## Linear dispersive effect on random lasing modes |

Optics Express, Vol. 19, Issue 14, pp. 13445-13453 (2011)

http://dx.doi.org/10.1364/OE.19.013445

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### 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 OSA

## 1. Introduction

1. N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, and E. Sauvain, “Laser action in strongly scattering media,” Nature **368**(6470), 436–438 (1994). [CrossRef]

16. O. Zaitsev and L. Deych, “Recent developments in the theory of multimode random lasers,” J. Opt. **12**(2), 024001–024013 (2010). [CrossRef]

3. X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. **85**(1), 70–73 (2000). [CrossRef] [PubMed]

14. H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science **320**(5876), 643–646 (2008). [CrossRef] [PubMed]

*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

3. X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. **85**(1), 70–73 (2000). [CrossRef] [PubMed]

5. C. M. Soukoulis, X. Jiang, J. Y. Xu, and H. Cao, “Dynamic response and relaxation oscillations in random lasers,” Phys. Rev. B **65**(4), 041103–041106 (2002). [CrossRef]

8. X. Jiang, S. Feng, C. M. Soukoulis, J. Zi, J. D. Joannopoulos, and H. Cao, “Coupling, competition, and stability of modes in random lasers,” Phys. Rev. B **69**(10), 104202 (2004). [CrossRef]

14. H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science **320**(5876), 643–646 (2008). [CrossRef] [PubMed]

_{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

*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*= 180

*nm*,

*w*is the strength of randomness, and

*γ*is a random value in the range [-0.5, 0.5].

### 2.1 Optical gain material

*P*

_{gain}is the polarization density component from which the amplification or gain can be obtained in z direction,

*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

*E*

_{3}transfer to

*E*

_{2}quickly in the form of spontaneous activity emission, the factor of probability is

*E*

_{2}transfer to

*E*

_{1}mostly in the form of spontaneous activity emission, the factor of probability is

*P*obeys the following equation:This equation links Maxwell’s equations with rate equations.

_{gain}*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

*t*,

*x*) and (ω,

*x*) to (

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

*P*

_{lorentz}of Eq. (4d) is defined aswhere the Fourier transform of all values is expressed with a tilde and

17. R. M. Joseph and A. Taflove, “FDTD Maxwell’s equations modes for nonlinear electrodynamics and optics,” IEEE Trans. Antenn. Propag. **45**(3), 364–374 (1997). [CrossRef]

*(ω) is expressed bywhere ω*

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

*B*

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

16. O. Zaitsev and L. Deych, “Recent developments in the theory of multimode random lasers,” J. Opt. **12**(2), 024001–024013 (2010). [CrossRef]

_{i}, the following system of differential equations is developed: In our simulation, the parameters [18] for Al

_{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..

*P*

_{i}(

*i*= 1, 2, 3) which is used to calculate

*E*:

_{z}#### 2.2.2 Non-dispersive dielectrics

#### 2.2.3 The computational algorithm

*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).

*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).

*E*

_{z}. Next, we update

*H*to time-step n + 3/2 by applying Eq. (13).

*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}^{24}/m

^{3}, and

^{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

_{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*, as shown in Fig. 3(b). This suggests that the mode λ

_{p}_{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.

*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.

_{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

6. T. Ito and M. Tomita, “Polarization-dependent laser action in a two-dimensional random medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **66**(2), 027601–027604 (2002). [CrossRef] [PubMed]

10. J. S. Liu and Z. Xiong, “Theoretical investigation on the threshold property of localized modes based on spectral width in two-dimensional random media,” Opt. Commun. **268**(2), 294–299 (2006). [CrossRef]

*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.

_{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

^{10}s

^{−1}, respectively. The two methods indicate that the mode

_{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

## Acknowledgments

## References and links

1. | N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, and E. Sauvain, “Laser action in strongly scattering media,” Nature |

2. | H. Cao, Y. Zhao, S. Ho, E. Seelig, Q. Wang, and R. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. |

3. | X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. |

4. | P. Sebbah and C. Vanneste, “Random laser in the localized regime,” Phys. Rev. B |

5. | C. M. Soukoulis, X. Jiang, J. Y. Xu, and H. Cao, “Dynamic response and relaxation oscillations in random lasers,” Phys. Rev. B |

6. | T. Ito and M. Tomita, “Polarization-dependent laser action in a two-dimensional random medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

7. | S. Mujumdar, M. Ricci, R. Torre, and D. S. Wiersma, “Amplified extended modes in random lasers,” Phys. Rev. Lett. |

8. | X. Jiang, S. Feng, C. M. Soukoulis, J. Zi, J. D. Joannopoulos, and H. Cao, “Coupling, competition, and stability of modes in random lasers,” Phys. Rev. B |

9. | J. S. Liu, H. Lu, and C. Wang, “Spectral time evolution of quasistate modes in two-dimensional random media,” Acta Phys. Sin. |

10. | J. S. Liu and Z. Xiong, “Theoretical investigation on the threshold property of localized modes based on spectral width in two-dimensional random media,” Opt. Commun. |

11. | J. S. Liu, H. Wang, and Z. Xiong, “Origin of light localization from orientational disorder in one and two-dimensional random media with uniaxial scatterers,” Phys. Rev. B |

12. | C. Wang and J. S. Liu, “Polarization dependence of lasing modes in two-dimensional random lasers,” Phys. Lett. A |

13. | C. Vanneste, P. Sebbah, and H. Cao, “Lasing with resonant feedback in weakly scattering random systems,” Phys. Rev. Lett. |

14. | H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science |

15. | D. S. Wiersma, “The physics and applications of random lasers,” Nat. Phys. |

16. | O. Zaitsev and L. Deych, “Recent developments in the theory of multimode random lasers,” J. Opt. |

17. | R. M. Joseph and A. Taflove, “FDTD Maxwell’s equations modes for nonlinear electrodynamics and optics,” IEEE Trans. Antenn. Propag. |

18. | M. J. Weber, |

19. | A. Taflove and S. C. Hagness |

**OCIS Codes**

(140.3430) Lasers and laser optics : Laser theory

(190.5890) Nonlinear optics : Scattering, stimulated

(260.5740) Physical optics : Resonance

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: April 18, 2011

Revised Manuscript: May 24, 2011

Manuscript Accepted: June 13, 2011

Published: June 27, 2011

**Citation**

Yong Liu, Jinsong Liu, and Kejia Wang, "Linear dispersive effect on random lasing modes," Opt. Express **19**, 13445-13453 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-14-13445

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### References

- N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, and E. Sauvain, “Laser action in strongly scattering media,” Nature 368(6470), 436–438 (1994). [CrossRef]
- H. Cao, Y. Zhao, S. Ho, E. Seelig, Q. Wang, and R. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82(11), 2278–2281 (1999). [CrossRef]
- X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85(1), 70–73 (2000). [CrossRef] [PubMed]
- P. Sebbah and C. Vanneste, “Random laser in the localized regime,” Phys. Rev. B 66(14), 144202 (2002). [CrossRef]
- C. M. Soukoulis, X. Jiang, J. Y. Xu, and H. Cao, “Dynamic response and relaxation oscillations in random lasers,” Phys. Rev. B 65(4), 041103–041106 (2002). [CrossRef]
- T. Ito and M. Tomita, “Polarization-dependent laser action in a two-dimensional random medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(2), 027601–027604 (2002). [CrossRef] [PubMed]
- S. Mujumdar, M. Ricci, R. Torre, and D. S. Wiersma, “Amplified extended modes in random lasers,” Phys. Rev. Lett. 93(5), 053903–053907 (2004). [CrossRef] [PubMed]
- X. Jiang, S. Feng, C. M. Soukoulis, J. Zi, J. D. Joannopoulos, and H. Cao, “Coupling, competition, and stability of modes in random lasers,” Phys. Rev. B 69(10), 104202 (2004). [CrossRef]
- J. S. Liu, H. Lu, and C. Wang, “Spectral time evolution of quasistate modes in two-dimensional random media,” Acta Phys. Sin. 54, 3116 (2005) (in Chinese).
- J. S. Liu and Z. Xiong, “Theoretical investigation on the threshold property of localized modes based on spectral width in two-dimensional random media,” Opt. Commun. 268(2), 294–299 (2006). [CrossRef]
- J. S. Liu, H. Wang, and Z. Xiong, “Origin of light localization from orientational disorder in one and two-dimensional random media with uniaxial scatterers,” Phys. Rev. B 73(19), 195110 (2006). [CrossRef]
- C. Wang and J. S. Liu, “Polarization dependence of lasing modes in two-dimensional random lasers,” Phys. Lett. A 353(2-3), 269–272 (2006). [CrossRef]
- C. Vanneste, P. Sebbah, and H. Cao, “Lasing with resonant feedback in weakly scattering random systems,” Phys. Rev. Lett. 98(14), 143902 (2007). [CrossRef] [PubMed]
- H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320(5876), 643–646 (2008). [CrossRef] [PubMed]
- D. S. Wiersma, “The physics and applications of random lasers,” Nat. Phys. 4(5), 359–367 (2008). [CrossRef]
- O. Zaitsev and L. Deych, “Recent developments in the theory of multimode random lasers,” J. Opt. 12(2), 024001–024013 (2010). [CrossRef]
- R. M. Joseph and A. Taflove, “FDTD Maxwell’s equations modes for nonlinear electrodynamics and optics,” IEEE Trans. Antenn. Propag. 45(3), 364–374 (1997). [CrossRef]
- M. J. Weber, CRC Handbook of Optical Materials (CRC Press, 2003).
- A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

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