Abstract

With two nonoverlapping beams incident at different angles on a heavily scattering medium, the spatial correlation of speckle patterns over source position has a beat that is related to the incident angle difference. A model presented explains the measurement. The spatial correlation is shown to decorrelate faster than the beam intensity correlation function and to be sensitive to the incident field profile. Increased scatter results in more rapid decorrelation. This work suggests new opportunities for imaging through scattering media.

© 2010 Optical Society of America

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    [CrossRef] [PubMed]
  2. I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
    [CrossRef] [PubMed]
  3. P. Sebbah, B. Hu, A. Z. Genack, R. Pnini, and B. Shapiro, “Spatial-field correlation: the building block of mesoscopic fluctutations,” Phys. Rev. Lett. 88, 123901 (2002).
    [CrossRef] [PubMed]
  4. P.-E. Wolf and G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
    [CrossRef] [PubMed]
  5. R. Berkovits and M. Kaveh, “Time-reversed memory effects,” Phys. Rev. B 41, 2635–2638 (1990).
    [CrossRef]
  6. R. Berkovits and M. Kaveh, “The vector memory effect for waves,” Europhys. Lett. 13, 97–101 (1990).
    [CrossRef]
  7. I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32, 2309–2311 (2007).
    [CrossRef] [PubMed]
  8. Z. Wang, M. A. Webster, A. M. Weiner, and K. J. Webb, “Polarized temporal impulse response for scattering media from third-order frequency correlations of speckle intensity patterns,” J. Opt. Soc. Am. A 23, 3045–3053 (2006).
    [CrossRef]
  9. J. Goodman, Statistical Optics (Wiley, 1985).
  10. I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory 8, 194–195 (1962).
    [CrossRef]
  11. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
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    [CrossRef]
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2007 (2)

2006 (1)

2002 (1)

P. Sebbah, B. Hu, A. Z. Genack, R. Pnini, and B. Shapiro, “Spatial-field correlation: the building block of mesoscopic fluctutations,” Phys. Rev. Lett. 88, 123901 (2002).
[CrossRef] [PubMed]

1990 (3)

I. Freund, “Looking through walls and around corners,” Physica A (Amsterdam) 168, 49–65 (1990).
[CrossRef]

R. Berkovits and M. Kaveh, “Time-reversed memory effects,” Phys. Rev. B 41, 2635–2638 (1990).
[CrossRef]

R. Berkovits and M. Kaveh, “The vector memory effect for waves,” Europhys. Lett. 13, 97–101 (1990).
[CrossRef]

1988 (2)

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmissions through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[CrossRef] [PubMed]

1985 (1)

P.-E. Wolf and G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

1962 (1)

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory 8, 194–195 (1962).
[CrossRef]

Berkovits, R.

R. Berkovits and M. Kaveh, “Time-reversed memory effects,” Phys. Rev. B 41, 2635–2638 (1990).
[CrossRef]

R. Berkovits and M. Kaveh, “The vector memory effect for waves,” Europhys. Lett. 13, 97–101 (1990).
[CrossRef]

Feng, S.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmissions through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[CrossRef] [PubMed]

Freund, I.

I. Freund, “Looking through walls and around corners,” Physica A (Amsterdam) 168, 49–65 (1990).
[CrossRef]

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[CrossRef] [PubMed]

Genack, A. Z.

P. Sebbah, B. Hu, A. Z. Genack, R. Pnini, and B. Shapiro, “Spatial-field correlation: the building block of mesoscopic fluctutations,” Phys. Rev. Lett. 88, 123901 (2002).
[CrossRef] [PubMed]

Goodman, J.

J. Goodman, Statistical Optics (Wiley, 1985).

Hu, B.

P. Sebbah, B. Hu, A. Z. Genack, R. Pnini, and B. Shapiro, “Spatial-field correlation: the building block of mesoscopic fluctutations,” Phys. Rev. Lett. 88, 123901 (2002).
[CrossRef] [PubMed]

Kane, C.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmissions through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

Kaveh, M.

R. Berkovits and M. Kaveh, “The vector memory effect for waves,” Europhys. Lett. 13, 97–101 (1990).
[CrossRef]

R. Berkovits and M. Kaveh, “Time-reversed memory effects,” Phys. Rev. B 41, 2635–2638 (1990).
[CrossRef]

Lee, P. A.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmissions through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Maret, G.

P.-E. Wolf and G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

Mosk, A. P.

Pnini, R.

P. Sebbah, B. Hu, A. Z. Genack, R. Pnini, and B. Shapiro, “Spatial-field correlation: the building block of mesoscopic fluctutations,” Phys. Rev. Lett. 88, 123901 (2002).
[CrossRef] [PubMed]

Reed, I. S.

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory 8, 194–195 (1962).
[CrossRef]

Rosenbluh, M.

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[CrossRef] [PubMed]

Sebbah, P.

P. Sebbah, B. Hu, A. Z. Genack, R. Pnini, and B. Shapiro, “Spatial-field correlation: the building block of mesoscopic fluctutations,” Phys. Rev. Lett. 88, 123901 (2002).
[CrossRef] [PubMed]

Shapiro, B.

P. Sebbah, B. Hu, A. Z. Genack, R. Pnini, and B. Shapiro, “Spatial-field correlation: the building block of mesoscopic fluctutations,” Phys. Rev. Lett. 88, 123901 (2002).
[CrossRef] [PubMed]

Stone, A. D.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmissions through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

Vellekoop, I. M.

Wang, Z.

Webb, K. J.

Webster, M. A.

Weiner, A. M.

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Wolf, P.-E.

P.-E. Wolf and G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

Europhys. Lett. (1)

R. Berkovits and M. Kaveh, “The vector memory effect for waves,” Europhys. Lett. 13, 97–101 (1990).
[CrossRef]

IRE Trans. Inf. Theory (1)

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory 8, 194–195 (1962).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Lett. (2)

Phys. Rev. B (1)

R. Berkovits and M. Kaveh, “Time-reversed memory effects,” Phys. Rev. B 41, 2635–2638 (1990).
[CrossRef]

Phys. Rev. Lett. (4)

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmissions through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[CrossRef] [PubMed]

P. Sebbah, B. Hu, A. Z. Genack, R. Pnini, and B. Shapiro, “Spatial-field correlation: the building block of mesoscopic fluctutations,” Phys. Rev. Lett. 88, 123901 (2002).
[CrossRef] [PubMed]

P.-E. Wolf and G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

Physica A (Amsterdam) (1)

I. Freund, “Looking through walls and around corners,” Physica A (Amsterdam) 168, 49–65 (1990).
[CrossRef]

Other (2)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

J. Goodman, Statistical Optics (Wiley, 1985).

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Figures (6)

Fig. 1
Fig. 1

(a) Scanned input beam arrangement, showing the joint support that contributes to the intensity correlation at a detector point on the right side of the scattering medium, and the disjoint region, which does not. U ( x ) is the incident field and U ( x + Δ x ) is the same field translated by Δ x . (b) Two spatially separated input beams, with different incident angles, produce a beat in the measured intensity correlation as a function of Δ x . The four speckle patterns correspond to a portion of the image for the measurement of Fig. 3a: upper left, reference position; upper right, decorrelated result (Point A in Fig. 3a]; lower left, second correlation peak (Point B in Fig. 3a]; lower right, second decorrelated valley (Point C in Fig. 3a]. The Δ x = 0 and Δ x = 0.24 mm patterns are virtually identical.

Fig. 2
Fig. 2

Experimental setup for the correlation over source position with two beams from an 850 nm laser. The distance between M2 and the left surface of the scattering medium is D (1422, 1118, and 934 mm used), the center-to-center distance between the beams at this surface is d (6.86 and 4.71 mm , for example), and the angle between the beams is θ. In the single-beam experiments, the dashed beam was removed. The unrestricted beam had a FWHM of about 1 mm . The spatial filter controls the speckle size and L2 images a small spot on the right surface of the scattering medium onto the CCD camera. A small region on the right side of the scattering medium is imaged to P1 and then magnified by L2 for imaging at the CCD camera. The “Scan over x” results in Δ x .

Fig. 3
Fig. 3

Spatial correlation for two incident beams with an angular difference: referring to Fig. 2, D = 1118 mm , d = 4.32 mm , and a thickness of (a) 9 mm and (b) 12 mm ; (c) D = 934 mm , with d = 6.86 mm and d = 4.71 mm , and a 9 mm sample thickness. The unrestricted beam had a FWHM of about 1 mm , and μ s = 4 cm 1 . The labels A, B, and C indicate points where the speckle patterns of Fig. 1b were captured.

Fig. 4
Fig. 4

(a) Measured modulation period as a function of λ / d and for different D (points) and straight line fits. (b) Measured modulation period as a function of λ / θ (points) and the predicted value (solid line).

Fig. 5
Fig. 5

(a) Spatial speckle correlation over source position for illumination through a circular aperture of varying size. For reference, the calculated aperture/beam autocorrelation function is plotted. (b) Comparison of a single circular aperture of diameter 0.35 mm and two apertures of this size with a 0.5 mm center-to-center distance. The scattering sample was 9 mm thick and had μ s = 4 cm 1 .

Fig. 6
Fig. 6

(a) Spatial correlation over the source coordinate for linear co- and cross-polarized light and various sample thicknesses for a scattering sample with μ s = 4 cm 1 . (b) Copolarized intensity correlation for samples having μ s = 4 cm 1 and μ s = 14 cm 1 and thicknesses of 6 and 12 mm .

Equations (6)

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I ˜ [ U ( r ) ] I ˜ [ U ( r + Δ r ) ] = | E ˜ [ U ( r ) ] E ˜ * [ U ( r + Δ r ) ] | 2 ,
E ˜ [ U ( r ) ] E ˜ * [ U ( r + Δ r ) ] = r d r G ( r d , r ) U ( r ) r d r G * ( r d , r ) U * ( r + Δ r ) ,
E ˜ [ U ( r ) ] E ˜ * [ U ( r + Δ r ) ] = j s d r U ( r ) U * ( r + Δ r ) | G ( r d , r ) | 2 ,
E ˜ [ U ( r ) ] E ˜ * [ U ( r + Δ r ) ] = e i k · Δ r d r U 0 ( r ) U 0 * ( r + Δ r ) | G ( r d , r ) | 2 = e i k · Δ r P ( Δ r ) ,
E ˜ [ U ( x ) ] E ˜ * [ U ( x + Δ x ) ] = 1 2 ( E ˜ 1 [ U 1 ( x ) ] + E ˜ 2 [ U 2 ( x ) ] ) × ( E ˜ 1 * [ U 1 ( x + Δ x ) ] + E ˜ 2 * [ U 2 ( x + Δ x ) ] ) = 1 2 { E ˜ 1 [ U 1 ( x ) ] E ˜ 1 * [ U 1 ( x + Δ x ) ] + E ˜ 2 [ U 2 ( x ) ] E ˜ 2 * [ U 2 ( x + Δ x ) ] } ,
| E ˜ [ U ( x ) ] E ˜ * [ U ( x + Δ x ) ] | 2 = 1 4 | P ( Δ x ) | 2 | e i k x 1 Δ x + e i k x 2 Δ x | 2 = | P ( Δ x ) | 2 [ 1 + cos ( Δ k x Δ x ) ] 2 ,

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