Abstract

Polarization-difference (PD) imaging techniques have been demonstrated to improve the detectability of target features that are embedded in scattering media. The improved detectability occurs for both passive imaging in moderately scattering media (<5 optical depths) and active imaging in more highly scattering media. These improvements are relative to what is possible with equivalent polarization-blind, polarization-sum (PS) imaging under the same conditions. In this investigation, the point-spread functions (PSF’s) for passive PS and PD imaging in single-scattering media are studied analytically, and Monte Carlo simulations are used to study the PSF’s in single- and moderately multiple-scattering media. The results indicate that the PD PSF can be significantly narrower than the corresponding PS PSF, implying that better images of target features with high-spatial-frequency information can be obtained by using differential polarimetry in scattering media. Although the analysis was performed for passive imaging at moderate optical depths, the results lend insight into experiments that have been performed in more highly scattering media with active imaging methods to help mitigate the effects of multiple scattering.

© 2000 Optical Society of America

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  1. B. A. Swartz, J. D. Cummings, “Laser range-gated underwater imaging including polarization discrimination,” in Underwater Imaging, Photography, and Visibility, R. W. Spinrad, ed., Proc. SPIE1537, 42–56 (1991).
    [CrossRef]
  2. S. K. Gayen, R. R. Alfano, “Emerging optical biomedical techniques,” Opt. Photon. News 7, 16–22 (March1996).
    [CrossRef]
  3. S. C. W. Hyde, N. P. Barry, R. Jones, J. C. Dainty, P. M. W. French, “High resolution depth resolved imaging through scattering media using time resolved holography,” Opt. Commun. 122, 111–116 (1996).
    [CrossRef]
  4. M. Kempe, W. Rudolph, E. Welsch, “Comparative study of confocal and heterodyne microscopy for imaging through scattering media,” J. Opt. Soc. Am. A 13, 46–52 (1996).
    [CrossRef]
  5. A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (March1995).
    [CrossRef]
  6. G. D. Gilbert, J. C. Pernicka, “Improvement of underwater visibility by reduction of backscatter with a circular polarization technique,” in Underwater Photo Optics I, A. B. Dember, ed., Proc. SPIE7, A-III-1–A-III-10 (1966).
  7. J. S. Tyo, M. P. Rowe, E. N. Pugh, N. Engheta, “Target detection in optically scattering media by polarization-difference imaging,” Appl. Opt. 35, 1855–1870 (1996).
    [CrossRef] [PubMed]
  8. S. G. Demos, R. R. Alfano, “Temporal gating in highly scattering media by the degree of optical polarization,” Opt. Lett. 21, 161–163 (1996).
    [CrossRef] [PubMed]
  9. W. B. Wang, S. G. Demos, J. Ali, R. R. Alfano, “Imaging fluorescent objects embedded inside animal tissues using polarization difference technique,” Opt. Commun. 142, 161–166 (1997).
    [CrossRef]
  10. J. F. de Boer, T. E. Milner, M. J. C. van Gemert, J. S. Nelson, “Two-dimensional birefringence imaging in biological tissue by polarization-sensitive optical coherence tomography,” Opt. Lett. 22, 934–936 (1997).
    [CrossRef] [PubMed]
  11. J. M. Harris, “The influence of random media on the propagation and depolarization of electromagnetic waves,” Ph.D. dissertation (California Institute of Technology, Pasadena, Calif., 1980).
  12. Y. Kuga, A. Ishimaru, “Modulation transfer function and image transmission through randomly distributed spherical particles,” J. Opt. Soc. Am. A 2, 2330–2335 (1985).
    [CrossRef]
  13. Y. Kuga, A. Ishimaru, “Modulation transfer function of layered inhomogeneous random media using the small-angle approximation,” Appl. Opt. 25, 4382–4385 (1986).
    [CrossRef]
  14. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  15. Q. Ma, A. Ishimaru, “Propagation and depolarization of an arbitrarily polarized wave obliquely incident on a slab of random medium,” IEEE Trans. Antennas Propag. 39, 1626–1632 (1991).
    [CrossRef]
  16. C. Brüning, R. Schmidt, W. Alpers, “Estimation of the ocean wave-radar modulation transfer function from synthetic aperture radar imagery,” J. Geophys. Res. C 99, 9803–9815 (1994).
    [CrossRef]
  17. A. Schmidt, V. Wismann, R. Romeiser, W. Alpers, “Simultaneous measurements of the ocean wave-radar modulation transfer function at L, C, and X bands from the research platform Nordsee,” J. Geophys. Res. C 100, 8815–8827 (1995).
    [CrossRef]
  18. K. Bhattacharya, A. Ghosh, A. K. Chakraborty, “Vector wave imagery with a lens masked by polarizers,” J. Mod. Opt. 40, 379–390 (1993).
    [CrossRef]
  19. J. S. Tyo, “Polarization difference imaging,” Ph.D. dissertation (University of Pennsylvania, Philadelphia, Pa., 1997).
  20. S. G. Demos, W. B. Wang, R. R. Alfano, “Imaging objects hidden in scattering media with fluorescence polarization preservation of contrast agents,” Appl. Opt. 37, 792–797 (1998).
    [CrossRef]
  21. M. P. Rowe, E. N. Pugh, J. S. Tyo, N. Engheta, “Polarization-difference imaging: a biologically inspired technique for imaging in scattering media,” Opt. Lett. 20, 608–610 (1995).
    [CrossRef] [PubMed]
  22. M. P. Silverman, W. Strange, “Light scattering from optically active and inactive turbid media,” in Proceedings of the IS&T/OSA Conference on Optics and Imaging in the Information Age (Society for Image Science and Technology, Springfield, Va., 1996), pp. 172–180.
  23. M. P. Silverman, W. Strange, “Object delineation within turbid media by backscattering of phase-modulated light,” Opt. Commun. 144, 7–11 (1997).
    [CrossRef]
  24. A. Ishimaru, Wave Propagation in Random Media (Academic, San Diego, Calif., 1978), Vol. 1, Chap. 4.
  25. G. E. Anderson, F. Liu, R. R. Alfano, “Microscope imaging through highly scattering media,” Opt. Lett. 19, 981–983 (1994).
    [CrossRef] [PubMed]
  26. M. Gu, T. Tannous, J. R. Sheppard, “Effect of an annular pupil on confocal imaging through highly scattering media,” Opt. Lett. 21, 312–314 (1996).
    [CrossRef] [PubMed]
  27. The apertures can be projected onto the intermediate plane without changing the results as long as the extent of the object is much smaller than the projected size of the limiting aperture of the system. When this condition is met, the vignetting can be ignored (see Ref. 14, chap. 5).
  28. For simplicity, the host medium in which the scatterers are embedded is assumed to be free space, although in general it is some other medium.
  29. E⇀sc(sˆ) must fall off as 1/r as one travels along the direction sˆ; the formulation in Eq. (2) gives the relative amplitude scattered in any direction at some constant distance from the scatterer. The 1/r fall-off will be taken into account, along with the system parameters, in the analysis that follows.
  30. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Chaps. 7 and 9.
  31. rˆ and θˆ refer to the spherical unit vectors at the position rˆn with respect to the Cartesian coordinate system shown in Fig. 2.
  32. Strictly speaking, Eq. (12) does not follow directly from Eq. (11). Equation (11) states that there is an ideal E-field point source given by κ2E⇀δ(xf+xn)δ(zf+zn). This source can also be thought of as an ideal intensity point source given by κ2|E⇀|2δ(xf+xn)δ(zf+zn).
  33. The term polarization sum was introduced in Ref. 21. It is meant to differentiate between a true, polarization-blind image where intensity alone is measured and a sum image formed by adding the intensities obtained at orthogonal polarizations. The two concepts are completely equivalent, and the term PS image is retained to provide the reader with information concerning how a specific image was formed.
  34. L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  35. M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
    [CrossRef] [PubMed]
  36. L. L. Carter, E. D. Cashwell, Particle Transport Simulation with the Monte-Carlo Method (Technical Information Center, Energy Research and Development Association, Oak Ridge, Tenn., 1975).
  37. B. R. Frieden, Probability, Statistical Optics, and Data Testing, 2nd ed., (Springer-Verlag, New York, 1991), Chap. 7.
  38. This experiment investigates the PSF that is due to a linearly polarized source. The portion of the radiation that is unpolarized will not be imaged by PDI.
  39. L. M. Lampert, Modern Dairy Products (Chemical Publishing Co., New York, 1965).

1998

1997

M. P. Silverman, W. Strange, “Object delineation within turbid media by backscattering of phase-modulated light,” Opt. Commun. 144, 7–11 (1997).
[CrossRef]

W. B. Wang, S. G. Demos, J. Ali, R. R. Alfano, “Imaging fluorescent objects embedded inside animal tissues using polarization difference technique,” Opt. Commun. 142, 161–166 (1997).
[CrossRef]

J. F. de Boer, T. E. Milner, M. J. C. van Gemert, J. S. Nelson, “Two-dimensional birefringence imaging in biological tissue by polarization-sensitive optical coherence tomography,” Opt. Lett. 22, 934–936 (1997).
[CrossRef] [PubMed]

1996

1995

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (March1995).
[CrossRef]

M. P. Rowe, E. N. Pugh, J. S. Tyo, N. Engheta, “Polarization-difference imaging: a biologically inspired technique for imaging in scattering media,” Opt. Lett. 20, 608–610 (1995).
[CrossRef] [PubMed]

A. Schmidt, V. Wismann, R. Romeiser, W. Alpers, “Simultaneous measurements of the ocean wave-radar modulation transfer function at L, C, and X bands from the research platform Nordsee,” J. Geophys. Res. C 100, 8815–8827 (1995).
[CrossRef]

1994

G. E. Anderson, F. Liu, R. R. Alfano, “Microscope imaging through highly scattering media,” Opt. Lett. 19, 981–983 (1994).
[CrossRef] [PubMed]

C. Brüning, R. Schmidt, W. Alpers, “Estimation of the ocean wave-radar modulation transfer function from synthetic aperture radar imagery,” J. Geophys. Res. C 99, 9803–9815 (1994).
[CrossRef]

1993

K. Bhattacharya, A. Ghosh, A. K. Chakraborty, “Vector wave imagery with a lens masked by polarizers,” J. Mod. Opt. 40, 379–390 (1993).
[CrossRef]

1991

Q. Ma, A. Ishimaru, “Propagation and depolarization of an arbitrarily polarized wave obliquely incident on a slab of random medium,” IEEE Trans. Antennas Propag. 39, 1626–1632 (1991).
[CrossRef]

1989

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

1986

1985

1941

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Alfano, R. R.

Ali, J.

W. B. Wang, S. G. Demos, J. Ali, R. R. Alfano, “Imaging fluorescent objects embedded inside animal tissues using polarization difference technique,” Opt. Commun. 142, 161–166 (1997).
[CrossRef]

Alpers, W.

A. Schmidt, V. Wismann, R. Romeiser, W. Alpers, “Simultaneous measurements of the ocean wave-radar modulation transfer function at L, C, and X bands from the research platform Nordsee,” J. Geophys. Res. C 100, 8815–8827 (1995).
[CrossRef]

C. Brüning, R. Schmidt, W. Alpers, “Estimation of the ocean wave-radar modulation transfer function from synthetic aperture radar imagery,” J. Geophys. Res. C 99, 9803–9815 (1994).
[CrossRef]

Anderson, G. E.

Barry, N. P.

S. C. W. Hyde, N. P. Barry, R. Jones, J. C. Dainty, P. M. W. French, “High resolution depth resolved imaging through scattering media using time resolved holography,” Opt. Commun. 122, 111–116 (1996).
[CrossRef]

Bhattacharya, K.

K. Bhattacharya, A. Ghosh, A. K. Chakraborty, “Vector wave imagery with a lens masked by polarizers,” J. Mod. Opt. 40, 379–390 (1993).
[CrossRef]

Brüning, C.

C. Brüning, R. Schmidt, W. Alpers, “Estimation of the ocean wave-radar modulation transfer function from synthetic aperture radar imagery,” J. Geophys. Res. C 99, 9803–9815 (1994).
[CrossRef]

Carter, L. L.

L. L. Carter, E. D. Cashwell, Particle Transport Simulation with the Monte-Carlo Method (Technical Information Center, Energy Research and Development Association, Oak Ridge, Tenn., 1975).

Cashwell, E. D.

L. L. Carter, E. D. Cashwell, Particle Transport Simulation with the Monte-Carlo Method (Technical Information Center, Energy Research and Development Association, Oak Ridge, Tenn., 1975).

Chakraborty, A. K.

K. Bhattacharya, A. Ghosh, A. K. Chakraborty, “Vector wave imagery with a lens masked by polarizers,” J. Mod. Opt. 40, 379–390 (1993).
[CrossRef]

Chance, B.

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (March1995).
[CrossRef]

Cummings, J. D.

B. A. Swartz, J. D. Cummings, “Laser range-gated underwater imaging including polarization discrimination,” in Underwater Imaging, Photography, and Visibility, R. W. Spinrad, ed., Proc. SPIE1537, 42–56 (1991).
[CrossRef]

Dainty, J. C.

S. C. W. Hyde, N. P. Barry, R. Jones, J. C. Dainty, P. M. W. French, “High resolution depth resolved imaging through scattering media using time resolved holography,” Opt. Commun. 122, 111–116 (1996).
[CrossRef]

de Boer, J. F.

Demos, S. G.

Engheta, N.

French, P. M. W.

S. C. W. Hyde, N. P. Barry, R. Jones, J. C. Dainty, P. M. W. French, “High resolution depth resolved imaging through scattering media using time resolved holography,” Opt. Commun. 122, 111–116 (1996).
[CrossRef]

Frieden, B. R.

B. R. Frieden, Probability, Statistical Optics, and Data Testing, 2nd ed., (Springer-Verlag, New York, 1991), Chap. 7.

Gayen, S. K.

S. K. Gayen, R. R. Alfano, “Emerging optical biomedical techniques,” Opt. Photon. News 7, 16–22 (March1996).
[CrossRef]

Ghosh, A.

K. Bhattacharya, A. Ghosh, A. K. Chakraborty, “Vector wave imagery with a lens masked by polarizers,” J. Mod. Opt. 40, 379–390 (1993).
[CrossRef]

Gilbert, G. D.

G. D. Gilbert, J. C. Pernicka, “Improvement of underwater visibility by reduction of backscatter with a circular polarization technique,” in Underwater Photo Optics I, A. B. Dember, ed., Proc. SPIE7, A-III-1–A-III-10 (1966).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Greenstein, J. L.

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Gu, M.

Harris, J. M.

J. M. Harris, “The influence of random media on the propagation and depolarization of electromagnetic waves,” Ph.D. dissertation (California Institute of Technology, Pasadena, Calif., 1980).

Henyey, L. G.

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Hyde, S. C. W.

S. C. W. Hyde, N. P. Barry, R. Jones, J. C. Dainty, P. M. W. French, “High resolution depth resolved imaging through scattering media using time resolved holography,” Opt. Commun. 122, 111–116 (1996).
[CrossRef]

Ishimaru, A.

Q. Ma, A. Ishimaru, “Propagation and depolarization of an arbitrarily polarized wave obliquely incident on a slab of random medium,” IEEE Trans. Antennas Propag. 39, 1626–1632 (1991).
[CrossRef]

Y. Kuga, A. Ishimaru, “Modulation transfer function of layered inhomogeneous random media using the small-angle approximation,” Appl. Opt. 25, 4382–4385 (1986).
[CrossRef]

Y. Kuga, A. Ishimaru, “Modulation transfer function and image transmission through randomly distributed spherical particles,” J. Opt. Soc. Am. A 2, 2330–2335 (1985).
[CrossRef]

A. Ishimaru, Wave Propagation in Random Media (Academic, San Diego, Calif., 1978), Vol. 1, Chap. 4.

Jacques, S. L.

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

Jones, R.

S. C. W. Hyde, N. P. Barry, R. Jones, J. C. Dainty, P. M. W. French, “High resolution depth resolved imaging through scattering media using time resolved holography,” Opt. Commun. 122, 111–116 (1996).
[CrossRef]

Kempe, M.

Kuga, Y.

Lampert, L. M.

L. M. Lampert, Modern Dairy Products (Chemical Publishing Co., New York, 1965).

Liu, F.

Ma, Q.

Q. Ma, A. Ishimaru, “Propagation and depolarization of an arbitrarily polarized wave obliquely incident on a slab of random medium,” IEEE Trans. Antennas Propag. 39, 1626–1632 (1991).
[CrossRef]

Milner, T. E.

Nelson, J. S.

Pernicka, J. C.

G. D. Gilbert, J. C. Pernicka, “Improvement of underwater visibility by reduction of backscatter with a circular polarization technique,” in Underwater Photo Optics I, A. B. Dember, ed., Proc. SPIE7, A-III-1–A-III-10 (1966).

Pugh, E. N.

Romeiser, R.

A. Schmidt, V. Wismann, R. Romeiser, W. Alpers, “Simultaneous measurements of the ocean wave-radar modulation transfer function at L, C, and X bands from the research platform Nordsee,” J. Geophys. Res. C 100, 8815–8827 (1995).
[CrossRef]

Rowe, M. P.

Rudolph, W.

Schmidt, A.

A. Schmidt, V. Wismann, R. Romeiser, W. Alpers, “Simultaneous measurements of the ocean wave-radar modulation transfer function at L, C, and X bands from the research platform Nordsee,” J. Geophys. Res. C 100, 8815–8827 (1995).
[CrossRef]

Schmidt, R.

C. Brüning, R. Schmidt, W. Alpers, “Estimation of the ocean wave-radar modulation transfer function from synthetic aperture radar imagery,” J. Geophys. Res. C 99, 9803–9815 (1994).
[CrossRef]

Sheppard, J. R.

Silverman, M. P.

M. P. Silverman, W. Strange, “Object delineation within turbid media by backscattering of phase-modulated light,” Opt. Commun. 144, 7–11 (1997).
[CrossRef]

M. P. Silverman, W. Strange, “Light scattering from optically active and inactive turbid media,” in Proceedings of the IS&T/OSA Conference on Optics and Imaging in the Information Age (Society for Image Science and Technology, Springfield, Va., 1996), pp. 172–180.

Star, W. M.

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

Sterenborg, H. J. C. M.

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

Strange, W.

M. P. Silverman, W. Strange, “Object delineation within turbid media by backscattering of phase-modulated light,” Opt. Commun. 144, 7–11 (1997).
[CrossRef]

M. P. Silverman, W. Strange, “Light scattering from optically active and inactive turbid media,” in Proceedings of the IS&T/OSA Conference on Optics and Imaging in the Information Age (Society for Image Science and Technology, Springfield, Va., 1996), pp. 172–180.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Chaps. 7 and 9.

Swartz, B. A.

B. A. Swartz, J. D. Cummings, “Laser range-gated underwater imaging including polarization discrimination,” in Underwater Imaging, Photography, and Visibility, R. W. Spinrad, ed., Proc. SPIE1537, 42–56 (1991).
[CrossRef]

Tannous, T.

Tyo, J. S.

van Gemert, M. J. C.

Wang, W. B.

S. G. Demos, W. B. Wang, R. R. Alfano, “Imaging objects hidden in scattering media with fluorescence polarization preservation of contrast agents,” Appl. Opt. 37, 792–797 (1998).
[CrossRef]

W. B. Wang, S. G. Demos, J. Ali, R. R. Alfano, “Imaging fluorescent objects embedded inside animal tissues using polarization difference technique,” Opt. Commun. 142, 161–166 (1997).
[CrossRef]

Welsch, E.

Wismann, V.

A. Schmidt, V. Wismann, R. Romeiser, W. Alpers, “Simultaneous measurements of the ocean wave-radar modulation transfer function at L, C, and X bands from the research platform Nordsee,” J. Geophys. Res. C 100, 8815–8827 (1995).
[CrossRef]

Yodh, A.

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (March1995).
[CrossRef]

Appl. Opt.

Astrophys. J.

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

IEEE Trans. Antennas Propag.

Q. Ma, A. Ishimaru, “Propagation and depolarization of an arbitrarily polarized wave obliquely incident on a slab of random medium,” IEEE Trans. Antennas Propag. 39, 1626–1632 (1991).
[CrossRef]

IEEE Trans. Biomed. Eng.

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

J. Geophys. Res. C

C. Brüning, R. Schmidt, W. Alpers, “Estimation of the ocean wave-radar modulation transfer function from synthetic aperture radar imagery,” J. Geophys. Res. C 99, 9803–9815 (1994).
[CrossRef]

A. Schmidt, V. Wismann, R. Romeiser, W. Alpers, “Simultaneous measurements of the ocean wave-radar modulation transfer function at L, C, and X bands from the research platform Nordsee,” J. Geophys. Res. C 100, 8815–8827 (1995).
[CrossRef]

J. Mod. Opt.

K. Bhattacharya, A. Ghosh, A. K. Chakraborty, “Vector wave imagery with a lens masked by polarizers,” J. Mod. Opt. 40, 379–390 (1993).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

W. B. Wang, S. G. Demos, J. Ali, R. R. Alfano, “Imaging fluorescent objects embedded inside animal tissues using polarization difference technique,” Opt. Commun. 142, 161–166 (1997).
[CrossRef]

S. C. W. Hyde, N. P. Barry, R. Jones, J. C. Dainty, P. M. W. French, “High resolution depth resolved imaging through scattering media using time resolved holography,” Opt. Commun. 122, 111–116 (1996).
[CrossRef]

M. P. Silverman, W. Strange, “Object delineation within turbid media by backscattering of phase-modulated light,” Opt. Commun. 144, 7–11 (1997).
[CrossRef]

Opt. Lett.

Opt. Photon. News

S. K. Gayen, R. R. Alfano, “Emerging optical biomedical techniques,” Opt. Photon. News 7, 16–22 (March1996).
[CrossRef]

Phys. Today

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (March1995).
[CrossRef]

Other

G. D. Gilbert, J. C. Pernicka, “Improvement of underwater visibility by reduction of backscatter with a circular polarization technique,” in Underwater Photo Optics I, A. B. Dember, ed., Proc. SPIE7, A-III-1–A-III-10 (1966).

B. A. Swartz, J. D. Cummings, “Laser range-gated underwater imaging including polarization discrimination,” in Underwater Imaging, Photography, and Visibility, R. W. Spinrad, ed., Proc. SPIE1537, 42–56 (1991).
[CrossRef]

J. M. Harris, “The influence of random media on the propagation and depolarization of electromagnetic waves,” Ph.D. dissertation (California Institute of Technology, Pasadena, Calif., 1980).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

J. S. Tyo, “Polarization difference imaging,” Ph.D. dissertation (University of Pennsylvania, Philadelphia, Pa., 1997).

The apertures can be projected onto the intermediate plane without changing the results as long as the extent of the object is much smaller than the projected size of the limiting aperture of the system. When this condition is met, the vignetting can be ignored (see Ref. 14, chap. 5).

For simplicity, the host medium in which the scatterers are embedded is assumed to be free space, although in general it is some other medium.

E⇀sc(sˆ) must fall off as 1/r as one travels along the direction sˆ; the formulation in Eq. (2) gives the relative amplitude scattered in any direction at some constant distance from the scatterer. The 1/r fall-off will be taken into account, along with the system parameters, in the analysis that follows.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Chaps. 7 and 9.

rˆ and θˆ refer to the spherical unit vectors at the position rˆn with respect to the Cartesian coordinate system shown in Fig. 2.

Strictly speaking, Eq. (12) does not follow directly from Eq. (11). Equation (11) states that there is an ideal E-field point source given by κ2E⇀δ(xf+xn)δ(zf+zn). This source can also be thought of as an ideal intensity point source given by κ2|E⇀|2δ(xf+xn)δ(zf+zn).

The term polarization sum was introduced in Ref. 21. It is meant to differentiate between a true, polarization-blind image where intensity alone is measured and a sum image formed by adding the intensities obtained at orthogonal polarizations. The two concepts are completely equivalent, and the term PS image is retained to provide the reader with information concerning how a specific image was formed.

M. P. Silverman, W. Strange, “Light scattering from optically active and inactive turbid media,” in Proceedings of the IS&T/OSA Conference on Optics and Imaging in the Information Age (Society for Image Science and Technology, Springfield, Va., 1996), pp. 172–180.

A. Ishimaru, Wave Propagation in Random Media (Academic, San Diego, Calif., 1978), Vol. 1, Chap. 4.

L. L. Carter, E. D. Cashwell, Particle Transport Simulation with the Monte-Carlo Method (Technical Information Center, Energy Research and Development Association, Oak Ridge, Tenn., 1975).

B. R. Frieden, Probability, Statistical Optics, and Data Testing, 2nd ed., (Springer-Verlag, New York, 1991), Chap. 7.

This experiment investigates the PSF that is due to a linearly polarized source. The portion of the radiation that is unpolarized will not be imaged by PDI.

L. M. Lampert, Modern Dairy Products (Chemical Publishing Co., New York, 1965).

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Fig. 1
Fig. 1

PSF geometry considered. An elemental dipole is parallel to the z axis is located at the origin in a scattering medium of thickness 2T in the y direction. The medium is infinite in the x and z directions. The radiation is captured by an imaging system made up of two lenses separated by twice the focal length f. The first lens is f from the dipole and the second lens is f from the image plane. A focal-plane filter can be included at the Fourier transform plane between the two lenses. For simplicity, Tf is assumed.

Fig. 2
Fig. 2

Geometry for a single scatterer. The nth scatterer is located at the position rn. The polarization is in the θ direction. At the scatterer the primed coordinates are used to solve for the scattering amplitude with Rayleigh or Mie theory. It is assumed that all scatterers in the medium are identical and that all scatterers are in the far field of the dipole source. In the SSA it is further assumed that all the scatterers are also noninteracting.

Fig. 3
Fig. 3

Corrupting portions of the PS and PD PSF’s for a medium of Rayleigh scatterers calculated by using the SSA. These plots show that the PD PSF is narrower than the PS PSF in such a medium.

Fig. 4
Fig. 4

Corrupting portions of the PS and PD PSF’s presented in Fig. 3 with radial variation suppressed by multiplying by νf2. With this presentation scheme the narrowing with respect to the x axis is more pronounced. The null can now clearly be seen at 45°. The actual PD PSF has a zero crossing at 45°, but the magnitude of the PD PSF is presented in this figure (as well as in Fig. 3).

Fig. 5
Fig. 5

Corrupting portions of the PS and PD PSF’s for a medium of Rayleigh scatterers that total 0.4 attenuation lengths thick. The mean number of scattering events per measured photon was 0.26. These PSF’s were obtained by Monte Carlo simulation and are meant to verify the results shown in Figs. 3 and 4. The SSA predicts that the two PSF’s should have the same absolute maximum value, but as is evident above, they do not. This is due to the finite acceptance solid angle in the Monte Carlo simulation as discussed in the text. A narrowing of the PD PSF is apparent along slices taken parallel to the z axis, as was predicted by the SSA theory. This narrowing is more apparent in Fig. 6. Both PSF’s are normalized to the value of the PSF for the PS image, and the two plots are presented on the same vertical scale.

Fig. 6
Fig. 6

Slices of the PS and PD PSF’s shown in Fig. 5 taken parallel to the z axis at varying x positions given in terms of the attenuation length τ. In each of the plots for |x/τ|>0, two sets of curves are plotted that correspond to ±|x/τ|. The solid curves are slices of the PS PSF, and the dotted curves are slices of the PD PSF. All curves have been normalized to their peak values to facilitate comparison with Fig. 4. At |x|=0, the PD and PS PSF’s are at their narrowest point as expected, and they are approximately the same width; however, as |x| increases, the width of the PS PSF parallel to the dipole increases faster than does the width of the PD PSF.

Fig. 7
Fig. 7

Corrupting portion of the PS and PD PSF’s computed, with Monte Carlo simulations for a medium of Mie scatterers that total 4 attenuation lengths thick. The mean number of scattering events per photon collected was 2.05. The Mie-scattering function used is that for spherical particles that have a radius equal to twice the wavelength of the incident radiation. The index of refraction of the sphere relative to the surrounding medium is 1.20, nearly the same as that of latex in water. The Mie-scattering function has an anisotropy factor (mean cosine) of g=0.8031, which is typical of the anisotropy factor found in human tissue.35 The point image intensity is the intensity due to unscattered light at the location of the source in the image plane. Since the medium depolarizes the radiation, the corrupting portion of the PD PSF is significantly lower than that of the PS PSF. The complete PSF is obtained by adding a peak of unit magnitude at the origin corresponding to the source.

Equations (21)

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E(rn)=Aexp(ikr)rsin θ θˆ,
Esc(sˆ)=f¯¯(sˆ,sˆ)Ein(sˆ),
Ef(n)(xf, zf)=B exp[ik(4f-yn)](λf)2OPEsc(n)(yˆ)κ2×δ(xo-xn)δ(zo-zn)AG(xa, za)×expiπλfynf(xa2+za2)×exp-i2πλf[xa(xo+xf)+za(zo+zf)]dxadzadxodzo,
Ef(n)(xf, zf)=Bκ2 exp[ik(4f-yn)](λf)2PEsc(n)×F2G(xa, za)×expiπλfynf(xa2+za2)fx,fz,
fx=xn+xfλf;fz=zn+zfλf,
Ef(xf, zf)=n=1NEf(n)(xf, zf),
|Ef(xf, zf)|2=V|Ef(xf, zf; xo, yo, zo)|2ρdxodyodzo,
Vρdxodyodzo=N.
Esc(n)(sˆ)=|Ein(rn)|exp(ik|rn|)CRexp(ikR)sin θ θˆ,
θˆ=(zˆ×rˆ)×rˆ|(zˆ×rˆ)×rˆ|=1(1-cos2 θ sin2 ϕ)1/2×[-cos θ cos ϕxˆ+sin θ zˆ]=1sin θ[-cos θ cos ϕxˆ+sin θzˆ],
Esc(n)(sˆ)=|Ein(rn)|exp(ik|rn|)CRexp(ikR)×[-cos θ cos ϕxˆ+sin θzˆ].
Ef(n)(xf, zf)=Bκ2(λf)2exp[ik(4f-yn)]PEsc(n)(sˆ)δ(fx)δ(fz)Bκ2PEsc(n)(sˆ)δ(xf+xn)δ(zf+zn).
|Ef(xf, zf)|2=yo=-TTρ|PE(xf, zf; -xf, yo, -zf)|2dyo=ABCκR2yo=-TTρ sin2 θr2|P[-cos θ cos ϕxˆ+sin θzˆ]|2dyo,
ρρ(r)=0,r<r0ρ0,otherwise,
|Efx(xf, zf)|2=ρABCκR2-Tzf42νf2(T2+νf2)2+Tzf2(4xf2+zf2)4νf4(T2+νf2)+zf2(4xf2+zf2)tan-1Tνf4νf5,
|Efz(xf, zf)|2=ρABCκR2Tzf42νf2(T2+νf2)2-Tzf2(8xf2+5zf2)4νf4(T2+νf2)+(8xf4+8xf2zf2+3zf4)tan-1Tνf4νf5.
tan-1TνfTνf,
|EfPS(xf, zf)|2=|Efx(xf, zf)|2+|Efz(xf, zf)|2,
|EfPD(xf, zf)|2=|Efx(xf, zf)|2-|Efz(xf, zf)|2.
|Efx(xf, zf)|21r2sin2 θ cos2 θ cos2 ϕ,
|Efz(xf, zf)|21r2sin4 θ,

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