Optics and Photonics Journal, 2013, 3, 15-20

doi:10.4236/opj.2013.32B004 Published Online June 2013 (http://www.scirp.org/journal/opj)

Study on the Gain Material with Four Energy Level

Model Using FDTD Method

Hui Xue1, Zhixiang Huang1, Xianliang Wu1,2

1Key Laboratory of Intelligent Computing and Signal Processing, Anhui University, Hefei, China

2Department of Physics and Electronic Engineering Hefei Normal University, Hefei, China

Email: xuehui@ahu.edu.cn

Received 2013

ABSTRACT

A faster numerical method based on FDTD for the four energy level atomic system is present here. The initial condi-

tions for the electrons of each level are achieving while the fields are in steady state. Polarization equation, rate equa-

tions of electronic population and Maxwell’s equations were used to describe the coupling between the atoms and elec-

tromagnetic wave. Numerical simulations, based on a finite-difference time-domain (FDTD) method, were utilized to

obtain the population inversion and lasing threshold. The validity of the model and its theory is confirmed. The time,

which we can observe the lasing phenomenon, is much shorter in our new model. Our model can be put into using in

large scale simulations in mutiph ysics to reduce the total simulated time.

Keywords: Finite-Difference Time-Domain; Gain Material; Lasing

1. Introduction

The system is always treated either semi classical or fully

quantum mechanical [2, 3] while a high-frequency light

is incident on a medium. Because of metallic nature of

metamaterials constituent metamolecules, they suffer from

high dissipative losses in the range of optical frequencies.

Losses are too large in the real applications. It is better to

incorporate the ga in media into matematerial to compen-

sate the losses. When an electromagnetic wave propa-

gates in a medium, the dipole moment in the individual

atom changes, in turn changing the total field coupling to

the medium until a steady state established. A full-vec-

torial time domain approach is utilized to do self-consistent

calculations. Fin ite-difference time-domain (FDTD) method

[4] is used as a powerful tool in modeling linear disper-

sive media [5, 6]. In an attempt to achieve more realistic

simulations two-level Maxwell-Bloch equations can be

solved using iterative predictor-corrector finite difference

time-domain FDTD methods to demonstrate saturation

and self-induced transparency [7].

To simulate lasing dynamics, we present here a faster

numerical simulation model for the four energy level

atomic system. We use the populations of each level

while the fields are achieving steady state as the initial

value, in this way, the simulation time will sharply be

reduced. The electromagnetic fields and atomic energy

level populations at any time step can be calculated in

terms of known quantities. Comparing the results of this

model with those which putting all the electrons on the

ground state level (E0). We can find that the results of

two methods are similar. So the validity of the model and

its theory is confirmed.

2. Theoretical and Numerical Model

2.1. Rate Equations

A simplified four level atomic system with energy levels

E0, E1, E2, E3 and populations upon each level N0, N1, N2,

N3 is depicted in Figure 1.

In our model, the gain atoms are embedded in the each

level of host medium aforehand. The electrons of the

ground state level are pumped to the third level by the

some external pumping mechanism (Pr). After a very

short time period

32, the electrons of the third level (E3)

fall into the second level (E2) by a non-radioactively

transition. A population accumulates in the second level

Figure 1. Schematic of the four-level atomic system model.

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