## Numerical study on terahertz random lasing in disordered ruby with three-level atomic system |

Optics Express, Vol. 18, Issue 22, pp. 22880-22885 (2010)

http://dx.doi.org/10.1364/OE.18.022880

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### Abstract

A scheme to generate terahertz (THz) emission using active disordered medium made of ruby grains with a three-level atomic system is proposed via a one-dimensional model. Our computed results reveal that THz random lasing phenomenon could occur under suitable conditions. The proposed scheme is based on the pumping of the

© 2010 OSA

## 1. Introduction

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

14. 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]

15. D. S. Wiersma, “The physics and applications of random lasers,” Nat. Phys. **4**(5), 359–367 (2008). [CrossRef]

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

6. P. Sebbah and C. Vanneste, “Random laser in the localized regime,” Phys. Rev. B **66**(14), 144202 (2002). [CrossRef]

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

14. 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]

17. N. M. Lawandy, “Far-infrared lasing in ruby,” IEEE J. Quantum Electron. **6**(15), 401–403 (1979). [CrossRef]

^{3+}ion (

^{−1}or 0.87 THz apart), forming two primary lines often denoted by R1 (694.3 nm) and R2 (693.9 nm), respectively. We hence can suggest that 0.87 THz random lasing could be operated in ruby grains using the levels

^{−1}far-infrared lasing based on the pumping of the

17. N. M. Lawandy, “Far-infrared lasing in ruby,” IEEE J. Quantum Electron. **6**(15), 401–403 (1979). [CrossRef]

18. E. Kuznetsova, Y. Rostovtsev, N. G. Kalugin, R. Kolesov, O. Kocharovskaya, and M. O. Scully, “Generation of coherent terahertz pulses in ruby at room temperature,” Phys. Rev. A **74**(2), 023819 (2006). [CrossRef]

^{−1}photons, this scheme also supports the interaction of the electronics states

^{3+}with the resonant 29 cm

^{−1}phonons, such as the 29 cm

^{−1}phonon “bottlenecked” effect at low temperatures [19

19. S. Geschwind, G. E. Devlin, R. L. Cohen, and S. R. Chinn, “Orbach relaxation and hyperfine structure in the excited, ” Phys. Rev. **137**(4A), A1087–A1100 (1965). [CrossRef]

^{−1}phonons [20]. Besides, the generation of THz radiation by difference-frequency mixing of the R1 and R2 lines in ZnTe was reported [21

21. T. Yajima and K. Inoue, “Submillimeter-wave generation by difference-frequency mixing of ruby lines in ZnTe,” IEEE J. Quantum Electron. **5**(3), 140–146 (1969). [CrossRef]

22. L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, H. E. Beere, D. A. Ritchie, and D. S. Wiersma, “Quasi-periodic distributed feedback laser,” Nat. Photonics **4**(3), 165–169 (2010). [CrossRef]

## 2. Theoretical model

_{1}= ε

_{0}simulates the air, while the black layer with a fixed thickness

*b*=90 μm and a dielectric constant ε

_{2}= 4ε

_{0}simulates the scatters that are also the gain media with a three-level atomic system, and the scheme of the energy levels is shown in Fig. 1(b). The random variable

*a*=100 μm,

*w*is the strength of randomness, and

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

*z*direction,

*N*(

_{i}*i*=1, 2, 3) is the population density in level

*i*; The electrons in the level 1 are transferred to the upper level 3 by an external optical pump at 693.9 nm with a fixed rate

*T*is the time of flight between emission and re-absorption of a 29 cm

^{−1}phonon and is on the order of 10

^{−9}—10

^{−11}s [17

17. N. M. Lawandy, “Far-infrared lasing in ruby,” IEEE J. Quantum Electron. **6**(15), 401–403 (1979). [CrossRef]

^{−1}phonons undergo reabsorption [20]. It was reported that the 29 cm

^{−1}phonons were absorbed and reemitted about 1000 times before they decay [20]. Meanwhile the lifetime of the 29 cm

^{−1}phonon is on the order of 10

^{−6}s. Accordingly,

*T*is on the order of 10

^{−9}s. As a result,

*T*is chosen as 1× 10

^{−9}s in our system. The frequency between levels 3 and 2 is

*κ*is given by

^{−14}s,

^{−9}s,

^{−3}s,

^{−6}s, and

^{25}/m

^{3}.

^{−6}m and

^{−15}s, respectively. The pulse response is recorded during a time window of length

^{−10}s at all nodes in the system and Fourier transformed in order to obtain the intensity spectrum.

## 3. Numerical results

^{−1}), there are many discrete peaks near 344 μm (or 0.87 THz), each having quite weak peak intensity in the spectrum. Each peak corresponds to a lasing mode supported by the disordered medium, for which four peaks are indicated by their central wavelengths λ

_{0}, λ

_{1}, λ

_{2}and λ

_{3}, respectively. As the pump rate increases, these peak intensities are nearly unchanged until the pump rate increases to a value (

^{−1}) at which only those indicated modes are effectively amplified, as shown in Figs. 2(b) and 2(c). When the pump rate

_{0}dominates the spectrum, as shown in Fig. 2(d). This means the mode λ

_{0}perhaps has the minimum lasing threshold among the modes. Meanwhile, let us pay attention to the spectral width of the modes. The width becomes quite broader at

^{−1}than those at lower

^{−1},

^{−1},

^{−1}and

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

_{0}has the minimum lasing threshold, which is easy to understand because the central wavelength of the mode λ

_{0}is very near the transition wavelength of the active medium. The above numerical results clearly indicate that this 1D ruby disordered system could support random lasing phenomenon in THz domain.

_{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 [11

11. 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 (2002). [CrossRef] [PubMed]

8. 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]

^{−1},

^{−1},

^{−1}and

^{−1}. Obviously, the results from the two methods are consistent. Note that for a given mode, the specific value for its pump threshold depends on the defining fashion.

^{3+}:BeAl

_{2}O

_{4}(alexandrite), having R line splittings of 41 cm

^{−1}(1.23 THz) [24

24. J. C. Walling, O. G. Peterson, H. P. Jenssen, R. C. Morris, and E. W. O’Dell, “Tunable alexandrite lasers,” IEEE J. Quantum Electron. **16**(12), 1302–1315 (1980). [CrossRef]

18. E. Kuznetsova, Y. Rostovtsev, N. G. Kalugin, R. Kolesov, O. Kocharovskaya, and M. O. Scully, “Generation of coherent terahertz pulses in ruby at room temperature,” Phys. Rev. A **74**(2), 023819 (2006). [CrossRef]

^{−1}for the ordinary ray and 0.5 cm

^{−1}for the extraordinary ray at 300K, and less than 0.01 cm

^{−1}at 4.2K [17

**6**(15), 401–403 (1979). [CrossRef]

## Conclusions

## Acknowledgments

## References and links

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

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

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

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

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

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

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

8. | 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. |

9. | 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 |

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

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

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

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

14. | 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 |

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

16. | X.-C. Zhang and J. Z. Xu, |

17. | N. M. Lawandy, “Far-infrared lasing in ruby,” IEEE J. Quantum Electron. |

18. | E. Kuznetsova, Y. Rostovtsev, N. G. Kalugin, R. Kolesov, O. Kocharovskaya, and M. O. Scully, “Generation of coherent terahertz pulses in ruby at room temperature,” Phys. Rev. A |

19. | S. Geschwind, G. E. Devlin, R. L. Cohen, and S. R. Chinn, “Orbach relaxation and hyperfine structure in the excited, ” Phys. Rev. |

20. | K. F. Renk and J. Deisenhonefer, “Imprisonment of resonant phonons observed with a new technique for the detection of 10l2Hz phonons,” Phys. Rev. Lett. |

21. | T. Yajima and K. Inoue, “Submillimeter-wave generation by difference-frequency mixing of ruby lines in ZnTe,” IEEE J. Quantum Electron. |

22. | L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, H. E. Beere, D. A. Ritchie, and D. S. Wiersma, “Quasi-periodic distributed feedback laser,” Nat. Photonics |

23. | A. E. Siegman, |

24. | J. C. Walling, O. G. Peterson, H. P. Jenssen, R. C. Morris, and E. W. O’Dell, “Tunable alexandrite lasers,” IEEE J. Quantum Electron. |

**OCIS Codes**

(140.3070) Lasers and laser optics : Infrared and far-infrared lasers

(140.3430) Lasers and laser optics : Laser theory

(190.5890) Nonlinear optics : Scattering, stimulated

(260.5740) Physical optics : Resonance

(300.6495) Spectroscopy : Spectroscopy, teraherz

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: June 1, 2010

Revised Manuscript: August 12, 2010

Manuscript Accepted: October 4, 2010

Published: October 14, 2010

**Citation**

Jinsong Liu, Yong Liu, Jiantao Lü, and Kejia Wang, "Numerical study on terahertz random lasing in disordered ruby with three-level atomic system," Opt. Express **18**, 22880-22885 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-22-22880

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### References

- N. M. Lawandy, R. M. Balachandra, A. S. L. Gomes, and E. Sauvain, “Laser action in strongly scattering media,” Nature 368(6470), 436–438 (1994). [CrossRef]
- H. Cao, Y. G. Zhao, H. C. Ong, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82(11), 2278–2281 (1999). [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]
- S. Mujumdar, M. Ricci, R. Torre, and D. S. Wiersma, “Amplified extended modes in random lasers,” Phys. Rev. Lett. 93(5), 053903 (2004). [CrossRef] [PubMed]
- 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 (2002). [CrossRef]
- 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]
- 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]
- 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 (2002). [CrossRef] [PubMed]
- 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]
- J. S. Liu, H. Lu, and C. Wang, “Spectral time evolution of quasistate modes in two-dimensional random media,” Acta Phys. Sin. 54, 3116–3122 (2005) (in Chinese).
- 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]
- D. S. Wiersma, “The physics and applications of random lasers,” Nat. Phys. 4(5), 359–367 (2008). [CrossRef]
- X.-C. Zhang and J. Z. Xu, Introduction to THz Wave Photonics (Springer, Berlin, 2010).
- N. M. Lawandy, “Far-infrared lasing in ruby,” IEEE J. Quantum Electron. 6(15), 401–403 (1979). [CrossRef]
- E. Kuznetsova, Y. Rostovtsev, N. G. Kalugin, R. Kolesov, O. Kocharovskaya, and M. O. Scully, “Generation of coherent terahertz pulses in ruby at room temperature,” Phys. Rev. A 74(2), 023819 (2006). [CrossRef]
- S. Geschwind, G. E. Devlin, R. L. Cohen, and S. R. Chinn, “Orbach relaxation and hyperfine structure in the excited, ” Phys. Rev. 137(4A), A1087–A1100 (1965). [CrossRef]
- K. F. Renk and J. Deisenhonefer, “Imprisonment of resonant phonons observed with a new technique for the detection of 10l2Hz phonons,” Phys. Rev. Lett. 26, 164–166 (1971).
- T. Yajima and K. Inoue, “Submillimeter-wave generation by difference-frequency mixing of ruby lines in ZnTe,” IEEE J. Quantum Electron. 5(3), 140–146 (1969). [CrossRef]
- L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, H. E. Beere, D. A. Ritchie, and D. S. Wiersma, “Quasi-periodic distributed feedback laser,” Nat. Photonics 4(3), 165–169 (2010). [CrossRef]
- A. E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986).
- J. C. Walling, O. G. Peterson, H. P. Jenssen, R. C. Morris, and E. W. O’Dell, “Tunable alexandrite lasers,” IEEE J. Quantum Electron. 16(12), 1302–1315 (1980). [CrossRef]

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