## Random stack of resonant dielectric layers as a laser system

Optics Express, Vol. 12, Issue 15, pp. 3307-3312 (2004)

http://dx.doi.org/10.1364/OPEX.12.003307

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### Abstract

We propose a random stack of resonant dielectric layers as a system for random-laser study. Owing to the Fabry–Perot resonance of the dielectric layers, the propagation of light in such systems is frequency dependent (a band structure). As a consequence, if the system is designed such that pump light is in passband while optical gain is in stop band, the laser threshold can be reduced dramatically compared with those of completely disordered systems.

© 2004 Optical Society of America

## 1. Introduction

2. H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. **82**, 2278–2281 (1999). [CrossRef]

4. Y. Feng, J.-F. Bisson, J. Lu, S. Huang, K. Takaichi, A. Shirakawa, M. Musha, and K.-I. Ueda, “Thermal effects in quasi-continuous-wave Nd^{3+}:Y_{3}Al_{5}O_{12} nanocrystalline-powder random laser,” Appl. Phys. Lett. **84**, 1040–1042 (2004). [CrossRef]

6. X. H. Wu, A. Yamilov, H. Noh, H. Cao, E. W. Seelig, and R. P. H. Chang, “Random lasing in closely packed resonant scatterers,” J. Opt. Soc. Am. B **21**, 159–169 (2004). [CrossRef]

5. J. Ripoll, C. M. Soukoulis, and E. N. Economou, “Optimal tuning of lasing modes through collective particle resonance,” J. Opt. Soc. Am. B **21**, 141–149 (2004). [CrossRef]

6. X. H. Wu, A. Yamilov, H. Noh, H. Cao, E. W. Seelig, and R. P. H. Chang, “Random lasing in closely packed resonant scatterers,” J. Opt. Soc. Am. B **21**, 159–169 (2004). [CrossRef]

## 2. Band structure in a transmission spectrum

7. S. A. Bulgakov and M. Nieto-Vesperinas, “Field distribution inside one-dimensional random photonic lattices,” J. Opt. Soc. Am. A **15**, 503–510 (1998). [CrossRef]

8. Z.-Q. Zhang, “Light amplification and localization in randomly layered media with gain,” Phys. Rev. B **52**, 7960–7964 (1995). [CrossRef]

*N*=15 dielectric layers, with dielectric constant

*ε*=9 and with thickness and spacing uniformly distributed in the range 0–2 µm, embedded in a homogeneous background with dielectric constant

*ε*=1. Figure 1(a) shows a typical transmission spectrum. The spikes indicate the resonant modes of the sample, and the linewidth reflects the decay rate of the modes. The transmission peaks are distributed randomly.

*σ*, respectively. Figure 2 shows transmission spectra averaged over 5000 realizations for systems of

*N*=15 with

*σ*=0.01, 0.03, 0.09. The transmission spectra are periodically modulated with wave vector k. The period is

*π*√

*εd*, where

*d*=1 µm is the thickness of identical layers. The wave vectors at the transmission maxima correspond to the Fabry–Perot resonances of the identical layers. The transmission peak value decreases gradually for higher-order Fabry–Perot resonances. The larger the variance of the thickness is, the quicker the decrease is. This is because the thickness variance is stricter for light of short wavelength than for light of long wavelength. One can see that the band structures are preserved even with a thickness variance of 0.09. At the long-wavelength part of the spectra, where the wavelength is of the order of the system length, there are sharp peaks, which relate to the resonance of the whole system, as one can see in the transmission spectra of a single realization [for example, in Fig.1(b)]. These resonance peaks survive after averaging.

9. A. A. Chabanov and A. Z. Genack, “Photon localization in resonant media,” Phys. Rev. Lett. **87**, 153901 (2001). [CrossRef] [PubMed]

*T*), provides a decisive test for localization. They applied this idea in searching for photon localization in quasi-one-dimensional systems. In one-dimensional disordered systems the wave is always localized, but the variance of transmission is still a good manifestation of the localization effect. Therefore var(

*T*) is also calculated, as shown in Fig. 2. We can see that var(

*T*) is frequency dependent too. There are low-variance windows at the Fabry–Perot resonances and at the first stop band. This means that in these regions the effect of the randomness on a light wave is small. The fluctuation of the transmission from sample to sample at these regions is relatively small.

## 3. Application to random lasers

10. Y. Feng and K.-I. Ueda, “One-mirror random laser,” Phys. Rev. A **68**, 025803 (2003). [CrossRef]

11. A. L. Burin, H. Cao, and M. A. Ratner, “Two-photon pumping of a random laser,” IEEE J. Sel. Top. Quantum Electron. **9**, 124–127 (2003). [CrossRef]

### 3.1 Qualitative model

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

10. Y. Feng and K.-I. Ueda, “One-mirror random laser,” Phys. Rev. A **68**, 025803 (2003). [CrossRef]

*g*(

*x*) is proportional to the pump distribution; therefore

*G*is a structure-independent parameter, which represents gain and is proportional to the pump power.

*A*(

*x*) is a randomly varying function.

*ξ*is the localization length of the pump light, which is due to scattering as well as to absorption [8

_{p}8. Z.-Q. Zhang, “Light amplification and localization in randomly layered media with gain,” Phys. Rev. B **52**, 7960–7964 (1995). [CrossRef]

*B*(

*x*,

*x*

_{0}) is a randomly varying function.

*ξ*is the localization length of the lasing light.

_{l}*x*

_{0}represents the distance from mode center to the left-hand surface. Losses of modes can be described as

*L*is the length of the sample and

*δ*

_{0}is the mode loss that is due to absorption inside the sample. At threshold, gain equals loss, so one has

*A*(

*x*) and

*B*(

*x*,

*x*

_{0}) is arbitrarily replaced by a constant

*C*, for our objective is a qualitative analysis.

*G*is the threshold gain of mode centering at

_{t}*x*

_{0}, which depends on

*x*

_{0}. The threshold for the system is the minimum of

*G*, which we cannot get in analytical form.

_{t}*ξ*(

*λ*) by fitting averaged transmission as a function of system length

*L*to the following formula [13

13. M. Patra, “Decay rate distributions of disordered slabs and application to random lasers,” Phys. Rev. E **67**, 016603 (2003). [CrossRef]

*ξ*(

*λ*), we use more than 1000 samples. In addition, in numerical simulations the depletion of pump light by absorption is assumed always to be much smaller than that by scattering; therefore the depletion by absorption is ignored, and

*δ*

_{0}is set to zero.

### 3.2 Numerical results

*σ*, respectively, and the dielectric constant is changed to 4 to be more practical.

^{-1}, which is near the best-localized wavelength in simulation, and let the pump wavelength vary. The samples are set to have 30 layers and 30 spacings, which correspond to a length of 60 µm. The results are shown in Fig. 3 (bottom). One can see that the lasing threshold for the systems studied is generally much smaller than that of a completely random system. The threshold is modulated according to the localization length. For the specific system simulated with

*σ*=0.01, a maximum 10

^{8}-times-smaller threshold can be achieved.

### 3.3. Discussion

*n*×

*d*of the monodisperse layers to fit the absorption and gain spectra. We see that near Fabry–Perot resonance the transmission bandwidth is large [Fig. 1(c)] and the fluctuation of transmission from sample to sample is relatively small [Fig. 2], so the requirement for control of thickness is not great. It may be worth noting that we consider here random stacks of resonant layers; in fact, for systems with uniform spacing between layers of random thickness the proposal described above is also applicable.

*Q*factor can be enhanced dramatically in two and three coupled microspheres [5

5. J. Ripoll, C. M. Soukoulis, and E. N. Economou, “Optimal tuning of lasing modes through collective particle resonance,” J. Opt. Soc. Am. B **21**, 141–149 (2004). [CrossRef]

## 4. Summary

## References

1. | V. S. Letokhov, “Generation of light by a scattering medium with negative resonance absorption,” Sov. Phys. JETP |

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

3. | H. Cao in |

4. | Y. Feng, J.-F. Bisson, J. Lu, S. Huang, K. Takaichi, A. Shirakawa, M. Musha, and K.-I. Ueda, “Thermal effects in quasi-continuous-wave Nd |

5. | J. Ripoll, C. M. Soukoulis, and E. N. Economou, “Optimal tuning of lasing modes through collective particle resonance,” J. Opt. Soc. Am. B |

6. | X. H. Wu, A. Yamilov, H. Noh, H. Cao, E. W. Seelig, and R. P. H. Chang, “Random lasing in closely packed resonant scatterers,” J. Opt. Soc. Am. B |

7. | S. A. Bulgakov and M. Nieto-Vesperinas, “Field distribution inside one-dimensional random photonic lattices,” J. Opt. Soc. Am. A |

8. | Z.-Q. Zhang, “Light amplification and localization in randomly layered media with gain,” Phys. Rev. B |

9. | A. A. Chabanov and A. Z. Genack, “Photon localization in resonant media,” Phys. Rev. Lett. |

10. | Y. Feng and K.-I. Ueda, “One-mirror random laser,” Phys. Rev. A |

11. | A. L. Burin, H. Cao, and M. A. Ratner, “Two-photon pumping of a random laser,” IEEE J. Sel. Top. Quantum Electron. |

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

13. | M. Patra, “Decay rate distributions of disordered slabs and application to random lasers,” Phys. Rev. E |

**OCIS Codes**

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

(290.4210) Scattering : Multiple scattering

**ToC Category:**

Research Papers

**History**

Original Manuscript: May 17, 2004

Revised Manuscript: July 6, 2004

Published: July 26, 2004

**Citation**

Yan Feng and Ken-ichi Ueda, "Random stack of resonant dielectric layers as a laser system," Opt. Express **12**, 3307-3312 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-15-3307

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### References

- V. S. Letokhov, "Generation of light by a scattering medium with negative resonance absorption," Sov. Phys. JETP 26, 835-840 (1968).
- H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, "Random laser action in semiconductor powder," Phys. Rev. Lett. 82, 2278-2281 (1999). [CrossRef]
- H. Cao, in Optical Properties of Nanostructured Random Media, V. M. Shalaev, ed. (Springer-Verlag, Berlin, 2002), pp. 303-329
- Y. Feng, J.-F. Bisson, J. Lu, S. Huang, K. Takaichi, A. Shirakawa, M. Musha, and K.-I. Ueda, "Thermal effects in quasi-continuous-wave Nd3+:Y3Al5O12 nanocrystalline-powder random laser," Appl. Phys. Lett. 84, 1040-1042 (2004). [CrossRef]
- J. Ripoll, C. M. Soukoulis, and E. N. Economou, "Optimal tuning of lasing modes through collective particle resonance," J. Opt. Soc. Am. B 21, 141-149 (2004). [CrossRef]
- X. H. Wu, A. Yamilov, H. Noh, H. Cao, E. W. Seelig, and R. P. H. Chang, "Random lasing in closely packed resonant scatterers," J. Opt. Soc. Am. B 21, 159-169 (2004). [CrossRef]
- S. A. Bulgakov and M. Nieto-Vesperinas, "Field distribution inside one-dimensional random photonic lattices," J. Opt. Soc. Am. A 15, 503-510 (1998). [CrossRef]
- Z.-Q. Zhang, "Light amplification and localization in randomly layered media with gain," Phys. Rev. B 52, 7960-7964 (1995). [CrossRef]
- A. A. Chabanov and A. Z. Genack, "Photon localization in resonant media," Phys. Rev. Lett. 87, 153901 (2001). [CrossRef] [PubMed]
- Y. Feng and K.-I. Ueda, "One-mirror random laser," Phys. Rev. A 68, 025803 (2003). [CrossRef]
- A. L. Burin, H. Cao, and M. A. Ratner, "Two-photon pumping of a random laser," IEEE J. Sel. Top. Quantum Electron. 9, 124-127 (2003). [CrossRef]
- P. Sebbah and C. Vanneste, "Random laser in the localized regime," Phys. Rev. B 66, 144202 (2002). [CrossRef]
- M. Patra, "Decay rate distributions of disordered slabs and application to random lasers," Phys. Rev. E 67, 016603 (2003). [CrossRef]

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