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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 15 — Jul. 26, 2004
  • pp: 3307–3312
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Random stack of resonant dielectric layers as a laser system

Yan Feng and Ken-ichi Ueda  »View Author Affiliations


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

When optical gain is introduced into random dielectric media, interesting phenomena appear. One is amplified spontaneous emission, which results from the increase of photon path length owing to multiple scattering; it was treated theoretically more than 30 years ago by Letokhov [1

1. V. S. Letokhov, “Generation of light by a scattering medium with negative resonance absorption,” Sov. Phys. JETP 26, 835–840 (1968).

] with a diffusion formalism. Another exciting consequence is lasing with coherent feedback, which was experimentally demonstrated recently [2

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]

, 3

3. H. Cao in Optical Properties of Nanostructured Random MediaV. M. Shalaev, ed. (Springer-Verlag, Berlin, 2002), pp. 303–329

].

In this paper we investigate propagation of light in random stacks of resonant dielectric layers and show that by special design the lasing threshold in such systems can be reduced dramatically compared with those in completely random media. Such systems could be good candidates for experimental study of random lasers.

2. Band structure in a transmission spectrum

We consider the propagation of vertically incident plane-wave light in random stacks of resonant dielectric layers. The transmission, reflection, and optical fields inside the media can be readily calculated by the transfer matrix method [7

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

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

].

First we simulate several sample stacks with different degrees of order. The first sample is a completely random one; it consists of 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.

By randomly replacing the layers with others of identical thickness, one introduces order into the stack. Figure 1(b) shows the transmission spectrum of a stack containing 10 identical layers of 1-µm thickness, where spacing between dielectric layers remains random and uniformly distributed in the range 0–2 µm. The transmission spectrum of a single dielectric layer of 1-µm thickness is also shown in Fig. 1(b) for comparison. It is evident that the transmission lines of a layered system have the tendency to form bands, which relates to Fabry–Periot resonance of the identical dielectric layers. Figure 1(c) shows the transmission spectrum of a stack all of whose dielectric layers have the identical thickness of 1 µm. The band structure is clearer. Figure 1(d) plots the transmission spectrum of a fully ordered system for which the thickness of all dielectric layers and spaces is 1 µm; the resonance peaks are regular now. So propagation of light in random dielectric stacks that contain resonant layers is substantially different from that in either completely disordered or ordered media.

The band structure of the spectra in the partially ordered systems is more evident when the spectra are assembly averaged. In performing this averaging, we consider stacks formed by monodisperse dielectric layers with normal distribution, and the spacing remains uniformly distributed over 0–2 µm. These kinds of random stacks of resonant dielectric layers are experimentally obtainable. The mean value and the variance of the dielectric layer thickness are 1µm and 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 πεd, 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.

Fig. 1. Transmission spectra for systems with number of layers N=15. In (a) the system is completely random; in (b) it contains 10 identical layers of 1-µm thickness and a transmission spectrum for a single dielectric layer of 1-µm thickness; in (c) all dielectric layers are identical but the spacing remains random; in (d) the dielectric layers and the spacing are identical.
Fig. 2. Transmission spectra averaged over 5000 realizations and the variance for systems formed by dielectric layers of thickness with normal distribution. System length N=15; ε=9; variance σ=0.01, 0.03, 0.09.

Chabanov and Genack [9

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

] proved that the variance of transmission fluctuation, var(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

The band structure can influence lasing greatly in systems of this kind. In a bandgap, transmission is reduced, whereas in a passband transmission is enhanced compared with the transmission in a completely random system. There are more modes of higher quality, which are located in the bandgap or at the edges of passbands, than in corresponding random systems. Such modes are likely to lase first.

The band structure in these kinds of systems may also be useful for effectively pumping of random media, which is one of the practical issues of random-laser study [10

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

, 11

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]

]. A difficulty arises from the fact that lasing light as well as pump light is strongly scattered in completely random media. Pump light can penetrate into the random sample only to a limited depth, and the modes that have good overlap with the pumped region will also suffer large loss, for they are located close to the sample’s surface. The lasing threshold is high and difficult to approach. For the kind of system studied in this paper, one can let pump light be in the passband, for example, in the first passband, and lasing light can be at the edge of the passband or in the bandgap, for it is usually longer than the pump light. With such a scheme, one can avoid the difficulty in effective pumping mentioned above.

3.1 Qualitative model

To model lasing in a random medium in detail, one may combine Maxwell’s equations with the rate equations of electronic population in the disordered system and apply the finite-difference time-domain (FDTD) method to simulate the lasing system [12

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

]. But the finite-difference time-domain method is time consuming, so here we choose to analyze the threshold reduction by using a qualitative approach [10

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

].

We study one-dimensional systems in a localized regime. The system is pumped from the left side. Gain distribution g(x) is proportional to the pump distribution; therefore

g(x)=GA(x)exp(xξp).
(1)

G is a structure-independent parameter, which represents gain and is proportional to the pump power. A(x) is a randomly varying function. ξp is the localization length of the pump light, which is due to scattering as well as to absorption [8

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

]. The intensity pattern of the localized modes follows the formula

I(x,x0)=B(x,x0)exp(xx0ξl),
(2)

where B(x,x 0) is a randomly varying function. ξl is the localization length of the lasing light. x 0 represents the distance from mode center to the left-hand surface. Losses of modes can be described as

δexp(x0ξl)+exp(Lx0ξl)+δ0,
(3)

where 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

0Lg(x)I(x,x0)dx=0LGtA(x)B(x,x0)exp(xξpxx0ξl)dx
=GtC0Lexp(xξpxx0ξl)dx
=δ.
(4)

where the product of two randomly varying functions A(x) and B(x,x 0) is arbitrarily replaced by a constant C, for our objective is a qualitative analysis. Gt is the threshold gain of mode centering at x 0, which depends on x 0. The threshold for the system is the minimum of Gt, which we cannot get in analytical form.

To evaluate the threshold, localization lengths for the pump and the lasing light are required. One calculates localization lengths 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]

]:

Lξ(λ)=<lnT(λ,L)>.
(5)

In calculating localization lengths 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

We calculate the localization lengths of systems that are formed by dielectric layers with normal distribution and spacing uniformly distributed in the range 0–2 µm and of random systems for which both dielectric layers and spacing are uniformly distributed in the range 0–2 µm]. The mean value and the variance of the dielectric layer’s thickness are 1 µm and σ, respectively, and the dielectric constant is changed to 4 to be more practical.

The lasing threshold is then simulated. To show our proposal, we have set the lasing wavelength at 0.325 π µm-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 108-times-smaller threshold can be achieved.

3.3. Discussion

It should be noted that the simulation here is qualitative and that the lasing threshold in a real system fluctuates from sample to sample. In practice, for a given gain medium, one needs to design the optical thickness 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.

Fig. 3. Simulation results for systems with σ=0.1, 0.01, and random system. Top, the reciprocal of localization length as a function of wave number; bottom, the threshold of laser at 0.325π µm-1, which is near the best-localized wavelength, for a system of 60-µm length, as a function of wave number of pump light.

4. Summary

The kinds of systems studied in this paper are intermediate between ordered and disordered systems. Consequently, propagation of light in such systems behaves intermediately: It fluctuates from sample to sample because of the randomness but shows a band structure to a certain extent, as in an ordered system.

In summary, we have proposed random stacks of resonant dielectric layers as systems for random-laser study. Owing to the Fabry–Perot resonances of the dielectric layers, the propagation of light in such systems is frequency dependent; a band structure is formed. As a consequence, the system can be designed such that the pump light is in the passband whereas the optical gain is in a stop band, and the laser threshold can be reduced dramatically compared with those in completely random systems.

This work is supported by the 21st Century COE (Center of Excellence) program of Ministry of Education, Science and Culture of Japan.

References

1.

V. S. Letokhov, “Generation of light by a scattering medium with negative resonance absorption,” Sov. Phys. JETP 26, 835–840 (1968).

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]

3.

H. Cao in Optical Properties of Nanostructured Random MediaV. M. Shalaev, ed. (Springer-Verlag, Berlin, 2002), pp. 303–329

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 Nd3+:Y3Al5O12 nanocrystalline-powder random laser,” Appl. Phys. Lett. 84, 1040–1042 (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]

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]

9.

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

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]

12.

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

13.

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

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

  1. V. S. Letokhov, "Generation of light by a scattering medium with negative resonance absorption," Sov. Phys. JETP 26, 835-840 (1968).
  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]
  3. H. Cao, in Optical Properties of Nanostructured Random Media, V. M. Shalaev, ed. (Springer-Verlag, Berlin, 2002), pp. 303-329
  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 Nd3+:Y3Al5O12 nanocrystalline-powder random laser," Appl. Phys. Lett. 84, 1040-1042 (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]
  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]
  9. A. A. Chabanov and A. Z. Genack, "Photon localization in resonant media," Phys. Rev. Lett. 87, 153901 (2001). [CrossRef] [PubMed]
  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]
  12. P. Sebbah and C. Vanneste, "Random laser in the localized regime," Phys. Rev. B 66, 144202 (2002). [CrossRef]
  13. M. Patra, "Decay rate distributions of disordered slabs and application to random lasers," Phys. Rev. E 67, 016603 (2003). [CrossRef]

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