## Numerical analysis of resonant and lasing properties at a defect region within a random structure

Optics Express, Vol. 17, Issue 5, pp. 3970-3977 (2009)

http://dx.doi.org/10.1364/OE.17.003970

Acrobat PDF (418 KB)

### Abstract

We propose a simple structure for manipulating resonant conditions in random structures, in which a “defect” region where no scatterer is set is deliberately made in the structure. By employing a two-dimensional finite-difference time-domain method including rate equations, we examine the resonant and lasing properties observed at the defect region by changing the filling factor of scatterers. From the numerical results, we confirm that a distinct localized spot at the defect can be realized by determining an optimal filling factor and scatterer size and selecting the appropriate defect size.

© 2009 Optical Society of America

## 1. Introduction

1. C. Gouedard, D. Husson, C. Sauteret, F. Auzel, and A. Migus, “Generation of spatially incoherent short pulses in laser-pumped neodymium stoichiometric crystals and powders,” J. Opt. Soc. Am. B **10**, 2358–2363 (1993). [CrossRef]

*e.g.*, for decreasing the threshold of nonlinear phenomena), technological developments in manipulating resonant properties are essential for applications.

5. D. S. Wiersma and S. Cavalieri, “Light emission: A temperature-tunable random laser,” Nature **414**, 708–709 (2001). [CrossRef] [PubMed]

12. H. Cao, J. Y. Xu, D. Z. Zhang, S.-H. Chang, S. T. Ho, E. W. Seeling, X. Liu, and R. P. H. Chang, “Spatial Confinement of Laser Light in Active Random Media,” Phys. Rev. Lett. **84**, 5584–5587 (2000). [CrossRef] [PubMed]

7. G. V. Soest, M. Tomita, and A. Lagendijk, “Amplifying volume in scattering media,” Opt. Lett. **24**, 306–308 (1999). [CrossRef]

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

12. H. Cao, J. Y. Xu, D. Z. Zhang, S.-H. Chang, S. T. Ho, E. W. Seeling, X. Liu, and R. P. H. Chang, “Spatial Confinement of Laser Light in Active Random Media,” Phys. Rev. Lett. **84**, 5584–5587 (2000). [CrossRef] [PubMed]

13. H. Cao, J. Y. Xu, E. W. Seeling, and R. P. H. Chang, “Microlaser made of disordered media,” Appl. Phys. Lett. **76**, 2997–2999 (2000). [CrossRef]

7. G. V. Soest, M. Tomita, and A. Lagendijk, “Amplifying volume in scattering media,” Opt. Lett. **24**, 306–308 (1999). [CrossRef]

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

*et al*. demonstrated that they introduced the shape disorder of airholes in photonic crystal waveguides and experimentally confirmed that the strong localization was able to be achieved in the waveguide [19

19. J. Topolancik, F. Vollmer, and B. Llic, “Random high-Q cavities in disordered photonic crystal waveguides,” Appl. Phys. Lett. **91**, 201102 (2007). [CrossRef]

20. J. Topolancik, B. Llic, and F. Vollmer, “Experimental Observation of Strong Photon Localization in Disordered Photonic Crystal Waveguides,” Phys. Rev. Lett. **99**, 253901 (2007). [CrossRef]

14. C. Vanneste and P. Sebbah, “Localized modes in random arrays of cylinders,” Phys. Rev. E **71**, 026612 (2005). [CrossRef]

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

21. X. Jiang and C. M. Soukoulis, “Time Dependent Theory for Random Lasers,” Phys. Rev. Lett. **85**, 70–73 (2000). [CrossRef] [PubMed]

*et al*. [23

23. H. Miyazaki, M. Hase, H. T. Miyazaki, Y. Kurokawa, and N. Shinya, “Photonic material for designing arbitrarily shaped waveguides in two dimensions,” Phys. Rev. B **67**, 235109 (2003). [CrossRef]

*et al*. [24

24. C. Rockstuhl, U. Peschel, and F. Lederer, “Correlation between single-cylinder properties and bandgap formation in photonic structures,” Opt. Lett. **31**, 1741–1743 (2006). [CrossRef] [PubMed]

**66**, 144202 (2002). [CrossRef]

21. X. Jiang and C. M. Soukoulis, “Time Dependent Theory for Random Lasers,” Phys. Rev. Lett. **85**, 70–73 (2000). [CrossRef] [PubMed]

25. A. S. Nagra and R. A. York, “FDTD analysis of Wave Propagation in Nonlinear Absorbing and Gain Media,” IEEE Trans. Antennas. Propagat. **46**, 334–340 (1998). [CrossRef]

## 2. Simulation and analysis

^{2}, respectively. For making random structure with a defect region, we first fixed scatterer size (diameter 400 nm) and calculated the frequency windows of surrounding random structures. Then, the defect size was determined to be larger than the scatter size and to roughly form a half-wavelength resonator at frequencies in the frequency window by taking into account of the average refractive index of surrounding structures. Then, we calculated resonant and lasing properties by changing filling factors in the following section. Note that, when making random structures, we allowed to contacting scatterers, but prohibited overlapping of scatterers. Filling factors of scatterers were controlled by changing the number of scatterers randomly dispersed in the dispersion area.

^{-17}s and 50 nm, respectively, were chosen. Mur’s second-order absorbing boundary condition was used, and each calculation was performed for 530000 steps (37 ps). Light sources were set at individual cells in the surrounding medium of the dispersion area in the structures. To obtain intensity distributions, the light sources were excited homogeneously by cosine waves at a given frequency until a steady state was achieved. For the calculation of resonant spectra, a short Gaussian pulses (duration time about 10

^{-15}s) were launched at individual light sources and electric field at the center within the defect was recorded during a whole calculation time (37 ps) and resonant spectra were calculated from Fourier transform of the signal. For the calculation of resonant intensity distributions and spectra, random structures were assumed to be passive systems (without gain or absorption).

**66**, 144202 (2002). [CrossRef]

21. X. Jiang and C. M. Soukoulis, “Time Dependent Theory for Random Lasers,” Phys. Rev. Lett. **85**, 70–73 (2000). [CrossRef] [PubMed]

25. A. S. Nagra and R. A. York, “FDTD analysis of Wave Propagation in Nonlinear Absorbing and Gain Media,” IEEE Trans. Antennas. Propagat. **46**, 334–340 (1998). [CrossRef]

*r*

_{32}= 10

^{12}s

^{-1},

*r*

_{21}= 10

^{9}s

^{-1},

*r*

_{10}= 10

^{11}s

^{-1}, and

*N*= 3.31 × 10

^{24}m

^{-3}. The levels represented by subscripts 0 and 3 indicate the ground and excited states of atoms, respectively, and those represented by 1 and 2 denote the lower and upper laser levels, respectively. In the polarization equation, the gain spectrum was assumed to have a center frequency of

*ω*

_{a}= 280 THz and a bandwidth of

*Δω*

_{a}= 16 THz. The pumping rate was sufficiently large (

*r*

_{p}= 10

^{9}s

^{-1}) to simultaneously excite many lasing modes appearing in each structure. The electromagnetic field was calculated by numerically solving Maxwell’s equations and was absorbed or stimulated by the rate equations depending on the pumping rate. From the population inversion of the rate equations, polarization was calculated and changes in the electromagnetic field were induced via the polarization term of Maxwell’s equations. By repeating the calculation until laser oscillations achieved steady state, we calculated the lasing properties. From the steady state, we obtained the lasing emission intensity distributions and also their lasing emission spectra by Fourier transformation of the recorded time series data of electromagnetic field at the defect region.

## 3. Results and discussion

^{2}), which were different distributions of scatterers from Fig. 1 and without a defect region. For the calculation of the transmitted intensity spectra in Fig. 2, different from the procedure described in the previous section, a point light source was set at the center of structures (within a defect region) and 40 detection points set around the edge of the structures. Then, by launching a short Gaussian pulse at the light source, the transmitted electric fields were recorded and averaged all of the transmitted intensity spectra calculated at individual detection points. The results are shown in Fig. 2. The filling factors vary from 40 to 60% (from top to bottom), and the dotted curve indicates the spectrum profile of the incident pulse. Individual spectra are offset for clarity. The transmitted intensities clearly exhibited several sharp dips (frequency windows) at similar frequencies for all structures with different filling factors. The visibilities of the frequency windows at 40 and 60% were lower, and their transmittances became higher compared with those at 50%. This tendency was similar to the numerical results reported in previous papers [23

23. H. Miyazaki, M. Hase, H. T. Miyazaki, Y. Kurokawa, and N. Shinya, “Photonic material for designing arbitrarily shaped waveguides in two dimensions,” Phys. Rev. B **67**, 235109 (2003). [CrossRef]

24. C. Rockstuhl, U. Peschel, and F. Lederer, “Correlation between single-cylinder properties and bandgap formation in photonic structures,” Opt. Lett. **31**, 1741–1743 (2006). [CrossRef] [PubMed]

^{3}, 1 × 10

^{4}, and 5 × 10

^{2}for the structures with the filling factors of 40, 50, and 60%, respectively. These results suggest that the quality factors would strongly depend on the filling factors and observed modes at 60% would be more leaky rather than those at 40 and 50%.

**66**, 144202 (2002). [CrossRef]

**85**, 70–73 (2000). [CrossRef] [PubMed]

25. A. S. Nagra and R. A. York, “FDTD analysis of Wave Propagation in Nonlinear Absorbing and Gain Media,” IEEE Trans. Antennas. Propagat. **46**, 334–340 (1998). [CrossRef]

^{9}s

^{-1}, which was sufficiently high to simultaneously excite many lasing modes in the structure. From the results, we found that at 40%, several intense spots were also found in other interspaces, in addition to the defect region, and at 60%, mainly lasing modes bound in the connected scatterers were induced. In both cases, laser oscillations in interspaces or connected scatterers were dominantly induced rather than those in defect regions. In contrast, at 50%, an intense lasing spot was observed at the defect region, and unlike the result of Fig. 4, several bright spots were also found in the structure because the gain was so high that various modes with lower quality factors also appeared in the distribution. However, even under such conditions, the numerical result with the filling factor of 50% indicated that the lasing was distinctly induced in the defect, rather than any other intespaces or higher index regions. These results suggest the possibility that the resonant and lasing properties in random structures can be moderately controlled and limited in space and frequency domains by use of the defect region.

## 4. Conclusion

## Acknowledgments

## References and links

1. | C. Gouedard, D. Husson, C. Sauteret, F. Auzel, and A. Migus, “Generation of spatially incoherent short pulses in laser-pumped neodymium stoichiometric crystals and powders,” J. Opt. Soc. Am. B |

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

3. | M. A. Noginov, N. E. Noginova, H. J. Caulfield, P. Venkateswarlu, T. Thompson, M. Mahdi, and V. Ostroumov, “Short-pulsed stimulated emission in the powders of NdAl |

4. | D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature |

5. | D. S. Wiersma and S. Cavalieri, “Light emission: A temperature-tunable random laser,” Nature |

6. | G. A. Berger, M. Kempe, and A. Z. Genack, “Dynamics of stimulated emission from random media,” Phys. Rev. E |

7. | G. V. Soest, M. Tomita, and A. Lagendijk, “Amplifying volume in scattering media,” Opt. Lett. |

8. | G. V. Soest and A. Lagendijk, “β factor in a random laser,” Phys. Rev. E |

9. | A. Kurita, Y. Kanematsu, M. Watanabe, K. Hirata, and T. Kushida, “Wavelength- and Angle-Selective Optical Memory Effect by Interference of Multiple-Scattered Light,” Phys. Rev. Lett. |

10. | H. Cao, Y. G. Zhao, H. C. Ong, S. T. Ho, J. Y. Dai, J. Y. Wu, and R. P. H. Chang, “Ultraviolet lasing in resonators formed by scattering in semiconductor polycrystalline films,” Appl. Phys. Lett. |

11. | H. Cao, Y. Ling, J. Y. Xu, and C. Q. Cao, “Photon Statistics of Random Lasers with Resonant Feedback,” Phys. Rev. Lett. |

12. | H. Cao, J. Y. Xu, D. Z. Zhang, S.-H. Chang, S. T. Ho, E. W. Seeling, X. Liu, and R. P. H. Chang, “Spatial Confinement of Laser Light in Active Random Media,” Phys. Rev. Lett. |

13. | H. Cao, J. Y. Xu, E. W. Seeling, and R. P. H. Chang, “Microlaser made of disordered media,” Appl. Phys. Lett. |

14. | C. Vanneste and P. Sebbah, “Localized modes in random arrays of cylinders,” Phys. Rev. E |

15. | H. Fujiwara and K. Sasaki, “Observation of upconversion lasing within a thulium-ion-doped glass powder film containing titanium dioxide particles,” Jpn. J. Appl. Phys. |

16. | H. Fujiwara and K. Sasaki, “Observation of optical bistability in a ZnO powder random medium,” Appl. Phys. Lett. |

17. | S. I. Bozhevolnyi, V. S. Volkov, and K. Leosson, “Localization and Waveguiding of Surface Plasmon Polaritons in Random Nanostructures,” Phys. Rev. Lett. |

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

19. | J. Topolancik, F. Vollmer, and B. Llic, “Random high-Q cavities in disordered photonic crystal waveguides,” Appl. Phys. Lett. |

20. | J. Topolancik, B. Llic, and F. Vollmer, “Experimental Observation of Strong Photon Localization in Disordered Photonic Crystal Waveguides,” Phys. Rev. Lett. |

21. | X. Jiang and C. M. Soukoulis, “Time Dependent Theory for Random Lasers,” Phys. Rev. Lett. |

22. | J. Liu and H. Liu, “Theoretical investigation on the threshold properties of localized modes in two-dimensional random media,” J. Mod. Opt. |

23. | H. Miyazaki, M. Hase, H. T. Miyazaki, Y. Kurokawa, and N. Shinya, “Photonic material for designing arbitrarily shaped waveguides in two dimensions,” Phys. Rev. B |

24. | C. Rockstuhl, U. Peschel, and F. Lederer, “Correlation between single-cylinder properties and bandgap formation in photonic structures,” Opt. Lett. |

25. | A. S. Nagra and R. A. York, “FDTD analysis of Wave Propagation in Nonlinear Absorbing and Gain Media,” IEEE Trans. Antennas. Propagat. |

**OCIS Codes**

(140.0140) Lasers and laser optics : Lasers and laser optics

(290.4210) Scattering : Multiple scattering

(140.3945) Lasers and laser optics : Microcavities

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: January 21, 2009

Revised Manuscript: February 20, 2009

Manuscript Accepted: February 26, 2009

Published: February 27, 2009

**Citation**

Hideki Fujiwara, Yosuke Hamabata, and Keiji Sasaki, "Numerical analysis of resonant and lasing properties at a defect region within a random structure," Opt. Express **17**, 3970-3977 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-5-3970

Sort: Year | Journal | Reset

### References

- C. Gouedard, D. Husson, C. Sauteret, F. Auzel, and A. Migus, "Generation of spatially incoherent short pulses in laser-pumped neodymium stoichiometric crystals and powders," J. Opt. Soc. Am. B 10, 2358-2363 (1993). [CrossRef]
- N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, and E. Sauvain, "Laser action in strongly scattering media," Nature 368, 436-438 (1994). [CrossRef]
- M. A. Noginov, N. E. Noginova, H. J. Caulfield, P. Venkateswarlu, T. Thompson, M. Mahdi, and V. Ostroumov, "Short-pulsed stimulated emission in the powders of NdAl3(BO3)4, NdSc3(BO3)4, and Nd:Sr5(PO4)3F laser crystals," J. Opt. Soc. Am. B 13, 2024-2033 (1996). [CrossRef]
- D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, "Localization of light in a disordered medium," Nature 390, 671-673 (1997). [CrossRef]
- D. S. Wiersma and S. Cavalieri, "Light emission: A temperature-tunable random laser," Nature 414, 708-709 (2001). [CrossRef] [PubMed]
- G. A. Berger, M. Kempe, and A. Z. Genack, "Dynamics of stimulated emission from random media," Phys. Rev. E 56, 6118-6122 (1997). [CrossRef]
- G. V. Soest, M. Tomita, and A. Lagendijk, "Amplifying volume in scattering media," Opt. Lett. 24, 306-308 (1999). [CrossRef]
- G. V. Soest and A. Lagendijk, "β factor in a random laser," Phys. Rev. E 65, 047601 (2002). [CrossRef]
- A. Kurita, Y. Kanematsu, M. Watanabe, K. Hirata, and T. Kushida, "Wavelength- and Angle-Selective Optical Memory Effect by Interference of Multiple-Scattered Light," Phys. Rev. Lett. 83, 1582-1585 (1999). [CrossRef]
- H. Cao, Y. G. Zhao, H. C. Ong, S. T. Ho, J. Y. Dai, J. Y. Wu, and R. P. H. Chang, "Ultraviolet lasing in resonators formed by scattering in semiconductor polycrystalline films," Appl. Phys. Lett. 73, 3656-3658 (1998). [CrossRef]
- H. Cao, Y. Ling, J. Y. Xu, and C. Q. Cao, "Photon Statistics of Random Lasers with Resonant Feedback," Phys. Rev. Lett. 86, 4524-4527 (2001). [CrossRef] [PubMed]
- H. Cao, J. Y. Xu, D. Z. Zhang, S.-H. Chang, S. T. Ho, E. W. Seeling, X. Liu, and R. P. H. Chang, "Spatial Confinement of Laser Light in Active Random Media," Phys. Rev. Lett. 84, 5584-5587 (2000). [CrossRef] [PubMed]
- H. Cao, J. Y. Xu, E. W. Seeling, and R. P. H. Chang, "Microlaser made of disordered media," Appl. Phys. Lett. 76, 2997-2999 (2000). [CrossRef]
- C. Vanneste and P. Sebbah, "Localized modes in random arrays of cylinders," Phys. Rev. E 71, 026612 (2005). [CrossRef]
- H. Fujiwara and K. Sasaki, "Observation of upconversion lasing within a thulium-ion-doped glass powder film containing titanium dioxide particles," Jpn. J. Appl. Phys. 43, L1337-L1339 (2004). [CrossRef]
- H. Fujiwara and K. Sasaki, "Observation of optical bistability in a ZnO powder random medium," Appl. Phys. Lett. 89, 071115 (2006). [CrossRef]
- S. I. Bozhevolnyi, V. S. Volkov, and K. Leosson, "Localization and Waveguiding of Surface Plasmon Polaritons in Random Nanostructures," Phys. Rev. Lett. 89, 186801 (2002). [CrossRef] [PubMed]
- P. Sebbah and C. Vanneste, "Random laser in the localized regime," Phys. Rev. B 66, 144202 (2002). [CrossRef]
- J. Topolancik, F. Vollmer, and B. Llic, "Random high-Q cavities in disordered photonic crystal waveguides," Appl. Phys. Lett. 91, 201102 (2007). [CrossRef]
- J. Topolancik, B. Llic, and F. Vollmer, "Experimental Observation of Strong Photon Localization in Disordered Photonic Crystal Waveguides," Phys. Rev. Lett. 99, 253901 (2007). [CrossRef]
- X. Jiang and C. M. Soukoulis, "Time Dependent Theory for Random Lasers," Phys. Rev. Lett. 85, 70-73 (2000). [CrossRef] [PubMed]
- J. Liu and H. Liu, "Theoretical investigation on the threshold properties of localized modes in two-dimensional random media," J. Mod. Opt. 53, 1429-1439 (2006). [CrossRef]
- H. Miyazaki, M. Hase, H. T. Miyazaki, Y. Kurokawa, and N. Shinya, "Photonic material for designing arbitrarily shaped waveguides in two dimensions," Phys. Rev. B 67, 235109 (2003). [CrossRef]
- C. Rockstuhl, U. Peschel, and F. Lederer, "Correlation between single-cylinder properties and bandgap formation in photonic structures," Opt. Lett. 31, 1741-1743 (2006). [CrossRef] [PubMed]
- A. S. Nagra and R. A. York, "FDTD analysis of Wave Propagation in Nonlinear Absorbing and Gain Media," IEEE Trans. Antennas. Propagat. 46, 334-340 (1998). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.