## Finite-difference time-domain model of lasing action in a four-level two-electron atomic system

Optics Express, Vol. 12, Issue 16, pp. 3827-3833 (2004)

http://dx.doi.org/10.1364/OPEX.12.003827

Acrobat PDF (960 KB)

### 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

## 1. Introduction

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]

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]

## 2. Method

*P*_{a}and Levels 0 and 3 correspond to dipole

**P**_{b}.

*N̂*

_{u}=|

*u*><

*u*|=number operator for the upper-level |

*u*>

*N̂*

_{g}=|

*g*><

*g*|=number operator for the upper-level |

*g*>

*ħω*

_{a}=energy difference between |

*u*> and ground level |

*g*>

*a*â+

_{kσ}=photon-creation operator;

*k*=mode number; σ=polarization direction

*e*is the electron charge and

*V̂*

_{ll}

^{′}=|

*l*><

*l*

^{′}′| is the atomic transition operator. For a two-level system, the dipole operator can be written as

*=*

**µ**̂

**µ***V̂*†+

***

**µ***V̂*where

*=〈*

**µ***u*|

*e*|

**r**̂*g*〉. The first-order differential equations of the dipole operator with the empirical dephasing term

*γV̂*are given by:

*=*

**µ***µe*

_{𝑧}. Therefore, the dipole operators are

*d*/

*dt*(

*µV̂*†-

*µ**

*V̂*)=

*iω*

_{a}

*µ̂*, Eq. (5) becomes

*P*

_{a}(

*t*) between |1> and |2> is then

*ω*

_{12}is the resonance angular frequency;

*µ*

_{a}is the dipole matrix element between levels |1> and |2>;

*γ*

_{a}is the dephasing rate for

*P*

_{a}(

*t*);

*A*Â is the vector potential; and

*N*

_{i}is the atomic population density in level

*i*. A similar equation holds for the polarization density

*P*

_{b}(

*t*) between |0> and |3>:

_{30}is the pumping frequency.

*·*

**µ**b*A*)

^{2}/

*ħ*

^{2}in Eq. (8), which is important only when the external electric field is very high. This yields the governing equations

*and*

**P**_{a}*have the resonant frequencies*

**P**_{b}*ω*

_{a}and

*ω*

_{b};

*ħω*

_{a}is the energy difference between Levels 1 and 2;

*ħω*

_{b}is the energy difference between Levels 0 and 3; and

*ζ*

_{a}is 6

*πε*

_{0}

*c*

^{3}/(

*τ*

_{21}). In Eq. (9), the driving terms are proportional to the population differences and the damping coefficients

*γ*

_{a}and

*γ*

_{b}simulate the non-radiation loss. For the example discussed later,

*γ*

_{a}=

*γ*

_{b}=10

^{-13}s

^{-1}. Further note that the electric field E is an instantaneous value that is composed of contributions from both the pumping and emission signals.

*N*

_{u}and

*N*

_{g}are defined in Eq. (1). Here, we include the spontaneous decay to the lower level by the term -

*γ*(1-

*N*

_{g})

*N̂*

_{u}.

*=*

**µ**

**µ**_{0}(1-

*N*). Here,

**µ**_{0}is the quantum efficiency when there are no electrons in the active region, and

*N*is the electron population density probability. As a consequence, the quantum efficiency drops by a factor of (1-

*N*). Similar to interband relaxations, intraband transitions (3→2 or 1→0) are reduced by a factor (1-

*N*) due to the Fermi distribution of the electron population within the band.

_{b}(between level 0 and 3) and P

_{a}(between level 1 and 2), The preceding considerations lead to the following rate equations for electron populations within four levels:

*N*

_{i}is the electron population density probability in Level

*i*and

*τ*

_{ij}is the decay time constant between levels

*i*and

*j*. The electron populations vary with pumping

*·(*

**E***d*/

**P***dt*) and spontaneous emission decay (

*N*

_{i}-

*N*

_{j})/

*τ*

_{ij}. Electrons in Level 3 spontaneously decay to Levels 2 and 0 with decay time constants τ

_{32}and τ

_{30}respectively. Electrons in Level 2 spontaneously decay to Levels 1 with decay time constants τ

_{21}.

*n*+1, we first implement Eq. (9) to update

*and*

**P**a*. Here, the use of explicit second-order finite-differences centered at time-step*

**P**b*n*requires only knowledge of

*at*

**E***n*. Next, we apply Eq. (12) to update

*to time-step*

**E***n*+1. Next, we apply Eq. (11) to update

*N*

_{3}to time-step

*n*+1. This involves the Pauli exclusion term (1-

*N*

_{2}), and

*N*

_{2}is taken from time-step

*n*. Next, we apply in sequence Eq. (11) to update

*N*

_{2}and

*N*

_{1}. This allows

*N*

_{0}to be calculated by using the conservation of electron populations. Finally we update

*to time-step*

**H***n*+3/2 by applying the Maxwell-Faraday law. This algorithm avoids the need to use the predictor-corrector algorithm of [4

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]

## 3. Results

*τ*

_{32}=

*τ*

_{10}=100 fs and

*τ*

_{21}=

*τ*

_{30}=300 ps (which takes into account the dephasing time). The initial population density is

*N*

_{1}=

*N*

_{0}=5×10

^{23}/m

^{3}. With these parameters we estimate the required density probability of population inversion to be approximately 8.4×10

^{-4}.

## 4. Summary and discussion

## Acknowledgments

## References and links

1. | A. Taflove and S. C. Hagness, |

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. |

3. | A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Trans. Antennas Propgat. |

4. | R. W. Ziolkowski, J. M. Arnold, and D. M. Gogny, “Ultrafast pulse interactions with two-level atoms,” Phys. Rev. A |

5. | A. E. Siegman, |

6. | J. Singh, |

7. | M.O Scully and M.S. Zubairy, |

**OCIS Codes**

(140.0140) Lasers and laser optics : Lasers and laser optics

(190.0190) Nonlinear optics : Nonlinear optics

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: July 1, 2004

Revised Manuscript: July 26, 2004

Published: August 9, 2004

**Citation**

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

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-16-3827

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### References

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