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Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

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  • Vol. 7, Iss. 10 — Oct. 5, 2012
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Perfect transmission through Anderson localized systems mediated by a cluster of localized modes

Wonjun Choi, Q-Han Park, and Wonshik Choi  »View Author Affiliations


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

The interference of multiply scattered waves in the disordered media leads to interesting phenomena such as the Anderson localization [1

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

4

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]

] and the existence of open eigenchannels [5

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

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]

]. Unlike in the diffusion regime where transmission decreases linearly over the thickness, it decreases exponentially when the Anderson localization occurs. In the strongly scattering media, constructively interfering local waves dominate over the incoherently interfering propagating waves, which undermines the propagation of waves within the medium [3

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]

]. On the contrary, there exist so-called open eigenchannels that allow perfect transmission through a disordered medium. The well-controlled coherent combination of input channels induces constructive interference throughout the medium and subsequently at the exit of the medium. Recent experimental studies performed in the optical regime proved the existence of open eigenchannels [6

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]

]. And in our recent study, we experimentally demonstrated that the injection of light into high-transmission eigenchannels enhances energy transmission [7

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]

]. The random matrix theory [5

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

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

] predicts that open eigenchannels exist regardless of the degree of disorder. This implies that they will exist even in the Anderson localization regime. In fact, the existence of the high-transmission eigenchannels has been observed in a recent study performed in the microwave regime for the disordered wave guide exhibiting Anderson localization [9

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]

]. We investigate this rather intriguing combination of wave localization and transmission enhancement, both originated from the constructive interference, to study the way open eigenchannels manifest themselves in the Anderson localization regime.

In this Letter, we investigate the internal field distribution of open eigenchannels for the Anderson localized systems. In the optical regime, we numerically prepare a highly disordered 2D medium whose thickness exceeds the localization length of the medium. Using the finite-difference time-domain (FDTD) method, we compute electromagnetic wave propagation through the medium and construct a transmission matrix, 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

For a disordered medium, we numerically prepare a 2D medium made of absorption-free dielectric square particles with a side length of 200 nm (Fig. 1(a)
Fig. 1 (a) Geometry of a disordered medium and (b) the determination of localization length. (a) A medium composed of randomly distributed square particles is prepared in the x-z plane. The width of the medium in the x-direction is 130 μm and the thickness in the z-direction is varied. The incident wave covers the entire width while the output is sampled along the black dashed line whose width is W = 90 μm. (b) Transmittance values of random media as a function of thickness. Data represented by squares are acquired for np = 1.6 media and circles for np = 2.5 media. The error bar is obtained from the statistical distribution of the transmittances taken at various incident angles. Dashed and dotted lines are linear fits to the squares and circles, respectively.
). The width of the medium in the x-direction (transverse direction) and the thickness in the z-direction are 130 μm and 8 μm, respectively. The particles are randomly distributed in the vacuum with a fill factor of 30.0 ± 0.5%. The incident light is a monochromatic plane wave of wavelength λ = 600 nm and the polarization of the light is perpendicular to the x-z plane. The numerical cell size is set as 10 nm, which is small enough compared with the wavelength to ensure the accuracy of the computation. The propagation of the wave is computed until the steady state is established. To obtain the complex electric field both within and outside the medium, a phase-shifting interferometry method is used [13

13. C. Katherine, “V Phase-Measurement Interferometry Techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1988), 349–393.

]. For each position, we record the electric field four times with an interval of one-quarter of an optical oscillation and process them to acquire the amplitude and phase of the wave.

In order to control the degree of disorder, we vary the refractive index, np, 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, ξ, is defined by the equation, lnT=L/ξ, where T and L stand for the average transmittance and thickness, respectively. We calculate the transmittance of the disordered medium of a given np at various thicknesses (Fig. 1(b)). The medium of larger np has a steeper decrease in transmittance with increasing thickness. From the fitting, the localization length corresponding to np = 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

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]

] rather than the exponential function for np = 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 np = 2.5 medium and smaller than that of the medium with np = 1.6 [11

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]

].

For the two 8 μm-thick disordered media of np = 1.6 and 2.5, we obtain the spatial intensity map of the electromagnetic wave within each disordered medium (Figs. 2(a)
Fig. 2 Field distribution within the disordered media. (a), (b) Intensity maps inside random media with np = 1.6 and 2.5, respectively. The incident wave is a plane wave propagating along the z-direction. The color bar shows the amplitude in arbitrary units. Scale bar, 10 μm. (c), (d) Angular spectrum maps for (a) and (b), respectively. Horizontal and vertical axes are the wave numbers along the x and z directions, respectively, in units of the free space wave number, k0. The color bar indicates the amplitude in arbitrary units.
and 2(b)). The illumination source is a plane wave normally incident to the input surface. As predicted in the theory [15

15. M. A. Noginov, Tutorials in Complex Photonic Media (SPIE Press, 2009), 25, 696 p., 696 p. of plates.

], we observe that internal intensity decreases exponentially for the np = 2.5 medium while it decreases linearly for the np = 1.6 medium. In particular, prominent intensity-enhanced spots exist for np = 2.5 while the intensity is relatively uniformly distributed for np = 1.6. These results suggest that the np = 2.5 medium is in the strong localization regime while the np = 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

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

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]

]. Specifically, the statistics of the speckle intensity is expected to deviate from Rayleigh’s law in the localization regime, which was indeed the case for np = 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 np = 1.6 and np = 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 np = 2.5 medium is about 4.4 times larger than that of the np = 1.6 medium. Therefore, we can infer that the wave localization increases the heterogeneity of the internal intensity distribution.

In addition to the statistics of internal intensity, we characterize the properties of internal field and derive a clear condition to distinguish wave localization from diffusion. Since the FDTD method computes the complex amplitude of the field within the medium, we can take the discrete 2D Fourier transform of the internal field with respect to the spatial coordinates x and z for the field within the medium. This provides us with an angular spectrum of the internal waves in the spatial frequency coordinates [18

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

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

]. Figures 2(c) and 2(d) are the field amplitude in k-space for np = 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) k0, corresponds to the magnitude of the effective wave vector inside the medium. This agrees well with the effective wave number, 1.27k0, estimated by the effective medium theory [20

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]

]. It is larger than k0, 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.

3. Construction of a transmission matrix

For each of the plane waves in the basis set, we compute its propagation and then record the transmitted wave along the sampling line (dashed line in Fig. 1(a)). In order to use the wave vector space as output channels, we convert the measured electric field on the recording line into k-space representation through the discrete Fourier transform. By repeating this procedure for all the incident channels, we construct a transmission matrix whose dimensions are 299 × 299 (Fig. 3(a)
Fig. 3 (a) Transmission matrix and (b) its singular value distribution. The kxi and kxo are the wave numbers along the x-direction for the input and output channels, respectively. The color bar indicates the amplitude in arbitrary units. Singular values are sorted in descending order for the eigenchannel index. Singular value distribution for np = 1.6 medium is shown in blue for comparison.
). The phase part of the transmission matrix is acquired simultaneously (not shown in the figure).

4. Internal field distribution of transmission eigenchannels

For each singular value, we obtain a corresponding eigenchannel in the form of the coefficients of incident channels. The coefficient is determined from the corresponding column of the unitary matrix 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
Fig. 4 Internal field distribution of eigenchannels for the disordered medium of np = 2.5. For (a)-(e), eigenchannel indices are 1, 6, 11, 31, and 291, respectively, and their transmittance values are 80, 37, 12, 0,39, and 2.0 × 10-7%, respectively. The color bar indicates the base-10 logarithm of the amplitude. The aspect ratio is set to have different scales for the horizontal and vertical axes for better visibility. Both scale bars indicates 5 μm.
, 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

We found that there exists a striking difference between the Anderson localization regime and the diffusion regime. In the diffusion regime, the field energy of the open eigenchannel is spread over the entire transverse extent [10

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]

]. In the Anderson localization regime, however, it is highly concentrated in space such that a dense cluster of localized modes appears within the disordered medium (Fig. 4(a)). Moreover, the clustering is found to exist only for the eigenchannels of high transmittance. We repeatedly observe this appearance of the cluster for other media whose spatial organizations of constituting particles are different but with the same particle size and index, and the fill factor. As the transmittance becomes low, the internal waves gradually spread along the transverse direction (Fig. 4(b)-4(e)). For the eigenchannels of sufficiently low transmission, the internal wave covers the entire transverse range. These suggest that the dense cluster be formed in the process of enhancing transmission through a disordered medium where non-propagating localized waves are dominant.

We quantify the degree of transverse confinement of the internal waves, the unique characteristic of the cluster to enhance transmission in the Anderson localization regime. For a fixed z, we calculate the standard deviation of the field distribution along the transverse direction by using the local intensity as a weighting Eq. (2).

Δx(z)=x2I(x,z)dx(xI(x,z)dx)2(I(x,z)dx)2
(2)

We then define the average of the standard deviation for all z in the medium as the effective transverse width, Weff. Figure 5
Fig. 5 Effective width, Weff , of the internal field. Effective width, which is as a function of eigenchannel index, is defined through Eq. (2).
shows the plot of Weff for the eigenchannels of the medium. The Weff 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.

The existence of the dense cluster of internal modes in the Anderson localization regime can be explained as follows. According to Fig. 2(d), most waves exist in the form of non-propagating waves in the strongly scattering medium. In this case, the propagation of energy can mainly be mediated by the neighboring localized modes and by the coupling among them. This can be thought of as a kind of frustrated total internal reflection in which a neighboring pair of prisms induces the coupling of non-propagating evanescent waves and therefore enhances wave propagation. Studies in the 1D media also showed that neighboring of localized modes enhances the transmission [11

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

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

].

Finally, the square-shape particle used in the study exhibits weak resonance around np = 2.2 and 2.96 for the source wavelength of 600 nm. The strongly scattering medium (np = 2.5) considered in our study is in closer proximity to the single particle resonance than the weakly scattering medium (np = 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.

6. Conclusion

We studied the interplay between wave localization in a strongly scattering medium and the open eigenchannels of the disordered medium which induce perfect transmission. From the direct investigation of the internal field in the medium, we proved that the non-propagating localized waves dominate over the propagating waves in the Anderson localization regime. This has resulted in a unique consequence that open eigenchannels of high transmittance are manifested as a dense neighboring of localized modes with strong transverse confinement. In this way, the localized waves in the extended space are tightly coupled among each other and energy flow is enhanced from input to the output planes of the medium. Our study explicitly elucidates the mechanism of transmission enhancement in the Anderson localization regime. Future studies may include the investigation of phase transition from diffusion to localization regimes in terms of the statistical properties of internal fields.

Acknowledgments

This research was supported by the Basic Science Research Program through the National Research Foundation (NRF) of Korea funded by the Ministry of Education, Science and Technology (MEST) (2011-0005018, 2011-0029807, and 2011-0016568), the National R&D Program for Cancer Control, the Ministry of Health & Welfare, South Korea (1120290), a Korean University Grant, and Nano Material Technology Development Program through the NRF of Korea funded by MEST (2012-0006205 and 2012-0006655).

References and links

1.

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

2.

S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett. 53(22), 2169–2172 (1984). [CrossRef]

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]

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]

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. 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]

13.

C. Katherine, “V Phase-Measurement Interferometry Techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1988), 349–393.

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]

15.

M. A. Noginov, Tutorials in Complex Photonic Media (SPIE Press, 2009), 25, 696 p., 696 p. of plates.

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]

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.

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]

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. 82(5), 056603 (2010). [CrossRef] [PubMed]

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/vjbo/abstract.cfm?URI=oe-20-18-20721


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References

  1. P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev.109(5), 1492–1505 (1958). [CrossRef]
  2. S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett.53(22), 2169–2172 (1984). [CrossRef]
  3. D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature390(6661), 671–673 (1997). [CrossRef]
  4. 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]
  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]
  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.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. B83(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 State20(5), 733–742 (1987). [CrossRef]
  13. C. Katherine, “V Phase-Measurement Interferometry Techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1988), 349–393.
  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]
  15. M. A. Noginov, Tutorials in Complex Photonic Media (SPIE Press, 2009), 25, 696 p., 696 p. of plates.
  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]
  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 .
  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. Character205(387-401), 237–288 (1906). [CrossRef]
  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.82(5), 056603 (2010). [CrossRef] [PubMed]

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