## Perfect transmission through Anderson localized systems mediated by a cluster of localized modes |

Optics Express, Vol. 20, Issue 18, pp. 20721-20729 (2012)

http://dx.doi.org/10.1364/OE.20.020721

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### Abstract

In a strongly scattering medium where Anderson localization takes place, constructive interference of local non-propagating waves dominate over the incoherent addition of propagating waves. This results in the disappearance of propagating waves within the medium, which significantly attenuates energy transmission. In this numerical study performed in the optical regime, we systematically found resonance modes, called eigenchannels, of a 2-D Anderson localized system that allow for the near-perfect energy transmission. We observed that the internal field distribution of these eigenchannels exhibit dense clustering of localized modes. This strongly suggests that the clustered resonance modes facilitate long-range energy flow of local waves. Our study explicitly elucidates the interplay between wave localization and transmission enhancement in the Anderson localization regime.

© 2012 OSA

## 1. Introduction

1. P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. **109**(5), 1492–1505 (1958). [CrossRef]

4. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature **446**(7131), 52–55 (2007). [CrossRef] [PubMed]

5. O. N. Dorokhov, “On the coexistence of localized and extended electronic states in the metallic phase,” Solid State Commun. **51**(6), 381–384 (1984). [CrossRef]

6. I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. **101**(12), 120601 (2008). [CrossRef] [PubMed]

3. D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature **390**(6661), 671–673 (1997). [CrossRef]

6. I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. **101**(12), 120601 (2008). [CrossRef] [PubMed]

7. M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics, in press. (2012). [CrossRef] [PubMed]

5. O. N. Dorokhov, “On the coexistence of localized and extended electronic states in the metallic phase,” Solid State Commun. **51**(6), 381–384 (1984). [CrossRef]

8. C. W. J. Beenakker, “Random-matrix theory of quantum transport,” Rev. Mod. Phys. **69**(3), 731–808 (1997). [CrossRef]

9. Z. Shi and A. Z. Genack, “Transmission Eigenvalues and the bare conductance in the crossover to Anderson localization,” Phys. Rev. Lett. **108**(4), 043901 (2012). [CrossRef] [PubMed]

10. W. Choi, A. P. Mosk, Q. H. Park, and W. Choi, “Transmission eigenchannels in a disordered medium,” Phys. Rev. B **83**(13), 134207 (2011). [CrossRef]

11. J. Bertolotti, S. Gottardo, D. S. Wiersma, M. Ghulinyan, and L. Pavesi, “Optical necklace states in Anderson Localized 1D systems,” Phys. Rev. Lett. **94**(11), 113903 (2005). [CrossRef] [PubMed]

12. J. B. Pendry, “Quasi-extended electron-states in strongly disordered-systems,” J. Phys. C. Solid State **20**(5), 733–742 (1987). [CrossRef]

*t*. We observed that the eigenvalues of the matrix

*tt*, which are the transmittance of the corresponding eigenchannels, reaches to almost unity, thereby finding the open eigenchannels of the medium. Moreover, we compute the field distribution of open eigenchannels within the medium to find the mechanism of transmission enhancement. We found that clustering of localized modes mediates the coupling of waves such that the energy can efficiently propagate toward the output plane.

^{+}## 2. Preparation of disordered media exhibiting Anderson localization

*n*, of the particles constituting the medium. To check whether the medium is in the strong localization regime or not, we survey the relationship between the transmittance and the thickness of the medium. In the Anderson localization regime, the transmittance of the incident wave decreases exponentially with the increase of the thickness. Therefore, the localization length,

_{p}*T*and

*L*stand for the average transmittance and thickness, respectively. We calculate the transmittance of the disordered medium of a given

*n*at various thicknesses (Fig. 1(b)). The medium of larger

_{p}*n*has a steeper decrease in transmittance with increasing thickness. From the fitting, the localization length corresponding to

_{p}*n*= 2.5 (squares) and 1.6 (circles) are determined to be 3.46 ± 0.40 and 14.5 ± 0.91, respectively. We note that the transmittance can be better fitted by the modified Ohm’s law [14

_{p}14. S. E. Skipetrov and B. A. van Tiggelen, “Dynamics of Anderson Localization in Open 3D Media,” Phys. Rev. Lett. **96**(4), 043902 (2006). [CrossRef] [PubMed]

*n*= 1.6 medium at small thickness because the medium is in the diffusion regime. But at large thickness where localization takes place, the exponential fit was accurate enough to determine the localization length. In order to set up two distinctive settings of the disordered media, one in the strong localization regime and the other in the weak localization regime, we set the thickness of the medium to be studied in the following as 8 μm, such that it is larger than the localization length of the

_{p}*n*= 2.5 medium and smaller than that of the medium with

_{p}*n*= 1.6 [11

_{p}11. J. Bertolotti, S. Gottardo, D. S. Wiersma, M. Ghulinyan, and L. Pavesi, “Optical necklace states in Anderson Localized 1D systems,” Phys. Rev. Lett. **94**(11), 113903 (2005). [CrossRef] [PubMed]

*n*= 1.6 and 2.5, we obtain the spatial intensity map of the electromagnetic wave within each disordered medium (Figs. 2(a) and 2(b)). The illumination source is a plane wave normally incident to the input surface. As predicted in the theory [15], we observe that internal intensity decreases exponentially for the

_{p}*n*= 2.5 medium while it decreases linearly for the

_{p}*n*= 1.6 medium. In particular, prominent intensity-enhanced spots exist for

_{p}*n*= 2.5 while the intensity is relatively uniformly distributed for

_{p}*n*= 1.6. These results suggest that the

_{p}*n*= 2.5 medium is in the strong localization regime while the

_{p}*n*= 1.6 medium is in the weak localization regime. Another interesting consequence of the wave localization is the change in the statistics of the internal intensity distribution. Past studies have concerned the statistical properties of the speckle pattern only for the transmitted field [16

_{p}16. S. Zhang and A. Z. Genack, “Statistics of diffusive and localized fields in the vortex core,” Phys. Rev. Lett. **99**(20), 203901 (2007). [CrossRef] [PubMed]

17. R. Höhmann, U. Kuhl, H. J. Stöckmann, L. Kaplan, and E. J. Heller, “Freak waves in the linear regime: A microwave study,” Phys. Rev. Lett. **104**(9), 093901 (2010). [CrossRef] [PubMed]

*n*= 2.5 medium. Here we can also analyze the speckle pattern within the medium. Specifically, we calculate the speckledness, which is the ratio between the average of the intensity squared and the square of the average intensity, for both

_{p}*n*= 1.6 and

_{p}*n*= 2.5 media. It turns out that the speckledness is 5.8 for the strongly scattering medium and 2.1 for the weakly scattering medium. This means that the variance of the intensity distribution for the

_{p}*n*= 2.5 medium is about 4.4 times larger than that of the

_{p}*n*= 1.6 medium. Therefore, we can infer that the wave localization increases the heterogeneity of the internal intensity distribution.

_{p}18. R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, “Ultrafast evolution of photonic eigenstates in k-space,” Nat. Phys. **3**(6), 401–405 (2007). [CrossRef]

19. M. B. Kevin Vynck, F. Riboli, D. S. Wiersma, “Disordered optical modes for photon management.” http://arxiv.org/abs/1202.4601.

*n*= 1.6 and 2.5, respectively. A bright ring appears in the weakly scattering medium (Fig. 2(c)). The presence of the ring indicates that waves propagate in all different directions. The radius of the ring, (1.2 ± 0.5)

_{p}*k*, corresponds to the magnitude of the effective wave vector inside the medium. This agrees well with the effective wave number, 1.27

_{0}*k*, estimated by the effective medium theory [20

_{0}20. J. C. M. Garnett, “Colours in metal glasses, in metallic films, and in metallic solutions. II,” Philos. Trans. R. Soc. Lond., A Contain. Pap. Math. Phys. Character **205**(387-401), 237–288 (1906). [CrossRef]

*k*, the wave number in the free space, because of the high index particles. In the strongly scattering medium (Fig. 2(d)), on the other hand, the ring pattern disappears. This indicates that the propagating waves with pronounced oscillations of electric field over the extended space could be dominated by the localized waves with an abrupt amplitude variation over the wavelength scale. The spatial Fourier transform of the localized waves such as evanescent waves typically exhibits broadened spectra due to the abrupt decay of the field amplitude. This observation agrees well with the theory of Anderson localization in which the locally interfering waves dominate over the incoherently interfering propagating waves. Our analysis could possibly provide a direct observation of the disappearance of propagating waves within the medium and the prevalence of the local non-propagating waves.

_{0}## 3. Construction of a transmission matrix

*n*= 2.5 medium, we next search for its transmission eigenchannels. At first, a transmission matrix

_{p}**is constructed for the disordered medium whose element**

*H**k*satisfies the conditions,

_{x}*k*and

_{x}= 2mπ/W*-k*<

_{0}*k*<

_{x}*k*with

_{0}*W*= 90 μm at the output side of the medium for the sampling (black dashed lines in Fig. 1(a)). This is to avoid diffracted waves from the edges. The integer

*m*satisfying the above condition ranges from −149 to 149 such that the number of channels in the basis is 299.

## 4. Internal field distribution of transmission eigenchannels

**.where**

*H**τ*is a diagonal matrix with nonnegative real numbers on the diagonal, which are called singular values.

*V*and

*U*are unitary matrices mapping the input channels (

*k*) to eigenchannels and eigenchannels to output channels (

_{x}^{i}*k*), respectively. The square of a singular value, known as a transmission eigenvalue, corresponds to the intensity transmission coefficient of the eigenchannel. Figure 3(b) shows the plot of singular values after arranging them in descending order. The largest value is 0.90 which is quite close to unity. In terms of transmittance, which is the square of the singular values, the eigenchannel of the largest singular value is about 50 times higher than the average transmittance of the medium, 0.017. The transmittance of the largest singular value is smaller than unity partly because the medium is in the open slab geometry. Some of the energy entering the incident plane may not reach the detector due to the strong scattering in the medium. This may corrupt the solidarity of the transmission matrix. Also, the total number of channels is not large enough such that the probability of finding an eigenchannel of unity transmission may not be sufficient. We confirmed that the highest transmittance of the eigenchannels increases when the width of the sample is increased. This suggests that transmission of eigenchannels can reach to unity even for the Anderson localized medium.

_{x}^{o}*V*. By superposing incident channels of angular plane waves, the corresponding eigenchannel is constructed at the input plane. We then compute the propagation of the constructed eigenchannel and record the map of the field within the medium. In the Fig. 4 , the intensity maps of the first eigenchannel with maximum transmittance and the 23rd eigenchannel that has the same transmittance as the average transmittance are shown. The maximum intensity of the first eigenchannel (Fig. 4(a)) is enhanced by about 17,000 times in comparison with that of the plane wave illumination (Fig. 2(a)). The total stored energy within the medium is also enhanced in the open eigenchannel to about 18 times that of the plane wave illumination.

## 5. Discussion

10. W. Choi, A. P. Mosk, Q. H. Park, and W. Choi, “Transmission eigenchannels in a disordered medium,” Phys. Rev. B **83**(13), 134207 (2011). [CrossRef]

*z*, we calculate the standard deviation of the field distribution along the transverse direction by using the local intensity as a weighting Eq. (2).

*W*. Figure 5 shows the plot of

_{eff}*W*for the eigenchannels of the medium. The

_{eff}*W*is about 3 μm for the first eigenchannel, and gradually increases up to about the 70th eigenchannel and then saturates. The transverse width is thus in strong anti-correlation with the transmittance of the eigenchannels − the higher the transmittance, the narrower the width of the cluster. In the case of the weakly scattering medium, it is found that the effective transverse width is largely constant regardless of the eigenchannel index.

_{eff}11. J. Bertolotti, S. Gottardo, D. S. Wiersma, M. Ghulinyan, and L. Pavesi, “Optical necklace states in Anderson Localized 1D systems,” Phys. Rev. Lett. **94**(11), 113903 (2005). [CrossRef] [PubMed]

12. J. B. Pendry, “Quasi-extended electron-states in strongly disordered-systems,” J. Phys. C. Solid State **20**(5), 733–742 (1987). [CrossRef]

*n*= 2.2 and 2.96 for the source wavelength of 600 nm. The strongly scattering medium (

_{p}*n*= 2.5) considered in our study is in closer proximity to the single particle resonance than the weakly scattering medium (

_{p}*n*= 1.6). Therefore, the increased scattering cross-section of individual particle at near resonance has partly contributed to the wave localization. However, our observations including the disappearance of effective wavenumber and spatial confinement of highly transmitting eigenchannels in the localization regime are rather independent of the single particle resonance. This suggests that our observation is mainly determined by the collective behavior of the entire medium.

_{p}## 6. Conclusion

## Acknowledgments

## References and links

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

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

3. | D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature |

4. | T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature |

5. | O. N. Dorokhov, “On the coexistence of localized and extended electronic states in the metallic phase,” Solid State Commun. |

6. | I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. |

7. | M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics, in press. (2012). [CrossRef] [PubMed] |

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

9. | Z. Shi and A. Z. Genack, “Transmission Eigenvalues and the bare conductance in the crossover to Anderson localization,” Phys. Rev. Lett. |

10. | W. Choi, A. P. Mosk, Q. H. Park, and W. Choi, “Transmission eigenchannels in a disordered medium,” Phys. Rev. B |

11. | J. Bertolotti, S. Gottardo, D. S. Wiersma, M. Ghulinyan, and L. Pavesi, “Optical necklace states in Anderson Localized 1D systems,” Phys. Rev. Lett. |

12. | J. B. Pendry, “Quasi-extended electron-states in strongly disordered-systems,” J. Phys. C. Solid State |

13. | C. Katherine, “V Phase-Measurement Interferometry Techniques,” in |

14. | S. E. Skipetrov and B. A. van Tiggelen, “Dynamics of Anderson Localization in Open 3D Media,” Phys. Rev. Lett. |

15. | M. A. Noginov, |

16. | S. Zhang and A. Z. Genack, “Statistics of diffusive and localized fields in the vortex core,” Phys. Rev. Lett. |

17. | R. Höhmann, U. Kuhl, H. J. Stöckmann, L. Kaplan, and E. J. Heller, “Freak waves in the linear regime: A microwave study,” Phys. Rev. Lett. |

18. | R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, “Ultrafast evolution of photonic eigenstates in k-space,” Nat. Phys. |

19. | M. B. Kevin Vynck, F. Riboli, D. S. Wiersma, “Disordered optical modes for photon management.” http://arxiv.org/abs/1202.4601. |

20. | J. C. M. Garnett, “Colours in metal glasses, in metallic films, and in metallic solutions. II,” Philos. Trans. R. Soc. Lond., A Contain. Pap. Math. Phys. Character |

21. | N. Cherroret, S. E. Skipetrov, and B. A. van Tiggelen, “Transverse confinement of waves in three-dimensional random media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

**OCIS Codes**

(260.2160) Physical optics : Energy transfer

(290.7050) Scattering : Turbid media

**ToC Category:**

Scattering

**History**

Original Manuscript: June 27, 2012

Revised Manuscript: August 15, 2012

Manuscript Accepted: August 19, 2012

Published: August 24, 2012

**Virtual Issues**

Vol. 7, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

Wonjun Choi, Q-Han Park, and Wonshik Choi, "Perfect transmission through Anderson localized systems mediated by a cluster of localized modes," Opt. Express **20**, 20721-20729 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-18-20721

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### References

- P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev.109(5), 1492–1505 (1958). [CrossRef]
- S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett.53(22), 2169–2172 (1984). [CrossRef]
- D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature390(6661), 671–673 (1997). [CrossRef]
- T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature446(7131), 52–55 (2007). [CrossRef] [PubMed]
- O. N. Dorokhov, “On the coexistence of localized and extended electronic states in the metallic phase,” Solid State Commun.51(6), 381–384 (1984). [CrossRef]
- I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett.101(12), 120601 (2008). [CrossRef] [PubMed]
- M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics, in press. (2012). [CrossRef] [PubMed]
- C. W. J. Beenakker, “Random-matrix theory of quantum transport,” Rev. Mod. Phys.69(3), 731–808 (1997). [CrossRef]
- Z. Shi and A. Z. Genack, “Transmission Eigenvalues and the bare conductance in the crossover to Anderson localization,” Phys. Rev. Lett.108(4), 043901 (2012). [CrossRef] [PubMed]
- W. Choi, A. P. Mosk, Q. H. Park, and W. Choi, “Transmission eigenchannels in a disordered medium,” Phys. Rev. B83(13), 134207 (2011). [CrossRef]
- J. Bertolotti, S. Gottardo, D. S. Wiersma, M. Ghulinyan, and L. Pavesi, “Optical necklace states in Anderson Localized 1D systems,” Phys. Rev. Lett.94(11), 113903 (2005). [CrossRef] [PubMed]
- J. B. Pendry, “Quasi-extended electron-states in strongly disordered-systems,” J. Phys. C. Solid State20(5), 733–742 (1987). [CrossRef]
- C. Katherine, “V Phase-Measurement Interferometry Techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1988), 349–393.
- S. E. Skipetrov and B. A. van Tiggelen, “Dynamics of Anderson Localization in Open 3D Media,” Phys. Rev. Lett.96(4), 043902 (2006). [CrossRef] [PubMed]
- M. A. Noginov, Tutorials in Complex Photonic Media (SPIE Press, 2009), 25, 696 p., 696 p. of plates.
- S. Zhang and A. Z. Genack, “Statistics of diffusive and localized fields in the vortex core,” Phys. Rev. Lett.99(20), 203901 (2007). [CrossRef] [PubMed]
- R. Höhmann, U. Kuhl, H. J. Stöckmann, L. Kaplan, and E. J. Heller, “Freak waves in the linear regime: A microwave study,” Phys. Rev. Lett.104(9), 093901 (2010). [CrossRef] [PubMed]
- R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, “Ultrafast evolution of photonic eigenstates in k-space,” Nat. Phys.3(6), 401–405 (2007). [CrossRef]
- M. B. Kevin Vynck, F. Riboli, D. S. Wiersma, “Disordered optical modes for photon management.” http://arxiv.org/abs/1202.4601 .
- J. C. M. Garnett, “Colours in metal glasses, in metallic films, and in metallic solutions. II,” Philos. Trans. R. Soc. Lond., A Contain. Pap. Math. Phys. Character205(387-401), 237–288 (1906). [CrossRef]
- N. Cherroret, S. E. Skipetrov, and B. A. van Tiggelen, “Transverse confinement of waves in three-dimensional random media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.82(5), 056603 (2010). [CrossRef] [PubMed]

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