Abstract

We report a general computational model of complex material media for electrodynamics simulation using the Finite-Difference Time-Domain (FDTD) method. It is based on a multi-level multi-electron quantum system with electron dynamics governed by Pauli Exclusion Principle, state filling, and dynamical Fermi-Dirac Thermalization, enabling it to treat various solid-state, molecular, or atomic media. The formulation is valid at near or far off resonance as well as at high intensity. We show its FDTD application to a semiconductor in which the carriers’ intraband and interband dynamics, energy band filling, and thermal processes were all incorporated for the first time. The FDTD model is sufficiently complex and yet computationally efficient, enabling it to simulate nanophotonic devices with complex electromagnetic structures requiring simultaneous solution of the mediumfield dynamics in space and time. Applications to direct-gap semiconductors, ultrafast optical phenomena, and multimode microdisk lasers are illustrated.

© 2006 Optical Society of America

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References

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  1. K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in Isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966).
    [CrossRef]
  2. S. D. Gedney, "An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD Lattice," IEEE Trans. Antennas Propag. 44, 1630-1639 (1996), and references therein.
    [CrossRef]
  3. M. Okoniewski, M. Mrozowski, and M. A. Stuchly, "Simple treatment of multi-term dispersion in FDTD," IEEE Microwave Guid. Wave Lett. 7, 121-123 (1997), and references therein.
    [CrossRef]
  4. A. S. Nagra and R. A. York, "FDTD analysis of wave propagation in nonlinear absorbing and gain media," IEEE Trans. Antennas Propag. 46, 334-340 (1998).
    [CrossRef]
  5. Y. Huang, "Simulation of semiconductor material using FDTD method," Master Thesis, Northwestern University, June 2002. https://depot.northwestern.edu/yhu234/publish/YYHMS.pdf
  6. S. Chang, Y. Huang, G. Chang, and S. T. Ho, "THz all-optical shutter based on semiconductor transparency switching by two optical π-pulses," OSA Annual Meeting, TuY3, Long Beach, CA, 2001.
  7. S. T. Ho, research notes, 1998-1999.
  8. Y. Huang, "Simulation of semiconductor structure using FDTD method", presented to the Physics Department at Northwestern University, 15 Jan. 2002.
  9. W. W. Chow, S. Koch, and M. SargentIII, Semiconductor-Laser Physics, (Springer Verlag, Berlin, 1994).
    [CrossRef]
  10. J. Piprek, Optoelectronic Devices: Advanced Simulation and Analysis, (Springer Verlag, New York, 2005).
    [CrossRef]
  11. S. Park, "Development of InGaAsP/InP single-mode lasers using microring resonators for photonic integrated circuits," PhD Thesis, Northwestern University, Dec. 2000, and references therein.
  12. Y. Huang and S. T. Ho, "A numerically efficient semiconductor model with Fermi-Dirac thermalization dynamics (band-filling) for FDTD simulation of optoelectronic and photonic devices," 2005 Technical Digest of the Annual Conference on Lasers and Electro-Optics, Paper QTuD7, Baltimore, MD, May 2005.
  13. S. T. Ho, P. Kumar, and J. H. Shapiro, "Quantum theory of nondegenerate multiwave mixing (I) - General formulation," Phys. Rev. A 37, 2017-2032 (1988).
    [CrossRef] [PubMed]
  14. S. T. Ho and P. Kumar, "Quantum optics in a dielectric: Macroscopic electromagnetic-field and medium operators for a linear dispersive Lossy medium-A microscopic derivation of the operators and their commutation relations," J. Opt. Soc. Am. B 10, 1620-1636 (1993).
    [CrossRef]
  15. S. T. Ho, P. Kumar, and J. H. Shapiro, "Vector-field quantum model of degenerate four-wave mixing," Phys. Rev. A 34, 293-303 (July 1986).
    [CrossRef] [PubMed]
  16. J. J. Sakurai, Advanced Quantum Mechanics, (Addison Wesley, 1967).
  17. in semiconductor corresponds to the spatially localized operator.
  18. W. H. Louisell, Quantum Statistical Properties of Radiation, (Wiley-Interscience, New York, 1990).
  19. For example, if three upper levels can decay to a single ground level, then each upper level will be associated with a transition dipole so that the total number of dipoles involved will be three, which is equal to the number of the upper levels.
  20. R. F. Kazarinov, C. H. Henry, and R. A. Logan, "Longitudinal mode self-stabilization in semicondcutor lasers," J. Appl. Phys. 53, 4631-4644 (1982).
    [CrossRef]
  21. S. Marrin, B. Deveaud, F. Clerot, K. Fuliwara, and K. Mitsunaga, "Capture of photoexcited carriers in a single quantum well with different confinement structures," IEEE J. Quantum Electron. 27, 1669-1675 (1991).
    [CrossRef]
  22. L. A. Coldren and S. W. Corzine, Diode lasers and photonic integrated circuits, (Wiley, John & Sons. 1995).
  23. J. L. Oudar, D. Hulin, A. Migus, A. Antonetti, and F. Alexandre, "Subpicosecond spectral hole burning due to nonthermalized photoexcited carriers in GaAs," Phys. Rev. Lett. 55, 2074-2077 (1985).
    [CrossRef] [PubMed]
  24. D. Y. Chu, M. K. Chin, S. Z. Xu, T. Y. Chang, and S. T. Ho, "1.5 µm InGaAs/InAlGaAs Quantum-well microdisk lasers," IEEE Photon. Technol. Lett. 5, 1353-1355 (1993).
    [CrossRef]
  25. W. Fang, J. Y. Xu, A. Yamilov, H. Cao, Y. Ma, S. T. Ho, and G. S. Solomon, "Large enhancement of spontaneous emission rates of InAs quantum dots in GaAs microdisks," Opt. Lett. 27, 948-950 (2002).
    [CrossRef]
  26. J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, C. W. Tu, and S. T. Ho, "Photonic-wire laser," Phys. Rev. Lett. 75, 2678-2681 (1995).
    [CrossRef] [PubMed]

2002 (1)

1998 (1)

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

1997 (1)

M. Okoniewski, M. Mrozowski, and M. A. Stuchly, "Simple treatment of multi-term dispersion in FDTD," IEEE Microwave Guid. Wave Lett. 7, 121-123 (1997), and references therein.
[CrossRef]

1996 (1)

S. D. Gedney, "An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD Lattice," IEEE Trans. Antennas Propag. 44, 1630-1639 (1996), and references therein.
[CrossRef]

1995 (1)

J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, C. W. Tu, and S. T. Ho, "Photonic-wire laser," Phys. Rev. Lett. 75, 2678-2681 (1995).
[CrossRef] [PubMed]

1993 (2)

1991 (1)

S. Marrin, B. Deveaud, F. Clerot, K. Fuliwara, and K. Mitsunaga, "Capture of photoexcited carriers in a single quantum well with different confinement structures," IEEE J. Quantum Electron. 27, 1669-1675 (1991).
[CrossRef]

1988 (1)

S. T. Ho, P. Kumar, and J. H. Shapiro, "Quantum theory of nondegenerate multiwave mixing (I) - General formulation," Phys. Rev. A 37, 2017-2032 (1988).
[CrossRef] [PubMed]

1986 (1)

S. T. Ho, P. Kumar, and J. H. Shapiro, "Vector-field quantum model of degenerate four-wave mixing," Phys. Rev. A 34, 293-303 (July 1986).
[CrossRef] [PubMed]

1985 (1)

J. L. Oudar, D. Hulin, A. Migus, A. Antonetti, and F. Alexandre, "Subpicosecond spectral hole burning due to nonthermalized photoexcited carriers in GaAs," Phys. Rev. Lett. 55, 2074-2077 (1985).
[CrossRef] [PubMed]

1982 (1)

R. F. Kazarinov, C. H. Henry, and R. A. Logan, "Longitudinal mode self-stabilization in semicondcutor lasers," J. Appl. Phys. 53, 4631-4644 (1982).
[CrossRef]

1966 (1)

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in Isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

Alexandre, F.

J. L. Oudar, D. Hulin, A. Migus, A. Antonetti, and F. Alexandre, "Subpicosecond spectral hole burning due to nonthermalized photoexcited carriers in GaAs," Phys. Rev. Lett. 55, 2074-2077 (1985).
[CrossRef] [PubMed]

Antonetti, A.

J. L. Oudar, D. Hulin, A. Migus, A. Antonetti, and F. Alexandre, "Subpicosecond spectral hole burning due to nonthermalized photoexcited carriers in GaAs," Phys. Rev. Lett. 55, 2074-2077 (1985).
[CrossRef] [PubMed]

Bi, W. G.

J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, C. W. Tu, and S. T. Ho, "Photonic-wire laser," Phys. Rev. Lett. 75, 2678-2681 (1995).
[CrossRef] [PubMed]

Cao, H.

Chang, T. Y.

D. Y. Chu, M. K. Chin, S. Z. Xu, T. Y. Chang, and S. T. Ho, "1.5 µm InGaAs/InAlGaAs Quantum-well microdisk lasers," IEEE Photon. Technol. Lett. 5, 1353-1355 (1993).
[CrossRef]

Chin, M. K.

D. Y. Chu, M. K. Chin, S. Z. Xu, T. Y. Chang, and S. T. Ho, "1.5 µm InGaAs/InAlGaAs Quantum-well microdisk lasers," IEEE Photon. Technol. Lett. 5, 1353-1355 (1993).
[CrossRef]

Chu, D. Y.

J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, C. W. Tu, and S. T. Ho, "Photonic-wire laser," Phys. Rev. Lett. 75, 2678-2681 (1995).
[CrossRef] [PubMed]

D. Y. Chu, M. K. Chin, S. Z. Xu, T. Y. Chang, and S. T. Ho, "1.5 µm InGaAs/InAlGaAs Quantum-well microdisk lasers," IEEE Photon. Technol. Lett. 5, 1353-1355 (1993).
[CrossRef]

Clerot, F.

S. Marrin, B. Deveaud, F. Clerot, K. Fuliwara, and K. Mitsunaga, "Capture of photoexcited carriers in a single quantum well with different confinement structures," IEEE J. Quantum Electron. 27, 1669-1675 (1991).
[CrossRef]

Deveaud, B.

S. Marrin, B. Deveaud, F. Clerot, K. Fuliwara, and K. Mitsunaga, "Capture of photoexcited carriers in a single quantum well with different confinement structures," IEEE J. Quantum Electron. 27, 1669-1675 (1991).
[CrossRef]

Fang, W.

Fuliwara, K.

S. Marrin, B. Deveaud, F. Clerot, K. Fuliwara, and K. Mitsunaga, "Capture of photoexcited carriers in a single quantum well with different confinement structures," IEEE J. Quantum Electron. 27, 1669-1675 (1991).
[CrossRef]

Gedney, S. D.

S. D. Gedney, "An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD Lattice," IEEE Trans. Antennas Propag. 44, 1630-1639 (1996), and references therein.
[CrossRef]

Henry, C. H.

R. F. Kazarinov, C. H. Henry, and R. A. Logan, "Longitudinal mode self-stabilization in semicondcutor lasers," J. Appl. Phys. 53, 4631-4644 (1982).
[CrossRef]

Ho, S. T.

W. Fang, J. Y. Xu, A. Yamilov, H. Cao, Y. Ma, S. T. Ho, and G. S. Solomon, "Large enhancement of spontaneous emission rates of InAs quantum dots in GaAs microdisks," Opt. Lett. 27, 948-950 (2002).
[CrossRef]

J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, C. W. Tu, and S. T. Ho, "Photonic-wire laser," Phys. Rev. Lett. 75, 2678-2681 (1995).
[CrossRef] [PubMed]

S. T. Ho and P. Kumar, "Quantum optics in a dielectric: Macroscopic electromagnetic-field and medium operators for a linear dispersive Lossy medium-A microscopic derivation of the operators and their commutation relations," J. Opt. Soc. Am. B 10, 1620-1636 (1993).
[CrossRef]

D. Y. Chu, M. K. Chin, S. Z. Xu, T. Y. Chang, and S. T. Ho, "1.5 µm InGaAs/InAlGaAs Quantum-well microdisk lasers," IEEE Photon. Technol. Lett. 5, 1353-1355 (1993).
[CrossRef]

S. T. Ho, P. Kumar, and J. H. Shapiro, "Quantum theory of nondegenerate multiwave mixing (I) - General formulation," Phys. Rev. A 37, 2017-2032 (1988).
[CrossRef] [PubMed]

S. T. Ho, P. Kumar, and J. H. Shapiro, "Vector-field quantum model of degenerate four-wave mixing," Phys. Rev. A 34, 293-303 (July 1986).
[CrossRef] [PubMed]

Hulin, D.

J. L. Oudar, D. Hulin, A. Migus, A. Antonetti, and F. Alexandre, "Subpicosecond spectral hole burning due to nonthermalized photoexcited carriers in GaAs," Phys. Rev. Lett. 55, 2074-2077 (1985).
[CrossRef] [PubMed]

Kazarinov, R. F.

R. F. Kazarinov, C. H. Henry, and R. A. Logan, "Longitudinal mode self-stabilization in semicondcutor lasers," J. Appl. Phys. 53, 4631-4644 (1982).
[CrossRef]

Kumar, P.

S. T. Ho and P. Kumar, "Quantum optics in a dielectric: Macroscopic electromagnetic-field and medium operators for a linear dispersive Lossy medium-A microscopic derivation of the operators and their commutation relations," J. Opt. Soc. Am. B 10, 1620-1636 (1993).
[CrossRef]

S. T. Ho, P. Kumar, and J. H. Shapiro, "Quantum theory of nondegenerate multiwave mixing (I) - General formulation," Phys. Rev. A 37, 2017-2032 (1988).
[CrossRef] [PubMed]

S. T. Ho, P. Kumar, and J. H. Shapiro, "Vector-field quantum model of degenerate four-wave mixing," Phys. Rev. A 34, 293-303 (July 1986).
[CrossRef] [PubMed]

Logan, R. A.

R. F. Kazarinov, C. H. Henry, and R. A. Logan, "Longitudinal mode self-stabilization in semicondcutor lasers," J. Appl. Phys. 53, 4631-4644 (1982).
[CrossRef]

Ma, Y.

Marrin, S.

S. Marrin, B. Deveaud, F. Clerot, K. Fuliwara, and K. Mitsunaga, "Capture of photoexcited carriers in a single quantum well with different confinement structures," IEEE J. Quantum Electron. 27, 1669-1675 (1991).
[CrossRef]

Migus, A.

J. L. Oudar, D. Hulin, A. Migus, A. Antonetti, and F. Alexandre, "Subpicosecond spectral hole burning due to nonthermalized photoexcited carriers in GaAs," Phys. Rev. Lett. 55, 2074-2077 (1985).
[CrossRef] [PubMed]

Mitsunaga, K.

S. Marrin, B. Deveaud, F. Clerot, K. Fuliwara, and K. Mitsunaga, "Capture of photoexcited carriers in a single quantum well with different confinement structures," IEEE J. Quantum Electron. 27, 1669-1675 (1991).
[CrossRef]

Mrozowski, M.

M. Okoniewski, M. Mrozowski, and M. A. Stuchly, "Simple treatment of multi-term dispersion in FDTD," IEEE Microwave Guid. Wave Lett. 7, 121-123 (1997), and references therein.
[CrossRef]

Nagra, A. S.

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

Okoniewski, M.

M. Okoniewski, M. Mrozowski, and M. A. Stuchly, "Simple treatment of multi-term dispersion in FDTD," IEEE Microwave Guid. Wave Lett. 7, 121-123 (1997), and references therein.
[CrossRef]

Oudar, J. L.

J. L. Oudar, D. Hulin, A. Migus, A. Antonetti, and F. Alexandre, "Subpicosecond spectral hole burning due to nonthermalized photoexcited carriers in GaAs," Phys. Rev. Lett. 55, 2074-2077 (1985).
[CrossRef] [PubMed]

Shapiro, J. H.

S. T. Ho, P. Kumar, and J. H. Shapiro, "Quantum theory of nondegenerate multiwave mixing (I) - General formulation," Phys. Rev. A 37, 2017-2032 (1988).
[CrossRef] [PubMed]

S. T. Ho, P. Kumar, and J. H. Shapiro, "Vector-field quantum model of degenerate four-wave mixing," Phys. Rev. A 34, 293-303 (July 1986).
[CrossRef] [PubMed]

Solomon, G. S.

Stuchly, M. A.

M. Okoniewski, M. Mrozowski, and M. A. Stuchly, "Simple treatment of multi-term dispersion in FDTD," IEEE Microwave Guid. Wave Lett. 7, 121-123 (1997), and references therein.
[CrossRef]

Tiberio, R. C.

J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, C. W. Tu, and S. T. Ho, "Photonic-wire laser," Phys. Rev. Lett. 75, 2678-2681 (1995).
[CrossRef] [PubMed]

Tu, C. W.

J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, C. W. Tu, and S. T. Ho, "Photonic-wire laser," Phys. Rev. Lett. 75, 2678-2681 (1995).
[CrossRef] [PubMed]

Wu, S. L.

J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, C. W. Tu, and S. T. Ho, "Photonic-wire laser," Phys. Rev. Lett. 75, 2678-2681 (1995).
[CrossRef] [PubMed]

Xu, J. Y.

Xu, S. Z.

D. Y. Chu, M. K. Chin, S. Z. Xu, T. Y. Chang, and S. T. Ho, "1.5 µm InGaAs/InAlGaAs Quantum-well microdisk lasers," IEEE Photon. Technol. Lett. 5, 1353-1355 (1993).
[CrossRef]

Yamilov, A.

Yee, K. S.

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in Isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

York, R. A.

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

Zhang, J. P.

J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, C. W. Tu, and S. T. Ho, "Photonic-wire laser," Phys. Rev. Lett. 75, 2678-2681 (1995).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

S. Marrin, B. Deveaud, F. Clerot, K. Fuliwara, and K. Mitsunaga, "Capture of photoexcited carriers in a single quantum well with different confinement structures," IEEE J. Quantum Electron. 27, 1669-1675 (1991).
[CrossRef]

IEEE Microwave Guid. Wave Lett. (1)

M. Okoniewski, M. Mrozowski, and M. A. Stuchly, "Simple treatment of multi-term dispersion in FDTD," IEEE Microwave Guid. Wave Lett. 7, 121-123 (1997), and references therein.
[CrossRef]

IEEE Photon. Technol. Lett. (1)

D. Y. Chu, M. K. Chin, S. Z. Xu, T. Y. Chang, and S. T. Ho, "1.5 µm InGaAs/InAlGaAs Quantum-well microdisk lasers," IEEE Photon. Technol. Lett. 5, 1353-1355 (1993).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

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

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in Isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

S. D. Gedney, "An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD Lattice," IEEE Trans. Antennas Propag. 44, 1630-1639 (1996), and references therein.
[CrossRef]

J. Appl. Phys. (1)

R. F. Kazarinov, C. H. Henry, and R. A. Logan, "Longitudinal mode self-stabilization in semicondcutor lasers," J. Appl. Phys. 53, 4631-4644 (1982).
[CrossRef]

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

Opt. Lett. (1)

Phys. Rev. A (2)

S. T. Ho, P. Kumar, and J. H. Shapiro, "Vector-field quantum model of degenerate four-wave mixing," Phys. Rev. A 34, 293-303 (July 1986).
[CrossRef] [PubMed]

S. T. Ho, P. Kumar, and J. H. Shapiro, "Quantum theory of nondegenerate multiwave mixing (I) - General formulation," Phys. Rev. A 37, 2017-2032 (1988).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, C. W. Tu, and S. T. Ho, "Photonic-wire laser," Phys. Rev. Lett. 75, 2678-2681 (1995).
[CrossRef] [PubMed]

J. L. Oudar, D. Hulin, A. Migus, A. Antonetti, and F. Alexandre, "Subpicosecond spectral hole burning due to nonthermalized photoexcited carriers in GaAs," Phys. Rev. Lett. 55, 2074-2077 (1985).
[CrossRef] [PubMed]

Other (13)

L. A. Coldren and S. W. Corzine, Diode lasers and photonic integrated circuits, (Wiley, John & Sons. 1995).

J. J. Sakurai, Advanced Quantum Mechanics, (Addison Wesley, 1967).

in semiconductor corresponds to the spatially localized operator.

W. H. Louisell, Quantum Statistical Properties of Radiation, (Wiley-Interscience, New York, 1990).

For example, if three upper levels can decay to a single ground level, then each upper level will be associated with a transition dipole so that the total number of dipoles involved will be three, which is equal to the number of the upper levels.

Y. Huang, "Simulation of semiconductor material using FDTD method," Master Thesis, Northwestern University, June 2002. https://depot.northwestern.edu/yhu234/publish/YYHMS.pdf

S. Chang, Y. Huang, G. Chang, and S. T. Ho, "THz all-optical shutter based on semiconductor transparency switching by two optical π-pulses," OSA Annual Meeting, TuY3, Long Beach, CA, 2001.

S. T. Ho, research notes, 1998-1999.

Y. Huang, "Simulation of semiconductor structure using FDTD method", presented to the Physics Department at Northwestern University, 15 Jan. 2002.

W. W. Chow, S. Koch, and M. SargentIII, Semiconductor-Laser Physics, (Springer Verlag, Berlin, 1994).
[CrossRef]

J. Piprek, Optoelectronic Devices: Advanced Simulation and Analysis, (Springer Verlag, New York, 2005).
[CrossRef]

S. Park, "Development of InGaAsP/InP single-mode lasers using microring resonators for photonic integrated circuits," PhD Thesis, Northwestern University, Dec. 2000, and references therein.

Y. Huang and S. T. Ho, "A numerically efficient semiconductor model with Fermi-Dirac thermalization dynamics (band-filling) for FDTD simulation of optoelectronic and photonic devices," 2005 Technical Digest of the Annual Conference on Lasers and Electro-Optics, Paper QTuD7, Baltimore, MD, May 2005.

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Figures (7)

Fig. 1.
Fig. 1.

electron dynamics in our 4-level 2-electron model [Refs. 5–8]: (a) electron interband and intraband dynamics in semiconductor medium under excitation of photon with above-bandgap energy; (b) representation by four energy levels and two electrons.

Fig. 2.
Fig. 2.

The multi-level multi-electron model for FDTD simulation of semiconductor material.

Fig. 3.
Fig. 3.

The multi-energy-level model for the FDTD simulation of semiconductor material.

Fig. 4.
Fig. 4.

Absorption spectra at different carrier densities obtained by using different number of energy level pairs: (a) 5 level pairs, (b) 10 level pairs.

Fig. 5.
Fig. 5.

Medium’s transient response under strong optical pumping: (a) input optical pulse; (b) normalized volume density of states at each of the 5 energy levels in the conduction and valence band as a function of time.

Fig. 6.
Fig. 6.

Simulation of microdisk laser: (a) dimension and refractive index of the microdisk laser; (b) simulated electrical field pattern when lasing; (c) optical intensity inside the microdisk laser at different injection current densities.

Fig. 7.
Fig. 7.

(a) Lasing spectra of the 2μm diameter microdisk laser at different injection current densities showing multimode lasing at high current of 43kA/cm2 (b) zoom in to show the wavelength shift in the first lasing mode.

Equations (52)

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

H ̂ = 1 2 m e j = 1 N 0 [ P ̂ ej ( t ) e A ̂ ( r ̂ ej ( t ) , t ) ] 2 + j = 1 N 0 H ̂ Cj
+ 1 2 V Q d 3 r 1 2 [ d ( Ω m ε m ) d Ω m E ̂ 2 ( r , t ) + μ 0 H ̂ 2 ( r , t ) ] + H ̂ ph ,
A ̂ ( r ̂ ej ( t ) , t ) = g m e [ a ̂ ( t ) e i k m · r ̂ ej ( t ) + a ̂ ( t ) e i k m · r ̂ ej ( t ) ] ,
E ̂ ( r ̂ ej ( t ) , t ) = i Ω m g m e [ a ̂ ( t ) e i k m · r ̂ ej ( t ) a ̂ ( t ) e i k m · r ̂ ej ( t ) ] = E ̂ ( r ̂ ej ( t ) , t ) ,
μ 0 H ̂ ( r ̂ ej ( t ) , t ) = i g m ( K × e ) [ a ̂ ( t ) e i k m · r ̂ ej ( t ) + a ̂ ( t ) e i k m · r ̂ ej ( t ) ] = H ̂ ( r ̂ ej ( t ) , t ) ,
d a ̂ ( t ) dt = i ħ [ H ( t ) , a ̂ ( t ) ] = i Ω m a ̂ ( t ) + i g m j e m e [ P ̂ ej ( t ) e A ̂ ( r ̂ ej ( t ) , t ) ] · e e i k m · r ̂ ej ( t ) ,
d e r ̂ ej ( t ) d t = i ħ [ H ( t ) , e r ̂ ej ( t ) ] = e m e [ P ̂ ej ( t ) e A ̂ ( r ̂ ej ( t ) , t ) ] = d μ ̂ j ( t ) d t ,
a ̂ ( t ) t = i Ω m a ̂ ( t ) + i g m j μ ̂ j ( t ) t · e e i k m · r ̂ ej ( t ) .
H ̂ Atom = jg ħ ω g n ̂ gj ( t ) + js ħ ω s n ̂ sj ( t ) ,
H ̂ AF = js i ω a [ μ sj V ̂ sj ( t ) μ sj * V ̂ sj ( t ) ] · A ̂ ( r nj , t ) + j e 2 2 m e A ̂ 2 ( r nj , t ) ,
H ̂ F = ħ Ω m [ a ̂ ( t ) a ̂ ( t ) + 1 2 ] ,
d V ̂ sj ( t ) d t = i ω a V ̂ sj ( t ) + ω a μ sj ħ ( n ̂ sj ( t ) n ̂ gj ( t ) ) A ̂ s ( r nj , t ) ,
d V ̂ sj ( t ) d t = i ω a V ̂ sj ( t ) + ω a μ sj * ħ ( n ̂ sj ( t ) n ̂ gj ( t ) ) A ̂ s ( r nj , t ) ,
d n ̂ sj ( t ) d t = ω a ħ [ μ sj * V ̂ sj ( t ) + μ sj V ̂ sj ( t ) ] A ̂ s ( r nj , t ) = ω a ħ μ ̂ sj ( t ) A ̂ s ( r nj , t ) ,
d n ̂ sj ( t ) d t = s ω a ħ [ μ sj * V ̂ sj ( t ) + μ sj V ̂ sj ( t ) ] A ̂ s ( r nj , t ) = s ω a ħ μ ̂ sj ( t ) A ̂ s ( r nj , t ) ,
{ c ̂ k 1 j ( t ) , c ̂ k 2 j ( t ) } = c ̂ k 1 j ( t ) c ̂ k 2 j ( t ) + c ̂ k 2 j ( t ) c ̂ k 1 j ( t ) = δ k 1 , k 2 ,
{ c ̂ k 1 j ( t ) , c ̂ k 2 j ( t ) } = { c ̂ k 1 j ( t ) , c ̂ k 2 j ( t ) } = 0 ,
H ̂ Atom = jg ħ ω g c ̂ gj ( t ) c ̂ gj ( t ) + js ħ ω s c ̂ sj ( t ) c ̂ sj ( t ) ,
H ̂ AF = js i ω a [ μ sj c ̂ sj ( t ) c ̂ gj ( t ) μ sj * c ̂ gj ( t ) c ̂ sj ( t ) ] · A ̂ ( r nj , t ) + j e 2 2 m e A ̂ 2 ( r nj , t ) .
d V ̂ sj ( t ) d t = i ω a V ̂ sj ( t ) γ Vs V ̂ sj ( t ) + ω a μ sj ħ ( n ̂ sj ( t ) n ̂ gj ( t ) ) A ̂ s ( r nj , t ) + Γ ̂ V sj ( t ) ,
d V ̂ sj ( t ) d t = i ω a V ̂ sj ( t ) γ Vs V ̂ sj ( t ) + ω a μ sj * ħ ( n ̂ sj ( t ) n ̂ gj ( t ) ) A ̂ s ( r nj , t ) + Γ ̂ V sj ( t ) ,
d n ̂ sj ( t ) d t = γ Ns n ̂ sj ( t ) [ 1 n ̂ gj ( t ) ] ω a ħ μ ̂ sj ( t ) A ̂ s ( r nj , t ) + Γ ̂ n sj ( t ) ,
d n ̂ gj ( t ) d t = s γ Ns n ̂ sj ( t ) [ 1 n ̂ gj ( t ) ] + s ω a ħ μ ̂ sj ( t ) A ̂ s ( r nj , t ) Γ ̂ n sj ( t ) ,
2 μ ̂ sj ( t ) t 2 + 2 γ Vs μ ̂ sj ( t ) t + [ ω a 2 + ( 2 ω a ) 2 ħ 2 μ sj 2 A ̂ s 2 ( r nj , t ) ] μ ̂ sj ( t )
= 2 ω a ħ μ sj 2 [ n ̂ sj ( t ) n ̂ gj ( t ) ] E ̂ s ( r nj . t ) ,
d H ( r , t ) d t = 1 μ 0 × E ( r , t ) ; d E ( r , t ) d t = 1 ε 0 n 2 × H ( r , t ) 1 ε 0 n 2 d P ( r , t ) d t .
p z ( r , t ) | r ( r nj , δV ) = N 0 μ zj ( t ) δV = μ zj ( t ) N dip i ( r ) ,
d 2 P iz ( r , t ) d t 2 + γ i d P iz ( r , t ) dt + [ ω ai 2 + ( 2 ω ai ) 2 ħ 2 μ zi 2 A z 2 ( r , t ) ] P iz ( r , t )
= 2 ω ai ħ μ zi 2 [ N dip i ( r ) N Vi 0 ( r ) N Vi ( r , t ) N dip i ( r ) N Ci 0 ( r ) N Ci ( r , t ) ] E z ( r , t ) ,
d N Ci ( r , t ) d t = Δ N i ( r , t ) Δ N ( i , , i 1 ) C ( r , t ) + Δ N ( i + 1 , i ) C ( r , t ) + W pump ( r , t ) ,
d N Vi ( r , t ) d t = Δ N i ( r , t ) + Δ N ( i + 1 , i ) V ( r , t ) Δ N ( i , i 1 ) V ( r , t ) W pump ( r , t ) ,
Δ N ( i , i 1 ) C ( r , t ) = N Ci ( r , t ) [ 1 N C ( i 1 ) ( r , t ) / N C ( i 1 ) 0 ( r ) ] τ ( i , i 1 ) C N C ( i 1 ) ( r , t ) [ 1 N C i ( r , t ) / N C i 0 ( r ) ] τ ( i 1 , i ) C
Δ N ( i , i 1 ) V ( r , t ) = N Vi ( r , t ) [ 1 N V ( i 1 ) ( r , t ) / N V ( i 1 ) 0 ( r ) ] τ ( i , i 1 ) V N V ( i 1 ) ( r , t ) [ 1 N Vi ( r , t ) / N Vi 0 ( r ) ] τ ( i 1 , i ) V
Δ N i ( r , t ) = ω ai ħ A z ( r , t ) · P iz ( r , t ) + N Ci ( r , t ) [ 1 N Vi ( r , t ) / N Vi 0 ( r ) ] τ i .
ρ ( E ) d E = 1 2 π 2 [ 2 m * ħ 2 ] 3 / 2 Δ E 1 / 2 dE ,
Δ E Ci = ( E i + B E G ) m V m V + m C , Δ E Vi = ( E i + B E G ) m C m V + m C ,
N dip i = N C , Vi 0 ( r ) = Δ E C , V ( i 1 ) Δ E C , Vi ρ ( Δ E ) d E = 16 2 m C 3 / 2 m V 3 / 2 [ ( E i + B E G ) 3 / 2 ( E i B E G ) 3 / 2 ] 3 π 2 ħ 3 ( m C + m V ) 3 / 2
Δ N ( i , i 1 ) C ( r , t ) = N Ci ( r , t ) [ 1 N C ( i 1 ) ( r , t ) / N C ( i 1 ) 0 ( r ) ] τ ( i , i 1 ) C N C ( i 1 ) ( r , t ) [ 1 N C i ( r , t ) / N C i 0 ( r ) ] τ ( i 1 , i ) C
τ ( i 1 , i ) C τ ( i , i 1 ) C = N C ( i 1 ) _ S ( r ) [ 1 N Ci _ S ( r ) / N Ci 0 ( r ) ] N Ci _ S ( r ) [ 1 N C ( i 1 ) _ S ( r ) / N C ( i 1 ) 0 ( r ) ] ,
N Ci _ S = f ( Δ E Ci ) · N Ci 0 ( r ) = 1 e [ ( E Ci E FC k B T ) ] + 1 · N Ci 0 ( r ) ,
τ ( i 1 , i ) C τ ( i , i 1 ) C = 1 e ( ( E C ( i 1 ) E F C ) / k B T ) + 1 · N C ( i 1 ) 0 ( r ) e ( ( E Ci E F C ) / k B T ) e ( ( E Ci E F C ) / k B T ) + 1 1 e ( ( E Ci E F C ) / k B T ) + 1 · N Ci 0 ( r ) e ( ( E C ( i 1 ) E F C ) / k B T ) e ( ( E C ( i 1 ) E F C ) / k B T ) + 1 = N C ( i 1 ) 0 ( r ) N Ci 0 ( r ) e ( E Ci E C ( i 1 ) ) / k B T .
N Ci | u 1 2 , v + 1 2 , w n + 1 = N Ci | u 1 2 , v + 1 2 , w n 1 + 2 Δ t ( Δ N i | u 1 2 , v + 1 2 , w n Δ N ( i , i 1 ) C | u 1 2 , v + 1 2 , w n + Δ N ( i + 1 , i ) C | u 1 2 , v + 1 2 , w n + W pump ) ,
N Vi | u 1 2 , v + 1 2 , w n + 1 = N Vi | u 1 2 , v + 1 2 , w n 1 + 2 Δ t ( Δ N i | u 1 2 , v + 1 2 , w n Δ N ( i , i 1 ) V | u 1 2 , v + 1 2 , w n + Δ N ( i + 1 , i ) V | u 1 2 , v + 1 2 , w n + W pump ) ,
Δ N i | u 1 2 , v + 1 2 , w n = ω ai ħ A z | u 1 2 , v + 1 2 , w n P iz | u 1 2 , v + 1 2 , w n + N Ci | u 1 2 , v + 1 2 , w n ( 1 N Vi | u 1 2 , v + 1 2 , w n N iV 0 | u 1 2 , v + 1 2 , w ) τ i ,
Δ N ( i , i 1 ) C | u 1 2 , v + 1 2 , w n = N Ci | u 1 2 , v + 1 2 , w n [ 1 N C ( i 1 ) | u 1 2 , v + 1 2 , w n N ( i 1 ) C 0 | u 1 2 , v + 1 2 , w ] τ ( i 1 , i ) C N C ( i 1 ) | u 1 2 , v + 1 2 , w n [ 1 N C i | u 1 2 , v + 1 2 , w n N i C 0 | u 1 2 , v + 1 2 , w ] τ ( i , i 1 ) C ,
Δ N ( i , i 1 ) V | u 1 2 , v + 1 2 , w n = N Vi | u 1 2 , v + 1 2 , w n [ 1 N V ( i 1 ) | u 1 2 , v + 1 2 , w n N ( i 1 ) V 0 | u 1 2 , v + 1 2 , w ] τ ( i , i 1 ) V N V ( i 1 ) | u 1 2 , v + 1 2 , w n [ 1 N V i | u 1 2 , v + 1 2 , w n N i V 0 | u 1 2 , v + 1 2 , w ] τ ( i 1 , i ) V ,
i = 1 M N Ci | u 1 2 , v + 1 2 , w n + 1 + i = 1 M N Ci | u 1 2 , v + 1 2 , w n + 1 = i = 1 M N iC 0 | u 1 2 , v + 1 2 , w = i = 1 M N iV 0 | u 1 2 , v + 1 2 , w .
P i , z u 1 2 , ν + 1 2 , w n + 1
= 4 2 Δ t 2 ( ω ai 2 + 4 ω ai 2 ħ 2 μ i 2 A z 2 u 1 2 , ν + 1 2 , w n ) 2 + Δt· γ i P i , z u 1 2 , ν + 1 2 , w n + Δ γ i 2 2 + Δt· γ i P i , z u 1 2 , ν + 1 2 , w n - 1
4 Δ t 2 ω ai ħ ( 2 + Δt· γ i ) μ i 2 ( N Ci u 1 2 , ν + 1 2 , w n N z u 1 2 , ν + 1 2 , w n ) E z u 1 2 , ν + 1 2 , w n .
E z u 1 2 , ν + 1 2 , w n + 1 = E z u 1 2 , ν + 1 2 , w n + 1 1 ε i = 1 M ( P i , z u 1 2 , ν + 1 2 , w n + 1 P i , z u 1 2 , ν + 1 2 , w n )
+ Δ t ε Δ x ( H y u , ν + 1 2 , w n + 1 2 H y u 1 , , ν + 1 2 , w n + 1 2 ) Δ t ε Δ y ( H x u 1 2 , ν + 1 2 , w n + 1 2 H x u 1 2 , ν , w n + 1 2 ) .

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