Scaling Behavior of the Anderson Localization Transition of Light C.M.Aegerter, M. Störzer, S. Fiebig, W. Bührer and G.Maret Dept. of Physics, University of Konstanz, 78457 Konstanz, Germany
[email protected]
Abstract: Pulsed transmission measurements on colloidal titania reveal a time dependent photon diffusion constant D(t) ~ t-α. The scaling behavior of α and of the localization length with kl* provides evidence for a transition to Anderson localization of light. ©2007 Optical Society of America OCIS codes: (290.1990) Diffusion; (260.3160) Interference; (350.5500) Propagation
1.Introduction The experimental verification of Anderson localization [1] of light has a long history. An exponential decay of the transmission T(L) [2] with sample thickness L [3] and a small photon diffusion constant D [4] have been used as indications for the onset of localization. However, absorption similarly leads to an exponential decrease of the total transmission [5] and a decrease in the diffusion coefficient may be obtained by resonant scattering from particles with size comparable to the wavelength [6]. Therefore, a more clear-cut signature of localization is needed. One way to distinguish localization from absorption is time resolved measurements [7], since the photon diffusion coefficient near the onset of localization is expected [2,8] to be time dependent. This follows a power law, D(t) ~ t−α while absorption only leads to an exponential decay of T(t). In bulk systems the control-parameter of localization is the product of the wavenumber k and mean free path l* as introduced by Ioffe and Regel [9]. Thus, a quantitative test of localization theory requires independent measurements of the absorption length, a time dependent diffusion constant, the transport mean free path l* and the effective transport velocity all as a function of kl*. We have carried out time resolved measurements of transmission at optical wavelengths of slab-shaped samples (typical dimensions are 10 5x10 5x104 in units of l*) at various thicknesses L in the mm range [10]. The values of kl* in our samples varies between 2.5 and 30. Deviations from classical diffusion due to localization are found at very long times, i.e. on very long multiple scattering paths. Furthermore, the transmission profiles T(t) are well described by a phenomenological theory incorporating a temporally varying diffusion coefficient [8], where, at the smallest values of kl*, the diffusion coefficient decreases inversely proportional to time [11]. 2. Experimental Our samples consist of commercial (DuPont) powders of TiO2 with an average grain size ranging from 220-540 nm, a polydispersity of ~ 20% and varying packing fractions up to 45% between two parallel glass plates. At the wavelength of the experiment, 590 nm, TiO2 has a high refractive index of 2.73, such that these samples have very small transport mean free paths l*, typically well below 1µm. In order to experimentally determine the turbidity 1/l* of the samples, they are characterized by coherent backscattering [12], since the angular width of the coherent backscattering cone is given by (kl*)-1 [13]. In our strongly scattering samples the effective refractive index of the medium [14] is high so that internal reflections occur, which lead to a narrowing of the backscattering cone [15]. This was corrected for by evaluating the effective refractive index from the energy coherent potential approximation [16]. The angle resolved intensity was measured using a novel setup consisting of 256 sensitive photodiodes placed around an arc of a diameter of 1.2 m, covering a very wide angular range up to +/-70° [17]. Time resolved transmission measurements were done using 20 ps pulses from a dye (Rhodamin 6G) laser and start-stop electronics measuring the delay time between a sample photon and a delayed reference photon [10]. At long times, T(t) is given by [7] T(t) ~ exp[-(π2D/L2 +1/τabs)t],
(1)
where the characteristic time scale is given by the thickness L, the diffusion coefficient D and the absorption time τabs. The simultaneous measurement of kl* using coherent backscattering and the diffusion coefficient D from time of flight measurements also allows an experimental determination of the transport velocity vT [18].
3. Results
Fig. 1. The inverse localization length as determined from the deviations of time of flight measurements from classical diffusion as a function of the control parameter kl* [11] is shown on the left. At a critical value of kl* ~4, the localization length diverges and thus exceeds the sample thickness resulting in a constant value of L/Lloc. On the right, the time dependence of the diffusion coefficient is described by an exponent α, which is zero for classical diffusion and unity in the case of localization [11]. Just at the transition, a value of α = 1/3 is predicted from one parameter scaling theory [8,20].
In contrast to the theoretical prediction of Eq. 1 for classical diffusion, the time of flight distributions for a samples with kl* < 5 show a slow algebraic decay at long times revealing a time dependent diffusion constant [10]. This deviation scales with kl* [11] indicating a phase transition as demanded by the Ioffe-Regel criterion. In addition, we can determine the absorption length La = (Dτabs)1/2, which does not show any systematic dependence on kl* [19]. From a fit of T(t) with a constant diffusion constant at short times crossing over to a power law decay D(t) ~ t-α at long times we can determine the localization length, Lloc which shows the same qualitative scaling behavior with kl* (Fig. 1) [19]. The resulting exponent α for all samples investigated is also shown in Fig. 1. As can be seen, the behavior changes from classical at high kl* (with α = 0) to localized at kl* below 4 (with α = 1). Right at the transition, the prediction from classical scaling theory [8,20] of α = 1/3 is consistent with the data.
Fig. 2. The static transmission of both a classical (a, kl* = 25) and a localizing (b, kl* = 2.5) sample as a function of thickness. While in both cases the decrease of the intensity is exponential and hence cannot be explained by pure classical diffusion (dotted line), an unknown absorption could still explain the data. For kl* = 25 this is indeed the case, and including the measured absorption length from time resolved measurements yields the dashed line, which is in good agreement with the data. For the localizing sample however, this still contradicts the data. Only a description including the measured localization length as well gives an excellent fit to the data without a single adjustable parameter over twelve orders of magnitude. Note that there are no fitting parameters in any of the curves [19].
Finally, measurements of static transmissions on our most localizing samples at various thicknesses are shown in Fig. 2b. A nearly exponential decrease of the T(L) can be seen. Indeed localization theory predicts an exponential
decrease of the static transmission in the localized state, in contrast to the 1/L dependence of a purely diffusive sample (dotted line in Fig. 2). We observe a significant difference in the characteristic length scales of absorption and localization: When comparing our static transmission measurements with diffusion theory including absorption, where the absorption length has been determined from the time resolved measurements we obtain the dashed lines in Fig. 2. The shaded area indicates the error in the determination of the absorption length from the time resolved measurements. This still greatly overestimates the transmission T(L) found in the localizing samples. Therefore a correct description of the experimental T(L) is only possible when the localization length as determined from the time of flight measurements is included. This yields the full line in Fig.2, which completely describes the data without adjustable parameters over twelve decades in intensity. For a sample with kl* above the localization transition, shown in Fig. 2a, a description using diffusion and absorption works very well, even though such a sample does show an exponential decrease in transmission with thickness [19]. 4. Conclusion We observe deviations from classical diffusive transport in time of flight distributions of multiple scattered light at thick samples and very small values of kl*. These deviations go together with the decrease of the localization length and indicate a transition to localization of waves taking place at kl* ~ 4. The time resolved measurements allow a determination of the absorption length, which combined with the localization length give a quantitative account of the thickness dependence of the static transmission. These observations strongly support the conclusion that we have indeed observed Anderson localization of visible light in three-dimensional strongly disordered samples. 5. References [1] P.W. Anderson, "Absence of diffusion in certain random lattices," Phys. Rev. 109, 1492 (1958). [2] P.W. Anderson, "The question of classical localization: a theory of white paint?" Phil.Mag.B 52, 505 (1985); S. John, "Electromagnetic Absorption in a Disordered Medium near a Photon Mobility Edge," Phys. Rev. Lett. 53, 2169 (1984). [3] D.S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, "Localization of light in a disordered medium," Nature (London) 390, 671 (1997). [4] J.M. Drake and A.Z. Genack, "Observation of nonclassical optical diffusion," Phys. Rev. Lett. 63, 259 (1989). [5] F. Scheffold, R. Lenke, R. Tweer, and G. Maret, "Localization or classical diffusion of light?," Nature (London) 398, 206 (1999); see also D.S. Wiersma, J. Gómez Rivas, P.Bartolini, A. Lagendijk, and R. Righini, reply ibid 398, 207 (1999); A.A. Chabanov, M. Stoytchev, and A.Z. Genack, "Statistical signatures of photon localization," Nature (London) 404, 850-853 (2000). [6] M.P. van Albada, B.A. van Tiggelen, A. Lagendijk, and A. Tip, "Speed of propagation of classical waves in strongly scattering media," Phys. Rev. Lett. 66, 3132 (1991). [7] G.H. Watson, P.A. Fleury, and S.L. McCall, "Searching for photon localization in the time domain," Phys. Rev. Lett. 58, 945 (1987). P.M. Johnson, A. Imhof, B.P.J. Bret, J.G. Rivas, and A. Lagendijk, "Time resolved pulse propagation in a strongly scattering material," Phys. Rev. E 68, 016604 (2003). [8] R. Berkovitz and M. Kaveh, "Propagation of waves through a slab near the Anderson transition: a local scaling approach," J. Phys.:Cond Mat. 2, 307 (1990); R. Berkovitz and M. Kaveh, "Backscattering of light near the optical Anderson transition," Phys. Rev. B 36, 9322 (1987). [9] A.F. Ioffe, and A.R. Regel, "Non-crystalline, amorphous and liquid electronic semiconductors," Progress in Semiconductors 4, 237 (1960). [10] M. Störzer, P. Gross, C.M. Aegerter, and G.Maret, "Observation of the critical regime near Anderson localization of light," Phys. Rev. Lett. 96, 063904 (2006). [11] C.M. Aegerter, M. Störzer, and G.Maret, "Experimental determination of critical exponents in Anderson localization of light." Europhys. Lett. 75, 562 (2006). [12] P.E Wolf and G. Maret, "Weak localization and coherent backscattering of photons in disordered media," Phys. Rev. Lett 55, 2696 (1985); M.P. van Albada and A. Lagendijk, "Observation of weak localization of light in a random medium," Phys. Rev. Lett 55, 2692 (1985). [13] E. Akkermans, P.E. Wolf, and R. Maynard, "Coherent Backscattering of Light by Disordered Media: Analysis of the Peak Line Shape," Phys. Rev. Lett. 56, 1471 (1986). [14] J. C. M. Garnett, "Colours in metal glasses and in metallic films," Phil. Trans. R. Soc. A 203, 385 (1904). [15] J.X. Zhu, D.J. Pine, and D.A. Weitz, "Internal reflection of diffusive light in random media," Phys. Rev. A 44, 3948 (1991). [16] K. Busch and C.M. Soukoulis, "Transport properties of random media: An energy-density CPA approach," Phys. Rev. B 54, 893 (1996). [17] P. Gross, M. Störzer, S. Fiebig, M. Clausen, G. Maret, and C.M. Aegerter, "A precise method to determine the angular distribution of backscattered light to high angles," Rev.Sci. Instrum, in press (2007). [18] M. Störzer, C.M. Aegerter, and G. Maret, "Reduced transport velocity of multiply scattered light due to resonant scattering," Phys. Rev. E 73, 065602(R) (2006). [19] C.M. Aegerter, M. Störzer, S. Fiebig, W. Bührer, and G. Maret, "Observation of Anderson localization of light in three dimensions.", J. Opt. Soc. Am. To be published (2007). [20] E. Abrahams, P.W. Anderson, D. Licciardello, T.V. Ramakrishnan, "Scaling theory of localization: absence of quantum diffusion in two dimensions," Phys. Rev. Lett. 42, 673 (1979).
