Abstract

Enhanced backscattering (EBS) is investigated for a converging-beam geometry. We find that the functional form of the enhanced backscattering cone is preserved under a change of variables that involves the focal length of the lens and the lens-to-sample distance. In absolute terms, the effect of the lens is that the EBS enhancement cone is lowered in magnitude and narrowed in angle. The experimental data show good agreement with the theory presented, even for tightly focused beams, as long as the illuminated area is larger than the transport mean free path for the random medium.

© 1996 Optical Society of America

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References

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  1. P. Sheng, Scattering and Localization of Classical Waves in Random Media (World Scientific, Singapore, 1990).
    [CrossRef]
  2. S. Etemad, R. Thompson, M. J. Andrejo, Phys. Rev. Lett. 59, 1420 (1987).
    [CrossRef] [PubMed]
  3. I. Edrei, M. Kaveh, Phys. Rev. B 35, 6461 (1987).
    [CrossRef]
  4. P. M. Saulnier, G. H. Watson, Opt. Lett. 17, 946 (1992).
    [CrossRef] [PubMed]
  5. F. C. MacKintosh, S. John, Phys. Rev. B 37, 1884 (1988).
    [CrossRef]
  6. M. Tomita, H. Ikari, Phys. Rev. B 43, 3716 (1991).
    [CrossRef]
  7. A. Dogariu, G. D. Boreman, M. Dogariu, Opt. Lett. 20, 1665 (1995).
    [CrossRef] [PubMed]
  8. M. Kaveh, M. Rosenbluh, I. Edrei, I. Freund, Phys. Rev. Lett. 57, 2049 (1986).
    [CrossRef] [PubMed]
  9. E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, J. Phys. (Paris) 49, 77 (1988).
    [CrossRef]
  10. Spectralon, a diffuse reflectance material; Labsphere, Inc., North Sutton, New Hampshire.

1995 (1)

1992 (1)

1991 (1)

M. Tomita, H. Ikari, Phys. Rev. B 43, 3716 (1991).
[CrossRef]

1988 (2)

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, J. Phys. (Paris) 49, 77 (1988).
[CrossRef]

F. C. MacKintosh, S. John, Phys. Rev. B 37, 1884 (1988).
[CrossRef]

1987 (2)

S. Etemad, R. Thompson, M. J. Andrejo, Phys. Rev. Lett. 59, 1420 (1987).
[CrossRef] [PubMed]

I. Edrei, M. Kaveh, Phys. Rev. B 35, 6461 (1987).
[CrossRef]

1986 (1)

M. Kaveh, M. Rosenbluh, I. Edrei, I. Freund, Phys. Rev. Lett. 57, 2049 (1986).
[CrossRef] [PubMed]

Akkermans, E.

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, J. Phys. (Paris) 49, 77 (1988).
[CrossRef]

Andrejo, M. J.

S. Etemad, R. Thompson, M. J. Andrejo, Phys. Rev. Lett. 59, 1420 (1987).
[CrossRef] [PubMed]

Boreman, G. D.

Dogariu, A.

Dogariu, M.

Edrei, I.

I. Edrei, M. Kaveh, Phys. Rev. B 35, 6461 (1987).
[CrossRef]

M. Kaveh, M. Rosenbluh, I. Edrei, I. Freund, Phys. Rev. Lett. 57, 2049 (1986).
[CrossRef] [PubMed]

Etemad, S.

S. Etemad, R. Thompson, M. J. Andrejo, Phys. Rev. Lett. 59, 1420 (1987).
[CrossRef] [PubMed]

Freund, I.

M. Kaveh, M. Rosenbluh, I. Edrei, I. Freund, Phys. Rev. Lett. 57, 2049 (1986).
[CrossRef] [PubMed]

Ikari, H.

M. Tomita, H. Ikari, Phys. Rev. B 43, 3716 (1991).
[CrossRef]

John, S.

F. C. MacKintosh, S. John, Phys. Rev. B 37, 1884 (1988).
[CrossRef]

Kaveh, M.

I. Edrei, M. Kaveh, Phys. Rev. B 35, 6461 (1987).
[CrossRef]

M. Kaveh, M. Rosenbluh, I. Edrei, I. Freund, Phys. Rev. Lett. 57, 2049 (1986).
[CrossRef] [PubMed]

MacKintosh, F. C.

F. C. MacKintosh, S. John, Phys. Rev. B 37, 1884 (1988).
[CrossRef]

Maret, G.

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, J. Phys. (Paris) 49, 77 (1988).
[CrossRef]

Maynard, R.

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, J. Phys. (Paris) 49, 77 (1988).
[CrossRef]

Rosenbluh, M.

M. Kaveh, M. Rosenbluh, I. Edrei, I. Freund, Phys. Rev. Lett. 57, 2049 (1986).
[CrossRef] [PubMed]

Saulnier, P. M.

Sheng, P.

P. Sheng, Scattering and Localization of Classical Waves in Random Media (World Scientific, Singapore, 1990).
[CrossRef]

Thompson, R.

S. Etemad, R. Thompson, M. J. Andrejo, Phys. Rev. Lett. 59, 1420 (1987).
[CrossRef] [PubMed]

Tomita, M.

M. Tomita, H. Ikari, Phys. Rev. B 43, 3716 (1991).
[CrossRef]

Watson, G. H.

Wolf, P. E.

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, J. Phys. (Paris) 49, 77 (1988).
[CrossRef]

J. Phys. (1)

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, J. Phys. (Paris) 49, 77 (1988).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. B (3)

F. C. MacKintosh, S. John, Phys. Rev. B 37, 1884 (1988).
[CrossRef]

M. Tomita, H. Ikari, Phys. Rev. B 43, 3716 (1991).
[CrossRef]

I. Edrei, M. Kaveh, Phys. Rev. B 35, 6461 (1987).
[CrossRef]

Phys. Rev. Lett. (2)

S. Etemad, R. Thompson, M. J. Andrejo, Phys. Rev. Lett. 59, 1420 (1987).
[CrossRef] [PubMed]

M. Kaveh, M. Rosenbluh, I. Edrei, I. Freund, Phys. Rev. Lett. 57, 2049 (1986).
[CrossRef] [PubMed]

Other (2)

Spectralon, a diffuse reflectance material; Labsphere, Inc., North Sutton, New Hampshire.

P. Sheng, Scattering and Localization of Classical Waves in Random Media (World Scientific, Singapore, 1990).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

EBS experiment in a converging-beam configuration.

Fig. 2
Fig. 2

Calculated EBS line shapes for various diameters of the illuminated area and for the case of infinitely extended plane-wave illumination.

Fig. 3
Fig. 3

(a) Coherent enhancement data corresponding to the Spectralon sample placed at various distances d behind a lens with F = 50 mm and (b) the same data scaled as indicated. Also shown, by the continuous curve, is the EBS shape predicted by the diffusion approximation for l * = l 0 * = 23 μ m.

Fig. 4
Fig. 4

Values of the apparent transport mean free path as obtained by fitting the experimental EBS cones with the theoretical dependence of Eq. (9).

Equations (9)

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I = I 0 + I fluc + I ebs ,
I 0 ( k i , k f ) = l , m S 0 | A l m | 2 ,
I fluc ( k i , k f ) = ( l , m ) S 0 ( l , m ) S 0 A l m * A l m × exp [ i k i ( r l r l ) ] exp [ i k f ( r m r m ) ] ,
I ebs ( k i , k f ) = l , m S | A l m | 2 cos ( k i + k f ) ( r l r m ) .
I ( k i , k f ) = l , m S 0 S P l , m + l , m S P l , m + l , m S P l , m cos ( k i + k f ) ( r l r m ) .
I ( q ) = 1 + ( F d F ) 2 0 D P ( ρ ) exp ( i q ρ ) ρ d ρ ,
I ( q ) = 1 + η 0 D 0 ( F d / F ) P ( ρ ) exp ( i q ρ F F d ) ρ d ρ .
P ( ρ ) = 1 4 π 2 l * l 0 x 1 + l * x ( 1 1 + l x + l l * 1 ) × J 0 ( ρ x ) d x ,
I ( q ) = η { 2 z 0 l * + 1 ( 1 + q l * ) 2 [ 1 + 1 exp ( 2 q z 0 ) q l * ] } ,

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