## Effects of resonant tunneling and dynamics of coherent interaction on intrinsic linewidth of quantum cascade lasers |

Optics Express, Vol. 20, Issue 15, pp. 17145-17159 (2012)

http://dx.doi.org/10.1364/OE.20.017145

Acrobat PDF (1357 KB)

### Abstract

A theoretical model for calculation of the intrinsic linewidth of QCLs is built on the basis of the quantum Langevin approach. It differs from the traditional rate equation model in that the resonant tunneling and the dynamics of coherent interaction can be considered. Results show that the coupling strength and the dephasing rate associated with resonant tunneling strongly affect the linewidth of THz QCLs in the incoherent resonant-tunneling transport regime but only induce little influence in the coherent regime. The dynamics of coherent interaction and resonant-tunneling transport show insignificant effects on the linewidth calculation of mid-infrared QCLs due to strong coupling in resonant tunneling. We also demonstrate that by properly designing the active regions of QCLs, one can reduce the intrinsic linewidth according to our model.

© 2012 OSA

## 1. Introduction

10. S. Kumar and Q. Hu, “Coherence of resonant-tunneling transport in terahertz quantum-cascade lasers,” Phys. Rev. B **80**(24), 245316 (2009). [CrossRef]

12. H. Callebaut and Q. Hu, “Importance of coherence for electron transport in terahertz quantum cascade lasers,” J. Appl. Phys. **98**(10), 104505 (2005). [CrossRef]

13. L. Davidovich, “Sub-Poissonian processes in quantum optics,” Rev. Mod. Phys. **68**(1), 127–173 (1996). [CrossRef]

14. G. P. Agrawal and C. M. Bowden, “Concept of linewidth enhancement factor in semiconductor lasers: its usefulness and limitations,” IEEE Photon. Technol. Lett. **5**(6), 640–642 (1993). [CrossRef]

## 2. Quantum Langevin equations

*rotating-wave approximation*can be obtained from Ref. 15 by adding the resonant-tunneling term (the fourth item on the right side of the equation):where

*j*,

*j*, respectively. The coupling constant

*g*is given bywhere

*j*. By including the noise operators

*γ*is the rate of spontaneous emission coupling into all the field modes except for the laser mode,

_{sp}*j*,

10. S. Kumar and Q. Hu, “Coherence of resonant-tunneling transport in terahertz quantum-cascade lasers,” Phys. Rev. B **80**(24), 245316 (2009). [CrossRef]

*T*(approximate electron temperature at the quasi-thermal equilibrium) and can be written as

_{e}## 3. Laser intrinsic linewidths

### 3.1. Equivalent c-number Langevin equations

*c-*number equations [15]. For this we have to choose certain particular ordering for field and electron density operators, because the

*c*-number variables commute with each other while the operators do not. Here we choose the normal ordering of field and electron density operators to be

*c*-number variables corresponding to the operators

*c*-number Langevin equations of Eqs. (3) can be written as Here, the Langevin forces

*c*-number Langevin forces may be different from the corresponding diffusion coefficients

*c*-number equations agree with those calculated from the operator equations as explained in Appendix B.

### 3.2. Steady-state solution for above-threshold operation

*N*is the electron number of one module in the cavity.

_{0}### 3.3. Dynamics of fluctuations around steady state

### 3.4. Noise spectra

13. L. Davidovich, “Sub-Poissonian processes in quantum optics,” Rev. Mod. Phys. **68**(1), 127–173 (1996). [CrossRef]

*S*, therefore, can be derived as

_{f}18. C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. **18**(2), 259–264 (1982). [CrossRef]

^{2}if this factor is considered. This parameter is not included in our model since the α-value is negligible in QCLs, which has been confirmed by experiments [19

19. 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]

20. B. Daino, P. Spano, M. Tamburrini, and S. Piazzolla, “Phase noise and spectral line shape in semiconductor lasers,” IEEE J. Quantum Electron. **19**(3), 266–270 (1983). [CrossRef]

## 4. Results and discussions

*i.e.*thermal photons (2

*κn*), vacuum fluctuations (

_{th}*κ*), and spontaneous emission processes (2

*gg**/

*γ*

_{23}). These sources induce linewidth broadening in lasers, which cannot be overcome due to fundamental quantum limitations. It needs to be mentioned that the vacuum fluctuations as one of noise sources are not included in rate equation model. Although the fact that vacuum fluctuations cannot be detected directly, the interaction of this vacuum field and the laser field can result in a modulation of the photon flux, which causes noise in the cavity [21

21. C. H. Henry and R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. **68**(3), 801–853 (1996). [CrossRef]

9. M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron. **44**(1), 12–29 (2008). [CrossRef]

*τ =*(

*γ*

_{23}+

*κ*/2)

^{−1},

*i.e.*using ultrafast photodetectors, the phase fluctuation caused by spontaneous emission will disappear. This is attributed to suppression of spontaneous emission quantum noise caused by the electron memory effect associated with the transient behavior of the polarization. Therefore, for short-time measurements, the laser linewidth will reduce due to the disappearance of the spontaneous emission contribution. This property is not shown in the rate equation model. Since the rate equation model cannot include the dynamics of coherent interaction, the linewidth of the laser can only be based on the long-time measurement. Owing to the limit of bandwidth of the photodetector, we will not consider the cases of short-time measurements, and only investigate the laser linewidth of a long-time measurement in the following discussion.

9. M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron. **44**(1), 12–29 (2008). [CrossRef]

*κ*/2

*γ*

_{23})

^{2}. In the rate equation model,

*κ*/2

*γ*

_{23})

^{2}if the coherent time is comparable to the cavity loss according to Eq. (30). The intrinsic difference between these two models is from the resonant-tunneling effects. Electrons in the ground injector level are injected into the upper laser level in QCLs through resonant tunneling. Since coherence plays an important role in the resonant-tunneling mechanism, which can significantly influence the electron transport [10

10. S. Kumar and Q. Hu, “Coherence of resonant-tunneling transport in terahertz quantum-cascade lasers,” Phys. Rev. B **80**(24), 245316 (2009). [CrossRef]

12. H. Callebaut and Q. Hu, “Importance of coherence for electron transport in terahertz quantum cascade lasers,” J. Appl. Phys. **98**(10), 104505 (2005). [CrossRef]

**80**(24), 245316 (2009). [CrossRef]

12. H. Callebaut and Q. Hu, “Importance of coherence for electron transport in terahertz quantum cascade lasers,” J. Appl. Phys. **98**(10), 104505 (2005). [CrossRef]

**80**(24), 245316 (2009). [CrossRef]

**98**(10), 104505 (2005). [CrossRef]

**80**(24), 245316 (2009). [CrossRef]

_{31′}increases, only a small fraction of the electrons tunnels through the injector barrier into the upper laser level. As a result, the current density and photon number decreases (Fig. (4b)) and the noise associated with spontaneous emission becomes strong. Therefore, the linewidth increases as the dephasing rate increases.

_{31′}can only induce few changes of photon numbers, and hence only introduce little influence on the linewidth of mid-IR QCLs.

22. A. M. Andrews, A. Benz, C. Deutsch, G. Fasching, K. Unterrainer, P. Klang, W. Schrenk, and G. Strasser, “Doping dependence of LO-phonon depletion scheme THz quantum-cascade lasers,” Mater. Sci. Eng. B **147**(2-3), 152–155 (2008). [CrossRef]

## 5. Conclusions

*e.g.*thermal photons, vacuum fluctuations and spontaneous emission processes. The results show that for the short time measurement, the effects of electron memory can lead to suppression of spontaneous emission quantum noise; and the intrinsic linewidth of QCLs are reduced with the consideration of dynamics of coherent interaction. We also demonstrate that the coupling strength and dephasing rate have significant effects on the linewidth of THz QCLs in the incoherent resonant-tunneling regime, but small effects on that of mid-IR QCLs due to their strong coherent resonant-tunneling. The linewidth decreases with the increase of the injection coupling strength and reduction of the dephasing rate associated with resonant tunneling. According to our model, a reduced intrinsic linewidth can be easily designed through optimization of the injector barrier and doping density of the active region of QCLs.

## Appendix A: operator diffusion coefficients

*generalized Einstein relations*[15,16]

## Appendix B: *c-*number diffusion coefficients

*c-*number

*c-*number quantum Langevin equations

*generalized Einstein relations*

*c*-number product [15]. Therefore, we have

*c-*number second moments have

## Acknowledgments

## References and links

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

2. | T. Liu and Q. J. Wang, “Fundamental frequency noise and linewidth broadening caused by intrinsic temperature fluctuations in quantum cascade lasers,” Phys. Rev. B |

3. | R. F. Curl, F. Capsso, C. Gmachl, A. A. Kosterev, B. McManus, R. Lewicki, M. Pusharsky, G. Wysocki, and F. K. Tittel, “Quantum cascade lasers in chemical physics,” Chem. Phys. Lett. |

4. | M. S. Vitiello, L. Consolino, S. Bartalini, A. Tredicucci, M. Inguscio, and P. De Natale, “The intrinsic linewidth of THz quantum cascade lasers,” CLEO: Science and Innovations, (Optical Society of America, 2012), paper CTu2B.2. |

5. | C. Jirauschek, “Monte Carlo study of intrinsic linewidths in terahertz quantum cascade lasers,” Opt. Express |

6. | S. Bartalini, S. Borri, I. Galli, G. Giusfredi, D. Mazzotti, T. Edamura, N. Akikusa, M. Yamanishi, and P. De Natale, “Measuring frequency noise and intrinsic linewidth of a room-temperature DFB quantum cascade laser,” Opt. Express |

7. | L. Tombez, S. Schilt, J. Di Francesco, P. Thomann, and D. Hofstetter, “Temperature dependence of the frequency noise in a mid-IR DFB quantum cascade laser from cryogenic to room temperature,” Opt. Express |

8. | S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow-Townes limit,” Phys. Rev. Lett. |

9. | M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron. |

10. | S. Kumar and Q. Hu, “Coherence of resonant-tunneling transport in terahertz quantum-cascade lasers,” Phys. Rev. B |

11. | C. Sirtori, F. Capasso, J. Faist, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “Resonant tunneling in quantum cascade lasers,” IEEE J. Quantum Electron. |

12. | H. Callebaut and Q. Hu, “Importance of coherence for electron transport in terahertz quantum cascade lasers,” J. Appl. Phys. |

13. | L. Davidovich, “Sub-Poissonian processes in quantum optics,” Rev. Mod. Phys. |

14. | G. P. Agrawal and C. M. Bowden, “Concept of linewidth enhancement factor in semiconductor lasers: its usefulness and limitations,” IEEE Photon. Technol. Lett. |

15. | M. O. Scully and M. S. Zubairy, |

16. | W. H. Louisell, |

17. | K. Petermann, |

18. | C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. |

19. | 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. |

20. | B. Daino, P. Spano, M. Tamburrini, and S. Piazzolla, “Phase noise and spectral line shape in semiconductor lasers,” IEEE J. Quantum Electron. |

21. | C. H. Henry and R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. |

22. | A. M. Andrews, A. Benz, C. Deutsch, G. Fasching, K. Unterrainer, P. Klang, W. Schrenk, and G. Strasser, “Doping dependence of LO-phonon depletion scheme THz quantum-cascade lasers,” Mater. Sci. Eng. B |

23. | S. Kumar, Q. Hu, and J. L. Reno, “186 K operation of terahertz quantum-cascade lasers based on a diagonal design,” Appl. Phys. Lett. |

24. | D. Hofstetter, M. Beck, T. Aellen, and J. Faist, “High-temperature operation of distributed feedback quantum-cascade lasers at 5.3 μm,” Appl. Phys. Lett. |

**OCIS Codes**

(140.3070) Lasers and laser optics : Infrared and far-infrared lasers

(270.2500) Quantum optics : Fluctuations, relaxations, and noise

(300.3700) Spectroscopy : Linewidth

(140.5965) Lasers and laser optics : Semiconductor lasers, quantum cascade

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: May 23, 2012

Revised Manuscript: June 30, 2012

Manuscript Accepted: July 9, 2012

Published: July 12, 2012

**Citation**

Tao Liu, Kenneth E. Lee, and Qi Jie Wang, "Effects of resonant tunneling and dynamics of coherent interaction on intrinsic linewidth of quantum cascade lasers," Opt. Express **20**, 17145-17159 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-15-17145

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### References

- 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]
- T. Liu and Q. J. Wang, “Fundamental frequency noise and linewidth broadening caused by intrinsic temperature fluctuations in quantum cascade lasers,” Phys. Rev. B 84(12), 125322 (2011). [CrossRef]
- R. F. Curl, F. Capsso, C. Gmachl, A. A. Kosterev, B. McManus, R. Lewicki, M. Pusharsky, G. Wysocki, and F. K. Tittel, “Quantum cascade lasers in chemical physics,” Chem. Phys. Lett. 487(1-3), 1–18 (2010). [CrossRef]
- M. S. Vitiello, L. Consolino, S. Bartalini, A. Tredicucci, M. Inguscio, and P. De Natale, “The intrinsic linewidth of THz quantum cascade lasers,” CLEO: Science and Innovations, (Optical Society of America, 2012), paper CTu2B.2.
- C. Jirauschek, “Monte Carlo study of intrinsic linewidths in terahertz quantum cascade lasers,” Opt. Express 18(25), 25922–25927 (2010). [CrossRef] [PubMed]
- S. Bartalini, S. Borri, I. Galli, G. Giusfredi, D. Mazzotti, T. Edamura, N. Akikusa, M. Yamanishi, and P. De Natale, “Measuring frequency noise and intrinsic linewidth of a room-temperature DFB quantum cascade laser,” Opt. Express 19(19), 17996–18003 (2011). [CrossRef] [PubMed]
- L. Tombez, S. Schilt, J. Di Francesco, P. Thomann, and D. Hofstetter, “Temperature dependence of the frequency noise in a mid-IR DFB quantum cascade laser from cryogenic to room temperature,” Opt. Express 20(7), 6851–6859 (2012). [CrossRef] [PubMed]
- S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow-Townes limit,” Phys. Rev. Lett. 104(8), 083904 (2010). [CrossRef] [PubMed]
- M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron. 44(1), 12–29 (2008). [CrossRef]
- S. Kumar and Q. Hu, “Coherence of resonant-tunneling transport in terahertz quantum-cascade lasers,” Phys. Rev. B 80(24), 245316 (2009). [CrossRef]
- C. Sirtori, F. Capasso, J. Faist, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “Resonant tunneling in quantum cascade lasers,” IEEE J. Quantum Electron. 34(9), 1722–1729 (1998). [CrossRef]
- H. Callebaut and Q. Hu, “Importance of coherence for electron transport in terahertz quantum cascade lasers,” J. Appl. Phys. 98(10), 104505 (2005). [CrossRef]
- L. Davidovich, “Sub-Poissonian processes in quantum optics,” Rev. Mod. Phys. 68(1), 127–173 (1996). [CrossRef]
- G. P. Agrawal and C. M. Bowden, “Concept of linewidth enhancement factor in semiconductor lasers: its usefulness and limitations,” IEEE Photon. Technol. Lett. 5(6), 640–642 (1993). [CrossRef]
- M. O. Scully and M. S. Zubairy, Quantum optics (Cambridge University Press, 1997).
- W. H. Louisell, Quantum statistical properties of radiation (Wiley, 1973).
- K. Petermann, Laser diode modulation and noise (Kluwar Academic, 1988).
- C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18(2), 259–264 (1982). [CrossRef]
- 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]
- B. Daino, P. Spano, M. Tamburrini, and S. Piazzolla, “Phase noise and spectral line shape in semiconductor lasers,” IEEE J. Quantum Electron. 19(3), 266–270 (1983). [CrossRef]
- C. H. Henry and R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68(3), 801–853 (1996). [CrossRef]
- A. M. Andrews, A. Benz, C. Deutsch, G. Fasching, K. Unterrainer, P. Klang, W. Schrenk, and G. Strasser, “Doping dependence of LO-phonon depletion scheme THz quantum-cascade lasers,” Mater. Sci. Eng. B 147(2-3), 152–155 (2008). [CrossRef]
- S. Kumar, Q. Hu, and J. L. Reno, “186 K operation of terahertz quantum-cascade lasers based on a diagonal design,” Appl. Phys. Lett. 94(13), 131105 (2009). [CrossRef]
- D. Hofstetter, M. Beck, T. Aellen, and J. Faist, “High-temperature operation of distributed feedback quantum-cascade lasers at 5.3 μm,” Appl. Phys. Lett. 78(4), 396 (2001). [CrossRef]

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