Crossref]

Hutchinson, A.

J. Faist, F. Capasso, D. Sivco, C. Sirtori, A. Hutchinson, and A. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[Crossref] [PubMed]

Jirauschek, C.

Kaertner, F. X.

C. Y. Wang, L. Kuznetsova, V. M. Gkortsas, L. Diehl, F. X. Kaertner, M. A. Belkin, A. Belyanin, X. Li, D. Ham, H. Schneider, P. Grant, C. Y. Song, S. Haffouz, Z. R. Wasilewski, H. C. Liu, and F. Capasso, “Mode-locked pulses from mid-infrared quantum cascade lasers,” Opt. Express 17(15), 12929–12943 (2009).
[Crossref] [PubMed]

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77(5), 053804 (2008).
[Crossref]

Kartner, F. X.

Khanna, S. P.

S. Barbieri, M. Ravaro, P. Gellie, G. Santarelli, C. Manquest, C. Sirtori, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Coherent sampling of active mode-locked terahertz quantum cascade lasers and frequency synthesis (vol 5, pg 306, 2011),” Nat. Photonics 5(6), 378 (2011).

Khurgin, J. B.

J. B. Khurgin, Y. Dikmelik, A. Hugi, and J. Faist, “Coherent frequency combs produced by self frequency modulation in quantum cascade lasers,” Appl. Phys. Lett. 104(8), 081118 (2014).
[Crossref]

Kozlov, V V

V V Kozlov, “Self-induced transparency soliton laser via coherent mode locking,” Phys. Rev. A 56(2), 1607– 1612 (1997).
[Crossref]

Kuizenga, D. J.

D. J. Kuizenga and A. Siegman, “FM and AM mode locking of homogeneous Laser-Part I: Theory,” IEEE J. Quantum Electron. 6(11), 694 (1970).
[Crossref]

Kuznetsova, L.

Lamb, W. E.

W. E. Lamb, “Theory of an optical maser,” Phys. Rev. 134(6A), A1429–A1450 (1964).
[Crossref]

Li, X.

Linfield, E. H.

S. Barbieri, M. Ravaro, P. Gellie, G. Santarelli, C. Manquest, C. Sirtori, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Coherent sampling of active mode-locked terahertz quantum cascade lasers and frequency synthesis (vol 5, pg 306, 2011),” Nat. Photonics 5(6), 378 (2011).

Liu, H. C.

M. R. St-Jean, M. I. Amanti, A. Bernard, A. Calvar, A. Bismuto, E. Gini, M. Beck, J. Faist, H. C. Liu, and C. Sirtori, “Injection locking of mid-infrared quantum cascade laser at 14 GHz, by direct microwave modulation,” Laser Photon. Rev. 8(3), 443–449 (2014).
[Crossref]

A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature (London) 492(7428), 229–233 (2012).
[Crossref]

C. Y. Wang, L. Kuznetsova, V. M. Gkortsas, L. Diehl, F. X. Kaertner, M. A. Belkin, A. Belyanin, X. Li, D. Ham, H. Schneider, P. Grant, C. Y. Song, S. Haffouz, Z. R. Wasilewski, H. C. Liu, and F. Capasso, “Mode-locked pulses from mid-infrared quantum cascade lasers,” Opt. Express 17(15), 12929–12943 (2009).
[Crossref] [PubMed]

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77(5), 053804 (2008).
[Crossref]

Maier, T.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77(5), 053804 (2008).
[Crossref]

Maineult, W.

S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nat. Photonics 4(9), 636–640 (2010).
[Crossref]

Manquest, C.

S. Barbieri, M. Ravaro, P. Gellie, G. Santarelli, C. Manquest, C. Sirtori, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Coherent sampling of active mode-locked terahertz quantum cascade lasers and frequency synthesis (vol 5, pg 306, 2011),” Nat. Photonics 5(6), 378 (2011).

Menyuk, C.

C. Menyuk and M. Talukder, “Self-induced transparency modelocking of quantum cascade lasers,” Phys. Rev. Lett. 102(2), 023903 (2009).
[Crossref] [PubMed]

Norris, T.

H. Choi, L. Diehl, Z.-K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. Norris, “Time-resolved investigations of electronic transport dynamics in quantum cascade lasers based on diagonal lasing transition,” IEEE J. Quantum Electron. 45(4), 307–321 (2009).
[Crossref]

Ravaro, M.

S. Barbieri, M. Ravaro, P. Gellie, G. Santarelli, C. Manquest, C. Sirtori, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Coherent sampling of active mode-locked terahertz quantum cascade lasers and frequency synthesis (vol 5, pg 306, 2011),” Nat. Photonics 5(6), 378 (2011).

Riedi, S.

P. Friedli, H. Sigg, B. Hinkov, A. Hugi, S. Riedi, M. Beck, and J. Faist, “Four-wave mixing in a quantum cascade laser amplifier,” Appl. Phys. Lett. 102(22), 222104 (2013).
[Crossref]

Ritchie, D.

S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nat. Photonics 4(9), 636–640 (2010).
[Crossref]

Ryan, J. F.

M. C. Tatham, J. F. Ryan, and C. T. Foxon, “Time-resolved Raman measurements of intersubband relaxation in GaAs quantum qells,” Phys. Rev. Lett. 63(15), 1637–1640 (1989).
[Crossref] [PubMed]

Santarelli, G.

S. Barbieri, M. Ravaro, P. Gellie, G. Santarelli, C. Manquest, C. Sirtori, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Coherent sampling of active mode-locked terahertz quantum cascade lasers and frequency synthesis (vol 5, pg 306, 2011),” Nat. Photonics 5(6), 378 (2011).

S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nat. Photonics 4(9), 636–640 (2010).
[Crossref]

Schneider, H.

C. Y. Wang, L. Kuznetsova, V. M. Gkortsas, L. Diehl, F. X. Kaertner, M. A. Belkin, A. Belyanin, X. Li, D. Ham, H. Schneider, P. Grant, C. Y. Song, S. Haffouz, Z. R. Wasilewski, H. C. Liu, and F. Capasso, “Mode-locked pulses from mid-infrared quantum cascade lasers,” Opt. Express 17(15), 12929–12943 (2009).
[Crossref] [PubMed]

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77(5), 053804 (2008).
[Crossref]

Siegman, A.

D. J. Kuizenga and A. Siegman, “FM and AM mode locking of homogeneous Laser-Part I: Theory,” IEEE J. Quantum Electron. 6(11), 694 (1970).
[Crossref]

Sigg, H.

P. Friedli, H. Sigg, B. Hinkov, A. Hugi, S. Riedi, M. Beck, and J. Faist, “Four-wave mixing in a quantum cascade laser amplifier,” Appl. Phys. Lett. 102(22), 222104 (2013).
[Crossref]

Sirtori, C.

M. R. St-Jean, M. I. Amanti, A. Bernard, A. Calvar, A. Bismuto, E. Gini, M. Beck, J. Faist, H. C. Liu, and C. Sirtori, “Injection locking of mid-infrared quantum cascade laser at 14 GHz, by direct microwave modulation,” Laser Photon. Rev. 8(3), 443–449 (2014).
[Crossref]

S. Barbieri, M. Ravaro, P. Gellie, G. Santarelli, C. Manquest, C. Sirtori, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Coherent sampling of active mode-locked terahertz quantum cascade lasers and frequency synthesis (vol 5, pg 306, 2011),” Nat. Photonics 5(6), 378 (2011).

S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nat. Photonics 4(9), 636–640 (2010).
[Crossref]

J. Faist, F. Capasso, D. Sivco, C. Sirtori, A. Hutchinson, and A. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[Crossref] [PubMed]

Sivco, D.

J. Faist, F. Capasso, D. Sivco, C. Sirtori, A. Hutchinson, and A. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[Crossref] [PubMed]

Song, C. Y.

St-Jean, M. R.

M. R. St-Jean, M. I. Amanti, A. Bernard, A. Calvar, A. Bismuto, E. Gini, M. Beck, J. Faist, H. C. Liu, and C. Sirtori, “Injection locking of mid-infrared quantum cascade laser at 14 GHz, by direct microwave modulation,” Laser Photon. Rev. 8(3), 443–449 (2014).
[Crossref]

Talukder, M.

C. Menyuk and M. Talukder, “Self-induced transparency modelocking of quantum cascade lasers,” Phys. Rev. Lett. 102(2), 023903 (2009).
[Crossref] [PubMed]

Tatham, M. C.

M. C. Tatham, J. F. Ryan, and C. T. Foxon, “Time-resolved Raman measurements of intersubband relaxation in GaAs quantum qells,” Phys. Rev. Lett. 63(15), 1637–1640 (1989).
[Crossref] [PubMed]

Terazzi, R.

R. Terazzi and J. Faist, “A density matrix model of transport and radiation in quantum cascade lasers,” New Journal of Physics 12, 033045 (2010).
[Crossref]

Troccoli, M.

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Supplementary Material (6)

» Media 1: MP4 (3522 KB)     
» Media 2: MP4 (1717 KB)     
» Media 3: MOV (3522 KB)     
» Media 4: MOV (1717 KB)     
» Media 5: MP4 (3522 KB)     
» Media 6: MP4 (1717 KB)     

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Figures (5)

Fig. 1
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.
Fig. 2
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.
Fig. 3
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 ν n ¯ for three values of simulated dispersion (GVD = −500 fs2mm−1 and GVD = ±30000 fs2mm−1).
Fig. 4
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.
Fig. 5
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.

Equations (93)

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τ = 2 g M ω 2 Ω g 2 4 ,
ρ ˙ = i [ , ρ ] + ρ t ) coll
= ( ω 1 Ω ˜ 12 ( t ) Ω ˜ 21 ( t ) ω 2 )
Ω ˜ 12 ( t ) = Ω ˜ 21 ( t ) = μ 21 E ( t ) /
ρ t ) coll = ( γ 11 ( ρ 11 ρ 11 , p ) γ 12 ρ 12 γ 12 ρ 21 γ 22 ( ρ 22 ρ 22 , p ) )
Δ ρ ˙ = i Ω ˜ 12 ( ρ 21 ρ 12 ) γ 22 ( Δ ρ Δ ρ p ) ρ ˙ 21 = i ω 21 ρ 21 + i Ω ˜ 12 Δ ρ γ 12 ρ 21 ρ 21 = ρ 12 *
ω l = ω 21 + l ω l { n m , n m + 1 , . , 0 , . , n m }
E ( t ) = 1 2 e i ω 21 t l = n m l = n m A l e i l ω t + 1 2 e i ω 21 t l = n m l = n m A l * e i l ω t Ω ˜ 12 ( t ) = 1 2 e i ω 21 t l = n m l = n m Ω l e i l ω t + 1 2 e i ω 21 t l = n m l = n m Ω l * e i l ω t
Ω l = μ 21 A l
ρ 21 = e i ω 21 t k = n m k = n m σ 21 , k e i k ω t ρ 12 = e i ω 21 t k = n m k = n m σ 21 , k * e i k ω t Δ ρ = r = n m r = n m Δ ρ r e i r ω t
Δ ρ p = Δ ρ p , 0 + 1 2 ( Δ ρ p , M e i ω t + Δ ρ p , M * e i ω t )
r ( i r ω + γ 22 ) Δ ρ r e i r ω t = γ 22 ( Δ ρ p , 0 + 1 2 ( Δ ρ p , M e i ω t + Δ ρ p , M * e i ω t ) ) + i 2 k l ( Ω l * σ 21 , k e i ( k l ) ω t Ω l σ 21 , k * e i ( k l ) ω t )
k ( γ 12 i k ω ) σ 21 , k e i k ω t = i 2 l r Ω l Δ ρ r e i ( l + r ) ω t
Δ ρ r = 1 i r ω + γ 22 [ γ 22 Δ ρ p , 0 δ r , 0 + γ 22 Δ ρ p , M * 2 δ r , 1 + γ 22 Δ ρ p , M 2 δ r , 1 + i 2 k ( Ω k r * σ 21 , k Ω k + r σ 21 , k * ) ]
σ 21 , k = i 2 ( γ 12 i k ω ) l Ω l Δ ρ k l
Δ ρ r = 0 ( 0 ) = Δ ρ p , 0
Δ ρ r = 1 ( 0 ) = γ 22 γ 22 i ω Δ ρ p , M 2
Δ ρ r = 1 ( 0 ) = γ 22 γ 22 + i ω Δ ρ p , M * 2
σ 21 , k ( 1 ) = 1 2 ( i γ 12 + k ω ) [ Ω k Δ ρ p , 0 + Ω k + 1 γ 22 γ 22 + i ω Δ ρ p , M * 2 + Ω k 1 γ 22 γ 22 i ω Δ ρ p , M 2 ]
Δ ρ r ( 2 ) = i 2 ( γ 22 i r ω ) k ( Ω k r * σ 21 , k ( 1 ) Ω k + r σ 21 , k ( 1 ) * )
Δ ρ r ( 2 ) = i 2 ( γ 22 i r ω ) k ( Ω k r * 2 1 ( i γ 12 + k ω ) [ Ω k Δ ρ p , 0 + Ω k + 1 γ 22 γ 22 + i ω Δ ρ p , M * 2 + Ω k 1 γ 22 γ 22 i ω Δ ρ p , M 2 ] Ω k + r 2 1 ( i γ 12 + k ω ) [ Ω k * Δ ρ p , 0 + Ω k + 1 * γ 22 γ 22 i ω Δ ρ p , M 2 + Ω k 1 * γ 22 γ 22 + i ω Δ ρ p , M * 2 ] )
Δ ρ r ( 2 ) = i 2 ( γ 22 i r ω ) k { ( Ω k r * Ω k Δ ρ p , 0 2 ( i γ 12 k ω ) Ω k + r Ω k * Δ ρ p , 0 2 ( i γ 12 k ω ) ) + ( γ 22 γ 22 + i ω Ω k r * Ω k + 1 Δ ρ p , M * / 2 2 ( i γ 12 k ω ) γ 22 γ 22 i ω Ω k + r Ω k + 1 * Δ ρ p , M / 2 2 ( i γ 12 k ω ) ) + ( γ 22 γ 22 i ω Ω k r * Ω k 1 Δ ρ p , M / 2 2 ( i γ 12 k ω ) γ 22 γ 22 + i ω Ω k + r Ω k 1 * Δ ρ p , M * / 2 2 ( i γ 12 k ω ) ) }
Δ ρ r ( 2 ) = 1 4 i ( γ 22 i r ω ) k Ω k + r { Ω k * Δ ρ p , 0 ( 1 i γ 12 ( k + r ) ω 1 i γ 12 k ω ) + Ω k 1 * γ 22 γ 22 + i ω Δ ρ p , M * 2 ( 1 i γ 12 ( k 1 + r ) ω 1 i γ 12 k ω ) + Ω k + 1 * γ 22 γ 22 i ω Δ ρ p , M 2 ( 1 i γ 12 ( k + 1 + r ) ω 1 i γ 12 k ω ) }
σ 21 , n ( 3 ) = i 2 ( γ 12 i n ω ) l Ω l Δ ρ n l ( 2 )
σ 21 , n ( 3 ) = i 2 ( γ 12 i n ω ) 1 4 l k Ω l i ( γ 22 i ( n l ) ω ) Ω k + n l { Ω k * Δ ρ p , 0 ( 1 i γ 12 ( k + n l ) ω 1 i γ 12 k ω ) + Ω k 1 * γ 22 γ 22 + i ω Δ ρ p , M * 2 ( 1 i γ 12 ( k 1 + n l ) ω 1 i γ 12 k ω ) + Ω k + 1 * γ 22 γ 22 i ω Δ ρ p , M 2 ( 1 i γ 12 ( k + 1 + n l ) ω ) 1 i γ 12 k ω ) }
σ 21 , n ( 3 ) = i 2 1 γ 12 i n ω 1 4 l k Ω n + k l i ( γ 22 i ( l k ) ω ) Ω l { Ω k * Δ ρ p , 0 ( 1 i γ 12 l ω 1 i γ 12 k ω ) + Ω k 1 * γ 22 γ 22 + i ω Δ ρ p , M * 2 ( 1 i γ 12 ( l 1 ) ω 1 i γ 12 k ω ) + Ω k + 1 * γ 22 γ 22 i ω Δ ρ p , M 2 ( 1 i γ 12 ( l + 1 ) ω ) 1 i γ 12 k ω ) }
1 i γ 12 ( l ± 1 ) ω 1 i γ 12 l ω
M ± = γ 22 γ 22 ± i ω
C k l = γ 22 γ 22 i ( l k ) ω
G ˜ n = i γ 12 n ω + i γ 12
B k l = γ 12 2 i ( 1 i γ 12 l ω 1 i γ 12 k ω )
σ 21 , n ( 3 ) = i 4 γ 22 γ 12 2 G ˜ n l k C k l B k l Ω n + k l Ω l { Ω k * Δ ρ p , 0 + Ω k 1 * M + Δ ρ p , M * 2 + Ω k + 1 * M Δ ρ p , M 2 }
σ 21 , n ( 1 ) = i 2 γ 12 G ˜ n [ Ω n Δ ρ p , 0 + Ω n + 1 M + Δ ρ p , M * 2 + Ω n 1 M Δ ρ p , M 2 ]
2 E μ 0 σ E t ε r c 2 2 E t 2 = μ 0 2 P t 2
E ( t ) = 1 2 l = n m l = n m A l e i ω l t + 1 2 l = n m l = n m A l * e i ω l t P ( t ) = 1 2 l = n m l = n m P l e i ω l t + 1 2 l = n m l = n m P l * e i ω l t
A n ( z ) = A n sin ( k n z )
P n = N μ = N Tr [ ρ μ ] = N μ 21 σ 21 , n
k n l c = ( N 0 + n ) π
ω n c = k n c n 0
1 2 sin ( k n z ) { k n 2 A n μ 0 σ ( A ˙ n i ω n A n ) ε r c 2 ( ω n 2 A n 2 i ω n A ˙ n ) } = ω n 2 μ 0 N μ 21 1 2 σ 21 , n ( z )
{ ( ( ω n 2 ω n c 2 ) + i ω n σ ε 0 ε r ) A n + ( 2 i ω n σ ε 0 ε r ) A ˙ n } sin ( k n z ) = 1 ε 0 ε r ω n 2 N μ 21 σ 21 , n ( z )
2 i ω n A ˙ n = ( ( ω n 2 ω n c 2 ) + i ω n σ ε 0 ε r ) A n + ω n 2 N μ 21 σ 21 , n ( z ) ε 0 n 0 2 sin ( k n z )
A ˙ = σ 2 ε 0 ε r A
| A ˙ | = | A | 2 τ c
A ˙ n i ( ω n 2 ω n c 2 2 ω n ) A n = i ω n N μ 21 σ 21 , n ( z ) 2 ε 0 n 0 2 sin ( k n z ) 1 2 τ c A n
A ˙ n i ( ω n 2 ω n c 2 2 ω n ) A n = i ω n N μ 21 l c ε 0 n 0 2 0 l c σ 21 , n ( z ) sin ( k n z ) d z 1 2 τ c A n
σ 21 , n ( z ) = σ 21 , n ( 1 ) ( z ) + σ 21 , n ( 3 ) ( z )
1 l c 0 l c A n sin ( k n z ) 2 d z = A n κ n , n = A n 1 2
1 l c 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
κ n , l , k , m = 1 l c 0 l c sin ( k n z ) sin ( k l z ) sin ( k k z ) sin ( k m z ) d z
κ n , n , n , n = 3 8 ( k = l = n ) κ n , n , m , n = 1 4 ( k = l n ) or ( l = n ) κ n , l , k , n + k l = 1 8 otherwise
Δ ρ p , M ( z ) = Θ ( l m l c z ) Δ ρ p , M
κ n , l , k , m ( l m ) = 1 l c 0 l m l c sin ( k n z ) sin ( k l z ) sin ( k k z ) sin ( k m z ) d z
κ n , n + 1 = 1 l c 0 l m l c sin ( k n z ) sin ( k n + 1 z ) d z = 1 2 π { sin ( π l m ) sin ( π ( 2 ( n + N 0 ) + 1 ) l m ) ( 2 ( n + N 0 ) + 1 ) }
κ n , l , k , n + k l = κ n , l , k , n + k l ( l m )
A ˙ n i ( ω n 2 ω n c 2 2 ω n ) A n = 1 2 τ c A n i ω n N μ 21 ε 0 n 0 2 i 2 γ 12 G ˜ n [ Ω n Δ ρ p , 0 κ n , n + Ω n + 1 M + κ n , n + 1 Δ ρ p , M * 2 + Ω n 1 M κ n , n 1 Δ ρ p , M 2 ] i ω n N μ 21 ε 0 n 0 2 i 4 γ 22 γ 12 2 G ˜ n l 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 Δ ρ p , M * 2 + Ω k + 1 * M κ n , k + 1 , l , n + k + 1 l Δ ρ p , M 2 }
g 0 = ω n τ c N μ 21 2 Δ ρ p , 0 2 ε 0 n 0 2 γ 12
δ = Δ ρ p , m 2 Δ ρ p , 0
2 τ c A ˙ n 2 i τ c ( ω n 2 ω n c 2 2 ω n ) A n = A n + g 0 G ˜ n A n + 2 g 0 G ˜ n [ A n + 1 M + δ κ n , n + 1 + A n 1 M δ * κ n , n 1 ] μ 21 2 2 γ 22 γ 12 g 0 G ˜ n 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 } .
m = n + k l
m + = n + k 1 l
m = n + k + 1 l
A sat = γ 12 γ 22 μ 21
2 τ c A ˙ n = { G n 1 + i 2 τ c ( 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 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 } .
A ˙ n = { G n 1 Net gain + i ( ω n 2 ω n c 2 2 ω n ) Cavity dispersion } A n + 2 G n [ A n + 1 M + δ * κ n , n + 1 + A n 1 M δ κ n , n 1 ] Modulation G n 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 } FWM term .
A ˙ n = { G n 1 + i ( ω n 2 ω n c 2 2 ω n ) } A n G n 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 }
S n = k , l C k l B k l A m A l A k * κ n , k , l , m
C k ± 1 , l C k l B k ± 1 , l B k l
A ˙ n = { G n 1 + i ( 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 ) .
S n = k , l C k l B k l A m A l A k * κ n , k , l , m ,
A ˙ n = { G n 1 + i ( 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 )
δ ω n = d ( arg ( A n ) ) d t
D n = ( ω n 2 ω n c 2 2 ω n )
k ( ω n ) = ( N 0 + n ) π l c .
k ( ω n c ) = k ( ω 0 ) + k ω ( ω n c ω 0 ) + 1 2 2 k ω 2 ( ω n c ω 0 ) 2
n g = c k ω
GVD = 2 k ω 2
δ ω n c 2 + 2 n g c GVD δ ω n c 2 n π l c GVD = 0 .
ω c = π c n g l c
β GVD = ω c c GVD n g
δ ω n c = ω c β GVD ( 1 + 1 + 2 n β GVD )
δ ω n c = n ω c ( 1 n 2 β GVD )
D n = ( ω n 2 ω n c 2 2 ω n ) = ( ( ω 21 + n ω ) 2 ( ω 21 + δ ω n c ) 2 2 ( ω 21 + n ω ) )
D n = ω { n N 0 N 0 + n [ 1 ω ˜ c + 1 2 n ω ˜ c β GVD ] + n 2 2 ( N 0 + n ) ( 1 ω ˜ c 2 ) }
ω ˜ c = ω c ω
G n ( 1 + 2 ) = g 0 2 ( 1 1 i ( n n off ) ω τ coh + 1 1 i ( n + n off ) ω τ coh )
B k l ( 1 ) = 1 2 ( 1 1 i ( l n off ) ω τ coh + 1 1 + i ( k n off ) ω τ coh ) ,
G n ( 1 ) k , l C k l B k l ( 1 ) A m A l A k * + G n ( 2 ) k , l C k l B k l ( 2 ) A m A l A k * G n ( 1 + 2 ) k , l C k l B k l ( 1 + 2 ) A m A l A k *
A ˙ n = { G n 1 + i D n } A n G n k , l C k l B k l A m A l A k * κ n , k , l , m
ν n ¯ = τ r t δ t 1 N steps i φ n i + 1 φ n i δ t
S = 3 8 n | A n | 2
S g 0 = 1 .
g m = μ m n A n A n + 1 *

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