Abstract

We present a simple scheme to determine the diffusion properties of a thin slab of strongly scattering material by measuring the speckle contrast resulting from the transmission of a femtosecond pulse with controlled bandwidth. In contrast with previous methods, our scheme does not require time measurements nor interferometry. It is well adapted to the characterization of samples for pulse shaping, nonlinear excitation through scattering media, and biological imaging.

© 2011 Optical Society of America

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References

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  1. P. Sebbah, Waves and Imaging through Complex Media (Kluwer Academic, 2001).
  2. J. W. Goodman, J. Opt. Soc. Am. 66, 1145 (1976).
    [CrossRef]
  3. P. Wolf, G. Maret, E. Akkermans, and R. Maynard, J. Phys. 49, 63 (1988).
    [CrossRef]
  4. A. Z. Genack, Phys. Rev. Lett. 58, 2043 (1987).
    [CrossRef] [PubMed]
  5. D. S. Wiersma, A. Muzzi, M. Colocci, and R. Righini, Phys. Rev. E 62, 6681 (2000).
    [CrossRef]
  6. J. M. Drake and A. Z. Genack, Phys. Rev. Lett. 63, 259(1989).
    [CrossRef] [PubMed]
  7. W. Cai, B. Das, F. Liu, M. Zevallos, M. Lax, and R. Alfano, Proc. Natl. Acad. Sci. USA 93, 13561 (1996).
    [CrossRef] [PubMed]
  8. I. M. Vellekoop, P. Lodahl, and A. Lagendijk, Phys. Rev. E 71, 056604 (2005).
    [CrossRef]
  9. S. Faez, P. M. Johnson, and A. Lagendijk, Phys. Rev. Lett. 103, 053903 (2009).
    [CrossRef] [PubMed]
  10. M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, and A. Tip, Phys. Rev. Lett. 66, 3132 (1991).
    [CrossRef] [PubMed]
  11. R. Sapienza, P. D. García, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, Phys. Rev. Lett. 99, 233902 (2007).
    [CrossRef]
  12. D. J. Thouless, Phys. Rev. Lett. 39, 1167 (1977).
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  13. R. Landauer and M. Buttiker, Phys. Rev. B 36, 6255(1987).
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  15. J. D. McKinney, M. A. Webster, K. J. Webb, and A. M. Weiner, Opt. Lett. 25, 4 (2000).
    [CrossRef]
  16. J.-M. Tualle, H. L. Nghiêm, C. Schäfauer, P. Berthaud, E. Tinet, D. Ettori, and S. Avrillier, Opt. Lett. 30, 50 (2005).
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  18. W.-L. Liu, R.-X. Gao, and S.-L. Qu, Chin. Phys. B 19, 024204(2010).
    [CrossRef]
  19. P. M. Johnson, A. Imhof, B. P. J. Bret, J. G. Rivas, and A. Lagendijk, Phys. Rev. E 68, 016604 (2003).
    [CrossRef]
  20. E. Tal and Y. Silberberg, Opt. Lett. 31, 3529 (2006).
    [CrossRef] [PubMed]
  21. P. García, R. Sapienza, Á. Blanco, and C. López, Adv. Mater. 19, 2597 (2007).
    [CrossRef]
  22. A. Monmayrant, S. Weber, and B. Chatel, J. Phys. B 43, 103001 (2010).
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  24. N. Cherroret and A. Buchleitner, Phys. Rev. A 83, 033827(2011).
    [CrossRef]
  25. J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, and A. Lagendijk, Phys. Rev. Lett. 106, 103901 (2011).
    [CrossRef] [PubMed]
  26. O. Katz, Y. Bromberg, E. Small, and Y. Silberberg, Nat. Photon. 5, 372 (2011).
    [CrossRef]
  27. D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, “Shaping speckles: spatio-temporal focussing of an ultrafast pulse through a multiply scattering medium,” arXiv:1101.0976v1 (2011).

2011

N. Cherroret and A. Buchleitner, Phys. Rev. A 83, 033827(2011).
[CrossRef]

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

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

2010

W.-L. Liu, R.-X. Gao, and S.-L. Qu, Chin. Phys. B 19, 024204(2010).
[CrossRef]

A. Monmayrant, S. Weber, and B. Chatel, J. Phys. B 43, 103001 (2010).
[CrossRef]

2009

S. Faez, P. M. Johnson, and A. Lagendijk, Phys. Rev. Lett. 103, 053903 (2009).
[CrossRef] [PubMed]

2007

R. Sapienza, P. D. García, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, Phys. Rev. Lett. 99, 233902 (2007).
[CrossRef]

P. García, R. Sapienza, Á. Blanco, and C. López, Adv. Mater. 19, 2597 (2007).
[CrossRef]

2006

2005

2003

P. M. Johnson, A. Imhof, B. P. J. Bret, J. G. Rivas, and A. Lagendijk, Phys. Rev. E 68, 016604 (2003).
[CrossRef]

2000

J. D. McKinney, M. A. Webster, K. J. Webb, and A. M. Weiner, Opt. Lett. 25, 4 (2000).
[CrossRef]

D. S. Wiersma, A. Muzzi, M. Colocci, and R. Righini, Phys. Rev. E 62, 6681 (2000).
[CrossRef]

1997

1996

W. Cai, B. Das, F. Liu, M. Zevallos, M. Lax, and R. Alfano, Proc. Natl. Acad. Sci. USA 93, 13561 (1996).
[CrossRef] [PubMed]

1995

1991

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, and A. Tip, Phys. Rev. Lett. 66, 3132 (1991).
[CrossRef] [PubMed]

1989

1988

P. Wolf, G. Maret, E. Akkermans, and R. Maynard, J. Phys. 49, 63 (1988).
[CrossRef]

1987

A. Z. Genack, Phys. Rev. Lett. 58, 2043 (1987).
[CrossRef] [PubMed]

R. Landauer and M. Buttiker, Phys. Rev. B 36, 6255(1987).
[CrossRef]

1977

D. J. Thouless, Phys. Rev. Lett. 39, 1167 (1977).
[CrossRef]

1976

Akkermans, E.

P. Wolf, G. Maret, E. Akkermans, and R. Maynard, J. Phys. 49, 63 (1988).
[CrossRef]

Alfano, R.

W. Cai, B. Das, F. Liu, M. Zevallos, M. Lax, and R. Alfano, Proc. Natl. Acad. Sci. USA 93, 13561 (1996).
[CrossRef] [PubMed]

Aulbach, J.

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

Austin, D. R.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, “Shaping speckles: spatio-temporal focussing of an ultrafast pulse through a multiply scattering medium,” arXiv:1101.0976v1 (2011).

Avrillier, S.

Berthaud, P.

Bertolotti, J.

R. Sapienza, P. D. García, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, Phys. Rev. Lett. 99, 233902 (2007).
[CrossRef]

Blanco, A.

R. Sapienza, P. D. García, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, Phys. Rev. Lett. 99, 233902 (2007).
[CrossRef]

Blanco, Á.

P. García, R. Sapienza, Á. Blanco, and C. López, Adv. Mater. 19, 2597 (2007).
[CrossRef]

Bondareff, P.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, “Shaping speckles: spatio-temporal focussing of an ultrafast pulse through a multiply scattering medium,” arXiv:1101.0976v1 (2011).

Bret, B. P. J.

P. M. Johnson, A. Imhof, B. P. J. Bret, J. G. Rivas, and A. Lagendijk, Phys. Rev. E 68, 016604 (2003).
[CrossRef]

Bromberg, Y.

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

Buchleitner, A.

N. Cherroret and A. Buchleitner, Phys. Rev. A 83, 033827(2011).
[CrossRef]

Buttiker, M.

R. Landauer and M. Buttiker, Phys. Rev. B 36, 6255(1987).
[CrossRef]

Cai, W.

W. Cai, B. Das, F. Liu, M. Zevallos, M. Lax, and R. Alfano, Proc. Natl. Acad. Sci. USA 93, 13561 (1996).
[CrossRef] [PubMed]

Chance, B.

Chatel, B.

A. Monmayrant, S. Weber, and B. Chatel, J. Phys. B 43, 103001 (2010).
[CrossRef]

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, “Shaping speckles: spatio-temporal focussing of an ultrafast pulse through a multiply scattering medium,” arXiv:1101.0976v1 (2011).

Cherroret, N.

N. Cherroret and A. Buchleitner, Phys. Rev. A 83, 033827(2011).
[CrossRef]

Colocci, M.

D. S. Wiersma, A. Muzzi, M. Colocci, and R. Righini, Phys. Rev. E 62, 6681 (2000).
[CrossRef]

Das, B.

W. Cai, B. Das, F. Liu, M. Zevallos, M. Lax, and R. Alfano, Proc. Natl. Acad. Sci. USA 93, 13561 (1996).
[CrossRef] [PubMed]

Drake, J. M.

J. M. Drake and A. Z. Genack, Phys. Rev. Lett. 63, 259(1989).
[CrossRef] [PubMed]

Ettori, D.

Faez, S.

S. Faez, P. M. Johnson, and A. Lagendijk, Phys. Rev. Lett. 103, 053903 (2009).
[CrossRef] [PubMed]

Gao, R.-X.

W.-L. Liu, R.-X. Gao, and S.-L. Qu, Chin. Phys. B 19, 024204(2010).
[CrossRef]

García, P.

P. García, R. Sapienza, Á. Blanco, and C. López, Adv. Mater. 19, 2597 (2007).
[CrossRef]

García, P. D.

R. Sapienza, P. D. García, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, Phys. Rev. Lett. 99, 233902 (2007).
[CrossRef]

Genack, A. Z.

J. M. Drake and A. Z. Genack, Phys. Rev. Lett. 63, 259(1989).
[CrossRef] [PubMed]

A. Z. Genack, Phys. Rev. Lett. 58, 2043 (1987).
[CrossRef] [PubMed]

Gigan, S.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, “Shaping speckles: spatio-temporal focussing of an ultrafast pulse through a multiply scattering medium,” arXiv:1101.0976v1 (2011).

Gjonaj, B.

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

Goodman, J. W.

Imhof, A.

P. M. Johnson, A. Imhof, B. P. J. Bret, J. G. Rivas, and A. Lagendijk, Phys. Rev. E 68, 016604 (2003).
[CrossRef]

Johnson, P. M.

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

S. Faez, P. M. Johnson, and A. Lagendijk, Phys. Rev. Lett. 103, 053903 (2009).
[CrossRef] [PubMed]

P. M. Johnson, A. Imhof, B. P. J. Bret, J. G. Rivas, and A. Lagendijk, Phys. Rev. E 68, 016604 (2003).
[CrossRef]

Katz, O.

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

Lagendijk, A.

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

S. Faez, P. M. Johnson, and A. Lagendijk, Phys. Rev. Lett. 103, 053903 (2009).
[CrossRef] [PubMed]

I. M. Vellekoop, P. Lodahl, and A. Lagendijk, Phys. Rev. E 71, 056604 (2005).
[CrossRef]

P. M. Johnson, A. Imhof, B. P. J. Bret, J. G. Rivas, and A. Lagendijk, Phys. Rev. E 68, 016604 (2003).
[CrossRef]

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, and A. Tip, Phys. Rev. Lett. 66, 3132 (1991).
[CrossRef] [PubMed]

Landauer, R.

R. Landauer and M. Buttiker, Phys. Rev. B 36, 6255(1987).
[CrossRef]

Lax, M.

W. Cai, B. Das, F. Liu, M. Zevallos, M. Lax, and R. Alfano, Proc. Natl. Acad. Sci. USA 93, 13561 (1996).
[CrossRef] [PubMed]

Liu, F.

W. Cai, B. Das, F. Liu, M. Zevallos, M. Lax, and R. Alfano, Proc. Natl. Acad. Sci. USA 93, 13561 (1996).
[CrossRef] [PubMed]

Liu, W.-L.

W.-L. Liu, R.-X. Gao, and S.-L. Qu, Chin. Phys. B 19, 024204(2010).
[CrossRef]

Lodahl, P.

I. M. Vellekoop, P. Lodahl, and A. Lagendijk, Phys. Rev. E 71, 056604 (2005).
[CrossRef]

López, C.

P. García, R. Sapienza, Á. Blanco, and C. López, Adv. Mater. 19, 2597 (2007).
[CrossRef]

R. Sapienza, P. D. García, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, Phys. Rev. Lett. 99, 233902 (2007).
[CrossRef]

Maret, G.

P. Wolf, G. Maret, E. Akkermans, and R. Maynard, J. Phys. 49, 63 (1988).
[CrossRef]

Martín, M. D.

R. Sapienza, P. D. García, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, Phys. Rev. Lett. 99, 233902 (2007).
[CrossRef]

Matsumoto, T.

Maynard, R.

P. Wolf, G. Maret, E. Akkermans, and R. Maynard, J. Phys. 49, 63 (1988).
[CrossRef]

McCabe, D. J.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, “Shaping speckles: spatio-temporal focussing of an ultrafast pulse through a multiply scattering medium,” arXiv:1101.0976v1 (2011).

McKinney, J. D.

Monmayrant, A.

A. Monmayrant, S. Weber, and B. Chatel, J. Phys. B 43, 103001 (2010).
[CrossRef]

Mosk, A. P.

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

Muzzi, A.

D. S. Wiersma, A. Muzzi, M. Colocci, and R. Righini, Phys. Rev. E 62, 6681 (2000).
[CrossRef]

Nghiêm, H. L.

Patterson, M. S.

Qu, S.-L.

W.-L. Liu, R.-X. Gao, and S.-L. Qu, Chin. Phys. B 19, 024204(2010).
[CrossRef]

Righini, R.

D. S. Wiersma, A. Muzzi, M. Colocci, and R. Righini, Phys. Rev. E 62, 6681 (2000).
[CrossRef]

Rivas, J. G.

P. M. Johnson, A. Imhof, B. P. J. Bret, J. G. Rivas, and A. Lagendijk, Phys. Rev. E 68, 016604 (2003).
[CrossRef]

Sapienza, R.

P. García, R. Sapienza, Á. Blanco, and C. López, Adv. Mater. 19, 2597 (2007).
[CrossRef]

R. Sapienza, P. D. García, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, Phys. Rev. Lett. 99, 233902 (2007).
[CrossRef]

Schäfauer, C.

Sebbah, P.

P. Sebbah, Waves and Imaging through Complex Media (Kluwer Academic, 2001).

Silberberg, Y.

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

E. Tal and Y. Silberberg, Opt. Lett. 31, 3529 (2006).
[CrossRef] [PubMed]

Small, E.

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

Tajalli, A.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, “Shaping speckles: spatio-temporal focussing of an ultrafast pulse through a multiply scattering medium,” arXiv:1101.0976v1 (2011).

Tal, E.

Thompson, C. A.

Thouless, D. J.

D. J. Thouless, Phys. Rev. Lett. 39, 1167 (1977).
[CrossRef]

Tinet, E.

Tip, A.

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, and A. Tip, Phys. Rev. Lett. 66, 3132 (1991).
[CrossRef] [PubMed]

Tomita, M.

Tualle, J.-M.

van Albada, M. P.

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, and A. Tip, Phys. Rev. Lett. 66, 3132 (1991).
[CrossRef] [PubMed]

van Tiggelen, B. A.

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, and A. Tip, Phys. Rev. Lett. 66, 3132 (1991).
[CrossRef] [PubMed]

Vellekoop, I. M.

I. M. Vellekoop, P. Lodahl, and A. Lagendijk, Phys. Rev. E 71, 056604 (2005).
[CrossRef]

Viña, L.

R. Sapienza, P. D. García, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, Phys. Rev. Lett. 99, 233902 (2007).
[CrossRef]

Walmsley, I. A.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, “Shaping speckles: spatio-temporal focussing of an ultrafast pulse through a multiply scattering medium,” arXiv:1101.0976v1 (2011).

Webb, K. J.

Weber, S.

A. Monmayrant, S. Weber, and B. Chatel, J. Phys. B 43, 103001 (2010).
[CrossRef]

Webster, M. A.

Weiner, A. M.

Wiersma, D. S.

R. Sapienza, P. D. García, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, Phys. Rev. Lett. 99, 233902 (2007).
[CrossRef]

D. S. Wiersma, A. Muzzi, M. Colocci, and R. Righini, Phys. Rev. E 62, 6681 (2000).
[CrossRef]

Wilson, B. C.

Wolf, P.

P. Wolf, G. Maret, E. Akkermans, and R. Maynard, J. Phys. 49, 63 (1988).
[CrossRef]

Zevallos, M.

W. Cai, B. Das, F. Liu, M. Zevallos, M. Lax, and R. Alfano, Proc. Natl. Acad. Sci. USA 93, 13561 (1996).
[CrossRef] [PubMed]

Adv. Mater.

P. García, R. Sapienza, Á. Blanco, and C. López, Adv. Mater. 19, 2597 (2007).
[CrossRef]

Appl. Opt.

Chin. Phys. B

W.-L. Liu, R.-X. Gao, and S.-L. Qu, Chin. Phys. B 19, 024204(2010).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

J. Phys.

P. Wolf, G. Maret, E. Akkermans, and R. Maynard, J. Phys. 49, 63 (1988).
[CrossRef]

J. Phys. B

A. Monmayrant, S. Weber, and B. Chatel, J. Phys. B 43, 103001 (2010).
[CrossRef]

Nat. Photon.

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

Opt. Lett.

Phys. Rev. A

N. Cherroret and A. Buchleitner, Phys. Rev. A 83, 033827(2011).
[CrossRef]

Phys. Rev. B

R. Landauer and M. Buttiker, Phys. Rev. B 36, 6255(1987).
[CrossRef]

Phys. Rev. E

P. M. Johnson, A. Imhof, B. P. J. Bret, J. G. Rivas, and A. Lagendijk, Phys. Rev. E 68, 016604 (2003).
[CrossRef]

D. S. Wiersma, A. Muzzi, M. Colocci, and R. Righini, Phys. Rev. E 62, 6681 (2000).
[CrossRef]

I. M. Vellekoop, P. Lodahl, and A. Lagendijk, Phys. Rev. E 71, 056604 (2005).
[CrossRef]

Phys. Rev. Lett.

S. Faez, P. M. Johnson, and A. Lagendijk, Phys. Rev. Lett. 103, 053903 (2009).
[CrossRef] [PubMed]

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, and A. Tip, Phys. Rev. Lett. 66, 3132 (1991).
[CrossRef] [PubMed]

R. Sapienza, P. D. García, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, Phys. Rev. Lett. 99, 233902 (2007).
[CrossRef]

D. J. Thouless, Phys. Rev. Lett. 39, 1167 (1977).
[CrossRef]

J. M. Drake and A. Z. Genack, Phys. Rev. Lett. 63, 259(1989).
[CrossRef] [PubMed]

A. Z. Genack, Phys. Rev. Lett. 58, 2043 (1987).
[CrossRef] [PubMed]

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

Proc. Natl. Acad. Sci. USA

W. Cai, B. Das, F. Liu, M. Zevallos, M. Lax, and R. Alfano, Proc. Natl. Acad. Sci. USA 93, 13561 (1996).
[CrossRef] [PubMed]

Other

P. Sebbah, Waves and Imaging through Complex Media (Kluwer Academic, 2001).

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, “Shaping speckles: spatio-temporal focussing of an ultrafast pulse through a multiply scattering medium,” arXiv:1101.0976v1 (2011).

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Figures (2)

Fig. 1
Fig. 1

Experimental setup with variable slit to control the spectral bandwidth, typical measured spectrum, and speckle image on the CCD camera.

Fig. 2
Fig. 2

Contrast of the speckle for three different thicknesses of ZnO. Dots are the experimental measurements; solid curves are from calculation following the model developed by [23]. Inset: inverse of the transmission for ZnO layers of different thicknesses L in micrometers.

Equations (3)

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

T L ( t ) = ( 4 π D ) 1 / 2 × t 3 / 2 × { ( L t ) e ( L t ) 2 4 D t ( L + t ) e ( L + t ) 2 4 D t + ( 3 L t ) e ( 3 L t ) 2 4 D t ( 3 L + t ) e ( 3 L + t ) 2 4 D t } .
C = 0 1 S ( λ ) { 0 0 S ( λ ) S ( λ ) × | f ( λ , λ , T L ˜ ( t ) ) | 2 d λ d λ } 1 / 2 d λ ,
f = 0 c T L ˜ ( t ) × exp [ i 2 π c t ( 1 λ 1 λ ) ] d t .

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