Abstract

Vertical external cavity surface emitting lasers are ideal testbeds for studying nonlinear many-body systems driven far from equilibrium. The classical laser gain picture fails, however, when a high peak intensity optical pulse of duration shorter than the intrinsic carrier scattering time interacts with electrons in the conduction and holes in the valence band, and the non-equilibrium carrier distributions cannot recover during the presence of the exciting pulse. We present the optimization of ultrashort mode-locked pulses in a vertical external cavity surface emitting laser cavity with a saturable absorber mirror by modelling non-equilibrium quantum dynamics of the electron-hole excitations in the semiconductor quantum-well gain and absorber medium via the semiconductor Bloch equations and treating the field propagation at the level of Maxwell’s wave equation. We introduce a systematic design that predicts the generation of stable mode-locked pulses of duration less than twenty femtoseconds. This factor of five improvement is of interest for mode-locking and ultrafast semiconductor dynamics applications.

© 2017 Optical Society of America

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References

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  1. A. Garnache, S. Hoogland, A. C. Tropper, I. Sagnes, G. Saint-Girons, and J. S. Roberts, “Sub-500-fs soliton-like pulse in a passively mode-locked broadband surface-emitting laser with 100  mW average power,” Appl. Phys. Lett. 80, 3892 (2002).
    [Crossref]
  2. U. Keller and A. C. Tropper, “Passively modelocked surface-emitting semiconductor lasers,” Phys. Rep. 429, 67–120 (2006).
    [Crossref]
  3. A. H. Quarterman, K. G. Wilcox, V. Apostolopoulos, Z. Mihoubi, S. P. Elsmere, I. Farrer, D. A. Ritchie, and A. Tropper, “A passively mode-locked external-cavity semiconductor laser emitting 60-fs pulses,” Nat. Photonics 3, 729–731 (2009).
    [Crossref]
  4. M. Hoffmann, O. D. Sieber, V. J. Wittwer, I. L. Krestnikov, D. A. Livshits, Y. Barbarin, T. Südmeyer, and U. Keller, “Femtosecond high-power quantum dot vertical external cavity surface emitting laser,” Opt. Express 19, 8108–8116 (2011).
    [Crossref]
  5. A. Tropper, A. H. Quarterman, and K. G. Wilcox, “Ultrafast vertical-external-cavity surface-emitting semiconductor lasers,” in Semiconductors and Semimetals (Elsevier, 2012), Vol. 86.
  6. B. Heinen, T.-L. Wang, M. Sparenberg, A. Weber, B. Kunert, J. Hader, S. W. Koch, J. V. Moloney, M. Koch, and W. Stolz, “106  W continuous-wave output power from vertical-external-cavity surface-emitting laser,” Electron. Lett. 48, 516–517 (2012).
    [Crossref]
  7. T.-L. Wang, B. Heinen, J. Hader, C. Dineen, M. Sparenberg, A. Weber, B. Kunert, S. W. Koch, J. V. Moloney, M. Koch, and W. Stolz, “Quantum design strategy pushes high-power vertical-external-cavity surface-emitting lasers beyond 100  W,” Laser Photon. Rev. 6, L12–L14 (2012).
    [Crossref]
  8. M. Scheller, T.-L. Wang, B. Kunert, W. Stolz, S. W. Koch, and J. V. Moloney, “Passively modelocked VECSEL emitting 682  fs pulses with 5.1  W of average output power,” Electron. Lett. 48, 588–589 (2012).
    [Crossref]
  9. K. G. Wilcox, A. C. Tropper, H. E. Beere, D. A. Ritchie, B. Kunert, B. Heinen, and W. Stolz, “4.35  kW peak power femtosecond pulse mode-locked VECSEL for supercontinuum generation,” Opt. Express 21, 1599–1605 (2013).
    [Crossref]
  10. C. Zaugg, Z. Sun, V. J. Wittwer, D. Popa, S. Milana, T. S. Kulmala, R. S. Sundaram, M. Mangold, O. D. Sieber, M. Golling, Y. Lee, J. H. Ahn, A. C. Ferrari, and U. Keller, “Ultrafast and widely tunable vertical-external-cavity surface-emitting laser, mode-locked by a graphene-integrated distributed Bragg reflector,” Opt. Express 21, 31548–31559 (2013).
    [Crossref]
  11. S. Husaini and R. A. Bedford, Antiresonant Graphene Saturable Absorber Mirror for Mode-Locking VECSELs (personal communication, 2013).
  12. K. Seger, N. Meiser, S. Y. Choi, B. H. Jung, D.-I. Yeom, F. Rotermund, O. Okhotnikov, F. Laurell, and V. Pasiskevicius, “Carbon nanotube mode-locked optically-pumped semiconductor disk laser,” Opt. Express 21, 17806–17813 (2013).
    [Crossref]
  13. P. Klopp, U. Griebner, M. Zorn, and M. Weyers, “Pulse repetition rate up to 92  GHz or pulse duration shorter than 110  fs from a mode-locked semiconductor disk laser,” Appl. Phys. Lett. 98, 071103 (2011).
    [Crossref]
  14. A. Laurian, C. Mart, J. Hader, J. V. Moloney, B. Kunert, and W. Stolz, “15  W Single frequency optically pumped semiconductor laser with sub-megaHertz linewidth,” IEEE Photon. Technol. Lett. 26, 131–133 (2014).
    [Crossref]
  15. D. Waldburger, S. M. Link, M. Mangold, C. G. E. Alfieri, E. Gini, M. Golling, B. W. Tilma, and U. Keller, “High-power 100  fs semiconductor disk lasers,” Optica 3, 844–852 (2016).
    [Crossref]
  16. O. D. Sieber, M. Hoffmann, V. J. Wittwer, M. Mangold, M. Golling, B. W. Tilma, T. Südmeyer, and U. Keller, “Experimentally verified pulse formation model for high-power femtosecond VECSELs,” Appl. Phys. B 113, 133–145 (2013).
    [Crossref]
  17. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 5th ed. (World Scientific, 2009).
  18. I. Kilen, J. Hader, J. V. Moloney, and S. W. Koch, “Fully microscopic modeling of mode locking in microcavity lasers,” J. Opt. Soc. Am. B 33, 75–80 (2016).
    [Crossref]
  19. J. Hader, S. W. Koch, and J. V. Moloney, “Microscopic theory of gain and spontaneous emission in GaInNAs laser material,” Solid-State Electron. 47, 513–521 (2003).
    [Crossref]
  20. I. Kilen, J. Hader, J. V. Moloney, and S. W. Koch, “Ultrafast nonequilibrium carrier dynamics in semiconductor laser mode locking,” Optica 1, 192–197 (2014).
    [Crossref]
  21. P. Klopp, U. Griebner, M. Zorn, A. Klehr, A. Liero, M. Weyers, and G. Erbert, “Mode-locked InGaAs-AlGaAs disk laser generating sub-200-fs pulses, pulse picking and amplification by a tapered diode amplifier,” Opt. Express 17, 10820–10834 (2009).
    [Crossref]
  22. S. Backus, C. G. Durfee, and M. M. Murnane, “High power ultrafast lasers,” Rev. Sci. Instrum. 69, 1207–1223 (1998).
  23. S. J. Orfanidis, Electromagnetic Waves and Antennas (Rutgers University, 2008).
  24. D. Waldburger, S. M. Link, C. G. Alfieri, M. Golling, and U. Keller, “High-power 100-fs SESAM-modelocked VECSEL,” in Lasers Congress 2016 (ASSL, LSC, LAC) (Optical Society of America, 2016), paper. ATu1A.8.

2016 (2)

2014 (2)

A. Laurian, C. Mart, J. Hader, J. V. Moloney, B. Kunert, and W. Stolz, “15  W Single frequency optically pumped semiconductor laser with sub-megaHertz linewidth,” IEEE Photon. Technol. Lett. 26, 131–133 (2014).
[Crossref]

I. Kilen, J. Hader, J. V. Moloney, and S. W. Koch, “Ultrafast nonequilibrium carrier dynamics in semiconductor laser mode locking,” Optica 1, 192–197 (2014).
[Crossref]

2013 (4)

2012 (3)

B. Heinen, T.-L. Wang, M. Sparenberg, A. Weber, B. Kunert, J. Hader, S. W. Koch, J. V. Moloney, M. Koch, and W. Stolz, “106  W continuous-wave output power from vertical-external-cavity surface-emitting laser,” Electron. Lett. 48, 516–517 (2012).
[Crossref]

T.-L. Wang, B. Heinen, J. Hader, C. Dineen, M. Sparenberg, A. Weber, B. Kunert, S. W. Koch, J. V. Moloney, M. Koch, and W. Stolz, “Quantum design strategy pushes high-power vertical-external-cavity surface-emitting lasers beyond 100  W,” Laser Photon. Rev. 6, L12–L14 (2012).
[Crossref]

M. Scheller, T.-L. Wang, B. Kunert, W. Stolz, S. W. Koch, and J. V. Moloney, “Passively modelocked VECSEL emitting 682  fs pulses with 5.1  W of average output power,” Electron. Lett. 48, 588–589 (2012).
[Crossref]

2011 (2)

P. Klopp, U. Griebner, M. Zorn, and M. Weyers, “Pulse repetition rate up to 92  GHz or pulse duration shorter than 110  fs from a mode-locked semiconductor disk laser,” Appl. Phys. Lett. 98, 071103 (2011).
[Crossref]

M. Hoffmann, O. D. Sieber, V. J. Wittwer, I. L. Krestnikov, D. A. Livshits, Y. Barbarin, T. Südmeyer, and U. Keller, “Femtosecond high-power quantum dot vertical external cavity surface emitting laser,” Opt. Express 19, 8108–8116 (2011).
[Crossref]

2009 (2)

A. H. Quarterman, K. G. Wilcox, V. Apostolopoulos, Z. Mihoubi, S. P. Elsmere, I. Farrer, D. A. Ritchie, and A. Tropper, “A passively mode-locked external-cavity semiconductor laser emitting 60-fs pulses,” Nat. Photonics 3, 729–731 (2009).
[Crossref]

P. Klopp, U. Griebner, M. Zorn, A. Klehr, A. Liero, M. Weyers, and G. Erbert, “Mode-locked InGaAs-AlGaAs disk laser generating sub-200-fs pulses, pulse picking and amplification by a tapered diode amplifier,” Opt. Express 17, 10820–10834 (2009).
[Crossref]

2006 (1)

U. Keller and A. C. Tropper, “Passively modelocked surface-emitting semiconductor lasers,” Phys. Rep. 429, 67–120 (2006).
[Crossref]

2003 (1)

J. Hader, S. W. Koch, and J. V. Moloney, “Microscopic theory of gain and spontaneous emission in GaInNAs laser material,” Solid-State Electron. 47, 513–521 (2003).
[Crossref]

2002 (1)

A. Garnache, S. Hoogland, A. C. Tropper, I. Sagnes, G. Saint-Girons, and J. S. Roberts, “Sub-500-fs soliton-like pulse in a passively mode-locked broadband surface-emitting laser with 100  mW average power,” Appl. Phys. Lett. 80, 3892 (2002).
[Crossref]

1998 (1)

S. Backus, C. G. Durfee, and M. M. Murnane, “High power ultrafast lasers,” Rev. Sci. Instrum. 69, 1207–1223 (1998).

Ahn, J. H.

Alfieri, C. G.

D. Waldburger, S. M. Link, C. G. Alfieri, M. Golling, and U. Keller, “High-power 100-fs SESAM-modelocked VECSEL,” in Lasers Congress 2016 (ASSL, LSC, LAC) (Optical Society of America, 2016), paper. ATu1A.8.

Alfieri, C. G. E.

Apostolopoulos, V.

A. H. Quarterman, K. G. Wilcox, V. Apostolopoulos, Z. Mihoubi, S. P. Elsmere, I. Farrer, D. A. Ritchie, and A. Tropper, “A passively mode-locked external-cavity semiconductor laser emitting 60-fs pulses,” Nat. Photonics 3, 729–731 (2009).
[Crossref]

Backus, S.

S. Backus, C. G. Durfee, and M. M. Murnane, “High power ultrafast lasers,” Rev. Sci. Instrum. 69, 1207–1223 (1998).

Barbarin, Y.

Bedford, R. A.

S. Husaini and R. A. Bedford, Antiresonant Graphene Saturable Absorber Mirror for Mode-Locking VECSELs (personal communication, 2013).

Beere, H. E.

Choi, S. Y.

Dineen, C.

T.-L. Wang, B. Heinen, J. Hader, C. Dineen, M. Sparenberg, A. Weber, B. Kunert, S. W. Koch, J. V. Moloney, M. Koch, and W. Stolz, “Quantum design strategy pushes high-power vertical-external-cavity surface-emitting lasers beyond 100  W,” Laser Photon. Rev. 6, L12–L14 (2012).
[Crossref]

Durfee, C. G.

S. Backus, C. G. Durfee, and M. M. Murnane, “High power ultrafast lasers,” Rev. Sci. Instrum. 69, 1207–1223 (1998).

Elsmere, S. P.

A. H. Quarterman, K. G. Wilcox, V. Apostolopoulos, Z. Mihoubi, S. P. Elsmere, I. Farrer, D. A. Ritchie, and A. Tropper, “A passively mode-locked external-cavity semiconductor laser emitting 60-fs pulses,” Nat. Photonics 3, 729–731 (2009).
[Crossref]

Erbert, G.

Farrer, I.

A. H. Quarterman, K. G. Wilcox, V. Apostolopoulos, Z. Mihoubi, S. P. Elsmere, I. Farrer, D. A. Ritchie, and A. Tropper, “A passively mode-locked external-cavity semiconductor laser emitting 60-fs pulses,” Nat. Photonics 3, 729–731 (2009).
[Crossref]

Ferrari, A. C.

Garnache, A.

A. Garnache, S. Hoogland, A. C. Tropper, I. Sagnes, G. Saint-Girons, and J. S. Roberts, “Sub-500-fs soliton-like pulse in a passively mode-locked broadband surface-emitting laser with 100  mW average power,” Appl. Phys. Lett. 80, 3892 (2002).
[Crossref]

Gini, E.

Golling, M.

D. Waldburger, S. M. Link, M. Mangold, C. G. E. Alfieri, E. Gini, M. Golling, B. W. Tilma, and U. Keller, “High-power 100  fs semiconductor disk lasers,” Optica 3, 844–852 (2016).
[Crossref]

C. Zaugg, Z. Sun, V. J. Wittwer, D. Popa, S. Milana, T. S. Kulmala, R. S. Sundaram, M. Mangold, O. D. Sieber, M. Golling, Y. Lee, J. H. Ahn, A. C. Ferrari, and U. Keller, “Ultrafast and widely tunable vertical-external-cavity surface-emitting laser, mode-locked by a graphene-integrated distributed Bragg reflector,” Opt. Express 21, 31548–31559 (2013).
[Crossref]

O. D. Sieber, M. Hoffmann, V. J. Wittwer, M. Mangold, M. Golling, B. W. Tilma, T. Südmeyer, and U. Keller, “Experimentally verified pulse formation model for high-power femtosecond VECSELs,” Appl. Phys. B 113, 133–145 (2013).
[Crossref]

D. Waldburger, S. M. Link, C. G. Alfieri, M. Golling, and U. Keller, “High-power 100-fs SESAM-modelocked VECSEL,” in Lasers Congress 2016 (ASSL, LSC, LAC) (Optical Society of America, 2016), paper. ATu1A.8.

Griebner, U.

P. Klopp, U. Griebner, M. Zorn, and M. Weyers, “Pulse repetition rate up to 92  GHz or pulse duration shorter than 110  fs from a mode-locked semiconductor disk laser,” Appl. Phys. Lett. 98, 071103 (2011).
[Crossref]

P. Klopp, U. Griebner, M. Zorn, A. Klehr, A. Liero, M. Weyers, and G. Erbert, “Mode-locked InGaAs-AlGaAs disk laser generating sub-200-fs pulses, pulse picking and amplification by a tapered diode amplifier,” Opt. Express 17, 10820–10834 (2009).
[Crossref]

Hader, J.

I. Kilen, J. Hader, J. V. Moloney, and S. W. Koch, “Fully microscopic modeling of mode locking in microcavity lasers,” J. Opt. Soc. Am. B 33, 75–80 (2016).
[Crossref]

I. Kilen, J. Hader, J. V. Moloney, and S. W. Koch, “Ultrafast nonequilibrium carrier dynamics in semiconductor laser mode locking,” Optica 1, 192–197 (2014).
[Crossref]

A. Laurian, C. Mart, J. Hader, J. V. Moloney, B. Kunert, and W. Stolz, “15  W Single frequency optically pumped semiconductor laser with sub-megaHertz linewidth,” IEEE Photon. Technol. Lett. 26, 131–133 (2014).
[Crossref]

T.-L. Wang, B. Heinen, J. Hader, C. Dineen, M. Sparenberg, A. Weber, B. Kunert, S. W. Koch, J. V. Moloney, M. Koch, and W. Stolz, “Quantum design strategy pushes high-power vertical-external-cavity surface-emitting lasers beyond 100  W,” Laser Photon. Rev. 6, L12–L14 (2012).
[Crossref]

B. Heinen, T.-L. Wang, M. Sparenberg, A. Weber, B. Kunert, J. Hader, S. W. Koch, J. V. Moloney, M. Koch, and W. Stolz, “106  W continuous-wave output power from vertical-external-cavity surface-emitting laser,” Electron. Lett. 48, 516–517 (2012).
[Crossref]

J. Hader, S. W. Koch, and J. V. Moloney, “Microscopic theory of gain and spontaneous emission in GaInNAs laser material,” Solid-State Electron. 47, 513–521 (2003).
[Crossref]

Haug, H.

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 5th ed. (World Scientific, 2009).

Heinen, B.

K. G. Wilcox, A. C. Tropper, H. E. Beere, D. A. Ritchie, B. Kunert, B. Heinen, and W. Stolz, “4.35  kW peak power femtosecond pulse mode-locked VECSEL for supercontinuum generation,” Opt. Express 21, 1599–1605 (2013).
[Crossref]

B. Heinen, T.-L. Wang, M. Sparenberg, A. Weber, B. Kunert, J. Hader, S. W. Koch, J. V. Moloney, M. Koch, and W. Stolz, “106  W continuous-wave output power from vertical-external-cavity surface-emitting laser,” Electron. Lett. 48, 516–517 (2012).
[Crossref]

T.-L. Wang, B. Heinen, J. Hader, C. Dineen, M. Sparenberg, A. Weber, B. Kunert, S. W. Koch, J. V. Moloney, M. Koch, and W. Stolz, “Quantum design strategy pushes high-power vertical-external-cavity surface-emitting lasers beyond 100  W,” Laser Photon. Rev. 6, L12–L14 (2012).
[Crossref]

Hoffmann, M.

O. D. Sieber, M. Hoffmann, V. J. Wittwer, M. Mangold, M. Golling, B. W. Tilma, T. Südmeyer, and U. Keller, “Experimentally verified pulse formation model for high-power femtosecond VECSELs,” Appl. Phys. B 113, 133–145 (2013).
[Crossref]

M. Hoffmann, O. D. Sieber, V. J. Wittwer, I. L. Krestnikov, D. A. Livshits, Y. Barbarin, T. Südmeyer, and U. Keller, “Femtosecond high-power quantum dot vertical external cavity surface emitting laser,” Opt. Express 19, 8108–8116 (2011).
[Crossref]

Hoogland, S.

A. Garnache, S. Hoogland, A. C. Tropper, I. Sagnes, G. Saint-Girons, and J. S. Roberts, “Sub-500-fs soliton-like pulse in a passively mode-locked broadband surface-emitting laser with 100  mW average power,” Appl. Phys. Lett. 80, 3892 (2002).
[Crossref]

Husaini, S.

S. Husaini and R. A. Bedford, Antiresonant Graphene Saturable Absorber Mirror for Mode-Locking VECSELs (personal communication, 2013).

Jung, B. H.

Keller, U.

D. Waldburger, S. M. Link, M. Mangold, C. G. E. Alfieri, E. Gini, M. Golling, B. W. Tilma, and U. Keller, “High-power 100  fs semiconductor disk lasers,” Optica 3, 844–852 (2016).
[Crossref]

C. Zaugg, Z. Sun, V. J. Wittwer, D. Popa, S. Milana, T. S. Kulmala, R. S. Sundaram, M. Mangold, O. D. Sieber, M. Golling, Y. Lee, J. H. Ahn, A. C. Ferrari, and U. Keller, “Ultrafast and widely tunable vertical-external-cavity surface-emitting laser, mode-locked by a graphene-integrated distributed Bragg reflector,” Opt. Express 21, 31548–31559 (2013).
[Crossref]

O. D. Sieber, M. Hoffmann, V. J. Wittwer, M. Mangold, M. Golling, B. W. Tilma, T. Südmeyer, and U. Keller, “Experimentally verified pulse formation model for high-power femtosecond VECSELs,” Appl. Phys. B 113, 133–145 (2013).
[Crossref]

M. Hoffmann, O. D. Sieber, V. J. Wittwer, I. L. Krestnikov, D. A. Livshits, Y. Barbarin, T. Südmeyer, and U. Keller, “Femtosecond high-power quantum dot vertical external cavity surface emitting laser,” Opt. Express 19, 8108–8116 (2011).
[Crossref]

U. Keller and A. C. Tropper, “Passively modelocked surface-emitting semiconductor lasers,” Phys. Rep. 429, 67–120 (2006).
[Crossref]

D. Waldburger, S. M. Link, C. G. Alfieri, M. Golling, and U. Keller, “High-power 100-fs SESAM-modelocked VECSEL,” in Lasers Congress 2016 (ASSL, LSC, LAC) (Optical Society of America, 2016), paper. ATu1A.8.

Kilen, I.

Klehr, A.

Klopp, P.

P. Klopp, U. Griebner, M. Zorn, and M. Weyers, “Pulse repetition rate up to 92  GHz or pulse duration shorter than 110  fs from a mode-locked semiconductor disk laser,” Appl. Phys. Lett. 98, 071103 (2011).
[Crossref]

P. Klopp, U. Griebner, M. Zorn, A. Klehr, A. Liero, M. Weyers, and G. Erbert, “Mode-locked InGaAs-AlGaAs disk laser generating sub-200-fs pulses, pulse picking and amplification by a tapered diode amplifier,” Opt. Express 17, 10820–10834 (2009).
[Crossref]

Koch, M.

T.-L. Wang, B. Heinen, J. Hader, C. Dineen, M. Sparenberg, A. Weber, B. Kunert, S. W. Koch, J. V. Moloney, M. Koch, and W. Stolz, “Quantum design strategy pushes high-power vertical-external-cavity surface-emitting lasers beyond 100  W,” Laser Photon. Rev. 6, L12–L14 (2012).
[Crossref]

B. Heinen, T.-L. Wang, M. Sparenberg, A. Weber, B. Kunert, J. Hader, S. W. Koch, J. V. Moloney, M. Koch, and W. Stolz, “106  W continuous-wave output power from vertical-external-cavity surface-emitting laser,” Electron. Lett. 48, 516–517 (2012).
[Crossref]

Koch, S. W.

I. Kilen, J. Hader, J. V. Moloney, and S. W. Koch, “Fully microscopic modeling of mode locking in microcavity lasers,” J. Opt. Soc. Am. B 33, 75–80 (2016).
[Crossref]

I. Kilen, J. Hader, J. V. Moloney, and S. W. Koch, “Ultrafast nonequilibrium carrier dynamics in semiconductor laser mode locking,” Optica 1, 192–197 (2014).
[Crossref]

T.-L. Wang, B. Heinen, J. Hader, C. Dineen, M. Sparenberg, A. Weber, B. Kunert, S. W. Koch, J. V. Moloney, M. Koch, and W. Stolz, “Quantum design strategy pushes high-power vertical-external-cavity surface-emitting lasers beyond 100  W,” Laser Photon. Rev. 6, L12–L14 (2012).
[Crossref]

B. Heinen, T.-L. Wang, M. Sparenberg, A. Weber, B. Kunert, J. Hader, S. W. Koch, J. V. Moloney, M. Koch, and W. Stolz, “106  W continuous-wave output power from vertical-external-cavity surface-emitting laser,” Electron. Lett. 48, 516–517 (2012).
[Crossref]

M. Scheller, T.-L. Wang, B. Kunert, W. Stolz, S. W. Koch, and J. V. Moloney, “Passively modelocked VECSEL emitting 682  fs pulses with 5.1  W of average output power,” Electron. Lett. 48, 588–589 (2012).
[Crossref]

J. Hader, S. W. Koch, and J. V. Moloney, “Microscopic theory of gain and spontaneous emission in GaInNAs laser material,” Solid-State Electron. 47, 513–521 (2003).
[Crossref]

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 5th ed. (World Scientific, 2009).

Krestnikov, I. L.

Kulmala, T. S.

Kunert, B.

A. Laurian, C. Mart, J. Hader, J. V. Moloney, B. Kunert, and W. Stolz, “15  W Single frequency optically pumped semiconductor laser with sub-megaHertz linewidth,” IEEE Photon. Technol. Lett. 26, 131–133 (2014).
[Crossref]

K. G. Wilcox, A. C. Tropper, H. E. Beere, D. A. Ritchie, B. Kunert, B. Heinen, and W. Stolz, “4.35  kW peak power femtosecond pulse mode-locked VECSEL for supercontinuum generation,” Opt. Express 21, 1599–1605 (2013).
[Crossref]

M. Scheller, T.-L. Wang, B. Kunert, W. Stolz, S. W. Koch, and J. V. Moloney, “Passively modelocked VECSEL emitting 682  fs pulses with 5.1  W of average output power,” Electron. Lett. 48, 588–589 (2012).
[Crossref]

T.-L. Wang, B. Heinen, J. Hader, C. Dineen, M. Sparenberg, A. Weber, B. Kunert, S. W. Koch, J. V. Moloney, M. Koch, and W. Stolz, “Quantum design strategy pushes high-power vertical-external-cavity surface-emitting lasers beyond 100  W,” Laser Photon. Rev. 6, L12–L14 (2012).
[Crossref]

B. Heinen, T.-L. Wang, M. Sparenberg, A. Weber, B. Kunert, J. Hader, S. W. Koch, J. V. Moloney, M. Koch, and W. Stolz, “106  W continuous-wave output power from vertical-external-cavity surface-emitting laser,” Electron. Lett. 48, 516–517 (2012).
[Crossref]

Laurell, F.

Laurian, A.

A. Laurian, C. Mart, J. Hader, J. V. Moloney, B. Kunert, and W. Stolz, “15  W Single frequency optically pumped semiconductor laser with sub-megaHertz linewidth,” IEEE Photon. Technol. Lett. 26, 131–133 (2014).
[Crossref]

Lee, Y.

Liero, A.

Link, S. M.

D. Waldburger, S. M. Link, M. Mangold, C. G. E. Alfieri, E. Gini, M. Golling, B. W. Tilma, and U. Keller, “High-power 100  fs semiconductor disk lasers,” Optica 3, 844–852 (2016).
[Crossref]

D. Waldburger, S. M. Link, C. G. Alfieri, M. Golling, and U. Keller, “High-power 100-fs SESAM-modelocked VECSEL,” in Lasers Congress 2016 (ASSL, LSC, LAC) (Optical Society of America, 2016), paper. ATu1A.8.

Livshits, D. A.

Mangold, M.

Mart, C.

A. Laurian, C. Mart, J. Hader, J. V. Moloney, B. Kunert, and W. Stolz, “15  W Single frequency optically pumped semiconductor laser with sub-megaHertz linewidth,” IEEE Photon. Technol. Lett. 26, 131–133 (2014).
[Crossref]

Meiser, N.

Mihoubi, Z.

A. H. Quarterman, K. G. Wilcox, V. Apostolopoulos, Z. Mihoubi, S. P. Elsmere, I. Farrer, D. A. Ritchie, and A. Tropper, “A passively mode-locked external-cavity semiconductor laser emitting 60-fs pulses,” Nat. Photonics 3, 729–731 (2009).
[Crossref]

Milana, S.

Moloney, J. V.

I. Kilen, J. Hader, J. V. Moloney, and S. W. Koch, “Fully microscopic modeling of mode locking in microcavity lasers,” J. Opt. Soc. Am. B 33, 75–80 (2016).
[Crossref]

I. Kilen, J. Hader, J. V. Moloney, and S. W. Koch, “Ultrafast nonequilibrium carrier dynamics in semiconductor laser mode locking,” Optica 1, 192–197 (2014).
[Crossref]

A. Laurian, C. Mart, J. Hader, J. V. Moloney, B. Kunert, and W. Stolz, “15  W Single frequency optically pumped semiconductor laser with sub-megaHertz linewidth,” IEEE Photon. Technol. Lett. 26, 131–133 (2014).
[Crossref]

M. Scheller, T.-L. Wang, B. Kunert, W. Stolz, S. W. Koch, and J. V. Moloney, “Passively modelocked VECSEL emitting 682  fs pulses with 5.1  W of average output power,” Electron. Lett. 48, 588–589 (2012).
[Crossref]

B. Heinen, T.-L. Wang, M. Sparenberg, A. Weber, B. Kunert, J. Hader, S. W. Koch, J. V. Moloney, M. Koch, and W. Stolz, “106  W continuous-wave output power from vertical-external-cavity surface-emitting laser,” Electron. Lett. 48, 516–517 (2012).
[Crossref]

T.-L. Wang, B. Heinen, J. Hader, C. Dineen, M. Sparenberg, A. Weber, B. Kunert, S. W. Koch, J. V. Moloney, M. Koch, and W. Stolz, “Quantum design strategy pushes high-power vertical-external-cavity surface-emitting lasers beyond 100  W,” Laser Photon. Rev. 6, L12–L14 (2012).
[Crossref]

J. Hader, S. W. Koch, and J. V. Moloney, “Microscopic theory of gain and spontaneous emission in GaInNAs laser material,” Solid-State Electron. 47, 513–521 (2003).
[Crossref]

Murnane, M. M.

S. Backus, C. G. Durfee, and M. M. Murnane, “High power ultrafast lasers,” Rev. Sci. Instrum. 69, 1207–1223 (1998).

Okhotnikov, O.

Orfanidis, S. J.

S. J. Orfanidis, Electromagnetic Waves and Antennas (Rutgers University, 2008).

Pasiskevicius, V.

Popa, D.

Quarterman, A. H.

A. H. Quarterman, K. G. Wilcox, V. Apostolopoulos, Z. Mihoubi, S. P. Elsmere, I. Farrer, D. A. Ritchie, and A. Tropper, “A passively mode-locked external-cavity semiconductor laser emitting 60-fs pulses,” Nat. Photonics 3, 729–731 (2009).
[Crossref]

A. Tropper, A. H. Quarterman, and K. G. Wilcox, “Ultrafast vertical-external-cavity surface-emitting semiconductor lasers,” in Semiconductors and Semimetals (Elsevier, 2012), Vol. 86.

Ritchie, D. A.

K. G. Wilcox, A. C. Tropper, H. E. Beere, D. A. Ritchie, B. Kunert, B. Heinen, and W. Stolz, “4.35  kW peak power femtosecond pulse mode-locked VECSEL for supercontinuum generation,” Opt. Express 21, 1599–1605 (2013).
[Crossref]

A. H. Quarterman, K. G. Wilcox, V. Apostolopoulos, Z. Mihoubi, S. P. Elsmere, I. Farrer, D. A. Ritchie, and A. Tropper, “A passively mode-locked external-cavity semiconductor laser emitting 60-fs pulses,” Nat. Photonics 3, 729–731 (2009).
[Crossref]

Roberts, J. S.

A. Garnache, S. Hoogland, A. C. Tropper, I. Sagnes, G. Saint-Girons, and J. S. Roberts, “Sub-500-fs soliton-like pulse in a passively mode-locked broadband surface-emitting laser with 100  mW average power,” Appl. Phys. Lett. 80, 3892 (2002).
[Crossref]

Rotermund, F.

Sagnes, I.

A. Garnache, S. Hoogland, A. C. Tropper, I. Sagnes, G. Saint-Girons, and J. S. Roberts, “Sub-500-fs soliton-like pulse in a passively mode-locked broadband surface-emitting laser with 100  mW average power,” Appl. Phys. Lett. 80, 3892 (2002).
[Crossref]

Saint-Girons, G.

A. Garnache, S. Hoogland, A. C. Tropper, I. Sagnes, G. Saint-Girons, and J. S. Roberts, “Sub-500-fs soliton-like pulse in a passively mode-locked broadband surface-emitting laser with 100  mW average power,” Appl. Phys. Lett. 80, 3892 (2002).
[Crossref]

Scheller, M.

M. Scheller, T.-L. Wang, B. Kunert, W. Stolz, S. W. Koch, and J. V. Moloney, “Passively modelocked VECSEL emitting 682  fs pulses with 5.1  W of average output power,” Electron. Lett. 48, 588–589 (2012).
[Crossref]

Seger, K.

Sieber, O. D.

Sparenberg, M.

B. Heinen, T.-L. Wang, M. Sparenberg, A. Weber, B. Kunert, J. Hader, S. W. Koch, J. V. Moloney, M. Koch, and W. Stolz, “106  W continuous-wave output power from vertical-external-cavity surface-emitting laser,” Electron. Lett. 48, 516–517 (2012).
[Crossref]

T.-L. Wang, B. Heinen, J. Hader, C. Dineen, M. Sparenberg, A. Weber, B. Kunert, S. W. Koch, J. V. Moloney, M. Koch, and W. Stolz, “Quantum design strategy pushes high-power vertical-external-cavity surface-emitting lasers beyond 100  W,” Laser Photon. Rev. 6, L12–L14 (2012).
[Crossref]

Stolz, W.

A. Laurian, C. Mart, J. Hader, J. V. Moloney, B. Kunert, and W. Stolz, “15  W Single frequency optically pumped semiconductor laser with sub-megaHertz linewidth,” IEEE Photon. Technol. Lett. 26, 131–133 (2014).
[Crossref]

K. G. Wilcox, A. C. Tropper, H. E. Beere, D. A. Ritchie, B. Kunert, B. Heinen, and W. Stolz, “4.35  kW peak power femtosecond pulse mode-locked VECSEL for supercontinuum generation,” Opt. Express 21, 1599–1605 (2013).
[Crossref]

T.-L. Wang, B. Heinen, J. Hader, C. Dineen, M. Sparenberg, A. Weber, B. Kunert, S. W. Koch, J. V. Moloney, M. Koch, and W. Stolz, “Quantum design strategy pushes high-power vertical-external-cavity surface-emitting lasers beyond 100  W,” Laser Photon. Rev. 6, L12–L14 (2012).
[Crossref]

M. Scheller, T.-L. Wang, B. Kunert, W. Stolz, S. W. Koch, and J. V. Moloney, “Passively modelocked VECSEL emitting 682  fs pulses with 5.1  W of average output power,” Electron. Lett. 48, 588–589 (2012).
[Crossref]

B. Heinen, T.-L. Wang, M. Sparenberg, A. Weber, B. Kunert, J. Hader, S. W. Koch, J. V. Moloney, M. Koch, and W. Stolz, “106  W continuous-wave output power from vertical-external-cavity surface-emitting laser,” Electron. Lett. 48, 516–517 (2012).
[Crossref]

Südmeyer, T.

O. D. Sieber, M. Hoffmann, V. J. Wittwer, M. Mangold, M. Golling, B. W. Tilma, T. Südmeyer, and U. Keller, “Experimentally verified pulse formation model for high-power femtosecond VECSELs,” Appl. Phys. B 113, 133–145 (2013).
[Crossref]

M. Hoffmann, O. D. Sieber, V. J. Wittwer, I. L. Krestnikov, D. A. Livshits, Y. Barbarin, T. Südmeyer, and U. Keller, “Femtosecond high-power quantum dot vertical external cavity surface emitting laser,” Opt. Express 19, 8108–8116 (2011).
[Crossref]

Sun, Z.

Sundaram, R. S.

Tilma, B. W.

D. Waldburger, S. M. Link, M. Mangold, C. G. E. Alfieri, E. Gini, M. Golling, B. W. Tilma, and U. Keller, “High-power 100  fs semiconductor disk lasers,” Optica 3, 844–852 (2016).
[Crossref]

O. D. Sieber, M. Hoffmann, V. J. Wittwer, M. Mangold, M. Golling, B. W. Tilma, T. Südmeyer, and U. Keller, “Experimentally verified pulse formation model for high-power femtosecond VECSELs,” Appl. Phys. B 113, 133–145 (2013).
[Crossref]

Tropper, A.

A. H. Quarterman, K. G. Wilcox, V. Apostolopoulos, Z. Mihoubi, S. P. Elsmere, I. Farrer, D. A. Ritchie, and A. Tropper, “A passively mode-locked external-cavity semiconductor laser emitting 60-fs pulses,” Nat. Photonics 3, 729–731 (2009).
[Crossref]

A. Tropper, A. H. Quarterman, and K. G. Wilcox, “Ultrafast vertical-external-cavity surface-emitting semiconductor lasers,” in Semiconductors and Semimetals (Elsevier, 2012), Vol. 86.

Tropper, A. C.

K. G. Wilcox, A. C. Tropper, H. E. Beere, D. A. Ritchie, B. Kunert, B. Heinen, and W. Stolz, “4.35  kW peak power femtosecond pulse mode-locked VECSEL for supercontinuum generation,” Opt. Express 21, 1599–1605 (2013).
[Crossref]

U. Keller and A. C. Tropper, “Passively modelocked surface-emitting semiconductor lasers,” Phys. Rep. 429, 67–120 (2006).
[Crossref]

A. Garnache, S. Hoogland, A. C. Tropper, I. Sagnes, G. Saint-Girons, and J. S. Roberts, “Sub-500-fs soliton-like pulse in a passively mode-locked broadband surface-emitting laser with 100  mW average power,” Appl. Phys. Lett. 80, 3892 (2002).
[Crossref]

Waldburger, D.

D. Waldburger, S. M. Link, M. Mangold, C. G. E. Alfieri, E. Gini, M. Golling, B. W. Tilma, and U. Keller, “High-power 100  fs semiconductor disk lasers,” Optica 3, 844–852 (2016).
[Crossref]

D. Waldburger, S. M. Link, C. G. Alfieri, M. Golling, and U. Keller, “High-power 100-fs SESAM-modelocked VECSEL,” in Lasers Congress 2016 (ASSL, LSC, LAC) (Optical Society of America, 2016), paper. ATu1A.8.

Wang, T.-L.

T.-L. Wang, B. Heinen, J. Hader, C. Dineen, M. Sparenberg, A. Weber, B. Kunert, S. W. Koch, J. V. Moloney, M. Koch, and W. Stolz, “Quantum design strategy pushes high-power vertical-external-cavity surface-emitting lasers beyond 100  W,” Laser Photon. Rev. 6, L12–L14 (2012).
[Crossref]

B. Heinen, T.-L. Wang, M. Sparenberg, A. Weber, B. Kunert, J. Hader, S. W. Koch, J. V. Moloney, M. Koch, and W. Stolz, “106  W continuous-wave output power from vertical-external-cavity surface-emitting laser,” Electron. Lett. 48, 516–517 (2012).
[Crossref]

M. Scheller, T.-L. Wang, B. Kunert, W. Stolz, S. W. Koch, and J. V. Moloney, “Passively modelocked VECSEL emitting 682  fs pulses with 5.1  W of average output power,” Electron. Lett. 48, 588–589 (2012).
[Crossref]

Weber, A.

B. Heinen, T.-L. Wang, M. Sparenberg, A. Weber, B. Kunert, J. Hader, S. W. Koch, J. V. Moloney, M. Koch, and W. Stolz, “106  W continuous-wave output power from vertical-external-cavity surface-emitting laser,” Electron. Lett. 48, 516–517 (2012).
[Crossref]

T.-L. Wang, B. Heinen, J. Hader, C. Dineen, M. Sparenberg, A. Weber, B. Kunert, S. W. Koch, J. V. Moloney, M. Koch, and W. Stolz, “Quantum design strategy pushes high-power vertical-external-cavity surface-emitting lasers beyond 100  W,” Laser Photon. Rev. 6, L12–L14 (2012).
[Crossref]

Weyers, M.

P. Klopp, U. Griebner, M. Zorn, and M. Weyers, “Pulse repetition rate up to 92  GHz or pulse duration shorter than 110  fs from a mode-locked semiconductor disk laser,” Appl. Phys. Lett. 98, 071103 (2011).
[Crossref]

P. Klopp, U. Griebner, M. Zorn, A. Klehr, A. Liero, M. Weyers, and G. Erbert, “Mode-locked InGaAs-AlGaAs disk laser generating sub-200-fs pulses, pulse picking and amplification by a tapered diode amplifier,” Opt. Express 17, 10820–10834 (2009).
[Crossref]

Wilcox, K. G.

K. G. Wilcox, A. C. Tropper, H. E. Beere, D. A. Ritchie, B. Kunert, B. Heinen, and W. Stolz, “4.35  kW peak power femtosecond pulse mode-locked VECSEL for supercontinuum generation,” Opt. Express 21, 1599–1605 (2013).
[Crossref]

A. H. Quarterman, K. G. Wilcox, V. Apostolopoulos, Z. Mihoubi, S. P. Elsmere, I. Farrer, D. A. Ritchie, and A. Tropper, “A passively mode-locked external-cavity semiconductor laser emitting 60-fs pulses,” Nat. Photonics 3, 729–731 (2009).
[Crossref]

A. Tropper, A. H. Quarterman, and K. G. Wilcox, “Ultrafast vertical-external-cavity surface-emitting semiconductor lasers,” in Semiconductors and Semimetals (Elsevier, 2012), Vol. 86.

Wittwer, V. J.

Yeom, D.-I.

Zaugg, C.

Zorn, M.

P. Klopp, U. Griebner, M. Zorn, and M. Weyers, “Pulse repetition rate up to 92  GHz or pulse duration shorter than 110  fs from a mode-locked semiconductor disk laser,” Appl. Phys. Lett. 98, 071103 (2011).
[Crossref]

P. Klopp, U. Griebner, M. Zorn, A. Klehr, A. Liero, M. Weyers, and G. Erbert, “Mode-locked InGaAs-AlGaAs disk laser generating sub-200-fs pulses, pulse picking and amplification by a tapered diode amplifier,” Opt. Express 17, 10820–10834 (2009).
[Crossref]

Appl. Phys. B (1)

O. D. Sieber, M. Hoffmann, V. J. Wittwer, M. Mangold, M. Golling, B. W. Tilma, T. Südmeyer, and U. Keller, “Experimentally verified pulse formation model for high-power femtosecond VECSELs,” Appl. Phys. B 113, 133–145 (2013).
[Crossref]

Appl. Phys. Lett. (2)

P. Klopp, U. Griebner, M. Zorn, and M. Weyers, “Pulse repetition rate up to 92  GHz or pulse duration shorter than 110  fs from a mode-locked semiconductor disk laser,” Appl. Phys. Lett. 98, 071103 (2011).
[Crossref]

A. Garnache, S. Hoogland, A. C. Tropper, I. Sagnes, G. Saint-Girons, and J. S. Roberts, “Sub-500-fs soliton-like pulse in a passively mode-locked broadband surface-emitting laser with 100  mW average power,” Appl. Phys. Lett. 80, 3892 (2002).
[Crossref]

Electron. Lett. (2)

B. Heinen, T.-L. Wang, M. Sparenberg, A. Weber, B. Kunert, J. Hader, S. W. Koch, J. V. Moloney, M. Koch, and W. Stolz, “106  W continuous-wave output power from vertical-external-cavity surface-emitting laser,” Electron. Lett. 48, 516–517 (2012).
[Crossref]

M. Scheller, T.-L. Wang, B. Kunert, W. Stolz, S. W. Koch, and J. V. Moloney, “Passively modelocked VECSEL emitting 682  fs pulses with 5.1  W of average output power,” Electron. Lett. 48, 588–589 (2012).
[Crossref]

IEEE Photon. Technol. Lett. (1)

A. Laurian, C. Mart, J. Hader, J. V. Moloney, B. Kunert, and W. Stolz, “15  W Single frequency optically pumped semiconductor laser with sub-megaHertz linewidth,” IEEE Photon. Technol. Lett. 26, 131–133 (2014).
[Crossref]

J. Opt. Soc. Am. B (1)

Laser Photon. Rev. (1)

T.-L. Wang, B. Heinen, J. Hader, C. Dineen, M. Sparenberg, A. Weber, B. Kunert, S. W. Koch, J. V. Moloney, M. Koch, and W. Stolz, “Quantum design strategy pushes high-power vertical-external-cavity surface-emitting lasers beyond 100  W,” Laser Photon. Rev. 6, L12–L14 (2012).
[Crossref]

Nat. Photonics (1)

A. H. Quarterman, K. G. Wilcox, V. Apostolopoulos, Z. Mihoubi, S. P. Elsmere, I. Farrer, D. A. Ritchie, and A. Tropper, “A passively mode-locked external-cavity semiconductor laser emitting 60-fs pulses,” Nat. Photonics 3, 729–731 (2009).
[Crossref]

Opt. Express (5)

Optica (2)

Phys. Rep. (1)

U. Keller and A. C. Tropper, “Passively modelocked surface-emitting semiconductor lasers,” Phys. Rep. 429, 67–120 (2006).
[Crossref]

Rev. Sci. Instrum. (1)

S. Backus, C. G. Durfee, and M. M. Murnane, “High power ultrafast lasers,” Rev. Sci. Instrum. 69, 1207–1223 (1998).

Solid-State Electron. (1)

J. Hader, S. W. Koch, and J. V. Moloney, “Microscopic theory of gain and spontaneous emission in GaInNAs laser material,” Solid-State Electron. 47, 513–521 (2003).
[Crossref]

Other (5)

S. J. Orfanidis, Electromagnetic Waves and Antennas (Rutgers University, 2008).

D. Waldburger, S. M. Link, C. G. Alfieri, M. Golling, and U. Keller, “High-power 100-fs SESAM-modelocked VECSEL,” in Lasers Congress 2016 (ASSL, LSC, LAC) (Optical Society of America, 2016), paper. ATu1A.8.

A. Tropper, A. H. Quarterman, and K. G. Wilcox, “Ultrafast vertical-external-cavity surface-emitting semiconductor lasers,” in Semiconductors and Semimetals (Elsevier, 2012), Vol. 86.

S. Husaini and R. A. Bedford, Antiresonant Graphene Saturable Absorber Mirror for Mode-Locking VECSELs (personal communication, 2013).

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 5th ed. (World Scientific, 2009).

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Figures (5)

Fig. 1.
Fig. 1. Diagram of the material layers in the MQW structure. The vertical axis shows the refractive index of each material layer and the vertical grey lines represent the position of the QWs. The black, blue and red lines are standing waves at 980 nm, 930 nm, and 1030 nm, respectively, that have been rescaled for visibility. Not shown is a DBR on the left.
Fig. 2.
Fig. 2. Simultaneous snapshot of the mode-locked pulse and inversions in all the QWs. (a) Black dashed line is the inversion of the background Fermi distribution at a density n = 3.25 · 10 16    m 2 , and each blue curve corresponds to the instantaneous inversion in separate QWs. (b) Solid (dashed) curve is the optimized mode-locked output pulse with intensity FWHM of 19 fs (46 fs) that results from using a wide (narrow) DBR stopband.
Fig. 3.
Fig. 3. Comparison of the modal gain and GDD at the carrier density n = 3.25 · 10 16    m 2 ( 2.22 · 10 16    m 2 ) for the MQW (RPG) structure. (a) Modal gain of the two structures when they are clamped at the same peak gain. (b) Influence of inverted QWs on the GDD relative to absorbing QWs at a carrier density n = 5.0 · 10 14    m 2 .
Fig. 4.
Fig. 4. Overview of the peak intensity and intensity FWHM of the mode-locked pulses that result from varying the initial QW background carrier density of the MQW structure. The solid black curves corresponds to simulation results that start from noise, and the dashed red curves are results that come from reducing the background density. (a) Peak pulse intensity. (b) Pulse intensity FWHM.
Fig. 5.
Fig. 5. Total cavity absorption in black and the modal gain of the MQW structure for select QW carrier densities.

Equations (4)

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

[ 2 z 2 n 2 c 0 2 2 t 2 ] E ( z , t ) = μ 0 2 t 2 P ( z , t ) .
t p λ , ν , k = i λ 1 , ν 1 ( e λ , λ 1 , k e δ ν , ν 1 + e ν , ν 1 , k h δ λ , λ 1 ) p λ 1 , ν 1 , k i ( n λ , k e + n ν , k h 1 ) Ω λ , ν , k + Γ λ , ν , deph , t n λ ( ν ) , k e ( h ) = 2 Im ( Ω λ , ν , k ( p λ , ν , k ) * ) + Γ λ ( ν ) , scatt e ( h ) + Γ λ ( ν ) , fill e ( h ) .
e λ , λ 1 , k e = ε λ , k e δ λ , λ 1 λ 2 , q V | k q | λ , λ 2 , λ 1 , λ 2 n λ 2 , q e , e ν , ν 1 , k h = ε ν , k h δ ν , ν 1 ν 2 , q V | k q | ν , ν 2 , ν 1 , ν 2 n ν 2 , q h
Ω λ , ν , k = d k λ , ν E ( z , t ) + λ 1 , ν 1 , q k V | k q | λ , ν 1 , ν , λ 1 p λ 1 , ν 1 , q ,

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