Abstract

When illuminated by temporally coherent light, multiply scattering media produce speckle patterns that in many situations are unpolarized on spatial averaging. As a result, the underlying field statistics are assumed to be Gaussian and information about them can be extracted from intensity-intensity correlations. However, such an approach cannot be applied to any scattering medium where the interaction leads to partially developed speckle patterns. We present a general procedure to directly measure the field transfer matrix of a linear medium without regard to the scattering regime. Experimental results demonstrate the ability of our procedure to correctly measure field transfer matrices and use them to recover the polarization state of incident illumination.

© 2008 Optical Society of America

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References

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  1. J. W. Goodman, Speckle Phenomena in Optics, 1st ed. (Roberts & Co., Englewood, Co, 2007).
  2. I. Freund, “Stokes-vector reconstruction,” Opt. Lett. 15, 1425–1427 (1990).
    [Crossref] [PubMed]
  3. P. K. Rastogi, ed., Digital Speckle Pattern Interferometry & Related Techniques, 1st ed. (Wiley, New York, 2001).
  4. S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
    [Crossref] [PubMed]
  5. 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]
  6. I. Freund, “Looking through walls and around corners,” Physica A 168, 49–65 (1990).
    [Crossref]
  7. R. H. Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light. I. basic theory: the correlation between photons in coherent beams of radiation,” P. Roy. Soc. L. and A. Mat. 242, 300–324 (1957).
    [Crossref]
  8. C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (John Wiley & Sons, Inc., New York, 1998).
  9. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge UP, New York, 1995).

1990 (2)

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

I. Freund, “Stokes-vector reconstruction,” Opt. Lett. 15, 1425–1427 (1990).
[Crossref] [PubMed]

1988 (2)

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission 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]

1957 (1)

R. H. Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light. I. basic theory: the correlation between photons in coherent beams of radiation,” P. Roy. Soc. L. and A. Mat. 242, 300–324 (1957).
[Crossref]

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (John Wiley & Sons, Inc., New York, 1998).

Brown, R. H.

R. H. Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light. I. basic theory: the correlation between photons in coherent beams of radiation,” P. Roy. Soc. L. and A. Mat. 242, 300–324 (1957).
[Crossref]

Feng, S.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission 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 168, 49–65 (1990).
[Crossref]

I. Freund, “Stokes-vector reconstruction,” Opt. Lett. 15, 1425–1427 (1990).
[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]

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics, 1st ed. (Roberts & Co., Englewood, Co, 2007).

Kane, C.

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

Lee, P. A.

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

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge UP, New York, 1995).

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]

Stone, A. D.

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

Twiss, R. Q.

R. H. Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light. I. basic theory: the correlation between photons in coherent beams of radiation,” P. Roy. Soc. L. and A. Mat. 242, 300–324 (1957).
[Crossref]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge UP, New York, 1995).

Opt. Lett. (1)

P. Roy. Soc. L. and A. Mat. (1)

R. H. Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light. I. basic theory: the correlation between photons in coherent beams of radiation,” P. Roy. Soc. L. and A. Mat. 242, 300–324 (1957).
[Crossref]

Phys. Rev. Lett. (2)

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission 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]

Physica A (1)

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

Other (4)

P. K. Rastogi, ed., Digital Speckle Pattern Interferometry & Related Techniques, 1st ed. (Wiley, New York, 2001).

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (John Wiley & Sons, Inc., New York, 1998).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge UP, New York, 1995).

J. W. Goodman, Speckle Phenomena in Optics, 1st ed. (Roberts & Co., Englewood, Co, 2007).

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Figures (4)

Fig. 1.
Fig. 1.

Random walks through a static random medium and their resulting change in polarization state.

Fig. 2.
Fig. 2.

Example of groups of transfer matrices that can be used to recover an unknown incident field.

Fig. 3.
Fig. 3.

Effective transfer matrices measured for (a) A polarizer oriented at roughly 45° and 135°. (b) A quarter-wave plate rotated by 90° in 15° increments. (c) A multiply scattering solid sample.

Fig. 4.
Fig. 4.

Poincare sphere representations of the polarization states recovered by individual pixel groups (white dots), the geometric centers of the clouds of white points projected onto the surfaces of the spheres (red dots), and the expected polarization states (blue dots).

Equations (10)

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E ( r 2 ) scat , 1 = α ¯ ( r 2 , r 1 ) E ( r 1 ) inc
E ( r ) det = [ Π i α ¯ ( r i + 1 , r i ) ] E ( r 1 ) inc = α ¯ ( r , r 1 ) eff E ( r 1 ) inc
E ( r ) total = n eff α ¯ ( r , r n ) eff E ( r n ) inc = n α ˜ ( r , r n ) E ( r n ) inc = α ˜ ( r ) e ̂ inc
I ( r ) = [ α ˜ 11 ( r ) E x ] 2 + [ α ˜ 12 ( r ) E y ] 2 + 2 α ˜ 11 ( r ) α ˜ 12 ( r ) E x E y cos [ θ + ϕ ˜ ( r ) ] ,
Δ = ( α ˜ 11 2 α ˜ 12 2 ) ( α ˜ 11 2 + α ˜ 12 2 )
σ = 2 α ˜ 11 α ˜ 12 cos ( ϕ ˜ ) ( α ˜ 11 2 + α ˜ 12 2 )
ϕ = 2 α ˜ 11 α ˜ 12 sin ( ϕ ˜ ) ( α ˜ 11 2 + α ˜ 12 2 )
I ( r 1 ) = [ α ˜ 11 ( r 1 ) E x ] 2
I ( r 2 ) = [ α ˜ 12 ( r 2 ) E y ] 2
I ( r 3 ) = [ α ˜ 11 ( r 3 ) E x ] 2 + [ α ˜ 12 ( r 3 ) E y ] 2 + 2 α ˜ 11 ( r 3 ) α ˜ 12 ( r 3 ) E x E y cos [ θ + ϕ ˜ ( r 3 ) ]

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