Abstract

We have obtained a closed-form expression for the threshold of Risken-Nummedal-Graham-Haken (RNGH) multimode instability in a Fabry-Pérot (FP) cavity quantum cascade laser (QCL). This simple analytical expression is a versatile tool that can easily be applied in practical situations which require analysis of QCL dynamic behavior and estimation of its RNGH multimode instability threshold. Our model for a FP cavity laser accounts for the carrier coherence grating and carrier population grating as well as their relaxation due to carrier diffusion. In the model, the RNGH instability threshold is analyzed using a second-order bi-orthogonal perturbation theory and we confirm our analytical solution by a comparison with the numerical simulations. In particular, the model predicts a low RNGH instability threshold in QCLs. This agrees very well with experimental data available in the literature.

© 2016 Optical Society of America

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References

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  3. 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(15), 12929–12943 (2009).
    [Crossref] [PubMed]
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    [Crossref]
  9. H. Knapp, H. Risken, and H. D. Vollmer, “Instability of laser cw oscillations in presence of a passive medium,” Appl. Phys. (Berl.) 15(3), 265–270 (1978).
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2016 (1)

N. Vukovic, J. Radovanovic, V. Milanovic, and D. L. Boiko, “Multimode RNGH instabilities of Fabry-Pérot cavity QCLs: impact of diffusion,” Opt. Quantum Electron. 48(4), 254 (2016).
[Crossref]

2014 (1)

L. Gil and G. L. Lippi, “Phase instability in semiconductor lasers,” Phys. Rev. Lett. 113(21), 213902 (2014).
[Crossref] [PubMed]

2013 (1)

M. Bugajski, K. Pierściński, D. Pierścińska, A. Szerling, and K. Kosiel, “Multimode instabilities in mid-infrared quantum cascade lasers,” Photonics Lett. Pol. 5(3), 85–87 (2013).
[Crossref]

2012 (2)

A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature 492(7428), 229–233 (2012).
[Crossref] [PubMed]

D. L. Boiko and P. P. Vasil’ev, “Superradiance dynamics in semiconductor laser diode structures,” Opt. Express 20(9), 9501–9515 (2012).
[Crossref] [PubMed]

2009 (1)

2008 (1)

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(5), 053804 (2008).
[Crossref]

2007 (2)

F. Prati and L. Columbo, “Long-wavelength instability in broad-area semiconductor lasers,” Phys. Rev. A 75(5), 053811 (2007).
[Crossref]

Ch. 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(3), 031802 (2007).
[Crossref]

2006 (1)

C. Serrat and C. Masoller, “Modeling spatial effects in multi-longitudinal-mode semiconductor lasers,” Phys. Rev. A 73(4), 043812 (2006).
[Crossref]

1993 (1)

N. V. Kravtsov, E. G. Lariontsev, and A. N. Shelaev, “Oscillation regimes of ring solid-state lasers and possibilities for their stabilization,” Laser Phys. 3, 21 (1993).

1978 (1)

H. Knapp, H. Risken, and H. D. Vollmer, “Instability of laser cw oscillations in presence of a passive medium,” Appl. Phys. (Berl.) 15(3), 265–270 (1978).
[Crossref]

1973 (1)

M. D. Crisp, “Adiabatic-following approximation,” Phys. Rev. A 8(4), 2128–2135 (1973).
[Crossref]

1969 (1)

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183(2), 457–485 (1969).
[Crossref]

1968 (2)

H. Risken and K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39(10), 4663 (1968).
[Crossref]

R. Graham and H. Haken, “Quantum theory of light propagation in a fluctuating laser-active medium,” Z. Phys. 213(5), 420–450 (1968).
[Crossref]

Belkin, M. A.

Belyanin, A.

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(15), 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, and F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77(5), 053804 (2008).
[Crossref]

Ch. 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(3), 031802 (2007).
[Crossref]

Blaser, S.

A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature 492(7428), 229–233 (2012).
[Crossref] [PubMed]

Boiko, D. L.

N. Vukovic, J. Radovanovic, V. Milanovic, and D. L. Boiko, “Multimode RNGH instabilities of Fabry-Pérot cavity QCLs: impact of diffusion,” Opt. Quantum Electron. 48(4), 254 (2016).
[Crossref]

D. L. Boiko and P. P. Vasil’ev, “Superradiance dynamics in semiconductor laser diode structures,” Opt. Express 20(9), 9501–9515 (2012).
[Crossref] [PubMed]

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(5), 053804 (2008).
[Crossref]

Ch. 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(3), 031802 (2007).
[Crossref]

Bugajski, M.

M. Bugajski, K. Pierściński, D. Pierścińska, A. Szerling, and K. Kosiel, “Multimode instabilities in mid-infrared quantum cascade lasers,” Photonics Lett. Pol. 5(3), 85–87 (2013).
[Crossref]

Capasso, F.

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(15), 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, and F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77(5), 053804 (2008).
[Crossref]

Ch. 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(3), 031802 (2007).
[Crossref]

Columbo, L.

F. Prati and L. Columbo, “Long-wavelength instability in broad-area semiconductor lasers,” Phys. Rev. A 75(5), 053811 (2007).
[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(5), 053804 (2008).
[Crossref]

Ch. 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(3), 031802 (2007).
[Crossref]

Crisp, M. D.

M. D. Crisp, “Adiabatic-following approximation,” Phys. Rev. A 8(4), 2128–2135 (1973).
[Crossref]

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, and F. Capasso, “Mode-locked pulses from mid-infrared quantum cascade lasers,” Opt. Express 17(15), 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, and F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77(5), 053804 (2008).
[Crossref]

Ch. 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(3), 031802 (2007).
[Crossref]

Faist, J.

A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature 492(7428), 229–233 (2012).
[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, and F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77(5), 053804 (2008).
[Crossref]

Ch. 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(3), 031802 (2007).
[Crossref]

Gil, L.

L. Gil and G. L. Lippi, “Phase instability in semiconductor lasers,” Phys. Rev. Lett. 113(21), 213902 (2014).
[Crossref] [PubMed]

Gkortsas, V. M.

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, and F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77(5), 053804 (2008).
[Crossref]

Ch. 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(3), 031802 (2007).
[Crossref]

Graham, R.

R. Graham and H. Haken, “Quantum theory of light propagation in a fluctuating laser-active medium,” Z. Phys. 213(5), 420–450 (1968).
[Crossref]

Grant, P.

Haffouz, S.

Hahn, E. L.

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183(2), 457–485 (1969).
[Crossref]

Haken, H.

R. Graham and H. Haken, “Quantum theory of light propagation in a fluctuating laser-active medium,” Z. Phys. 213(5), 420–450 (1968).
[Crossref]

Ham, D.

Höfler, G.

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(5), 053804 (2008).
[Crossref]

Ch. 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(3), 031802 (2007).
[Crossref]

Hugi, A.

A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature 492(7428), 229–233 (2012).
[Crossref] [PubMed]

Jirauschek, C.

Ch. 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(3), 031802 (2007).
[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, and F. Capasso, “Mode-locked pulses from mid-infrared quantum cascade lasers,” Opt. Express 17(15), 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, and F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77(5), 053804 (2008).
[Crossref]

Ch. 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(3), 031802 (2007).
[Crossref]

Knapp, H.

H. Knapp, H. Risken, and H. D. Vollmer, “Instability of laser cw oscillations in presence of a passive medium,” Appl. Phys. (Berl.) 15(3), 265–270 (1978).
[Crossref]

Kosiel, K.

M. Bugajski, K. Pierściński, D. Pierścińska, A. Szerling, and K. Kosiel, “Multimode instabilities in mid-infrared quantum cascade lasers,” Photonics Lett. Pol. 5(3), 85–87 (2013).
[Crossref]

Kravtsov, N. V.

N. V. Kravtsov, E. G. Lariontsev, and A. N. Shelaev, “Oscillation regimes of ring solid-state lasers and possibilities for their stabilization,” Laser Phys. 3, 21 (1993).

Kuznetsova, L.

Lariontsev, E. G.

N. V. Kravtsov, E. G. Lariontsev, and A. N. Shelaev, “Oscillation regimes of ring solid-state lasers and possibilities for their stabilization,” Laser Phys. 3, 21 (1993).

Li, X.

Lippi, G. L.

L. Gil and G. L. Lippi, “Phase instability in semiconductor lasers,” Phys. Rev. Lett. 113(21), 213902 (2014).
[Crossref] [PubMed]

Liu, H. C.

A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature 492(7428), 229–233 (2012).
[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, and F. Capasso, “Mode-locked pulses from mid-infrared quantum cascade lasers,” Opt. Express 17(15), 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, and F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77(5), 053804 (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, and F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77(5), 053804 (2008).
[Crossref]

Masoller, C.

C. Serrat and C. Masoller, “Modeling spatial effects in multi-longitudinal-mode semiconductor lasers,” Phys. Rev. A 73(4), 043812 (2006).
[Crossref]

McCall, S. L.

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183(2), 457–485 (1969).
[Crossref]

Milanovic, V.

N. Vukovic, J. Radovanovic, V. Milanovic, and D. L. Boiko, “Multimode RNGH instabilities of Fabry-Pérot cavity QCLs: impact of diffusion,” Opt. Quantum Electron. 48(4), 254 (2016).
[Crossref]

Nummedal, K.

H. Risken and K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39(10), 4663 (1968).
[Crossref]

Pierscinska, D.

M. Bugajski, K. Pierściński, D. Pierścińska, A. Szerling, and K. Kosiel, “Multimode instabilities in mid-infrared quantum cascade lasers,” Photonics Lett. Pol. 5(3), 85–87 (2013).
[Crossref]

Pierscinski, K.

M. Bugajski, K. Pierściński, D. Pierścińska, A. Szerling, and K. Kosiel, “Multimode instabilities in mid-infrared quantum cascade lasers,” Photonics Lett. Pol. 5(3), 85–87 (2013).
[Crossref]

Prati, F.

F. Prati and L. Columbo, “Long-wavelength instability in broad-area semiconductor lasers,” Phys. Rev. A 75(5), 053811 (2007).
[Crossref]

Radovanovic, J.

N. Vukovic, J. Radovanovic, V. Milanovic, and D. L. Boiko, “Multimode RNGH instabilities of Fabry-Pérot cavity QCLs: impact of diffusion,” Opt. Quantum Electron. 48(4), 254 (2016).
[Crossref]

Risken, H.

H. Knapp, H. Risken, and H. D. Vollmer, “Instability of laser cw oscillations in presence of a passive medium,” Appl. Phys. (Berl.) 15(3), 265–270 (1978).
[Crossref]

H. Risken and K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39(10), 4663 (1968).
[Crossref]

Schneider, H.

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M. Bugajski, K. Pierściński, D. Pierścińska, A. Szerling, and K. Kosiel, “Multimode instabilities in mid-infrared quantum cascade lasers,” Photonics Lett. Pol. 5(3), 85–87 (2013).
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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(5), 053804 (2008).
[Crossref]

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A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature 492(7428), 229–233 (2012).
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[Crossref]

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N. Vukovic, J. Radovanovic, V. Milanovic, and D. L. Boiko, “Multimode RNGH instabilities of Fabry-Pérot cavity QCLs: impact of diffusion,” Opt. Quantum Electron. 48(4), 254 (2016).
[Crossref]

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, and F. Capasso, “Mode-locked pulses from mid-infrared quantum cascade lasers,” Opt. Express 17(15), 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, and F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77(5), 053804 (2008).
[Crossref]

Wang, Ch. Y.

Ch. 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(3), 031802 (2007).
[Crossref]

Wasilewski, Z. R.

Appl. Phys. (Berl.) (1)

H. Knapp, H. Risken, and H. D. Vollmer, “Instability of laser cw oscillations in presence of a passive medium,” Appl. Phys. (Berl.) 15(3), 265–270 (1978).
[Crossref]

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[Crossref]

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N. V. Kravtsov, E. G. Lariontsev, and A. N. Shelaev, “Oscillation regimes of ring solid-state lasers and possibilities for their stabilization,” Laser Phys. 3, 21 (1993).

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(7428), 229–233 (2012).
[Crossref] [PubMed]

Opt. Express (2)

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N. Vukovic, J. Radovanovic, V. Milanovic, and D. L. Boiko, “Multimode RNGH instabilities of Fabry-Pérot cavity QCLs: impact of diffusion,” Opt. Quantum Electron. 48(4), 254 (2016).
[Crossref]

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M. Bugajski, K. Pierściński, D. Pierścińska, A. Szerling, and K. Kosiel, “Multimode instabilities in mid-infrared quantum cascade lasers,” Photonics Lett. Pol. 5(3), 85–87 (2013).
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[Crossref]

C. Serrat and C. Masoller, “Modeling spatial effects in multi-longitudinal-mode semiconductor lasers,” Phys. Rev. A 73(4), 043812 (2006).
[Crossref]

Ch. 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(3), 031802 (2007).
[Crossref]

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L. Lugiato, F. Prati, and M. Brambilla, Nonlinear Optical Systems (Cambridge University, 2015), Chap. 22.

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D. L. Boiko, “Paraxial Hamiltonian for photons in two-dimensional photonic crystal microstructures,” http://arxiv.org/abs/0710.5287

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Figures (7)

Fig. 1
Fig. 1

(a) Threshold parameter ν0(p) plotted as a function of the pump normalized to the lasing threshold for two different sets of QCL parameters (solid black and blue curves) and for a laser diode (green curve). Dashed curves depict ν0(p) behavior without the coherence grating (T2,g→0, see Eq. (5)′) in Section 3.2). (b) Their corresponding Rabi frequencies are calculated from Eq. (4) (solid curves) and with ν0(p) = 1 (dashed curves). The cavity length is 4 mm in all considered examples. All other parameters are listed in Table 1.

Fig. 2
Fig. 2

(a) Round-trip gain spectra responsible for multimode Risken-Numedal-Graham-Haken instability in a short-cavity QCL (L = 100 µm) are plotted for different pump rates. The zero offset frequency corresponds to the initially lasing cavity mode (the mode with index n) when the laser still operates in the single-mode regime. The cavity mode separation is c/2ngL = 0.45 THz (b) The probability of occurrence for RNGH instability is plotted as a function of the pump normalized to lasing threshold p. Each data point is based on 80 realizations and therefore the uncertainty is of ± 0.056 (vertical bars). Dashed vertical line indicates the second threshold obtained from Lyapunov stability analysis. Example of numerical simulations: (c) Output power waveform, (d) corresponding RF power spectra and (e) P-N attractor for one of the realizations at p = 3. The parameters used in these simulations are listed in the column labeled “QCL 1” in Table 1.

Fig. 3
Fig. 3

Real part of eigenvalues of matrices M and M(0) from Eq. (2) for QCL with the cavity length L = 4 mm at pump rate p = 1.5 times the lasing threshold. The red dashed curves show the real part of the eigenvalues of matrix M(0) and the black solid curves represent real part of eigenvalues of matrix M. The parameters used in simulations are listed in the column “QCL 1” in Table 1.

Fig. 4
Fig. 4

The spectral behavior of the round-trip gain coefficient for multimode instabilities in QCL with the cavity length L = 4 mm (The cavity mode separation c/2Lng = 11 GHz) at different pump rates p in the range from 1 to 2.5 times above the lasing threshold. The red curves show the gain spectra obtained from the analytical expression (9) and the black curves are obtained by solving numerically the eigenproblem of the matrix (2). The parameters used in simulations are listed in the column “QCL 1” in Table 1.

Fig. 5
Fig. 5

Peak gain frequency of RNGH instability calculated numerically from our linear stability matrix (black curve) and from our analytic expression Eq. (10) (red curve) is plotted vs. pump normalized to lasing threshold p. It is shown in comparison with the behavior of Rabi oscillations ΩRabi/2π (blue curve). According to Eq. (6), the intersection between horizontal dotted line at c/2Lng and the peak gain frequency defines the value of RNGH threshold (vertical dotted line) in case of the 100 µm long cavity.

Fig. 6
Fig. 6

RNGH instability threshold (represented as a relative pump excess above the lasing threshold pth2-1) vs. cavity length. We compare pth2 calculated from our Eq. (14) (curves) and calculated by numerical solving of the eigenproblem of the matrix (2) (squares and triangles). The parameters for QCL 1 and QCL 2 are listed in Table 1.

Fig. 7
Fig. 7

Using parameters of QCL2 from Table 1, we plot squares of the following frequencies as a function of the pump rate: Ω max frequency for multimode RNGH instability from Eq. (11) obtained for T 2,g 0 (solid blue curve) and the corresponding Rabi frequency Ω Rabi (4) (dashed blue curve), the frequency Ω max (SHB) for multimode instability occurring without coherence grating (when T 2,g 0 , solid red curve) and its corresponding Rabi frequency Ω Rabi (SHB) (dashed red curve) as well as the frequency Ω SHB [ 7 ] for this instability calculated in [7] (green curve).

Tables (1)

Tables Icon

Table 1 Dynamic model parameters for QCLs and QW LDs considered in this paper.

Equations (39)

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det( M 9×9 (Ω)ΛI)=0
M = M (0) + M (1) =[ 1 T 2,eff (2 ν 0 1) 2 T 2,eff 0 0 c n g l 0 c 2 n g l 0 iΩ 0 0 0 0 1 T g 2E 0 0 E 2 1 T 2,g ]+[ 0 0 E 2 0 0 0 0 0 2E E[ 1+( ν 0 1) T 2,g T 2,eff ] 0 0 0 ( ν 0 1) 2 T 2,eff 0 0 ]
E= p ν 0 (p) 2 T 1 T 2,eff ,
Ω Rabi = 2 E= p ν 0 T 1 T 2,eff
ν 0 (p)= 1 2 ( p+1+ T 2,eff T 2,g + 2 T 1 T 2,eff T g T 2,g ) 1 4 ( p+1+ T 2,eff T 2,g + 2 T 1 T 2,eff T g T 2,g ) 2 p( 1+ T 2,eff T 2,g ) 2 T 1 T 2,eff T g T 2,g
| Ω max (th2) |/2π=c/2L n g
Λ max (0) = 1 2 T 2,eff 1 4τ i Ω 2 + 1 2 ( 1 T 2,eff + 1 2τ +iΩ ) 2 4iΩ T 2,eff +4 ν 0 1 T 2,eff τ
Λ i = Λ i (0) + U i (0) | M (1) | V i (0) U i (0) | V i (0) + m=1, mi 4 U m (0) | M (1) | V i (0) U i (0) | M (1) | V m (0) ( Λ i (0) Λ m (0) ) U m (0) | V m (0) U i (0) | V i (0) ,
Re( Λ max )= 1 2τ + C 0 (p) Ω 2 +1/ T 2,eff 2 + C 1 (p) Ω 2 +A (p) 2 + C 2 (p) Ω 2 +1/ T 2,eff 2 + C 3 (p) ( Ω 2 +1/ T 2,eff 2 ) 2
Ω max 2 T 2,eff 2 C 1 / C 0 A 2 .
Ω max 2 Ω Rabi 2 1 2 T 2,g T 2,eff ( 1+ 2 Ω Rabi 2 T g T 2,g ) 1 T g 2
pν(p)= T 1 T 2,eff T g T 2,g [ 1+2 T 2,g T 2,eff T g 2 (1+ T g 2 Ω max 2 ) 2 1 ]
p min =1+ T 2,eff T 2,g +( T 1 T 2,eff T g T 2,g T g 2 T 2,g 2 )[ 1+2 T 2,g T 2,eff T g 2 1 ]1+ T 2,eff 2 T g 2 [ 1 2 + T 1 T g ( 1 T 2,g T 2,eff 2 T g 2 ) ]
p th2 =1+ θ th2 ( 1+ [ 2 T 1 T g + T 2,g T 2,eff θ th2 ] 1 ), θ th2 = T 1 T 2,eff T g T 2,g [ 1+2 T 2,g T 2,eff T g 2 ( 1+ c 2 π 2 T g 2 L 2 n g 2 ) 2 1 ]
Re( Λ max ) | Ω= Ω max = 1 2τ ( 1 3 2 Ω Rabi 2 T 2,eff 2 2 )+ 2 ν 0 1 2τ [ 1 Ω max 2 T 2,eff 2 T 2,eff 2 T g 2 ] 2 .
p min 1+ T 2,eff 2 T g 2 [ 1 2 + T 1 T g ( 1 T 2,g T 2,eff 2 T g 2 ) ].
Ω SHB [ 7 ] = 1 T 1 p1 3 T 1 T 2
ν 0 (SHB) =1+ T g (p1)/(2 T 1 + T g ),
( Ω max (SHB) ) 2 ( Ω Rabi (SHB) ) 2 1 2 ν 0 (SHB) (2 ν 0 (SHB) 1)( ν 0 (SHB) 1) 1 T g 2 , Ω Rabi (SHB) = 2(p1) (2 T 1 + T g ) T 2,eff .
Re( Λ max ) | Ω= Ω max 3 8τ Ω Rabi 2 T 2,eff 2 1 τ Ω max 2 T 2,eff 2 ,
Re( Λ max (RNGH) ) | Ω= Ω max (RNGH) 3 2τ Ω Rabi 2 T 2 2 1 τ ( Ω max (RNGH) ) 2 T 2 2 ,
Ω max (RNGH) Ω Rabi 2 4 , Ω Rabi = p1 T 1 T 2 .
for the field amplitudes: n g c t E ± = z E ± i NμΓω c n g ε 0 η ± 1 2 l 0 E ± ,
for the medium polarization: t η ± = iμ 2 ( Δ 0 E ± + Δ 2 E ) η ± T 2 k 2 D η ± ,
for the coherence grating amplitude: t η ±± = iμ 2 E ± Δ 2 η ±± T 2 9D k 2 η ±± ,
for the carrier population density: t Δ 0 = Δ pump Δ 0 T 1 + iμ ( E + * η + + E * η c.c ),
and population grating: t Δ 2 ± = iμ ( E ± * η E η ± * E ± η ±± * + E * η ) Δ 2 ± T 1 4 k 2 D Δ 2 ± .
C 0 = ρ τ T 2,eff
C 1 = 1 τ E A 2 T 2,eff 2 ( AEρ+ Aα T 2,eff + 2Eρ T 2,eff + A 2 α )+ 2 τ E 2 ρ(A+ T 2,eff 1 ) ( A 2 T 2,eff 2 ) 2 T 2,eff 2 ,
C 2 = 1 τ E A 2 T 2,eff 2 ( AEρ+ Aα T 2,eff + 2Eρ T 2,eff + α T 2,eff 2 ) 2 τ E 2 ρ(A+ T 2,eff 1 ) ( A 2 T 2,eff 2 ) 2 T 2,eff 2
C 3 = 2 τ E 2 ρ(A+ T 2,eff 1 ) ( A 2 T 2,eff 2 ) T 2,eff 2
ρ= 2 ν 0 1 2 T 2,eff ,α= E 2 [ 1+( ν 0 1) T 2,g T 2,eff ],A= 1 2 T g + 1 2 T 2,g 1 2 ( 1 T 2,g 1 T g ) 2 4 E 2
C 1 1 2τ T g T 2,g E 2 ( T g T 2,eff ) 2 [ ν 0 ( 1+2 T 2,eff T 2,g )1 ]( 1+ E 2 T g 3 T 2,g ( T g T 2,g )( T g T 2,eff ) )
C 2 1 2τ T g 2 T 2,g E 2 T 2,eff ( T g T 2,eff ) 2 [ ν 0 ( 1+ T 2,eff T g )1+ T 2,eff T 2,g ]
C 3 1 τ E 2 T g (2 ν 0 1) T 2,eff 2 ( T g T 2,eff ) ,A 1 T g + T g T 2,g E 2 T g T 2,g 1 T g + 1 2 T 2,g Ω Rabi 2
C 1 C 0 T 2,g 2 T 2,eff 2 Ω Rabi 4 1 4 ν 0 ( 1+2 T 2,eff T 2,g 1 )1 2 ν 0 1 ( 1+ 2 Ω Rabi 2 T g T 2,g ).
( 2θp+ϕ 2 ) 2 = ( p+ϕ 2 ) 2 p ( ϕ 2 T 1 T 2,eff T g T 2,g ) 2 2 T 1 T 2,eff T g T 2,g
φ=1+ T 2,eff T 2,g + 2 T 1 T 2,eff T g T 2,g
p=1+θ( 1+ [ 2 T 1 T g + T 2,g T 2,eff θ ] 1 )

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