Abstract

The wave transport through disordered media, although a random process, has some universal physical properties. One of these properties that has been investigated in this report is the relation between transmission eigenchannels and the so-called single-channel optimizing mode, which maximizes the intensity of the transmitted wave at a single specific output channel. Since single-channel optimizing modes have higher transmittance than the uncontrolled waves, it has been predicted before that transmission eigenchannels with higher transmittance preferentially contribute to the single-channel optimizing modes in proportion to the square of eigenvalues. In this Letter, we report the experimental validation of this prediction by measuring cross-correlation between the single-channel optimizing modes and the transmission eigenchannels.

© 2013 Optical Society of America

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References

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  1. M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. H. Park, and W. Choi, Nat. Photonics 6, 581 (2012).
    [CrossRef]
  2. O. N. Dorokhov, Sol. State Commun. 51, 381 (1984).
    [CrossRef]
  3. Z. Shi and A. Z. Genack, Phys. Rev. Lett. 108, 043901 (2012).
    [CrossRef]
  4. S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
    [CrossRef]
  5. I. M. Vellekoop and A. P. Mosk, Phys. Rev. Lett. 101, 120601 (2008).
    [CrossRef]
  6. W. Choi, A. P. Mosk, Q. H. Park, and W. Choi, Phys. Rev. B 83, 134207 (2011).
    [CrossRef]

2012 (2)

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. H. Park, and W. Choi, Nat. Photonics 6, 581 (2012).
[CrossRef]

Z. Shi and A. Z. Genack, Phys. Rev. Lett. 108, 043901 (2012).
[CrossRef]

2011 (1)

W. Choi, A. P. Mosk, Q. H. Park, and W. Choi, Phys. Rev. B 83, 134207 (2011).
[CrossRef]

2010 (1)

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef]

2008 (1)

I. M. Vellekoop and A. P. Mosk, Phys. Rev. Lett. 101, 120601 (2008).
[CrossRef]

1984 (1)

O. N. Dorokhov, Sol. State Commun. 51, 381 (1984).
[CrossRef]

Boccara, A. C.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef]

Carminati, R.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef]

Choi, W.

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. H. Park, and W. Choi, Nat. Photonics 6, 581 (2012).
[CrossRef]

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. H. Park, and W. Choi, Nat. Photonics 6, 581 (2012).
[CrossRef]

W. Choi, A. P. Mosk, Q. H. Park, and W. Choi, Phys. Rev. B 83, 134207 (2011).
[CrossRef]

W. Choi, A. P. Mosk, Q. H. Park, and W. Choi, Phys. Rev. B 83, 134207 (2011).
[CrossRef]

Choi, Y.

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. H. Park, and W. Choi, Nat. Photonics 6, 581 (2012).
[CrossRef]

Dorokhov, O. N.

O. N. Dorokhov, Sol. State Commun. 51, 381 (1984).
[CrossRef]

Fink, M.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef]

Genack, A. Z.

Z. Shi and A. Z. Genack, Phys. Rev. Lett. 108, 043901 (2012).
[CrossRef]

Gigan, S.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef]

Kim, J.

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. H. Park, and W. Choi, Nat. Photonics 6, 581 (2012).
[CrossRef]

Kim, M.

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. H. Park, and W. Choi, Nat. Photonics 6, 581 (2012).
[CrossRef]

Lerosey, G.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef]

Mosk, A. P.

W. Choi, A. P. Mosk, Q. H. Park, and W. Choi, Phys. Rev. B 83, 134207 (2011).
[CrossRef]

I. M. Vellekoop and A. P. Mosk, Phys. Rev. Lett. 101, 120601 (2008).
[CrossRef]

Park, Q. H.

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. H. Park, and W. Choi, Nat. Photonics 6, 581 (2012).
[CrossRef]

W. Choi, A. P. Mosk, Q. H. Park, and W. Choi, Phys. Rev. B 83, 134207 (2011).
[CrossRef]

Popoff, S. M.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef]

Shi, Z.

Z. Shi and A. Z. Genack, Phys. Rev. Lett. 108, 043901 (2012).
[CrossRef]

Vellekoop, I. M.

I. M. Vellekoop and A. P. Mosk, Phys. Rev. Lett. 101, 120601 (2008).
[CrossRef]

Yoon, C.

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. H. Park, and W. Choi, Nat. Photonics 6, 581 (2012).
[CrossRef]

Nat. Photonics (1)

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. H. Park, and W. Choi, Nat. Photonics 6, 581 (2012).
[CrossRef]

Phys. Rev. B (1)

W. Choi, A. P. Mosk, Q. H. Park, and W. Choi, Phys. Rev. B 83, 134207 (2011).
[CrossRef]

Phys. Rev. Lett. (3)

Z. Shi and A. Z. Genack, Phys. Rev. Lett. 108, 043901 (2012).
[CrossRef]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef]

I. M. Vellekoop and A. P. Mosk, Phys. Rev. Lett. 101, 120601 (2008).
[CrossRef]

Sol. State Commun. (1)

O. N. Dorokhov, Sol. State Commun. 51, 381 (1984).
[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

Experimental setup for recording a transmission matrix and generating a transmission eigenchannel. BS1, BS2, and BS3, beam splitters; SLM, spatial light modulator; C, condenser lens; IP, input plane, OP, output plane, OL, objective lens; θξ andθη, incident angle of a plane wave onto the sample plane.

Fig. 2.
Fig. 2.

Transmission matrix, single-channel optimizing mode and transmission eigenchannels. (a) The amplitude part of a transmission matrix. Horizontal and vertical axes are the position vectors at IP and OP, respectively. (b) Phase part of the transmission matrix. Color bar, phase in radians. (c) A single-channel optimizing mode generated by SLM at the input plane. (d) Transmitted wave when experimentally generated single-channel optimizing mode shown in (c) was sent to the disordered medium. (e), (f) Output intensity maps for 81st and 501st eigenchannels, respectively, obtained from t. Color bar, intensity in arbitrary unit. Scale bar, 10 μm.

Fig. 3.
Fig. 3.

Contribution of each transmission eigenchannels to the single-channel optimizing modes. (a) Absolute square of cross-correlation coefficients calculated from the transmission matrix (triangular dots) and from the transmission matrix after accounting for the phase-only control (circular dots). Square dots: absolute square of the cross-correlation coefficients between experimentally measured single-channel optimizing modes and outputs of transmission eigenchannels. (b) Triangular dots: square of eigenvalues with a constant factor. Square dots: experimentally measured absolute square of the cross-correlation coefficients after accounting for the effect of phase-only control explained in the text.

Equations (1)

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Emo=n=1NtmnEni,

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