Abstract

A time-domain approach based on the finite difference time-domain (FDTD) method is described for vertical-extended-cavity surface-emitting lasers (VECSELs). To permit the simulation of realistic devices with large extended cavities a combined analytical-FDTD model is developed. This approach uses plane wave propagation in the linear extended cavity to couple the active mirror and the semiconductor saturable absorber mirror (SESAM) in which the standard FDTD method is applied. This allows for a significant computational time reduction. The material response in the active quantum wells (QWs) and the absorber is incorporated by an infinite impulse response digital filter. The model is validated by the simulation of a passively mode-locked VECSEL with a multiple QW active region and a single QW SESAM.

© 2008 Optical Society of America

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  1. A. C. Tropper, H. D. Foreman, A. Garnache, K. G. Wilcox, and S. H. Hoogland, “Vertical-external-cavity semiconductor lasers,” J. Phys. D 37, R75-R85 (2004).
    [CrossRef]
  2. U. Keller and A. C. Tropper, “Passively modelocked surface-emitting semiconductor lasers,” Phys. Rep. 429, 67-120 (2006).
    [CrossRef]
  3. U. Keller, K. J. Weingarten, F. X. Kartner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Honninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAMs) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Sel. Top. Quantum Electron. 2, 435-453 (1996).
    [CrossRef]
  4. 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, 3892-3894 (2002).
    [CrossRef]
  5. H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049-3058 (1975).
    [CrossRef]
  6. R. Paschotta, R. Häring, A. Garnache, S. Hoogland, A. C. Tropper, and U. Keller, “Soliton-like pulse-shaping mechanism in passively mode-locked surface-emitting semiconductor lasers,” Appl. Phys. B 75, 445-451 (2002).
    [CrossRef]
  7. J. Mulet and S. Balle, “Mode-locking dynamics in electrically driven vertical-external-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 41, 1148-1156 (2005).
    [CrossRef]
  8. M. Bahl, N. C. Panoiu, and R. M. Osgood, “Modeling ultrashort field dynamics in surface emitting lasers by using finite-difference time-domain method,” IEEE J. Quantum Electron. 41, 1244-1252 (2005).
    [CrossRef]
  9. C. Z. Ning, R. A. Indik, and J. V. Moloney, “Self-consistent approach to thermal effects in vertical-cavity surface-emitting lasers,” J. Opt. Soc. Am. B 12, 1993-2004 (1995).
    [CrossRef]
  10. M. Bahl, N. C. Panoiu, and R. M. Osgood, “Novel optical and thermal modeling for modelocked VCSELS,” in Proceedings of the IEEE Lightwave Technologies in Instrumentation and Measurement Conference (IEEE, 2004), pp. 17-22.
    [CrossRef]
  11. C. Z. Ning, R. A. Indik, and J. V. Moloney, “Effective bloch equations for semiconductor lasers and amplifiers,” IEEE J. Quantum Electron. 33, 1543-1550 (1997).
    [CrossRef]
  12. W. W. Chow and S. W. Koch, Semiconductor-Laser Fundamentals: Physics of the Gain Materials (Springer-Verlag, 1999).
  13. R. Binder, D. Scott, A. E. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, “Carrier-carrier scattering and optical dephasing in highly excited semiconductors,” Phys. Rev. B 45, 1107-1115 (1992).
    [CrossRef]
  14. W. H. Weedon and C. M. Rappaport, “A general method for FDTD modeling of wave propagation in arbitrary frequency-dispersive media,” IEEE Trans. Antennas Propag. 45, 401-410 (1997).
    [CrossRef]
  15. J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Series in Computational Mathematics (Prentice-Hall, 1983).
    [PubMed]
  16. D. M. Sullivan, Electromagnetic Simulation using the FDTD Method, IEEE Press Series on RF and Microwave Technology (IEEE, 2000).
    [CrossRef] [PubMed]
  17. L. R. Rabiner and B. Gold, Theory and Application of Digital Signal Processing, Computational Microelectronics (Prentice-Hall, 1975).
  18. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 1995).
  19. C. S. Shin and R. Nevels, “Optimizing the Gaussian excitation function in the finite difference time domain method,” IEEE Trans. Educ. 45, 15-18 (2002).
    [CrossRef]
  20. D. Sullivan and J. L. Young, “Far-field time-domain calculation from aperture radiators using the FDTD method,” IEEE Trans. Antennas Propag. 49, 464-469 (2001).
    [CrossRef]
  21. O. M. Ramahi, “Near- and far-field calculations in FDTD simulations using Kirchhoff surface integral representation,” IEEE Trans. Antennas Propag. 45, 753-759 (1997).
    [CrossRef]
  22. I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, “Band parameters for III-V compound semiconductors and their alloys,” J. Appl. Phys. 89, 5815-5875 (2001).
    [CrossRef]
  23. A. Christ, Analysis and Improvement of the Numerical Properties of the FDTD Algorithm, Series in Microelectronics (Hartung-Gorre, 2005, Vol. 160).

2006 (1)

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

2005 (2)

J. Mulet and S. Balle, “Mode-locking dynamics in electrically driven vertical-external-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 41, 1148-1156 (2005).
[CrossRef]

M. Bahl, N. C. Panoiu, and R. M. Osgood, “Modeling ultrashort field dynamics in surface emitting lasers by using finite-difference time-domain method,” IEEE J. Quantum Electron. 41, 1244-1252 (2005).
[CrossRef]

2004 (1)

A. C. Tropper, H. D. Foreman, A. Garnache, K. G. Wilcox, and S. H. Hoogland, “Vertical-external-cavity semiconductor lasers,” J. Phys. D 37, R75-R85 (2004).
[CrossRef]

2002 (3)

R. Paschotta, R. Häring, A. Garnache, S. Hoogland, A. C. Tropper, and U. Keller, “Soliton-like pulse-shaping mechanism in passively mode-locked surface-emitting semiconductor lasers,” Appl. Phys. B 75, 445-451 (2002).
[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, 3892-3894 (2002).
[CrossRef]

C. S. Shin and R. Nevels, “Optimizing the Gaussian excitation function in the finite difference time domain method,” IEEE Trans. Educ. 45, 15-18 (2002).
[CrossRef]

2001 (2)

D. Sullivan and J. L. Young, “Far-field time-domain calculation from aperture radiators using the FDTD method,” IEEE Trans. Antennas Propag. 49, 464-469 (2001).
[CrossRef]

I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, “Band parameters for III-V compound semiconductors and their alloys,” J. Appl. Phys. 89, 5815-5875 (2001).
[CrossRef]

1997 (3)

W. H. Weedon and C. M. Rappaport, “A general method for FDTD modeling of wave propagation in arbitrary frequency-dispersive media,” IEEE Trans. Antennas Propag. 45, 401-410 (1997).
[CrossRef]

O. M. Ramahi, “Near- and far-field calculations in FDTD simulations using Kirchhoff surface integral representation,” IEEE Trans. Antennas Propag. 45, 753-759 (1997).
[CrossRef]

C. Z. Ning, R. A. Indik, and J. V. Moloney, “Effective bloch equations for semiconductor lasers and amplifiers,” IEEE J. Quantum Electron. 33, 1543-1550 (1997).
[CrossRef]

1996 (1)

U. Keller, K. J. Weingarten, F. X. Kartner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Honninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAMs) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Sel. Top. Quantum Electron. 2, 435-453 (1996).
[CrossRef]

1995 (1)

1992 (1)

R. Binder, D. Scott, A. E. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, “Carrier-carrier scattering and optical dephasing in highly excited semiconductors,” Phys. Rev. B 45, 1107-1115 (1992).
[CrossRef]

1975 (1)

H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049-3058 (1975).
[CrossRef]

Aus der Au, J.

U. Keller, K. J. Weingarten, F. X. Kartner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Honninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAMs) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Sel. Top. Quantum Electron. 2, 435-453 (1996).
[CrossRef]

Bahl, M.

M. Bahl, N. C. Panoiu, and R. M. Osgood, “Modeling ultrashort field dynamics in surface emitting lasers by using finite-difference time-domain method,” IEEE J. Quantum Electron. 41, 1244-1252 (2005).
[CrossRef]

M. Bahl, N. C. Panoiu, and R. M. Osgood, “Novel optical and thermal modeling for modelocked VCSELS,” in Proceedings of the IEEE Lightwave Technologies in Instrumentation and Measurement Conference (IEEE, 2004), pp. 17-22.
[CrossRef]

Balle, S.

J. Mulet and S. Balle, “Mode-locking dynamics in electrically driven vertical-external-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 41, 1148-1156 (2005).
[CrossRef]

Binder, R.

R. Binder, D. Scott, A. E. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, “Carrier-carrier scattering and optical dephasing in highly excited semiconductors,” Phys. Rev. B 45, 1107-1115 (1992).
[CrossRef]

Braun, B.

U. Keller, K. J. Weingarten, F. X. Kartner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Honninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAMs) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Sel. Top. Quantum Electron. 2, 435-453 (1996).
[CrossRef]

Chow, W. W.

W. W. Chow and S. W. Koch, Semiconductor-Laser Fundamentals: Physics of the Gain Materials (Springer-Verlag, 1999).

Christ, A.

A. Christ, Analysis and Improvement of the Numerical Properties of the FDTD Algorithm, Series in Microelectronics (Hartung-Gorre, 2005, Vol. 160).

Dennis, J. E.

J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Series in Computational Mathematics (Prentice-Hall, 1983).
[PubMed]

Fluck, R.

U. Keller, K. J. Weingarten, F. X. Kartner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Honninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAMs) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Sel. Top. Quantum Electron. 2, 435-453 (1996).
[CrossRef]

Foreman, H. D.

A. C. Tropper, H. D. Foreman, A. Garnache, K. G. Wilcox, and S. H. Hoogland, “Vertical-external-cavity semiconductor lasers,” J. Phys. D 37, R75-R85 (2004).
[CrossRef]

Garnache, A.

A. C. Tropper, H. D. Foreman, A. Garnache, K. G. Wilcox, and S. H. Hoogland, “Vertical-external-cavity semiconductor lasers,” J. Phys. D 37, R75-R85 (2004).
[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, 3892-3894 (2002).
[CrossRef]

R. Paschotta, R. Häring, A. Garnache, S. Hoogland, A. C. Tropper, and U. Keller, “Soliton-like pulse-shaping mechanism in passively mode-locked surface-emitting semiconductor lasers,” Appl. Phys. B 75, 445-451 (2002).
[CrossRef]

Gold, B.

L. R. Rabiner and B. Gold, Theory and Application of Digital Signal Processing, Computational Microelectronics (Prentice-Hall, 1975).

Häring, R.

R. Paschotta, R. Häring, A. Garnache, S. Hoogland, A. C. Tropper, and U. Keller, “Soliton-like pulse-shaping mechanism in passively mode-locked surface-emitting semiconductor lasers,” Appl. Phys. B 75, 445-451 (2002).
[CrossRef]

Haus, H. A.

H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049-3058 (1975).
[CrossRef]

Henneberger, K.

R. Binder, D. Scott, A. E. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, “Carrier-carrier scattering and optical dephasing in highly excited semiconductors,” Phys. Rev. B 45, 1107-1115 (1992).
[CrossRef]

Honninger, C.

U. Keller, K. J. Weingarten, F. X. Kartner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Honninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAMs) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Sel. Top. Quantum Electron. 2, 435-453 (1996).
[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, 3892-3894 (2002).
[CrossRef]

R. Paschotta, R. Häring, A. Garnache, S. Hoogland, A. C. Tropper, and U. Keller, “Soliton-like pulse-shaping mechanism in passively mode-locked surface-emitting semiconductor lasers,” Appl. Phys. B 75, 445-451 (2002).
[CrossRef]

Hoogland, S. H.

A. C. Tropper, H. D. Foreman, A. Garnache, K. G. Wilcox, and S. H. Hoogland, “Vertical-external-cavity semiconductor lasers,” J. Phys. D 37, R75-R85 (2004).
[CrossRef]

Indik, R. A.

C. Z. Ning, R. A. Indik, and J. V. Moloney, “Effective bloch equations for semiconductor lasers and amplifiers,” IEEE J. Quantum Electron. 33, 1543-1550 (1997).
[CrossRef]

C. Z. Ning, R. A. Indik, and J. V. Moloney, “Self-consistent approach to thermal effects in vertical-cavity surface-emitting lasers,” J. Opt. Soc. Am. B 12, 1993-2004 (1995).
[CrossRef]

Jung, I. D.

U. Keller, K. J. Weingarten, F. X. Kartner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Honninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAMs) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Sel. Top. Quantum Electron. 2, 435-453 (1996).
[CrossRef]

Kartner, F. X.

U. Keller, K. J. Weingarten, F. X. Kartner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Honninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAMs) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Sel. Top. Quantum Electron. 2, 435-453 (1996).
[CrossRef]

Keller, U.

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

R. Paschotta, R. Häring, A. Garnache, S. Hoogland, A. C. Tropper, and U. Keller, “Soliton-like pulse-shaping mechanism in passively mode-locked surface-emitting semiconductor lasers,” Appl. Phys. B 75, 445-451 (2002).
[CrossRef]

U. Keller, K. J. Weingarten, F. X. Kartner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Honninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAMs) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Sel. Top. Quantum Electron. 2, 435-453 (1996).
[CrossRef]

Koch, S. W.

R. Binder, D. Scott, A. E. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, “Carrier-carrier scattering and optical dephasing in highly excited semiconductors,” Phys. Rev. B 45, 1107-1115 (1992).
[CrossRef]

W. W. Chow and S. W. Koch, Semiconductor-Laser Fundamentals: Physics of the Gain Materials (Springer-Verlag, 1999).

Kopf, D.

U. Keller, K. J. Weingarten, F. X. Kartner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Honninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAMs) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Sel. Top. Quantum Electron. 2, 435-453 (1996).
[CrossRef]

Lindberg, M.

R. Binder, D. Scott, A. E. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, “Carrier-carrier scattering and optical dephasing in highly excited semiconductors,” Phys. Rev. B 45, 1107-1115 (1992).
[CrossRef]

Matuschek, N.

U. Keller, K. J. Weingarten, F. X. Kartner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Honninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAMs) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Sel. Top. Quantum Electron. 2, 435-453 (1996).
[CrossRef]

Meyer, J. R.

I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, “Band parameters for III-V compound semiconductors and their alloys,” J. Appl. Phys. 89, 5815-5875 (2001).
[CrossRef]

Moloney, J. V.

C. Z. Ning, R. A. Indik, and J. V. Moloney, “Effective bloch equations for semiconductor lasers and amplifiers,” IEEE J. Quantum Electron. 33, 1543-1550 (1997).
[CrossRef]

C. Z. Ning, R. A. Indik, and J. V. Moloney, “Self-consistent approach to thermal effects in vertical-cavity surface-emitting lasers,” J. Opt. Soc. Am. B 12, 1993-2004 (1995).
[CrossRef]

Mulet, J.

J. Mulet and S. Balle, “Mode-locking dynamics in electrically driven vertical-external-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 41, 1148-1156 (2005).
[CrossRef]

Nevels, R.

C. S. Shin and R. Nevels, “Optimizing the Gaussian excitation function in the finite difference time domain method,” IEEE Trans. Educ. 45, 15-18 (2002).
[CrossRef]

Ning, C. Z.

C. Z. Ning, R. A. Indik, and J. V. Moloney, “Effective bloch equations for semiconductor lasers and amplifiers,” IEEE J. Quantum Electron. 33, 1543-1550 (1997).
[CrossRef]

C. Z. Ning, R. A. Indik, and J. V. Moloney, “Self-consistent approach to thermal effects in vertical-cavity surface-emitting lasers,” J. Opt. Soc. Am. B 12, 1993-2004 (1995).
[CrossRef]

Osgood, R. M.

M. Bahl, N. C. Panoiu, and R. M. Osgood, “Modeling ultrashort field dynamics in surface emitting lasers by using finite-difference time-domain method,” IEEE J. Quantum Electron. 41, 1244-1252 (2005).
[CrossRef]

M. Bahl, N. C. Panoiu, and R. M. Osgood, “Novel optical and thermal modeling for modelocked VCSELS,” in Proceedings of the IEEE Lightwave Technologies in Instrumentation and Measurement Conference (IEEE, 2004), pp. 17-22.
[CrossRef]

Panoiu, N. C.

M. Bahl, N. C. Panoiu, and R. M. Osgood, “Modeling ultrashort field dynamics in surface emitting lasers by using finite-difference time-domain method,” IEEE J. Quantum Electron. 41, 1244-1252 (2005).
[CrossRef]

M. Bahl, N. C. Panoiu, and R. M. Osgood, “Novel optical and thermal modeling for modelocked VCSELS,” in Proceedings of the IEEE Lightwave Technologies in Instrumentation and Measurement Conference (IEEE, 2004), pp. 17-22.
[CrossRef]

Paschotta, R.

R. Paschotta, R. Häring, A. Garnache, S. Hoogland, A. C. Tropper, and U. Keller, “Soliton-like pulse-shaping mechanism in passively mode-locked surface-emitting semiconductor lasers,” Appl. Phys. B 75, 445-451 (2002).
[CrossRef]

Paul, A. E.

R. Binder, D. Scott, A. E. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, “Carrier-carrier scattering and optical dephasing in highly excited semiconductors,” Phys. Rev. B 45, 1107-1115 (1992).
[CrossRef]

Rabiner, L. R.

L. R. Rabiner and B. Gold, Theory and Application of Digital Signal Processing, Computational Microelectronics (Prentice-Hall, 1975).

Ramahi, O. M.

O. M. Ramahi, “Near- and far-field calculations in FDTD simulations using Kirchhoff surface integral representation,” IEEE Trans. Antennas Propag. 45, 753-759 (1997).
[CrossRef]

Ram-Mohan, L. R.

I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, “Band parameters for III-V compound semiconductors and their alloys,” J. Appl. Phys. 89, 5815-5875 (2001).
[CrossRef]

Rappaport, C. M.

W. H. Weedon and C. M. Rappaport, “A general method for FDTD modeling of wave propagation in arbitrary frequency-dispersive media,” IEEE Trans. Antennas Propag. 45, 401-410 (1997).
[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, 3892-3894 (2002).
[CrossRef]

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, 3892-3894 (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, 3892-3894 (2002).
[CrossRef]

Schnabel, R. B.

J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Series in Computational Mathematics (Prentice-Hall, 1983).
[PubMed]

Scott, D.

R. Binder, D. Scott, A. E. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, “Carrier-carrier scattering and optical dephasing in highly excited semiconductors,” Phys. Rev. B 45, 1107-1115 (1992).
[CrossRef]

Shin, C. S.

C. S. Shin and R. Nevels, “Optimizing the Gaussian excitation function in the finite difference time domain method,” IEEE Trans. Educ. 45, 15-18 (2002).
[CrossRef]

Sullivan, D.

D. Sullivan and J. L. Young, “Far-field time-domain calculation from aperture radiators using the FDTD method,” IEEE Trans. Antennas Propag. 49, 464-469 (2001).
[CrossRef]

Sullivan, D. M.

D. M. Sullivan, Electromagnetic Simulation using the FDTD Method, IEEE Press Series on RF and Microwave Technology (IEEE, 2000).
[CrossRef] [PubMed]

Taflove, A.

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 1995).

Tropper, A. C.

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

A. C. Tropper, H. D. Foreman, A. Garnache, K. G. Wilcox, and S. H. Hoogland, “Vertical-external-cavity semiconductor lasers,” J. Phys. D 37, R75-R85 (2004).
[CrossRef]

R. Paschotta, R. Häring, A. Garnache, S. Hoogland, A. C. Tropper, and U. Keller, “Soliton-like pulse-shaping mechanism in passively mode-locked surface-emitting semiconductor lasers,” Appl. Phys. B 75, 445-451 (2002).
[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, 3892-3894 (2002).
[CrossRef]

Vurgaftman, I.

I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, “Band parameters for III-V compound semiconductors and their alloys,” J. Appl. Phys. 89, 5815-5875 (2001).
[CrossRef]

Weedon, W. H.

W. H. Weedon and C. M. Rappaport, “A general method for FDTD modeling of wave propagation in arbitrary frequency-dispersive media,” IEEE Trans. Antennas Propag. 45, 401-410 (1997).
[CrossRef]

Weingarten, K. J.

U. Keller, K. J. Weingarten, F. X. Kartner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Honninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAMs) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Sel. Top. Quantum Electron. 2, 435-453 (1996).
[CrossRef]

Wilcox, K. G.

A. C. Tropper, H. D. Foreman, A. Garnache, K. G. Wilcox, and S. H. Hoogland, “Vertical-external-cavity semiconductor lasers,” J. Phys. D 37, R75-R85 (2004).
[CrossRef]

Young, J. L.

D. Sullivan and J. L. Young, “Far-field time-domain calculation from aperture radiators using the FDTD method,” IEEE Trans. Antennas Propag. 49, 464-469 (2001).
[CrossRef]

Appl. Phys. B (1)

R. Paschotta, R. Häring, A. Garnache, S. Hoogland, A. C. Tropper, and U. Keller, “Soliton-like pulse-shaping mechanism in passively mode-locked surface-emitting semiconductor lasers,” Appl. Phys. B 75, 445-451 (2002).
[CrossRef]

Appl. Phys. Lett. (1)

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Figures (8)

Fig. 1
Fig. 1

(a) Real and (b) imaginary parts of the susceptibility spectrum for carrier densities ranging from 1 × 10 17 to 6 × 10 18 cm 3 in an In 0.13 Ga 0.87 As gain QW at 300 K . Solid curves show the microscopically calculated values and the circles are a [ 4 4 ] Padé approximation.

Fig. 2
Fig. 2

(a) Denominator coefficients of the [ 4 4 ] Padé approximation as a function of carrier density. The markers represent the values obtained by fitting to the calculated susceptibility curves and the curves show the linear interpolation. In the lower subplot, a 1 is depicted by dots and a solid curve and a 3 is depicted by crosses and a dashed curve, respectively. a 0 is not shown since it is 1 by default. (b) Gain curves obtained by linear interpolation of the filter coefficients (curves) between three adjacent carrier density sampling points (circles).

Fig. 3
Fig. 3

(a) Schematic of the VECSEL structure (not to scale). (b) Standard FDTD method resolving the complete structure and (c) combined analytical-FDTD method separately resolving the active mirror and the SESAM on a Yee grid and coupling the two regions via a delayed mutual injection of the backscattered fields.

Fig. 4
Fig. 4

Staggered E and H grids for the two separate FDTD domains and their coupling by injection of the delayed (SFs). The PMLs are used to prevent reflections from the perfect electron conductor (PEC) boundary condition.

Fig. 5
Fig. 5

Frequency response of a Kaiser window low pass filter with a filter order 50 .

Fig. 6
Fig. 6

Output of a VCSEL. In the insets the frequency spectrum and a zoom of the time evolution of the electric field can be seen. The dashed curves depict the result of the new model and the solid curves depict that of a standard FDTD simulation.

Fig. 7
Fig. 7

(a) Output pulse train of the VECSEL recorded above the top mirror. (b) Modulus squared of the output electric field envelope compared to a hyperbolic secant.

Fig. 8
Fig. 8

Electric field output spectrum of the mode-locked device showing the discrete longitudinal modes.

Tables (1)

Tables Icon

Table 1 Simulation Parameters ( 300 K )

Equations (18)

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D ( N , ω ) = ϵ ( N , ω ) E ( ω ) = ϵ 0 ϵ b E ( ω ) + P ( N , ω ) = ϵ 0 ϵ b ( 1 + χ ( N , ω ) ) E ( ω ) ,
H t = 1 μ 0 E z ,
E t = 1 ϵ 0 ϵ b ( H z P t ) .
d N d t = Λ N τ + 1 ω 0 E P t ,
P ( N , ω ) = ϵ 0 ϵ b χ ( N , ω ) E ( ω ) .
χ ( ω ) = i ϵ 0 n opt 2 V n , m k μ k n , m 2 ( f e , k n + f h , k m 1 ) i ( ω k ω ) + γ Q k n , m ( ω ) ,
N = 1 V i = 1 n k f e , k i = 1 V j = 1 m k f h , k j
χ ( N , ω ) B ( Z ) A ( Z ) = b 0 ( N ) + b 1 ( N ) Z 1 + + b L ( N ) Z L a 0 ( N ) + a 1 ( N ) Z 1 + + a M ( N ) Z M ,
{ a , b } = arg min a , b j = 1 n χ ( ω ( j ) ) B ( ω ( j ) ) A ( ω ( j ) ) 2 ,
H j + 1 2 n + 1 2 = H j + 1 2 n 1 2 + Δ t μ 0 Δ z j ( E j + 1 n E j n ) ,
E j n + 1 = E j n + 2 Δ t ϵ 0 ϵ b ( Δ z j 1 + Δ z j ) ( H j + 1 2 n + 1 2 H j 1 2 n + 1 2 ) 1 ϵ 0 ϵ b ( P j n + 1 P j n )
N j n + 1 2 = 1 Δ t ( 2 τ ) 1 + Δ t ( 2 τ ) N j n 1 2 + Δ t 1 + Δ t ( 2 τ ) ( Λ + E j n P j n ω 0 ) .
P j n + 1 = 1 a 4 + b 4 { ( b 4 a 3 ) P j n a 2 P j n 1 a 1 P j n 2 a 0 P j n 3 + 2 b 4 Δ t Δ z j 1 + Δ z j × ( H j + 1 2 n + 1 2 H j 1 2 n + 1 2 ) + ϵ 0 ϵ b [ ( b 4 + b 3 ) E j n + b 2 E j n 1 + b 1 E j n 2 + b 0 E j n 3 ] } ,
( P ) j n + 1 = ( P ) j n + 2 Δ t ( P j n + 1 P j n ) .
E tot , R n + 1 = E tot , R n + 2 Δ t ϵ 0 ϵ b ( Δ z R 1 + Δ z R ) ( H tot , R + 1 2 n + 1 2 H scat , R 1 2 n + 1 2 ) 2 Δ t ϵ 0 ϵ b ( Δ z R 1 + Δ z R ) H inc , R 1 2 n + 1 2 ,
H scat , R 1 2 n + 1 2 = H scat , R 1 2 n 1 2 + Δ t μ 0 Δ z R 1 ( E tot , R n E scat , R 1 n ) Δ t μ 0 Δ z R 1 E inc , R n ,
E tot , L n + 1 = E tot , L n + 2 Δ t ϵ 0 ϵ b ( Δ z L 1 + Δ z L ) ( H scat , L + 1 2 n + 1 2 H tot , L 1 2 n + 1 2 ) + 2 Δ t ϵ 0 ϵ b ( Δ z L 1 + Δ z L ) H inc , L + 1 2 n + 1 2 ,
H scat , L + 1 2 n + 1 2 = H scat , L + 1 2 n 1 2 + Δ t μ 0 Δ z L ( E scat , L + 1 n E tot , L n ) + Δ t μ 0 Δ z R 1 E inc , L n ,

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