Abstract

Frequency comb formation in quantum cascade lasers is studied theoretically using a Maxwell-Bloch formalism based on a modal decomposition, where dispersion is considered. In the mid-infrared, comb formation persists in the presence of weak cavity dispersion (500 fs2 mm−1) but disappears when much larger values are used (30’000 fs2 mm−1). Active modulation at the round-trip frequency is found to induce mode-locking in THz devices, where the upper state lifetime is in the tens of picoseconds. Our results show that mode-locking based on four-wave mixing in broadband gain, low dispersion cavities is the most promising way of achieving broadband quantum cascade laser frequency combs.

© 2015 Optical Society of America

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References

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  1. H. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1173–1185 (2000).
    [Crossref]
  2. J. Faist, F. Capasso, D. Sivco, C. Sirtori, A. Hutchinson, and A. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
    [Crossref] [PubMed]
  3. M. C. Tatham, J. F. Ryan, and C. T. Foxon, “Time-resolved Raman measurements of intersubband relaxation in GaAs quantum qells,” Phys. Rev. Lett. 63(15), 1637–1640 (1989).
    [Crossref] [PubMed]
  4. S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nat. Photonics 4(9), 636–640 (2010).
    [Crossref]
  5. 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 (vol 5, pg 306, 2011),” Nat. Photonics 5(6), 378 (2011).
  6. C. Y. Wang, L. Kuznetsova, V. M. Gkortsas, L. Diehl, F. X. Kaertner, 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]
  7. D. J. Kuizenga and A. Siegman, “FM and AM mode locking of homogeneous Laser-Part I: Theory,” IEEE J. Quantum Electron. 6(11), 694 (1970).
    [Crossref]
  8. A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature (London) 492(7428), 229–233 (2012).
    [Crossref]
  9. G. Villares, A. Hugi, S. Blaser, and J. Faist, “Dual-comb spectroscopy based on quantum-cascade-laser frequency combs,” Nat. Commun. 5, 1–9 (2014).
    [Crossref]
  10. P. Friedli, H. Sigg, B. Hinkov, A. Hugi, S. Riedi, M. Beck, and J. Faist, “Four-wave mixing in a quantum cascade laser amplifier,” Appl. Phys. Lett. 102(22), 222104 (2013).
    [Crossref]
  11. W. E. Lamb, “Theory of an optical maser,” Phys. Rev. 134(6A), A1429–A1450 (1964).
    [Crossref]
  12. J. B. Khurgin, Y. Dikmelik, A. Hugi, and J. Faist, “Coherent frequency combs produced by self frequency modulation in quantum cascade lasers,” Appl. Phys. Lett. 104(8), 081118 (2014).
    [Crossref]
  13. J. Faist, Quantum Cascade Lasers, 1st ed. (Oxford University, 2013).
    [Crossref]
  14. H. Choi, L. Diehl, Z.-K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. Norris, “Time-resolved investigations of electronic transport dynamics in quantum cascade lasers based on diagonal lasing transition,” IEEE J. Quantum Electron. 45(4), 307–321 (2009).
    [Crossref]
  15. A. Gordon, C. Y. Wang, L. Diehl, F. X. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, 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]
  16. V. M. Gkortsas, C. Wang, L. Kuznetsova, L. Diehl, A. Gordon, C. Jirauschek, M. A. Belkin, A. Belyanin, F. Capasso, and F. X. Kartner, “Dynamics of actively mode-locked quantum cascade lasers,” Opt. Express 18(13), 13616–13630 (2010).
    [Crossref] [PubMed]
  17. M. R. St-Jean, M. I. Amanti, A. Bernard, A. Calvar, A. Bismuto, E. Gini, M. Beck, J. Faist, H. C. Liu, and C. Sirtori, “Injection locking of mid-infrared quantum cascade laser at 14 GHz, by direct microwave modulation,” Laser Photon. Rev. 8(3), 443–449 (2014).
    [Crossref]
  18. R. Terazzi and J. Faist, “A density matrix model of transport and radiation in quantum cascade lasers,” New Journal of Physics 12, 033045 (2010).
    [Crossref]
  19. V V Kozlov, “Self-induced transparency soliton laser via coherent mode locking,” Phys. Rev. A 56(2), 1607– 1612 (1997).
    [Crossref]
  20. C. Menyuk and M. Talukder, “Self-induced transparency modelocking of quantum cascade lasers,” Phys. Rev. Lett. 102(2), 023903 (2009).
    [Crossref] [PubMed]

2014 (3)

G. Villares, A. Hugi, S. Blaser, and J. Faist, “Dual-comb spectroscopy based on quantum-cascade-laser frequency combs,” Nat. Commun. 5, 1–9 (2014).
[Crossref]

J. B. Khurgin, Y. Dikmelik, A. Hugi, and J. Faist, “Coherent frequency combs produced by self frequency modulation in quantum cascade lasers,” Appl. Phys. Lett. 104(8), 081118 (2014).
[Crossref]

M. R. St-Jean, M. I. Amanti, A. Bernard, A. Calvar, A. Bismuto, E. Gini, M. Beck, J. Faist, H. C. Liu, and C. Sirtori, “Injection locking of mid-infrared quantum cascade laser at 14 GHz, by direct microwave modulation,” Laser Photon. Rev. 8(3), 443–449 (2014).
[Crossref]

2013 (1)

P. Friedli, H. Sigg, B. Hinkov, A. Hugi, S. Riedi, M. Beck, and J. Faist, “Four-wave mixing in a quantum cascade laser amplifier,” Appl. Phys. Lett. 102(22), 222104 (2013).
[Crossref]

2012 (1)

A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature (London) 492(7428), 229–233 (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 (vol 5, pg 306, 2011),” Nat. Photonics 5(6), 378 (2011).

2010 (3)

S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nat. Photonics 4(9), 636–640 (2010).
[Crossref]

R. Terazzi and J. Faist, “A density matrix model of transport and radiation in quantum cascade lasers,” New Journal of Physics 12, 033045 (2010).
[Crossref]

V. M. Gkortsas, C. Wang, L. Kuznetsova, L. Diehl, A. Gordon, C. Jirauschek, M. A. Belkin, A. Belyanin, F. Capasso, and F. X. Kartner, “Dynamics of actively mode-locked quantum cascade lasers,” Opt. Express 18(13), 13616–13630 (2010).
[Crossref] [PubMed]

2009 (3)

C. Menyuk and M. Talukder, “Self-induced transparency modelocking of quantum cascade lasers,” Phys. Rev. Lett. 102(2), 023903 (2009).
[Crossref] [PubMed]

C. Y. Wang, L. Kuznetsova, V. M. Gkortsas, L. Diehl, F. X. Kaertner, 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]

H. Choi, L. Diehl, Z.-K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. Norris, “Time-resolved investigations of electronic transport dynamics in quantum cascade lasers based on diagonal lasing transition,” IEEE J. Quantum Electron. 45(4), 307–321 (2009).
[Crossref]

2008 (1)

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, 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]

2000 (1)

H. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1173–1185 (2000).
[Crossref]

1997 (1)

V V Kozlov, “Self-induced transparency soliton laser via coherent mode locking,” Phys. Rev. A 56(2), 1607– 1612 (1997).
[Crossref]

1994 (1)

J. Faist, F. Capasso, D. Sivco, C. Sirtori, A. Hutchinson, and A. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[Crossref] [PubMed]

1989 (1)

M. C. Tatham, J. F. Ryan, and C. T. Foxon, “Time-resolved Raman measurements of intersubband relaxation in GaAs quantum qells,” Phys. Rev. Lett. 63(15), 1637–1640 (1989).
[Crossref] [PubMed]

1970 (1)

D. J. Kuizenga and A. Siegman, “FM and AM mode locking of homogeneous Laser-Part I: Theory,” IEEE J. Quantum Electron. 6(11), 694 (1970).
[Crossref]

1964 (1)

W. E. Lamb, “Theory of an optical maser,” Phys. Rev. 134(6A), A1429–A1450 (1964).
[Crossref]

Amanti, M. I.

M. R. St-Jean, M. I. Amanti, A. Bernard, A. Calvar, A. Bismuto, E. Gini, M. Beck, J. Faist, H. C. Liu, and C. Sirtori, “Injection locking of mid-infrared quantum cascade laser at 14 GHz, by direct microwave modulation,” Laser Photon. Rev. 8(3), 443–449 (2014).
[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 (vol 5, pg 306, 2011),” Nat. Photonics 5(6), 378 (2011).

S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nat. Photonics 4(9), 636–640 (2010).
[Crossref]

Beck, M.

M. R. St-Jean, M. I. Amanti, A. Bernard, A. Calvar, A. Bismuto, E. Gini, M. Beck, J. Faist, H. C. Liu, and C. Sirtori, “Injection locking of mid-infrared quantum cascade laser at 14 GHz, by direct microwave modulation,” Laser Photon. Rev. 8(3), 443–449 (2014).
[Crossref]

P. Friedli, H. Sigg, B. Hinkov, A. Hugi, S. Riedi, M. Beck, and J. Faist, “Four-wave mixing in a quantum cascade laser amplifier,” Appl. Phys. Lett. 102(22), 222104 (2013).
[Crossref]

Beere, H.

S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nat. Photonics 4(9), 636–640 (2010).
[Crossref]

Belkin, M. A.

Belyanin, A.

Bernard, A.

M. R. St-Jean, M. I. Amanti, A. Bernard, A. Calvar, A. Bismuto, E. Gini, M. Beck, J. Faist, H. C. Liu, and C. Sirtori, “Injection locking of mid-infrared quantum cascade laser at 14 GHz, by direct microwave modulation,” Laser Photon. Rev. 8(3), 443–449 (2014).
[Crossref]

Bismuto, A.

M. R. St-Jean, M. I. Amanti, A. Bernard, A. Calvar, A. Bismuto, E. Gini, M. Beck, J. Faist, H. C. Liu, and C. Sirtori, “Injection locking of mid-infrared quantum cascade laser at 14 GHz, by direct microwave modulation,” Laser Photon. Rev. 8(3), 443–449 (2014).
[Crossref]

Blaser, S.

G. Villares, A. Hugi, S. Blaser, and J. Faist, “Dual-comb spectroscopy based on quantum-cascade-laser frequency combs,” Nat. Commun. 5, 1–9 (2014).
[Crossref]

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

Bour, D.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, 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]

Calvar, A.

M. R. St-Jean, M. I. Amanti, A. Bernard, A. Calvar, A. Bismuto, E. Gini, M. Beck, J. Faist, H. C. Liu, and C. Sirtori, “Injection locking of mid-infrared quantum cascade laser at 14 GHz, by direct microwave modulation,” Laser Photon. Rev. 8(3), 443–449 (2014).
[Crossref]

Capasso, F.

V. M. Gkortsas, C. Wang, L. Kuznetsova, L. Diehl, A. Gordon, C. Jirauschek, M. A. Belkin, A. Belyanin, F. Capasso, and F. X. Kartner, “Dynamics of actively mode-locked quantum cascade lasers,” Opt. Express 18(13), 13616–13630 (2010).
[Crossref] [PubMed]

H. Choi, L. Diehl, Z.-K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. Norris, “Time-resolved investigations of electronic transport dynamics in quantum cascade lasers based on diagonal lasing transition,” IEEE J. Quantum Electron. 45(4), 307–321 (2009).
[Crossref]

C. Y. Wang, L. Kuznetsova, V. M. Gkortsas, L. Diehl, F. X. Kaertner, 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. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, 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]

J. Faist, F. Capasso, D. Sivco, C. Sirtori, A. Hutchinson, and A. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[Crossref] [PubMed]

Cho, A.

J. Faist, F. Capasso, D. Sivco, C. Sirtori, A. Hutchinson, and A. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[Crossref] [PubMed]

Choi, H.

H. Choi, L. Diehl, Z.-K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. Norris, “Time-resolved investigations of electronic transport dynamics in quantum cascade lasers based on diagonal lasing transition,” IEEE J. Quantum Electron. 45(4), 307–321 (2009).
[Crossref]

Colombelli, R.

S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nat. Photonics 4(9), 636–640 (2010).
[Crossref]

Corzine, S.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, 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]

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 (vol 5, pg 306, 2011),” Nat. Photonics 5(6), 378 (2011).

Diehl, L.

V. M. Gkortsas, C. Wang, L. Kuznetsova, L. Diehl, A. Gordon, C. Jirauschek, M. A. Belkin, A. Belyanin, F. Capasso, and F. X. Kartner, “Dynamics of actively mode-locked quantum cascade lasers,” Opt. Express 18(13), 13616–13630 (2010).
[Crossref] [PubMed]

H. Choi, L. Diehl, Z.-K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. Norris, “Time-resolved investigations of electronic transport dynamics in quantum cascade lasers based on diagonal lasing transition,” IEEE J. Quantum Electron. 45(4), 307–321 (2009).
[Crossref]

C. Y. Wang, L. Kuznetsova, V. M. Gkortsas, L. Diehl, F. X. Kaertner, 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. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, 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]

Dikmelik, Y.

J. B. Khurgin, Y. Dikmelik, A. Hugi, and J. Faist, “Coherent frequency combs produced by self frequency modulation in quantum cascade lasers,” Appl. Phys. Lett. 104(8), 081118 (2014).
[Crossref]

Ding, L.

S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nat. Photonics 4(9), 636–640 (2010).
[Crossref]

Faist, J.

M. R. St-Jean, M. I. Amanti, A. Bernard, A. Calvar, A. Bismuto, E. Gini, M. Beck, J. Faist, H. C. Liu, and C. Sirtori, “Injection locking of mid-infrared quantum cascade laser at 14 GHz, by direct microwave modulation,” Laser Photon. Rev. 8(3), 443–449 (2014).
[Crossref]

J. B. Khurgin, Y. Dikmelik, A. Hugi, and J. Faist, “Coherent frequency combs produced by self frequency modulation in quantum cascade lasers,” Appl. Phys. Lett. 104(8), 081118 (2014).
[Crossref]

G. Villares, A. Hugi, S. Blaser, and J. Faist, “Dual-comb spectroscopy based on quantum-cascade-laser frequency combs,” Nat. Commun. 5, 1–9 (2014).
[Crossref]

P. Friedli, H. Sigg, B. Hinkov, A. Hugi, S. Riedi, M. Beck, and J. Faist, “Four-wave mixing in a quantum cascade laser amplifier,” Appl. Phys. Lett. 102(22), 222104 (2013).
[Crossref]

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

R. Terazzi and J. Faist, “A density matrix model of transport and radiation in quantum cascade lasers,” New Journal of Physics 12, 033045 (2010).
[Crossref]

H. Choi, L. Diehl, Z.-K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. Norris, “Time-resolved investigations of electronic transport dynamics in quantum cascade lasers based on diagonal lasing transition,” IEEE J. Quantum Electron. 45(4), 307–321 (2009).
[Crossref]

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, 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]

J. Faist, F. Capasso, D. Sivco, C. Sirtori, A. Hutchinson, and A. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[Crossref] [PubMed]

J. Faist, Quantum Cascade Lasers, 1st ed. (Oxford University, 2013).
[Crossref]

Foxon, C. T.

M. C. Tatham, J. F. Ryan, and C. T. Foxon, “Time-resolved Raman measurements of intersubband relaxation in GaAs quantum qells,” Phys. Rev. Lett. 63(15), 1637–1640 (1989).
[Crossref] [PubMed]

Friedli, P.

P. Friedli, H. Sigg, B. Hinkov, A. Hugi, S. Riedi, M. Beck, and J. Faist, “Four-wave mixing in a quantum cascade laser amplifier,” Appl. Phys. Lett. 102(22), 222104 (2013).
[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 (vol 5, pg 306, 2011),” Nat. Photonics 5(6), 378 (2011).

S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nat. Photonics 4(9), 636–640 (2010).
[Crossref]

Gini, E.

M. R. St-Jean, M. I. Amanti, A. Bernard, A. Calvar, A. Bismuto, E. Gini, M. Beck, J. Faist, H. C. Liu, and C. Sirtori, “Injection locking of mid-infrared quantum cascade laser at 14 GHz, by direct microwave modulation,” Laser Photon. Rev. 8(3), 443–449 (2014).
[Crossref]

Giovannini, M.

H. Choi, L. Diehl, Z.-K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. Norris, “Time-resolved investigations of electronic transport dynamics in quantum cascade lasers based on diagonal lasing transition,” IEEE J. Quantum Electron. 45(4), 307–321 (2009).
[Crossref]

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. Kartner, “Dynamics of actively mode-locked quantum cascade lasers,” Opt. Express 18(13), 13616–13630 (2010).
[Crossref] [PubMed]

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, 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]

Grant, P.

Haffouz, S.

Ham, D.

Haus, H.

H. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1173–1185 (2000).
[Crossref]

Hinkov, B.

P. Friedli, H. Sigg, B. Hinkov, A. Hugi, S. Riedi, M. Beck, and J. Faist, “Four-wave mixing in a quantum cascade laser amplifier,” Appl. Phys. Lett. 102(22), 222104 (2013).
[Crossref]

Hoefler, G.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, 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]

Hugi, A.

J. B. Khurgin, Y. Dikmelik, A. Hugi, and J. Faist, “Coherent frequency combs produced by self frequency modulation in quantum cascade lasers,” Appl. Phys. Lett. 104(8), 081118 (2014).
[Crossref]

G. Villares, A. Hugi, S. Blaser, and J. Faist, “Dual-comb spectroscopy based on quantum-cascade-laser frequency combs,” Nat. Commun. 5, 1–9 (2014).
[Crossref]

P. Friedli, H. Sigg, B. Hinkov, A. Hugi, S. Riedi, M. Beck, and J. Faist, “Four-wave mixing in a quantum cascade laser amplifier,” Appl. Phys. Lett. 102(22), 222104 (2013).
[Crossref]

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

Hutchinson, A.

J. Faist, F. Capasso, D. Sivco, C. Sirtori, A. Hutchinson, and A. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[Crossref] [PubMed]

Jirauschek, C.

Kaertner, F. X.

C. Y. Wang, L. Kuznetsova, V. M. Gkortsas, L. Diehl, F. X. Kaertner, 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. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, 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]

Kartner, 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 (vol 5, pg 306, 2011),” Nat. Photonics 5(6), 378 (2011).

Khurgin, J. B.

J. B. Khurgin, Y. Dikmelik, A. Hugi, and J. Faist, “Coherent frequency combs produced by self frequency modulation in quantum cascade lasers,” Appl. Phys. Lett. 104(8), 081118 (2014).
[Crossref]

Kozlov, V V

V V Kozlov, “Self-induced transparency soliton laser via coherent mode locking,” Phys. Rev. A 56(2), 1607– 1612 (1997).
[Crossref]

Kuizenga, D. J.

D. J. Kuizenga and A. Siegman, “FM and AM mode locking of homogeneous Laser-Part I: Theory,” IEEE J. Quantum Electron. 6(11), 694 (1970).
[Crossref]

Kuznetsova, L.

Lamb, W. E.

W. E. Lamb, “Theory of an optical maser,” Phys. Rev. 134(6A), A1429–A1450 (1964).
[Crossref]

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 (vol 5, pg 306, 2011),” Nat. Photonics 5(6), 378 (2011).

Liu, H. C.

M. R. St-Jean, M. I. Amanti, A. Bernard, A. Calvar, A. Bismuto, E. Gini, M. Beck, J. Faist, H. C. Liu, and C. Sirtori, “Injection locking of mid-infrared quantum cascade laser at 14 GHz, by direct microwave modulation,” Laser Photon. Rev. 8(3), 443–449 (2014).
[Crossref]

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

C. Y. Wang, L. Kuznetsova, V. M. Gkortsas, L. Diehl, F. X. Kaertner, 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. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, 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. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, 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]

Maineult, W.

S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nat. Photonics 4(9), 636–640 (2010).
[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 (vol 5, pg 306, 2011),” Nat. Photonics 5(6), 378 (2011).

Menyuk, C.

C. Menyuk and M. Talukder, “Self-induced transparency modelocking of quantum cascade lasers,” Phys. Rev. Lett. 102(2), 023903 (2009).
[Crossref] [PubMed]

Norris, T.

H. Choi, L. Diehl, Z.-K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. Norris, “Time-resolved investigations of electronic transport dynamics in quantum cascade lasers based on diagonal lasing transition,” IEEE J. Quantum Electron. 45(4), 307–321 (2009).
[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 (vol 5, pg 306, 2011),” Nat. Photonics 5(6), 378 (2011).

Riedi, S.

P. Friedli, H. Sigg, B. Hinkov, A. Hugi, S. Riedi, M. Beck, and J. Faist, “Four-wave mixing in a quantum cascade laser amplifier,” Appl. Phys. Lett. 102(22), 222104 (2013).
[Crossref]

Ritchie, D.

S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nat. Photonics 4(9), 636–640 (2010).
[Crossref]

Ryan, J. F.

M. C. Tatham, J. F. Ryan, and C. T. Foxon, “Time-resolved Raman measurements of intersubband relaxation in GaAs quantum qells,” Phys. Rev. Lett. 63(15), 1637–1640 (1989).
[Crossref] [PubMed]

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 (vol 5, pg 306, 2011),” Nat. Photonics 5(6), 378 (2011).

S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nat. Photonics 4(9), 636–640 (2010).
[Crossref]

Schneider, H.

C. Y. Wang, L. Kuznetsova, V. M. Gkortsas, L. Diehl, F. X. Kaertner, 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. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, 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]

Siegman, A.

D. J. Kuizenga and A. Siegman, “FM and AM mode locking of homogeneous Laser-Part I: Theory,” IEEE J. Quantum Electron. 6(11), 694 (1970).
[Crossref]

Sigg, H.

P. Friedli, H. Sigg, B. Hinkov, A. Hugi, S. Riedi, M. Beck, and J. Faist, “Four-wave mixing in a quantum cascade laser amplifier,” Appl. Phys. Lett. 102(22), 222104 (2013).
[Crossref]

Sirtori, C.

M. R. St-Jean, M. I. Amanti, A. Bernard, A. Calvar, A. Bismuto, E. Gini, M. Beck, J. Faist, H. C. Liu, and C. Sirtori, “Injection locking of mid-infrared quantum cascade laser at 14 GHz, by direct microwave modulation,” Laser Photon. Rev. 8(3), 443–449 (2014).
[Crossref]

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 (vol 5, pg 306, 2011),” Nat. Photonics 5(6), 378 (2011).

S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nat. Photonics 4(9), 636–640 (2010).
[Crossref]

J. Faist, F. Capasso, D. Sivco, C. Sirtori, A. Hutchinson, and A. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[Crossref] [PubMed]

Sivco, D.

J. Faist, F. Capasso, D. Sivco, C. Sirtori, A. Hutchinson, and A. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[Crossref] [PubMed]

Song, C. Y.

St-Jean, M. R.

M. R. St-Jean, M. I. Amanti, A. Bernard, A. Calvar, A. Bismuto, E. Gini, M. Beck, J. Faist, H. C. Liu, and C. Sirtori, “Injection locking of mid-infrared quantum cascade laser at 14 GHz, by direct microwave modulation,” Laser Photon. Rev. 8(3), 443–449 (2014).
[Crossref]

Talukder, M.

C. Menyuk and M. Talukder, “Self-induced transparency modelocking of quantum cascade lasers,” Phys. Rev. Lett. 102(2), 023903 (2009).
[Crossref] [PubMed]

Tatham, M. C.

M. C. Tatham, J. F. Ryan, and C. T. Foxon, “Time-resolved Raman measurements of intersubband relaxation in GaAs quantum qells,” Phys. Rev. Lett. 63(15), 1637–1640 (1989).
[Crossref] [PubMed]

Terazzi, R.

R. Terazzi and J. Faist, “A density matrix model of transport and radiation in quantum cascade lasers,” New Journal of Physics 12, 033045 (2010).
[Crossref]

Troccoli, M.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, 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]

Villares, G.

G. Villares, A. Hugi, S. Blaser, and J. Faist, “Dual-comb spectroscopy based on quantum-cascade-laser frequency combs,” Nat. Commun. 5, 1–9 (2014).
[Crossref]

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

Wang, C.

Wang, C. Y.

C. Y. Wang, L. Kuznetsova, V. M. Gkortsas, L. Diehl, F. X. Kaertner, 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. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, 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]

Wasilewski, Z. R.

Wu, Z.-K.

H. Choi, L. Diehl, Z.-K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. Norris, “Time-resolved investigations of electronic transport dynamics in quantum cascade lasers based on diagonal lasing transition,” IEEE J. Quantum Electron. 45(4), 307–321 (2009).
[Crossref]

Appl. Phys. Lett. (2)

P. Friedli, H. Sigg, B. Hinkov, A. Hugi, S. Riedi, M. Beck, and J. Faist, “Four-wave mixing in a quantum cascade laser amplifier,” Appl. Phys. Lett. 102(22), 222104 (2013).
[Crossref]

J. B. Khurgin, Y. Dikmelik, A. Hugi, and J. Faist, “Coherent frequency combs produced by self frequency modulation in quantum cascade lasers,” Appl. Phys. Lett. 104(8), 081118 (2014).
[Crossref]

IEEE J. Quantum Electron. (2)

H. Choi, L. Diehl, Z.-K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. Norris, “Time-resolved investigations of electronic transport dynamics in quantum cascade lasers based on diagonal lasing transition,” IEEE J. Quantum Electron. 45(4), 307–321 (2009).
[Crossref]

D. J. Kuizenga and A. Siegman, “FM and AM mode locking of homogeneous Laser-Part I: Theory,” IEEE J. Quantum Electron. 6(11), 694 (1970).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

H. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1173–1185 (2000).
[Crossref]

Laser Photon. Rev. (1)

M. R. St-Jean, M. I. Amanti, A. Bernard, A. Calvar, A. Bismuto, E. Gini, M. Beck, J. Faist, H. C. Liu, and C. Sirtori, “Injection locking of mid-infrared quantum cascade laser at 14 GHz, by direct microwave modulation,” Laser Photon. Rev. 8(3), 443–449 (2014).
[Crossref]

Nat. Commun. (1)

G. Villares, A. Hugi, S. Blaser, and J. Faist, “Dual-comb spectroscopy based on quantum-cascade-laser frequency combs,” Nat. Commun. 5, 1–9 (2014).
[Crossref]

Nat. Photonics (2)

S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nat. Photonics 4(9), 636–640 (2010).
[Crossref]

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 (vol 5, pg 306, 2011),” Nat. Photonics 5(6), 378 (2011).

Nature (London) (1)

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

New Journal of Physics (1)

R. Terazzi and J. Faist, “A density matrix model of transport and radiation in quantum cascade lasers,” New Journal of Physics 12, 033045 (2010).
[Crossref]

Opt. Express (2)

Phys. Rev. (1)

W. E. Lamb, “Theory of an optical maser,” Phys. Rev. 134(6A), A1429–A1450 (1964).
[Crossref]

Phys. Rev. A (2)

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, 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]

V V Kozlov, “Self-induced transparency soliton laser via coherent mode locking,” Phys. Rev. A 56(2), 1607– 1612 (1997).
[Crossref]

Phys. Rev. Lett. (2)

C. Menyuk and M. Talukder, “Self-induced transparency modelocking of quantum cascade lasers,” Phys. Rev. Lett. 102(2), 023903 (2009).
[Crossref] [PubMed]

M. C. Tatham, J. F. Ryan, and C. T. Foxon, “Time-resolved Raman measurements of intersubband relaxation in GaAs quantum qells,” Phys. Rev. Lett. 63(15), 1637–1640 (1989).
[Crossref] [PubMed]

Science (1)

J. Faist, F. Capasso, D. Sivco, C. Sirtori, A. Hutchinson, and A. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[Crossref] [PubMed]

Other (1)

J. Faist, Quantum Cascade Lasers, 1st ed. (Oxford University, 2013).
[Crossref]

Supplementary Material (6)

» Media 1: MP4 (3522 KB)     
» Media 2: MP4 (1717 KB)     
» Media 3: MOV (3522 KB)     
» Media 4: MOV (1717 KB)     
» Media 5: MP4 (3522 KB)     
» Media 6: MP4 (1717 KB)     

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Figures (5)

Fig. 1
Fig. 1 Pulse propagation in QCL and theoretical framework used to study the laser dynamics. a) Simulation, based on rate equations, of a 2ps long pulse propagating in a quantum cascade laser. Because of the very fast gain relaxation time, the pulse is damped after only a few millimeter of propagation. b) Maxwell-Bloch equations for the evolution of the populations ρ11 and ρ22 as well as the coherences ρ12 and ρ21. A modal decomposition is used in order to study lasers exhibiting a frequency modulated output.

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Fig. 2
Fig. 2 Free-running MIR QCL (see Media 1). a) Optical amplitude and phase spectrum. b) Instantaneous power as function of time. c) Instantaneous frequency as function of time. d) Radio frequency amplitude spectrum of the detected power, equivalent to the photocurrent spectrum measured with a spectrum analyzer.

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Fig. 3
Fig. 3 Impact of the cavity dispersion on a free-running MIR QCL. a) Schematic description of the impact of dispersion on the laser modes. b) Average of the frequency of the mode n ν n ¯ for three values of simulated dispersion (GVD = −500 fs2mm−1 and GVD = ±30000 fs2mm−1).

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Fig. 4
Fig. 4 Broad gain MIR QCL. a) Gain profile as well as optical amplitude spectrum of a MIR QCL containing two active regions centered at 1375 cm−1 and 1436 cm−1, corresponding to an offset of 61 cm−1. b) Radio frequency amplitude spectrum of the detected power, equivalent to the photocurrent spectrum measured with a spectrum analyzer. c) Optical spectrum of a MIR QCL frequency comb, measured with a Fourier transform infrared spectrometer (0.12 cm−1 resolution), centered at 1430 cm−1 and covering 60 cm−1. d) Radio frequency amplitude spectrum of the detected power, measured with fast quantum well infrared detector (QWIP) and a spectrum analyzer (RBW = 10 Hz), showing an narrow beat note at the round trip frequency with an extremely low amplitude.

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Fig. 5
Fig. 5 Active mode-locked THz QCL (see Media 2). a Optical amplitude and phase spectrum. b Instantaneous power as function of time. c Radio frequency amplitude spectrum of the detected power, equivalent to the photocurrent spectrum measured with a spectrum analyzer. d Time domain evolution of the modes amplitude.

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Equations (93)

Equations on this page are rendered with MathJax. Learn more.

τ = 2 g M ω 2 Ω g 2 4 ,
ρ ˙ = − i ℏ [ ℋ , ρ ] + ∂ ρ ∂ t ) coll
ℋ = ( ℏ ω 1 ℏ Ω ˜ 12 ( t ) ℏ Ω ˜ 21 ( t ) ℏ ω 2 )
Ω ˜ 12 ( t ) = Ω ˜ 21 ( t ) = μ 21 E ( t ) / ℏ
∂ ρ ∂ t ) coll = − ( γ 11 ( ρ 11 − ρ 11 , p ) γ 12 ρ 12 γ 12 ρ 21 γ 22 ( ρ 22 − ρ 22 , p ) )
Δ ρ ˙ = i Ω ˜ 12 ( ρ 21 − ρ 12 ) − γ 22 ( Δ ρ − Δ ρ p ) ρ ˙ 21 = − i ω 21 ρ 21 + i Ω ˜ 12 Δ ρ − γ 12 ρ 21 ρ 21 = ρ 12 *
ω l = ω 21 + l ω                         l ∈ { − n m , − n m + 1 , . , 0 , . , n m }
E ( t ) = 1 2 e − i ω 21 t ∑ l = − n m l = n m A l e − i l ω t + 1 2 e i ω 21 t ∑ l = − n m l = n m A l * e i l ω t Ω ˜ 12 ( t ) = 1 2 e − i ω 21 t ∑ l = − n m l = n m Ω l e − i l ω t + 1 2 e i ω 21 t ∑ l = − n m l = n m Ω l * e i l ω t
Ω l = μ 21 A l ℏ
ρ 21 = e − i ω 21 t ∑ k = − n m k = n m σ 21 , k e − i k ω t ρ 12 = e i ω 21 t ∑ k = − n m k = n m σ 21 , k * e i k ω t Δ ρ = ∑ r = − n m r = n m Δ ρ r e − i r ω t
Δ ρ p = Δ ρ p , 0 + 1 2 ( Δ ρ p , M e − i ω t + Δ ρ p , M * e i ω t )
∑ r ( − i r ω + γ 22 ) Δ ρ r e − i r ω t = γ 22 ( Δ ρ p , 0 + 1 2 ( Δ ρ p , M e − i ω t + Δ ρ p , M * e i ω t ) ) + i 2 ∑ k ∑ l ( Ω l * σ 21 , k e − i ( k − l ) ω t − Ω l σ 21 , k * e i ( k − l ) ω t )
∑ k ( γ 12 − i k ω ) σ 21 , k e − i k ω t = i 2 ∑ l ∑ r Ω l Δ ρ r e − i ( l + r ) ω t
Δ ρ r = 1 − i r ω + γ 22 [ γ 22 Δ ρ p , 0 δ r , 0 + γ 22 Δ ρ p , M * 2 δ r , − 1 + γ 22 Δ ρ p , M 2 δ r , 1 + i 2 ∑ k ( Ω k − r * σ 21 , k − Ω k + r σ 21 , k * ) ]
σ 21 , k = i 2 ( γ 12 − i k ω ) ∑ l Ω l Δ ρ k − l
Δ ρ r = 0 ( 0 ) = Δ ρ p , 0
Δ ρ r = 1 ( 0 ) = γ 22 γ 22 − i ω Δ ρ p , M 2
Δ ρ r = − 1 ( 0 ) = γ 22 γ 22 + i ω Δ ρ p , M * 2
σ 21 , k ( 1 ) = − 1 2 ( i γ 12 + k ω ) [ Ω k Δ ρ p , 0 + Ω k + 1 γ 22 γ 22 + i ω Δ ρ p , M * 2 + Ω k − 1 γ 22 γ 22 − i ω Δ ρ p , M 2 ]
Δ ρ r ( 2 ) = i 2 ( γ 22 − i r ω ) ∑ k ( Ω k − r * σ 21 , k ( 1 ) − Ω k + r σ 21 , k ( 1 ) * )
Δ ρ r ( 2 ) = i 2 ( γ 22 − i r ω ) ∑ k ( Ω k − r * 2 − 1 ( i γ 12 + k ω ) [ Ω k Δ ρ p , 0 + Ω k + 1 γ 22 γ 22 + i ω Δ ρ p , M * 2 + Ω k − 1 γ 22 γ 22 − i ω Δ ρ p , M 2 ] − Ω k + r 2 − 1 ( − i γ 12 + k ω ) [ Ω k * Δ ρ p , 0 + Ω k + 1 * γ 22 γ 22 − i ω Δ ρ p , M 2 + Ω k − 1 * γ 22 γ 22 + i ω Δ ρ p , M * 2 ] )
Δ ρ r ( 2 ) = i 2 ( γ 22 − i r ω ) ∑ k { ( Ω k − r * Ω k Δ ρ p , 0 2 ( − i γ 12 − k ω ) − Ω k + r Ω k * Δ ρ p , 0 2 ( i γ 12 − k ω ) ) + ( γ 22 γ 22 + i ω Ω k − r * Ω k + 1 Δ ρ p , M * / 2 2 ( − i γ 12 − k ω ) − γ 22 γ 22 − i ω Ω k + r Ω k + 1 * Δ ρ p , M / 2 2 ( i γ 12 − k ω ) ) + ( γ 22 γ 22 − i ω Ω k − r * Ω k − 1 Δ ρ p , M / 2 2 ( − i γ 12 − k ω ) − γ 22 γ 22 + i ω Ω k + r Ω k − 1 * Δ ρ p , M * / 2 2 ( i γ 12 − k ω ) ) }
Δ ρ r ( 2 ) = 1 4 i ( γ 22 − i r ω ) ∑ k Ω k + r { Ω k * Δ ρ p , 0 ( 1 − i γ 12 − ( k + r ) ω − 1 i γ 12 − k ω ) + Ω k − 1 * γ 22 γ 22 + i ω Δ ρ p , M * 2 ( 1 − i γ 12 − ( k − 1 + r ) ω − 1 i γ 12 − k ω ) + Ω k + 1 * γ 22 γ 22 − i ω Δ ρ p , M 2 ( 1 − i γ 12 − ( k + 1 + r ) ω − 1 i γ 12 − k ω ) }
σ 21 , n ( 3 ) = i 2 ( γ 12 − i n ω ) ∑ l Ω l Δ ρ n − l ( 2 )
σ 21 , n ( 3 ) = i 2 ( γ 12 − i n ω ) 1 4 ∑ l ∑ k Ω l i ( γ 22 − i ( n − l ) ω ) Ω k + n − l { Ω k * Δ ρ p , 0 ( 1 − i γ 12 − ( k + n − l ) ω − 1 i γ 12 − k ω ) + Ω k − 1 * γ 22 γ 22 + i ω Δ ρ p , M * 2 ( 1 − i γ 12 − ( k − 1 + n − l ) ω − 1 i γ 12 − k ω ) + Ω k + 1 * γ 22 γ 22 − i ω Δ ρ p , M 2 ( 1 − i γ 12 − ( k + 1 + n − l ) ω ) − 1 i γ 12 − k ω ) }
σ 21 , n ( 3 ) = i 2 1 γ 12 − i n ω 1 4 ∑ l ∑ k Ω n + k − l i ( γ 22 − i ( l − k ) ω ) Ω l { Ω k * Δ ρ p , 0 ( 1 − i γ 12 − l ω − 1 i γ 12 − k ω ) + Ω k − 1 * γ 22 γ 22 + i ω Δ ρ p , M * 2 ( 1 − i γ 12 − ( l − 1 ) ω − 1 i γ 12 − k ω ) + Ω k + 1 * γ 22 γ 22 − i ω Δ ρ p , M 2 ( 1 − i γ 12 − ( l + 1 ) ω ) − 1 i γ 12 − k ω ) }
1 − i γ 12 − ( l ± 1 ) ω ≈ 1 − i γ 12 − l ω
M ± = γ 22 γ 22 ± i ω
C k l = γ 22 γ 22 − i ( l − k ) ω
G ˜ n = i γ 12 n ω + i γ 12
B k l = γ 12 2 i ( 1 − i γ 12 − l ω − 1 i γ 12 − k ω )
σ 21 , n ( 3 ) = − i 4 γ 22 γ 12 2 G ˜ n ∑ l ∑ k C k l B k l Ω n + k − l Ω l { Ω k * Δ ρ p , 0 + Ω k − 1 * M + Δ ρ p , M * 2 + Ω k + 1 * M − Δ ρ p , M 2 }
σ 21 , n ( 1 ) = i 2 γ 12 G ˜ n [ Ω n Δ ρ p , 0 + Ω n + 1 M + Δ ρ p , M * 2 + Ω n − 1 M − Δ ρ p , M 2 ]
∇ 2 E − μ 0 σ ∂ E ∂ t − ε r c 2 ∂ 2 E ∂ t 2 = μ 0 ∂ 2 P ∂ t 2
E ( t ) = 1 2 ∑ l = − n m l = n m A l e − i ω l t + 1 2 ∑ l = − n m l = n m A l * e i ω l t P ( t ) = 1 2 ∑ l = − n m l = n m P l e − i ω l t + 1 2 ∑ l = − n m l = n m P l * e i ω l t
A n ( z ) = A n sin ( k n z )
P n = − N 〈 μ 〉 = − N Tr [ ρ μ ] = − N μ 21 σ 21 , n
k n l c = ( N 0 + n ) π
ω n c = k n c n 0
1 2 sin ( k n z )   { − k n 2 A n − μ 0 σ ( A ˙ n − i ω n A n ) − ε r c 2 ( − ω n 2 A n − 2 i ω n A ˙ n ) } = ω n 2 μ 0 N μ 21 1 2 σ 21 , n ( z )
{ ( ( ω n 2 − ω n c 2 ) + i ω n σ ε 0 ε r ) A n + ( 2 i ω n − σ ε 0 ε r ) A ˙ n } sin ( k n z ) = 1 ε 0 ε r ω n 2 N μ 21 σ 21 , n ( z )
2 i ω n A ˙ n = − ( ( ω n 2 − ω n c 2 ) + i ω n σ ε 0 ε r ) A n + ω n 2 N μ 21 σ 21 , n ( z ) ε 0 n 0 2 sin ( k n z )
A ˙ = − σ 2 ε 0 ε r A
| A ˙ | = − | A | 2 τ c
A ˙ n − i ( ω n 2 − ω n c 2 2 ω n ) A n = − i ω n N μ 21 σ 21 , n ( z ) 2 ε 0 n 0 2 sin ( k n z ) − 1 2 τ c A n
A ˙ n − i ( ω n 2 − ω n c 2 2 ω n ) A n = − i ω n N μ 21 l c ε 0 n 0 2 ∫ 0 l c σ 21 , n ( z ) sin ( k n z ) d z − 1 2 τ c A n
σ 21 , n ( z ) = σ 21 , n ( 1 ) ( z ) + σ 21 , n ( 3 ) ( z )
1 l c ∫ 0 l c A n sin ( k n z ) 2 d z = A n κ n , n = A n 1 2
1 l c ∫ 0 l c A n + k − l sin ( k n + k − l z ) A l sin ( k l z ) A k * sin ( k k z ) sin ( k n z ) d z = A n + k − l A l A k * κ n , l , k , n + k − l
κ n , l , k , m = 1 l c ∫ 0 l c sin ( k n z ) sin ( k l z ) sin ( k k z ) sin ( k m z ) d z
κ n , n , n , n = 3 8 ( k = l = n ) κ n , n , m , n = 1 4 ( k = l ≠ n )   or   ( l = n ) κ n , l , k , n + k − l = 1 8 otherwise
Δ ρ p , M ( z ) = Θ ( l m l c − z ) Δ ρ p , M
κ n , l , k , m ′ ( l m ) = 1 l c ∫ 0 l m l c sin ( k n z ) sin ( k l z ) sin ( k k z ) sin ( k m z ) d z
κ n , n + 1 ′ = 1 l c ∫ 0 l m l c sin ( k n z ) sin ( k n + 1 z ) d z = 1 2 π { sin ( π l m ) − sin ( π ( 2 ( n + N 0 ) + 1 ) l m ) ( 2 ( n + N 0 ) + 1 ) }
κ n , l , k , n + k − l ′ = κ n , l , k , n + k − l ′ ( l m )
A ˙ n − i ( ω n 2 − ω n c 2 2 ω n ) A n = − 1 2 τ c A n − i ω n N μ 21 ε 0 n 0 2 i 2 γ 12 G ˜ n [ Ω n Δ ρ p , 0 κ n , n + Ω n + 1 M + κ n , n + 1 ′ Δ ρ p , M * 2 + Ω n − 1 M − κ n , n − 1 ′ Δ ρ p , M 2 ] − i ω n N μ 21 ε 0 n 0 2 − i 4 γ 22 γ 12 2 G ˜ n ∑ l ∑ k C k l B k l Ω n + k − l Ω l { Ω k * κ n , k , l , n + k − l Δ ρ p , 0 + Ω k − 1 * M + κ n , k − 1 , l , n + k − 1 − l ′ Δ ρ p , M * 2 + Ω k + 1 * M − κ n , k + 1 , l , n + k + 1 − l ′ Δ ρ p , M 2 }
g 0 = ω n τ c N μ 21 2 Δ ρ p , 0 2 ε 0 n 0 2 ℏ γ 12
δ = Δ ρ p , m 2 Δ ρ p , 0
2 τ c A ˙ n − 2 i τ c ( ω n 2 − ω n c 2 2 ω n ) A n = − A n + g 0 G ˜ n A n + 2 g 0 G ˜ n [ A n + 1 M + δ κ n , n + 1 ′ + A n − 1 M − δ * κ n , n − 1 ′ ] − μ 21 2 ℏ 2 γ 22 γ 12 g 0 G ˜ n ∑ k , l C k l B k l A m A l { A k * κ n , k , l , m + A k − 1 * M + δ * κ n , k − 1 , l , m + ′ + A k + 1 * M − δ κ n , k + 1 , l , m − ′ } .
m = n + k − l
m + = n + k − 1 − l
m − = n + k + 1 − l
A sat = ℏ γ 12 γ 22 μ 21
2 τ c A ˙ n = { G n − 1 + i 2 τ c ( ω n 2 − ω n c 2 2 ω n ) } A n + 2 G n [ A n + 1 M + δ * κ n , n + 1 ′ + A n − 1 M − δ κ n , n − 1 ′ ] − G n ∑ k , l C k l B k l A m A l { A k * κ n , k , l , m + A k − 1 * M + δ * κ n , k − 1 , l , m + ′ + A k + 1 * M − δ κ n , k + 1 , l , m − ′ } .
A ˙ n = { G n − 1 ︸ Net   gain + i ( ω n 2 − ω n c 2 2 ω n ) ︸ Cavity   dispersion } A n + 2 G n [ A n + 1 M + δ * κ n , n + 1 ′ + A n − 1 M − δ κ n , n − 1 ′ ] ︸ Modulation − G n ∑ k , l C k l B k l A m A l { A k * κ n , k , l , m + A k − 1 * M + δ * κ n , k − 1 , l , m + ′ + A k + 1 * M − δ κ n , k + 1 , l , m − ′ } ︸ FWM   term .
A ˙ n = { G n − 1 + i ( ω n 2 − ω n c 2 2 ω n ) } A n − G n ∑ k , l C k l B k l A m A l { A k * κ n , k , l , m + A k − 1 * M + δ * κ n , k − 1 , l , m + + A k + 1 * M − δ κ n , k + 1 , l , m − }
S n = ∑ k , l C k l B k l A m A l A k * κ n , k , l , m
C k ± 1 , l ≈ C k l B k ± 1 , l ≈ B k l
A ˙ n = { G n − 1 + i ( ω n 2 − ω n c 2 2 ω n ) } A n − G n ( S n + δ * M + S n + 1 + δ M − S n − 1 ) .
S n ′ = ∑ k , l C k l B k l A m A l A k * κ n , k , l , m ′ ,
A ˙ n = { G n − 1 + i ( ω n 2 − ω n c 2 2 ω n ) } A n + 2 G n [ A n + 1 M + δ * κ n , n + 1 ′ + A n − 1 M − δ κ n , n − 1 ′ ] − G n ( S n + δ * M + S n + 1 ′ + δ M − S n − 1 ′ )
δ ω n = d ( arg ( A n ) ) d t
D n = ( ω n 2 − ω n c 2 2 ω n )
k ( ω n ) = ( N 0 + n ) π l c .
k ( ω n c ) = k ( ω 0 ) + ∂ k ∂ ω ( ω n c − ω 0 ) + 1 2 ∂ 2 k ∂ ω 2 ( ω n c − ω 0 ) 2
n g = c ∂ k ∂ ω
GVD = ∂ 2 k ∂ ω 2
δ ω n c 2 + 2 n g c GVD δ ω n c − 2 n π l c GVD = 0 .
ω c = π c n g l c
β GVD = ω c c GVD n g
δ ω n c = ω c β GVD ( − 1 + 1 + 2 n β GVD )
δ ω n c = n ω c ( 1 − n 2 β GVD )
D n = ( ω n 2 − ω n c 2 2 ω n ) = ( ( ω 21 + n ω ) 2 − ( ω 21 + δ ω n c ) 2 2 ( ω 21 + n ω ) )
D n = ω { n N 0 N 0 + n [ 1 − ω ˜ c + 1 2 n ω ˜ c β GVD ] + n 2 2 ( N 0 + n ) ( 1 − ω ˜ c 2 ) }
ω ˜ c = ω c ω
G n ( 1 + 2 ) = g 0 2 ( 1 1 − i ( n − n off ) ω τ coh + 1 1 − i ( n + n off ) ω τ coh )
B k l ( 1 ) = 1 2 ( 1 1 − i ( l − n off ) ω τ coh + 1 1 + i ( k − n off ) ω τ coh ) ,
G n ( 1 ) ∑ k , l C k l B k l ( 1 ) A m A l A k * + G n ( 2 ) ∑ k , l C k l B k l ( 2 ) A m A l A k * ≈ G n ( 1 + 2 ) ∑ k , l C k l B k l ( 1 + 2 ) A m A l A k *
A ˙ n = { G n − 1 + i D n } A n − G n ∑ k , l C k l B k l A m A l A k * κ n , k , l , m
ν n ¯ = τ r t δ t 1 N steps ∑ i φ n i + 1 − φ n i δ t
S = 3 8 ∑ n | A n | 2
∂ S ∂ g 0 = 1 .
g m = μ m ∑ n A n A n + 1 *

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