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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 2 — Jan. 16, 2012
  • pp: 1450–1464
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Theoretical investigation of injection-locked high modulation bandwidth quantum cascade lasers

Bo Meng and Qi Jie Wang  »View Author Affiliations


Optics Express, Vol. 20, Issue 2, pp. 1450-1464 (2012)
http://dx.doi.org/10.1364/OE.20.001450


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Abstract

In this study, we report for the first time to our knowledge theoretical investigation of modulation responses of injection-locked mid-infrared quantum cascade lasers (QCLs) at wavelengths of 4.6 μm and 9 μm, respectively. It is shown through a three-level rate equations model that the direct intensity modulation of QCLs gives the maximum modulation bandwidths of ~7 GHz at 4.6 μm and ~20 GHz at 9 μm. By applying the injection locking scheme, we find that the modulation bandwidths of up to ~30 GHz and ~70 GHz can be achieved for QCLs at 4.6 μm and 9 μm, respectively, with an injection ratio of 5 dB. The result also shows that an ultrawide modulation bandwidth of more than 200 GHz is possible with a 10 dB injection ratio for QCLs at 9 μm. An important characteristic of injection-locked QCLs is the nonexistence of unstable locking region in the locking map, in contrast to their diode laser counterparts. We attribute this to the ultra-short upper laser state lifetimes of QCLs.

© 2012 OSA

1. Introduction

Semiconductor laser sources with high modulation bandwidths are always desirable for high speed data transmission systems. However, the modulation bandwidth of direct modulated semiconductor laser is largely limited by relaxation resonance frequency determined by the interactions of the carriers and photons in the laser cavity. Optical injection locking scheme has been shown theoretically [1

1. A. Murakami, K. Kawashima, and K. Atsuki, “Cavity resonance shift and bandwidth enhancement in semiconductor lasers with strong light injection,” IEEE J. Quantum Electron. 39(10), 1196–1204 (2003). [CrossRef]

] and experimentally [2

2. E. K. Lau, H.-K. Sung, and M. C. Wu, “Frequency response enhancement of optical injection-locked lasers,” IEEE J. Quantum Electron. 44(1), 90–99 (2008). [CrossRef]

] as an effective approach to enhance the relaxation resonance frequency of semiconductor lasers, making it a useful method to increase the modulation bandwidth of lasers. With this technique, vertical-cavity surface-emitting lasers (VCSELs) with a 80 GHz intrinsic 3-dB bandwidth [3

3. E. K. Lau, X. Zhao, H.-K. Sung, D. Parekh, C. Chang-Hasnain, and M. C. Wu, “Strong optical injection-locked semiconductor lasers demonstrating > 100-GHz resonance frequencies and 80-GHz intrinsic bandwidths,” Opt. Express 16(9), 6609–6618 (2008). [CrossRef] [PubMed]

], quantum-dot (QD) DFB lasers with a 16.3 GHz bandwidth [4

4. N. B. Terry, N. A. Naderi, M. Pochet, A. J. Moscho, L. F. Lester, and V. Kovanis, “Bandwidth enhancement of injection-locked 1.3 μm quantum-dot DFB laser,” Electron. Lett. 44(15), 904–905 (2008). [CrossRef]

], and distributed reflector (DR) lasers with wirelike active regions [5

5. S. H. Lee, D. Parekh, T. Shindo, W. J. Yang, P. Guo, D. Takahashi, N. Nishiyama, C. J. Chang-Hasnain, and S. Arai, “Bandwidth enhancement of injection-locked distributed reflector lasers with wirelike active regions,” Opt. Express 18(16), 16370–16378 (2010). [CrossRef] [PubMed]

] with a 15 GHz bandwidth have been demonstrated recently.

In the paper, we theoretically and systematically investigate the injection-locking of QCLs, for the first time to our knowledge, at emitting wavelengths of 4.6 μm [17

17. A. Lyakh, R. Maulini, A. Tsekoun, R. Go, C. Pflügl, L. Diehl, Q. J. Wang, F. Capasso, C. Kumar, and N. Patel, “3 W continuous-wave room temperature single-facet emission from quantum cascade lasers based on nonresonant extraction design approach,” Appl. Phys. Lett. 95(14), 141113 (2009).

] and 9 μm [18

18. Q. J. Wang, C. Pflügl, L. Diehl, F. Capasso, T. Edamura, S. Furuta, M. Yamanishi, and H. Kan, “High performance quantum cascade lasers based on three-phonon-resonance design,” Appl. Phys. Lett. 94(1), 011103 (2009). [CrossRef]

] corresponding to the two atmospheric transmission windows, respectively. We first examine the direct intensity modulation characteristics of QCLs, using a three-level rate equations model. The results show that no resonant frequency appears in the frequency modulation response, and the maximum modulation bandwidths of round 7 GHz for QCLs at 4.6 μm and 20 GHz for QCLs at 9 μm are obtained. The theoretical analysis is in good agreement with the experimental observations [15

15. R. Paiella, R. Martini, F. Capasso, C. Gmachl, H. Y. Hwang, D. L. Sivco, J. N. Baillargeon, A. Y. Cho, E. A. Whittaker, and H. C. Liu, “High-frequency modulation without the relaxation oscillation resonance in quantum cascade lasers,” Appl. Phys. Lett. 79(16), 2526–2528 (2001). [CrossRef]

]. In addition, we apply the optical injection locking scheme to increase the frequency modulation bandwidth of QCL. Rate equation analysis shows that the obtained modulation bandwidth of injection locked QCLs is about three times higher than that of the directly modulated QCLs for both lasers at 4.6 μm and 9 μm under a 5 dB injection ratio, with modulation bandwidths up to ~30 GHz and ~70 GHz, respectively. With a 10 dB injection ratio, a modulation bandwidth of over 200 GHz can be achieved with the injection locking scheme. A unique feature of injection locked QCLs is that there is no unstable locking range, as opposed to other semiconductor lasers [19

19. F. Mogensen, H. Olesen, and G. Jacobsen, “Locking conditions and stability properties for a semiconductor laser with external light injection,” IEEE J. Quantum Electron. 21(7), 784–793 (1985). [CrossRef]

]. We attribute this effect to the ultra-short lifetime of the upper laser state.

2. Theory

2.1 Direct intensity modulation of QCLs

2.1.1 Steady state analysis

We set the time derivatives of Eqs. (1)-(3) to zero, and denote the steady state values of J, N2, N3, and P as J0, N20, N30 and P0, then the steady state rate equations can be expressed as:

J0N30τ3G0(N30N20)P0=0
(4)
N30τ3+G0(N30N20)P0N20τ2=0
(5)
G(N30N20)P0P0τP=0
(6)

The steady state population of lower lasing level, N20 is given as
N20=J0τ2
(7)
with Eqs. (4) and (5), P0 can be obtained in the following equation,
J0(1η)N30N20τ3G0(N30N20)P0=0
(8)
where η=τ2/ τ3. Also, we note that at threshold,
(N30N20)=1NpG0τp
(9)
where Jth is the threshold current (s−1), given by Jth = 1/3τP (1-η). Substituting Eq. (9) into Eq. (8), one gets the steady state photon number P0,
P0=1G0τ3(J0Jth1)
(10)
with Eqs. (7) and (9), the upper laser level population N30 can be written as,

N30=1NPG0τP+J0τ2
(11)

2.1.2 Small-signal modulation analysis

The small-signal analysis is necessary in order to investigate the modulation response of QCLs. Assuming small variations ∆N2, ∆N3, ∆P and ∆J around the steady state values, and substituting N2=N20+∆N2, N3=N30+∆N3, P=P0=+∆P and J=J0+∆J into Eqs. (1)(3), we get the rate equations for the small deviations as follows:

dΔN3dt=ΔJΔN3τ3G0(ΔN3ΔN2)P0G0(N30N20)ΔP
(12)
dΔN2dt=ΔN3τ3+G0(ΔN3ΔN2)P0+G0(N30N20)ΔPΔN2τ2
(13)
dΔPdt=G(ΔN3ΔN2)P0+G(N30N20)ΔPΔPτP
(14)

Taking the Laplace transform of Eqs. (12)(14), and placing them into a matrix form, we have the following transformed matrix,
[s+f11f12f13f21s+f22f23f31f32s+f33][ΔN3ΔN2ΔP]=[ΔJ00]
(15)
where the matrix terms are expressed as

f11=1τ3+G0P0f21=(1τ3+G0P0)f31=GP0f12=G0P0f22=1τ2+G0P0f32=GP0f13=1τPNPf23=1τPNPf33=0
(16)

The normalized modulation response is
H(s)=ΔPΔJ(1η)τPNP=sA+1Bs3+Cs2+Ds+1
(17)
Where

A=(1η1)1τ3B=τPτ2NPGP0C=B(2G0P0+1τ3(1+1η))D=B(1τ2G0P0+1τ3τ2+2GP01τPNP)
(18)

2.2 Injection locking of quantum cascade lasers (QCLs)

2.2.1 Steady state analysis

The injection locking scheme involves two lasers: the slave laser (SL) is optically locked by the master laser (ML) [2

2. E. K. Lau, H.-K. Sung, and M. C. Wu, “Frequency response enhancement of optical injection-locked lasers,” IEEE J. Quantum Electron. 44(1), 90–99 (2008). [CrossRef]

]. The differential equation describing electric field inside the slave laser under injection locking was first proposed by Lang and Kobayashi, given as
ddtESL(t){jω(N)+12[NPG0(N)1τp]}ESL(t)=fdEML(t)
(19)
where fd is the coupling rate between the master laser and the slave laser, approximately expressed as: fd = νg(1-R)/(2LR1/2) where R stands for the power reflectivity of the injected cavity facet. We denote the electromagnetic fields in the slave laser and the master laser as,
ESL(t)=E0(t)ej(ω0t+ϕ0(t))
(20a)
EML(t)=E1(t)ej(ω1t+ϕ1(t))
(20b)
where E1(t) is taken as a constant, ϕ0(t) and ϕ1(t) are the phases of the two electromagnetic fields, (ϕ1(t) is usually set as zero for computing convenience); ω0 and ω1 are the angular frequencies of the slave laser and the master laser, respectively. Substituting Eq. (20) into (19), the differential equation can be split into field magnitude and phase rate equations.

ddtE0(t)=12NPG0ΔN(t)E0(t)+fdE1cosϕ(t)
(21)
ddtϕ(t)=12αNPG0ΔN(t)fdE1E0sinϕ(t)Δωinj
(22)

In the above equations, ∆N is the carrier number change due to light injection from the master laser, given by N-Nth. Different from rate equations for diode lasers, the carrier number in the above equations is the carrier number difference between the upper and the lower laser levels, i.e. N = N3-N2. From Eqs. (12) and (13), and taking into account the carrier number in the lower laser state, the differential equations governing the carrier dynamics are given as,
dN(t)dt=J(N+N2)2τ32E02G0N+N2τ2
(23)
dN2(t)dt=(N+N2)1τ3+E02G0NN2τ2
(24)
where E0(t), ϕ0(t), N(t), and N2(t) represent the slave laser’s field magnitude, phase, carrier number difference between the transition states, and carrier number in the lower laser state, respectively. The field magnitude E0(t) has been normalized, so that |E0(t)|2=P(t), where P(t) is regarded as the total photon number for a single longitudinal mode inside the cavity. ϕ(t) is defined as the phase difference between the master and the slave lasers, i.e. ϕ(t)=ϕSL(t)−ϕML(t), G0, NP, α, J, τ2, and τ3 are the slave laser’s gain coefficient for one stage, the number of period, linewidth enhancement factor, the injection current, the lower laser state lifetime, and the upper laser state lifetime, respectively. While fd, E1 and ∆ωinj represent the coupling rate, injected electrical field magnitude, and frequency detuning, respectively. The latter is expressed as ∆ωinj = ω1-ω0. For free-running laser, N = Nth, N2 = N2th, E = Efr, with Eq. (24), the carrier number in the lower laser state for free-running slave laser can be obtained,
N2th=(Efr2γP+Nthτ3)τ2τ3τ3τ2
(25)
where we define γP as,

γP=G0Nth
(26)

Remembering that cavity photon decay rate γP0 is given by GNth, so that γP = γP0/NP. With Eq. (23) and the expression of γP, the threshold current, defined as I/e (where I is current, and e is the electronic charge), and the free-running field magnitude of slave laser are found to be,

Jth=Nth1τ3τ2
(27)
Efr2=JNth1τ3τ2γPτ3τ3τ2
(28)

Under the injection-locking region, the carrier number difference N in the slave laser should decrease, due to the enhanced stimulated mission in the gain medium. The steady state values for E, ∆N, ϕ, and N2 under injection locking are denoted as E0, ∆N0, ϕ0 and N20 respectively. The corresponding expressions of ∆N0, ϕ0 and N20 are given below,

ΔN0=2fdE1NpG0E0cosϕ0
(29)
ϕ0=sin1(ΔωinjE0fdE11+α2)tan1α
(30)
N20=[E02(γP+G0ΔN0)+ΔN0τ3+Nthτ3]τ2τ3τ3τ2
(31)

Applying Eqs. (29)-(31) into Eqs. (23), and setting the time derivative to zero, one gets a cubic equation for the steady state value E0. Defining two new parameters,
T1=2τ2τ3τ3τ2
(32)
T2=2τ3+T1τ3
(33)
the equation is shown as,

E03(2G0NPγP+G0NPγPT1)E022G0fdE1cosϕ0(2+T1)E0(G0JNPNthG0NPT2)2fdE1cosϕ0T2=0
(34)

The steady state value of E0 can be solved by various numerical methods, given that the phase difference ϕ0 varies approximately from -π/2 to cot−1α in the injection locking range, and ∆N0 should be negative.

2.2.2 Small-signal modulation analysis

Assuming a small variances ∆E, ∆ϕ, ∆N and ∆N2 around the corresponding steady state value, and substituting E=E0+∆E, ϕ=ϕ0+ϕ, N=N0+∆N, N2=N20+∆N2 into Eqs. (21)(24), one gets the rate equations for the small signal modulation:

ddtΔE=12NPG0ΔN0ΔEfdE1sinϕ0Δϕ+12NpG0E0ΔN
(35)
ddtΔϕ=fdE1E02sinϕ0ΔEfdE1E0cosϕ0Δϕ+α2NPG0ΔN
(36)
ddtΔN=ΔJ(4E0γP+4E0G0ΔN0)ΔE(2τ3+2E02G0)ΔN+(1τ22τ3)ΔN2
(37)
ddtΔN2=(2E0γP+2G0ΔN0E0)ΔE+(1τ3+E02G0)ΔN+(1τ31τ2)ΔN2
(38)

Converting the rate equations into the matrix form and taking Laplace transform, we have the following matrix:

[Q11+sQ12Q13Q14Q21Q22+sQ23Q24Q31Q32Q33+sQ34Q41Q42Q43Q44+s][ΔEΔϕΔNΔN2]=[00ΔJ0]
(39)

In which the matrix elements are

Q11=12G0ΔN0NPQ12=fdE1sinϕ0Q13=12G0E0NPQ14=0Q21=fdE1E02sinϕ0Q22=fdE1E0cosϕ0Q23=α2G0NPQ24=0Q31=4E0γP+4E0G0ΔN0Q32=0Q33=2τ3+2E02G0Q34=2τ31τ2Q41=(2E0γP+2G0ΔN0E0)Q42=0Q43=(1τ3+E02G0)Q44=(1τ21τ3)
(40)

The direct modulation frequency response is given by,
H(s)=Gs2+Es+Fs4+As3+Bs2+Cs+D
(41)
where

A=Q11+Q22+Q33+Q44B=Q33Q44+Q11Q33+Q11Q44+Q22Q33+Q22Q44+Q11Q22Q21Q12Q31Q13Q34Q43C=Q11Q33Q44+Q22Q33Q44+Q33Q11Q22+Q44Q11Q22Q33Q21Q12Q44Q21Q12Q13Q31Q22Q13Q31Q44+Q12Q31Q23+Q13Q34Q41Q22Q43Q34Q11Q34Q43D=Q11Q22Q33Q44Q12Q21Q33Q44+Q12Q21Q43Q34Q13Q31Q22Q44+Q12Q31Q23Q44Q12Q23Q34Q41+Q13Q22Q34Q41Q11Q22Q34Q43E=(Q12Q23Q13Q22Q13Q44)/Q13F=(Q12Q23Q44Q13Q22Q44)/Q13G=Q13
(42)

3. Results and discussions

3.1 Direct modulations

First, we examine the direct modulation bandwidths of QCLs at different wavelengths in the mid-IR regime. Table 1

Table 1. Characteristic parameters of the state-of-the-art QCLs at 4.6 μm [17] and 9 μm [18] for direct modulation

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shows the parameters used in the simulations, of the state-of-the-art mid-infrared QCLs emitting at around 4.6 μm [17

17. A. Lyakh, R. Maulini, A. Tsekoun, R. Go, C. Pflügl, L. Diehl, Q. J. Wang, F. Capasso, C. Kumar, and N. Patel, “3 W continuous-wave room temperature single-facet emission from quantum cascade lasers based on nonresonant extraction design approach,” Appl. Phys. Lett. 95(14), 141113 (2009).

] and 9 μm [18

18. Q. J. Wang, C. Pflügl, L. Diehl, F. Capasso, T. Edamura, S. Furuta, M. Yamanishi, and H. Kan, “High performance quantum cascade lasers based on three-phonon-resonance design,” Appl. Phys. Lett. 94(1), 011103 (2009). [CrossRef]

], which correspond to the two atmospheric windows in the mid-infrared spectrum range. A simple way to analyze the modulation behavior of QCLs can be done by analyzing the zeros and poles of the normalized frequency response H(s). Tables 2

Table 2. Poles, zeros, and f3dB for QCLs at 4.6 μm

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and 3

Table 3. Poles, zeros, and f3dB for QCLs at 9 μm

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list the corresponding zeros and poles for the QCLs at 4.6 μm and 9 μm under different injection currents. Three poles can be obtained from the denominator of the frequency response, which are expressed as p1, p2 and p3 respectively. One zero will be obtained from the numerator. Also shown is the calculated 3-dB bandwidth of the modulation response. We also notice that the phenomena that no resonance frequency appears and the calculated direct modulation bandwidth are in good agreement with the experimental observations [15

15. R. Paiella, R. Martini, F. Capasso, C. Gmachl, H. Y. Hwang, D. L. Sivco, J. N. Baillargeon, A. Y. Cho, E. A. Whittaker, and H. C. Liu, “High-frequency modulation without the relaxation oscillation resonance in quantum cascade lasers,” Appl. Phys. Lett. 79(16), 2526–2528 (2001). [CrossRef]

], if the experimental parameters [15

15. R. Paiella, R. Martini, F. Capasso, C. Gmachl, H. Y. Hwang, D. L. Sivco, J. N. Baillargeon, A. Y. Cho, E. A. Whittaker, and H. C. Liu, “High-frequency modulation without the relaxation oscillation resonance in quantum cascade lasers,” Appl. Phys. Lett. 79(16), 2526–2528 (2001). [CrossRef]

] are used in the calculations.

Figures 1(a) and (b)
Fig. 1 Normalized modulation response of mid-infrared QCLs. (a) QCLs emitting at 4.6 μm with parameters shown in Table.1 (b) QCLs emitting at 9 μm with parameters shown in Table.1. The 3-dB bandwidth is indicated by the black dashed line.
show the normalized frequency responses of QCLs at 4.6 μm and 9 μm, for I = 1.5Ith, 2Ith, 3Ith, and 4Ith, respectively, corresponding to the injection currents from just above threshold to roll-over, respectively. Obtaining the poles and zeros of the frequency response, we can transfer it into the following form,

H(s)=(s|z1|+1)(s|p1|+1)(s|p2|+1)(s|p3|+1)
(43)

This is the normal form of the Bode plot. Because the first pole is three to four magnitudes smaller than the other poles and zeros, it will mainly determine the modulation bandwidth, so H(s)≈1/(s/|p1|+1) in the Bode plot. For cubic equations, we have p1p2+p2p3+p1p3=D/B, and p1p2p3=−1/B, leading to 1/p1+1/p2+1/p3=−1/D. For p1 is much smaller than p2 and p3, we get |1/p1|≈|D|. Substituting |1/p1| with |D| in the expression of H(s), and keeping in mind that D is positive in the whole locking region; one can approximate the frequency response as,
H(s)=1sD+1
(44)
where D is given by (τP + 2τ2) + τP/(G0P0τ3). The definition of 3-dB bandwidth is defined as |H(s)| at half of its zero value, i.e. |H(s)|f3dB| = 1/2 in the case of normalized frequency response. An approximated expression for 3-dB bandwidth is given as,

f3dB32πD=32π[(τP+2τ2)+τpG0P0τ3]
(45)

Combining the D and f3dB, we can easily see that a larger value of τp leads to a decreased modulation bandwidth. This can also be seen from the simulated frequency responses of QCLs at 4.6 μm and 9 μm. Due to larger optical loss, e.g. free carrier absorption (proportional to λ2) and intersubband absorption in the waveguide, the photon lifetime is much shorter for longer emission wavelengths, leading to an increased modulation bandwidth. However, a larger τp means a decreased optical loss, leading to a higher optical output. Thus, the tradeoff between the modulation bandwidth and the output optical power has to be taken into account when designing QCLs for high speed modulation applications. Another feature of QCLs is the absence of resonance peak as normally shown in conventional diode lasers. The physics behind it lies in the ultrafast upper state lifetime compared with the photon lifetime, making QCLs an overdamped system, showing no resonance peak in the frequency modulation response.

3.2 Injection locking modulations

To further enhance the modulation bandwidth of QCLs in the mid-IR regime, we can employ injection locking scheme for QCLs, the theoretical analysis of which is shown in section 2.2. All the additional parameters used in injection locking calculation are listed in Table 4

Table 4. Characteristic parameters of the state-of-the-art QCL at 4.6 μm [17] and 9 μm [18] for injection locking modulation

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, for QCLs emitting at 4.6 μm and 9 μm, respectively. The rest of the parameters which are common to direct modulated QCLs are shown in Tables 1. Though the linewidth enhance factor of QCLs is expected to be zero [22

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

], the experiments showed different values (0.02 ± 0.2 in [23

23. T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett. 89(9), 091121 (2006). [CrossRef]

],-0.44 to 2.29 in [24

24. J. von Staden, T. Gensty, W. Elsäßer, G. Giuliani, and C. Mann, “Measurements of the α factor of a distributed-feedback quantum cascade laser by an optical feedback self-mixing technique,” Opt. Lett. 31(17), 2574–2576 (2006). [CrossRef] [PubMed]

], and −1.8 to −1.7 in [25

25. M. Ishihara, T. Morimoto, S. Furuta, K. Kasahara, N. Akikusa, K. Fujita, and T. Edamura, “Linewidth enhancement factor of quantum cascade lasers with single phonon resonance-continuum depopulation structure on Peltier cooler,” Electron. Lett. 45(23), 1168–1169 (2009). [CrossRef]

]). Without loss of generality, we set the linewidth enhancement factor as unity in the calculations.

Under injection locking scheme, four equations are involved in the calculations. This results in a four-order denominator and a two-order numerator in the frequency response. The typical locking maps for both of QCLs at 4.6 μm and 9μm are illustrated in Fig. 2
Fig. 2 Locking maps for QCLs emitting at (a) 4.6 μm and (b) 9 μm. The parameters used in the simulations are shown in Tables 1 and 4.
.(a) and (b), respectively, with J = 4Jth. In our simulations, the boundaries of phase in the injection locking range are approximately cot−1α to -π/2, from negative to positive detuning edge. This is derived from rate equations for the field magnitude E and phase difference ϕ, where the noise terms and spontaneous terms are neglected. The calculated locking maps of QCLs are similar to those of diode lasers, where the locking range increases linearly with the increase of the amplitude of the injected optical field, as demonstrated in Ref [26

26. M. S. Taubman, T. L. Myers, B. D. Cannon, and R. M. Williams, “Stabilization, injection and control of quantum cascade lasers, and their application to chemical sensing in the infrared,” Spectrochim. Acta A Mol. Biomol. Spectrosc. 60(14), 3457–3468 (2004). [CrossRef] [PubMed]

].

To examine the frequency response of the injection locked QCLs, we calculate the response curves of QCLs at 9 μm with an injection ratio of 5 dB across the locking range and a frequency spacing of 1 GHz. The corresponding waterfall plot is shown in Fig. 3
Fig. 3 Normalized frequency response curves versus frequency detuning at a fixed injection ratio R = 5 dB, for the 9 μm QCL.
. Table 5

Table 5. Poles, Zeros and 3-dB bandwidth f3dB for the QCL in Fig. 3.

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lists the accompanying poles, zeros and 3 dB bandwidth of the responses.

The frequency response exhibits resonance-like behaviors close to both edges of the locking range (e.g. the green and the black curves, respectively, in Fig. 3). However, the reasons behind these two are different. A further explanation can be sought by examining the poles of the frequency response. As illustrated in Table 5, when the frequency comes close to the positive frequency detuning edge (black curve in Fig. 3), two complex conjugate poles appear. In conventional semiconductor lasers, the imaginary part of the complex conjugate poles gives the resonance frequency of the response, while the real part defines the damping term, so that a peak appears in the frequency response. This is also the case for QCLs. At the negative detuning edge (black curve in Fig. 3), because all the poles are real (and negative), the peak in the frequency response has to be analyzed by the Bode Plots. According to the Bode Plots theory, zeros will increase the magnitude of the frequency response from its critical frequency and beyond 10 dB per decade, while poles decrease the magnitude at the same rate. For QCLs, the first zero z1 is smaller than any of the poles, making the peak in the frequency response apparent, even though all the poles are real.

For conventional semiconductor lasers, increasing slave laser’s bias current is an effective way to enhance the bandwidth of the modulation system [12

12. C. Y. Cheung and K. A. Shore, “Self-consistent analysis of dc modulation response of unipolar semiconductor lasers,” J. Mod. Opt. 45(6), 1219–1229 (1998). [CrossRef]

]. Here we calculate the frequency response at a fixed injection ratio of 5 dB, with the injection current varying 1.5x, 2x, 3x, and 4xJth, respectively, and a frequency detuning 0.1 GHz away from the negative detuning edge, as shown in Fig. 4
Fig. 4 Normalized modulation response of mid-infrared QCLs under injection locking scheme. (a) QCLs emitting at 4.6 μm. (b) QCLs emitting at 9 μm. The inset show the corresponding poles (x) and zeros (), with the arrow indicating direction of increasing the injection current of the slave laser. The 3 dB bandwidth is indicated by the dashed line.
. The inset is the corresponding pole/zero diagrams. Tables 6

Table 6. Poles, zeros, and f3dB for the QCL in Fig. 4 (a)

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and 7

Table 7. Poles, zeros, and f3dB for Fig. 4 (b)

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list the associated poles and zeros of the modulation responses. The increased bandwidth can be attributed to the increased values of poles and zeros of the frequency response, and the much smaller z1 than any of the poles. Similar effects can also be seen by increasing the injection ratio, for which a much broader modulation bandwidth can be achieved. Our calculation shows that for injection ratio R = 10 dB and frequency detuning of 0.1 GHz away from the edge, over 200 GHz modulation bandwidth can be obtained. However, there is a tradeoff between the injection ratio and the slave laser’s injection current, even though both of which can effectively increase the modulation bandwidth. This is because the increased injection current leads to an increased slave laser output power, so that in order to achieve a high injection ratio, a master laser with a higher output power is required, which in some cases is limited in real applications. We noticed that by direct modulating the injection current using an RF source, Pierre et al demonstrated that the cavity resonance frequency of terahertz QCLs can be injection-locked [27

27. P. Gellie, S. Barbieri, J. F. Lampin, P. Filloux, C. Manquest, C. Sirtori, I. Sagnes, S. P. Khanna, E. H. Linfield, A. G. Davies, H. Beere, and D. Ritchie, “Injection-locking of terahertz quantum cascade lasers up to 35 GHz using RF amplitude modulation,” Opt. Express 18(20), 20799–20816 (2010). [CrossRef]

]. However, their scheme is different to ours in that the injection signal in our case is the optical signal from the master laser.

It is also noted that there is no unstable locking range in QCLs based on the above rate equation analysis, compared to other types of semiconductor lasers. Mathematically, the stable locking condition is satisfied when the eigenvalues of the frequency response function’s denominators are located at the left half of the complex s-plane. In the entire locking region of QCLs, we find that the roots of the denominator are always negative, making the whole locking region a stable locking system. One of the experimental evidence is that the injected light induced pulsation effect [19

19. F. Mogensen, H. Olesen, and G. Jacobsen, “Locking conditions and stability properties for a semiconductor laser with external light injection,” IEEE J. Quantum Electron. 21(7), 784–793 (1985). [CrossRef]

], which is a unique sign of the unstable locking condition in diode lasers [28

28. C. H. Henry, N. A. Olsson, and N. K. Dutta, “Locking range and stability of injection locked 1.54 μm InGaAsP semiconductor laser,” IEEE J. Quantum Electron. 21(8), 1152–1156 (1985). [CrossRef]

] caused by the reduced damping and the strong correlation between the phase and intensity of photons, has not been observed in QCLs. The reasons are as follows: first, there is no relaxation oscillation in QCLs which has been experimentally verified, owing to the picosecond carrier lifetime of the laser states [15

15. R. Paiella, R. Martini, F. Capasso, C. Gmachl, H. Y. Hwang, D. L. Sivco, J. N. Baillargeon, A. Y. Cho, E. A. Whittaker, and H. C. Liu, “High-frequency modulation without the relaxation oscillation resonance in quantum cascade lasers,” Appl. Phys. Lett. 79(16), 2526–2528 (2001). [CrossRef]

]. It is equivalent to say that the damping for the relaxation oscillation is so high such that the carrier numbers and photons come to a steady state in a much shorter time than those in conventional diode lasers. Second, using the phasor diagram and small signal analysis (here we express the small change of N, P, and ϕ as δN, δP and δϕ), Henry pointed out [28

28. C. H. Henry, N. A. Olsson, and N. K. Dutta, “Locking range and stability of injection locked 1.54 μm InGaAsP semiconductor laser,” IEEE J. Quantum Electron. 21(8), 1152–1156 (1985). [CrossRef]

] that the increase of δP causes an increased δϕ in relaxation oscillations. On the other hand, at the negative detuning edge, the increase of δϕ causes an enhanced mismatch of E0 and E1, leading to a larger cavity intensity change, i.e. larger δP. This forms a positive feedback between δϕ and δP, which is enhanced as the relaxation oscillations increase. However, there is no relaxation oscillation in QCLs, thus the interaction between δϕ and δP is reduced. Therefore the large damping effect and the reduced interactions between ϕ and P, make the unstable locking range in QCLs disappear, as shown in Fig. 2.

4. Conclusions

Theoretical investigation of injection-locked QCLs at wavelengths of 4.6 μm and 9 μm are carried out in the paper. By using a three-level rate equations model, we find that the maximum modulation bandwidths of round 7 GHz for QCLs at 4.6 μm and 20 GHz for QCLs at 9 μm were obtained. It is shown that by applying the injection locking scheme, the modulation bandwidths can be increased by around three times under a 5 dB injection ratio, compared to the direct modulation scheme. The frequency modulation responses were analyzed using the Bode diagram, based on which it shows that increasing the injection ratio and the injected current of the slave laser are effective ways to enhance the modulation bandwidth of the master QCLs. With a 10 dB injection ratio, more than 200 GHz modulation bandwidth can be obtained for QCLs at 9 μm. Unlike conventional semiconductor lasers, no unstable locking ranges appear in the locking map, which we attribute to the ultra-short lifetime of the upper laser state of QCLs, due to the nature of intersubband transitions.

References and links

1.

A. Murakami, K. Kawashima, and K. Atsuki, “Cavity resonance shift and bandwidth enhancement in semiconductor lasers with strong light injection,” IEEE J. Quantum Electron. 39(10), 1196–1204 (2003). [CrossRef]

2.

E. K. Lau, H.-K. Sung, and M. C. Wu, “Frequency response enhancement of optical injection-locked lasers,” IEEE J. Quantum Electron. 44(1), 90–99 (2008). [CrossRef]

3.

E. K. Lau, X. Zhao, H.-K. Sung, D. Parekh, C. Chang-Hasnain, and M. C. Wu, “Strong optical injection-locked semiconductor lasers demonstrating > 100-GHz resonance frequencies and 80-GHz intrinsic bandwidths,” Opt. Express 16(9), 6609–6618 (2008). [CrossRef] [PubMed]

4.

N. B. Terry, N. A. Naderi, M. Pochet, A. J. Moscho, L. F. Lester, and V. Kovanis, “Bandwidth enhancement of injection-locked 1.3 μm quantum-dot DFB laser,” Electron. Lett. 44(15), 904–905 (2008). [CrossRef]

5.

S. H. Lee, D. Parekh, T. Shindo, W. J. Yang, P. Guo, D. Takahashi, N. Nishiyama, C. J. Chang-Hasnain, and S. Arai, “Bandwidth enhancement of injection-locked distributed reflector lasers with wirelike active regions,” Opt. Express 18(16), 16370–16378 (2010). [CrossRef] [PubMed]

6.

R. Martini, R. Paiella, C. Gmachl, F. Capasso, E. A. Whittaker, H. C. Liu, H. Y. Hwang, D. L. Sivco, J. N. Baillargeon, and A. Y. Cho, “High-speed digital data transmission using mid-infrared quantum cascade lasers,” Electron. Lett. 37(21), 1290–1291 (2001). [CrossRef]

7.

R. Martini, C. Bethea, F. Capasso, C. Gmachl, R. Paiella, E. A. Whittaker, H. Y. Hwang, D. L. Sivco, J. N. Baillargeon, and A. Y. Cho, “Free-space optical transmission of multimedia satellite data streams using mid-infrared quantum cascade lasers,” Electron. Lett. 38(4), 181–183 (2002). [CrossRef]

8.

A. Lyakh, C. Pflügl, L. Diehl, Q. J. Wang, F. Capasso, X. J. Wang, J. Y. Fan, T. Tanban-Ek, R. Maulini, A. Tsekoun, R. Go, and C. K. N. Patel, “1.6 W high wall plug efficiency, continuous-wave room temperature quantum cascade laser emitting at 4.6μm,” Appl. Phys. Lett. 92, 111110 (2008).

9.

Y. Bai, S. R. Darvish, S. Slivken, W. Zhang, A. Evans, J. Nguyen, and M. Razeghi, “Room temperature continuous wave operation of quantum cascade lasers with watt-level optical power,” Appl. Phys. Lett. 92(10), 101105 (2008). [CrossRef]

10.

P. Corrigan, R. Martini, E. A. Whittaker, and C. Bethea, “Quantum cascade lasers and the Kruse model in free space optical communication,” Opt. Express 17(6), 4355–4359 (2009). [CrossRef] [PubMed]

11.

C. Y. L. Cheung, P. S. Spencer, and K. A. Shore, “Modulation bandwidth optimization for unipolar intersubband semiconductor lasers,” IEE Proc.: Optoelectron. 144, 44–47 (1997). [CrossRef]

12.

C. Y. Cheung and K. A. Shore, “Self-consistent analysis of dc modulation response of unipolar semiconductor lasers,” J. Mod. Opt. 45(6), 1219–1229 (1998). [CrossRef]

13.

N. Mustafa, L. Pesquera, C. Y. L. Cheung, and K. A. Shore, “Terahertz bandwidth prediction for amplitude modulation response of unipolar intersubband semiconductor lasers,” IEEE Photon. Technol. Lett. 11(5), 527–529 (1999). [CrossRef]

14.

M. K. Haldar, “A simplified analysis of direct intensity modulation of quantum cascade lasers,” IEEE J. Quantum Electron. 41(11), 1349–1355 (2005). [CrossRef]

15.

R. Paiella, R. Martini, F. Capasso, C. Gmachl, H. Y. Hwang, D. L. Sivco, J. N. Baillargeon, A. Y. Cho, E. A. Whittaker, and H. C. Liu, “High-frequency modulation without the relaxation oscillation resonance in quantum cascade lasers,” Appl. Phys. Lett. 79(16), 2526–2528 (2001). [CrossRef]

16.

S. Barbieri, W. Maineult, S. S. Dhillon, C. Sirtori, J. Alton, N. Breuil, H. E. Beere, and D. A. Ritchie, “13 GHz direct modulation of terahertz quantum cascade lasers,” Appl. Phys. Lett. 91(14), 143510 (2007). [CrossRef]

17.

A. Lyakh, R. Maulini, A. Tsekoun, R. Go, C. Pflügl, L. Diehl, Q. J. Wang, F. Capasso, C. Kumar, and N. Patel, “3 W continuous-wave room temperature single-facet emission from quantum cascade lasers based on nonresonant extraction design approach,” Appl. Phys. Lett. 95(14), 141113 (2009).

18.

Q. J. Wang, C. Pflügl, L. Diehl, F. Capasso, T. Edamura, S. Furuta, M. Yamanishi, and H. Kan, “High performance quantum cascade lasers based on three-phonon-resonance design,” Appl. Phys. Lett. 94(1), 011103 (2009). [CrossRef]

19.

F. Mogensen, H. Olesen, and G. Jacobsen, “Locking conditions and stability properties for a semiconductor laser with external light injection,” IEEE J. Quantum Electron. 21(7), 784–793 (1985). [CrossRef]

20.

D. Indjin, P. Harrison, R. W. Kelsall, and Z. Ikonić, “Self-consistent scattering theory of transport and output characteristics of quantum cascade lasers,” J. Appl. Phys. 91(11), 9019–9026 (2002). [CrossRef]

21.

F. Rana and R. J. Ram, “Current noise and photon noise in quantum cascade lasers,” Phys. Rev. B 65(12), 125313 (2002). [CrossRef]

22.

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

23.

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett. 89(9), 091121 (2006). [CrossRef]

24.

J. von Staden, T. Gensty, W. Elsäßer, G. Giuliani, and C. Mann, “Measurements of the α factor of a distributed-feedback quantum cascade laser by an optical feedback self-mixing technique,” Opt. Lett. 31(17), 2574–2576 (2006). [CrossRef] [PubMed]

25.

M. Ishihara, T. Morimoto, S. Furuta, K. Kasahara, N. Akikusa, K. Fujita, and T. Edamura, “Linewidth enhancement factor of quantum cascade lasers with single phonon resonance-continuum depopulation structure on Peltier cooler,” Electron. Lett. 45(23), 1168–1169 (2009). [CrossRef]

26.

M. S. Taubman, T. L. Myers, B. D. Cannon, and R. M. Williams, “Stabilization, injection and control of quantum cascade lasers, and their application to chemical sensing in the infrared,” Spectrochim. Acta A Mol. Biomol. Spectrosc. 60(14), 3457–3468 (2004). [CrossRef] [PubMed]

27.

P. Gellie, S. Barbieri, J. F. Lampin, P. Filloux, C. Manquest, C. Sirtori, I. Sagnes, S. P. Khanna, E. H. Linfield, A. G. Davies, H. Beere, and D. Ritchie, “Injection-locking of terahertz quantum cascade lasers up to 35 GHz using RF amplitude modulation,” Opt. Express 18(20), 20799–20816 (2010). [CrossRef]

28.

C. H. Henry, N. A. Olsson, and N. K. Dutta, “Locking range and stability of injection locked 1.54 μm InGaAsP semiconductor laser,” IEEE J. Quantum Electron. 21(8), 1152–1156 (1985). [CrossRef]

OCIS Codes
(060.4080) Fiber optics and optical communications : Modulation
(140.5965) Lasers and laser optics : Semiconductor lasers, quantum cascade

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: November 15, 2011
Manuscript Accepted: December 1, 2011
Published: January 9, 2012

Citation
Bo Meng and Qi Jie Wang, "Theoretical investigation of injection-locked high modulation bandwidth quantum cascade lasers," Opt. Express 20, 1450-1464 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-2-1450


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References

  1. A. Murakami, K. Kawashima, and K. Atsuki, “Cavity resonance shift and bandwidth enhancement in semiconductor lasers with strong light injection,” IEEE J. Quantum Electron.39(10), 1196–1204 (2003). [CrossRef]
  2. E. K. Lau, H.-K. Sung, and M. C. Wu, “Frequency response enhancement of optical injection-locked lasers,” IEEE J. Quantum Electron.44(1), 90–99 (2008). [CrossRef]
  3. E. K. Lau, X. Zhao, H.-K. Sung, D. Parekh, C. Chang-Hasnain, and M. C. Wu, “Strong optical injection-locked semiconductor lasers demonstrating > 100-GHz resonance frequencies and 80-GHz intrinsic bandwidths,” Opt. Express16(9), 6609–6618 (2008). [CrossRef] [PubMed]
  4. N. B. Terry, N. A. Naderi, M. Pochet, A. J. Moscho, L. F. Lester, and V. Kovanis, “Bandwidth enhancement of injection-locked 1.3 μm quantum-dot DFB laser,” Electron. Lett.44(15), 904–905 (2008). [CrossRef]
  5. S. H. Lee, D. Parekh, T. Shindo, W. J. Yang, P. Guo, D. Takahashi, N. Nishiyama, C. J. Chang-Hasnain, and S. Arai, “Bandwidth enhancement of injection-locked distributed reflector lasers with wirelike active regions,” Opt. Express18(16), 16370–16378 (2010). [CrossRef] [PubMed]
  6. R. Martini, R. Paiella, C. Gmachl, F. Capasso, E. A. Whittaker, H. C. Liu, H. Y. Hwang, D. L. Sivco, J. N. Baillargeon, and A. Y. Cho, “High-speed digital data transmission using mid-infrared quantum cascade lasers,” Electron. Lett.37(21), 1290–1291 (2001). [CrossRef]
  7. R. Martini, C. Bethea, F. Capasso, C. Gmachl, R. Paiella, E. A. Whittaker, H. Y. Hwang, D. L. Sivco, J. N. Baillargeon, and A. Y. Cho, “Free-space optical transmission of multimedia satellite data streams using mid-infrared quantum cascade lasers,” Electron. Lett.38(4), 181–183 (2002). [CrossRef]
  8. A. Lyakh, C. Pflügl, L. Diehl, Q. J. Wang, F. Capasso, X. J. Wang, J. Y. Fan, T. Tanban-Ek, R. Maulini, A. Tsekoun, R. Go, and C. K. N. Patel, “1.6 W high wall plug efficiency, continuous-wave room temperature quantum cascade laser emitting at 4.6μm,” Appl. Phys. Lett.92, 111110 (2008).
  9. Y. Bai, S. R. Darvish, S. Slivken, W. Zhang, A. Evans, J. Nguyen, and M. Razeghi, “Room temperature continuous wave operation of quantum cascade lasers with watt-level optical power,” Appl. Phys. Lett.92(10), 101105 (2008). [CrossRef]
  10. P. Corrigan, R. Martini, E. A. Whittaker, and C. Bethea, “Quantum cascade lasers and the Kruse model in free space optical communication,” Opt. Express17(6), 4355–4359 (2009). [CrossRef] [PubMed]
  11. C. Y. L. Cheung, P. S. Spencer, and K. A. Shore, “Modulation bandwidth optimization for unipolar intersubband semiconductor lasers,” IEE Proc.: Optoelectron.144, 44–47 (1997). [CrossRef]
  12. C. Y. Cheung and K. A. Shore, “Self-consistent analysis of dc modulation response of unipolar semiconductor lasers,” J. Mod. Opt.45(6), 1219–1229 (1998). [CrossRef]
  13. N. Mustafa, L. Pesquera, C. Y. L. Cheung, and K. A. Shore, “Terahertz bandwidth prediction for amplitude modulation response of unipolar intersubband semiconductor lasers,” IEEE Photon. Technol. Lett.11(5), 527–529 (1999). [CrossRef]
  14. M. K. Haldar, “A simplified analysis of direct intensity modulation of quantum cascade lasers,” IEEE J. Quantum Electron.41(11), 1349–1355 (2005). [CrossRef]
  15. R. Paiella, R. Martini, F. Capasso, C. Gmachl, H. Y. Hwang, D. L. Sivco, J. N. Baillargeon, A. Y. Cho, E. A. Whittaker, and H. C. Liu, “High-frequency modulation without the relaxation oscillation resonance in quantum cascade lasers,” Appl. Phys. Lett.79(16), 2526–2528 (2001). [CrossRef]
  16. S. Barbieri, W. Maineult, S. S. Dhillon, C. Sirtori, J. Alton, N. Breuil, H. E. Beere, and D. A. Ritchie, “13 GHz direct modulation of terahertz quantum cascade lasers,” Appl. Phys. Lett.91(14), 143510 (2007). [CrossRef]
  17. A. Lyakh, R. Maulini, A. Tsekoun, R. Go, C. Pflügl, L. Diehl, Q. J. Wang, F. Capasso, C. Kumar, and N. Patel, “3 W continuous-wave room temperature single-facet emission from quantum cascade lasers based on nonresonant extraction design approach,” Appl. Phys. Lett.95(14), 141113 (2009).
  18. Q. J. Wang, C. Pflügl, L. Diehl, F. Capasso, T. Edamura, S. Furuta, M. Yamanishi, and H. Kan, “High performance quantum cascade lasers based on three-phonon-resonance design,” Appl. Phys. Lett.94(1), 011103 (2009). [CrossRef]
  19. F. Mogensen, H. Olesen, and G. Jacobsen, “Locking conditions and stability properties for a semiconductor laser with external light injection,” IEEE J. Quantum Electron.21(7), 784–793 (1985). [CrossRef]
  20. D. Indjin, P. Harrison, R. W. Kelsall, and Z. Ikonić, “Self-consistent scattering theory of transport and output characteristics of quantum cascade lasers,” J. Appl. Phys.91(11), 9019–9026 (2002). [CrossRef]
  21. F. Rana and R. J. Ram, “Current noise and photon noise in quantum cascade lasers,” Phys. Rev. B65(12), 125313 (2002). [CrossRef]
  22. J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science264(5158), 553–556 (1994). [CrossRef] [PubMed]
  23. T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett.89(9), 091121 (2006). [CrossRef]
  24. J. von Staden, T. Gensty, W. Elsäßer, G. Giuliani, and C. Mann, “Measurements of the α factor of a distributed-feedback quantum cascade laser by an optical feedback self-mixing technique,” Opt. Lett.31(17), 2574–2576 (2006). [CrossRef] [PubMed]
  25. M. Ishihara, T. Morimoto, S. Furuta, K. Kasahara, N. Akikusa, K. Fujita, and T. Edamura, “Linewidth enhancement factor of quantum cascade lasers with single phonon resonance-continuum depopulation structure on Peltier cooler,” Electron. Lett.45(23), 1168–1169 (2009). [CrossRef]
  26. M. S. Taubman, T. L. Myers, B. D. Cannon, and R. M. Williams, “Stabilization, injection and control of quantum cascade lasers, and their application to chemical sensing in the infrared,” Spectrochim. Acta A Mol. Biomol. Spectrosc.60(14), 3457–3468 (2004). [CrossRef] [PubMed]
  27. P. Gellie, S. Barbieri, J. F. Lampin, P. Filloux, C. Manquest, C. Sirtori, I. Sagnes, S. P. Khanna, E. H. Linfield, A. G. Davies, H. Beere, and D. Ritchie, “Injection-locking of terahertz quantum cascade lasers up to 35 GHz using RF amplitude modulation,” Opt. Express18(20), 20799–20816 (2010). [CrossRef]
  28. C. H. Henry, N. A. Olsson, and N. K. Dutta, “Locking range and stability of injection locked 1.54 μm InGaAsP semiconductor laser,” IEEE J. Quantum Electron.21(8), 1152–1156 (1985). [CrossRef]

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