Abstract

We exploit the evolution in time of the transmission matrix following pulse excitation of a random medium to focus radiation at a selected time delay t and position r. The temporal profile of a focused microwave pulse is the same as the incident Gaussian pulse. The contrast in space at time t of the focused wave is determined by the participation number of transmission eigenvalues M and the size N of the measured transmission matrix. The initial rise and subsequent decay of the contrast observed reflect the distribution of decay rates of the quasi-normal modes within the sample.

© 2013 Optical Society of America

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  1. A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, Nat. Photonics 6, 283 (2012).
    [CrossRef]
  2. A. Vogel, J. Noack, G. Huttmann, and G. Paultauf, Appl. Phys. B 81, 1015 (2005).
    [CrossRef]
  3. W. R. Zipfel, R. M. Williams, and W. W. Webb, Nat. Biotechnol. 21, 1369 (2003).
    [CrossRef]
  4. A. M. Tulino and S. Verdú, Random Matrix Theory and Wireless Communications (Now Publishers, 2004).
  5. M. Fink, Phys. Today 50(3), 34 (1997).
    [CrossRef]
  6. I. M. Vellekoop and A. P. Mosk, Opt. Lett. 32, 2309 (2007).
    [CrossRef]
  7. J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, and A. Lagendijk, Phys. Rev. Lett. 106, 103901 (2011).
    [CrossRef]
  8. O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, Nat. Photonics 5, 372 (2011).
    [CrossRef]
  9. O. N. Dorokhov, Solid State Commun. 51, 381 (1984).
    [CrossRef]
  10. P. A. Mello, P. Pereyra, and N. Kumar, Ann. Phys. 181, 290 (1988).
    [CrossRef]
  11. S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
    [CrossRef]
  12. M. Davy, Z. Shi, and A. Z. Genack, Phys. Rev. B 85, 035105 (2012).
    [CrossRef]
  13. I. M. Vellekoop and A. P. Mosk, Phys. Rev. Lett. 101, 120601 (2008).
    [CrossRef]
  14. M. Davy, Z. Shi, J. Wang, and A. Z. Genack, Opt. Express 21, 10367 (2013).
    [CrossRef]
  15. A. A. Chabanov, B. Hu, and A. Z. Genack, Phys. Rev. Lett. 93, 123901 (2004).
    [CrossRef]
  16. I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, Nat. Photonics 4, 320 (2010).
    [CrossRef]
  17. Z. Shi and A. Z. Genack, Phys. Rev. Lett. 108, 043901 (2012).
    [CrossRef]
  18. M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q-H. Park, and W. Choi, Nat. Photonics 6, 581 (2012).
  19. A. Goetschy and A. D. Stone, arxiv.org/abs/1304.5562.
  20. A. Mirlin, Phys. Rep. 326, 259 (2000).
    [CrossRef]
  21. A. A. Chabanov, Z. Q. Zhang, and A. Z. Genack, Phys. Rev. Lett. 90, 203903 (2003).
    [CrossRef]
  22. J. Wang and A. Z. Genack, Nature 471, 345 (2011).
    [CrossRef]
  23. J. B. Pendry, Adv. Phys. 43, 461 (1994).
    [CrossRef]
  24. J. Bertolotti, S. Gottardo, D. S. Wiersma, M. Ghulinyan, and L. Pavesi, Phys. Rev. Lett. 94, 113903 (2005).
    [CrossRef]
  25. P. Sebbah, B. Hu, J. M. Klosner, and A. Z. Genack, Phys. Rev. Lett. 96, 183902 (2006).
    [CrossRef]

2013 (1)

2012 (4)

M. Davy, Z. Shi, and A. Z. Genack, Phys. Rev. B 85, 035105 (2012).
[CrossRef]

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

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

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, Nat. Photonics 6, 283 (2012).
[CrossRef]

2011 (3)

J. Wang and A. Z. Genack, Nature 471, 345 (2011).
[CrossRef]

J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, and A. Lagendijk, Phys. Rev. Lett. 106, 103901 (2011).
[CrossRef]

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, Nat. Photonics 5, 372 (2011).
[CrossRef]

2010 (2)

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, A. Lagendijk, and A. P. Mosk, Nat. Photonics 4, 320 (2010).
[CrossRef]

2008 (1)

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

2007 (1)

2006 (1)

P. Sebbah, B. Hu, J. M. Klosner, and A. Z. Genack, Phys. Rev. Lett. 96, 183902 (2006).
[CrossRef]

2005 (2)

J. Bertolotti, S. Gottardo, D. S. Wiersma, M. Ghulinyan, and L. Pavesi, Phys. Rev. Lett. 94, 113903 (2005).
[CrossRef]

A. Vogel, J. Noack, G. Huttmann, and G. Paultauf, Appl. Phys. B 81, 1015 (2005).
[CrossRef]

2004 (1)

A. A. Chabanov, B. Hu, and A. Z. Genack, Phys. Rev. Lett. 93, 123901 (2004).
[CrossRef]

2003 (2)

W. R. Zipfel, R. M. Williams, and W. W. Webb, Nat. Biotechnol. 21, 1369 (2003).
[CrossRef]

A. A. Chabanov, Z. Q. Zhang, and A. Z. Genack, Phys. Rev. Lett. 90, 203903 (2003).
[CrossRef]

2000 (1)

A. Mirlin, Phys. Rep. 326, 259 (2000).
[CrossRef]

1997 (1)

M. Fink, Phys. Today 50(3), 34 (1997).
[CrossRef]

1994 (1)

J. B. Pendry, Adv. Phys. 43, 461 (1994).
[CrossRef]

1988 (1)

P. A. Mello, P. Pereyra, and N. Kumar, Ann. Phys. 181, 290 (1988).
[CrossRef]

1984 (1)

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

Aulbach, J.

J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, and A. Lagendijk, Phys. Rev. Lett. 106, 103901 (2011).
[CrossRef]

Bertolotti, J.

J. Bertolotti, S. Gottardo, D. S. Wiersma, M. Ghulinyan, and L. Pavesi, Phys. Rev. Lett. 94, 113903 (2005).
[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]

Bromberg, Y.

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, Nat. Photonics 5, 372 (2011).
[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]

Chabanov, A. A.

A. A. Chabanov, B. Hu, and A. Z. Genack, Phys. Rev. Lett. 93, 123901 (2004).
[CrossRef]

A. A. Chabanov, Z. Q. Zhang, and A. Z. Genack, Phys. Rev. Lett. 90, 203903 (2003).
[CrossRef]

Choi, W.

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

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

Choi, Y.

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

Davy, M.

M. Davy, Z. Shi, J. Wang, and A. Z. Genack, Opt. Express 21, 10367 (2013).
[CrossRef]

M. Davy, Z. Shi, and A. Z. Genack, Phys. Rev. B 85, 035105 (2012).
[CrossRef]

Dorokhov, O. N.

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

Fink, M.

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, Nat. Photonics 6, 283 (2012).
[CrossRef]

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

M. Fink, Phys. Today 50(3), 34 (1997).
[CrossRef]

Genack, A. Z.

M. Davy, Z. Shi, J. Wang, and A. Z. Genack, Opt. Express 21, 10367 (2013).
[CrossRef]

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

M. Davy, Z. Shi, and A. Z. Genack, Phys. Rev. B 85, 035105 (2012).
[CrossRef]

J. Wang and A. Z. Genack, Nature 471, 345 (2011).
[CrossRef]

P. Sebbah, B. Hu, J. M. Klosner, and A. Z. Genack, Phys. Rev. Lett. 96, 183902 (2006).
[CrossRef]

A. A. Chabanov, B. Hu, and A. Z. Genack, Phys. Rev. Lett. 93, 123901 (2004).
[CrossRef]

A. A. Chabanov, Z. Q. Zhang, and A. Z. Genack, Phys. Rev. Lett. 90, 203903 (2003).
[CrossRef]

Ghulinyan, M.

J. Bertolotti, S. Gottardo, D. S. Wiersma, M. Ghulinyan, and L. Pavesi, Phys. Rev. Lett. 94, 113903 (2005).
[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]

Gjonaj, B.

J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, and A. Lagendijk, Phys. Rev. Lett. 106, 103901 (2011).
[CrossRef]

Goetschy, A.

A. Goetschy and A. D. Stone, arxiv.org/abs/1304.5562.

Gottardo, S.

J. Bertolotti, S. Gottardo, D. S. Wiersma, M. Ghulinyan, and L. Pavesi, Phys. Rev. Lett. 94, 113903 (2005).
[CrossRef]

Hu, B.

P. Sebbah, B. Hu, J. M. Klosner, and A. Z. Genack, Phys. Rev. Lett. 96, 183902 (2006).
[CrossRef]

A. A. Chabanov, B. Hu, and A. Z. Genack, Phys. Rev. Lett. 93, 123901 (2004).
[CrossRef]

Huttmann, G.

A. Vogel, J. Noack, G. Huttmann, and G. Paultauf, Appl. Phys. B 81, 1015 (2005).
[CrossRef]

Johnson, P. M.

J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, and A. Lagendijk, Phys. Rev. Lett. 106, 103901 (2011).
[CrossRef]

Katz, O.

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, Nat. Photonics 5, 372 (2011).
[CrossRef]

Kim, J.

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

Kim, M.

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

Klosner, J. M.

P. Sebbah, B. Hu, J. M. Klosner, and A. Z. Genack, Phys. Rev. Lett. 96, 183902 (2006).
[CrossRef]

Kumar, N.

P. A. Mello, P. Pereyra, and N. Kumar, Ann. Phys. 181, 290 (1988).
[CrossRef]

Lagendijk, A.

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, Nat. Photonics 6, 283 (2012).
[CrossRef]

J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, and A. Lagendijk, Phys. Rev. Lett. 106, 103901 (2011).
[CrossRef]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, Nat. Photonics 4, 320 (2010).
[CrossRef]

Lerosey, G.

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, Nat. Photonics 6, 283 (2012).
[CrossRef]

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

Mello, P. A.

P. A. Mello, P. Pereyra, and N. Kumar, Ann. Phys. 181, 290 (1988).
[CrossRef]

Mirlin, A.

A. Mirlin, Phys. Rep. 326, 259 (2000).
[CrossRef]

Mosk, A. P.

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, Nat. Photonics 6, 283 (2012).
[CrossRef]

J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, and A. Lagendijk, Phys. Rev. Lett. 106, 103901 (2011).
[CrossRef]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, Nat. Photonics 4, 320 (2010).
[CrossRef]

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

I. M. Vellekoop and A. P. Mosk, Opt. Lett. 32, 2309 (2007).
[CrossRef]

Noack, J.

A. Vogel, J. Noack, G. Huttmann, and G. Paultauf, Appl. Phys. B 81, 1015 (2005).
[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).

Paultauf, G.

A. Vogel, J. Noack, G. Huttmann, and G. Paultauf, Appl. Phys. B 81, 1015 (2005).
[CrossRef]

Pavesi, L.

J. Bertolotti, S. Gottardo, D. S. Wiersma, M. Ghulinyan, and L. Pavesi, Phys. Rev. Lett. 94, 113903 (2005).
[CrossRef]

Pendry, J. B.

J. B. Pendry, Adv. Phys. 43, 461 (1994).
[CrossRef]

Pereyra, P.

P. A. Mello, P. Pereyra, and N. Kumar, Ann. Phys. 181, 290 (1988).
[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]

Sebbah, P.

P. Sebbah, B. Hu, J. M. Klosner, and A. Z. Genack, Phys. Rev. Lett. 96, 183902 (2006).
[CrossRef]

Shi, Z.

M. Davy, Z. Shi, J. Wang, and A. Z. Genack, Opt. Express 21, 10367 (2013).
[CrossRef]

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

M. Davy, Z. Shi, and A. Z. Genack, Phys. Rev. B 85, 035105 (2012).
[CrossRef]

Silberberg, Y.

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, Nat. Photonics 5, 372 (2011).
[CrossRef]

Small, E.

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, Nat. Photonics 5, 372 (2011).
[CrossRef]

Stone, A. D.

A. Goetschy and A. D. Stone, arxiv.org/abs/1304.5562.

Tulino, A. M.

A. M. Tulino and S. Verdú, Random Matrix Theory and Wireless Communications (Now Publishers, 2004).

Vellekoop, I. M.

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, Nat. Photonics 4, 320 (2010).
[CrossRef]

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

I. M. Vellekoop and A. P. Mosk, Opt. Lett. 32, 2309 (2007).
[CrossRef]

Verdú, S.

A. M. Tulino and S. Verdú, Random Matrix Theory and Wireless Communications (Now Publishers, 2004).

Vogel, A.

A. Vogel, J. Noack, G. Huttmann, and G. Paultauf, Appl. Phys. B 81, 1015 (2005).
[CrossRef]

Wang, J.

Webb, W. W.

W. R. Zipfel, R. M. Williams, and W. W. Webb, Nat. Biotechnol. 21, 1369 (2003).
[CrossRef]

Wiersma, D. S.

J. Bertolotti, S. Gottardo, D. S. Wiersma, M. Ghulinyan, and L. Pavesi, Phys. Rev. Lett. 94, 113903 (2005).
[CrossRef]

Williams, R. M.

W. R. Zipfel, R. M. Williams, and W. W. Webb, Nat. Biotechnol. 21, 1369 (2003).
[CrossRef]

Yoon, C.

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

Zhang, Z. Q.

A. A. Chabanov, Z. Q. Zhang, and A. Z. Genack, Phys. Rev. Lett. 90, 203903 (2003).
[CrossRef]

Zipfel, W. R.

W. R. Zipfel, R. M. Williams, and W. W. Webb, Nat. Biotechnol. 21, 1369 (2003).
[CrossRef]

Adv. Phys. (1)

J. B. Pendry, Adv. Phys. 43, 461 (1994).
[CrossRef]

Ann. Phys. (1)

P. A. Mello, P. Pereyra, and N. Kumar, Ann. Phys. 181, 290 (1988).
[CrossRef]

Appl. Phys. B (1)

A. Vogel, J. Noack, G. Huttmann, and G. Paultauf, Appl. Phys. B 81, 1015 (2005).
[CrossRef]

Nat. Biotechnol. (1)

W. R. Zipfel, R. M. Williams, and W. W. Webb, Nat. Biotechnol. 21, 1369 (2003).
[CrossRef]

Nat. Photonics (4)

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, Nat. Photonics 6, 283 (2012).
[CrossRef]

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, Nat. Photonics 5, 372 (2011).
[CrossRef]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, Nat. Photonics 4, 320 (2010).
[CrossRef]

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

Nature (1)

J. Wang and A. Z. Genack, Nature 471, 345 (2011).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rep. (1)

A. Mirlin, Phys. Rep. 326, 259 (2000).
[CrossRef]

Phys. Rev. B (1)

M. Davy, Z. Shi, and A. Z. Genack, Phys. Rev. B 85, 035105 (2012).
[CrossRef]

Phys. Rev. Lett. (8)

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

A. A. Chabanov, B. Hu, and A. Z. Genack, Phys. Rev. Lett. 93, 123901 (2004).
[CrossRef]

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]

A. A. Chabanov, Z. Q. Zhang, and A. Z. Genack, Phys. Rev. Lett. 90, 203903 (2003).
[CrossRef]

J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, and A. Lagendijk, Phys. Rev. Lett. 106, 103901 (2011).
[CrossRef]

J. Bertolotti, S. Gottardo, D. S. Wiersma, M. Ghulinyan, and L. Pavesi, Phys. Rev. Lett. 94, 113903 (2005).
[CrossRef]

P. Sebbah, B. Hu, J. M. Klosner, and A. Z. Genack, Phys. Rev. Lett. 96, 183902 (2006).
[CrossRef]

Phys. Today (1)

M. Fink, Phys. Today 50(3), 34 (1997).
[CrossRef]

Solid State Commun. (1)

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

Other (2)

A. Goetschy and A. D. Stone, arxiv.org/abs/1304.5562.

A. M. Tulino and S. Verdú, Random Matrix Theory and Wireless Communications (Now Publishers, 2004).

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Figures (3)

Fig. 1.
Fig. 1.

Spatiotemporal control of wave propagation through a random waveguide. (a) Typical resposne of Iba(t) and the time-of-flight distribution I(t) found by averaging over an ensemble of random samples. The incident pulse is sketched in the dashed blue curve. Phase conjugation is applied numerically to the same configuration as in (a) to focus at t=33 and 40 ns at the center of the output surface in (b) and (c), respectively. The peak intensity in (b) is enhanced by a factor of 26 over the unfocused intensity background at that time, which is in line with the number of independent waveguide modes ~33 for single polarization. The Whittaker–Shannon sampling theorem is used to obtain the high-resolution spatial intensity patterns shown in the insets of (b) and (c).

Fig. 2.
Fig. 2.

Profile of the focused pulse compared with the square modulus of the field correlation function in time and the profile of the incident Gaussian pulse. The focused pulse in Fig. 1(b) has been plotted relative to the time of the peak. All curves are normalized to unity at Δt=0ns.

Fig. 3.
Fig. 3.

Time evolution of M (lower solid curve) and the maximal focusing contrast μ. μ is well described by Eq. (1) after the time of the ballistic arrival, t21ns. At early times, the signal-to-noise ratio is too low to analyze the transmission matrix.

Equations (1)

Equations on this page are rendered with MathJax. Learn more.

μ=1/(1/M1/N),

Metrics