## Abstract

A model, by combining Maxwell’s equations with all-parameters of Sellmeier’s fitting equations and four-level rate equations, is built to investigate linear dispersive effect on the property of random lasing modes. Computed results show that the first excited modes for both dispersive and non-dispersive scattering cases have almost the same resonant frequency but the spectral intensity for dispersive case is lower than that for non-dispersive case, and there exist more modes in the whole spectra for dispersive case. Further analysis demonstrates that threshold of random lasing in dispersive case is higher than that of the non-dispersive case.

©2011 Optical Society of America

## 1. Introduction

Over the past few years, laser action in random media (random lasing) have attracted considerable attention [1–16]. The feedback of laser action for random lasing is not provided by an external resonator, but by scatterers which are randomly distributed in an active medium or which by themselves act as optical amplifiers. It is noting that the sharp peaks in the emission spectra of semi-conductor powders, first observed in 1999, has therefore lead to an intense debate about the nature of the lasing modes in these so-called lasers with resonant feedback. Many experimental and theoretical studies aimed at clarifying the nature of the lasing modes in disordered scattering systems with gain [3–14].

For a random lasing system different modes supported by the disordered medium have different resonant wavelengths, for which the wavelength interval between modes usually has only a few nanometers. Therefore, in the previous theoretical models of random lasing, the electric permittivity *ε* of the medium, to the best of our knowledge, was treated as a constant because the dispersion of the medium employed was usually quite small in optical band [3–5,8–14]. However, if the dispersive effect is strong enough, what will happen? For example, the value of ε for Al_{2}O_{3} takes from 3.6 to 3.1 while the wavelength λ varies from 0.2 µm to 0.8 µm. How such a dispersion effect will influence the property of random lasing supported by the disordered medium made by Al_{2}O_{3} scattering particles? In this paper, we devote to study the linear dispersive effect on the character of random lasing. To do so, we build a new model via combining Maxwell’s equations with all-parameters of Sellmeier’s fitting equations and four-level rate equations to investigate the random lasing phenomenon in the disordered system with dye as active medium and Al_{2}O_{3} grains as scattering particles. Our numerical results show that the peak intensity of the first excited mode for dispersive case is lower than that for non-dispersive case, but the resonant wavelengths of the modes are nearly the same, and there are more modes in the whole spectra for dispersive case. Further analysis demonstrates that the threshold of random lasing in dispersive case is higher than that of the non-dispersive case.

## 2. Theoretical model

We start our analysis with constructing a one-dimension (1-D) system, which consists of two dielectric materials, as shown in Fig. 1
. The white layer with a random variable thickness *a*
_{n} and a dielectric constant ε_{1} = ε_{0} simulates the dye, while the black layer with a fixed thickness b = 300 *nm* simulates the scatters. For sake of contrast, two scattering medium models of a linear dispersive material and a non-dispersive material are selected, respectively. Note that the linear dispersive materials are characterized by Sellmeier’s equation while the permittivity of the non-dispersive material is chosen as a constant ε_{2}. The random variable *a*
_{n} is described as ${a}_{n}=a(1+w\gamma )$, where *a* = 180 *nm*, *w* is the strength of randomness, and *γ* is a random value in the range [-0.5, 0.5].

In the following, we will discuss the computational theory and algorithm for gain and scattering medium, respectively.

#### 2.1 Optical gain material

For optical gain and non-magnetic medium, the 1D time-dependent Maxwell equations read

*P*

_{gain}is the polarization density component from which the amplification or gain can be obtained in z direction, ${\epsilon}_{0}$ and ${\mu}_{0}$are the electric permittivity and the magnetic permeability of vacuum, respectively. ${\epsilon}_{1}$ is the relative electric permittivity of gain medium. For dye, ${\epsilon}_{1}$is chosen as 1.

For the four-level atomic system, the rate equations of the gain medium read

*N*

_{4},

*N*

_{3},

*N*

_{2}and

*N*

_{1}individually. The pumping rate from

*E*

_{1}to

*E*

_{4}is

*W*

_{p;}The particles arrived

*E*

_{4}transfer to

*E*

_{3}quickly in the form of radiationless transition, the factor of probability is $1/{\tau}_{43}$. Before the population inversed,

*E*

_{3}transfer to

*E*

_{2}quickly in the form of spontaneous activity emission, the factor of probability is $1/{\tau}_{32}$.

*E*

_{2}transfer to

*E*

_{1}mostly in the form of spontaneous activity emission, the factor of probability is $1/{\tau}_{21}$.

The polarization *P _{gain}* obeys the following equation:

*N*<0. The linewidth of the atomic transition is Δω

*=1/τ*

_{l}_{21}+2/

*T*

_{2}where the collision time

*T*

_{2}is usually much smaller than the lifetime τ

_{21}. The constant κ is given by κ=6πε

_{0}

*c*

^{3}/ω

_{l}^{2}τ

_{21}.

#### 2.2 Scattering medium

### 2.2.1 Linear dispersive dielectrics

In the linear dispersive scattering medium, Maxwell’s equations for the linear dispersive scattering medium read

Here,${P}_{lorentz}$ is the linear polarization density component in z direction. For simplicity, we omit hereafter the expression of x dependence, that is, we simplify (*t*,

*x*) and (ω,

*x*) to (

*t*) and (ω), respectively. In the frequency domain,

*P*

_{lorentz}of Eq. (4d) is defined as

*(ω) is expressed by*

_{r}_{i}is the resonance frequency and

*B*

_{i}is the strength of the ith resonance.

Equations (4b), (5) and (6) lead to the following differential equations:

_{i}, the following system of differential equations is developed:

_{2}O

_{3}in Eq. (6) are set as

*B*

_{1}= 1.43134936,

*B*

_{2}= 0.65054713,

*B*

_{3}= 5.3414021, λ

_{1}= 0.0726631 µm, λ

_{2}= 0.1193242µm, and λ

_{3}= 18.028251µm, where λ

_{i}= 2πc/ω

_{i}and c is the velocity of light in vacuum. Figure 2 depicts the permittivity of Al

_{2}O

_{3}in the visible wavelength range as calculated from Sellmeier fitting coefficients. This figure illustrates that the variation takes ε from 3.6 to 3.1 while the wavelength λ varies from 0.2 µm to 0.8 µm..

This system of three equations above is then solved to obtain *P*
_{i} (*i* = 1, 2, 3) which is used to calculate *E _{z}*:

### 2.2.2 Non-dispersive dielectrics

For the 1D time-dependent Maxwell equations and for a non-dispersive and non-magnetic medium, we have

*ε*

_{2}is a fixed constant.

### 2.2.3 The computational algorithm

We now briefly summarize the computational algorithm.

For the optical gain medium, at time step n + 1, we first implement Eq. (3) to update *P*
_{gain}. Here, the use of explicit second-order finite-differences centered at time-step n requires only knowledge of *E*
_{z} at n. Next, we apply Eq. (1a) to update *E*
_{z} to time-step n + 1. Next, we apply Eq. (2b)–(2d) to update *N*
_{i} (i = 2, 3, 4)to time-step n + 1. Next, *N*
_{0} are calculated by using the conservation of electron populations. Finally we update *H* to time-step n + 3/2 by applying the Maxwell-Faraday law Eq. (1b).

For the dispersive scattering medium, applying explicit second-order finite-differences centered at time-step n for Eqs. (8), (9), and (10), this system can be solved to update *P*
_{1}, *P*
_{2}, and *P*
_{3} at time step n + 1. Next, with the updated values *P*
_{1}, *P*
_{2}, and *P*
_{3}, we can update *D*
_{z} from Eq. (4a). Next, we apply Eq. (11) to update *E*
_{z}. Finally we update *H* to time-step n + 3/2 by applying Eq. (4b).

For the non-dispersive scattering medium, first we apply Eq. (12) to update *E*
_{z}. Next, we update *H* to time-step n + 3/2 by applying Eq. (13).

The values of those parameters in above equations that will be used in simulating the active part in the following numerical calculations are taken as: *T*
_{2} = 2.14 × 10^{−14} s, τ_{21} = 5 × 10^{−12} s, τ_{32} = 1 × 10^{−10} s, τ_{43} = 1 × 10^{−13} s, *N _{T}* = $\sum _{i=1}^{4}{N}_{i}$ = 3.313 × 10

^{24}/m

^{3}, and ${\omega}_{l}=\left({E}_{3}-{E}_{2}\right)/\hslash $ = 6 × 10

^{14}Hz (λ

*= 500 nm). When pumping is provided over the whole system, the electromagnetic fields can be calculated. In order to model such an open system, a Liao absorbing layer [19] is used to absorb the outward wave. The space and time increments have been chosen to be Δ*

_{l }*x*= 10 nm and Δ

*t*= 1.67 × 10

^{−17}s, respectively. The pulse response is recorded during a time window of length

*T*= 6 × 10

_{w}^{−12}s at all nodes in the system and Fourier transformed in order to obtain the intensity spectrum.

## 3. Numerical results

We start our analysis with calculating spectral intensities vs pumping rates in the case of dispersive scattering medium, as shown in Fig. 3
. When pumping rate is relatively low, there exist many discrete peaks, each denoting a mode supported by disordered medium, as shown in Fig. 3(a). Note that three peaks are indicated by their central wavelengths λ_{0}(497.5 nm), λ_{1}(491.2 nm), and λ_{2}(504.1 nm), respectively. When pumping rate increase to a special value (*W*
_{p} = 1 × 10^{9} s^{−1}), the mode λ_{0} dominates the whole spectra and its width becomes quitebroader than those at lower *W _{p}*, as shown in Fig. 3(b). This suggests that the mode λ

_{0}is perhaps the first excited mode. With pumping rates further increasing, more modes are also excited and their widths become narrower and narrower, as shown in Fig. 3(c), 3(d), 3(e), and 3(f). And we can see that there are obvious mode competitions in whole spectrum.

To explore the gain threshold behavior on random lasing in the case of dispersive scattering medium, numerical calculations are performed at different pump rates from which we can obtain the curves of the peak intensity and the spectral width vs the pump rate, as shown in Fig. 4
. According to the traditional method, the pump thresholds for the three modes can be measured from the intensity curves in Fig. 4(a) as *W _{I }*

_{0}= 1.1 × 10

^{10}s

^{−1},

*W*

_{I}_{ 1}= 2 × 10

^{10}s

^{−1}, and

*W*

_{I}_{ 2}= 4 × 10

^{10}s

^{−1}, thus indicating different modes have different pump thresholds. Note that the mode λ

_{0}has the minimum lasing threshold, that is to say the mode λ

_{0}is the first excited mode. This is due to the fact that the central wavelength of the mode λ

_{0}is very near the transition wavelength of the active medium.

As can be seen from Fig. 4(b), within a pump regime near the pump threshold, a jump occurs for the spectral width for the mode λ_{0}. The peak value of the jump appears at the point that is very near the threshold of the mode. A method was proposed to determine the lasing threshold for a 2D random laser based on the spectral width [6]. That is, the lasing threshold is defined as such a pumping energy at which the spectral width becomes half of its maximal value. This method was used to analyze the threshold gain behavior for random lasing in a 2D disordered medium [10]. Based on this method and the width curves in Fig. 4(c), the thresholds are measured as *W*
_{w0} = 5 × 10^{9} s^{−1}, *W*
_{w1} = 0.9 × 10^{10} s^{−1}, and *W*
_{w2} = 1.1 × 10^{10} s^{−1}. Obviously, the results from the two methods are consistent.

Compared with dispersive scattering case, we restart to calculate the non-dispersive random lasing by the use of fixed constant ε_{2} = 3.1487, which is obtained from Sellmeier Eq. (6) at λ_{0}(497.54 nm). The calculated spectral intensities vs pumping rates are plotted in Fig. 5
. Here two peaks are indicated by their central wavelengths λ_{0}
^{´}(497.3 nm) and λ_{1}
^{´}(493.1 nm), respectively. And the pump thresholds for the two modes via the above two methods can be measured from the intensity and width curves in Fig. 6(a)
as *W _{I }*

_{0}= 4 × 10

^{9}s

^{−1}and

*W*

_{I}_{ 1}= 4 × 10

^{10}s

^{−1}, and from the width curves in Fig. 6(c) as

*W'*

_{W }_{0}= 3 × 10

^{9}s

^{−1}, and ${W}_{W1}^{\text{'}}$ = 0.9 × 10

^{10}s

^{−1}, respectively. The two methods indicate that the mode ${\lambda}_{0}^{\text{'}}$ is the first excited mode in the non-dispersive scattering case. It is quoting that the first excited modes for both dispersive and non-dispersive cases are only slightly different, whereas threshold of random lasing in dispersive scattering case is higher than that in the non-dispersive case via the two measured methods. Further note that, as shown in Fig. 3 and Fig. 5, the excited modes in the case of non-dispersive scattering case are fewer than that in the case of dispersive scattering case.As discussed above, we now know that, although the linear dispersion in optical domain is relatively small, the linear dispersion for scattering nanoparticles has a strong effect on modes of random lasing, which lead to richer lasing modes. In order to check whether the results above are universal or not, we calculate five different disordered structures for Al

_{2}O

_{3}, ZnO and TiO

_{2}, as shown in Table 1 . One can see that all thresholds of random lasing in dispersive scattering case are higher than that in the non-dispersive case for three scattering medium. And the numbers of the spectral spikes for dispersive medium are more than that for non-dispersive medium. All the above results demonstrate that the conclusion we have obtained is universal.

## 4. Conclusion

This work reports a model to reveal linear dispersive effect on modes of random lasing. The computed results show that dispersion leads to more modes in the spectra and higher thresholds.

## Acknowledgments

The National Natural Science Foundation of China under Grant No. 10876010, and No. 60778003, and the Foundation Research Funds for the Central Universities under Grant No. 2010MS041 have supported this research.

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