Abstract

The generation of frequency combs in the mid-infrared and terahertz regimes from compact and potentially cheap sources could have a strong impact on spectroscopy, as many molecules have their rotovibrational bands in this spectral range. Thus, quantum cascade lasers (QCLs) are the perfect candidates for comb generation in these portions of the electromagnetic spectrum. Here we present a theoretical model based on a full numerical solution of Maxwell-Bloch equations suitable for the simulation of such devices. We show that our approach captures the intricate interplay between four wave mixing, spatial hole burning, coherent tunneling and chromatic dispersion which are present in free running QCLs. We investigate the premises for the generation of QCL based terahertz combs. The simulated comb spectrum is in good agreement with experiment, and also the observed temporal pulse switching between high and low frequency components is reproduced. Furthermore, non-comb operation resulting in a complex multimode dynamics is investigated.

© 2016 Optical Society of America

Full Article  |  PDF Article
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References

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2016 (2)

N. Vukovic, J. Radovanovic, V. Milanovic, and D. Boiko, “Multimode RNGH instabilities of Fabry-Pérot cavity QCLs: impact of diffusion,” Opt. Quant. Electron. 48, 1–10 (2016).
[Crossref]

Y. Yang, D. Burghoff, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Terahertz multiheterodyne spectroscopy using laser frequency combs,” Optica 3, 499–502 (2016).
[Crossref]

2015 (6)

2014 (5)

G. Villares, A. Hugi, S. Blaser, and J. Faist, “Dual-comb spectroscopy based on quantum-cascade-laser frequency combs,” Nat. Commun. 5, 5192 (2014).
[Crossref]

D. Burghoff, T.-Y. Kao, N. Han, C. W. I. Chan, X. Cai, Y. Yang, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Terahertz laser frequency combs,” Nature Photon. 8, 462–467 (2014).
[Crossref]

C. Jirauschek and T. Kubis, “Modeling techniques for quantum cascade lasers,” Appl. Phys. Rev. 1, 011307 (2014).
[Crossref]

J. Khurgin, Y. Dikmelik, A. Hugi, and J. Faist, “Coherent frequency combs produced by self frequency modulation in quantum cascade lasers,” Appl. Phys. Lett. 104, 081118 (2014).
[Crossref]

M. Wienold, B. Röben, L. Schrottke, and H. Grahn, “Evidence for frequency comb emission from a Fabry-Pérot terahertz quantum-cascade laser,” Opt. Express 22, 30410–30424 (2014).
[Crossref]

2013 (2)

P. Friedli, H. Sigg, B. Hinkov, A. Hugi, S. Riedi, M. Beck, and J. Faist, “Four-wave mixing in a quantum cascade laser amplifier,” Appl. Phys. Lett. 102, 222104 (2013).
[Crossref]

I. Knezevic and B. Novakovic, “Time-dependent transport in open systems based on quantum master equations,” J. Comput. Electron. 12, 363–374 (2013).
[Crossref]

2012 (1)

A. Hugi, G. Villares, S. Blaser, H. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature 492, 229–233 (2012).
[Crossref]

2011 (1)

2010 (5)

2009 (4)

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

C. Y. Wang, L. Kuznetsova, V. M. Gkortsas, L. Diehl, F. X. Kärtner, M. A. Belkin, A. Belyanin, X. Li, D. Ham, H. Schneider, P. Grant, C. Y. Song, S. Haffouz, Z. R. Wasilewski, H. C. Liu, and F. Capasso, “Mode-locked pulses from mid-infrared quantum cascade lasers,” Opt. Express 17, 12929–12943 (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]

C. Jirauschek, “Accuracy of transfer matrix approaches for solving the effective mass Schrödinger equation,” IEEE J. Quantum Electron. 45, 1059–1067 (2009).
[Crossref]

2008 (2)

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

N. Jukam, S. Dhillon, Z. Y. Zhao, G. Duerr, J. Armijo, N. Sirmons, S. Hameau, S. Barbieri, P. Filloux, C. Sirtori, X. Marcadet, and J. Tignon, “Gain measurements of THz quantum cascade lasers using THz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 14, 436–442 (2008).
[Crossref]

2007 (2)

C. Y. Wang, L. Diehl, A. Gordon, C. Jirauschek, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, M. Troccoli, J. Faist, and F. Capasso, “Coherent instabilities in a semiconductor laser with fast gain recovery,” Phys. Rev. A 75, 031802 (2007).
[Crossref]

C. Jirauschek, G. Scarpa, P. Lugli, M. S. Vitiello, and G. Scamarcio, “Comparative analysis of resonant phonon THz quantum cascade lasers,” J. Appl. Phys. 101, 086109 (2007).
[Crossref]

2006 (1)

C. Weber, F. Banit, S. Butscher, A. Knorr, and A. Wacker, “Theory of the ultrafast nonlinear response of terahertz quantum cascade laser structures,” Appl. Phys. Lett. 89, 091112 (2006).
[Crossref]

2005 (4)

R. C. Iotti and F. Rossi, “Microscopic theory of semiconductor-based optoelectronic devices,” Rep. Prog. Phys. 68, 2533 (2005).
[Crossref]

S. Butscher, J. Förstner, I. Waldmüller, and A. Knorr, “Ultrafast electron-phonon interaction of intersubband transitions: Quantum kinetics from adiabatic following to Rabi-oscillations,” Phys. Rev. B 72, 045314 (2005).
[Crossref]

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

S. Kohen, B. S. Williams, and Q. Hu, “Electromagnetic modeling of terahertz quantum cascade laser waveguides and resonators,” J. Appl. Phys. 97, 053106 (2005).
[Crossref]

1968 (1)

H. Risken and K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39, 4662–4672 (1968).
[Crossref]

Amanti, M.

Q. Lu, M. Razeghi, S. Slivken, N. Bandyopadhyay, Y. Bai, W. Zhou, M. Chen, D. Heydari, A. Haddadi, R. McClintock, M. Amanti, and C. Sirtori, “High power frequency comb based on mid-infrared quantum cascade laser at λ ~9 μ m,” Appl. Phys. Lett. 106, 051105 (2015).
[Crossref]

Andrews, A. M.

Armijo, J.

N. Jukam, S. Dhillon, Z. Y. Zhao, G. Duerr, J. Armijo, N. Sirmons, S. Hameau, S. Barbieri, P. Filloux, C. Sirtori, X. Marcadet, and J. Tignon, “Gain measurements of THz quantum cascade lasers using THz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 14, 436–442 (2008).
[Crossref]

Bai, Y.

Q. Lu, M. Razeghi, S. Slivken, N. Bandyopadhyay, Y. Bai, W. Zhou, M. Chen, D. Heydari, A. Haddadi, R. McClintock, M. Amanti, and C. Sirtori, “High power frequency comb based on mid-infrared quantum cascade laser at λ ~9 μ m,” Appl. Phys. Lett. 106, 051105 (2015).
[Crossref]

Bandyopadhyay, N.

Q. Lu, M. Razeghi, S. Slivken, N. Bandyopadhyay, Y. Bai, W. Zhou, M. Chen, D. Heydari, A. Haddadi, R. McClintock, M. Amanti, and C. Sirtori, “High power frequency comb based on mid-infrared quantum cascade laser at λ ~9 μ m,” Appl. Phys. Lett. 106, 051105 (2015).
[Crossref]

Banit, F.

C. Weber, F. Banit, S. Butscher, A. Knorr, and A. Wacker, “Theory of the ultrafast nonlinear response of terahertz quantum cascade laser structures,” Appl. Phys. Lett. 89, 091112 (2006).
[Crossref]

Barbieri, S.

N. Jukam, S. Dhillon, Z. Y. Zhao, G. Duerr, J. Armijo, N. Sirmons, S. Hameau, S. Barbieri, P. Filloux, C. Sirtori, X. Marcadet, and J. Tignon, “Gain measurements of THz quantum cascade lasers using THz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 14, 436–442 (2008).
[Crossref]

Bastard, G.

G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures (John Wiley & Sons, 1990).

Beck, M.

M. Rösch, G. Scalari, M. Beck, and J. Faist, “Octave-spanning semiconductor laser,” Nature Photon. 9, 42–47 (2015).
[Crossref]

P. Friedli, H. Sigg, B. Hinkov, A. Hugi, S. Riedi, M. Beck, and J. Faist, “Four-wave mixing in a quantum cascade laser amplifier,” Appl. Phys. Lett. 102, 222104 (2013).
[Crossref]

Belkin, M.

Belkin, M. A.

Belyanin, A.

Y. Wang and A. Belyanin, “Active mode-locking of mid-infrared quantum cascade lasers with short gain recovery time,” Opt. Express 23, 4173–4185 (2015).
[Crossref]

V.-M. Gkortsas, C. Wang, L. Kuznetsova, L. Diehl, A. Gordon, C. Jirauschek, M. Belkin, A. Belyanin, F. Capasso, and F. Kärtner, “Dynamics of actively mode-locked quantum cascade lasers,” Opt. Express 18, 13616–13630 (2010).
[Crossref]

C. Y. Wang, L. Kuznetsova, V. M. Gkortsas, L. Diehl, F. X. Kärtner, M. A. Belkin, A. Belyanin, X. Li, D. Ham, H. Schneider, P. Grant, C. Y. Song, S. Haffouz, Z. R. Wasilewski, H. C. Liu, and F. Capasso, “Mode-locked pulses from mid-infrared quantum cascade lasers,” Opt. Express 17, 12929–12943 (2009).
[Crossref]

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

C. Y. Wang, L. Diehl, A. Gordon, C. Jirauschek, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, M. Troccoli, J. Faist, and F. Capasso, “Coherent instabilities in a semiconductor laser with fast gain recovery,” Phys. Rev. A 75, 031802 (2007).
[Crossref]

Blaser, S.

G. Villares, A. Hugi, S. Blaser, and J. Faist, “Dual-comb spectroscopy based on quantum-cascade-laser frequency combs,” Nat. Commun. 5, 5192 (2014).
[Crossref]

A. Hugi, G. Villares, S. Blaser, H. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature 492, 229–233 (2012).
[Crossref]

Boiko, D.

N. Vukovic, J. Radovanovic, V. Milanovic, and D. Boiko, “Multimode RNGH instabilities of Fabry-Pérot cavity QCLs: impact of diffusion,” Opt. Quant. Electron. 48, 1–10 (2016).
[Crossref]

Bour, D.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

C. Y. Wang, L. Diehl, A. Gordon, C. Jirauschek, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, M. Troccoli, J. Faist, and F. Capasso, “Coherent instabilities in a semiconductor laser with fast gain recovery,” Phys. Rev. A 75, 031802 (2007).
[Crossref]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 2003).

Brandstetter, M.

Burghoff, D.

Burghoff, D. P.

D. P. Burghoff, “Broadband terahertz photonics,” Ph.D. thesis, Massachusetts Institute of Technology (2014).

Butcher, P. N.

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University, 1991).

Butscher, S.

C. Weber, F. Banit, S. Butscher, A. Knorr, and A. Wacker, “Theory of the ultrafast nonlinear response of terahertz quantum cascade laser structures,” Appl. Phys. Lett. 89, 091112 (2006).
[Crossref]

S. Butscher, J. Förstner, I. Waldmüller, and A. Knorr, “Ultrafast electron-phonon interaction of intersubband transitions: Quantum kinetics from adiabatic following to Rabi-oscillations,” Phys. Rev. B 72, 045314 (2005).
[Crossref]

Cai, X.

D. Burghoff, T.-Y. Kao, N. Han, C. W. I. Chan, X. Cai, Y. Yang, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Terahertz laser frequency combs,” Nature Photon. 8, 462–467 (2014).
[Crossref]

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]

Capasso, F.

V.-M. Gkortsas, C. Wang, L. Kuznetsova, L. Diehl, A. Gordon, C. Jirauschek, M. Belkin, A. Belyanin, F. Capasso, and F. Kärtner, “Dynamics of actively mode-locked quantum cascade lasers,” Opt. Express 18, 13616–13630 (2010).
[Crossref]

C. Y. Wang, L. Kuznetsova, V. M. Gkortsas, L. Diehl, F. X. Kärtner, M. A. Belkin, A. Belyanin, X. Li, D. Ham, H. Schneider, P. Grant, C. Y. Song, S. Haffouz, Z. R. Wasilewski, H. C. Liu, and F. Capasso, “Mode-locked pulses from mid-infrared quantum cascade lasers,” Opt. Express 17, 12929–12943 (2009).
[Crossref]

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

C. Y. Wang, L. Diehl, A. Gordon, C. Jirauschek, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, M. Troccoli, J. Faist, and F. Capasso, “Coherent instabilities in a semiconductor laser with fast gain recovery,” Phys. Rev. A 75, 031802 (2007).
[Crossref]

Cappelli, F.

Chan, C. W. I.

D. Burghoff, T.-Y. Kao, N. Han, C. W. I. Chan, X. Cai, Y. Yang, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Terahertz laser frequency combs,” Nature Photon. 8, 462–467 (2014).
[Crossref]

Chen, M.

Q. Lu, M. Razeghi, S. Slivken, N. Bandyopadhyay, Y. Bai, W. Zhou, M. Chen, D. Heydari, A. Haddadi, R. McClintock, M. Amanti, and C. Sirtori, “High power frequency comb based on mid-infrared quantum cascade laser at λ ~9 μ m,” Appl. Phys. Lett. 106, 051105 (2015).
[Crossref]

Corzine, S.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

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Ham, D.

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P. Friedli, H. Sigg, B. Hinkov, A. Hugi, S. Riedi, M. Beck, and J. Faist, “Four-wave mixing in a quantum cascade laser amplifier,” Appl. Phys. Lett. 102, 222104 (2013).
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J. Khurgin, Y. Dikmelik, A. Hugi, and J. Faist, “Coherent frequency combs produced by self frequency modulation in quantum cascade lasers,” Appl. Phys. Lett. 104, 081118 (2014).
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G. Villares, A. Hugi, S. Blaser, and J. Faist, “Dual-comb spectroscopy based on quantum-cascade-laser frequency combs,” Nat. Commun. 5, 5192 (2014).
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P. Friedli, H. Sigg, B. Hinkov, A. Hugi, S. Riedi, M. Beck, and J. Faist, “Four-wave mixing in a quantum cascade laser amplifier,” Appl. Phys. Lett. 102, 222104 (2013).
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A. Hugi, G. Villares, S. Blaser, H. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature 492, 229–233 (2012).
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C. Y. Wang, L. Diehl, A. Gordon, C. Jirauschek, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, M. Troccoli, J. Faist, and F. Capasso, “Coherent instabilities in a semiconductor laser with fast gain recovery,” Phys. Rev. A 75, 031802 (2007).
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Kärtner, F. X.

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S. Kohen, B. S. Williams, and Q. Hu, “Electromagnetic modeling of terahertz quantum cascade laser waveguides and resonators,” J. Appl. Phys. 97, 053106 (2005).
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C. Jirauschek and T. Kubis, “Modeling techniques for quantum cascade lasers,” Appl. Phys. Rev. 1, 011307 (2014).
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S. Kumar and Q. Hu, “Coherence of resonant-tunneling transport in terahertz quantum-cascade lasers,” Phys. Rev. B 80, 245316 (2009).
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Li, X.

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A. Hugi, G. Villares, S. Blaser, H. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature 492, 229–233 (2012).
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A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
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C. Jirauschek and P. Lugli, “Monte-Carlo-based spectral gain analysis for terahertz quantum cascade lasers,” J. Appl. Phys. 105, 123102 (2009).
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C. Jirauschek, G. Scarpa, P. Lugli, M. S. Vitiello, and G. Scamarcio, “Comparative analysis of resonant phonon THz quantum cascade lasers,” J. Appl. Phys. 101, 086109 (2007).
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Riedi, S.

F. Cappelli, G. Villares, S. Riedi, and J. Faist, “Intrinsic linewidth of quantum cascade laser frequency combs,” Optica 2, 836–840 (2015).
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P. Friedli, H. Sigg, B. Hinkov, A. Hugi, S. Riedi, M. Beck, and J. Faist, “Four-wave mixing in a quantum cascade laser amplifier,” Appl. Phys. Lett. 102, 222104 (2013).
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R. C. Iotti and F. Rossi, “Microscopic theory of semiconductor-based optoelectronic devices,” Rep. Prog. Phys. 68, 2533 (2005).
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M. Rösch, G. Scalari, M. Beck, and J. Faist, “Octave-spanning semiconductor laser,” Nature Photon. 9, 42–47 (2015).
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C. Jirauschek, G. Scarpa, P. Lugli, M. S. Vitiello, and G. Scamarcio, “Comparative analysis of resonant phonon THz quantum cascade lasers,” J. Appl. Phys. 101, 086109 (2007).
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C. Jirauschek, G. Scarpa, P. Lugli, M. S. Vitiello, and G. Scamarcio, “Comparative analysis of resonant phonon THz quantum cascade lasers,” J. Appl. Phys. 101, 086109 (2007).
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C. Y. Wang, L. Kuznetsova, V. M. Gkortsas, L. Diehl, F. X. Kärtner, M. A. Belkin, A. Belyanin, X. Li, D. Ham, H. Schneider, P. Grant, C. Y. Song, S. Haffouz, Z. R. Wasilewski, H. C. Liu, and F. Capasso, “Mode-locked pulses from mid-infrared quantum cascade lasers,” Opt. Express 17, 12929–12943 (2009).
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Sigg, H.

P. Friedli, H. Sigg, B. Hinkov, A. Hugi, S. Riedi, M. Beck, and J. Faist, “Four-wave mixing in a quantum cascade laser amplifier,” Appl. Phys. Lett. 102, 222104 (2013).
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N. Jukam, S. Dhillon, Z. Y. Zhao, G. Duerr, J. Armijo, N. Sirmons, S. Hameau, S. Barbieri, P. Filloux, C. Sirtori, X. Marcadet, and J. Tignon, “Gain measurements of THz quantum cascade lasers using THz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 14, 436–442 (2008).
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Q. Lu, M. Razeghi, S. Slivken, N. Bandyopadhyay, Y. Bai, W. Zhou, M. Chen, D. Heydari, A. Haddadi, R. McClintock, M. Amanti, and C. Sirtori, “High power frequency comb based on mid-infrared quantum cascade laser at λ ~9 μ m,” Appl. Phys. Lett. 106, 051105 (2015).
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N. Jukam, S. Dhillon, Z. Y. Zhao, G. Duerr, J. Armijo, N. Sirmons, S. Hameau, S. Barbieri, P. Filloux, C. Sirtori, X. Marcadet, and J. Tignon, “Gain measurements of THz quantum cascade lasers using THz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 14, 436–442 (2008).
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Q. Lu, M. Razeghi, S. Slivken, N. Bandyopadhyay, Y. Bai, W. Zhou, M. Chen, D. Heydari, A. Haddadi, R. McClintock, M. Amanti, and C. Sirtori, “High power frequency comb based on mid-infrared quantum cascade laser at λ ~9 μ m,” Appl. Phys. Lett. 106, 051105 (2015).
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N. Jukam, S. Dhillon, Z. Y. Zhao, G. Duerr, J. Armijo, N. Sirmons, S. Hameau, S. Barbieri, P. Filloux, C. Sirtori, X. Marcadet, and J. Tignon, “Gain measurements of THz quantum cascade lasers using THz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 14, 436–442 (2008).
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A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
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C. Y. Wang, L. Diehl, A. Gordon, C. Jirauschek, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, M. Troccoli, J. Faist, and F. Capasso, “Coherent instabilities in a semiconductor laser with fast gain recovery,” Phys. Rev. A 75, 031802 (2007).
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G. Villares, A. Hugi, S. Blaser, and J. Faist, “Dual-comb spectroscopy based on quantum-cascade-laser frequency combs,” Nat. Commun. 5, 5192 (2014).
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A. Hugi, G. Villares, S. Blaser, H. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature 492, 229–233 (2012).
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C. Jirauschek, G. Scarpa, P. Lugli, M. S. Vitiello, and G. Scamarcio, “Comparative analysis of resonant phonon THz quantum cascade lasers,” J. Appl. Phys. 101, 086109 (2007).
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N. Vukovic, J. Radovanovic, V. Milanovic, and D. Boiko, “Multimode RNGH instabilities of Fabry-Pérot cavity QCLs: impact of diffusion,” Opt. Quant. Electron. 48, 1–10 (2016).
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C. Weber, F. Banit, S. Butscher, A. Knorr, and A. Wacker, “Theory of the ultrafast nonlinear response of terahertz quantum cascade laser structures,” Appl. Phys. Lett. 89, 091112 (2006).
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A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
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C. Y. Wang, L. Diehl, A. Gordon, C. Jirauschek, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, M. Troccoli, J. Faist, and F. Capasso, “Coherent instabilities in a semiconductor laser with fast gain recovery,” Phys. Rev. A 75, 031802 (2007).
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C. Weber, F. Banit, S. Butscher, A. Knorr, and A. Wacker, “Theory of the ultrafast nonlinear response of terahertz quantum cascade laser structures,” Appl. Phys. Lett. 89, 091112 (2006).
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N. Jukam, S. Dhillon, Z. Y. Zhao, G. Duerr, J. Armijo, N. Sirmons, S. Hameau, S. Barbieri, P. Filloux, C. Sirtori, X. Marcadet, and J. Tignon, “Gain measurements of THz quantum cascade lasers using THz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 14, 436–442 (2008).
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Q. Lu, M. Razeghi, S. Slivken, N. Bandyopadhyay, Y. Bai, W. Zhou, M. Chen, D. Heydari, A. Haddadi, R. McClintock, M. Amanti, and C. Sirtori, “High power frequency comb based on mid-infrared quantum cascade laser at λ ~9 μ m,” Appl. Phys. Lett. 106, 051105 (2015).
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Appl. Phys. Lett. (5)

Q. Lu, M. Razeghi, S. Slivken, N. Bandyopadhyay, Y. Bai, W. Zhou, M. Chen, D. Heydari, A. Haddadi, R. McClintock, M. Amanti, and C. Sirtori, “High power frequency comb based on mid-infrared quantum cascade laser at λ ~9 μ m,” Appl. Phys. Lett. 106, 051105 (2015).
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P. Friedli, H. Sigg, B. Hinkov, A. Hugi, S. Riedi, M. Beck, and J. Faist, “Four-wave mixing in a quantum cascade laser amplifier,” Appl. Phys. Lett. 102, 222104 (2013).
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J. Khurgin, Y. Dikmelik, A. Hugi, and J. Faist, “Coherent frequency combs produced by self frequency modulation in quantum cascade lasers,” Appl. Phys. Lett. 104, 081118 (2014).
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C. Jirauschek, “Monte Carlo study of carrier-light coupling in terahertz quantum cascade lasers,” Appl. Phys. Lett. 96, 011103 (2010).
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C. Weber, F. Banit, S. Butscher, A. Knorr, and A. Wacker, “Theory of the ultrafast nonlinear response of terahertz quantum cascade laser structures,” Appl. Phys. Lett. 89, 091112 (2006).
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Appl. Phys. Rev. (1)

C. Jirauschek and T. Kubis, “Modeling techniques for quantum cascade lasers,” Appl. Phys. Rev. 1, 011307 (2014).
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IEEE J. Quantum Electron. (1)

C. Jirauschek, “Accuracy of transfer matrix approaches for solving the effective mass Schrödinger equation,” IEEE J. Quantum Electron. 45, 1059–1067 (2009).
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IEEE J. Sel. Top. Quantum Electron. (1)

N. Jukam, S. Dhillon, Z. Y. Zhao, G. Duerr, J. Armijo, N. Sirmons, S. Hameau, S. Barbieri, P. Filloux, C. Sirtori, X. Marcadet, and J. Tignon, “Gain measurements of THz quantum cascade lasers using THz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 14, 436–442 (2008).
[Crossref]

J. Appl. Phys. (5)

C. Jirauschek, G. Scarpa, P. Lugli, M. S. Vitiello, and G. Scamarcio, “Comparative analysis of resonant phonon THz quantum cascade lasers,” J. Appl. Phys. 101, 086109 (2007).
[Crossref]

C. Jirauschek and P. Lugli, “Monte-Carlo-based spectral gain analysis for terahertz quantum cascade lasers,” J. Appl. Phys. 105, 123102 (2009).
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H. Callebaut and Q. Hu, “Importance of coherence for electron transport in terahertz quantum cascade lasers,” J. Appl. Phys. 98, 104505 (2005).
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H. Risken and K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39, 4662–4672 (1968).
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S. Kohen, B. S. Williams, and Q. Hu, “Electromagnetic modeling of terahertz quantum cascade laser waveguides and resonators,” J. Appl. Phys. 97, 053106 (2005).
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J. Comput. Electron. (1)

I. Knezevic and B. Novakovic, “Time-dependent transport in open systems based on quantum master equations,” J. Comput. Electron. 12, 363–374 (2013).
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Nat. Commun. (1)

G. Villares, A. Hugi, S. Blaser, and J. Faist, “Dual-comb spectroscopy based on quantum-cascade-laser frequency combs,” Nat. Commun. 5, 5192 (2014).
[Crossref]

Nature (1)

A. Hugi, G. Villares, S. Blaser, H. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature 492, 229–233 (2012).
[Crossref]

Nature Photon. (2)

D. Burghoff, T.-Y. Kao, N. Han, C. W. I. Chan, X. Cai, Y. Yang, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Terahertz laser frequency combs,” Nature Photon. 8, 462–467 (2014).
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M. Rösch, G. Scalari, M. Beck, and J. Faist, “Octave-spanning semiconductor laser,” Nature Photon. 9, 42–47 (2015).
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Opt. Express (9)

M. Martl, J. Darmo, C. Deutsch, M. Brandstetter, A. M. Andrews, P. Klang, G. Strasser, and K. Unterrainer, “Gain and losses in THz quantum cascade laser with metal-metal waveguide,” Opt. Express 19, 733–738 (2011).
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D. Burghoff, Y. Yang, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Evaluating the coherence and time-domain profile of quantum cascade laser frequency combs,” Opt. Express 23, 1190–1202 (2015).
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C. Jirauschek, “Monte Carlo study of intrinsic linewidths in terahertz quantum cascade lasers,” Opt. Express 18, 25922–25927 (2010).
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M. Wienold, B. Röben, L. Schrottke, and H. Grahn, “Evidence for frequency comb emission from a Fabry-Pérot terahertz quantum-cascade laser,” Opt. Express 22, 30410–30424 (2014).
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G. Villares and J. Faist, “Quantum cascade laser combs: effects of modulation and dispersion,” Opt. Express 23, 1651–1669 (2015).
[Crossref]

V.-M. Gkortsas, C. Wang, L. Kuznetsova, L. Diehl, A. Gordon, C. Jirauschek, M. Belkin, A. Belyanin, F. Capasso, and F. Kärtner, “Dynamics of actively mode-locked quantum cascade lasers,” Opt. Express 18, 13616–13630 (2010).
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M. A. Talukder and C. R. Menyuk, “Self-induced transparency modelocking of quantum cascade lasers in the presence of saturable nonlinearity and group velocity dispersion,” Opt. Express 18, 5639–5653 (2010).
[Crossref]

Y. Wang and A. Belyanin, “Active mode-locking of mid-infrared quantum cascade lasers with short gain recovery time,” Opt. Express 23, 4173–4185 (2015).
[Crossref]

C. Y. Wang, L. Kuznetsova, V. M. Gkortsas, L. Diehl, F. X. Kärtner, M. A. Belkin, A. Belyanin, X. Li, D. Ham, H. Schneider, P. Grant, C. Y. Song, S. Haffouz, Z. R. Wasilewski, H. C. Liu, and F. Capasso, “Mode-locked pulses from mid-infrared quantum cascade lasers,” Opt. Express 17, 12929–12943 (2009).
[Crossref]

Opt. Quant. Electron. (1)

N. Vukovic, J. Radovanovic, V. Milanovic, and D. Boiko, “Multimode RNGH instabilities of Fabry-Pérot cavity QCLs: impact of diffusion,” Opt. Quant. Electron. 48, 1–10 (2016).
[Crossref]

Optica (2)

Phys. Rev. A (2)

C. Y. Wang, L. Diehl, A. Gordon, C. Jirauschek, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, M. Troccoli, J. Faist, and F. Capasso, “Coherent instabilities in a semiconductor laser with fast gain recovery,” Phys. Rev. A 75, 031802 (2007).
[Crossref]

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

Phys. Rev. B (3)

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. Liu, “Simplified density-matrix model applied to three-well terahertz quantum cascade lasers,” Phys. Rev. B 81, 205311 (2010).
[Crossref]

S. Butscher, J. Förstner, I. Waldmüller, and A. Knorr, “Ultrafast electron-phonon interaction of intersubband transitions: Quantum kinetics from adiabatic following to Rabi-oscillations,” Phys. Rev. B 72, 045314 (2005).
[Crossref]

Rep. Prog. Phys. (1)

R. C. Iotti and F. Rossi, “Microscopic theory of semiconductor-based optoelectronic devices,” Rep. Prog. Phys. 68, 2533 (2005).
[Crossref]

Other (7)

B. S. Williams, “Terahertz quantum-cascade lasers,” Ph.D. thesis, Massachusetts Institute of Technology (2003).

D. P. Burghoff, “Broadband terahertz photonics,” Ph.D. thesis, Massachusetts Institute of Technology (2014).

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University, 1991).

G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures (John Wiley & Sons, 1990).

J. Ye, Femtosecond Optical Frequency Comb: Principle, Operation and Applications (Springer Science & Business Media, 2005).
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R. W. Boyd, Nonlinear Optics (Academic, 2003).

P. Wesseling, Principles of Computational Fluid Dynamics (Springer Science & Business Media, 2009).

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Figures (7)

Fig. 1
Fig. 1 Schematic diagram of a simple three-level LO-phonon depopulation THz QCL, where the upper laser level is populated via resonant tunneling.
Fig. 2
Fig. 2 (a) The moduli squared of the wave functions of the THz QCL in [2] within the tight-binding approximation. (b), The anticrossing coupling strengths between the pair of injector levels, |IN J1〉, |IN J2〉, and the upper laser level |ULL〉, computed via the method outlined in [31]. (c), Dipole elements calculated for the same laser, obtained with the extended basis Hamiltonian, Ĥext.
Fig. 3
Fig. 3 Simulated spectral gain profile (blue curve, left y-axis) together with the normalized group velocity vgn0/c (red curve, right y-axis). (Left inset) The higher order phase acquired by the test pulse. (Right inset) The second derivative of the wave number with respect to the angular frequency.
Fig. 4
Fig. 4 (a) Optical spectrum obtained from THz-TDs simulations when the seed frequencies, ω1 and ω2, are separated by the free spectral range Δω. (b) Same as (a), however this time ω1ω2 = 2Δω. (c) Optical spectrum from simulations where the seed frequencies are distributed under both peaks of the spectral gain. (d) Schematic representation of degenerate FWM [37], where two pump modes ω1,ω2 combine to produce sidebands at frequencies ωa = 2ω1ω2 and ωs = 2ω2ω1.
Fig. 5
Fig. 5 Optical power spectra (left column) and beatnotes (right column) from experiment and simulations of the device in [2]. (a), (b) Experimental data for the dispersion compensated laser, at driving current of 0.9 A. Simulation results without dispersion compensation (c)–(d) and with dispersion compensation (e)–(f). The experimentally detected beatnote in (b) has a FWHM of approx. 0.553MHz, whereas the simulated beatnote in (f) has a resolution limited FWHM of 1.66MHz. The strongest peak in (d) has a FWHM of 3.04 MHz.
Fig. 6
Fig. 6 Time domain separation of the optical field into high and low frequency lobe components. (a) Measured intensity over time from [32] for the same THz comb device as in [2], driven with 0.9 A current. (b) Simulated intensity over time for a dispersion compensated QCL evolved for ~15000 round trips. (c) Time dependence of the electron populations in the injector and upper laser level (i.e. ρ1′1′, ρ33) as well as the dressed states (i.e. ρ++, ρ−−).
Fig. 7
Fig. 7 Simulated high (blue) and low (red) frequency pulses for a numerically dispersion compensated (left column) and an uncompensated (right column) THz QCL. Data from round trip 4615-4618 (top) and 4710-4713 (bottom) are shown.

Tables (2)

Tables Icon

Table 1 Simulation parameters for a THz QCL, modelled after the device in [2]. The modal overlap factor and the facet reflectivities are selected based on [43] for a metal-metal waveguide with thickness 10 μm and width of 20 μm.

Tables Icon

Table 2 Total scattering rates between each pair of relevant subbands of the device in [2] for a bias of 10.8 kV/cm. The rates are presented in ps−1.

Equations (27)

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d d t ( ρ 1 1 ρ 1 3 ρ 1 2 ρ 3 1 ρ 33 ρ 32 ρ 2 1 ρ 23 ρ 22 ) = i [ ( ρ 1 1 ρ 1 3 ρ 1 2 ρ 3 1 ρ 33 ρ 32 ρ 2 1 ρ 23 ρ 22 ) , ( ϵ 2 Ω 1 3 0 Ω 1 3 ϵ 2 Ω L ( t ) 0 Ω L ( t ) ϵ 2 ω 0 ) resonant tunneling and radiative coupling ] + ( ρ 1 1 τ 1 + ( 1 τ 3 1 + 1 τ 31 ) ρ 33 + ( 1 τ 2 1 + 1 τ 21 ) ρ 22 τ | | 1 3 1 ρ 1 3 τ | | 1 2 1 ρ 1 2 τ | | 1 3 1 ρ 3 1 ρ 1 1 τ 1 3 ρ 33 τ 3 + ρ 22 τ 23 τ | | 32 1 ρ 32 τ | | 1 2 1 ρ 2 1 τ | | 32 1 ρ 32 ρ 1 1 τ 1 2 + ρ 33 τ 32 ρ 22 τ 2 ) scattering rates matrix .
τ | | i j 1 = 1 2 ( 1 τ i + 1 τ j ) + 1 τ i j p u r e
J = e N L p ( j = 2 3 ρ i j τ j 1 + ρ 1 1 τ 1 1 ) .
P z = N Γ Tr { ρ ^ e z ^ } = N Γ ( e z 32 ρ 32 + e z 23 ρ 23 ) = N Γ e z 32 ( ρ 32 + ρ 23 ) ,
( x 2 n 0 2 c 2 t 2 ) E z = 1 ϵ 0 c 2 t 2 P z ,
n ( ω ) = c L ω { F o u t ( ω ω c ) / F i n ( ω ω c ) } + c k c ω ,
n ( ω ) = g ( ω ) c / ω = c L ω ln { | F o u t ( ω ω c ) | / | F i n ( ω ω c ) | } .
L Δ k = [ 2 k ( ω 1 ) k ( ω 2 ) k ( ω a ) ] L ,
L Δ k = L Δ ω 2 2 k ω 2 | ω 1 + O ( Δ ω 4 ) 1 L ( π c n 0 ) 2 2 k ω 2 | ω 1 .
E z ( x , t ) = 1 2 { f + ( x , t ) exp [ i ( k c x ω c t ) ] + f ( x , t ) exp [ i ( k c x + ω c t ) ] + c . c . } ,
ρ i i ( x , t ) = ρ i i 0 ( x , t ) + ρ i i + + ( x , t ) exp ( 2 i k c x ) + ρ i i ( x , t ) exp ( 2 i k c x ) ,
ρ 32 ( x , t ) = η 32 + ( x , t ) exp [ i ( k c x ω c t ) ] + η 32 ( x , t ) exp [ i ( k c x + ω c t ) ] ,
ρ 1 2 ( x , t ) = η 1 2 + ( x , t ) exp [ i ( k c x ω c t ) ] + η 1 2 ( x , t ) exp [ i ( k c x + ω c t ) ] ,
ρ 1 3 ( x , t ) = ρ 1 3 0 ( x , t ) + ρ 1 3 + ( x , t ) exp ( 2 i k c x ) + ρ 1 3 ( x , t ) exp ( 2 i k c x ) .
n 0 C t f ± ± x f ± = i N Γ e z 32 k c ϵ 0 n 0 2 η 32 ± l 0 f ± ,
d ρ 1 1 0 d t = i Ω 1 3 ( ρ 1 3 0 ρ 3 1 0 ) + ( 1 τ 3 1 + 1 τ 31 ) ρ 33 0 + ( 1 τ 2 1 + 1 τ 21 ) ρ 22 0 ρ 1 1 0 τ 1 ,
d ρ 1 1 + d t = i Ω 1 3 ( ρ 1 3 + ρ 3 1 + ) + ( 1 τ 3 1 + 1 τ 31 ) ρ 33 + + ( 1 τ 2 1 + 1 τ 21 ) ρ 22 + ( 1 τ 1 + 4 k c 2 D ) ρ 1 1 + ,
d ρ 33 0 d t = i Ω 1 3 ( ρ 3 1 0 ρ 1 3 0 ) + i e z 32 2 ( f * η 32 + f + * η 32 + c . c ) + 1 τ 1 3 ρ 1 1 0 + 1 τ 23 ρ 22 0 ρ 33 0 τ 3 ,
d ρ 33 + d t = i Ω 1 3 ( ρ 3 1 + ρ 1 3 + ) + i e z 32 2 [ f * η 32 + f + ( η 32 ) * ] + ρ 1 1 + τ 1 3 + ρ 22 + τ 23 ( 1 τ 3 + 4 k c 2 D ) ρ 33 + ,
d ρ 22 0 d t = i e z 32 2 ( f * η 32 + f + * η 32 + c . c ) + 1 τ 1 2 ρ 1 1 0 + 1 τ 32 ρ 33 0 ρ 22 0 τ 21 ,
d ρ 22 + d t = i e z 32 2 [ f * η 32 + f + ( η 32 ) * ] + 1 τ 1 2 ρ 1 1 + + 1 τ 32 ρ 33 + ( 1 τ 2 + 4 k c 2 D ) ρ 22 + ,
d ρ 1 3 0 d t = i ϵ ρ 1 3 0 + i Ω 1 3 ( ρ 1 1 0 ρ 33 0 ) + i e z 32 2 ( f + * η 1 2 + + f * η 1 2 ) τ | | 1 3 1 ρ 1 3 0 ,
d ρ 1 3 ± d t = i ϵ ρ 1 3 ± + i Ω 1 3 ( ρ 1 1 ± ρ 33 ± ) + i e z 32 2 f * η 1 2 ± ( τ | | 1 3 1 + 4 k c 2 D ) ρ 1 3 ± ,
d η 32 ± d t = i ( ω c ω 0 ) η 32 ± + i e z 32 2 [ f ± ( ρ 33 0 ρ 22 0 ) + f ( ρ 33 ± ρ 22 ± ) ] i Ω 1 3 η 1 2 ± τ | | 32 1 η 32 ± ,
d η 1 2 ± d t = i ( ω c ω 0 ϵ ) η 1 2 ± + i e z 32 2 ( f ± ρ 1 3 0 + f ρ 1 3 ± ) i Ω 1 3 η 32 ± τ | | 1 2 1 η 1 2 ± .
f ± t = c f ± x + η ± ( x , t ) + k f ± .
f ± ( m , n + 1 ) = f ± ( m 1 , n ) + Δ t [ η ± ( m , n ) + k f ± ( m , n ) ] + Δ t 2 2 { [ η ± t ] m n c [ η ± x ] m n 2 k c [ f ± x ] m n + k η ± ( m , n ) + k 2 f ± ( m , n ) } ,

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