Abstract

We study the instability thresholds of the stationary emission of a quantum cascade laser with optical feedback described by the Lang Kobayashi model. We introduce an exact linear stability analysis and an approximated one for an unipolar lasers, who does not exhibit relaxation oscillations, and investigate the regimes of the emitter beyond the continuous wave instability threshold, depending on the number and density of the external cavity modes. We then show that a unipolar laser with feedback can exhibit coherent multimode oscillations that indicate spontaneous phase-locking.

© 2014 Optical Society of America

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    [CrossRef] [PubMed]
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  4. 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, 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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    [CrossRef] [PubMed]
  6. S. Barbieri, M. Ravaro, P. Gellie, G. Santarelli, C. Manquest, C. Sirtori, S. P. Khanna, H. Linfield, A. G. Davies, “Coherent sampling of active mode-locked terahertz quantum cascade lasers and frequency synthesis,” Nat. Photonics 5, 306–313 (2011).
    [CrossRef]
  7. A. K. Wójcik, P. Malara, R. Blanchard, T. S. Mansuripur, F. Capasso, A. Belyanin, “Generation of picosecond pulses and frequency combs in actively mode locked external ring cavity quantum cascade lasers,” Appl. Phys. Lett. 103, 231102 (2013).
    [CrossRef]
  8. N. Yu, L. Diehl, E. Cubukcu, D. Bour, S. Corzine, G. Höfler, A. K. Wójcik, K. B. Crozier, A. Belyanin, F. Capasso, “Coherent coupling of multiple transverse modes in quantum cascade lasers,” Phys. Rev. Lett. 102, 013901 (2009).
    [CrossRef] [PubMed]
  9. P. Dean, Y. L. Lim, A. Valavanis, R. Kliese, M. Nikolić, S. P. Khanna, M. Lachab, D. Indjin, Z. Ikonić, P. Harrison, A. D. Rakić, E. H. Linfield, A. G. Davies, “Terahertz imaging through self-mixing in a quantum cascade laser,” Opt. Lett. 36, 2587–2589 (2011).
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    [CrossRef]
  26. T. Erneux, V. Kovanis, A. Gavrielides, “Nonlinear dynamics of an injected quantum cascade laser,” Phys. Rev. E 88, 032907 (2013).
    [CrossRef]
  27. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Fortran 77: The Art of Scientific Computing (Academic, 1992).
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2013 (4)

2012 (3)

2011 (3)

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

P. Dean, Y. L. Lim, A. Valavanis, R. Kliese, M. Nikolić, S. P. Khanna, M. Lachab, D. Indjin, Z. Ikonić, P. Harrison, A. D. Rakić, E. H. Linfield, A. G. Davies, “Terahertz imaging through self-mixing in a quantum cascade laser,” Opt. Lett. 36, 2587–2589 (2011).
[CrossRef] [PubMed]

Y. L. Lim, P. Dean, M. Nikolić, R. Kliese, S. P. Khanna, M. Lachab, A. Valavanis, D. Indjin, Z. Ikonić, P. Harrison, E. Linfield, A. G. Davies, S. J. Wilson, A. D. Rakić, “Demonstration of a self-mixing displacement sensor based on terahertz quantum cascade lasers,” Appl. Phys. Lett. 99, 081108 (2011).
[CrossRef]

2010 (1)

R. F. Curl, F. Capasso, C. Gmachl, A. A. Kosterev, B. Mc Manus, R. Lewicki, M. Pusharsky, G. Wysocki, F. K. Tittel, “Quantum cascade lasers in chemical physics,” Chem. Phys. Lett. 487, 1–18 (2010).
[CrossRef]

2009 (2)

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, F. Capasso, “Mode-locked pulses from mid-infrared quantum cascade lasers,” Opt. Express 17, 12929–12943 (2009).
[CrossRef] [PubMed]

N. Yu, L. Diehl, E. Cubukcu, D. Bour, S. Corzine, G. Höfler, A. K. Wójcik, K. B. Crozier, A. Belyanin, F. Capasso, “Coherent coupling of multiple transverse modes in quantum cascade lasers,” Phys. Rev. Lett. 102, 013901 (2009).
[CrossRef] [PubMed]

2008 (3)

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, F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[CrossRef]

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

R. P. Green, J. H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

2007 (1)

2006 (1)

2005 (1)

2001 (1)

R. Paiella, R. Martini, F. Capasso, C. Gmachl, H. Y. Hwang, “High-frequency modulation without the relaxation oscillation resonance in quantum cascade lasers,” Appl. Phys. Lett. 79, 2526–2528 (2001).
[CrossRef]

1995 (1)

A. M. Levine, G. H. M. van Tartwijk, D. Lenstra, T. Erneux, “Diode lasers with optical feedback: Stability of the maximum gain mode,” Phys. Rev. A 52, 3436–3439 (1995).
[CrossRef]

1994 (1)

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, A. Y. Cho, “Quantum cascade laser,” Science 264, 553–556 (1994).
[CrossRef] [PubMed]

1990 (1)

J. Helms, K. Petermann, “A simple analytic expression for the stable operation range of laser diodes with optical feedback,” IEEE J. Quantum Electron. 26, 833–836 (1990).
[CrossRef]

1980 (1)

R. Lang, K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980).
[CrossRef]

Akikusa, N.

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

Antonio, A.

F. P. Mezzapesa, V. Spagnolo, A. Antonio, G. Scamarcio, “Detection of ultrafast laser ablation using quantum cascade laser-based sensing,” Appl. Phys. Lett. 101, 171101 (2012).
[CrossRef]

Barbieri, S.

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

Beere, H. E.

F. P. Mezzapesa, L. L. Columbo, M. Brambilla, M. Dabbicco, S. Borri, M. S. Vitiello, H. E. Beere, D. A. Ritchie, G. Scamarcio, “Intrinsic stability of quantum cascade lasers against optical feedback,” Opt. Express 21, 13748–13757 (2013).
[CrossRef] [PubMed]

R. P. Green, J. H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

Belkin, M. A.

Beltram, F.

R. P. Green, J. H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

Belyanin, A.

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

N. Yu, L. Diehl, E. Cubukcu, D. Bour, S. Corzine, G. Höfler, A. K. Wójcik, K. B. Crozier, A. Belyanin, F. Capasso, “Coherent coupling of multiple transverse modes in quantum cascade lasers,” Phys. Rev. Lett. 102, 013901 (2009).
[CrossRef] [PubMed]

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, F. Capasso, “Mode-locked pulses from mid-infrared quantum cascade lasers,” Opt. Express 17, 12929–12943 (2009).
[CrossRef] [PubMed]

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, F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[CrossRef]

Bertling, K.

Blanchard, R.

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

Borri, S.

Bour, D.

N. Yu, L. Diehl, E. Cubukcu, D. Bour, S. Corzine, G. Höfler, A. K. Wójcik, K. B. Crozier, A. Belyanin, F. Capasso, “Coherent coupling of multiple transverse modes in quantum cascade lasers,” Phys. Rev. Lett. 102, 013901 (2009).
[CrossRef] [PubMed]

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, F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[CrossRef]

Brambilla, M.

Capasso, F.

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

R. F. Curl, F. Capasso, C. Gmachl, A. A. Kosterev, B. Mc Manus, R. Lewicki, M. Pusharsky, G. Wysocki, F. K. Tittel, “Quantum cascade lasers in chemical physics,” Chem. Phys. Lett. 487, 1–18 (2010).
[CrossRef]

N. Yu, L. Diehl, E. Cubukcu, D. Bour, S. Corzine, G. Höfler, A. K. Wójcik, K. B. Crozier, A. Belyanin, F. Capasso, “Coherent coupling of multiple transverse modes in quantum cascade lasers,” Phys. Rev. Lett. 102, 013901 (2009).
[CrossRef] [PubMed]

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, F. Capasso, “Mode-locked pulses from mid-infrared quantum cascade lasers,” Opt. Express 17, 12929–12943 (2009).
[CrossRef] [PubMed]

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, F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[CrossRef]

R. Paiella, R. Martini, F. Capasso, C. Gmachl, H. Y. Hwang, “High-frequency modulation without the relaxation oscillation resonance in quantum cascade lasers,” Appl. Phys. Lett. 79, 2526–2528 (2001).
[CrossRef]

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, A. Y. Cho, “Quantum cascade laser,” Science 264, 553–556 (1994).
[CrossRef] [PubMed]

Cho, A. Y.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, A. Y. Cho, “Quantum cascade laser,” Science 264, 553–556 (1994).
[CrossRef] [PubMed]

Columbo, L.

Columbo, L. L.

Corzine, S.

N. Yu, L. Diehl, E. Cubukcu, D. Bour, S. Corzine, G. Höfler, A. K. Wójcik, K. B. Crozier, A. Belyanin, F. Capasso, “Coherent coupling of multiple transverse modes in quantum cascade lasers,” Phys. Rev. Lett. 102, 013901 (2009).
[CrossRef] [PubMed]

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, F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[CrossRef]

Crozier, K. B.

N. Yu, L. Diehl, E. Cubukcu, D. Bour, S. Corzine, G. Höfler, A. K. Wójcik, K. B. Crozier, A. Belyanin, F. Capasso, “Coherent coupling of multiple transverse modes in quantum cascade lasers,” Phys. Rev. Lett. 102, 013901 (2009).
[CrossRef] [PubMed]

Cubukcu, E.

N. Yu, L. Diehl, E. Cubukcu, D. Bour, S. Corzine, G. Höfler, A. K. Wójcik, K. B. Crozier, A. Belyanin, F. Capasso, “Coherent coupling of multiple transverse modes in quantum cascade lasers,” Phys. Rev. Lett. 102, 013901 (2009).
[CrossRef] [PubMed]

Curl, R. F.

R. F. Curl, F. Capasso, C. Gmachl, A. A. Kosterev, B. Mc Manus, R. Lewicki, M. Pusharsky, G. Wysocki, F. K. Tittel, “Quantum cascade lasers in chemical physics,” Chem. Phys. Lett. 487, 1–18 (2010).
[CrossRef]

Dabbicco, M.

Davies, A. G.

A. D. Rakić, T. Taimre, K. Bertling, Y. L. Lim, P. Dean, D. Indjin, Z. Ikonić, P. Harrison, A. Valavanis, S. P. Khanna, M. Lachab, S. J. Wilson, E. H. Linfield, A. G. Davies, “Swept-frequency feedback interferometry using terahertz frequency QCLs: a method for imaging and materials analysis,” Opt. Express 21, 22194–22205 (2013).
[CrossRef]

P. Dean, Y. L. Lim, A. Valavanis, R. Kliese, M. Nikolić, S. P. Khanna, M. Lachab, D. Indjin, Z. Ikonić, P. Harrison, A. D. Rakić, E. H. Linfield, A. G. Davies, “Terahertz imaging through self-mixing in a quantum cascade laser,” Opt. Lett. 36, 2587–2589 (2011).
[CrossRef] [PubMed]

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

Y. L. Lim, P. Dean, M. Nikolić, R. Kliese, S. P. Khanna, M. Lachab, A. Valavanis, D. Indjin, Z. Ikonić, P. Harrison, E. Linfield, A. G. Davies, S. J. Wilson, A. D. Rakić, “Demonstration of a self-mixing displacement sensor based on terahertz quantum cascade lasers,” Appl. Phys. Lett. 99, 081108 (2011).
[CrossRef]

Dean, P.

Diehl, L.

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, F. Capasso, “Mode-locked pulses from mid-infrared quantum cascade lasers,” Opt. Express 17, 12929–12943 (2009).
[CrossRef] [PubMed]

N. Yu, L. Diehl, E. Cubukcu, D. Bour, S. Corzine, G. Höfler, A. K. Wójcik, K. B. Crozier, A. Belyanin, F. Capasso, “Coherent coupling of multiple transverse modes in quantum cascade lasers,” Phys. Rev. Lett. 102, 013901 (2009).
[CrossRef] [PubMed]

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, F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[CrossRef]

Edamura, T.

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

Ellison, B.

Elsäßer, W.

Erneux, T.

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

A. M. Levine, G. H. M. van Tartwijk, D. Lenstra, T. Erneux, “Diode lasers with optical feedback: Stability of the maximum gain mode,” Phys. Rev. A 52, 3436–3439 (1995).
[CrossRef]

Faist, J.

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, F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[CrossRef]

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, A. Y. Cho, “Quantum cascade laser,” Science 264, 553–556 (1994).
[CrossRef] [PubMed]

J. Faist, Quantum Cascade Lasers (Academic, 2013).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Fortran 77: The Art of Scientific Computing (Academic, 1992).

Fujita, K.

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

Gavrielides, A.

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

Gellie, P.

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

Gensty, T.

Giuliani, G.

R. P. Green, J. H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

J. Staden, T. Gensty, W. Elsäßer, G. Giuliani, C. Mann, “Measurements of the α factor of a distributed-feedback quantum cascade laser by an optical feedback self-mixing technique,” Opt. Lett. 31, 2574–2576 (2006).
[CrossRef]

Gkortsas, V. M.

Gmachl, C.

R. F. Curl, F. Capasso, C. Gmachl, A. A. Kosterev, B. Mc Manus, R. Lewicki, M. Pusharsky, G. Wysocki, F. K. Tittel, “Quantum cascade lasers in chemical physics,” Chem. Phys. Lett. 487, 1–18 (2010).
[CrossRef]

R. Paiella, R. Martini, F. Capasso, C. Gmachl, H. Y. Hwang, “High-frequency modulation without the relaxation oscillation resonance in quantum cascade lasers,” Appl. Phys. Lett. 79, 2526–2528 (2001).
[CrossRef]

Gordon, A.

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, F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[CrossRef]

Grant, P.

Green, R. P.

R. P. Green, J. H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

Haffouz, S.

Ham, D.

Harrison, P.

Helms, J.

J. Helms, K. Petermann, “A simple analytic expression for the stable operation range of laser diodes with optical feedback,” IEEE J. Quantum Electron. 26, 833–836 (1990).
[CrossRef]

Höfler, G.

N. Yu, L. Diehl, E. Cubukcu, D. Bour, S. Corzine, G. Höfler, A. K. Wójcik, K. B. Crozier, A. Belyanin, F. Capasso, “Coherent coupling of multiple transverse modes in quantum cascade lasers,” Phys. Rev. Lett. 102, 013901 (2009).
[CrossRef] [PubMed]

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, F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[CrossRef]

Hutchinson, A. L.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, A. Y. Cho, “Quantum cascade laser,” Science 264, 553–556 (1994).
[CrossRef] [PubMed]

Hwang, H. Y.

R. Paiella, R. Martini, F. Capasso, C. Gmachl, H. Y. Hwang, “High-frequency modulation without the relaxation oscillation resonance in quantum cascade lasers,” Appl. Phys. Lett. 79, 2526–2528 (2001).
[CrossRef]

Ikonic, Z.

Indjin, D.

Kan, H.

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

Kane, D. M.

D. M. Kane, K. A. Shore, Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor Diode Lasers (John Wiley, 2005).
[CrossRef]

Kärtner, F. X.

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, F. Capasso, “Mode-locked pulses from mid-infrared quantum cascade lasers,” Opt. Express 17, 12929–12943 (2009).
[CrossRef] [PubMed]

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, F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[CrossRef]

Khanna, S. P.

A. D. Rakić, T. Taimre, K. Bertling, Y. L. Lim, P. Dean, D. Indjin, Z. Ikonić, P. Harrison, A. Valavanis, S. P. Khanna, M. Lachab, S. J. Wilson, E. H. Linfield, A. G. Davies, “Swept-frequency feedback interferometry using terahertz frequency QCLs: a method for imaging and materials analysis,” Opt. Express 21, 22194–22205 (2013).
[CrossRef]

P. Dean, Y. L. Lim, A. Valavanis, R. Kliese, M. Nikolić, S. P. Khanna, M. Lachab, D. Indjin, Z. Ikonić, P. Harrison, A. D. Rakić, E. H. Linfield, A. G. Davies, “Terahertz imaging through self-mixing in a quantum cascade laser,” Opt. Lett. 36, 2587–2589 (2011).
[CrossRef] [PubMed]

Y. L. Lim, P. Dean, M. Nikolić, R. Kliese, S. P. Khanna, M. Lachab, A. Valavanis, D. Indjin, Z. Ikonić, P. Harrison, E. Linfield, A. G. Davies, S. J. Wilson, A. D. Rakić, “Demonstration of a self-mixing displacement sensor based on terahertz quantum cascade lasers,” Appl. Phys. Lett. 99, 081108 (2011).
[CrossRef]

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

Kliese, R.

Y. L. Lim, P. Dean, M. Nikolić, R. Kliese, S. P. Khanna, M. Lachab, A. Valavanis, D. Indjin, Z. Ikonić, P. Harrison, E. Linfield, A. G. Davies, S. J. Wilson, A. D. Rakić, “Demonstration of a self-mixing displacement sensor based on terahertz quantum cascade lasers,” Appl. Phys. Lett. 99, 081108 (2011).
[CrossRef]

P. Dean, Y. L. Lim, A. Valavanis, R. Kliese, M. Nikolić, S. P. Khanna, M. Lachab, D. Indjin, Z. Ikonić, P. Harrison, A. D. Rakić, E. H. Linfield, A. G. Davies, “Terahertz imaging through self-mixing in a quantum cascade laser,” Opt. Lett. 36, 2587–2589 (2011).
[CrossRef] [PubMed]

Kobayashi, K.

R. Lang, K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980).
[CrossRef]

Kosterev, A. A.

R. F. Curl, F. Capasso, C. Gmachl, A. A. Kosterev, B. Mc Manus, R. Lewicki, M. Pusharsky, G. Wysocki, F. K. Tittel, “Quantum cascade lasers in chemical physics,” Chem. Phys. Lett. 487, 1–18 (2010).
[CrossRef]

Kovanis, V.

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

Kuznetsova, L.

Lachab, M.

Lang, R.

R. Lang, K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980).
[CrossRef]

Lenstra, D.

A. M. Levine, G. H. M. van Tartwijk, D. Lenstra, T. Erneux, “Diode lasers with optical feedback: Stability of the maximum gain mode,” Phys. Rev. A 52, 3436–3439 (1995).
[CrossRef]

Levine, A. M.

A. M. Levine, G. H. M. van Tartwijk, D. Lenstra, T. Erneux, “Diode lasers with optical feedback: Stability of the maximum gain mode,” Phys. Rev. A 52, 3436–3439 (1995).
[CrossRef]

Lewicki, R.

R. F. Curl, F. Capasso, C. Gmachl, A. A. Kosterev, B. Mc Manus, R. Lewicki, M. Pusharsky, G. Wysocki, F. K. Tittel, “Quantum cascade lasers in chemical physics,” Chem. Phys. Lett. 487, 1–18 (2010).
[CrossRef]

Li, X.

Lim, Y. L.

Linfield, E.

Y. L. Lim, P. Dean, M. Nikolić, R. Kliese, S. P. Khanna, M. Lachab, A. Valavanis, D. Indjin, Z. Ikonić, P. Harrison, E. Linfield, A. G. Davies, S. J. Wilson, A. D. Rakić, “Demonstration of a self-mixing displacement sensor based on terahertz quantum cascade lasers,” Appl. Phys. Lett. 99, 081108 (2011).
[CrossRef]

Linfield, E. H.

Linfield, H.

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

Liu, H. C.

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, F. Capasso, “Mode-locked pulses from mid-infrared quantum cascade lasers,” Opt. Express 17, 12929–12943 (2009).
[CrossRef] [PubMed]

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, F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[CrossRef]

Mahler, L.

R. P. Green, J. H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

Maier, T.

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, F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[CrossRef]

Malara, P.

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

Mann, C.

Manquest, C.

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

Mansuripur, T. S.

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

Martini, R.

R. Paiella, R. Martini, F. Capasso, C. Gmachl, H. Y. Hwang, “High-frequency modulation without the relaxation oscillation resonance in quantum cascade lasers,” Appl. Phys. Lett. 79, 2526–2528 (2001).
[CrossRef]

Mc Manus, B.

R. F. Curl, F. Capasso, C. Gmachl, A. A. Kosterev, B. Mc Manus, R. Lewicki, M. Pusharsky, G. Wysocki, F. K. Tittel, “Quantum cascade lasers in chemical physics,” Chem. Phys. Lett. 487, 1–18 (2010).
[CrossRef]

Mezzapesa, F.

F. Mezzapesa, Internal CNR-IFN report (2013).

Mezzapesa, F. P.

F. P. Mezzapesa, L. L. Columbo, M. Brambilla, M. Dabbicco, S. Borri, M. S. Vitiello, H. E. Beere, D. A. Ritchie, G. Scamarcio, “Intrinsic stability of quantum cascade lasers against optical feedback,” Opt. Express 21, 13748–13757 (2013).
[CrossRef] [PubMed]

F. P. Mezzapesa, V. Spagnolo, A. Antonio, G. Scamarcio, “Detection of ultrafast laser ablation using quantum cascade laser-based sensing,” Appl. Phys. Lett. 101, 171101 (2012).
[CrossRef]

Nikolic, M.

Y. L. Lim, P. Dean, M. Nikolić, R. Kliese, S. P. Khanna, M. Lachab, A. Valavanis, D. Indjin, Z. Ikonić, P. Harrison, E. Linfield, A. G. Davies, S. J. Wilson, A. D. Rakić, “Demonstration of a self-mixing displacement sensor based on terahertz quantum cascade lasers,” Appl. Phys. Lett. 99, 081108 (2011).
[CrossRef]

P. Dean, Y. L. Lim, A. Valavanis, R. Kliese, M. Nikolić, S. P. Khanna, M. Lachab, D. Indjin, Z. Ikonić, P. Harrison, A. D. Rakić, E. H. Linfield, A. G. Davies, “Terahertz imaging through self-mixing in a quantum cascade laser,” Opt. Lett. 36, 2587–2589 (2011).
[CrossRef] [PubMed]

Paiella, R.

R. Paiella, R. Martini, F. Capasso, C. Gmachl, H. Y. Hwang, “High-frequency modulation without the relaxation oscillation resonance in quantum cascade lasers,” Appl. Phys. Lett. 79, 2526–2528 (2001).
[CrossRef]

Petermann, K.

J. Helms, K. Petermann, “A simple analytic expression for the stable operation range of laser diodes with optical feedback,” IEEE J. Quantum Electron. 26, 833–836 (1990).
[CrossRef]

Phillips, M. C.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Fortran 77: The Art of Scientific Computing (Academic, 1992).

Pusharsky, M.

R. F. Curl, F. Capasso, C. Gmachl, A. A. Kosterev, B. Mc Manus, R. Lewicki, M. Pusharsky, G. Wysocki, F. K. Tittel, “Quantum cascade lasers in chemical physics,” Chem. Phys. Lett. 487, 1–18 (2010).
[CrossRef]

Rakic, A. D.

Ravaro, M.

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

Ritchie, D. A.

F. P. Mezzapesa, L. L. Columbo, M. Brambilla, M. Dabbicco, S. Borri, M. S. Vitiello, H. E. Beere, D. A. Ritchie, G. Scamarcio, “Intrinsic stability of quantum cascade lasers against optical feedback,” Opt. Express 21, 13748–13757 (2013).
[CrossRef] [PubMed]

R. P. Green, J. H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

Santarelli, G.

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

Scamarcio, G.

Schneider, H

Schneider, H.

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, F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[CrossRef]

Shore, K. A.

D. M. Kane, K. A. Shore, Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor Diode Lasers (John Wiley, 2005).
[CrossRef]

Sirtori, C.

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

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, A. Y. Cho, “Quantum cascade laser,” Science 264, 553–556 (1994).
[CrossRef] [PubMed]

Sivco, D. L.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, A. Y. Cho, “Quantum cascade laser,” Science 264, 553–556 (1994).
[CrossRef] [PubMed]

Smith, K.

Song, C. Y.

Spagnolo, V.

F. P. Mezzapesa, V. Spagnolo, A. Antonio, G. Scamarcio, “Detection of ultrafast laser ablation using quantum cascade laser-based sensing,” Appl. Phys. Lett. 101, 171101 (2012).
[CrossRef]

Staden, J.

Taimre, T.

Taubman, S.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Fortran 77: The Art of Scientific Computing (Academic, 1992).

Tittel, F. K.

R. F. Curl, F. Capasso, C. Gmachl, A. A. Kosterev, B. Mc Manus, R. Lewicki, M. Pusharsky, G. Wysocki, F. K. Tittel, “Quantum cascade lasers in chemical physics,” Chem. Phys. Lett. 487, 1–18 (2010).
[CrossRef]

Tredicucci, A.

R. P. Green, J. H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

Troccoli, M.

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, F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[CrossRef]

Valavanis, A.

van Tartwijk, G. H. M.

A. M. Levine, G. H. M. van Tartwijk, D. Lenstra, T. Erneux, “Diode lasers with optical feedback: Stability of the maximum gain mode,” Phys. Rev. A 52, 3436–3439 (1995).
[CrossRef]

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Fortran 77: The Art of Scientific Computing (Academic, 1992).

Vitiello, M. S.

Wang, C. Y.

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, F. Capasso, “Mode-locked pulses from mid-infrared quantum cascade lasers,” Opt. Express 17, 12929–12943 (2009).
[CrossRef] [PubMed]

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, F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[CrossRef]

Wasilewski, Z. R.

Weidmann, D.

Wilson, S. J.

A. D. Rakić, T. Taimre, K. Bertling, Y. L. Lim, P. Dean, D. Indjin, Z. Ikonić, P. Harrison, A. Valavanis, S. P. Khanna, M. Lachab, S. J. Wilson, E. H. Linfield, A. G. Davies, “Swept-frequency feedback interferometry using terahertz frequency QCLs: a method for imaging and materials analysis,” Opt. Express 21, 22194–22205 (2013).
[CrossRef]

Y. L. Lim, P. Dean, M. Nikolić, R. Kliese, S. P. Khanna, M. Lachab, A. Valavanis, D. Indjin, Z. Ikonić, P. Harrison, E. Linfield, A. G. Davies, S. J. Wilson, A. D. Rakić, “Demonstration of a self-mixing displacement sensor based on terahertz quantum cascade lasers,” Appl. Phys. Lett. 99, 081108 (2011).
[CrossRef]

Wójcik, A. K.

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

N. Yu, L. Diehl, E. Cubukcu, D. Bour, S. Corzine, G. Höfler, A. K. Wójcik, K. B. Crozier, A. Belyanin, F. Capasso, “Coherent coupling of multiple transverse modes in quantum cascade lasers,” Phys. Rev. Lett. 102, 013901 (2009).
[CrossRef] [PubMed]

Wysocki, G.

R. F. Curl, F. Capasso, C. Gmachl, A. A. Kosterev, B. Mc Manus, R. Lewicki, M. Pusharsky, G. Wysocki, F. K. Tittel, “Quantum cascade lasers in chemical physics,” Chem. Phys. Lett. 487, 1–18 (2010).
[CrossRef]

Xu, J. H.

R. P. Green, J. H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

Yamanishi, M.

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

Yu, N.

N. Yu, L. Diehl, E. Cubukcu, D. Bour, S. Corzine, G. Höfler, A. K. Wójcik, K. B. Crozier, A. Belyanin, F. Capasso, “Coherent coupling of multiple transverse modes in quantum cascade lasers,” Phys. Rev. Lett. 102, 013901 (2009).
[CrossRef] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (5)

Y. L. Lim, P. Dean, M. Nikolić, R. Kliese, S. P. Khanna, M. Lachab, A. Valavanis, D. Indjin, Z. Ikonić, P. Harrison, E. Linfield, A. G. Davies, S. J. Wilson, A. D. Rakić, “Demonstration of a self-mixing displacement sensor based on terahertz quantum cascade lasers,” Appl. Phys. Lett. 99, 081108 (2011).
[CrossRef]

F. P. Mezzapesa, V. Spagnolo, A. Antonio, G. Scamarcio, “Detection of ultrafast laser ablation using quantum cascade laser-based sensing,” Appl. Phys. Lett. 101, 171101 (2012).
[CrossRef]

R. Paiella, R. Martini, F. Capasso, C. Gmachl, H. Y. Hwang, “High-frequency modulation without the relaxation oscillation resonance in quantum cascade lasers,” Appl. Phys. Lett. 79, 2526–2528 (2001).
[CrossRef]

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

R. P. Green, J. H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

Chem. Phys. Lett. (1)

R. F. Curl, F. Capasso, C. Gmachl, A. A. Kosterev, B. Mc Manus, R. Lewicki, M. Pusharsky, G. Wysocki, F. K. Tittel, “Quantum cascade lasers in chemical physics,” Chem. Phys. Lett. 487, 1–18 (2010).
[CrossRef]

IEEE J. Quantum Electron. (3)

J. Helms, K. Petermann, “A simple analytic expression for the stable operation range of laser diodes with optical feedback,” IEEE J. Quantum Electron. 26, 833–836 (1990).
[CrossRef]

R. Lang, K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980).
[CrossRef]

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

Nat. Photonics (1)

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

Opt. Express (5)

Opt. Lett. (3)

Phys. Rev. A (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, F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[CrossRef]

A. M. Levine, G. H. M. van Tartwijk, D. Lenstra, T. Erneux, “Diode lasers with optical feedback: Stability of the maximum gain mode,” Phys. Rev. A 52, 3436–3439 (1995).
[CrossRef]

Phys. Rev. E (1)

T. Erneux, V. Kovanis, A. Gavrielides, “Nonlinear dynamics of an injected quantum cascade laser,” Phys. Rev. E 88, 032907 (2013).
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N. Yu, L. Diehl, E. Cubukcu, D. Bour, S. Corzine, G. Höfler, A. K. Wójcik, K. B. Crozier, A. Belyanin, F. Capasso, “Coherent coupling of multiple transverse modes in quantum cascade lasers,” Phys. Rev. Lett. 102, 013901 (2009).
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[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Schematic layout of the self-mixing configuration. The QCL radiation is focused on the external target and re-injected into the cavity laser after a cavity round trip.

Fig. 2
Fig. 2

(Continuous wave solutions (a) in the (ωω0, |E|2) plane for k = 0.5 and three values of α; (b) in the (ωω0, k) plane for three different values of α. The dashed lines represents a set of adjacent ECMs. The other parameters are: Ip = 1.5, τ = 30.

Fig. 3
Fig. 3

Linear stability analysis of the CW solutions for Ip = 1.5, τ = 30.

Fig. 4
Fig. 4

Ip = 1.5. Numerically calculated values of αc against τ.

Fig. 5
Fig. 5

Ip = 1.5, τ = 30, α = 1.3. Maximum and minimum value of the intensity obtained by gradually increasing k during the system dynamical evolution. The corresponding values of k are reported on the right vertical axis.

Fig. 6
Fig. 6

Ip = 1.5, τ = 30, α = 3. (a), (b) Maximum and minimum value of the intensity and intensity power spectrum obtained by gradually increasing k during the system dynamical evolution. (c)–(e) Power spectrum and temporal variation of the intensity for k = 0.12 (c), k = 0.163 (d) and k = 0.2 (e). (f) Correlation plot for Δt = 1.2τ = 35 and three different values of k.

Fig. 7
Fig. 7

Ip = 1.5, τ = 60, α = 2. Power spectrum (a) and temporal variation (b) of the field intensity showing a regime regular multimode dynamics for k = 0.13.

Tables (1)

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Table 1 Physical parameters for a QCL in the LK model

Equations (22)

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d E ( t ) d t = 1 2 ( 1 + i α ) ( N ( t ) 1 ) E ( t ) + k τ p τ c E ( t τ ) e i ω 0 τ
d N ( t ) d t = γ ( I p N ( t ) ( 1 + | E ( t ) | 2 ) )
N s = 1 2 k τ p τ c cos ( ω F τ )
ω F = ω 0 k τ p τ c [ α cos ( ω F τ ) + sin ( ω F τ ) ]
| E s | 2 = I p 1 ( 2 k τ p / τ c ) cos ( ω F τ ) 1
G ( N ) = G t h + Δ N G N η ( N ) = η t h + Δ N η N
ω n ( N ) = n π c / L η ( N ) = ω 0 + α 2 ( N 1 )
ω F = ω n k τ p τ c sin ( ω F τ ) ,
Max ( | ω F ω 0 | ) k = ( α + 1 ) k τ p τ c .
δ E ( t ) = δ E s exp ( λ t ) exp [ i ( ω F ω 0 t ) ] , δ E E s
δ N ( t ) = δ N s exp ( λ t ) , δ N N s
λ 3 + a 2 ( λ ) λ 2 + a 1 ( λ ) λ + a 0 = 0
a 2 = 2 k τ p τ c ( 1 e λ τ ) cos ( ω F τ ) + γ I p N s
a 1 = [ k τ p τ c ( 1 e λ τ ) ] 2 + γ [ 2 k τ p I p N s τ c ( 1 e λ τ ) cos ( ω F τ ) + E s 2 N s ]
a 0 = γ { E s 2 N s k τ p τ c ( 1 e λ τ ) [ cos ( ω F τ ) α sin ( ω F τ ) ] + I p N s [ k τ p τ c ( 1 e λ τ ) ] 2 }
b 2 ( λ ) λ 2 + b 1 ( λ ) λ + b 0 = 0
b 1 = 2 k τ p τ c ( 1 e λ τ ) cos ( ω F τ ) + E s 2 N s 2 I p
b 0 = E s 2 N s 2 k τ p I p τ c ( 1 e λ τ ) [ cos ( ω F τ ) α sin ( ω F τ ) ] + [ k τ p τ c ( 1 e λ τ ) ] 2
Ω 2 + i Ω [ 2 C ( 1 e i Ω τ ) cos ( θ ) + A ] + C A ( 1 e i Ω τ ) [ cos ( θ ) α sin ( θ ) ] + C 2 ( 1 e i Ω τ ) 2
Ω τ = 2 n π + ε ; 1 e i Ω τ ε 2 2 + i ε
ε 2 [ 1 + 2 C ( cos ( θ ) + C 2 ) C A 2 ( cos ( θ ) α sin ( θ ) ) ] + 4 n π ε ( 1 + C cos ( θ ) ) + 4 n 2 π 2 = 0
ε 2 4 n π C cos ( θ ) + 2 ε A [ 1 + C ( cos ( θ ) α sin ( θ ) ) ] + 4 n π A = 0

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