Abstract

We report a new finite-difference time-domain (FDTD) computational model of the lasing dynamics of a four-level two-electron atomic system. Transitions between the energy levels are governed by coupled rate equations and the Pauli Exclusion Principle. This approach is an advance relative to earlier FDTD models that did not include the pumping dynamics, or the Pauli Exclusion Principle. Further, the method proposed in this paper is more versatile than the conventional modal expansion of the electromagnetic field for complex inhomogeneous laser geometries constructed in photonic crystals or light-localizing random media. For such complex geometries, the lasing modes are either difficult or impossible to calculate. The present work aims at the self-consistent treatment of the dynamics of the 4-level atomic system and the instantaneous ambient optical electromagnetic field. This permits in principle a much more robust treatment of the overall lasing dynamics of four-level gain systems integrated into virtually arbitrary electromagnetic field confinement geometries.

© 2004 Optical Society of America

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Errata

Allen Taflove, "Finite-difference time-domain model of lasing action in a four-level two-electron atomic system: erratum," Opt. Express 14, 1702-1702 (2006)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-14-4-1702

References

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  1. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Boston: Artech House, 2000).
  2. S. C. Hagness, R. M. Joseph, and A. Taflove, ???Subpicosecond electrodynamics of distributed Bragg reflector microlasers: Results from finite difference time domain simulation,??? Radio Sci. 31, 931, (1996)
    [CrossRef]
  3. A. S. Nagra and R. A. York, ???FDTD analysis of wave propagation in nonlinear absorbing and gain media,??? IEEE Trans. Antennas Propgat. 46, 334, (1998).
    [CrossRef]
  4. R. W. Ziolkowski, J. M. Arnold, and D. M. Gogny, ???Ultrafast pulse interactions with two-level atoms,??? Phys. Rev. A 52, 3082, (1995).
    [CrossRef] [PubMed]
  5. A. E. Siegman, Lasers (University Science Books, 1986).
  6. J. Singh, Semiconductor Optoelectronics Physics and Technology (New York: McGraw-Hill, 1995)
  7. M.O Scully, M.S. Zubairy, Quantum Optics (Cambridge, 1997)

IEEE Trans. Antennas Propgat.

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

Phys. Rev. A

R. W. Ziolkowski, J. M. Arnold, and D. M. Gogny, ???Ultrafast pulse interactions with two-level atoms,??? Phys. Rev. A 52, 3082, (1995).
[CrossRef] [PubMed]

Radio Sci.

S. C. Hagness, R. M. Joseph, and A. Taflove, ???Subpicosecond electrodynamics of distributed Bragg reflector microlasers: Results from finite difference time domain simulation,??? Radio Sci. 31, 931, (1996)
[CrossRef]

Other

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

A. E. Siegman, Lasers (University Science Books, 1986).

J. Singh, Semiconductor Optoelectronics Physics and Technology (New York: McGraw-Hill, 1995)

M.O Scully, M.S. Zubairy, Quantum Optics (Cambridge, 1997)

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Figures (4)

Fig. 1.
Fig. 1.

four-level two-electron model

Fig. 2.
Fig. 2.

Electron population density probability. Left shows the inversion between levels 1 and 2.

Fig. 3.
Fig. 3.

Intensity output of pumping and lasing signals.

Fig. 4.
Fig. 4.

Output intensity vs. pump intensity.

Equations (19)

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H = H Atom + H Field + H AF
E ̂ = A ̂ t = i k σ ħ ω k 2 ε 0 V e k σ ( a ̂ k σ e ik · x a ̂ k σ + e ik · x )
μ ̂ = e r ̂ = ll μ ll V ̂ ll
d V ̂ dt = i [ H ̂ , V ̂ ] = i ω a V ̂ γ V ̂ + ω a [ N ̂ 2 N ̂ 1 ] μ · A ̂
d V ̂ + dt = i [ H ̂ , V ̂ ] = i ω a V ̂ + γ V ̂ + + ω a [ N ̂ 2 N ̂ 1 ] μ * · A ̂
d μ ̂ dt = μ d V ̂ dt + μ * d V ̂ dt
= i ω a ( μ V ̂ μ * V ̂ ) γ μ ̂ + 2 ω a [ N ̂ 2 N ̂ 1 ] μ 2 ( e z · A ̂ )
d 2 μ ̂ dt 2 + 2 γ d μ ̂ dt + ( ω a 2 + 2 ω a 2 2 μ 2 A ̂ z 2 ) μ ̂ = 2 ω a μ 2 E ̂ z ( N ̂ 2 N ̂ 1 )
d 2 P a dt 2 + γ a dP a dt + [ ω 12 2 + 2 ω 12 2 2 ( μ a · A ) 2 ] P a = 2 ω 12 μ a 2 E z ( N 1 N 2 )
d 2 P b dt 2 + γ 30 dP b dt + [ ω 30 2 + 2 ω 30 2 2 ( μ b · A ) 2 ] P b = 2 ω 30 μ b 2 E z ( N 0 N 3 )
d 2 P a dt 2 + γ a d P a dt + ω a 2 P a = ζ a ( N 2 N 1 ) E
d 2 P b dt 2 + γ b d P b dt + ω b 2 P b = ζ b ( N 3 N 0 ) E
d N ̂ u dt = i [ H ̂ , N ̂ u ] = γ ( 1 N g ) N ̂ u ω a μ ̂ · A ̂
= γ ( 1 N g ) N ̂ u + 1 ω a μ ̂ t · E ̂
dN 3 dt = N 3 ( 1 N 2 ) τ 32 N 3 ( 1 N 0 ) τ 30 + 1 ω b E · d P b dt
dN 2 dt = N 3 ( 1 N 2 ) τ 32 N 2 ( 1 N 1 ) τ 21 + 1 ω a E · d P a dt
dN 1 dt = N 2 ( 1 N 1 ) τ 21 N 1 ( 1 N 0 ) τ 10 1 ω a E · d P a dt
dN 0 dt = N 3 ( 1 N 0 ) τ 30 + N 1 ( 1 N 0 ) τ 10 1 ω b E · d P b dt
d E dt = 1 ε × H 1 ε N density ( d P a dt + d P b dt )

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