Abstract

We investigate the dynamics of actively modulated mid-infrared quantum cascade lasers (QCLs) using space- and time-domain simulations of coupled density matrix and Maxwell equations with resonant tunneling current taken into account. We show that it is possible to achieve active mode locking and stable generation of picosecond pulses in high performance QCLs with a vertical laser transition and a short gain recovery time by bias modulation of a short section of a monolithic Fabry-Perot cavity. In fact, active mode locking in QCLs with a short gain recovery time turns out to be more robust to the variation of parameters as compared to previously studied lasers with a long gain recovery time. We investigate the effects of spatial hole burning and phase locking on the laser output.

© 2015 Optical Society of America

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References

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  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,” Nat. Photonics 5, 306–313 (2011).
    [Crossref]
  2. J. Freeman, J. Maysonnave, N. Jukam, P. Cavali, K. Maussang, H. Beere, D. Ritchie, J. Mangeney, S. Dhillon, and J. Tignon, “Direct intensity sampling of a modelocked terahertz quantum cascade laser,” Appl. Phys. Lett. 101, 181115 (2012).
    [Crossref]
  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. Liu, and F. Capasso, “Mode-locked pulses from mid-infrared quantum cascade lasers,” Opt. Express 17, 12929–12943 (2009).
    [Crossref] [PubMed]
  4. V.-M. Gkortsas, C. Wang, L. Kuznetsova, L. Diehl, A. Gordon, C. Jirauschek, M. A. Belkin, A. Belyanin, F. Capasso, and F. X. Kärtner, “Dynamics of actively mode-locked quantum cascade lasers,” Opt. Express 18, 13616–13630 (2010).
    [Crossref] [PubMed]
  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]
  6. 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,” Appl. Phys. Lett. 103, 231102 (2013).
    [Crossref]
  7. R. Terazzi, T. Gresch, A. Wittmann, and J. Faist, “Sequential resonant tunneling in quantum cascade lasers,” Phys. Rev. B 78, 155328 (2008).
    [Crossref]

2013 (1)

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,” Appl. Phys. Lett. 103, 231102 (2013).
[Crossref]

2012 (1)

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

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,” Nat. Photonics 5, 306–313 (2011).
[Crossref]

2010 (1)

2009 (1)

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. Terazzi, T. Gresch, A. Wittmann, and J. Faist, “Sequential resonant tunneling in quantum cascade lasers,” Phys. Rev. B 78, 155328 (2008).
[Crossref]

Barbieri, S.

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,” Nat. Photonics 5, 306–313 (2011).
[Crossref]

Beere, H.

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

Belkin, M. A.

Belyanin, A.

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,” Appl. Phys. Lett. 103, 231102 (2013).
[Crossref]

V.-M. Gkortsas, C. Wang, L. Kuznetsova, L. Diehl, A. Gordon, C. Jirauschek, M. A. Belkin, A. Belyanin, F. Capasso, and F. X. Kärtner, “Dynamics of actively mode-locked quantum cascade lasers,” Opt. Express 18, 13616–13630 (2010).
[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. Liu, and 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, and F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

Blanchard, R.

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,” Appl. Phys. Lett. 103, 231102 (2013).
[Crossref]

Bour, D.

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

Capasso, F.

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,” Appl. Phys. Lett. 103, 231102 (2013).
[Crossref]

V.-M. Gkortsas, C. Wang, L. Kuznetsova, L. Diehl, A. Gordon, C. Jirauschek, M. A. Belkin, A. Belyanin, F. Capasso, and F. X. Kärtner, “Dynamics of actively mode-locked quantum cascade lasers,” Opt. Express 18, 13616–13630 (2010).
[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. Liu, and 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, and F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

Cavali, P.

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

Corzine, S.

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

Davies, A. G.

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,” Nat. Photonics 5, 306–313 (2011).
[Crossref]

Dhillon, S.

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

Diehl, L.

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

R. Terazzi, T. Gresch, A. Wittmann, and J. Faist, “Sequential resonant tunneling in quantum cascade lasers,” Phys. Rev. B 78, 155328 (2008).
[Crossref]

Freeman, J.

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

Gellie, P.

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,” Nat. Photonics 5, 306–313 (2011).
[Crossref]

Gkortsas, V. M.

Gkortsas, V.-M.

Gordon, A.

V.-M. Gkortsas, C. Wang, L. Kuznetsova, L. Diehl, A. Gordon, C. Jirauschek, M. A. Belkin, A. Belyanin, F. Capasso, and F. X. Kärtner, “Dynamics of actively mode-locked quantum cascade lasers,” Opt. Express 18, 13616–13630 (2010).
[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, 053804 (2008).
[Crossref]

Grant, P.

Gresch, T.

R. Terazzi, T. Gresch, A. Wittmann, and J. Faist, “Sequential resonant tunneling in quantum cascade lasers,” Phys. Rev. B 78, 155328 (2008).
[Crossref]

Haffouz, S.

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, 053804 (2008).
[Crossref]

Jirauschek, C.

Jukam, N.

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

Kärtner, F. X.

Khanna, S. P.

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,” Nat. Photonics 5, 306–313 (2011).
[Crossref]

Kuznetsova, L.

Li, X.

Linfield, E. H.

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,” Nat. Photonics 5, 306–313 (2011).
[Crossref]

Liu, H.

Liu, H. C.

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]

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, 053804 (2008).
[Crossref]

Malara, P.

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,” Appl. Phys. Lett. 103, 231102 (2013).
[Crossref]

Mangeney, J.

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

Manquest, C.

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,” Nat. Photonics 5, 306–313 (2011).
[Crossref]

Mansuripur, T. S.

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,” Appl. Phys. Lett. 103, 231102 (2013).
[Crossref]

Maussang, K.

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

Maysonnave, J.

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

Ravaro, M.

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,” Nat. Photonics 5, 306–313 (2011).
[Crossref]

Ritchie, D.

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

Santarelli, G.

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,” Nat. Photonics 5, 306–313 (2011).
[Crossref]

Schneider, H.

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

Sirtori, C.

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,” Nat. Photonics 5, 306–313 (2011).
[Crossref]

Song, C. Y.

Terazzi, R.

R. Terazzi, T. Gresch, A. Wittmann, and J. Faist, “Sequential resonant tunneling in quantum cascade lasers,” Phys. Rev. B 78, 155328 (2008).
[Crossref]

Tignon, J.

J. Freeman, J. Maysonnave, N. Jukam, P. Cavali, K. Maussang, H. Beere, D. Ritchie, J. Mangeney, S. Dhillon, and J. Tignon, “Direct intensity sampling of a modelocked terahertz quantum cascade laser,” Appl. Phys. Lett. 101, 181115 (2012).
[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, and F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

Wang, C.

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. Liu, and 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, and 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.

Wittmann, A.

R. Terazzi, T. Gresch, A. Wittmann, and J. Faist, “Sequential resonant tunneling in quantum cascade lasers,” Phys. Rev. B 78, 155328 (2008).
[Crossref]

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,” Appl. Phys. Lett. 103, 231102 (2013).
[Crossref]

Appl. Phys. Lett. (2)

J. Freeman, J. Maysonnave, N. Jukam, P. Cavali, K. Maussang, H. Beere, D. Ritchie, J. Mangeney, S. Dhillon, and J. Tignon, “Direct intensity sampling of a modelocked terahertz quantum cascade laser,” Appl. Phys. Lett. 101, 181115 (2012).
[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,” Appl. Phys. Lett. 103, 231102 (2013).
[Crossref]

Nat. Photonics (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,” Nat. Photonics 5, 306–313 (2011).
[Crossref]

Opt. Express (2)

Phys. Rev. A (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, 053804 (2008).
[Crossref]

Phys. Rev. B (1)

R. Terazzi, T. Gresch, A. Wittmann, and J. Faist, “Sequential resonant tunneling in quantum cascade lasers,” Phys. Rev. B 78, 155328 (2008).
[Crossref]

Supplementary Material (1)

» Media 1: AVI (3589 KB)     

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Figures (10)

Fig. 1
Fig. 1 (a) A schematic of the active region model. The current J (thick horizontal line) is due to resonant tunneling between aligned k states of injector g and the upper laser subband u. The distance δ is the separation between the centroids of electron states in g and u, and Δ is the detuning between the bottoms of these two subbands. The bias electric field is defined to be zero when g and u are aligned. Dashed arrows denote the non-radiative transitions, mostly due to LO phonons. (b) A schematic of a two-section cavity with an RF modulation at the cavity round-trip time applied to a shorter section.
Fig. 2
Fig. 2 The spectrum of the modulated bias (a), injection current (b), and gain (c) at the point adjacent to the left facet. All parameters are taken at base values except for a higher modulation amplitude VMod,Amp = 0.9. Although the modulation of the bias is at a single frequency, the injection current and gain contain higher harmonics.
Fig. 3
Fig. 3 The output field amplitude over one roundtrip time taken close to the end of the simulation time for different values of the DC bias VDC in units of threshold bias for lasers with (a) Tul = 1 ps and (b) Tul = 50 ps. The DC biases in the modulated section and DC section are set to be equal.
Fig. 4
Fig. 4 Same as Fig. 3, but without population grating.
Fig. 5
Fig. 5 (a) The output field amplitude and (b) real part of the field at the left facet for the whole simulation range of 10,000 roundtrips and the base set of parameters. After the output is stable, there are still some small oscillations in the pulse amplitude. The real part of the field is not periodic in the modulation period, indicating that the phase is not locked.
Fig. 6
Fig. 6 (a) The amplitude of the output laser field on the left facet. (b) The injection current at the point adjacent to the left facet. (c) The difference between the gain and waveguide loss glw at the point adjacent to the left facet. The gain follows the injection current almost instantaneously, due to the short gain recovery time. The peak of the pulse has a delay with respect to the maximum of the gain. (d, e) A frame in the movie ( Media 1, size: 3.50 MB) of the evolution of field intensity (d) and glw (e) in the same laser.
Fig. 7
Fig. 7 (a) The output field amplitude and (b) real part of the field at the left facet over 1,000 roundtrips in a laser with a short gain recovery time for the base set of parameters but a slightly longer modulation period Tmod = 1.003, which matches the group roundtrip time of the pulse. Both the field amplitude and the real part of the field become strictly periodic after about 200 roundtrips.
Fig. 8
Fig. 8 The output field amplitude over one roundtrip time taken close to the end of the simulation time for different values of the modulation period Tmod measured in units of phase roundtrip time for lasers with (a) Tul = 1 ps and (b) Tul = 50 ps and the base set of parameters.
Fig. 9
Fig. 9 The output field amplitude over one roundtrip time taken close to the end of the simulation time for different values of the modulation amplitude VMod,Amp in Eq. (16), measured in units of threshold bias for lasers with (a) Tul = 1 ps and (b) Tul = 50 ps and the base set of parameters. No significant output is generated when VMod,Amp ≤ 0.1.
Fig. 10
Fig. 10 The output field amplitude over one roundtrip time taken close to the end of the simulation time for different lengths of the modulated section lmod measured in units of the total cavity length Lact for lasers with (a) Tul = 1 ps and (b) Tul = 50 ps and the base set of parameters.

Equations (17)

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t n g = n u T u g + n l T lg J + D 2 n g z 2 , t n u = J n u T u l n u T u g i d E ( ρ u l ρ u l * ) + D 2 n u z 2 , t n l = n u T u l n l T lg + i d E ( ρ u l ρ u l * ) + D 2 n l z 2 , t ρ u l = ( i ω + 1 T 2 ) ρ u l i d E ( n u n l ) , z 2 E n 2 c 2 t 2 E = Γ d ε 0 c 2 L p t 2 ( ρ u l + ρ u l * ) ,
J = e Ω 2 γ ( Δ 2 + γ 2 ) { θ ( Δ ) ( n g n u e | Δ | / k B T ) + θ ( Δ ) ( n g e | Δ | / k B T n u ) } .
E ( z , t ) = 1 2 [ E + ( z , t ) e i ( ω t k z ) + E + * ( z , t ) e i ( ω t k z ) ] + 1 2 [ E ( z , t ) e i ( ω t + k z ) + E + * ( z , t ) e i ( ω t + k z ) ] , ρ u l ( z , t ) = η + e i ( ω t k z ) + η e i ( ω t + k z ) , n g ( z , t ) = n g 0 + n g 2 e 2 i k z + n g 2 * e 2 i k z , n u ( z , t ) = n u 0 + n u 2 e 2 i k z + n u 2 * e 2 i k z , n l ( z , t ) = n l 0 + n l 2 e 2 i k z + n l 2 * e 2 i k z ,
J ( z , t ) = J 0 + J 2 e 2 i k z + J 2 * e 2 i k z .
t n g 0 = n u 0 T u g + n l 0 T lg J 0 ,
t n g 2 = n u 2 T u g + n l 2 T lg J 2 4 k 2 D n g 2 ,
t n u 0 = J 0 n u 0 T u l n u 0 T u g + i d 2 [ E + η + * + E η * c . c . ] ,
t n u 2 = J 2 n u 2 T u l n u 2 T u g 4 k 2 D n u 2 + i d 2 [ E + η * E * η + ] ,
t n l 0 = n u 0 T u l n l 0 T lg i d 2 [ E + η + * + E η * c . c . ] ,
t n l 2 = n u 2 T u l n l 2 T u g 4 k 2 D n l 2 i d 2 [ E + η * E * η + ] ,
t η + = i d 2 [ ( n u 0 n l 0 ) E + + ( n u 2 n l 2 ) E ] η + T 2 ,
t η = i d 2 [ ( n u 0 n l 0 ) E + ( n u 2 * n l 2 * ) E + ] η T 2 ,
( n c t + z ) E + = i Γ d ω n ε 0 c L p η + l w E + ,
( n c t z ) E = i Γ d ω n ε 0 c L p η l w E .
f ( ( n + 1 ) Δ t ) = f ( n Δ t ) + Δ t f ˙ ( n Δ t ) + 1 2 ( Δ t ) 2 f ¨ ( n Δ t ) .
V Mod = V Mod , DC + V Mod , Amp sin ( 2 π t / T mod ) .
R E ( τ ) = t 1 t 2 Re [ E ( t ) ] Re [ E ( t + τ ) ] d t / t 1 t 2 ( Re [ E ( t ) ] 2 ) d t ,

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