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Chinese Science Bulletin © 2009
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Polarization-dependent difference of the power spectra from two-dimensional random media with different shapes LIU Hai1,2, LIU JinSong1†, LÜ JianTao1 & WANG KeJia1 1
Wuhan National Laboratory for Optoelectronics, School of Optoelectronic Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China; 2 School of Information and Electrical Engineering, China University of Mining and Technology, Xuzhou 221116, China
random lasers, laser spectroscopy, polarization, morphology
The optical feedback generated by multiple light-scattering in a disordered medium was first predicted by Letokhov in 1968[1], and the random lasing phenomena was first observed by Lawandy et al. in 1994[2]. Since then the research on the random media has become a hotspot in this field[3–17]. Up to present, many different models have been built for the investigation. There-into, the latest model was built by Türeci et al. in 2008[3], they introduced a formulation of semi-classical laser theory in terms of biorthogonal modes, and treated lasing media with any degree of out-coupling and included the effects of nonlinear modal interactions to all orders. In our research, direct resolution of Maxwell’s equation has been used[4–8] in order to reproduce the key aspects of the experiment observations of coherent feedback random lasers. The effective model combining Maxwell’s equation with the rate equations of a four-level atomic system was built for one-dimensional case[5] and was extended to two-dimensional (2D) case[6]. Based on such a model, a time dependent theory for random lasers was built, in which finite difference time domain (FDTD) method
was used to obtain the field pattern and the spectra of localized lasing modes inside the disorder system, and the results qualitatively agree with the experimental results obtained. However, random lasers can be made of random media with arbitrary shapes of their own property and the influence by the shapes is ever neglected, as each lasing mode has its own spatial distribution and the corresponding region in the 2D random medium has much effect on this mode[6,7]. The characteristics may result in the emission spectra of random lasers of morphology dependent. As we know, there exist two polarized states simultaneously in a 2D random medium. One is the transverse magnetic (TM) field (TM fields mean that the electric and magnetic components of the light wave are perpendicular to and within the 2D plan respectively), and the other is the transverse electric (TE) field (TE fields mean that the electric and magnetic components are within and perpendicular to the 2D plan Received March 9, 2009; accepted May 13, 2009 doi: 10.1007/s11434-009-0521-8 † Corresponding author (email:
[email protected]) Supported by the National Natural Science Foundation of China (Grant Nos. 60778003, 60378001 and 10876080)
Citation: Liu H, Liu J S, Lü J T, et al. Polarization-dependent difference of the power spectra from two-dimensional random media with different shapes. Chinese Sci Bull, 2009, 54: 3215―3219, doi: 10.1007/s11434-009-0521-8
OPTICS
Polarization-dependent difference of the power spectra from a set of two-dimensional (2D) passive random media is investigated by simultaneously solving Maxwell’s equations for both transverse magnetic (TM) and transverse electric (TE) fields. The random media have the same random constitution but different shapes. Results show that both two polarized states are morphology dependent, and the variety of the shapes has more influence on the selection of TM polarized modes than that of TE polarized modes. Such polarization-dependent difference of morphology property presents a new modeselecting technique for random lasers.
respectively). In previous research, the two polarized states were studied independently[8–15]. However, if the two polarized state exist in random media together and share the same inversed population, the research for each polarized state cannot be studied independently any more. There exists a competition in the inversed population between the two states. The competition between two polarized states in 2D planar media has been researched before, and the results indicate that TM state is suppressed strongly by TE state in the competition[13]. In the same way, the morphology dependence of power spectrum with the competition on inversed population between two polarized states would be much different from the investigation for independent polarized state before[14]. That is the purpose of this paper. We calculated the power spectra from a set of 2D random media with the same random constitution but different shapes, and the results demonstrated that the shapes have different influence for different polarized states. Appropriate design of the shapes has obvious effect on the selection of TM polarized modes and can improve the adverse situation of TM polarized state in the competition. Based on these results, the experiment by Ito and Tomita in 2002[15] can be further extended, such as changing the gain materials or arranging the fibers with different shapes, which are easy to actualize in the experiment. It can be expected that the results will be much different.
1 Theoretical model and method
random media is shown in Figure 1(a), and l=5.5 m, r=60 nm, Ф=40%, n1=1, n2=2.3. In order to clear up the influence of the random constitution, we construct a set of 2D random media with the same random constitution but different shapes via inscribing an arbitrary shape into the square one, as shown in Figures 1(b)―(d). That is, the shapes of these media are a set of arbitrary inscribed figures of the square with size l2. We use FDTD method to solve numerically Maxwell’s equations combined with the rate equation to analyze the properties of 2D random media. For transverse magnetic (TM) field, Maxwell’s equations read as follows:
0
H x E z , t y H y
Ez , t x E P H y H x i 0 z z . t t x y
0
(1a) (1b)
(1c)
For TE field, Maxwell’s equations read as follows: E P H z 0 i x x , y t dt E y Py H z 0 i , x t dt E y Ex H z 0 , x y t
(2a) (2b)
(2c)
where ε0 and μ0 are the electric permittivity and the mag2
We begin with a 2D square disordered medium of size l that is essentially a 2D simplification of real experiments and consists of circular particles with a radius r, an optical index n2 and a surface-filling fraction Ф. These particles are randomly distributed in a background medium with an optical index n1. When the system parameters (i.e. r, Ф, n1 and n2) are given, they can create huge amount of random media. A typical pattern of these
netic permeability of vacuum, respectively, and i ni2 , i 1, 2. The four-level rate equations are shown as eq. (3). In these equations, the parameters in the above equations have the same meaning as prior research[11–14], the stimuE dP lated transition rate is given by the term . , where l dt
l is the transition frequency between levels 2 and 3.
Figure 1 A set of 2D random medium with different shapes and the same random constitution created by r=60 nm, Ф=40%, n1 =1, n2=2.3, l =5.5 m. 3216
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dN 2 N3 N 2 E dP , dt 32 21 l dt
(3b)
dN3 N 4 N3 E dP , 43 32 l dt dt
(3c)
dN 4 N 4 W p N1. dt 43
(3d)
Eventually, P is described as obeying the quantum population equation of motion[6,7] d2 P dP l l 2 P NE , (4) 2 t d dt where E Ex e x E y e y Ez e z , P Px e x Py e y Pz e z with e x , e y , ez being the unit vectors in the x, y and z directions, respectively; N N3 N 2 ; 6π 0 c3 / 2 (32 32 ) ; 32 1 32 2 T2 with T2 being the colli-
sion time. The values of those parameters in the above equations are the same as those in previous research[11–14], which are close to those used in several experiments[17].
2 Results and discussion As is known, the shapes have remarkable influence on
the power spectra from random media. In order to find out the difference between two different polarized states, we compare the power spectra from passive random media for both two polarized states at first. The designed shapes are shown as Figure 1. From the research before[11–14], we know that the TM polarized modes hold higher energy density than TE modes. Since the spatial distribution area of the 2D random medium has much effect on the corresponding polarized mode, the morphology dependence of the power spectrums for the two polarized states should be different. The corresponding power spectrums for TM polarized states and TE polarized states with the shapes in Figure 1 at 6 ps are shown as Figures 2 and 3 respectively. The wavelength of the marked TM polarized modes is λM1=415.1 nm, λM2= 446.6 nm, λM3=460.2 nm, λM4=470.9 nm, λM5=479.5 nm and λM6=540.9 nm. From Figures 2 and 3, we find that if we cut out different area, the TM polarized modes in the spectrum may suffer different damage and some corresponding TM modes may disappear. The more damage to the random medium, the fewer TM modes can exist. Different from TM polarized modes, there are much more long-life TE polarized modes existing in passive random media because the TM modes have dispersed energy distribution. From Figures 3 and 4, although there are
Figure 2 Corresponding power spectra for TM polarized states from 2D passive random media with the same random constitution and different shapes as shown in Figure 1.
Figure 3 Corresponding power spectra for TE polarized states from 2D passive random media with the same random constitution and different shapes as shown in Figure 1. Liu H et al. Chinese Science Bulletin | September 2009 | vol. 54 | no. 18
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(3a)
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dN1 N 2 W p N1 , dt 21
3
1
Figure 4 The spectral intensity in arbitrary units vs. the wavelength λ for TM fields corresponding to Figure 1(b). (a) Wp=10 s ; (b) Wp=10 12 1 1 s ; (c) Wp=10 s .
much more TE polarized modes existing in passive random media at the time 6 ps, the mainly peak plots are not as visible as TM modes. For further investigation on the polarization-dependent difference under pumping, we select a fixed random media shape shown in Figure 1(b) to calculate the power spectra for both two polarized states with different pumping rates. Obviously, the selected random medium is badly damaged and would influence the existed polarized modes in random media. The corresponding power spectrums for TM polarized state and TE polarized state are shown in Figures 4 and 5 respectively. As we know, the TM polarized modes are suppressed badly by the TE polarized modes in the competition in square random media[14]. From Figures 4 and 5, we find that the adverse situation of TM modes in the competition on inversed population with TE modes has been improved by designing appropriate shapes of the random media with the same constitution. Via inscribing different shapes into the square one, the spectral intensity of the TE modes fall to the same level of TM modes. It can be seen from the threshold curve of the marked polarized modes which is shown in Figure 6. Contrast with the case in square random medium, the gap of the spectral intensity between the two polarized states has been improved. In addition, different from the case in passive random media, few TE modes can be excited when the
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random medium is badly damaged. If we want to obtain a TM polarized mode with adverse situation in the competition, appropriate design of the shapes of the random media is an effective method.
3 Conclusions We calculated the spectra for TE polarized state and TM polarized stated from both passive random media and active media. Results demonstrate that the lasing modes
Figure 6 Plot of the peak intensity of the marked polarized modes vs. the pump rate with the shape as shown in Figure 1(b).
3
1
Figure 5 The spectral intensity in arbitrary units vs. the wavelength λ for TE fields corresponding to Figure 1(b). (a) Wp=10 s ; (b) Wp=10 12 1 1 s ; (c) Wp=10 s . 3218
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power spectra from 2D random media is investigated, and the 2D random media with different shapes will have different morphology-dependent property for different polarized states. In future, we should use the new model built by Türeci et al.[3] for further research. The new model is applicable to the study of the optical property of some novel laser cavity systems.
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in Figures 4 and 5 come from the modes of the passive system for each shape shown in Figure 1. As a result, the emission spectrum of 2D random laser for each polarized state is morphology dependent. Via inscribing different shapes into the square one, the adverse situation of TM polarized state in the competition can be improved. In conclusion, the polarization-dependent difference of the
