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

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    [CrossRef]
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    [CrossRef]
  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 79, 195323 (2009).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  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. 15, 952 (2009).
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    [CrossRef]

2014

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]

2013

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. 99, 1200611 (2013).

2012

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

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

2011

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]

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]

2010

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

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

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]

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

2009

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

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

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 79, 195323 (2009).
[CrossRef]

T. Schmielau and M. Pereira, “Nonequilibrium many body theory for quantum transport in terahertz quantum cascade lasers,” Appl. Phys. Lett. 95, 231111 (2009).
[CrossRef]

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. 15, 952 (2009).
[CrossRef]

2007

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]

B. S. Williams, “Terahertz quantum-cascade lasers,” Nat. Photonics 1, 517 (2007).
[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. Photonics 1, 288–292 (2007).
[CrossRef]

2005

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

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]

F. Banit, S.-C. Lee, A. Knorr, and A. Wacker, “Self-consistent theory of the gain linewidth for quantum cascade lasers,” Appl. Phys. Lett. 86, 041108 (2005).
[CrossRef]

2004

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. 84, 2721–2723 (2004).
[CrossRef]

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]

2003

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]

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]

2002

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]

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]

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]

2001

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

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]

1998

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]

1997

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

1996

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

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

1995

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]

1994

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]

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]

1989

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]

Bai, J.

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]

Banit, F.

F. Banit, S.-C. Lee, A. Knorr, and A. Wacker, “Self-consistent theory of the gain linewidth for quantum cascade lasers,” Appl. Phys. Lett. 86, 041108 (2005).
[CrossRef]

Belkin, M.

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. 15, 952 (2009).
[CrossRef]

Belkin, M. A.

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]

Belyanin, A.

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

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. 15, 952 (2009).
[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. Photonics 1, 288–292 (2007).
[CrossRef]

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. 84, 2721–2723 (2004).
[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]

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]

Bengloan, J.-Y.

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]

Benz, A.

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 79, 195323 (2009).
[CrossRef]

Berger, V.

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

Bethea, C.

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]

Bois, P.

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

Boyd, R.

R. Boyd, Nonlinear Optics (Academic, 1992).

Brandes, T.

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

Byer, R. L.

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]

Callebaut, H.

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

Calligaro, M.

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]

Capasso, F.

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. 15, 952 (2009).
[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. Photonics 1, 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]

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]

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]

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]

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]

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

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]

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]

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]

Cho, A. Y.

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]

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. 84, 2721–2723 (2004).
[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]

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]

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]

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]

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]

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]

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]

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]

Cho, Y.-H.

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

Chu, S.-N. G.

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]

Ciancio, E.

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]

Citrin, D.

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]

Colombelli, R.

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]

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]

Davies, A.

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. 15, 952 (2009).
[CrossRef]

De Rossi, A.

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]

Destic, F.

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]

Deutsch, C.

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 79, 195323 (2009).
[CrossRef]

Dupont, E.

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]

Faist, J.

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

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

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]

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

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]

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]

Fasching, G.

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 79, 195323 (2009).
[CrossRef]

Fathololoumi, S.

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]

Fejer, M. M.

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]

Fiore, A.

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

Gajic, A.

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]

Gmachl, C.

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. 84, 2721–2723 (2004).
[CrossRef]

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]

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]

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]

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]

Gmachl, C. F.

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

Haldas, G.

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]

Harris, J. S.

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]

Harrison, P.

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]

Harwit, A.

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]

Haug, H.

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

Hoffman, J. A.

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

Hu, Q.

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

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

Hugi, A.

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

Hutchinson, A. L.

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]

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]

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]

Hwang, H. Y.

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]

Ikonic, Z.

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]

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]

Indjin, D.

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]

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]

Iotti, R. C.

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]

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

Jirauschek, C.

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

Kelsall, R. W.

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]

Khanna, S.

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. 15, 952 (2009).
[CrossRef]

Knorr, A.

F. Banit, S.-C. Lee, A. Knorr, and A. Wacker, “Self-consistent theory of the gain linewidth for quantum cascade lasers,” Appl. Phys. Lett. 86, 041108 (2005).
[CrossRef]

Koch, S.

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

Kocharovsky, V.

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]

Kolek, A.

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]

Kubis, T.

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 79, 195323 (2009).
[CrossRef]

Kumar, S.

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

Lee, S.-C.

F. Banit, S.-C. Lee, A. Knorr, and A. Wacker, “Self-consistent theory of the gain linewidth for quantum cascade lasers,” Appl. Phys. Lett. 86, 041108 (2005).
[CrossRef]

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]

Lindskog, M.

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. 99, 1200611 (2013).

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

Linfield, E.

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. 15, 952 (2009).
[CrossRef]

Liu, H.

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]

Liu, H. C.

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]

Lugli, P.

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

Malis, O.

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. 84, 2721–2723 (2004).
[CrossRef]

Marcadet, X.

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]

Martini, R.

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]

Maulini, R.

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

Maurin, I.

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]

Milanovic, V.

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]

Mollier, J.

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]

Myers, T.

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]

Nagle, J.

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

Oakley, D. C.

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]

Ortiz, V.

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]

Owschimikow, N.

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]

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]

Paiella, R.

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]

Peabody, M. L.

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. 84, 2721–2723 (2004).
[CrossRef]

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]

Pereira, M.

T. Schmielau and M. Pereira, “Nonequilibrium many body theory for quantum transport in terahertz quantum cascade lasers,” Appl. Phys. Lett. 95, 231111 (2009).
[CrossRef]

Petitjean, Y.

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]

Pflugl, C.

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. 15, 952 (2009).
[CrossRef]

Radovanovic, J.

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]

Rosencher, E.

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

Rossi, F.

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]

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

Schmielau, T.

T. Schmielau and M. Pereira, “Nonequilibrium many body theory for quantum transport in terahertz quantum cascade lasers,” Appl. Phys. Lett. 95, 231111 (2009).
[CrossRef]

Sergent, A.

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]

Sergent, A. M.

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. 84, 2721–2723 (2004).
[CrossRef]

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]

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).

Sirtori, C.

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]

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]

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

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]

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]

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]

Sivco, D. L.

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]

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. 84, 2721–2723 (2004).
[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]

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]

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]

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]

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]

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]

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]

Taubman, M. S.

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]

Terazzi, R.

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

Tralle, I.

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]

Turner, G. W.

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]

Unterrainer, K.

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]

Vineis, C. J.

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]

Vogl, P.

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 79, 195323 (2009).
[CrossRef]

Wacker, A.

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. 99, 1200611 (2013).

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

F. Banit, S.-C. Lee, A. Knorr, and A. Wacker, “Self-consistent theory of the gain linewidth for quantum cascade lasers,” Appl. Phys. Lett. 86, 041108 (2005).
[CrossRef]

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]

A. Wacker, “Quantum cascade laser: An emerging technology,” in Nonlinear Laser Dynamics, K. Lüdge, ed. (Wiley-VCH, 2011).

Wang, Q. J.

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. 15, 952 (2009).
[CrossRef]

Whittaker, E. A.

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]

Williams, B. S.

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

Williams, R.

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]

Winge, D.

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. 99, 1200611 (2013).

Winge, D. O.

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

Winter, B.

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

Yao, Y.

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

Yeh, C.

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 79, 195323 (2009).
[CrossRef]

Yoo, S. J. B.

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]

Appl. Phys. Lett.

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]

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]

T. Schmielau and M. Pereira, “Nonequilibrium many body theory for quantum transport in terahertz quantum cascade lasers,” Appl. Phys. Lett. 95, 231111 (2009).
[CrossRef]

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]

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

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. 84, 2721–2723 (2004).
[CrossRef]

F. Banit, S.-C. Lee, A. Knorr, and A. Wacker, “Self-consistent theory of the gain linewidth for quantum cascade lasers,” Appl. Phys. Lett. 86, 041108 (2005).
[CrossRef]

IEEE J. Quantum Electron.

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]

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]

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]

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]

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]

IEEE J. Sel. Top. Quantum Electron.

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. 99, 1200611 (2013).

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]

IEEE J. Sel. Topics Quantum Electron.

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. 15, 952 (2009).
[CrossRef]

J. Appl. Phys.

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]

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

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]

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

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

J. Math. Phys.

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

Nat. Photonics

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]

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

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

New J. Phys.

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

Phys. Rev. B

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

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]

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]

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 79, 195323 (2009).
[CrossRef]

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

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]

Phys. Rev. Lett.

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

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]

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]

Rep. Prog. Phys.

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]

Science

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]

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

Semicond. Sci. Technol.

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

Other

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

A. Wacker, “Quantum cascade laser: An emerging technology,” in Nonlinear Laser Dynamics, K. Lüdge, ed. (Wiley-VCH, 2011).

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).

R. Boyd, Nonlinear Optics (Academic, 1992).

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Figures (6)

Fig. 1
Fig. 1

One period of the structure from [15] with the two-well active region in the middle. Carriers are injected into the upper laser state B ensuring inversion between B and A, whereas levels A,C and B,D produce SH generation. The states and meanfield potential bending the conduction band structure are calculated at a bias drop of 190 mV per period. The period length d is 49.5 nm.

Fig. 2
Fig. 2

Simulated bias-current relation of the structure studied for both the off- and on-state. The level of losses used to simulate the laser under operation was 40 cm−1. Also shown is the output power at the pump and second harmonic frequency, respectively, using the TW model, which constitutes an upper bound. The second harmonic signal is scaled by a factor of 105. At the marked points 180, 190, 200, 210, 230 and 250 mV further analyses of gain and SH generation were carried out. A lattice temperature of 100 K was used.

Fig. 3
Fig. 3

Gain at weak ac field simulated at the bias points marked in Fig. 2. The orange arrow indicates the photon energy 136.25 meV corresponding to the fundamental wavelength of 9.5 μm, as well as increasing bias.

Fig. 4
Fig. 4

Gain saturation with increasing ac field strength for the photon energy of 136.25 meV. The bias points examined are the ones indicated in Fig. 2. The ac field in the cavity increases as long as gain surpasses gthreshold indicated as a dotted line.

Fig. 5
Fig. 5

Current oscillations | J 2 | = ( ( J 2 cos ) 2 + ( J 2 sin ) 2 ) 1 / 2, at the second harmonic frequency plotted versus ac field strength for the bias points indicated in Fig. 2. Simulations were made with nmax = 3. The points inserted along the 230 mV data are results with nmax = 4 in order to verify the convergence. Along eFacd = 70 meV there is a dashed line indicating the region studied in more detail in Fig. 6. The inset shows a close-up on small values of the ac field strength, emphasizing the quadratic behavior.

Fig. 6
Fig. 6

Current response at 2ω in terms of the cosine and sine part respectively, from the NEGF model (solid lines). These are compared with the results from Eq. (5) using occupations and dipole matrix elements from our nonequilibrium states (connected dots). To compare directly to Eq. 4, the negative particle current, appropriate for electrons, is shown. The ac field strength is held constant at eFacd = 70 meV.

Tables (1)

Tables Icon

Table 1 Intensities for the different waveguide models outside the waveguide. These are calculated for a transmission coefficient of 71%. For comparison the experimental results from [15] are shown.

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

J ( t ) = J 0 + n = 1 n max ( J n cos cos ( n ω t ) + J n sin sin ( n ω t ) )
g threshold = α W + α M Γ .
F ( x , t ) = F 0 e z cos ( k ω x ω t ) + F 0 e z cos ( k ω x + ω t ) = 2 F 0 cos ( k ω x ) cos ( ω t ) e z .
I 2 ω = μ 0 | J 2 | 2 c 8 n 2 ω | 1 e k 2 ω L e i Δ k L Δ k + i k 2 ω | 2
J ( t ) = P ˙ ( t ) = ω ε 0 F ac 2 [ { χ ( 2 ) ( 2 ω ) } sin ( 2 ω t ) { χ ( 2 ) ( 2 ω ) } cos ( 2 ω t ) ]
χ ( 2 ) ( 2 ω ) = e 3 h ¯ 2 d ε 0 m n v ( n m n v ) z m n z n v z v m ( ω n m 2 ω i γ n m ) ( ω v m ω i γ v m ) ( n v n n ) z m n z v m z n v ( ω n m 2 ω i γ n m ) ( ω n v ω i γ n v )
J ( t ) = J 0 + { J 1 e i ( ω t φ ) + J 2 e i ( 2 ω t 2 φ ) } for F ( t ) = F dc + F ac cos ( ω t φ ) .
F 0 cos ( k ω x ω t )
S = F 0 2 n ω c ε 0 2 e x
F 0 = F ac , 0
J sh ( x , t ) = { J ˜ sh ( x ) e 2 i ω t } e z
2 x 2 A z ( x , 2 ω ) + k 2 ω 2 A z ( x , 2 ω ) = μ 0 J ˜ sh ( x )
A z ( x , 2 ω ) = 0 L d x μ 0 J ˜ sh ( x ) i 2 k 2 ω e i k 2 ω ( x x ) .
P = | A 0 | 2 ω { k 2 ω } μ 0 e 2 k 2 ω ( x L ) e x

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