## Interplay between localization and absorption in disordered waveguides |

Optics Express, Vol. 21, Issue 10, pp. 11688-11697 (2013)

http://dx.doi.org/10.1364/OE.21.011688

Acrobat PDF (1120 KB)

### 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

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] .

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] .

*D*

_{0}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. 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] .

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] .

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] .

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] .

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

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] .

*g*∝ exp(−

*L*/

*ξ*

_{a}_{0}) 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] .

*ξ*is the localization length,

*τ*is the ballistic absorption time, and

_{a}*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] .

*τ*exceeds the radiative decay rate through the boundaries of the random medium, the resonant tunneling through the modes of the random medium [21

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

*L*, the amount of absorption needed to suppress the resonant tunneling becomes small.

*ξ*

_{a}_{0}≲

*L*) 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

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] .

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] .

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] .

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

*ℓ*<

*L*<

*ξ*and localization for

*L*>

*ξ*.

*ℓ*denotes the transport mean free path. Furthermore, as we argue in Sec. 5, the 2D random waveguides can be fabricated experimentally [27] to test the predictions of this work.

### 2.2. Position-dependent diffusion coefficient

*E*(

**r**)

*e*

^{−}

*propagating in a 2D volume-disordered waveguide of width*

^{iωt}*W*and length

*L*≫

*W*that supports

*N*propagating modes. The wave field

*E*(

**r**) obeys the 2D Helmholtz equation: Here

*k*=

*ω*/

*c*is the wavenumber,

*δε*(

**r**) = (1 +

*iα*)

*δε*(

_{r}**r**), where

*δε*(

_{r}**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] .

*E*(

**r**) we calculate the cross section-averaged energy density

*𝒲*(

*z*) and the longitudinal component of flux

*J*(

_{z}*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: The averages 〈...〉 are taken over a statistical ensemble of 10

^{6}disorder realizations simulated on a supercomputer.

### 2.3. D_{0} calculation

*D*(

*z*) with SCT without fitting parameters, we need to obtain the value of the diffusion coefficient unrenormalized by the wave interference effects

*D*

_{0}=

*vℓ*/2. Here

*v*is the diffusive speed and

*ℓ*is the transport mean free path.

*v*we use the definition of diffusive flux resolved with respect to the direction of propagation [1] where

### 2.4. τ_{a} calculation

*τ*can be determined numerically using the condition of flux continuity

_{a}*d*〈

*J*(

_{z}*z*)〉/

*dz*= 〈

*𝒲*(

*z*)〉/

*τ*. The desired diffusive absorption length

_{a}*α*in Eq. (1).

## 3. Self-consistent theory

*G*(

**r**,

**r′**) of Eq. (1) with

*δε*(

**r**) =

*δε*(

_{r}**r**)+

*iα*. The disorder-averaged function

*Ĉ*(

**r**,

**r′**) = (4

*πWD*

_{0}/

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

*d*(

*ζ*) =

*D*(

*ζ*)/

*D*

_{0}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: at

*ζ*= 0 and

*ζ*= 1. We obtain the numerical solution of Eqs. (5–7) 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

*D*(

*z*) (blue curves) and the prediction of SCT from Eqs. (5–7) (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.

*ℓ*and

*z*

_{0}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*≲

*ℓ*and

*L*−

*ℓ*≲

*z*<

*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] .

*z*∼

*L*/2 deep in the localization regime

*L*≫

*ξ*. 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] .

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

*D*(

*z*)/

*D*

_{0}falls below the value of ≃ 0.2, which also agree with the numerical simulations reported in [18

**105**, 263905 (2010) [CrossRef] .

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

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

*D*(

*z*)/

*D*

_{0}→ 1 with a decrease of

*ξ*

_{a}_{0}(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*) ≃

*D*in the spatial region

_{p}*ξ*

_{a}_{0}≲

*z*≲

*L*−

*ξ*

_{a}_{0}. 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

*z*−

*ξ*

_{a}_{0}to

*z*+

*ξ*

_{a}_{0}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] .

*∂d*(

*ζ*)/

*∂ζ*term in Eq. (5). We solve the self-consistent Eqs. (5–7) under the above assumption and obtain the following self-contained equation on

*d*≡

_{p}*D*/

_{p}*D*

_{0}: which, in the limit of strong absorptions

*ξ*

_{a}_{0}/

*ξ*→ 0, yields an even simpler result

*d*≃ [1 +

_{p}*ξ*

_{a}_{0}/

*ξ*]

^{−1}. In this expression the parameter

*g*≡

_{eff}*ξ*/

*ξ*

_{a}_{0}controls the extent of renormalization of the diffusion coefficient

*D*, similar to the role played by the conductance

_{p}*g*

_{0}≃

*ξ*/

*L*in passive systems [16

**82**, 024205 (2010) [CrossRef] .

*D*(

*z*) becomes asymmetric with respect to the middle of the sample

*D*(

*z*) ≠

*D*(

*L*−

*z*). 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] .

### 4.2. Minimum diffusion coefficient in absorbing systems

*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*/

*ξ*

_{a}_{0}= 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 ∼ 2

*ξ*

_{a}_{0}and develops a plateau where it is no longer sensitive to the boundaries of the system.

*D*(

*z*=

*L*/2) with an increase of the sample size

*L*. Importantly, the saturated value

*D*is determined by the ratio

_{p}*ξ*/

*ξ*

_{a}_{0}, 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.

### 4.3. Universal scaling of position-dependent diffusion

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] .

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] .

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] .

**82**, 024205 (2010) [CrossRef] .

**105**, 263905 (2010) [CrossRef] .

*D*(

*z*) is uniquely determined by the value of conductance.

*ξ*/

*ξ*

_{a}_{0}. 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.

## 5. Conclusions

*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].

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

## Acknowledgments

## References and links

1. | P. M. Morse and H. Feshbach, |

2. | M. C. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. |

3. | S. Chandresekhar, |

4. | L. V. Wang and H. Wu, |

5. | D. Vollhardt and P. Wölfle, “Diagrammatic, self-consistent treatment of the anderson localization problem in |

6. | J. Kroha, C. M. Soukoulis, and P. Wölfle, “Localization of classical waves in a random medium: A self-consistent theory,” Phys. Rev. B |

7. | P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. |

8. | B. A. van Tiggelen, A. Lagendijk, and D. S. Wiersma, “Reflection and transmission of waves near the localization threshold,” Phys. Rev. Lett. |

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 |

10. | C. Tian, “Supersymmetric field theory of local light diffusion in semi-infinite media,” Phys. Rev. B |

11. | C. Tian, “Hydrodynamic and field-theoretic approaches to light localization in open media,” Physica E |

12. | M. Störzer, P. Gross, C. Aegerter, and G. Maret, “Observation of the critical regime near anderson localization of light,” Phys. Rev. Lett. |

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. |

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 |

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. |

16. | B. Payne, A. Yamilov, and S. E. Skipetrov, “Anderson localization as position-dependent diffusion in disordered waveguides,” Phys. Rev. B |

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 |

18. | C. Tian, S. Cheung, and Z. Zhang, “Local diffusion theory for localized waves in open media,” Phys. Rev. Lett. |

19. | J. B. Pendry, “Symmetry and transport of waves in one-dimensional disordered systems,” Adv. Phys. |

20. | A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature |

21. | J. Wang and A. Z. Genack, “Transport through modes in random media,” Nature |

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. |

23. | L. I. Deych, A. Yamilov, and A. A. Lisyansky, “Scaling in one-dimensional localized absorbing systems,” Phys. Rev. B |

24. | L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with anderson-localized modes,” Science |

25. | C. W. Beenakker, “Random-matrix theory of quantum transport,” Rev. Mod. Phys. |

26. | P. W. Brouwer, “Transmission through a many-channel random waveguide with absorption,” Phys. Rev. B |

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. |

29. | E. Akkermans and G. Montambaux, |

30. | S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett. |

31. | A. Lagendijk, B. van Tiggelen, and D. S. Wiersma, “Fifty years of anderson localization,” Phys. Today |

32. | B. Payne, J. Andreasen, H. Cao, and A. Yamilov, “Relation between transmission and energy stored in random media with gain,” Phys. Rev. B |

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 |

34. | P. W. Anderson, D. J. Thouless, E. Abrahams, and D. S. Fisher, “New method for a scaling theory of localization,” Phys. Rev. B |

35. | B. L. Altshuler, P. A. Lee, and R. A. Webb, eds., |

36. | V. D. Freilikher, M. Pustilnik, and I. Yurkevich, “Effect of absorption on the wave transport in the strong localization regime,” Phys. Rev. Lett. |

37. | A. Yamilov and B. Payne, “Classification of regimes of wave transport in quasi-one-dimensional nonconservative random media,” J. Mod. Opt. |

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

- P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
- 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]
- S. Chandresekhar, Radiative Transfer (Dover, New York, 1960).
- L. V. Wang and H. Wu, Biomedical Optics: Principles and Imaging(Wiley-Interscience, 2007).
- 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]
- 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]
- P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev.109, 1492–1505 (1958). [CrossRef]
- 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]
- 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]
- C. Tian, “Supersymmetric field theory of local light diffusion in semi-infinite media,” Phys. Rev. B77, 064205 (2008). [CrossRef]
- C. Tian, “Hydrodynamic and field-theoretic approaches to light localization in open media,” Physica E49, 124–153 (2013). [CrossRef]
- 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]
- 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]
- 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]
- 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]
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