## Theoretical investigation of injection-locked high modulation bandwidth quantum cascade lasers |

Optics Express, Vol. 20, Issue 2, pp. 1450-1464 (2012)

http://dx.doi.org/10.1364/OE.20.001450

Acrobat PDF (1276 KB)

### 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

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]

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]

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]

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]

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]

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]

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]

## 2. Theory

### 2.1 Direct intensity modulation of QCLs

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]

*N*

_{2}and

*N*

_{3}, respectively, and the photon number by

*P*. Here the cavity is assumed to have only one longitudinal mode. We noticed that analysis of direct intensity modulation of QCLs with rate equation model has also been investigated in [14

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]

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]

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

*J*denotes the current injected into the active region divided by electronic charge

*e*,

*G*is the optical gain coefficient of the entire active region,

*N*is the number of stages,

_{p}*τ*and

_{2}*τ*represent the lifetime of lower and upper laser levels, respectively, and

_{3}*τ*is the lifetime of photon in the cavity, expressed as

_{P}*τ*where

_{P}= ν_{g}(α_{w}+α_{m})*α*,

_{w}*α*, and

_{m}*ν*are the waveguide loss, mirror loss, and group velocity, respectively. The mirror loss can be calculated using

_{g}*α*= -ln(

_{m}*R*)/(2

_{1}R_{2}*L*), where

*R*and

_{1}*R*are the power reflectivity of the facets 1 and 2, respectively. The group velocity

_{2}*ν*is given by

_{g}*ν*=

_{g}*c*/

*n*, where

_{eff}*c*and

*n*are the speed of light in vacuum and modal effective refractive index, respectively. In our analysis,

_{eff}*G*/

*N*instead of

_{P}*G*as in [14

**41**(11), 1349–1355 (2005). [CrossRef]

*G*=

_{0}*G*/

*N*, we have

_{P}*G*/

_{0}= Γ^{’}ν_{g}σ_{32}*V*, where

*Γ*is the optical mode confinement factor for a single period, and

^{’}*V*stands for the volume for a period, given by

*WLL*, where

_{P}*W*and

*L*are the width and length of the active region, and

*L*denotes the period length, respectively. The stimulated emission cross section

_{P}*σ*is,where

_{32}*e*is the electronic charge,

*z*is the dipole matrix element between the upper and the lower lasing levels,

_{32}*ε*is the vacuum dielectric constant,

_{0}*λ*is the free-space emission wavelength, and (2

_{0}*γ*) stands for the full width at half maximum (FWHM) of the optical transition spectrum.

_{32}#### 2.1.1 Steady state analysis

*J*,

*N*and

_{2}, N_{3},*P*as

*J*,

_{0}*N*,

_{20}*N*and

_{30}*P*then the steady state rate equations can be expressed as:

_{0},*N*is given aswith Eqs. (4) and (5),

_{20}*P*can be obtained in the following equation,where

_{0}*η=τ*. Also, we note that at threshold,where

_{2}/ τ_{3}*J*is the threshold current (s

_{th}^{−1}), given by

*J*= 1/

_{th}*Gτ*(

_{3}τ_{P}*1-η*). Substituting Eq. (9) into Eq. (8), one gets the steady state photon number

*P*

_{0},with Eqs. (7) and (9), the upper laser level population

*N*

_{30}can be written as,

#### 2.1.2 Small-signal modulation analysis

*N*, ∆

_{2}*N*, ∆

_{3}*P*and ∆

*J*around the steady state values, and substituting

*N*=

_{2}*N*+∆

_{20}*N*,

_{2}*N*=

_{3}*N*+∆

_{30}*N*,

_{3}*P*=

*P*=+∆

_{0}*P*and

*J*=

*J*+∆

_{0}*J*into Eqs. (1) – (3), we get the rate equations for the small deviations as follows:

### 2.2 Injection locking of quantum cascade lasers (QCLs)

#### 2.2.1 Steady state analysis

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]

*f*is the coupling rate between the master laser and the slave laser, approximately expressed as:

_{d}*f*= ν

_{d}*(1-*

_{g}*R*)/(2

*LR*

^{1/2}) 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, where

*E*(

_{1}*t*) is taken as a constant,

*ϕ*(

_{0}*t*) and

*ϕ*(

_{1}*t*) are the phases of the two electromagnetic fields, (

*ϕ*(

_{1}*t*) is usually set as zero for computing convenience);

*ω*and

_{0}*ω*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.

_{1}*N*is the carrier number change due to light injection from the master laser, given by

*N*-

*N*. 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.

_{th}*N*=

*N*-

_{3}*N*. 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, where

_{2}*E*(

_{0}*t*),

*ϕ*(

_{0}*t*),

*N*(

*t*), and

*N*(

_{2}*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

*E*

_{0}(

*t*) has been normalized, so that |

*E*(

_{0}*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*)=

*ϕ*(

_{SL}*t*)−

*ϕ*(

_{ML}*t*),

*G*,

_{0}*N*,

_{P}*α*,

*J*,

*τ*, and

_{2}*τ*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

_{3}*f*,

_{d}*E*and ∆

_{1}*ω*represent the coupling rate, injected electrical field magnitude, and frequency detuning, respectively. The latter is expressed as ∆

_{inj}*ω*=

_{inj}*ω*-

_{1}*ω*. For free-running laser,

_{0}*N = N*,

_{th}*N*

_{2}

*= N*

_{2}

*,*

_{th}*E = E*, with Eq. (24), the carrier number in the lower laser state for free-running slave laser can be obtained,where we define γ

_{fr}*as,*

_{P}*is given by*

_{P0}*GN*, so that γ

_{th}*= γ*

_{P}*/*

_{P0}*N*. With Eq. (23) and the expression of γ

_{P}*, the threshold current, defined as*

_{P}*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,

*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

*N*under injection locking are denoted as

_{2}*E*, ∆

_{0}*N*,

_{0}*ϕ*and

_{0}*N*respectively. The corresponding expressions of ∆

_{20}*N*,

_{0}*ϕ*and

_{0}*N*are given below,

_{20}*E*. Defining two new parameters, the equation is shown as,

_{0}*E*can be solved by various numerical methods, given that the phase difference

_{0}*ϕ*varies approximately from -π/2 to cot

_{0}^{−1}α in the injection locking range, and ∆

*N*should be negative.

_{0}#### 2.2.2 Small-signal modulation analysis

*E*, ∆

*ϕ*, ∆

*N*and ∆

*N*around the corresponding steady state value, and substituting

_{2}*E*=

*E*

_{0}+∆

*E*,

*ϕ*=

*ϕ*∆

_{0}+*ϕ*,

*N*=

*N*+∆

_{0}*N*,

*N*=

_{2}*N*+∆

_{20}*N*into Eqs. (21) – (24), one gets the rate equations for the small signal modulation:

_{2}## 3. Results and discussions

### 3.1 Direct modulations

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]

*H*(s). Tables 2 and 3 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

*p*

_{1},

*p*

_{2}and

*p*

_{3}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

**79**(16), 2526–2528 (2001). [CrossRef]

**79**(16), 2526–2528 (2001). [CrossRef]

*I*= 1.5

*I*, 2

_{th}*I*

_{th}, 3

*I*

_{th}, and 4

*I*

_{th}, 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*)≈1/(

*s*/|

*p*

_{1}|+1) in the Bode plot. For cubic equations, we have

*p*

_{1}

*p*

_{2}+

*p*

_{2}

*p*

_{3}+

*p*

_{1}

*p*

_{3}=

*D*/

*B*, and

*p*

_{1}

*p*

_{2}

*p*

_{3}=−1/

*B*, leading to 1/

*p*

_{1}+1/

*p*

_{2}+1/

*p*

_{3}=−1/

*D*. For

*p*

_{1}is much smaller than

*p*

_{2}and

*p*

_{3}, we get |1/

*p*

_{1}|≈|

*D*|. Substituting |1/

*p*

_{1}| 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,where

*D*is given by (

*τ*

_{P}+ 2

*τ*

_{2}) +

*τ*

_{P}/(

*G*

_{0}

*P*

_{0}

*τ*

_{3}). The definition of 3-dB bandwidth is defined as |

*H*(

*s*)| at half of its zero value, i.e. |H(s)|

_{f}_{3dB}| = 1/2 in the case of normalized frequency response. An approximated expression for 3-dB bandwidth is given as,

*D*and

*f*

_{3dB}, we can easily see that a larger value of

*τ*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

_{p}*λ*

^{2}) 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

*τ*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.

_{p}### 3.2 Injection locking modulations

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]

*J*= 4

*J*

_{th}. In our simulations, the boundaries of phase in the injection locking range are approximately cot

^{−1}

*α*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]

*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 z

_{1}is smaller than any of the poles, making the peak in the frequency response apparent, even though all the poles are real.

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]

*J*

_{th}, respectively, and a frequency detuning 0.1 GHz away from the negative detuning edge, as shown in Fig. 4 . The inset is the corresponding pole/zero diagrams. Tables 6 and 7 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 z

_{1}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]

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]

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]

**79**(16), 2526–2528 (2001). [CrossRef]

*N*,

*P*, and

*ϕ*as δ

*N,*δ

*P*and δ

*ϕ*), 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]

*P*causes an increased δ

*ϕ*in relaxation oscillations. On the other hand, at the negative detuning edge, the increase of δ

*ϕ*causes an enhanced mismatch of

*E*

_{0}and

*E*

_{1}, leading to a larger cavity intensity change,

*i.e.*larger δ

*P*. This forms a positive feedback between δ

*ϕ*and δ

*P*, which is enhanced as the relaxation oscillations increase. However, there is no relaxation oscillation in QCLs, thus the interaction between δ

*ϕ*and δ

*P*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

## 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. |

2. | E. K. Lau, H.-K. Sung, and M. C. Wu, “Frequency response enhancement of optical injection-locked lasers,” IEEE J. Quantum Electron. |

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 |

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. |

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 |

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. |

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. |

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. |

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. |

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 |

11. | C. Y. L. Cheung, P. S. Spencer, and K. A. Shore, “Modulation bandwidth optimization for unipolar intersubband semiconductor lasers,” IEE Proc.: Optoelectron. |

12. | C. Y. Cheung and K. A. Shore, “Self-consistent analysis of dc modulation response of unipolar semiconductor lasers,” J. Mod. Opt. |

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. |

14. | M. K. Haldar, “A simplified analysis of direct intensity modulation of quantum cascade lasers,” IEEE J. Quantum Electron. |

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. |

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. |

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. |

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. |

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. |

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. |

21. | F. Rana and R. J. Ram, “Current noise and photon noise in quantum cascade lasers,” Phys. Rev. B |

22. | J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science |

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. |

24. | J. von Staden, T. Gensty, W. Elsäßer, G. Giuliani, and C. Mann, “Measurements of the |

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. |

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. |

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 |

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. |

**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

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