научная статья по теме LIGHT LOCALIZATION SIGNATURES IN BACKSCATTERING FROM PERIODIC DISORDERED MEDIA Физика

Текст научной статьи на тему «LIGHT LOCALIZATION SIGNATURES IN BACKSCATTERING FROM PERIODIC DISORDERED MEDIA»

Pis'ma v ZhETF, vol.86, iss.9, pp. 651-656

© 2007 November 10

Light localization signatures in backscattering from periodic

disordered media

C. Tian

Institut für Theoretische Physik, Universität zu Köln, 50937 Köln, Germany Submitted 27 August 2007

The backscattering line shape is analytically predicted for thick disordered medium films where, remarkably, the medium configuration is periodic along the direction perpendicular to the incident light. A blunt triangular peak is found to emerge on the sharp top. The phenomenon roots in the coexistence of quasi-lD localization and 2D extended states.

PACS: 42.25.-p

Introduction. Since the mid-eighties [1] the coherent backscattering (CBS) has been one of the pilots of studies of Anderson localization of light. Indeed, manifestations of weak localization (WL) in the CBS line shape have been well documented for various disordered dielectric media [2], while how strong localization (SL) affects CBS has been a long term fascinating subject [3]. The last decade has witnessed spectacular progress on CBS near [4, 5] or far below [6] the localization transition, which undoubtedly is an intellectual challenge both experimentally and theoretically. Indeed, to prepare strong scattering media and to extract localization from medium absorption are highly restrictive [4], while the failure of perturbation theory [7] - crucially mapping the pictorial reciprocal paths into (diagrammatical) one-loop approximation [1, 2]-enforces the invention of a nonperturbative theory to allow a microscopic analysis. Despite of these difficulties a common belief is that SL is responsible for rounding the CBS sharp top [3-6].

In studies of light localization much attention has been paid to fully disordered media. There have also been increasing interests on other medium structure such as disordered photonic crystals [8] where the Bloch symmetry is slightly destroyed by impurities, and systems with perfect Bloch symmetry [9] where WL is analytically found. Most interestingly, the recent invention of so-called planar random laser introduces a novel medium structure [10], which consists of a random gain layer sandwiched by two mirrors. It was then conjectured that SL in the layer plane might be responsible for the laser emission [10]. To prove it is yet a nontrivial task which may be traced back to the striking feature of the partially disordered structure. Indeed, (from the geometrical optics view) two perfect reflection mirrors map the medium to an extended periodic one but fully disordered inside the primitive cell (apart from the mirror

symmetry). This immediately raises many important questions. For examples, does Anderson localization exist in such structure? If so, can it be probed by CBS measurements?

Unfortunately, the interplay between Anderson localization and the periodicity is a difficult issue. In general the common wisdom regarding localization may be drastically modified and very little has been known, among which are: a new scale essential to WL [9] may appear, and whether the constructive interference between reciprocal paths encompasses WL depends on periodic medium configuration [9, 11].

In this letter these two problems will be investigated for a simplified model-thick disordered medium film with a dielectric function periodic in the «-direction (Fig.l). The lattice constant is a and the primitive cell

Fig.l. Light backscattering from a periodic thick disordered medium film. Inset: the film section

consists of randomly positioned point-like scatterers filling the half space z > 0. The film section is uniformly

illuminated by a beam of stationary unpolarized light (with the wavelength A and the frequency 0) perpendicular to it. Analytical results are presented for the angular resolution of light intensity near the inverse incident direction for sufficiently large times.

Qualitatively, as shown in (Fig.2), the traditional line shape [1, 2] develops at \q±\ > 2ir/a and is sharpened

a - 1.1

1.0

0.8 : \\ 1.0 .......

: 0.9

0.6 0.01 0.02

0.4 1

0.2 " 1 1 1 1

Fig.2. The predicted (solid) versus traditional (dotted) line shape (symmetric with respect to q± =0). The parameters are a = 51, £o = 10 Z (setting 1 = 1). Inset: the blunt triangular peak at |gj_ | <C C,^1 = 0.1

then requires that the transverse («-direction) hydrody-namic wave number 2irN/a with N an integer satisfies q_l = 2irN/a — (kp + k'p). Hydrodynamic modes with N ^ 0 (N = 0) describe 2D (quasi-lD) motion.

As kp , k'p € [—7r/a, it/a) there are two contributions which correspond to two successive N responsible for the line shape. For \q±\ > 2ir/a both N ^ 0. Therefore, these two hydrodynamic modes are inhomogeneous in the transverse direction, and extended in the longitudinal (z-) direction because of a -C . As usual the diffusive 2D motion then leads to the traditional line shape.

For \q± | < 2ir/a the line shape is contributed by both 2D diffusive and quasi-lD motion since the two successive integers are now 1 (or — 1) and 0. The former (or latter) occupies a portion of a\q±\/2ir (or 1 — a\q±\/2ir). In the quasi-lD geometry there are two scales: I and An incident flux decays over the scale ~ I then diffuses inside the medium and eventually exit at z ~ I. The penetration length is ~ I-

For \q±\ closed to 2ir/a photons penetrate into the medium of a distance -C £o via quasi-lD diffusive motion. Upon penetration they may self-intersect then propagate around the formed loop along the same direction-so-called diffuson (Fig.3a). The probability

at io^1 \9±\ < 2n/a, eventually a blunt triangular peak emerges at \q±\ -C i^1. Here q± = (2ir/X)s'm0 and £o = itavD with v the photon density of states at 0 and D = 1/3 the bare diffusion constant (I the transport mean free path and the velocity c = 1). Quantitatively, analytical predictions are made for A -C I -C a -C I e'/A, where the last inequality ensures photon states to be far from 2D SL [7]. In particular, the line shape a(9) is singular at \q± \ = 0,2ir/a, around which

r (l + l\q±\)-2=Io(e), a(0)= {1 + 7 (0)}IO(0), (1)

1 (l + l/H0)^a/(2nZ0)l\q±l l^l«^1,

while a smooth interpolation between the last two lines is expected. Here the enhancement factor 7(6) = (3Z/2£o) [l^a\q±\/(27r)}.

Qualitative picture. The well known Bloch theorem allows us to reduce the photon motion into an effective one within a primitive cell dictated by the Bloch wave number kp. The enhanced backscattering finds its origin in the constructive interference-described by low-energy hydrodynamic modes-between two counter-propagating photons (so-called cooperon) each of which carries a Bloch wave number kp , k'p , respectively. kp + k'p plays the role of "Aharonov-Bohm flux". The gauge invariance

(a) ^^

Fig.3. One-loop (a) and typical two-loop (b) interference picture underlying quasi-lD weak localization

is larger in quasi-lD than in 2D. As a result two initially closed but counter-propagating photons have a larger probability to be brought back to their starting point and the backscattered light intensity is thereby enhanced. At z ~ Z the probability of forming such a loop is ~ l/D. Consequently, the inverse diffusivity at z ~ Z increases with an amount of 1/avD ~ Z/£o- Noticing that the backscattered light intensity is proportional to the inverse diffusivity D[1] and taking into account the weight of quasi-lD motion, we find that the traditional line shape is magnified by a factor of 1 + ^(9).

Notice that the quasi-lD cooperon and diffuson have different masses: D(kp + k'p)2 = Dq\ and D(kp - k'p)2, respectively. As \q±\ decreases the diffuson tends to acquire a larger massive and be damped. Consequently, all the constructive interference involving diffuson-cooperon coupling (e.g., Fig.3a) tends to be suppressed. Opposed to this the cooperon becomes less massive. Consequently higher order loop-wise interference paths involving solely cooperon-cooperon coupling (e.g., Fig.3b) accumulate and eventually dominate the backscattered light intensity at \q±\ < , where photons penetrate deeply into the medium forming SL in the bulk.

In this region from one-parameter scaling hypothesis we expect that the diffusion coefficient exponentially decays from the interface, i.e., — InD(z) oc o . Therefore, the average inverse diffusivity of the boundary layer increases also by an amount of ~ Z/£o • Thus for \q±\ -C C^1 the quasi-lD motion contributes to the line shape (1 + Z/£o)[l — a\l-l )/(27r)], together with the portion contributed by the 2D extended motion: Io(0)[a\q±\/(2ir)] « a|<?jJ/(27r), and leads to a blunt triangular peak.

General formalism. We then outline the proof [12]. Let us start from the retarded (advanced) Green's

O2 [1 + c(R)] ±

-[{&„ + ikp)2 + + dl] - O2 c(r) (r = (p,y, z)). The two sets of Green's functions are related through

G*lA(R,R') = E eik^x-^g*iA(r,r'-,kp). (2)

The albedo characterizing the radiation intensity in the direction s = (sin 0,0,— cos 9) [13] generally depends on the time t. With the single scattering event ignored the albedo at t oo, denoted as a(0), can be shown to be

„w = //

= Il cŒMR2e--"[G« (R1,R2)G^2(R2,R1)

eiOs. (R2 - Rl ) (Rl]R2)Gi(RliR2)] (3)

in the unit of A2£Jq/167t with (• • •) the average over the fluctuating dielectric field e(R). The first (second) term gives the background intensity «o (line shape &(#)). Notice that the y-dependence of Green's functions is irrelevant and will be ignored from now on. Substituting Eq. (2) into Eq. (3) gives (up to an irrelevant overall normalization factor)

/>/>00 , a0= dzdz'e^^y2yo(z,z';kp,kp), J Jo kp

m=![

(4)

a[tt) = ij dzdz'e

EE,NycN(z,z'-,kp,k'p), (5)

function: G$a( R,R') = (R|{V2 iO+}_1|R') describing the propagation of the electric field. The fluctuating dielectric field e(R) has zero mean and vanishes for z < 0, while for z > 0 is periodic in x and satisfies O4 (e(R)e(R')) = A 5(y — y') 5{z- z') Eivez s(p ~ p' ~ Na). where (3m G Z) p = x — ma € [0,o) stands for the re

Для дальнейшего прочтения статьи необходимо приобрести полный текст. Статьи высылаются в формате PDF на указанную при оплате почту. Время доставки составляет менее 10 минут. Стоимость одной статьи — 150 рублей.

Показать целиком