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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 10 — May. 20, 2013
  • pp: 11688–11697
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Interplay between localization and absorption in disordered waveguides

Alexey G. Yamilov and Ben Payne  »View Author Affiliations


Optics Express, Vol. 21, Issue 10, pp. 11688-11697 (2013)
http://dx.doi.org/10.1364/OE.21.011688


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Abstract

This work presents results of ab-initio simulations of continuous wave transport in disordered absorbing waveguides. Wave interference effects cause deviations from diffusive picture of wave transport and make the diffusion coefficient position- and absorption-dependent. As a consequence, the true limit of a zero diffusion coefficient is never reached in an absorbing random medium of infinite size, instead, the diffusion coefficient saturates at some finite constant value. Transition to this absorption-limited diffusion exhibits a universality which can be captured within the framework of the self-consistent theory (SCT) of localization. The results of this work (i) justify use of SCT in analyses of experiments in localized regime, provided that absorption is not weak; (ii) open the possibility of diffusive description of wave transport in the saturation regime even when localization effects are strong.

© 2013 OSA

1. Introduction

The diffusive description of electromagnetic wave propagation in inhomogeneous media [1

1. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

, 2

2. M. C. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999) [CrossRef] .

] is overwhelmingly successful with wide ranging applications from astrophysics [3

3. S. Chandresekhar, Radiative Transfer (Dover, New York, 1960).

] to biomedical optics [4

4. L. V. Wang and H. Wu, Biomedical Optics: Principles and Imaging(Wiley-Interscience, 2007).

]. Wave interference effects are usually negligible because the phases of the multiply scattered partial waves are random. However, for a wave that returns to its source location there is always the time-reversed path which necessarily yields the same phase. When the return probability is sufficiently high, the constructive interference between the wave propagated via the two paths can suppress long-range transport so that the diffusion coefficient in an infinitely large (passive) system turns to zero [5

5. D. Vollhardt and P. Wölfle, “Diagrammatic, self-consistent treatment of the anderson localization problem in d≤ 2 dimensions,” Phys. Rev. B 22, 4666–4679 (1980) [CrossRef] .

, 6

6. J. Kroha, C. M. Soukoulis, and P. Wölfle, “Localization of classical waves in a random medium: A self-consistent theory,” Phys. Rev. B 47, 11093–11096 (1993) [CrossRef] .

]. This is Anderson localization – the phenomenon first conceived in context of electronic (de Broglie) wave transport in condensed matter physics [7

7. P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958) [CrossRef] .

].

The question of how the diffusion coefficient evolves with an increase of system size from the unrenomalized value of D0 toward the limit of D = 0 has been addressed in [8

8. B. A. van Tiggelen, A. Lagendijk, and D. S. Wiersma, “Reflection and transmission of waves near the localization threshold,” Phys. Rev. Lett. 84, 4333–4336 (2000) [CrossRef] [PubMed] .

, 9

9. N. Cherroret and S. E. Skipetrov, “Microscopic derivation of self-consistent equations of anderson localization in a disordered medium of finite size,” Phys. Rev. E 77, 046608 (2008) [CrossRef] .

] by extending the original self-consistent theory of Vollhardt and Wölfle [5

5. D. Vollhardt and P. Wölfle, “Diagrammatic, self-consistent treatment of the anderson localization problem in d≤ 2 dimensions,” Phys. Rev. B 22, 4666–4679 (1980) [CrossRef] .

, 6

6. J. Kroha, C. M. Soukoulis, and P. Wölfle, “Localization of classical waves in a random medium: A self-consistent theory,” Phys. Rev. B 47, 11093–11096 (1993) [CrossRef] .

] to systems of finite size. The key prediction of the modified self-consistent theory (SCT) is that the diffusion coefficient is no longer a constant but varies spatially. This conclusion is also reached independently in the supersymmetric field theory [10

10. C. Tian, “Supersymmetric field theory of local light diffusion in semi-infinite media,” Phys. Rev. B 77, 064205 (2008) [CrossRef] .

]. The position dependence of the diffusion coefficient arises because the return probability responsible for the renormalization of diffusion is position-dependent in a system of finite size. As the wave explores the larger and larger neighborhood of a source point, the return probability becomes sensitive to the proximity of a boundary where the wave has a chance to escape from the system. In fact, in [11

11. C. Tian, “Hydrodynamic and field-theoretic approaches to light localization in open media,” Physica E 49, 124–153 (2013) [CrossRef] .

] it is pointed out that waves propagation in open media can be thought as a highly unconventional macroscopic diffusive phenomenon.

Although SCT has been employed in interpreting the results of several experiments [12

12. M. Störzer, P. Gross, C. Aegerter, and G. Maret, “Observation of the critical regime near anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006) [CrossRef] [PubMed] .

15

15. T. Sperling, W. Bührer, C. M. Aegerter, and G. Maret, “Direct determination of the transition to localization of light in three dimensions,” Nat. Phot. 7, 48–52 (2013) [CrossRef] .

], the key prediction of position-dependent diffusion has not been verified in an experiment because it requires the access to the inside of the random medium. Position-dependent diffusion has been also predicted by the super-symmetry approach [10

10. C. Tian, “Supersymmetric field theory of local light diffusion in semi-infinite media,” Phys. Rev. B 77, 064205 (2008) [CrossRef] .

], suggesting that this is more than just a convenient mathematical abstraction. Indeed, quantitative agreement between SCT and numerical simulation has been found in disordered waveguides at the onset of localization [16

16. B. Payne, A. Yamilov, and S. E. Skipetrov, “Anderson localization as position-dependent diffusion in disordered waveguides,” Phys. Rev. B 82, 024205 (2010) [CrossRef] .

, 17

17. B. Payne, T. Mahler, and A. Yamilov, “Effect of evanescent channels on position-dependent diffusion in disordered waveguides,” Waves in Random and Complex Media 23, 43–55 (2013) [CrossRef] .

]. Numerical simulations in [18

18. C. Tian, S. Cheung, and Z. Zhang, “Local diffusion theory for localized waves in open media,” Phys. Rev. Lett. 105, 263905 (2010) [CrossRef] .

], show that deeper in the localization regime, the prediction by SCT for the position-dependent diffusion coefficient is quantitatively incorrect. Instead, by using the supersymmetric field theory it was found that the diffusion coefficient exhibits novel scaling [18

18. C. Tian, S. Cheung, and Z. Zhang, “Local diffusion theory for localized waves in open media,” Phys. Rev. Lett. 105, 263905 (2010) [CrossRef] .

]. This is because SCT underestimates the energy density inside the bulk of medium that is strongly affected by the presence of necklace states [19

19. J. B. Pendry, “Symmetry and transport of waves in one-dimensional disordered systems,” Adv. Phys. 43, 461–542 (1994) [CrossRef] .

] formed via resonant tunneling.

Material absorption and other sources of dissipation are usually a nuisance in experiments with classical waves because it leads to the same exponential scaling of e.g. optical conductance g ∝ exp(−L/ξa0) as in a localized system g ∝ exp(−L/ξ) [20

20. A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature 404, 850–853 (2000) [CrossRef] [PubMed] .

]. Here ξa0=D0τa is the diffusive absorption length, ξ is the localization length, τa is the ballistic absorption time, and g is the total transmission through random medium under diffuse illumination [20

20. A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature 404, 850–853 (2000) [CrossRef] [PubMed] .

]. When the absorption rate 1/τa exceeds the radiative decay rate through the boundaries of the random medium, the resonant tunneling through the modes of the random medium [21

21. J. Wang and A. Z. Genack, “Transport through modes in random media,” Nature 471, 345–348 (2011) [CrossRef] [PubMed] .

] is suppressed. Because the radiative decay rate diminishes exponentially with an increase of the system size L, the amount of absorption needed to suppress the resonant tunneling becomes small.

In this work, we show that the applicability of SCT in the localized regime is restored when sufficiently strong (ξa0L) absorption is added. We also study the size scaling of the position-dependent diffusion coefficient and show that in presence of absorption it saturates at a non-zero value. This suggests that that the wave transport in absorbing random media can be viewed as diffusion, albeit with a renormalized diffusion coefficient, even when localization corrections are present and strong.

2. Model description

2.1. Motivation for the choice of the model

The goal of this work is to study the effect of an interplay between absorption and localization on the position-dependent diffusion coefficient. To achieve this goal we selected the model geometry based on the following considerations. Because the diffusion coefficient is modified due to localization corrections, the system must be in the localization regime in the limit of infinitely large system size. Scaling theory of localization predicts [22

22. E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: Absence of quantum diffusion in two dimensions,” Phys. Rev. Lett. 42, 673–676 (1979) [CrossRef] .

] that this condition is met in systems with physical dimension less than or equal to two. In three dimensional random media the disorder strength has to be sufficiently high to reach localization [15

15. T. Sperling, W. Bührer, C. M. Aegerter, and G. Maret, “Direct determination of the transition to localization of light in three dimensions,” Nat. Phot. 7, 48–52 (2013) [CrossRef] .

]. From a computational standpoint, it is desirable to consider a system with the least dimension. Although a one-dimensional (1D) random medium, e.g. a random stack of dielectric slabs [23

23. L. I. Deych, A. Yamilov, and A. A. Lisyansky, “Scaling in one-dimensional localized absorbing systems,” Phys. Rev. B 64, 024201 (2001) [CrossRef] .

] or a single mode random waveguide [24

24. L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with anderson-localized modes,” Science 327, 1352–1355 (2010) [CrossRef] [PubMed] .

], exhibits the phenomenon of localization it is not suitable for our study. This is because wave transport in 1D shows either ballistic or localized behavior; diffusion is not applicable in any parameter range. Instead, we perform numerical simulations for 2D disordered multimode waveguides. Because the volume of the system increases as the first power of the waveguide length L (for fixed width W), this geometry is often referred to as a quasi-1D [25

25. C. W. Beenakker, “Random-matrix theory of quantum transport,” Rev. Mod. Phys. 69, 731–808 (1997) [CrossRef] .

, 26

26. P. W. Brouwer, “Transmission through a many-channel random waveguide with absorption,” Phys. Rev. B 57, 10526–10536 (1998) [CrossRef] .

]. However, unlike 1D, random waveguides exhibit both a diffusive behavior for < L < ξ and localization for L > ξ. denotes the transport mean free path. Furthermore, as we argue in Sec. 5, the 2D random waveguides can be fabricated experimentally [27

27. A. Yamilov, R. Sarma, B. Redding, B. Payne, H. Noh, and H. Cao, “Position-dependent diffusion of light in disordered waveguides,” arXiv:1303.3244 (2013).

] to test the predictions of this work.

2.2. Position-dependent diffusion coefficient

To investigate the applicability of a diffusive description of wave transport, we perform the ab-initio numerical simulations where all wave interference effects are accounted for without any approximations. We consider a scalar, monochromatic wave E(r)eiωt propagating in a 2D volume-disordered waveguide of width W and length LW that supports N propagating modes. The wave field E(r) obeys the 2D Helmholtz equation:
{2+k2[1+δε(r)]}E(r)=0.
(1)
Here k = ω/c is the wavenumber, δε(r) = (1 + )δεr(r), where δεr(r) is the randomly fluctuating part of the dielectric constant, and α > 0 is the strength of absorption. The system is excited from the left by illuminating the waveguide with N unit fluxes and the wave field E(r) throughout the volume of the random medium is computed with a transfer matrix method for a given realization of disorder [17

17. B. Payne, T. Mahler, and A. Yamilov, “Effect of evanescent channels on position-dependent diffusion in disordered waveguides,” Waves in Random and Complex Media 23, 43–55 (2013) [CrossRef] .

]. From E(r) we calculate the cross section-averaged energy density 𝒲(z) and the longitudinal component of flux Jz(z). Here z is the axis along the length of the waveguide. Local values of the energy density and flux formally define via Fick’s law the diffusion coefficient D(z) which, in general, may be position-dependent:
D(z)=Jz(z)/[d𝒲(z)/dz].
(2)
The averages 〈...〉 are taken over a statistical ensemble of 106 disorder realizations simulated on a supercomputer.

2.3. D0 calculation

In order to compare our numerical results for D(z) with SCT without fitting parameters, we need to obtain the value of the diffusion coefficient unrenormalized by the wave interference effects D0 = vℓ/2. Here v is the diffusive speed and is the transport mean free path.

To find the diffusive speed v we use the definition of diffusive flux resolved with respect to the direction of propagation [1

1. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

]
Jz(±)(z)=(v/π)𝒲(z)(D(z)/2)d𝒲(z)/dz,
(3)
where Jz(±)(z) represent the forward (plus) and backward (minus) propagating components of the flux. Combining the two components we find
v=π2Jz(+)(z)+Jz()(z)𝒲(z).
(4)

2.4. τa calculation

The characteristic absorption time τa can be determined numerically using the condition of flux continuity dJz(z)〉/dz = 〈 𝒲(z)〉/τa. The desired diffusive absorption length ξa0=D0τa can be obtained by the proper choice of α in Eq. (1).

3. Self-consistent theory

Self-consistent theory is developed starting with the Green’s function G(r, r′) of Eq. (1) with δε(r) = δεr(r)+ . The disorder-averaged function Ĉ(r, r′) = (4πWD0/cL)〈| G(r, r′)|2〉 obeys self-consistent equations in a dimensionless form [9

9. N. Cherroret and S. E. Skipetrov, “Microscopic derivation of self-consistent equations of anderson localization in a disordered medium of finite size,” Phys. Rev. E 77, 046608 (2008) [CrossRef] .

, 16

16. B. Payne, A. Yamilov, and S. E. Skipetrov, “Anderson localization as position-dependent diffusion in disordered waveguides,” Phys. Rev. B 82, 024205 (2010) [CrossRef] .

]:
[(Lξa0)2ζd(ζ)ζ]C^(ζ,ζ)=δ(ζζ),
(5)
1d(ζ)=1+2LξC^(ζ,ζ),
(6)
where d(ζ) = D(ζ)/D0 and all position-dependent quantities are functions of the longitudinal coordinate ζ = z/L in the considered waveguide geometry. The quantity Ĉ(ζ, ζ) which renormalizes the diffusion coefficient is proportional to the return probability at ζ. Hence, Eq. (6) formally accounts for wave interference of the time-reversed paths. Equations (5,6) are solved with the following boundary conditions:
C^(ζ,ζ)z0Ld(ζ)ζC^(ζ,ζ)=0
(7)
at ζ = 0 and ζ = 1. We obtain the numerical solution of Eqs. (57) via an iterative procedure with an initial guess d(ζ) ≡ 1. The convergence is usually achieved within five iteration steps.

4. Analysis of numerical results

4.1. Interplay between localization and absorption

First, we consider the effect of localization on the local diffusion coefficient as defined by Eq. (2). Figure 1(a) compares the ab-initio D(z) (blue curves) and the prediction of SCT from Eqs. (57) (red curves) in passive systems (no absorption). The behavior of the systems is determined by the ratio of system length L to localization length ξ. The five samples shown in Fig. 1(a) correspond to L/ξ = 0.4, 0.7, 1.5, 3.0 and 4.4 covering both L < ξ and L > ξ regimes. We observe that without absorption SCT agrees well with the ab-initio simulations.

Fig. 1 Comparison between position-dependent diffusion coefficient d(z) = D(z)/D0 found from ab-initio numerical simulations (c.f. Eq. (2), blue curves) and self-consistent theory (Eqs. (57), red curves). (a) In passive systems (L/ξ = 0.4, 0.7, 1.5, 3.0 and 4.4) the diffusion coefficient diminishes in the interior of the system due to the enhanced return probability and, hence, stronger localization correction caused by wave interference. (b) For fixed length (L/ξ = 4.4), an increase of absorption suppresses the localization corrections to the position-dependent diffusion coefficient. The five curves correspond to L/ξa0 = 0.0, 3.3, 5.7, 9.8 and 21. (c) When absorption (ξ/ξa0 = 1.3) is added to the five samples shown in (a), the position-dependent diffusion coefficient no longer decreases below its saturated plateau value Dp, see Eq. (8). In all cases SCT agrees well with the ab-initio simulation of wave transport in disordered waveguides. The cause of the small deviations are discussed in the text.

The agreement is achieved for the same values of parameters and z0 found via the procedure described in Sec. 2. We would like to comment on two kinds of systematic discrepancies between the numerical data and SCT. (i) We attribute the discrepancy in the boundary regions 0 < z and Lz < L to the failure of the diffusion approximation at the boundaries. This is a well known effect which requires a more sophisticated description, such as that by the Milne equation [29

29. E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University Press, Cambridge, UK, 2007) [CrossRef] .

], even without localization corrections. (ii) The second discrepancy arises at zL/2 deep in the localization regime Lξ. In the most localized system (L/ξ = 4.4) accessible to us computationally, we find that SCT systematically underestimates the value of the position-dependent diffusion coefficient [16

16. B. Payne, A. Yamilov, and S. E. Skipetrov, “Anderson localization as position-dependent diffusion in disordered waveguides,” Phys. Rev. B 82, 024205 (2010) [CrossRef] .

]. An explanation was provided in the work of Tian et al [18

18. C. Tian, S. Cheung, and Z. Zhang, “Local diffusion theory for localized waves in open media,” Phys. Rev. Lett. 105, 263905 (2010) [CrossRef] .

] who pointed out the inability of SCT to describe the effects of resonant tunneling which becomes the dominant mechanism of wave transport in a localized system. Based on our simulation, we find significant deviations in systems where D(z)/D0 falls below the value of ≃ 0.2, which also agree with the numerical simulations reported in [18

18. C. Tian, S. Cheung, and Z. Zhang, “Local diffusion theory for localized waves in open media,” Phys. Rev. Lett. 105, 263905 (2010) [CrossRef] .

].

Absorption has long been considered a nuisance as it tends to both mask and suppress the effects of wave localization [20

20. A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature 404, 850–853 (2000) [CrossRef] [PubMed] .

, 26

26. P. W. Brouwer, “Transmission through a many-channel random waveguide with absorption,” Phys. Rev. B 57, 10526–10536 (1998) [CrossRef] .

, 30

30. S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett. 53, 2169–2172 (1984) [CrossRef] .

, 31

31. A. Lagendijk, B. van Tiggelen, and D. S. Wiersma, “Fifty years of anderson localization,” Phys. Today 62, 24–29 (2009) [CrossRef] .

]. Indeed, absorption suppresses the contribution of long propagation paths to the return probability which are responsible for localization effects. In Fig. 1(b) we observe an increase of D(z)/D0 → 1 with a decrease of ξa0 (stronger absorption) while the sample size L is fixed. The suppression occurs first in the middle of the sample (farthermost from the boundaries) where the longest paths are the most probable. We note the development of a plateau D(z) ≃ Dp in the spatial region ξa0zLξa0. This can be understood based on the following argument. Due to the presence of absorption, a wave that originates at position z in the plateau region can explore only the interval from zξa0 to z +ξa0 which does not include a boundary. Hence, as far as the return probability in Eq. (6) is concerned, the effective system size [26

26. P. W. Brouwer, “Transmission through a many-channel random waveguide with absorption,” Phys. Rev. B 57, 10526–10536 (1998) [CrossRef] .

] Leff~2×ξa=2×Dpτa. This argument can be made more quantitative by neglecting the ∂d(ζ)/∂ζ term in Eq. (5). We solve the self-consistent Eqs. (57) under the above assumption and obtain the following self-contained equation on dpDp/D0:
dp1=1+(ξa0/ξ)dp1/2,
(8)
which, in the limit of strong absorptions ξa0/ξ → 0, yields an even simpler result dp ≃ [1 +ξa0/ξ]−1. In this expression the parameter geffξ/ξa0 controls the extent of renormalization of the diffusion coefficient Dp, similar to the role played by the conductance g0ξ/L in passive systems [16

16. B. Payne, A. Yamilov, and S. E. Skipetrov, “Anderson localization as position-dependent diffusion in disordered waveguides,” Phys. Rev. B 82, 024205 (2010) [CrossRef] .

].

A careful comparison between between the numerical results and SCT in Fig. 1(b) shows a new kind of discrepancy which arises in the localized systems with weak absorption. We observe that unlike SCT, the ab-initio D(z) becomes asymmetric with respect to the middle of the sample D(z) ≠ D(Lz). We hypothesize that this occurs because the absorption tends to affect the energy density inside a localized system differently depending on the position of the localization center in a transmission measurement [32

32. B. Payne, J. Andreasen, H. Cao, and A. Yamilov, “Relation between transmission and energy stored in random media with gain,” Phys. Rev. B 82, 104204 (2010) [CrossRef] .

]. The detailed study of these weakly absorbing systems goes beyond the scope of this work and will be reported elsewhere.

4.2. Minimum diffusion coefficient in absorbing systems

As seen in Fig. 1(a), in the passive systems the value of the diffusion coefficient in the middle of the sample D(z = L/2) approaches zero with an increase of L. As the boundaries are further removed from z = L/2, it becomes progressively more difficult for a wave to escape and the return probability keeps increasing. As discussed above, the same argument no longer applies in an absorbing random medium. In Fig. 1(c) we plot the position-dependent diffusion coefficient in absorbing (ξ/ξa0 = 1.3) systems of different size. Unlike passive systems in Fig. 1(a), the diffusion coefficient stops decreasing after the system size L exceeds the length ∼ 2ξa0 and develops a plateau where it is no longer sensitive to the boundaries of the system.

Figure 2 shows saturation of the diffusion coefficient in the middle of the sample D(z = L/2) with an increase of the sample size L. Importantly, the saturated value Dp is determined by the ratio ξ/ξa0, see Eq. (8). As we discussed above, SCT underestimates the value of the diffusion coefficient when the latter becomes small. However, the phenomenon of saturation is observed in both ab-initio simulations and SCT.

Fig. 2 Existence of the minimum diffusion coefficient is seen from the evolution of d1/2D(z = L/2)/D0 with the increase of the system size. In the passive systems (ξ/ξa0 = 0, open circles) the limit is expected to be zero. In absorbing systems (ξ/ξa0 = 0.3, 0.7, 1.3, 2.9 shown as cross, diamond, upward and downward triangle symbols respectively) saturation corresponds to formation of the plateau region seen in Figs. 1(b,c). The saturation value Dp increases monotonically with an increase of ξ/ξa0. Five solid lines are obtained from the self-consistent theory Eqs. (57) for each value of the absorption strength. Qualitative prediction of the minimum value of position-dependent diffusion coefficient in SCT is supported by the numerical simulations. The agreement is also quantitative for Dp/D0 ≳ 0.2.

4.3. Universal scaling of position-dependent diffusion

Single parameter scaling [22

22. E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: Absence of quantum diffusion in two dimensions,” Phys. Rev. Lett. 42, 673–676 (1979) [CrossRef] .

] is a key concept in mesoscopic physics. It predicts that when the size of the system increases, the evolution of the average conductance, its variance [33

33. P. A. Lee, D. A. Stone, and H. Fukuyamak, “Universal conductance fluctuations in metals: Effects of finite temperature, interactions, and magnetic field,” Phys. Rev. B 35, 1039–1070 (1987) [CrossRef] .

] as well as its entire distribution [34

34. P. W. Anderson, D. J. Thouless, E. Abrahams, and D. S. Fisher, “New method for a scaling theory of localization,” Phys. Rev. B 22, 3519–3526 (1980) [CrossRef] .

] are determined by the average conductance. In other words, the mesoscopic electronic transport through a disordered system is universal [35

35. B. L. Altshuler, P. A. Lee, and R. A. Webb, eds., Mesoscopic Phenomena in Solids (North Holland, Amsterdam, 1991).

] – independent of the microscopic parameters of disorder. The concept of universality can be directly applied also to the electromagnetic wave transport [2

2. M. C. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999) [CrossRef] .

]. Recently, it was shown that in the absence of absorption the concept of position-dependent diffusion coefficient is also universal [16

16. B. Payne, A. Yamilov, and S. E. Skipetrov, “Anderson localization as position-dependent diffusion in disordered waveguides,” Phys. Rev. B 82, 024205 (2010) [CrossRef] .

18

18. C. Tian, S. Cheung, and Z. Zhang, “Local diffusion theory for localized waves in open media,” Phys. Rev. Lett. 105, 263905 (2010) [CrossRef] .

] – the shape of D(z) is uniquely determined by the value of conductance.

Equation (8) describing the plateau value of the diffusion coefficient in absorbing systems depends only on one parameter ξ/ξa0. This suggests a new universal behavior. In Fig. 3 we compare this prediction of SCT (solid line) to our numerical results in different disordered absorbing waveguides (symbols) and observe a good agreement.

Fig. 3 Universal scaling of the local diffusion coefficient in the middle of the sample d1/2D(z = L/2)/D0 is described by a single parameter. The ab-initio numerical simulations in disordered waveguides with absorption (symbols) agree well with Eq. (8), which we derived based on SCT.

5. Conclusions

In this work we compared the results of the ab-initio numerical simulation of wave transport through disordered absorbing waveguides to the prediction of self-consistent theory with position-dependent diffusion. This theory has the power to provide a simple intuitive description of an intricate interplay between localization effects due to wave interference and absorption. The following results were obtained. (i) We demonstrated that in localized regime (L > ξ) for weak/strong absorption SCT provides the correct qualitative/quantitative description of the renormalizations in position-dependent diffusion coefficient. (ii) We pointed out the key distinction in scaling of the position-dependent diffusion in passive and absorbing media. In the former, the diffusion coefficient vanishes in the limit of infinite size. In contrast, in absorbing random media, the local diffusion coefficient always remains finite so that true localization in the sense of vanishing diffusion is never reached. This conclusion appear to be borne out also by supersymmetric field theory [38

38. L. Y. Zhao, C. Tian, Z. Q. Zhang, and X. D. Zhang, “Unusual Brownian motion of photons in open absorbing media,” arXiv:1304.0516 (2013).

].

We would like to point out that a similar (but not equivalent) effect occurs in the mesoscopic electronic systems due to the effects of dephasing [29

29. E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University Press, Cambridge, UK, 2007) [CrossRef] .

] which introduces the cutoff length scale beyond which waves no longer contribute coherently to the return probability. Instead, these inelastically scattered (de Broglie) waves of the electrons contribute to the incoherent background which only obscures the localization phenomena. In contrast, optical absorption removes photons while the remaining component retains all information about interferences. This observation points to the possibility of observation of the phenomenon of position-dependent diffusion in optical system that we discuss next.

There are three main conditions that would allow the direct experimental observation position-dependent diffusion. First, without absorption the system must be in the localization regime in the limit of infinitely large size. Second, the absorption, or more generally loss, cannot be too strong so that the localization effects are not completely washed out. Third, because the renormalization of diffusion occurs away from the boundaries, the interior of the medium has to be experimentally (non-invasively) monitored. These conditions can be met in the two dimensional disordered optical waveguides where the position-dependent diffusion is indeed observed [27

27. A. Yamilov, R. Sarma, B. Redding, B. Payne, H. Noh, and H. Cao, “Position-dependent diffusion of light in disordered waveguides,” arXiv:1303.3244 (2013).

].

Acknowledgments

We gratefully acknowledge the discussions with H. Cao, R. Sarma, A. D. Stone and B. Shapiro. We also acknowledge comments by S. Skipetrov, P. Sebbah during the early stages of this work. We are thankful to C. Tian for reading and commenting on the manuscript. Financial support was provided by National Science Foundation under grants Nos. DMR-0704981 and DMR-1205223. Computational resources were provided under the Extreme Science and Engineering Discovery Environment (XSEDE) grant No. DMR-100030.

References and links

1.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

2.

M. C. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999) [CrossRef] .

3.

S. Chandresekhar, Radiative Transfer (Dover, New York, 1960).

4.

L. V. Wang and H. Wu, Biomedical Optics: Principles and Imaging(Wiley-Interscience, 2007).

5.

D. Vollhardt and P. Wölfle, “Diagrammatic, self-consistent treatment of the anderson localization problem in d≤ 2 dimensions,” Phys. Rev. B 22, 4666–4679 (1980) [CrossRef] .

6.

J. Kroha, C. M. Soukoulis, and P. Wölfle, “Localization of classical waves in a random medium: A self-consistent theory,” Phys. Rev. B 47, 11093–11096 (1993) [CrossRef] .

7.

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958) [CrossRef] .

8.

B. A. van Tiggelen, A. Lagendijk, and D. S. Wiersma, “Reflection and transmission of waves near the localization threshold,” Phys. Rev. Lett. 84, 4333–4336 (2000) [CrossRef] [PubMed] .

9.

N. Cherroret and S. E. Skipetrov, “Microscopic derivation of self-consistent equations of anderson localization in a disordered medium of finite size,” Phys. Rev. E 77, 046608 (2008) [CrossRef] .

10.

C. Tian, “Supersymmetric field theory of local light diffusion in semi-infinite media,” Phys. Rev. B 77, 064205 (2008) [CrossRef] .

11.

C. Tian, “Hydrodynamic and field-theoretic approaches to light localization in open media,” Physica E 49, 124–153 (2013) [CrossRef] .

12.

M. Störzer, P. Gross, C. Aegerter, and G. Maret, “Observation of the critical regime near anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006) [CrossRef] [PubMed] .

13.

H. Hu, A. Strybulevych, J. H. Page, S. E. Skipetrov, and B. A. van Tiggelen, “Localization of ultrasound in a three-dimensional elastic network,” Nat. Phys. 4, 945–948 (2008) [CrossRef] .

14.

Z. Q. Zhang, A. A. Chabanov, S. K. Cheung, C. H. Wong, and A. Z. Genack, “Dynamics of localized waves: Pulsed microwave transmissions in quasi-one-dimensional media,” Phys. Rev. B 79, 144203 (2009) [CrossRef] .

15.

T. Sperling, W. Bührer, C. M. Aegerter, and G. Maret, “Direct determination of the transition to localization of light in three dimensions,” Nat. Phot. 7, 48–52 (2013) [CrossRef] .

16.

B. Payne, A. Yamilov, and S. E. Skipetrov, “Anderson localization as position-dependent diffusion in disordered waveguides,” Phys. Rev. B 82, 024205 (2010) [CrossRef] .

17.

B. Payne, T. Mahler, and A. Yamilov, “Effect of evanescent channels on position-dependent diffusion in disordered waveguides,” Waves in Random and Complex Media 23, 43–55 (2013) [CrossRef] .

18.

C. Tian, S. Cheung, and Z. Zhang, “Local diffusion theory for localized waves in open media,” Phys. Rev. Lett. 105, 263905 (2010) [CrossRef] .

19.

J. B. Pendry, “Symmetry and transport of waves in one-dimensional disordered systems,” Adv. Phys. 43, 461–542 (1994) [CrossRef] .

20.

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature 404, 850–853 (2000) [CrossRef] [PubMed] .

21.

J. Wang and A. Z. Genack, “Transport through modes in random media,” Nature 471, 345–348 (2011) [CrossRef] [PubMed] .

22.

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: Absence of quantum diffusion in two dimensions,” Phys. Rev. Lett. 42, 673–676 (1979) [CrossRef] .

23.

L. I. Deych, A. Yamilov, and A. A. Lisyansky, “Scaling in one-dimensional localized absorbing systems,” Phys. Rev. B 64, 024201 (2001) [CrossRef] .

24.

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with anderson-localized modes,” Science 327, 1352–1355 (2010) [CrossRef] [PubMed] .

25.

C. W. Beenakker, “Random-matrix theory of quantum transport,” Rev. Mod. Phys. 69, 731–808 (1997) [CrossRef] .

26.

P. W. Brouwer, “Transmission through a many-channel random waveguide with absorption,” Phys. Rev. B 57, 10526–10536 (1998) [CrossRef] .

27.

A. Yamilov, R. Sarma, B. Redding, B. Payne, H. Noh, and H. Cao, “Position-dependent diffusion of light in disordered waveguides,” arXiv:1303.3244 (2013).

28.

A. Mirlin, “Statistics of energy levels and eigen-functions in disordered systems,” Phys. Rep. 326, 259–382 (2000) [CrossRef] .

29.

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University Press, Cambridge, UK, 2007) [CrossRef] .

30.

S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett. 53, 2169–2172 (1984) [CrossRef] .

31.

A. Lagendijk, B. van Tiggelen, and D. S. Wiersma, “Fifty years of anderson localization,” Phys. Today 62, 24–29 (2009) [CrossRef] .

32.

B. Payne, J. Andreasen, H. Cao, and A. Yamilov, “Relation between transmission and energy stored in random media with gain,” Phys. Rev. B 82, 104204 (2010) [CrossRef] .

33.

P. A. Lee, D. A. Stone, and H. Fukuyamak, “Universal conductance fluctuations in metals: Effects of finite temperature, interactions, and magnetic field,” Phys. Rev. B 35, 1039–1070 (1987) [CrossRef] .

34.

P. W. Anderson, D. J. Thouless, E. Abrahams, and D. S. Fisher, “New method for a scaling theory of localization,” Phys. Rev. B 22, 3519–3526 (1980) [CrossRef] .

35.

B. L. Altshuler, P. A. Lee, and R. A. Webb, eds., Mesoscopic Phenomena in Solids (North Holland, Amsterdam, 1991).

36.

V. D. Freilikher, M. Pustilnik, and I. Yurkevich, “Effect of absorption on the wave transport in the strong localization regime,” Phys. Rev. Lett. 73, 810–813 (1994) [CrossRef] [PubMed] .

37.

A. Yamilov and B. Payne, “Classification of regimes of wave transport in quasi-one-dimensional nonconservative random media,” J. Mod. Opt. 57, 1916–1921 (2010) [CrossRef] .

38.

L. Y. Zhao, C. Tian, Z. Q. Zhang, and X. D. Zhang, “Unusual Brownian motion of photons in open absorbing media,” arXiv:1304.0516 (2013).

OCIS Codes
(030.1670) Coherence and statistical optics : Coherent optical effects
(290.4210) Scattering : Multiple scattering
(160.2710) Materials : Inhomogeneous optical media

ToC Category:
Scattering

History
Original Manuscript: March 12, 2013
Manuscript Accepted: April 26, 2013
Published: May 6, 2013

Citation
Alexey G. Yamilov and Ben Payne, "Interplay between localization and absorption in disordered waveguides," Opt. Express 21, 11688-11697 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-10-11688


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References

  1. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
  2. M. C. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys.71, 313–371 (1999). [CrossRef]
  3. S. Chandresekhar, Radiative Transfer (Dover, New York, 1960).
  4. L. V. Wang and H. Wu, Biomedical Optics: Principles and Imaging(Wiley-Interscience, 2007).
  5. D. Vollhardt and P. Wölfle, “Diagrammatic, self-consistent treatment of the anderson localization problem in d≤ 2 dimensions,” Phys. Rev. B22, 4666–4679 (1980). [CrossRef]
  6. J. Kroha, C. M. Soukoulis, and P. Wölfle, “Localization of classical waves in a random medium: A self-consistent theory,” Phys. Rev. B47, 11093–11096 (1993). [CrossRef]
  7. P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev.109, 1492–1505 (1958). [CrossRef]
  8. B. A. van Tiggelen, A. Lagendijk, and D. S. Wiersma, “Reflection and transmission of waves near the localization threshold,” Phys. Rev. Lett.84, 4333–4336 (2000). [CrossRef] [PubMed]
  9. N. Cherroret and S. E. Skipetrov, “Microscopic derivation of self-consistent equations of anderson localization in a disordered medium of finite size,” Phys. Rev. E77, 046608 (2008). [CrossRef]
  10. C. Tian, “Supersymmetric field theory of local light diffusion in semi-infinite media,” Phys. Rev. B77, 064205 (2008). [CrossRef]
  11. C. Tian, “Hydrodynamic and field-theoretic approaches to light localization in open media,” Physica E49, 124–153 (2013). [CrossRef]
  12. M. Störzer, P. Gross, C. Aegerter, and G. Maret, “Observation of the critical regime near anderson localization of light,” Phys. Rev. Lett.96, 063904 (2006). [CrossRef] [PubMed]
  13. H. Hu, A. Strybulevych, J. H. Page, S. E. Skipetrov, and B. A. van Tiggelen, “Localization of ultrasound in a three-dimensional elastic network,” Nat. Phys.4, 945–948 (2008). [CrossRef]
  14. Z. Q. Zhang, A. A. Chabanov, S. K. Cheung, C. H. Wong, and A. Z. Genack, “Dynamics of localized waves: Pulsed microwave transmissions in quasi-one-dimensional media,” Phys. Rev. B79, 144203 (2009). [CrossRef]
  15. T. Sperling, W. Bührer, C. M. Aegerter, and G. Maret, “Direct determination of the transition to localization of light in three dimensions,” Nat. Phot.7, 48–52 (2013). [CrossRef]
  16. B. Payne, A. Yamilov, and S. E. Skipetrov, “Anderson localization as position-dependent diffusion in disordered waveguides,” Phys. Rev. B82, 024205 (2010). [CrossRef]
  17. B. Payne, T. Mahler, and A. Yamilov, “Effect of evanescent channels on position-dependent diffusion in disordered waveguides,” Waves in Random and Complex Media23, 43–55 (2013). [CrossRef]
  18. C. Tian, S. Cheung, and Z. Zhang, “Local diffusion theory for localized waves in open media,” Phys. Rev. Lett.105, 263905 (2010). [CrossRef]
  19. J. B. Pendry, “Symmetry and transport of waves in one-dimensional disordered systems,” Adv. Phys.43, 461–542 (1994). [CrossRef]
  20. A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature404, 850–853 (2000). [CrossRef] [PubMed]
  21. J. Wang and A. Z. Genack, “Transport through modes in random media,” Nature471, 345–348 (2011). [CrossRef] [PubMed]
  22. E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: Absence of quantum diffusion in two dimensions,” Phys. Rev. Lett.42, 673–676 (1979). [CrossRef]
  23. L. I. Deych, A. Yamilov, and A. A. Lisyansky, “Scaling in one-dimensional localized absorbing systems,” Phys. Rev. B64, 024201 (2001). [CrossRef]
  24. L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with anderson-localized modes,” Science327, 1352–1355 (2010). [CrossRef] [PubMed]
  25. C. W. Beenakker, “Random-matrix theory of quantum transport,” Rev. Mod. Phys.69, 731–808 (1997). [CrossRef]
  26. P. W. Brouwer, “Transmission through a many-channel random waveguide with absorption,” Phys. Rev. B57, 10526–10536 (1998). [CrossRef]
  27. A. Yamilov, R. Sarma, B. Redding, B. Payne, H. Noh, and H. Cao, “Position-dependent diffusion of light in disordered waveguides,” arXiv:1303.3244 (2013).
  28. A. Mirlin, “Statistics of energy levels and eigen-functions in disordered systems,” Phys. Rep.326, 259–382 (2000). [CrossRef]
  29. E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University Press, Cambridge, UK, 2007). [CrossRef]
  30. S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett.53, 2169–2172 (1984). [CrossRef]
  31. A. Lagendijk, B. van Tiggelen, and D. S. Wiersma, “Fifty years of anderson localization,” Phys. Today62, 24–29 (2009). [CrossRef]
  32. B. Payne, J. Andreasen, H. Cao, and A. Yamilov, “Relation between transmission and energy stored in random media with gain,” Phys. Rev. B82, 104204 (2010). [CrossRef]
  33. P. A. Lee, D. A. Stone, and H. Fukuyamak, “Universal conductance fluctuations in metals: Effects of finite temperature, interactions, and magnetic field,” Phys. Rev. B35, 1039–1070 (1987). [CrossRef]
  34. P. W. Anderson, D. J. Thouless, E. Abrahams, and D. S. Fisher, “New method for a scaling theory of localization,” Phys. Rev. B22, 3519–3526 (1980). [CrossRef]
  35. B. L. Altshuler, P. A. Lee, and R. A. Webb, eds., Mesoscopic Phenomena in Solids (North Holland, Amsterdam, 1991).
  36. V. D. Freilikher, M. Pustilnik, and I. Yurkevich, “Effect of absorption on the wave transport in the strong localization regime,” Phys. Rev. Lett.73, 810–813 (1994). [CrossRef] [PubMed]
  37. A. Yamilov and B. Payne, “Classification of regimes of wave transport in quasi-one-dimensional nonconservative random media,” J. Mod. Opt.57, 1916–1921 (2010). [CrossRef]
  38. L. Y. Zhao, C. Tian, Z. Q. Zhang, and X. D. Zhang, “Unusual Brownian motion of photons in open absorbing media,” arXiv:1304.0516 (2013).

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