## Microscopic approach to second harmonic generation in quantum cascade lasers |

Optics Express, Vol. 22, Issue 15, pp. 18389-18400 (2014)

http://dx.doi.org/10.1364/OE.22.018389

Acrobat PDF (801 KB)

### Abstract

Second harmonic generation is analyzed from a microscopical point of view using a non-equilibrium Green’s function formalism. Through this approach the complete on-state of the laser can be modeled and results are compared to experiment with good agreement. In addition, higher order current response is extracted from the calculations and together with waveguide properties, these currents provide the intensity of the second harmonic in the structure considered. This power is compared to experimental results, also with good agreement. Furthermore, our results, which contain all coherences in the system, allow to check the validity of common simplified expressions.

© 2014 Optical Society of America

## 1. Introduction

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

2. M. M. Fejer, S. J. B. Yoo, R. L. Byer, A. Harwit, and J. S. Harris Jr., “Observation of extremely large quadratic susceptibility at 9.6 – 10.8μm in electric-field-biased AlGaAs quantum wells,” Phys. Rev. Lett. **62**, 1041–1044 (1989). [CrossRef] [PubMed]

3. E. Rosencher, A. Fiore, B. Winter, V. Berger, P. Bois, and J. Nagle, “Quantum engineering of optical nonlinearities,” Science **271**, 168–173 (1996). [CrossRef]

4. F. Capasso, C. Sirtori, and A. Y. Cho, “Coupled quantum well semiconductors with giant electric field tunable nonlinear optical properties in the infrared,” IEEE J. Quantum Electron. **30**, 1313–1326 (1994). [CrossRef]

5. M. A. Belkin, F. Capasso, A. Belyanin, D. L. Sivco, A. Y. Cho, D. C. Oakley, C. J. Vineis, and G. W. Turner, “Terahertz quantum-cascade-laser source based on intracavity difference-frequency generation,” Nat. Photonics **1**, 288–292 (2007). [CrossRef]

6. N. Owschimikow, C. Gmachl, A. Belyanin, V. Kocharovsky, D. L. Sivco, R. Colombelli, F. Capasso, and A. Y. Cho, “Resonant second-order nonlinear optical processes in quantum cascade lasers,” Phys. Rev. Lett. **90**, 043902 (2003). [CrossRef] [PubMed]

7. J.-Y. Bengloan, A. De Rossi, V. Ortiz, X. Marcadet, M. Calligaro, I. Maurin, and C. Sirtori, “Intracavity sum-frequency generation in GaAs quantum cascade lasers,” Appl. Phys. Lett. **84**, 2019–2021 (2004). [CrossRef]

8. Y. Yao, J. A. Hoffman, and C. F. Gmachl, “Mid-infrared quantum cascade lasers,” Nat. Photonics **6**, 432–439 (2012). [CrossRef]

9. B. S. Williams, “Terahertz quantum-cascade lasers,” Nat. Photonics **1**, 517 (2007). [CrossRef]

11. Y.-H. Cho and A. Belyanin, “Short-wavelength infrared second harmonic generation in quantum cascade lasers,” J. Appl. Phys. **107**, 053116 (2010). [CrossRef]

*μ*m [12

12. J. Faist, F. Capasso, D. L. Sivco, A. L. Hutchinson, S.-N. G. Chu, and A. Y. Cho, “Short wavelength (λ ∼ 3.4μm) quantum cascade laser based on strained compensated InGaAs/AlInAs,” Appl. Phys. Lett. **72**, 680 (1998). [CrossRef]

13. C. Gmachl, F. Capasso, D. L. Sivco, and A. Y. Cho, “Recent progress in quantum cascade lasers and applications,” Rep. Prog. Phys. **64**, 1533 (2001). [CrossRef]

14. F. Capasso, R. Paiella, R. Martini, R. Colombelli, C. Gmachl, T. Myers, M. S. Taubman, R. Williams, C. Bethea, K. Unterrainer, H. Y. Hwang, D. L. Sivco, A. Y. Cho, A. Sergent, H. Liu, and E. A. Whittaker, “Quantum cascade lasers: ultrahigh-speed operation, optical wireless communication, narrow linewidth, and far-infrared emission,” IEEE J. Quantum Electron. **38**, 511–532 (2002). [CrossRef]

11. Y.-H. Cho and A. Belyanin, “Short-wavelength infrared second harmonic generation in quantum cascade lasers,” J. Appl. Phys. **107**, 053116 (2010). [CrossRef]

15. C. Gmachl, A. Belyanin, D. L. Sivco, M. L. Peabody, N. Owschimikow, A. M. Sergent, F. Capasso, and A. Y. Cho, “Optimized second-harmonic generation in quantum cascade lasers,” IEEE J. Quantum Electron. **39**, 1345–1355 (2003). [CrossRef]

6. N. Owschimikow, C. Gmachl, A. Belyanin, V. Kocharovsky, D. L. Sivco, R. Colombelli, F. Capasso, and A. Y. Cho, “Resonant second-order nonlinear optical processes in quantum cascade lasers,” Phys. Rev. Lett. **90**, 043902 (2003). [CrossRef] [PubMed]

*et al.*[17

17. A. Gajić, J. Radovanović, V. Milanović, D. Indjin, and Z. Ikonić, “Genetic algorithm applied to the optimization of quantum cascade lasers with second harmonic generation,” J. Appl. Phys. **115**, 053712 (2014). [CrossRef]

*et al.*[18

18. J. Bai and D. Citrin, “Optical and transport characteristics of quantum-cascade lasers with optimized second-harmonic generation,” IEEE J. Quantum Electron. **43**, 391–398 (2007). [CrossRef]

## 2. The model

19. F. Capasso, J. Faist, and C. Sirtori, “Mesoscopic phenomena in semiconductor nanostructures by quantum design,” J. Math. Phys. **37**, 4775 (1996). [CrossRef]

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

21. A. Hugi, R. Maulini, and J. Faist, “External cavity quantum cascade laser,” Semicond. Sci. Technol. **25**, 083001 (2010). [CrossRef]

22. Y. Petitjean, F. Destic, J. Mollier, and C. Sirtori, “Dynamic modeling of terahertz quantum cascade lasers,” IEEE J. Sel. Top. Quantum Electron. **17**, 22–29 (2011). [CrossRef]

23. R. C. Iotti, E. Ciancio, and F. Rossi, “Quantum transport theory for semiconductor nanostructures: A density matrix formulation,” Phys. Rev. B **72**, 125347 (2005). [CrossRef]

26. E. Dupont, S. Fathololoumi, and H. C. Liu, “Simplified density-matrix model applied to three-well terahertz quantum cascade lasers,” Phys. Rev. B **81**, 205311 (2010). [CrossRef]

27. R. C. Iotti and F. Rossi, “Nature of charge transport in quantum-cascade lasers,” Phys. Rev. Lett. **87**, 146603 (2001). [CrossRef] [PubMed]

29. C. Jirauschek and P. Lugli, “Monte-carlo-based spectral gain analysis for terahertz quantum cascade lasers,” J. Appl. Phys. **105**, 123102 (2009). [CrossRef]

30. S.-C. Lee and A. Wacker, “Nonequilibrium Green’s function theory for transport and gain properties of quantum cascade structures,” Phys. Rev. B **66**, 245314 (2002). [CrossRef]

33. G. Haldaś, A. Kolek, and I. Tralle, “Modeling of mid-infrared quantum cascade laser by means of nonequilibrium green’s functions,” IEEE J. Quantum Electron. **47**, 878–885 (2011). [CrossRef]

35. H. Haug and S. Koch, *Quantum Theory of the Optical and Electronic Properties of Semiconductors* (World Scientific, 2009). [CrossRef]

*F*(

*t*) =

*F*

_{ac}cos(

*ωt*) where

*ω*is the driving field frequency and

*F*

_{ac}is the field amplitude. In the following we express the ac field strength as

*eF*

_{ac}

*d*in units of eV, where

*d*is the period length. High intensities inside the QCL require a model going beyond linear response to the external electromagnetic field. This is done in our simulations by decomposing the time dependence of observables of the system in a Fourier series of the fundamental frequency

*ω*and its higher harmonics. This procedure follows the concepts outlined in [34

34. T. Brandes, “Truncation method for green’s functions in time-dependent fields,” Phys. Rev. B **56**, 1213 (1997). [CrossRef]

15. C. Gmachl, A. Belyanin, D. L. Sivco, M. L. Peabody, N. Owschimikow, A. M. Sergent, F. Capasso, and A. Y. Cho, “Optimized second-harmonic generation in quantum cascade lasers,” IEEE J. Quantum Electron. **39**, 1345–1355 (2003). [CrossRef]

## 3. Laser operation analysis

*n*

_{max}= 0 provide the stationary response

*J*

_{0}from Eq. (1). The oscillating electromagnetic field is thus not taken into account, giving the current density in the off-state of the laser, as shown in Fig. 2. Here, a number of bias points are marked indicating the points were more extensive analyses were made.

*n*

_{max}= 1. In Fig. 3, gain spectra as a function of photon energies are shown for the different bias points marked in Fig. 2. In the experiment of [15

15. C. Gmachl, A. Belyanin, D. L. Sivco, M. L. Peabody, N. Owschimikow, A. M. Sergent, F. Capasso, and A. Y. Cho, “Optimized second-harmonic generation in quantum cascade lasers,” IEEE J. Quantum Electron. **39**, 1345–1355 (2003). [CrossRef]

36. C. Sirtori, J. Faist, F. Capasso, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser with plasmon-enhanced waveguide operating at 8.4μm wavelength,” Appl. Phys. Lett. **66**, 3242–3244 (1995). [CrossRef]

*α*= 15 cm

_{W}^{−1}, and by calculating the mirror loss from the reflectivity,

*α*= 5.6 cm

_{M}^{−1}, the gain in the QCL structure required to compensate the losses is In this work the overlap factor is set to Γ = 0.5 following [18

18. J. Bai and D. Citrin, “Optical and transport characteristics of quantum-cascade lasers with optimized second-harmonic generation,” IEEE J. Quantum Electron. **43**, 391–398 (2007). [CrossRef]

*g*

_{threshold}= 40 cm

^{−1}. Studying Fig. 3, it can be seen that gain well above the level of the losses is reached for a large range of bias points. It can also be seen how the gain has a two-peak structure (e.g. 131 and 142 meV at

*Fd*= 210 mV), where the lower energy transition becomes dominating with increasing bias. This indicates that the level structure is complex and that a crossing occurs at these bias points.

^{2}, corresponding to 190 mV, for the design laser wavelength of 9.5

*μm*, or

*h̄ω*= 136.25 meV. This value compares well to experiment, where a threshold current of 6.6 kA/cm

^{2}was reported. It can also be seen that the gain at the laser energy increases monotonically with bias, suggesting a steady increase in output power with bias. Provided that the nonlinear resonator in the active region is capable of sum frequency generation, conditions are thus fulfilled for observation of second harmonic generation in the structure. Experimental measurements stop at 15 kA/cm

^{2}, but it is clear from the simulations that gain persists even at higher bias points with higher currents.

37. D. O. Winge, M. Lindskog, and A. Wacker, “Nonlinear response of quantum cascade structures,” Appl. Phys. Lett. **101**, 211113 (2012). [CrossRef]

*F*

_{0}

**e**

*cos(*

_{z}*k*−

_{ω}x*ωt*) is traveling towards the facet with an intensity given by the Pointing vector. Neglecting the intricate mode structure, we assume a constant field over the active region of the waveguide. The corresponding facet area is given by 32.5

*μ*m

^{2}from experimental data (#periods ×

*d*× waveguidewidth). Furthermore 71% is transmitted due to the Fresnel losses at the fundamental frequency. This provides the output power from our data, which is proportional to

*F*

_{0}to

*F*

_{ac}used in our simulation. If the electric field is dominated by the travelling wave (TW)

*F*

_{0}

**e**

*cos(*

_{z}*kx*−

*ωt*), we can identify

*F*

_{0}=

*F*

_{ac}and the resulting power is shown in Fig. 2. On the other hand, there is a reflected wave, which is amplified and becomes of the same magnitude as the incoming wave further away from the facet. In the middle of the waveguide we have actually a standing wave (SW) As can be seen in Fig. 4 the gain saturation is roughly proportional to

*z*dependence of the electric field due to the mode profile will further complicate the situation. Thus, the TW and SW values should be taken as a confidence range for our results when we compare to experimental data in the following.

^{2}, we find an output power of 280 mW for the TW case and 140 mW for the SW case. Here the experimental value is about 100 mW [15

**39**, 1345–1355 (2003). [CrossRef]

*et al.*[17

17. A. Gajić, J. Radovanović, V. Milanović, D. Indjin, and Z. Ikonić, “Genetic algorithm applied to the optimization of quantum cascade lasers with second harmonic generation,” J. Appl. Phys. **115**, 053712 (2014). [CrossRef]

*et al.*[18

18. J. Bai and D. Citrin, “Optical and transport characteristics of quantum-cascade lasers with optimized second-harmonic generation,” IEEE J. Quantum Electron. **43**, 391–398 (2007). [CrossRef]

*et al.*finds an on-state current that shows NDR features already at biases that compare to 200 mV per period with a maximum current of 22 kA/cm

^{2}. The features of an early NDR is also observed by Gajić

*et al.*where the maximum current is however lower, at 8 kA/cm

^{2}. Both results differ from our findings and experimental observations. The linear increase in gain with respect to

*Fd*, below saturation, see Fig. 3, is also reported by Bai

*et al.*. The output is estimated by both groups and their results are in the range of 300–400 mW, showing a larger discrepancy with experiment than our results.

## 4. Second harmonic generation

*n*

_{max}≥ 2 is used. Its total amplitude

*n*

_{max}can also be studied in Fig. 5. Here calculations with

*n*

_{max}= 3 and

*n*

_{max}= 4 are compared for the bias of 230 mV per period. As seen, the inclusion of four-photon-processes slightly affects the result, especially at high ac field strengths. However, the main features are not changed (this is true for all bias points) and the values are very similar, which is the reason that the computationally lighter simulations with

*n*

_{max}= 3 were used to calculate the mainstay of the results.

*et al.*and Gajić

*et al.*which yielded powers of 90

*μ*W and 45

*μ*W respectively, our findings are significantly closer to the experimental results, where the maximum SH output power was reported to be 550 nW in total. As the mode structure in the waveguide is not taken into account in the calculation, it is expected to overestimate the experimental value. Although the quantitative agreement of the SH power is not perfect, the final estimation of the second harmonic signal so close to the experimental value should still be regarded as a good indication towards the validity of the approach and the robustness of the finite-intensity-model that we have applied to the problem.

## 5. Density matrix calculation

*Fd*= 230 mV, also visible in Fig. 5. In the density matrix calculation this is heavily damped, which indicates that the full model, where also higher than second order parts are included, shows additional features not pursueable with ordinary population- and rate-based calculations. It is our belief that these effects are better described in the full model as the nonlinear current response is directly extracted from the off-diagonal density matrix elements, i.e. the coherences in the system.

## 6. Conclusion

**39**, 1345–1355 (2003). [CrossRef]

## Appendix

## Waveguide models

*F*(

*t*) =

*F*

_{dc}+

*F*

_{ac}cos(

*ωt*) is assumed. If the electric field has a phase shift

*φ*, this corresponds to a local shift in time to

*t′*=

*t*−

*φ/ω*and the current response is In the waveguide a traveling wave has the form where

*k*=

_{ω}*n*is approximately real, as absorption and gain compensate (this is not entirely true, as the absorption contains waveguide losses). This wave has the intensity given by the time-averaged Poynting vector which is then subject to reflection and transmission when it reaches the facet at the end of the waveguide.

_{ω}ω/c## Second harmonic generation neglecting reflecting wave

*φ*=

*k*. Thus we can identify where

_{ω}x*F*

_{ac,0}is the ac field strength of a traveling wave in balance with the gain medium. This simple equality would not hold for a standing wave, as the back reflected wave would increase the gain saturation.

*x*= 0, and

*x*=

*L*, where the complex current

*J̃*(

_{sh}*x*) =

*J*

_{2}

*e*

^{i2kωx}can be found from inspection of Eq. (6). Next we evaluate the electromagnetic field radiated from this current distribution. We consider the Helmholtz equation for the

*z*-component of the vector potential

*A*at a frequency 2

_{z}*ω*where

*k*

_{2ω}is the complex wavevector at frequency 2

*ω*in the waveguide. With the outgoing Green’s function ie

^{ik2ω|x−x′|}/(2

*k*

_{2ω}) the solution for the Helmholtz equation reads for

*x*≥

*L*

**e**

_{z}*A*

_{0}e

^{i(k2ω(x−L)−2ωt)}} we obtain the time-averaged Poynting vector

**P**within a standard calculation as where

*k″*

_{2ω}= ℑ{

*k*

_{2ω}} is the waveguide attenuation at the second harmonic. Inserting

*A*

_{0}from Eq. (8) will yield Eq. (3) with

*x*=

*L*and assuming that ℜ{

*k*

_{2ω}} >> ℑ{

*k*

_{2ω}} holds.

## 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. | M. M. Fejer, S. J. B. Yoo, R. L. Byer, A. Harwit, and J. S. Harris Jr., “Observation of extremely large quadratic susceptibility at 9.6 – 10.8μm in electric-field-biased AlGaAs quantum wells,” Phys. Rev. Lett. |

3. | E. Rosencher, A. Fiore, B. Winter, V. Berger, P. Bois, and J. Nagle, “Quantum engineering of optical nonlinearities,” Science |

4. | F. Capasso, C. Sirtori, and A. Y. Cho, “Coupled quantum well semiconductors with giant electric field tunable nonlinear optical properties in the infrared,” IEEE J. Quantum Electron. |

5. | M. A. Belkin, F. Capasso, A. Belyanin, D. L. Sivco, A. Y. Cho, D. C. Oakley, C. J. Vineis, and G. W. Turner, “Terahertz quantum-cascade-laser source based on intracavity difference-frequency generation,” Nat. Photonics |

6. | N. Owschimikow, C. Gmachl, A. Belyanin, V. Kocharovsky, D. L. Sivco, R. Colombelli, F. Capasso, and A. Y. Cho, “Resonant second-order nonlinear optical processes in quantum cascade lasers,” Phys. Rev. Lett. |

7. | J.-Y. Bengloan, A. De Rossi, V. Ortiz, X. Marcadet, M. Calligaro, I. Maurin, and C. Sirtori, “Intracavity sum-frequency generation in GaAs quantum cascade lasers,” Appl. Phys. Lett. |

8. | Y. Yao, J. A. Hoffman, and C. F. Gmachl, “Mid-infrared quantum cascade lasers,” Nat. Photonics |

9. | B. S. Williams, “Terahertz quantum-cascade lasers,” Nat. Photonics |

10. | A. Wacker, “Quantum cascade laser: An emerging technology,” in |

11. | Y.-H. Cho and A. Belyanin, “Short-wavelength infrared second harmonic generation in quantum cascade lasers,” J. Appl. Phys. |

12. | J. Faist, F. Capasso, D. L. Sivco, A. L. Hutchinson, S.-N. G. Chu, and A. Y. Cho, “Short wavelength (λ ∼ 3.4μm) quantum cascade laser based on strained compensated InGaAs/AlInAs,” Appl. Phys. Lett. |

13. | C. Gmachl, F. Capasso, D. L. Sivco, and A. Y. Cho, “Recent progress in quantum cascade lasers and applications,” Rep. Prog. Phys. |

14. | F. Capasso, R. Paiella, R. Martini, R. Colombelli, C. Gmachl, T. Myers, M. S. Taubman, R. Williams, C. Bethea, K. Unterrainer, H. Y. Hwang, D. L. Sivco, A. Y. Cho, A. Sergent, H. Liu, and E. A. Whittaker, “Quantum cascade lasers: ultrahigh-speed operation, optical wireless communication, narrow linewidth, and far-infrared emission,” IEEE J. Quantum Electron. |

15. | C. Gmachl, A. Belyanin, D. L. Sivco, M. L. Peabody, N. Owschimikow, A. M. Sergent, F. Capasso, and A. Y. Cho, “Optimized second-harmonic generation in quantum cascade lasers,” IEEE J. Quantum Electron. |

16. | A. Wacker, M. Lindskog, and D. Winge, “Nonequilibrium green’s function model for simulation of quantum cascade laser devices under operating conditions,” IEEE J. Sel. Top. Quantum Electron. |

17. | A. Gajić, J. Radovanović, V. Milanović, D. Indjin, and Z. Ikonić, “Genetic algorithm applied to the optimization of quantum cascade lasers with second harmonic generation,” J. Appl. Phys. |

18. | J. Bai and D. Citrin, “Optical and transport characteristics of quantum-cascade lasers with optimized second-harmonic generation,” IEEE J. Quantum Electron. |

19. | F. Capasso, J. Faist, and C. Sirtori, “Mesoscopic phenomena in semiconductor nanostructures by quantum design,” J. Math. Phys. |

20. | D. Indjin, P. Harrison, R. W. Kelsall, and Z. Ikonic, “Self-consistent scattering theory of transport and output characteristics of quantum cascade lasers,” J. Appl. Phys. |

21. | A. Hugi, R. Maulini, and J. Faist, “External cavity quantum cascade laser,” Semicond. Sci. Technol. |

22. | Y. Petitjean, F. Destic, J. Mollier, and C. Sirtori, “Dynamic modeling of terahertz quantum cascade lasers,” IEEE J. Sel. Top. Quantum Electron. |

23. | R. C. Iotti, E. Ciancio, and F. Rossi, “Quantum transport theory for semiconductor nanostructures: A density matrix formulation,” Phys. Rev. B |

24. | R. Terazzi and J. Faist, “A density matrix model of transport and radiation in quantum cascade lasers,” New J. Phys. |

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

26. | E. Dupont, S. Fathololoumi, and H. C. Liu, “Simplified density-matrix model applied to three-well terahertz quantum cascade lasers,” Phys. Rev. B |

27. | R. C. Iotti and F. Rossi, “Nature of charge transport in quantum-cascade lasers,” Phys. Rev. Lett. |

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

29. | C. Jirauschek and P. Lugli, “Monte-carlo-based spectral gain analysis for terahertz quantum cascade lasers,” J. Appl. Phys. |

30. | S.-C. Lee and A. Wacker, “Nonequilibrium Green’s function theory for transport and gain properties of quantum cascade structures,” Phys. Rev. B |

31. | T. Kubis, C. Yeh, P. Vogl, A. Benz, G. Fasching, and C. Deutsch, “Theory of nonequilibrium quantum transport and energy dissipation in terahertz quantum cascade lasers,” Phys. Rev. B |

32. | T. Schmielau and M. Pereira, “Nonequilibrium many body theory for quantum transport in terahertz quantum cascade lasers,” Appl. Phys. Lett. |

33. | G. Haldaś, A. Kolek, and I. Tralle, “Modeling of mid-infrared quantum cascade laser by means of nonequilibrium green’s functions,” IEEE J. Quantum Electron. |

34. | T. Brandes, “Truncation method for green’s functions in time-dependent fields,” Phys. Rev. B |

35. | H. Haug and S. Koch, |

36. | C. Sirtori, J. Faist, F. Capasso, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser with plasmon-enhanced waveguide operating at 8.4μm wavelength,” Appl. Phys. Lett. |

37. | D. O. Winge, M. Lindskog, and A. Wacker, “Nonlinear response of quantum cascade structures,” Appl. Phys. Lett. |

38. | O. Malis, A. Belyanin, C. Gmachl, D. L. Sivco, M. L. Peabody, A. M. Sergent, and A. Y. Cho, “Improvement of second-harmonic generation in quantum-cascade lasers with true phase matching,” Appl. Phys. Lett. |

39. | M. Belkin, Q. J. Wang, C. Pflugl, A. Belyanin, S. Khanna, A. Davies, E. Linfield, and F. Capasso, “High-temperature operation of terahertz quantum cascade laser sources,” IEEE J. Sel. Topics Quantum Electron. |

40. | Y. R. Shen, |

41. | R. Boyd, |

42. | F. Banit, S.-C. Lee, A. Knorr, and A. Wacker, “Self-consistent theory of the gain linewidth for quantum cascade lasers,” Appl. Phys. Lett. |

**OCIS Codes**

(190.4160) Nonlinear optics : Multiharmonic generation

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

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: May 8, 2014

Revised Manuscript: June 17, 2014

Manuscript Accepted: June 18, 2014

Published: July 22, 2014

**Citation**

David O. Winge, Martin Lindskog, and Andreas Wacker, "Microscopic approach to second harmonic generation in quantum cascade lasers," Opt. Express **22**, 18389-18400 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-15-18389

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### References

- J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science264, 553 (1994). [CrossRef] [PubMed]
- M. M. Fejer, S. J. B. Yoo, R. L. Byer, A. Harwit, and J. S. Harris, “Observation of extremely large quadratic susceptibility at 9.6 – 10.8μm in electric-field-biased AlGaAs quantum wells,” Phys. Rev. Lett.62, 1041–1044 (1989). [CrossRef] [PubMed]
- E. Rosencher, A. Fiore, B. Winter, V. Berger, P. Bois, and J. Nagle, “Quantum engineering of optical nonlinearities,” Science271, 168–173 (1996). [CrossRef]
- F. Capasso, C. Sirtori, and A. Y. Cho, “Coupled quantum well semiconductors with giant electric field tunable nonlinear optical properties in the infrared,” IEEE J. Quantum Electron.30, 1313–1326 (1994). [CrossRef]
- M. A. Belkin, F. Capasso, A. Belyanin, D. L. Sivco, A. Y. Cho, D. C. Oakley, C. J. Vineis, and G. W. Turner, “Terahertz quantum-cascade-laser source based on intracavity difference-frequency generation,” Nat. Photonics1, 288–292 (2007). [CrossRef]
- N. Owschimikow, C. Gmachl, A. Belyanin, V. Kocharovsky, D. L. Sivco, R. Colombelli, F. Capasso, and A. Y. Cho, “Resonant second-order nonlinear optical processes in quantum cascade lasers,” Phys. Rev. Lett.90, 043902 (2003). [CrossRef] [PubMed]
- J.-Y. Bengloan, A. De Rossi, V. Ortiz, X. Marcadet, M. Calligaro, I. Maurin, and C. Sirtori, “Intracavity sum-frequency generation in GaAs quantum cascade lasers,” Appl. Phys. Lett.84, 2019–2021 (2004). [CrossRef]
- Y. Yao, J. A. Hoffman, and C. F. Gmachl, “Mid-infrared quantum cascade lasers,” Nat. Photonics6, 432–439 (2012). [CrossRef]
- B. S. Williams, “Terahertz quantum-cascade lasers,” Nat. Photonics1, 517 (2007). [CrossRef]
- A. Wacker, “Quantum cascade laser: An emerging technology,” in Nonlinear Laser Dynamics, K. Lüdge, ed. (Wiley-VCH, 2011).
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