Influence of gain localization in one-dimensional random media

ARTICLE IN PRESS Physica B 404 (2009) 1471–1476

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Influence of gain localization in one-dimensional random media Yulong Tang , Yong Yang, Jianqiu Xu Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Science, Shanghai 201800, China

a r t i c l e in fo

abstract

Article history: Received 16 October 2007 Received in revised form 14 July 2008 Accepted 3 January 2009

Based on the transfer matrix method, we studied the effects of gain distribution on the performance of light localization in one-dimensional completely random systems. Due to the scattering of the pump light, the gain can also be localized. Compared with homogenous gain distribution, the localized gain can induce light confinement more efficiently. The overlapping between the laser modes and the localized gain brings on the high-Q modes that are amplified preferentially, and reduces the lasing threshold. The threshold can be hundreds times lowered in some special cases. With increase of the random system size, the localization position goes deeper into the media first, and then saturates when the system is thicker than 550 layers. For a given kind of random laser systems, an optimal system size exists, for which the mode intensity is significantly enhanced. The influence of the system structure on the lasing intensity and localization position is also discussed. Crown Copyright & 2009 Published by Elsevier B.V. All rights reserved.

PACS: 42.55.Zz 42.25.Dd 78.90.+t Keywords: Random laser Light localization Scattering Disorder

1. Introduction The amplification of stimulated emission in random scattering media was first proposed by Letokhov in 1968 [1], but the experimental observation of random laser emission was carried out till 1986 from the polycrystalline powder [2], and 1994 from the dye solution [3], respectively. Recently, coherent random laser emission was also extensively observed [4–6]. Since being reported, random lasers have attracted more and more concerns for their unique properties and potential applications. Differing from the conventional lasers, the feedback in the random laser comes from multiple scattering in disordered media rather than from the traditional cavity that consists of mirrors. There are two categories of random lasers: incoherent and coherent. The incoherent random laser, characterized by optical power feedback, occurs in the weakly scattering regime. On the contrary, the coherent random emission, characterized by amplitude feedback, was observed only in the strongly scattering media, where the scattering of light can form closed loop paths [7]. Anderson localization of light was originally considered as the underlying mechanism of coherent random laser in strongly scattering media [8]. Recently, several models such as amplification of quasi-modes [9], random resonators [10], and enhanced amplified spontaneous

 Corresponding author. Tel.: +86 21 69918602; fax: +86 21 69918507.

E-mail addresses: [email protected] (Y. Tang), [email protected] (J. Xu).

emission (ASE) [11] have been suggested to explain the coherent random emission. Many theoretical analyses and numerical simulations have been constructed to understand the unusual behavior of emission observed in random systems [4,5,12]. The analysis methods include the diffusion equation with gain [13], the ring laser with non-resonant feedback [14], the analytical model based on quasistates [8], and the finite-difference time-domain (FDTD) method combined with the interplay between localization and amplification [15]. Compared with these methods, the transfer matrix (TM) method contains no simplified approximation, and can simulate exactly the wave propagation in random systems [16–18]. Besides, the TM method can calculate the quasi modes of weakly scattering systems that overlap spectrally and have short lifetimes. Due to the resonance of localized modes, the transmission spectra of localized 1-D systems exhibit many randomly distributed high-transmission peaks [19], which is related to both the system size L [20] and the particle density [21]. Besides localized modes, non-localized modes (so-called necklace states) can also exist in 1-D random systems [22]. For generating the confinement of light, the excitation methods can be optical or electrical [23]. In the research of random lasers, most of attention is concentrated on the localization of the lasing light, few researches considered the influence of the localization of pump light. In their simulations, the pump light and gain were assumed to be uniform and homogeneous inside the whole random media. Actually, because the pump light is scattered too by the random media,

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both the pump and lasing light can be trapped inside the strongly scattering media. In this paper, we explore the interaction between the localized gain and the random emission in one dimensional random media. Based on the TM method, the localizations of both the pump and signal light are calculated. Compared to the homogenous gain, the coupling between the localized pump and signal light enhances the amplification of the localized modes, and reduces the lasing threshold. Lasing features of the random structure with localized gain are examined, and the influence of random system thickness is analyzed.

2. Theoretical model As shown in Fig. 1, the 1-D random system is made up of a number of dielectric layers. Each layer is constituted by a couple of film a and b. The thickness of the film a and b are randomized by La ¼ La0 ð1 þ Aa r a Þ and Lb ¼ Lb0 ð1 þ Ab r b Þ, where ra and rb are random numbers distributed between (0.5, 0.5) uniformly, and the amplitudes of randomness are 0pAa p1 and 0pAb p1. The dielectric constants of the films are a ¼ 0  i00 ðoÞ and eb ¼ 1, respectively. The whole system is embedded in a homogeneous infinite material with the dielectric constant e0 ¼ 1. For the 1-D case, the time-independent Maxwell’s equations are given as (only considering one direction of the electromagnetic field) 8 2 > q EðzÞ o2 > > þ 2 ðzÞEðzÞ ¼ 0; < qz2 c (1) 2 > q HðzÞ o2 > > þ 2 ðzÞHðzÞ ¼ 0: : 2 qz c The corresponding solution can be written in the form ( EðzÞ ¼ UðzÞ expðikzÞ; HðzÞ ¼ VðzÞ expðikzÞ;

(2)

where U(z) and V(z) are slowly varying amplitudes of electric and magnetic fields, respectively. Substituting Eq. (2) into Eq. (1) and using the continuity boundary conditions of the electric and magnetic fields, we obtain [24] " # " # U nþ1 Un ¼ Mn . (3) V nþ1 Vn The characteristic TM of the nth dielectric film layer is " #  pi sinðkLn cos yÞ cosðkLn cos yÞ , (4) Mn ¼ ip sinðkLn cos yÞ cosðkLn cos yÞ pffiffiffiffiffiffiffiffi pffiffiffi where k ¼ ðo=cÞ , p ¼ =m cos y, and y denoting the angle between z axis and the direction of the co-phasal surface. In active media, the complex dielectric constant (e) has a passive imaginary part (00 40), originating from population inversion. When 00 40, the emission light is amplified by the stimulated polarization; while 00 o0, the emission light is absorbed. For the whole system, Q the TM will be MðLÞ ¼ M n . Using the TM, final electric and

Lb εb ε0

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Fig. 1. Diagram of one-dimensional random system.

magnetic fields intensities can be obtained for given initial fields after propagating in active media. When pump light travels in the random system, it undergoes scattering and absorption. The medium ions are thus excited by absorbed pump light. The pump rate Pr can be written as Pr ¼ Za P in =hnp , where Pin is the incident pump power, np is the frequency of pump beam, h is the Plank’s constant, and Za is the fraction of incident pump power absorbed by the random medium. Considering a four-level system, the second level (N2) and the first level (N1) are called the upper and the lower lasing levels. The lifetimes of up levels N3, N2 and N1 are t32, t21 and t10. The rate equations can be written as [25] 8 dN3 N3 > > ¼ Pr N0  ; > > dt t > 32 > > > > dN2 N3 N2 > > > < dt ¼ t  DN  sfc  t ; 32 21 (5) dN1 N2 N1 > > > ¼ þ D N  s f c  ; > > dt t21 t10 > > > > > dN0 N1 > > ¼  Pr N0 ; : dt t10 where s is the stimulated cross section, f is the photon density, c is the velocity of light, and DN ¼ N2  N1 is the population inversion per unit volume. In Eq. (5), the contribution of the spontaneous emission is included in the parameter t21. By solving the rate equations, the population inversion DN can be obtained. In the media, the polarization driven by absorbed pump light EðoÞ expðiotÞ can be described by the differential equation of an electric oscillator [26] with amplitude of x, frequency of o0, and damping constant of g 2

d x dx e þ o20 x ¼  EðoÞeiot , þ 2g dt m dt 2

(6)

where e and m are the electron charge and mass, respectively. From Eq. (6), the imaginary part of the complex dielectric constant can be solved as

00 ðoÞ ¼

DNe2 g  . 2o0 m ðo  o0 Þ2 þ g2

(7)

In terms of light wavelength, 00 ðoÞ is re-written as

00 ðlÞ ¼ C 0 

og l3g ðl  lg Þ2 þ o2g

,

(8)

where C 0 ¼ DNe2 =8p2 mc2 , l is light wavelength, lg is the light wavelength at the center of the gain line, and og is the half-width of the gain spectrum.

3. Numerical simulation and results Our simulation process is as follow. First, the pump light distribution in the medium is calculated with the TM. Different pump light intensities cause different pump rates in different film layers. Secondly, population inversion DN of different film layers is obtained by solving the rate equations of (5), leading to different DN dependent e00 of the complex dielectric constant. Finally, e00 is substituted into the TM to calculate signal light intensity. pffiffiffiffiffi Parameters taken in our simulations are: Re ð a Þ ¼ 1:82 and b ¼ 1 (corresponding to Nd:YAG and air, respectively); La0 ¼ 100 nm, Lb0 ¼ 50 nm; Aa ¼ 0:5, Ab ¼ 0:7; and lg ¼ 1064 nm, wg ¼ 0:45 nm. In the 1-D model shown in Fig. 1, light may experience significant reflection from multiple dielectric layers due to the wave interference effect. Collective multiple-scattering interference effects lead to large field enhancement in the random

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system. When the pump light is launched into the random stack, interference between the forward and backward light gives rise to inhomogeneous distribution. As a result, the pump intensities in some regions will be much higher than that in other regions. This is the well-known light localization in the 1-D random system. At the same time, the signal light arising from spontaneous emission will be localized in the similar manner. As shown in Fig. 2 (obtained from a 20-layer random system), the localization of the pump and signal light may be overlapped. In the overlapped regions, the amplification of the laser modes is enhanced. In most cases, there are many localized modes in a random medium. In contrast to the assumption of homogenous gain, the localized modes are selected by the localized gain. Only the modes overlapped with the localized gain can survive. Furthermore, the mode space, frequency and linewidth should depend on the properties of the gain localization. The intensity distributions of the plane-wave pump light inside various random media are depicted in Fig. 3. The random systems in Fig. 3 are constructed with different random seeds. Clearly, the pump intensity changes irregularly across the whole random media, and thus the laser gain is definitely inhomogeneous. Moreover, the pump intensity decays to zero when the light penetrates a short distance L into the random media. As a result, the light localization occurs mostly in the distance L close to the medium surface. The laser gain is related to the pump intensity through Eq. (8). For homogeneous gain distribution, C0 is constant in all film layers; when gain localization is considered, the pump light distribution Ip inside the random medium is calculated first, and then C0 is normalized by C 0  Ip =Ip, where Ip denotes the averaged value of Ip. We first calculated the emission intensity when a plane-wave pump light irradiates on a 100-layer random system. As shown in Fig. 4, the intensities of the localized modes with the localized gain are much higher than that with the homogeneous gain. It can be explained as follows: with the localized gain, some high quality modes overlap with the high gain region, thus extract more pump power than in the homogenous gain medium. Moreover, because of the mode selection effect by the localized gain, the mode number is less than that in the homogenous gain medium. With the homogeneous and localized gain separately to a 500-layer random system, the emission intensity was calculated

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Fig. 3. Intensity distribution of the same plane-wave pump light in four different kinds of random structures (a), (b), (c) and (d).

as a function of wavelength l. The results are presented in Fig. 5. The typical resonant peaks around 1064 nm (denoted by l1, l2) in Fig. 5(c) and (d) correspond to the resonant cavity modes in the random media. The quality factor (Q) of the random cavity determines the spectral linewidth and the lasing threshold [12]. The coupling between the localized mode and gain region leads to high-Q modes, and narrows the linewidth of the cavity mode l2. Because the optical gain predominantly benefits the mode with the higher quality factor Q, the cavity mode l2 is about tens of times higher than the cavity mode l1. It is the efficient coupling between high-Q modes and localized gain that gives rise to high intensity of the localized mode, and reduces the lasing threshold. In Fig. 6, we have also calculated the intensity spatial distribution inside the random medium for the wavelength l1 (with the homogeneous gain) and l2 (with the localized gain). For a given mode, the localization of the emission is found in several small regions, and the thickness of each localization region is about 10 layers (1.5 mm). Furthermore, there is at least one localization center, where the lasing intensity is much higher than the other localization regions. Further, the positions of localization centers are different with different gain distributions. In order to explore the influence of the system structure, the calculations were carried out for various random systems. The results of ten samples of 500-layer random systems with different structures are given in Fig. 7. It is found that the wavelengths for the homogenous and localized gain are nearly the same, but the wavelength changes in different random structures. This is in agreement with the observation of Vanneste and Sebbah [27]. The emission intensity with the localized gain is considerably larger than that with the homogeneous gain for the same structure. In some special cases, the intensity can be enhanced hundreds times with the localized gain. Meanwhile, in some other cases, the intensity is only several times higher than that with the homogenous gain. The position and intensity of the strongest localized modes as a function of the size of random systems are presented in Fig. 8. All data were averaged over 50 random samples. The averaged results were obtained by finding out the wavelength of the resonant

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Fig. 5. The pump distribution and the emission spectra for a 500-layer 1-D completely random system: (a) intensity distribution of the pump light, and of the signal light for (b) without gain (C0 ¼ 0), (c) with homogeneous gain (C0 ¼ 0.021), and (d) with localized gain (C 0 ¼ 0:021  Ip =Ip ).

mode in each random system first, and then calculating corresponding mode position and intensity. The final results were averaged over all simulations. Because the closed loop paths are stretched in larger systems, the position of the localization center goes deeper into the media with increasing the size of random system. The localization center moves from 20 to 400 layers when

the random system grows from 50 to 500 layers. When the system is much larger than the penetration depth of the pump light, the pump intensity has decayed to zero. Therefore, increasing the system size further does not change the localization position. In our simulations, the localization position keeps nearly unchanged when the system is thicker than 550 layers.

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As seen from Fig. 8(b), the mode intensity is greatly enhanced in the random system with optimal size. Optimum system size exists for a given kind of random laser systems. The maximum intensity of the localized mode in random systems with 450 layers originates from the best coupling between the localized gain and signal light. In these optimal sized random

systems, the localized position and the penetration depth of the pump light are well balanced, and the overlapping between the localized mode and the pump distribution is maximized. Consequently, the mode intensity in the optimized random system is much higher than that of the non-optimized systems.

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4. Conclusions We have presented the theoretical investigation of light confinement in 1-D completely random systems, and analyzed the influence of gain distribution on the lasing characteristics of the 1-D random lasers. Due to multiple scattering in the random media, the pump light should be inhomogeneous. The localization of the laser gain happened in the strongly scattering random system. In the meantime, the signal light arising from ASE can also be localized in the medium. The spatial overlapping between the localized emission and gain enhances the light amplification, and the lasing threshold is then reduced. The emission intensity and wavelength of the localized mode depend on the structure of random system. In some special structures, the lasing threshold can be greatly reduced. Additionally, the light behavior is influenced by the thickness of 1-D random system. In the optimal sized random system, the overlapping between the localized mode and gain is maximized, and the lasing intensity is significantly high. Optimization of the random system, where the localized mode can be overlapped with the gain more efficiently, will be interesting and meaningful for applications of random lasers.

Acknowledgment This work is supported by the National Science Foundation of China under Contract 60678016. References [1] V.S. Letokhov, Sov. Phys. JETP 26 (1968) 835. [2] V.M. Markushev, V.F. Zolin, Ch.M. Briskina, Sov. J. Quant. Electron. 16 (1986) 281.

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