Abstract

In this paper, we present a finite-difference time-domain formulation for active gain materials. Our scheme is based on a frequency-dependent conductivity. Experimental material gain is fitted with high accuracy to a multipole Lorentzian model using a semideterministic fitting algorithm. Because our model is an approximation to the full vectorial Maxwell's system of equations, we include carrier diffusion into the rate equations for a two-level system. The material gain is included into the standard set of Maxwell's equations by linking the frequency-dependent conductivity to the rate equations. Lasing is demonstrated for a vertical-cavity-surface-emitting-laser structure and photonic crystal lasers.

© 2007 IEEE

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  1. A. Taflove, S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).
  2. K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).
  3. T. Kashiwa, I. Fukai, "A treatment by the FDTD method of the dispersive characteristics associated with electronic polarization," Microw. Opt. Technol. Lett. 3, 203-205 (1990).
  4. J. L. Young, "Propagation in linear dispersive media: Finite difference time domain methodologies ," IEEE Trans. Antennas Propag. 43, 422-426 (1995).
  5. P. M. Goorjian, A. Taflove, "Direct time integration of Maxwell's equations in nonlinear dispersive media for propagation and scattering of femtosecond electromagnetic solitons," Opt. Lett. 17, 180-182 (1992).
  6. P. M. Goorjian, A. Taflove, R. M. Joseph, S. C. Hagness, "Computational modeling of femtosecond optical solitons from Maxwell's equations," IEEE J. Quantum Electron. 28, 2142-2146 (1992).
  7. M.-K. Seo, G. H. Song, I.-K. Hwang, Y.-H. Lee, "Nonlinear dispersive three-dimensional finite-difference time-domain analysis for photonic-crystal lasers," Opt. Express 13, 9645-9651 (2005).
  8. T. Shiozawa, A. Hirata, "Analysis of free-electron lasers via FDTD method," Electron. Commun. Jpn.—Part II: Electronics 86, 26-36 (2003).
  9. X. Song, S. Gong, S. Jin, Z. Xu, "Two-color interference effects for ultrashort laser pulses propagating in a two-level medium ," Phys. Lett. A 319, 150-156 (2003).
  10. M. Bahl, N. C. Panoiu, 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).
  11. M. Bahl, H. Rao, N. C. Panoiu, R. M. Osgood, "Modeling passively mode-locked VCSELs using FDTD," Proc. NUSOD (2004) pp. 59-60.
  12. J. A. Gruetzmacher, N. F. Scherer, "Finite-difference time-domain simulation of ultrashort pulse propagation incorporating quantum-mechanical response functions," Opt. Lett. 28, 573-575 (2003).
  13. A. Nagra, R. York, "FDTD analysis of wave propagation in nonlinear absorbing and gain media," IEEE Trans. Antennas Propag. 46, 334-340 (1998).
  14. S. Chang, A. Taflove, "Finite-difference time-domain model of lasing action in a four-level two-electron atomic system ," Opt. Express 12, 3827-3833 (2004).
  15. R. W. Ziolkowski, J. M. Arnold, D. M. Gogny, "Ultrafast pulse interactions with two-level atoms," Phys. Rev. A 52, 3082-3094 (1995).
  16. G. M. Slavcheva, J. M. Arnold, R. W. Ziolkowski, "FDTD simulation of the nonlinear gain dynamics in active optical waveguides and semiconductor microcavities," IEEE J. Sel. Topics Quantum Electron. 10, 1052-1062 (2004).
  17. S. C. Hagness, R. M. Joseph, A. Taflove, "Subpicosecond electrodynamics of distributed Bragg reflector microlasers: Results from finite difference time domain simulations," Radio Sci. 31, 931-941 (1996).
  18. S. C. Hagness, FDTD computational electromagnetics modeling of microcavity lasers and resonant optical structures Ph.D. dissertation Northwestern Univ. Dept. Electr. and Comput. Eng.EvanstonIL (1998).
  19. M. Kretschmann, A. A. Maradudin, "Lasing action in waveguide systems and the influence of rough walls," J. Opt. Soc. Amer. B, Opt. Phys. 21, 150-158 (2004).
  20. Harold, Photon Design http://www.photond.com.
  21. M. Jungo, D. Erni, W. Baechthold, "Alternative formulation of carrier transport in spatially-dependent laser rate equations ," Opt. Quantum Electron. 36, 881-891 (2004).
  22. R. W. Hockney, "A fast direct solution of Poisson's equation using Fourier analysis," J. Assoc. Comput. Mach. 8, 95-113 (1965).
  23. P. N. Swarztrauber, "The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle," SIAM Rev. 19, 490-501 (1977).
  24. P. N. Swarztrauber, "A direct method for the discrete solution of separable elliptic equations," SIAM J. Numer. Anal. 11, 1136-1150 (1974).
  25. H. Altug, J. Vuckovic, "Photonic crystal nanocavity array laser," Opt. Express 13, 8820-8828 (2005).
  26. W. H. P. Pernice, F. P. Payne, D. F. G. Gallagher, "A general framework for the finite-difference time-domain simulation of real metals ," IEEE Trans. Antennas Propag. 55, 2916-923 (2007).
  27. J. E. Dennis, Jr.D. M. Gay, R. E. Welsch, "An adaptive nonlinear least-squares algorithm," ACM Trans. Math. Softw. 7, 348-368 (1981).
  28. J. Carroll, J. Whiteaway, D. Plumb, Distributed Feedback Semiconductor Lasers (SPIE Press, 1998) pp. 243-244.
  29. S. W. Corzine, R. S. Geels, J. W. Scott, R. H. Yan, L. A. Coldren, "Design of Fabry–Pérot surface emitting lasers with a periodic gain structure ," IEEE J. Quantum Electron. 25, 1513-1524 (1989).
  30. S. V. Boriskina, "Theoretical prediction of a dramatic Q-factor enhancement and degeneracy removal of WG modes in symmetrical photonic molecules," Opt. Lett. 31, 338-340 (2006).
  31. S. V. Boriskina, T. M. Benson, P. Sewell, A. I. Nosich, "Directional emission, increased free spectral range and mode Q-factors in 2-D wavelength-scale optical microcavity structures," IEEE J. Sel. Topics Quantum Electron. 12, 1175-1182 (2006).

2007 (1)

W. H. P. Pernice, F. P. Payne, D. F. G. Gallagher, "A general framework for the finite-difference time-domain simulation of real metals ," IEEE Trans. Antennas Propag. 55, 2916-923 (2007).

2006 (2)

S. V. Boriskina, "Theoretical prediction of a dramatic Q-factor enhancement and degeneracy removal of WG modes in symmetrical photonic molecules," Opt. Lett. 31, 338-340 (2006).

S. V. Boriskina, T. M. Benson, P. Sewell, A. I. Nosich, "Directional emission, increased free spectral range and mode Q-factors in 2-D wavelength-scale optical microcavity structures," IEEE J. Sel. Topics Quantum Electron. 12, 1175-1182 (2006).

2005 (3)

H. Altug, J. Vuckovic, "Photonic crystal nanocavity array laser," Opt. Express 13, 8820-8828 (2005).

M.-K. Seo, G. H. Song, I.-K. Hwang, Y.-H. Lee, "Nonlinear dispersive three-dimensional finite-difference time-domain analysis for photonic-crystal lasers," Opt. Express 13, 9645-9651 (2005).

M. Bahl, N. C. Panoiu, 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).

2004 (4)

S. Chang, A. Taflove, "Finite-difference time-domain model of lasing action in a four-level two-electron atomic system ," Opt. Express 12, 3827-3833 (2004).

G. M. Slavcheva, J. M. Arnold, R. W. Ziolkowski, "FDTD simulation of the nonlinear gain dynamics in active optical waveguides and semiconductor microcavities," IEEE J. Sel. Topics Quantum Electron. 10, 1052-1062 (2004).

M. Kretschmann, A. A. Maradudin, "Lasing action in waveguide systems and the influence of rough walls," J. Opt. Soc. Amer. B, Opt. Phys. 21, 150-158 (2004).

M. Jungo, D. Erni, W. Baechthold, "Alternative formulation of carrier transport in spatially-dependent laser rate equations ," Opt. Quantum Electron. 36, 881-891 (2004).

2003 (3)

J. A. Gruetzmacher, N. F. Scherer, "Finite-difference time-domain simulation of ultrashort pulse propagation incorporating quantum-mechanical response functions," Opt. Lett. 28, 573-575 (2003).

T. Shiozawa, A. Hirata, "Analysis of free-electron lasers via FDTD method," Electron. Commun. Jpn.—Part II: Electronics 86, 26-36 (2003).

X. Song, S. Gong, S. Jin, Z. Xu, "Two-color interference effects for ultrashort laser pulses propagating in a two-level medium ," Phys. Lett. A 319, 150-156 (2003).

1998 (1)

A. Nagra, R. York, "FDTD analysis of wave propagation in nonlinear absorbing and gain media," IEEE Trans. Antennas Propag. 46, 334-340 (1998).

1996 (1)

S. C. Hagness, R. M. Joseph, A. Taflove, "Subpicosecond electrodynamics of distributed Bragg reflector microlasers: Results from finite difference time domain simulations," Radio Sci. 31, 931-941 (1996).

1995 (2)

R. W. Ziolkowski, J. M. Arnold, D. M. Gogny, "Ultrafast pulse interactions with two-level atoms," Phys. Rev. A 52, 3082-3094 (1995).

J. L. Young, "Propagation in linear dispersive media: Finite difference time domain methodologies ," IEEE Trans. Antennas Propag. 43, 422-426 (1995).

1992 (2)

P. M. Goorjian, A. Taflove, "Direct time integration of Maxwell's equations in nonlinear dispersive media for propagation and scattering of femtosecond electromagnetic solitons," Opt. Lett. 17, 180-182 (1992).

P. M. Goorjian, A. Taflove, R. M. Joseph, S. C. Hagness, "Computational modeling of femtosecond optical solitons from Maxwell's equations," IEEE J. Quantum Electron. 28, 2142-2146 (1992).

1990 (1)

T. Kashiwa, I. Fukai, "A treatment by the FDTD method of the dispersive characteristics associated with electronic polarization," Microw. Opt. Technol. Lett. 3, 203-205 (1990).

1989 (1)

S. W. Corzine, R. S. Geels, J. W. Scott, R. H. Yan, L. A. Coldren, "Design of Fabry–Pérot surface emitting lasers with a periodic gain structure ," IEEE J. Quantum Electron. 25, 1513-1524 (1989).

1981 (1)

J. E. Dennis, Jr.D. M. Gay, R. E. Welsch, "An adaptive nonlinear least-squares algorithm," ACM Trans. Math. Softw. 7, 348-368 (1981).

1977 (1)

P. N. Swarztrauber, "The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle," SIAM Rev. 19, 490-501 (1977).

1974 (1)

P. N. Swarztrauber, "A direct method for the discrete solution of separable elliptic equations," SIAM J. Numer. Anal. 11, 1136-1150 (1974).

1966 (1)

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).

1965 (1)

R. W. Hockney, "A fast direct solution of Poisson's equation using Fourier analysis," J. Assoc. Comput. Mach. 8, 95-113 (1965).

ACM Trans. Math. Softw. (1)

J. E. Dennis, Jr.D. M. Gay, R. E. Welsch, "An adaptive nonlinear least-squares algorithm," ACM Trans. Math. Softw. 7, 348-368 (1981).

Electron. Commun. Jpn.—Part II: Electronics (1)

T. Shiozawa, A. Hirata, "Analysis of free-electron lasers via FDTD method," Electron. Commun. Jpn.—Part II: Electronics 86, 26-36 (2003).

IEEE J. Quantum Electron. (3)

P. M. Goorjian, A. Taflove, R. M. Joseph, S. C. Hagness, "Computational modeling of femtosecond optical solitons from Maxwell's equations," IEEE J. Quantum Electron. 28, 2142-2146 (1992).

M. Bahl, N. C. Panoiu, 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).

S. W. Corzine, R. S. Geels, J. W. Scott, R. H. Yan, L. A. Coldren, "Design of Fabry–Pérot surface emitting lasers with a periodic gain structure ," IEEE J. Quantum Electron. 25, 1513-1524 (1989).

IEEE J. Sel. Topics Quantum Electron. (2)

S. V. Boriskina, T. M. Benson, P. Sewell, A. I. Nosich, "Directional emission, increased free spectral range and mode Q-factors in 2-D wavelength-scale optical microcavity structures," IEEE J. Sel. Topics Quantum Electron. 12, 1175-1182 (2006).

G. M. Slavcheva, J. M. Arnold, R. W. Ziolkowski, "FDTD simulation of the nonlinear gain dynamics in active optical waveguides and semiconductor microcavities," IEEE J. Sel. Topics Quantum Electron. 10, 1052-1062 (2004).

IEEE Trans. Antennas Propag. (4)

A. Nagra, R. York, "FDTD analysis of wave propagation in nonlinear absorbing and gain media," IEEE Trans. Antennas Propag. 46, 334-340 (1998).

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).

J. L. Young, "Propagation in linear dispersive media: Finite difference time domain methodologies ," IEEE Trans. Antennas Propag. 43, 422-426 (1995).

W. H. P. Pernice, F. P. Payne, D. F. G. Gallagher, "A general framework for the finite-difference time-domain simulation of real metals ," IEEE Trans. Antennas Propag. 55, 2916-923 (2007).

J. Assoc. Comput. Mach. (1)

R. W. Hockney, "A fast direct solution of Poisson's equation using Fourier analysis," J. Assoc. Comput. Mach. 8, 95-113 (1965).

J. Opt. Soc. Amer. B, Opt. Phys. (1)

M. Kretschmann, A. A. Maradudin, "Lasing action in waveguide systems and the influence of rough walls," J. Opt. Soc. Amer. B, Opt. Phys. 21, 150-158 (2004).

Microw. Opt. Technol. Lett. (1)

T. Kashiwa, I. Fukai, "A treatment by the FDTD method of the dispersive characteristics associated with electronic polarization," Microw. Opt. Technol. Lett. 3, 203-205 (1990).

Opt. Express (3)

Opt. Lett. (3)

Opt. Quantum Electron. (1)

M. Jungo, D. Erni, W. Baechthold, "Alternative formulation of carrier transport in spatially-dependent laser rate equations ," Opt. Quantum Electron. 36, 881-891 (2004).

Phys. Lett. A (1)

X. Song, S. Gong, S. Jin, Z. Xu, "Two-color interference effects for ultrashort laser pulses propagating in a two-level medium ," Phys. Lett. A 319, 150-156 (2003).

Phys. Rev. A (1)

R. W. Ziolkowski, J. M. Arnold, D. M. Gogny, "Ultrafast pulse interactions with two-level atoms," Phys. Rev. A 52, 3082-3094 (1995).

Radio Sci. (1)

S. C. Hagness, R. M. Joseph, A. Taflove, "Subpicosecond electrodynamics of distributed Bragg reflector microlasers: Results from finite difference time domain simulations," Radio Sci. 31, 931-941 (1996).

SIAM J. Numer. Anal. (1)

P. N. Swarztrauber, "A direct method for the discrete solution of separable elliptic equations," SIAM J. Numer. Anal. 11, 1136-1150 (1974).

SIAM Rev. (1)

P. N. Swarztrauber, "The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle," SIAM Rev. 19, 490-501 (1977).

Other (5)

Harold, Photon Design http://www.photond.com.

J. Carroll, J. Whiteaway, D. Plumb, Distributed Feedback Semiconductor Lasers (SPIE Press, 1998) pp. 243-244.

S. C. Hagness, FDTD computational electromagnetics modeling of microcavity lasers and resonant optical structures Ph.D. dissertation Northwestern Univ. Dept. Electr. and Comput. Eng.EvanstonIL (1998).

A. Taflove, S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).

M. Bahl, H. Rao, N. C. Panoiu, R. M. Osgood, "Modeling passively mode-locked VCSELs using FDTD," Proc. NUSOD (2004) pp. 59-60.

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