Abstract

The dynamics of a multimode quantum cascade laser, are studied in a model based on effective semiconductor Maxwell-Bloch equations, encompassing key features for the radiation-medium interaction such as an asymmetric frequency dependent gain and refractive index as well as the phase-amplitude coupling provided by the linewidth enhancement factor. By considering its role and that of the free spectral range, we find the conditions in which the traveling wave emitted by the laser at the threshold can be destabilized by adjacent modes, thus leading the laser emission towards chaotic or regular multimode dynamics. In the latter case our simulations show that the field oscillations are associated to self-confined structures which travel along the laser cavity, bridging mode-locking and solitary wave propagation. In addition, we show how a RF modulation of the bias current leads to active mode-locking yielding high-contrast, picosecond pulses. Our results compare well with recent experiments on broad-band THz-QCLs and may help in the understanding of the conditions for the generation of ultrashort pulses and comb operation in mid-IR and THz spectral regions.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (1)

2016 (5)

2015 (6)

F. Gustave, L. Columbo, G. Tissoni, M. Brambilla, F. Prati, B. Kelleher, B. Tykalewicz, and S. Barland, “Dissipative Phase Solitons in Semiconductor Lasers,” Phys. Rev. Lett. 115, 043902 (2015).
[Crossref] [PubMed]

H. Li, P. Laffaille, D. Gacemi, M. Apfel, C. Sirtori, J. Leonardon, G. Santarelli, Markus Rösch, G. Scalari, M. Beck, J. Faist, W. Hänsel, R. Holzwarth, and S. Barbieri, “Dynamics of ultra-broadband terahertz quantum cascade lasers for comb operation,” Opt. Express 23, 33270–33294 (2015).
[Crossref]

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

M. S. Vitiello, G. Scalari, B. Williams, and P. De Natale, “Quantum cascade lasers: 20 years of challenges,” Opt. Express 23, 5167–5182 (2015).
[Crossref] [PubMed]

C. Juretzka, H. Simos, A. Bogris, D. Syvridis, W. Elsäßer, and M. Carras, “Intensity Noise Properties of Midinfrared Injection Locked Quantum Cascade Lasers: II. Experiments,” IEEE J. Quantum Electron. 51, 2300208 (2015).
[Crossref]

G. Villares and J. Faist, “Quantum cascade laser combs: effects of modulation and dispersion,” Opt. Express 23, 1651–1669 (2015).
[Crossref] [PubMed]

2014 (3)

J. B. 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]

D. Bachmann, M. Rösch, C. Deutsch, M. Krall, G. Scalari, M. Beck, J. Faist, K. Unterrainer, and J. Darmo, “Spectral gain profile of a multi-stack terahertz quantum cascade laser,” App. Phys. Lett. 105, 181118 (2014).
[Crossref]

L. Gil and G. Lippi, “Phase Instability in Semiconductor Lasers,” Phys. Rev. Lett. 113, 213902 (2014).
[Crossref] [PubMed]

2013 (3)

P. Malara, R. Blanchard, T. S. Mansuripur, A. K. Wojcik, A. Belyanin, K. Fujita, T. Edamura, S. Furuta, M. Yamanishi, P. de Natale, and F. Capasso, “External ring-cavity quantum cascade lasers,” Appl. Phys. Lett. 102, 141105 (2013).
[Crossref]

T. Erneux, V. Kovanis, and A. Gavrielides, “Nonlinear dynamics of an injected quantum cascade laser,” Phys. Rev. E 88, 032907 (2013).
[Crossref]

A. K. Wójcik, P. Malara, R. Blanchard, T. S. Mansuripur, F. Capasso, and A. Belyanin, “Generation of picosecond pulses and frequency combs in actively mode locked external ring cavity quantum cascade lasers,” App. Phys. Lett. 103, 231102 (2013).
[Crossref]

2012 (3)

J. R. Freeman, J. Maysonnave, N. Jukam, P. Cavali, K. Maussang, H. E. Beere, D. A. Ritchie, J. Mangeney, S. S. Dhillon, and J. Tignon, “Direct intensity sampling of a modelocked terahertz quantum cascade laser,” Appl. Phys. Lett. 101, 181115 (2012).
[Crossref]

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

M. Ravaro, S. Barbieri, G. Santarelli, V. Jagtap, C. Manquest, C. Sirtori, S. P. Khanna, and E. H. Linfield, “Measurement of the intrinsic linewidth of terahertz quantum cascade lasers using a near infrared frequency comb,” Opt. Express 20, 25654–25661 (2012).
[Crossref] [PubMed]

2011 (1)

S. Barbieri, M. Ravaro, P. Gellie, G. Santarelli, C. Manquest, C. Sirtori, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Coherent sampling of active mode-locked terahertz quantum cascade lasers and frequency synthesis,” Nature Photon. 5, 306–313 (2011).
[Crossref]

2010 (5)

S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nature Photon. 4, 636–640 (2010).
[Crossref]

D. Oustinov, N. Jukam, R. Rungsawang, J. Madéo, S. Barbieri, P. Filloux, C. Sirtori, X. Marcadet, J. Tignon, and S. Dhillon, “Phase seeding of a terahertz quantum cascade laser,” Nat. Commun. 1, 69 (2010).
[Crossref] [PubMed]

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, Günther Roelkens, E.-Jan Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nature Photon. 4, 182–187 (2010).

F. Prati, G. Tissoni, C. McIntyre, and G. L. Oppo, “Static and dynamic properties of cavity solitons in VCSELs with optical injection,” Eur. Phys. J. D 59, 139–147 (2010).
[Crossref]

C. C. Nshii, C. N. Ironside, M. Sorel, T. J. Slight, S. Y. Zhang, D. G. Revin, and J. W. Cockburn, “A unidirectional quantum cascade ring laser,” App. Phys. Lett. 97, 231107 (2010).
[Crossref]

2009 (2)

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]

R. P. Green, Ji-H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade laser,” App. Phys. Lett. 92, 071106 (2008).
[Crossref]

2007 (1)

F. Prati and L. Columbo, “Long-wavelength instability in broad-area semiconductor lasers,” Phys. Rev. A 75, 053811 (2007).
[Crossref]

1999 (2)

J. K. White and J. Moloney, “Multichannel communication using an infinite dimensional spatiotemporal chaotic system,” Phys. Rev. A 59, 2422–2426 (1999).
[Crossref]

E. M. Pessina, F. Prati, J. Redondo, E. Roldán, and G. J. de Valcárcel, “Multimode instability in ring fiber lasers,” Phys. Rev. A 60, 2517–2528 (1999).
[Crossref]

1997 (1)

C. Z. Ning, R. A. Indik, and J. V. Moloney, “Effective Bloch Equations for Semiconductor Lasers and Amplifiers,” IEEE J. Quantum Electon. 33, 1543–1550 (1997).
[Crossref]

1968 (2)

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

P. Graham and H. Haken, “Quantum Theory of Light Propagation in a Fluctuating Laser-Active Medium,” Z. Phys. 213, 420–450 (1968).
[Crossref]

Amanti, M.

Apfel, M.

Bachmann, D.

D. Bachmann, M. Rösch, M. J. Süess, M. Beck, K. Unterrainer, J. Darmo, J. Faist, and G. Scalari, “Short pulse generation and mode control of broadband terahertz quantum cascade lasers,” Optica 3, 1087–1094 (2016).
[Crossref]

D. Bachmann, M. Rösch, C. Deutsch, M. Krall, G. Scalari, M. Beck, J. Faist, K. Unterrainer, and J. Darmo, “Spectral gain profile of a multi-stack terahertz quantum cascade laser,” App. Phys. Lett. 105, 181118 (2014).
[Crossref]

Baets, R.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, Günther Roelkens, E.-Jan Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nature Photon. 4, 182–187 (2010).

Barbieri, S.

A. Mottaghizadeh, D. Gacemi, P. Laffaille, H. Li, M. Amanti, C. Sirtori, G. Santarelli, W. Hänsel, R. Holzwart, L. H. Li, E. H. Linfield, and S. Barbieri, “5-ps-long terahertz pulses from an active-modelocked quantum cascade laser,” Optica 4, 168–171 (2017).
[Crossref]

H. Li, P. Laffaille, D. Gacemi, M. Apfel, C. Sirtori, J. Leonardon, G. Santarelli, Markus Rösch, G. Scalari, M. Beck, J. Faist, W. Hänsel, R. Holzwarth, and S. Barbieri, “Dynamics of ultra-broadband terahertz quantum cascade lasers for comb operation,” Opt. Express 23, 33270–33294 (2015).
[Crossref]

M. Ravaro, S. Barbieri, G. Santarelli, V. Jagtap, C. Manquest, C. Sirtori, S. P. Khanna, and E. H. Linfield, “Measurement of the intrinsic linewidth of terahertz quantum cascade lasers using a near infrared frequency comb,” Opt. Express 20, 25654–25661 (2012).
[Crossref] [PubMed]

S. Barbieri, M. Ravaro, P. Gellie, G. Santarelli, C. Manquest, C. Sirtori, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Coherent sampling of active mode-locked terahertz quantum cascade lasers and frequency synthesis,” Nature Photon. 5, 306–313 (2011).
[Crossref]

S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nature Photon. 4, 636–640 (2010).
[Crossref]

D. Oustinov, N. Jukam, R. Rungsawang, J. Madéo, S. Barbieri, P. Filloux, C. Sirtori, X. Marcadet, J. Tignon, and S. Dhillon, “Phase seeding of a terahertz quantum cascade laser,” Nat. Commun. 1, 69 (2010).
[Crossref] [PubMed]

Barland, S.

F. Gustave, L. Columbo, G. Tissoni, M. Brambilla, F. Prati, B. Kelleher, B. Tykalewicz, and S. Barland, “Dissipative Phase Solitons in Semiconductor Lasers,” Phys. Rev. Lett. 115, 043902 (2015).
[Crossref] [PubMed]

Beck, M.

J. Faist, G. Villares, G. Scalari, M. Rösch, C. Bonzon, A. Hugi, and M. Beck, “Quantum Cascade Laser Frequency Combs,” Nanophotonics 5, 272–291 (2016).
[Crossref]

D. Bachmann, M. Rösch, M. J. Süess, M. Beck, K. Unterrainer, J. Darmo, J. Faist, and G. Scalari, “Short pulse generation and mode control of broadband terahertz quantum cascade lasers,” Optica 3, 1087–1094 (2016).
[Crossref]

H. Li, P. Laffaille, D. Gacemi, M. Apfel, C. Sirtori, J. Leonardon, G. Santarelli, Markus Rösch, G. Scalari, M. Beck, J. Faist, W. Hänsel, R. Holzwarth, and S. Barbieri, “Dynamics of ultra-broadband terahertz quantum cascade lasers for comb operation,” Opt. Express 23, 33270–33294 (2015).
[Crossref]

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

D. Bachmann, M. Rösch, C. Deutsch, M. Krall, G. Scalari, M. Beck, J. Faist, K. Unterrainer, and J. Darmo, “Spectral gain profile of a multi-stack terahertz quantum cascade laser,” App. Phys. Lett. 105, 181118 (2014).
[Crossref]

Beere, H.

S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nature Photon. 4, 636–640 (2010).
[Crossref]

Beere, H. E.

J. R. Freeman, J. Maysonnave, N. Jukam, P. Cavali, K. Maussang, H. E. Beere, D. A. Ritchie, J. Mangeney, S. S. Dhillon, and J. Tignon, “Direct intensity sampling of a modelocked terahertz quantum cascade laser,” Appl. Phys. Lett. 101, 181115 (2012).
[Crossref]

R. P. Green, Ji-H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade laser,” App. Phys. Lett. 92, 071106 (2008).
[Crossref]

Belkin, M. A.

Beltram, F.

R. P. Green, Ji-H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade laser,” App. Phys. Lett. 92, 071106 (2008).
[Crossref]

Belyanin, A.

D. G. Revin, M. Hemingway, Y. Wang, J. W. Cockburn, and A. Belyanin, “Active mode locking of quantum cascade lasers in an external ring cavity,” Nat. Commun. 7, 11440 (2016).
[Crossref] [PubMed]

P. Malara, R. Blanchard, T. S. Mansuripur, A. K. Wojcik, A. Belyanin, K. Fujita, T. Edamura, S. Furuta, M. Yamanishi, P. de Natale, and F. Capasso, “External ring-cavity quantum cascade lasers,” Appl. Phys. Lett. 102, 141105 (2013).
[Crossref]

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D. Bachmann, M. Rösch, M. J. Süess, M. Beck, K. Unterrainer, J. Darmo, J. Faist, and G. Scalari, “Short pulse generation and mode control of broadband terahertz quantum cascade lasers,” Optica 3, 1087–1094 (2016).
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S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nature Photon. 4, 636–640 (2010).
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J. Faist, G. Villares, G. Scalari, M. Rösch, C. Bonzon, A. Hugi, and M. Beck, “Quantum Cascade Laser Frequency Combs,” Nanophotonics 5, 272–291 (2016).
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J. B. 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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Sorel, M.

C. C. Nshii, C. N. Ironside, M. Sorel, T. J. Slight, S. Y. Zhang, D. G. Revin, and J. W. Cockburn, “A unidirectional quantum cascade ring laser,” App. Phys. Lett. 97, 231107 (2010).
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J. Faist, G. Villares, G. Scalari, M. Rösch, C. Bonzon, A. Hugi, and M. Beck, “Quantum Cascade Laser Frequency Combs,” Nanophotonics 5, 272–291 (2016).
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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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Wang, Y.

D. G. Revin, M. Hemingway, Y. Wang, J. W. Cockburn, and A. Belyanin, “Active mode locking of quantum cascade lasers in an external ring cavity,” Nat. Commun. 7, 11440 (2016).
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P. Malara, R. Blanchard, T. S. Mansuripur, A. K. Wojcik, A. Belyanin, K. Fujita, T. Edamura, S. Furuta, M. Yamanishi, P. de Natale, and F. Capasso, “External ring-cavity quantum cascade lasers,” Appl. Phys. Lett. 102, 141105 (2013).
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Wójcik, A. K.

A. K. Wójcik, P. Malara, R. Blanchard, T. S. Mansuripur, F. Capasso, and A. Belyanin, “Generation of picosecond pulses and frequency combs in actively mode locked external ring cavity quantum cascade lasers,” App. Phys. Lett. 103, 231102 (2013).
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Xu, Ji-H.

R. P. Green, Ji-H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade laser,” App. Phys. Lett. 92, 071106 (2008).
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P. Malara, R. Blanchard, T. S. Mansuripur, A. K. Wojcik, A. Belyanin, K. Fujita, T. Edamura, S. Furuta, M. Yamanishi, P. de Natale, and F. Capasso, “External ring-cavity quantum cascade lasers,” Appl. Phys. Lett. 102, 141105 (2013).
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Yao, A. M.

G.-L. Oppo, A. M. Yao, F. Prati, and G. J. de Valcarcel, “Long-term spatiotemporal dynamics of solid-state lasers and vertical-cavity surface-emitting lasers,” Phys. Rev. A 79, 033824 (2009).
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C. C. Nshii, C. N. Ironside, M. Sorel, T. J. Slight, S. Y. Zhang, D. G. Revin, and J. W. Cockburn, “A unidirectional quantum cascade ring laser,” App. Phys. Lett. 97, 231107 (2010).
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A. K. Wójcik, P. Malara, R. Blanchard, T. S. Mansuripur, F. Capasso, and A. Belyanin, “Generation of picosecond pulses and frequency combs in actively mode locked external ring cavity quantum cascade lasers,” App. Phys. Lett. 103, 231102 (2013).
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C. C. Nshii, C. N. Ironside, M. Sorel, T. J. Slight, S. Y. Zhang, D. G. Revin, and J. W. Cockburn, “A unidirectional quantum cascade ring laser,” App. Phys. Lett. 97, 231107 (2010).
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D. Bachmann, M. Rösch, C. Deutsch, M. Krall, G. Scalari, M. Beck, J. Faist, K. Unterrainer, and J. Darmo, “Spectral gain profile of a multi-stack terahertz quantum cascade laser,” App. Phys. Lett. 105, 181118 (2014).
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R. P. Green, Ji-H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade laser,” App. Phys. Lett. 92, 071106 (2008).
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P. Malara, R. Blanchard, T. S. Mansuripur, A. K. Wojcik, A. Belyanin, K. Fujita, T. Edamura, S. Furuta, M. Yamanishi, P. de Natale, and F. Capasso, “External ring-cavity quantum cascade lasers,” Appl. Phys. Lett. 102, 141105 (2013).
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J. B. 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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J. Faist, G. Villares, G. Scalari, M. Rösch, C. Bonzon, A. Hugi, and M. Beck, “Quantum Cascade Laser Frequency Combs,” Nanophotonics 5, 272–291 (2016).
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Nat. Commun. (2)

D. G. Revin, M. Hemingway, Y. Wang, J. W. Cockburn, and A. Belyanin, “Active mode locking of quantum cascade lasers in an external ring cavity,” Nat. Commun. 7, 11440 (2016).
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D. Oustinov, N. Jukam, R. Rungsawang, J. Madéo, S. Barbieri, P. Filloux, C. Sirtori, X. Marcadet, J. Tignon, and S. Dhillon, “Phase seeding of a terahertz quantum cascade laser,” Nat. Commun. 1, 69 (2010).
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Nature (1)

A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature 492, 229–233 (2012).
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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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S. Barbieri, M. Ravaro, P. Gellie, G. Santarelli, C. Manquest, C. Sirtori, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Coherent sampling of active mode-locked terahertz quantum cascade lasers and frequency synthesis,” Nature Photon. 5, 306–313 (2011).
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S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nature Photon. 4, 636–640 (2010).
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L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, Günther Roelkens, E.-Jan Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nature Photon. 4, 182–187 (2010).

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H. Li, P. Laffaille, D. Gacemi, M. Apfel, C. Sirtori, J. Leonardon, G. Santarelli, Markus Rösch, G. Scalari, M. Beck, J. Faist, W. Hänsel, R. Holzwarth, and S. Barbieri, “Dynamics of ultra-broadband terahertz quantum cascade lasers for comb operation,” Opt. Express 23, 33270–33294 (2015).
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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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M. S. Vitiello, G. Scalari, B. Williams, and P. De Natale, “Quantum cascade lasers: 20 years of challenges,” Opt. Express 23, 5167–5182 (2015).
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G. Villares and J. Faist, “Quantum cascade laser combs: effects of modulation and dispersion,” Opt. Express 23, 1651–1669 (2015).
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N. Vukovic, J. Radovanovic, V. Milanovic, and D. L. Boiko, “Analytical expression for Risken-Nummedal-Graham-Haken instability threshold in quantum cascade lasers,” Opt. Express 24, 26911–26929 (2016).
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P. Tzenov, D. Burghoff, Q. Hu, and C. Jirauschek, “Time domain modeling of terahertz quantum cascade lasers for frequency comb generation,” Opt. Express 24, 23232–23247 (2016).
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M. Ravaro, S. Barbieri, G. Santarelli, V. Jagtap, C. Manquest, C. Sirtori, S. P. Khanna, and E. H. Linfield, “Measurement of the intrinsic linewidth of terahertz quantum cascade lasers using a near infrared frequency comb,” Opt. Express 20, 25654–25661 (2012).
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Phys. Rev. A (5)

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]

G.-L. Oppo, A. M. Yao, F. Prati, and G. J. de Valcarcel, “Long-term spatiotemporal dynamics of solid-state lasers and vertical-cavity surface-emitting lasers,” Phys. Rev. A 79, 033824 (2009).
[Crossref]

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[Crossref]

F. Prati and L. Columbo, “Long-wavelength instability in broad-area semiconductor lasers,” Phys. Rev. A 75, 053811 (2007).
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T. Erneux, V. Kovanis, and A. Gavrielides, “Nonlinear dynamics of an injected quantum cascade laser,” Phys. Rev. E 88, 032907 (2013).
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F. Gustave, L. Columbo, G. Tissoni, M. Brambilla, F. Prati, B. Kelleher, B. Tykalewicz, and S. Barland, “Dissipative Phase Solitons in Semiconductor Lasers,” Phys. Rev. Lett. 115, 043902 (2015).
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[Crossref]

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Figures (13)

Fig. 1
Fig. 1 (a) Sketch of a semiconductor laser in a unidirectional ring cavity. (b) Schematic conduction band diagram of a QCL. Each stage of the structure consists in an active region and a injection region. Lasing action occurs between level 3 and level 2. Thus, the lifetime of the nonradiative transition 3 → 2 denoted as τ32 has to be longer than the lifetime τ2 of the level 2. The lifetime of the level 1 is supposed to be 0.
Fig. 2
Fig. 2 (a) Phenomenological s.c. susceptibility. (b) Two-level system susceptibility. Dashed lines delimit frequency regions of interest in simulations.
Fig. 3
Fig. 3 The laser threshold plotted versus k (for Γ=1.1) for different values of α (a) and cavity length l (b). Since z is scaled on l the set of discrete k values compatible with the boundary conditions (empty cavity modes) are separated by 2π. They are indicated by continuous vertical lines in the figure. As expected, the mode TW0 corresponds to k = 0 and to a value of Ω given by the dispersion relation Eq. (29). (c) Laser threshold plotted as a function of Ω for α = 1.5, l = 1mm (free spectral range of 100GHz).
Fig. 4
Fig. 4 Beat note spectrum ((a), left panel) and its zoom on the first beatnote ((a), right panel) and optical spectrum (b) of the QCL field for different values of the normalized pump parameter. For an easier comparison with experimental data, the frequency in physical units is drawn on the upper horizontal scale. The color scale is logarithmic. All spectra have been shifted to have the same minimum, taken as zero and the maximum corresponds to the absolute maximum among all spectra. During the whole simulation µ is adiabatically increased by steps of 0.2µth leaving the dynamical system described by Eqs. (18)(20) to reach a regime. The dashed lines delimit the regions where the value of µ is kept constant. Color scale variations across these regions are due to graphical interpolation. The red rectangle highlights the region of the BNS where analogous experimental data are available [34]. Parameters as in Fig. 3(c).
Fig. 5
Fig. 5 Intensity at half the cavity versus time (a) and corresponding OS (b) in the irregular multimode regime corresponding to µ = 1.2µth. Time is scaled to τd = 0.1ps. Other parameters as in Fig. 4.
Fig. 6
Fig. 6 Intensity at half the cavity versus time (a) and corresponding OS (b) in the regular multimode regime for µ = 1.6µth. Note the main peaks at integer multiples of ≃ 2× (free spectral range). Other parameters as in Fig. 4.
Fig. 7
Fig. 7 Spatio-temporal evolution (z: cavity coordinate, t: time scaled on τd) of the intracavity intensity and (inset) spatial intensity profile at a given instant for µ = 1.6µth. Other parameters as in Fig. 4.
Fig. 8
Fig. 8 Intensity profile at a given instant of time obtained starting from two different initial conditions for µ = 2.0µth: a two peaked structure (upper panel) and a single peak structure coexisting with the former (lower panel). Other parameters as in Fig. 4.
Fig. 9
Fig. 9 Γ = 0.3. Other parameters and color scale definition as in Fig. 4. BNS of the emitted field above threshold for increasing µ/µth. µ is adiabatically increased by 0.02µth.
Fig. 10
Fig. 10 Beat note spectrum of the electric field emitted by the QCL in presence of a RF modulation of amplitude 10% applied to 2/5 of the QCL at an angular frequency of ΩM = 0.062. The BN, as evidenced by the red box, is now present throughout the whole spectrum. Other parameters and color scale definition as in Fig. 4.
Fig. 11
Fig. 11 Beat note spectrum of the electric field emitted by the QCL in presence (black line) and absence (red line) of RF pump modulation. In the simulations we used µ0 = 1.4µth and a RF modulation, applied only to a just 2/5 of the whole laser cavity, with an amplitude equal to 35% of the mean value and ΩM = 0.062. Other parameters as in Fig. 4.
Fig. 12
Fig. 12 α = 1.2. Other parameters as in Fig. 4. Intensity at half the cavity (a) and intracavity intensity (b) versus time at steady showing the formation of ultrashort pulses of FWHM ≃ 1.5 ps via active mode-locking for a sinusoidal modulation of the pump at the angular frequency ΩM = 0.0628 (µ0 = 1.5µth, µA = 4µth) and a DC biased in the remaining section of the QCL (µb = 0.6µth).
Fig. 13
Fig. 13 Parameters as in Fig. 12. Optical spectrum (a) and beat note spectrum (b) corresponding to the mode-locked pulses in Fig. 12.

Equations (56)

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E ˜ ( z , t ) = E ( z , t ) 2 exp [ i ( k 0 z ω 0 t ) ] + c . c .
P ˜ ( z , t ) = P ( z , t ) 2 exp [ i ( k 0 z ω 0 t ) ] + c . c .
E z + 1 ν E t = g P
N t = I e V N τ e i 4 ( E * P E P * )
P ^ ( ω ) = ϵ 0 ϵ b χ ( ω ) E ^ ( ω )
χ ( ω , N ) = A ( N ) B ( N ) i ω = R e ( A ) R e ( B ) + I m ( A ) ( I m ( B ) ω ) R e ( B ) 2 + ( I m ( B ) ω ) 2 + i I m ( A ) R e ( B ) R e ( A ) ( I m ( B ) ω ) R e ( B ) 2 + ( I m ( B ) ω ) 2
χ 1 , 2 l e v = I m ( A ) ( I m ( B ) ω ) R e ( B ) 2 + ( I m ( B ) ω ) 2 + i I m ( A ) R e ( B ) R e ( B ) 2 + ( I m ( B ) ω ) 2
P t = ϵ 0 ϵ b A ( N ) E B ( N ) P
B = ( Γ + i δ ) / τ d , ( Γ R + , δ R )
ω M = I m ( B ) R e ( B ) ( I m ( A ) R e ( A ) ( I m ( A ) R e ( A ) ) 2 + 1 )
A ( N ) B = f 0 ( α + i ) N , ( f 0 R )
A ( N ) = f 0 N ( α Γ δ τ d ) i f 0 N ( Γ + α δ τ d )
E z + 1 ν E t = g P
P t = 1 τ d Γ ( 1 i α ) [ P i f 0 ϵ 0 ϵ b ( 1 i α ) E N ]
N t = 1 τ e [ I τ e e V N i τ e 4 ( E * P E P * ) ]
η 1 E E , i η 2 P P , η 3 N D η 1 2 = η 3 τ e 2 , η 2 = η 1 , η 3 = ϵ 0 f 0 ϵ b , g ˜ = i g R , μ = I η 3 τ e e V
E z + 1 ν E t = g ˜ P
P t = 1 τ d Γ ( 1 i α ) [ ( 1 i α ) E D P ]
D t = 1 τ e [ μ D 1 2 ( E * P + E P * ) ]
c τ d l n E ˜ z + E ˜ t = σ ( E ˜ + P ˜ )
P ˜ t = Γ ( 1 i α ) [ ( 1 i α ) E ˜ D ˜ P ˜ ]
D ˜ t = b [ μ ˜ D ˜ 1 2 ( E ˜ * P ˜ + E ˜ P ˜ * ) ]
E ˜ = E s e i ( k z Ω t ) P ˜ = P s e i ( k z Ω t ) D ˜ = D s
c τ d l n i k E s i Ω E s = σ ( E s + P s )
i Ω P s = Γ ( 1 i α ) [ ( 1 i α ) E s D s P s ]
0 = μ D s 1 2 ( E s * P s + E s P s * )
P s = E s μ ( 1 + | E s | 2 G 1 ) ( G 1 + i G 2 )
D s = μ ( 1 + | E s | 2 G 1 )
G 1 = ( Γ 2 ( 1 α 2 ) + 2 Γ α ( Γ α + Ω ) ) / [ Γ 2 + ( α Γ + Ω ) 2 ) ] , G 2 = [ 2 Γ 2 α + Γ ( α Γ + Ω ) α 2 Γ ( α Γ + Ω ) ] / [ Γ 2 + ( α Γ + Ω ) 2 ) ]
E s ( c τ d l n i k i Ω ) = σ E s [ 1 + μ ( G 1 + i G 2 ) ( 1 + | E s | 2 G 1 ) ]
| E s | 2 = μ 1 G 1
Ω = c τ d l n k σ G 2 G 1
μ ( t ) = μ M ( t ) = μ 0 + μ A c o s ( Ω M t ) μ 0 , μ A > μ t h
E ( 0 , t ) = R E ( l , t ( L l ) / c ) e i k 0 l + i ω 0 ( L l ) / c E ( 0 , t ) = R E ( l , t Δ t ) e i ω 0 Λ / c
e i ω 0 Λ c = e i Λ c ( 2 π m c Λ ω 0 ) = e i Λ c ( ω c ω 0 ) = e i δ 0 , m = 0 , ± 1 , ± 2 , ..
E ( 0 , t ) = R E ( l , t Δ t )
η = z t = t + z l Δ t
E ( 0 , t ) = R E ( l , t )
z = η + Δ t l t , t = t
E η + Λ l c E t = g ˜ P
P t = 1 τ d Γ ( 1 i α ) [ ( 1 i α ) E D P ]
D t = 1 τ e [ μ D 1 2 ( E * P + E P * ) ]
E ( η , t ) = E ( η , t ) e [ ( ln R ) η / l ] , P ( η , t ) = P ( η , t ) e [ ( ln R ) η / l ]
E η + Λ l c E t = 1 l ( ln R ) E + g ˜ P
P t = 1 τ d Γ ( 1 i α ) [ ( 1 i α ) D E P ]
D t = 1 τ e { μ D 1 2 e ( 2 ln R ) η / l [ E * P + E P * ] }
E ( 0 , t ) = E ( l , t )
g ˜ l 1 , T 1
A = g ˜ l T = O ( i )
l c Λ E η + E t = 1 τ p [ E + A P ]
P t = 1 τ d Γ ( 1 i α ) [ ( 1 i α ) E D P ]
D t = 1 τ e [ μ D 1 2 ( E * P + E P * ) ]
E ( 0 , t ) = E ( l , t )
l c τ d Λ E ˜ η + E ˜ t = σ ( E ˜ + P ˜ )
P ˜ t = Γ ( 1 i α ) [ ( 1 i α ) E ˜ D ˜ P ˜ ]
D ˜ t = b [ μ ˜ D ˜ 1 2 ( E ˜ * P ˜ + E ˜ P ˜ * ) ]

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