Abstract

We consider the inverse problem of reconstructing the absorption and diffusion coefficients of an inhomogeneous highly scattering medium probed by diffuse light. Inversion formulas based on the Fourier–Laplace transform are used to establish the existence and uniqueness of solutions to this problem in planar, cylindrical, and spherical geometries.

© 2001 Optical Society of America

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  28. O. V. Kravtsenyuk, V. V. Lyubimov, “Specific features of statistical characteristics of photon trajectories in a strongly scattering medium near an object surface,” Opt. Spectrosc. 88, 608–614 (2000).
    [CrossRef]
  29. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  33. D. N. Pattanayak, A. G. Yodh, “Diffuse optical 3D-slice imaging of bounded turbid media using a new integro-differential equation,” Opt. Express 4, 231–240 (1999) http://www.opticsexpress.org .
    [CrossRef] [PubMed]
  34. X. Li, D. N. Pattanayak, T. Durduran, J. P. Culver, B. Chance, A. G. Yodh, “Near-field diffraction tomography with diffuse photon-density waves,” Phys. Rev. E 61, 4295–4309 (2000).
    [CrossRef]
  35. The right-hand side of this equation is the same in different versions of perturbation theory, such as the first Born approximation or the first Rytov approximation, but the definition of the data function ϕ is different. Definition (10) is obtained in the first Born approximation; reexponentiation according to the Rytov expansion leads to ϕ(r1, r2)=-G0(r1, r2)ln[G(r1, r2)/G0(r1, r2)].
  36. B. Chu, E. Gulari, E. Gulari, “Photon correlation measurements of colloidal size distributions. II. Details of histogram approach and comparison of methods of data analysis,” Phys. Scr. 19, 476–485 (1979).
    [CrossRef]
  37. I. Ersh, L. S. Muratov, S. Y. Novozhilov, B. M. Stockman, M. I. Stockman, “Kinetics of the immunological reaction of agglutination and rapid determination of bacteria using an automated laser photon-correlation spectrometer,” Dok. Biochem. 287, 125–129 (1986).
  38. R. Bauer, M. Hansen, S. Hansen, L. Ogendal, S. Lomholt, K. Qvist, “The structure of casein aggregates during renneting studied by indirect Fourier transformation and inverse Laplace transformation of static and dynamic light scattering data, respectively,” J. Chem. Phys. 103, 2725–2737 (1995).
    [CrossRef]
  39. P. K. Venkatesh, R. W. Carr, M. H. Cohen, A. M. Dean, “Microcanonical transition state theory rate coefficients from thermal rate constants via inverse Laplace transformation,” J. Phys. Chem. A 102, 8104–8115 (1998).
    [CrossRef]
  40. E. Schnedermann, “The inverse Laplace transform as the ultimate tool for transverse mass spectroscopy,” Z. Phys. C 64, 85–90 (1994).
    [CrossRef]

2000 (2)

O. V. Kravtsenyuk, V. V. Lyubimov, “Specific features of statistical characteristics of photon trajectories in a strongly scattering medium near an object surface,” Opt. Spectrosc. 88, 608–614 (2000).
[CrossRef]

X. Li, D. N. Pattanayak, T. Durduran, J. P. Culver, B. Chance, A. G. Yodh, “Near-field diffraction tomography with diffuse photon-density waves,” Phys. Rev. E 61, 4295–4309 (2000).
[CrossRef]

1999 (6)

V. B. Volkonskii, O. V. Kravtsenyuk, V. V. Lyubimov, E. P. Mironov, A. G. Murzin, “The use of statistical characteristics of photon trajectories for the tomographic studies of optical macroheterogeneities in strongly scattering objects,” Opt. Spectrosc. 86, 253–260 (1999).

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

D. N. Pattanayak, A. G. Yodh, “Diffuse optical 3D-slice imaging of bounded turbid media using a new integro-differential equation,” Opt. Express 4, 231–240 (1999) http://www.opticsexpress.org .
[CrossRef] [PubMed]

M. C. W. van Rossum, T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[CrossRef]

A. B. Davis, R. F. Cahalan, D. Spinehirne, M. J. McGill, S. P. Love, “Off-beam lidar: an emerging technique in cloud remote sensing based on radiative Green-function theory in the diffusion domain,” Phys. Chem. Earth B 24, 177–185 (1999).
[CrossRef]

V. V. Lyubimov, “On the spatial resolution of optical tomography of strongly scattering media with the use of the directly passing photons,” Opt. Spectrosc. 86, 251–252 (1999).

1998 (3)

1997 (3)

1996 (1)

V. V. Lyubimov, “Optics of photon density waves in strongly scattering media and spatial resolution in tomography,” Opt. Spectrosc. 81, 299–301 (1996).

1995 (4)

C. P. Gonatas, M. Ishii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
[CrossRef]

M. O’Leary, D. Boas, B. Chance, A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing photon tomography,” Opt. Lett. 20, 426–429 (1995).
[CrossRef]

V. V. Lyubimov, “Spatial resolution in probing a strongly scattering medium with a short optical pulse,” Opt. Spectrosc. 78, 259–260 (1995).

R. Bauer, M. Hansen, S. Hansen, L. Ogendal, S. Lomholt, K. Qvist, “The structure of casein aggregates during renneting studied by indirect Fourier transformation and inverse Laplace transformation of static and dynamic light scattering data, respectively,” J. Chem. Phys. 103, 2725–2737 (1995).
[CrossRef]

1994 (2)

E. Schnedermann, “The inverse Laplace transform as the ultimate tool for transverse mass spectroscopy,” Z. Phys. C 64, 85–90 (1994).
[CrossRef]

V. V. Lyubimov, “Image transfer in a plane layer of a scattering medium and estimation of the resolving power of optical tomography using 1st transmitted photons of ultrashort pulses,” Opt. Spectrosc. 76, 725–726 (1994).

1993 (2)

1992 (1)

J. Schotland, J. S. Leigh, “Photon diffusion imaging,” Biophys. J. 61, 446 (1992).

1991 (4)

1990 (1)

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

1986 (1)

I. Ersh, L. S. Muratov, S. Y. Novozhilov, B. M. Stockman, M. I. Stockman, “Kinetics of the immunological reaction of agglutination and rapid determination of bacteria using an automated laser photon-correlation spectrometer,” Dok. Biochem. 287, 125–129 (1986).

1979 (1)

B. Chu, E. Gulari, E. Gulari, “Photon correlation measurements of colloidal size distributions. II. Details of histogram approach and comparison of methods of data analysis,” Phys. Scr. 19, 476–485 (1979).
[CrossRef]

Alfano, R. R.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

K. M. Yoo, F. Liu, R. R. Alfano, “Imaging objects hidden in scattering media using an absorption technique,” Opt. Lett. 16, 1068–1070 (1991).
[CrossRef] [PubMed]

Aronson, R.

R. L. Barbour, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. SPIE1431, 192–203 (1991).
[CrossRef]

Arridge, S. R.

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

S. R. Arridge, W. R. B. Lionhart, “Nonuniqueness in diffusion-based optical tomography,” Opt. Lett. 23, 882–884 (1998).
[CrossRef]

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “New results for the development of infrared absorption imaging,” in Biomedical Image Processing, A. C. Bovik, W. E. Higgins, eds., Proc. SPIE1245, 92–103 (1991).
[CrossRef]

Barbour, R. L.

R. L. Barbour, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. SPIE1431, 192–203 (1991).
[CrossRef]

Bauer, R.

R. Bauer, M. Hansen, S. Hansen, L. Ogendal, S. Lomholt, K. Qvist, “The structure of casein aggregates during renneting studied by indirect Fourier transformation and inverse Laplace transformation of static and dynamic light scattering data, respectively,” J. Chem. Phys. 103, 2725–2737 (1995).
[CrossRef]

Benaron, D. A.

D. A. Benaron, D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science 259, 1463–1466 (1993).
[CrossRef] [PubMed]

Boas, D.

Cahalan, R. F.

A. B. Davis, R. F. Cahalan, D. Spinehirne, M. J. McGill, S. P. Love, “Off-beam lidar: an emerging technique in cloud remote sensing based on radiative Green-function theory in the diffusion domain,” Phys. Chem. Earth B 24, 177–185 (1999).
[CrossRef]

Carr, R. W.

P. K. Venkatesh, R. W. Carr, M. H. Cohen, A. M. Dean, “Microcanonical transition state theory rate coefficients from thermal rate constants via inverse Laplace transformation,” J. Phys. Chem. A 102, 8104–8115 (1998).
[CrossRef]

Chance, B.

Chen, H.

Chen, Y.

Chu, B.

B. Chu, E. Gulari, E. Gulari, “Photon correlation measurements of colloidal size distributions. II. Details of histogram approach and comparison of methods of data analysis,” Phys. Scr. 19, 476–485 (1979).
[CrossRef]

Cohen, M. H.

P. K. Venkatesh, R. W. Carr, M. H. Cohen, A. M. Dean, “Microcanonical transition state theory rate coefficients from thermal rate constants via inverse Laplace transformation,” J. Phys. Chem. A 102, 8104–8115 (1998).
[CrossRef]

Cope, M.

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “New results for the development of infrared absorption imaging,” in Biomedical Image Processing, A. C. Bovik, W. E. Higgins, eds., Proc. SPIE1245, 92–103 (1991).
[CrossRef]

Culver, J. P.

X. Li, D. N. Pattanayak, T. Durduran, J. P. Culver, B. Chance, A. G. Yodh, “Near-field diffraction tomography with diffuse photon-density waves,” Phys. Rev. E 61, 4295–4309 (2000).
[CrossRef]

Davis, A. B.

A. B. Davis, R. F. Cahalan, D. Spinehirne, M. J. McGill, S. P. Love, “Off-beam lidar: an emerging technique in cloud remote sensing based on radiative Green-function theory in the diffusion domain,” Phys. Chem. Earth B 24, 177–185 (1999).
[CrossRef]

Dean, A. M.

P. K. Venkatesh, R. W. Carr, M. H. Cohen, A. M. Dean, “Microcanonical transition state theory rate coefficients from thermal rate constants via inverse Laplace transformation,” J. Phys. Chem. A 102, 8104–8115 (1998).
[CrossRef]

Delpy, D. T.

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “New results for the development of infrared absorption imaging,” in Biomedical Image Processing, A. C. Bovik, W. E. Higgins, eds., Proc. SPIE1245, 92–103 (1991).
[CrossRef]

Dilworth, D.

Dilworth, D. S.

Durduran, T.

X. Li, D. N. Pattanayak, T. Durduran, J. P. Culver, B. Chance, A. G. Yodh, “Near-field diffraction tomography with diffuse photon-density waves,” Phys. Rev. E 61, 4295–4309 (2000).
[CrossRef]

X. D. Li, T. Durduran, A. G. Yodh, B. Chance, D. N. Pattanayak, “Diffraction tomography for biochemical imaging with diffuse-photon density waves,” Opt. Lett. 22, 573–575 (1997).
[CrossRef] [PubMed]

Ersh, I.

I. Ersh, L. S. Muratov, S. Y. Novozhilov, B. M. Stockman, M. I. Stockman, “Kinetics of the immunological reaction of agglutination and rapid determination of bacteria using an automated laser photon-correlation spectrometer,” Dok. Biochem. 287, 125–129 (1986).

Feinberg, J.

A. Rebane, J. Feinberg, “Time-resolved holography,” Nature 351, 378–380 (1991).
[CrossRef]

Fujimoto, J. G.

Gonatas, C. P.

C. P. Gonatas, M. Ishii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
[CrossRef]

Graber, H. L.

R. L. Barbour, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. SPIE1431, 192–203 (1991).
[CrossRef]

Grunbaum, F. A.

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Gulari, E.

B. Chu, E. Gulari, E. Gulari, “Photon correlation measurements of colloidal size distributions. II. Details of histogram approach and comparison of methods of data analysis,” Phys. Scr. 19, 476–485 (1979).
[CrossRef]

B. Chu, E. Gulari, E. Gulari, “Photon correlation measurements of colloidal size distributions. II. Details of histogram approach and comparison of methods of data analysis,” Phys. Scr. 19, 476–485 (1979).
[CrossRef]

Hansen, M.

R. Bauer, M. Hansen, S. Hansen, L. Ogendal, S. Lomholt, K. Qvist, “The structure of casein aggregates during renneting studied by indirect Fourier transformation and inverse Laplace transformation of static and dynamic light scattering data, respectively,” J. Chem. Phys. 103, 2725–2737 (1995).
[CrossRef]

Hansen, S.

R. Bauer, M. Hansen, S. Hansen, L. Ogendal, S. Lomholt, K. Qvist, “The structure of casein aggregates during renneting studied by indirect Fourier transformation and inverse Laplace transformation of static and dynamic light scattering data, respectively,” J. Chem. Phys. 103, 2725–2737 (1995).
[CrossRef]

Hee, M. R.

Ho, P. P.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Hoover, B. G.

Ishii, M.

C. P. Gonatas, M. Ishii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
[CrossRef]

M. Ishii, J. S. Leigh, J. C. Schotland, “Photon diffusion imaging of model and biological systems,” in Optical Tomography, Photon Migration and Spectroscopy of Tissue and Model Media: Theory, Human Studies, B. Chance, ed., Proc. SPIE2389, 312–317 (1995).

Izatt, J. A.

Jackobson, J. M.

Kohn, P.

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Kravtsenyuk, O. V.

O. V. Kravtsenyuk, V. V. Lyubimov, “Specific features of statistical characteristics of photon trajectories in a strongly scattering medium near an object surface,” Opt. Spectrosc. 88, 608–614 (2000).
[CrossRef]

V. B. Volkonskii, O. V. Kravtsenyuk, V. V. Lyubimov, E. P. Mironov, A. G. Murzin, “The use of statistical characteristics of photon trajectories for the tomographic studies of optical macroheterogeneities in strongly scattering objects,” Opt. Spectrosc. 86, 253–260 (1999).

Leigh, J. S.

C. P. Gonatas, M. Ishii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
[CrossRef]

J. Schotland, J. S. Leigh, “Photon diffusion imaging,” Biophys. J. 61, 446 (1992).

M. Ishii, J. S. Leigh, J. C. Schotland, “Photon diffusion imaging of model and biological systems,” in Optical Tomography, Photon Migration and Spectroscopy of Tissue and Model Media: Theory, Human Studies, B. Chance, ed., Proc. SPIE2389, 312–317 (1995).

Leith, E.

Leith, E. N.

Li, X.

X. Li, D. N. Pattanayak, T. Durduran, J. P. Culver, B. Chance, A. G. Yodh, “Near-field diffraction tomography with diffuse photon-density waves,” Phys. Rev. E 61, 4295–4309 (2000).
[CrossRef]

Li, X. D.

Lionhart, W. R. B.

Liu, C.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Liu, F.

Lomholt, S.

R. Bauer, M. Hansen, S. Hansen, L. Ogendal, S. Lomholt, K. Qvist, “The structure of casein aggregates during renneting studied by indirect Fourier transformation and inverse Laplace transformation of static and dynamic light scattering data, respectively,” J. Chem. Phys. 103, 2725–2737 (1995).
[CrossRef]

Lopez, J.

Love, S. P.

A. B. Davis, R. F. Cahalan, D. Spinehirne, M. J. McGill, S. P. Love, “Off-beam lidar: an emerging technique in cloud remote sensing based on radiative Green-function theory in the diffusion domain,” Phys. Chem. Earth B 24, 177–185 (1999).
[CrossRef]

Lubowsky, J.

R. L. Barbour, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. SPIE1431, 192–203 (1991).
[CrossRef]

Lyubimov, V. V.

O. V. Kravtsenyuk, V. V. Lyubimov, “Specific features of statistical characteristics of photon trajectories in a strongly scattering medium near an object surface,” Opt. Spectrosc. 88, 608–614 (2000).
[CrossRef]

V. V. Lyubimov, “On the spatial resolution of optical tomography of strongly scattering media with the use of the directly passing photons,” Opt. Spectrosc. 86, 251–252 (1999).

V. B. Volkonskii, O. V. Kravtsenyuk, V. V. Lyubimov, E. P. Mironov, A. G. Murzin, “The use of statistical characteristics of photon trajectories for the tomographic studies of optical macroheterogeneities in strongly scattering objects,” Opt. Spectrosc. 86, 253–260 (1999).

V. V. Lyubimov, “Optics of photon density waves in strongly scattering media and spatial resolution in tomography,” Opt. Spectrosc. 81, 299–301 (1996).

V. V. Lyubimov, “Spatial resolution in probing a strongly scattering medium with a short optical pulse,” Opt. Spectrosc. 78, 259–260 (1995).

V. V. Lyubimov, “Image transfer in a plane layer of a scattering medium and estimation of the resolving power of optical tomography using 1st transmitted photons of ultrashort pulses,” Opt. Spectrosc. 76, 725–726 (1994).

Masri, R.

Matson, C. L.

McGill, M. J.

A. B. Davis, R. F. Cahalan, D. Spinehirne, M. J. McGill, S. P. Love, “Off-beam lidar: an emerging technique in cloud remote sensing based on radiative Green-function theory in the diffusion domain,” Phys. Chem. Earth B 24, 177–185 (1999).
[CrossRef]

Mironov, E. P.

V. B. Volkonskii, O. V. Kravtsenyuk, V. V. Lyubimov, E. P. Mironov, A. G. Murzin, “The use of statistical characteristics of photon trajectories for the tomographic studies of optical macroheterogeneities in strongly scattering objects,” Opt. Spectrosc. 86, 253–260 (1999).

Muratov, L. S.

I. Ersh, L. S. Muratov, S. Y. Novozhilov, B. M. Stockman, M. I. Stockman, “Kinetics of the immunological reaction of agglutination and rapid determination of bacteria using an automated laser photon-correlation spectrometer,” Dok. Biochem. 287, 125–129 (1986).

Murzin, A. G.

V. B. Volkonskii, O. V. Kravtsenyuk, V. V. Lyubimov, E. P. Mironov, A. G. Murzin, “The use of statistical characteristics of photon trajectories for the tomographic studies of optical macroheterogeneities in strongly scattering objects,” Opt. Spectrosc. 86, 253–260 (1999).

Naulleau, P. P.

Nieuwenhuizen, T. M.

M. C. W. van Rossum, T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[CrossRef]

Novozhilov, S. Y.

I. Ersh, L. S. Muratov, S. Y. Novozhilov, B. M. Stockman, M. I. Stockman, “Kinetics of the immunological reaction of agglutination and rapid determination of bacteria using an automated laser photon-correlation spectrometer,” Dok. Biochem. 287, 125–129 (1986).

O’Leary, M.

Ogendal, L.

R. Bauer, M. Hansen, S. Hansen, L. Ogendal, S. Lomholt, K. Qvist, “The structure of casein aggregates during renneting studied by indirect Fourier transformation and inverse Laplace transformation of static and dynamic light scattering data, respectively,” J. Chem. Phys. 103, 2725–2737 (1995).
[CrossRef]

Pattanayak, D. N.

Qvist, K.

R. Bauer, M. Hansen, S. Hansen, L. Ogendal, S. Lomholt, K. Qvist, “The structure of casein aggregates during renneting studied by indirect Fourier transformation and inverse Laplace transformation of static and dynamic light scattering data, respectively,” J. Chem. Phys. 103, 2725–2737 (1995).
[CrossRef]

Rebane, A.

A. Rebane, J. Feinberg, “Time-resolved holography,” Nature 351, 378–380 (1991).
[CrossRef]

Rudd, J.

Schnedermann, E.

E. Schnedermann, “The inverse Laplace transform as the ultimate tool for transverse mass spectroscopy,” Z. Phys. C 64, 85–90 (1994).
[CrossRef]

Schotland, J.

J. Schotland, J. S. Leigh, “Photon diffusion imaging,” Biophys. J. 61, 446 (1992).

Schotland, J. C.

J. C. Schotland, “Continuous wave diffusion imaging,” J. Opt. Soc. Am. A 14, 275–279 (1997).
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C. P. Gonatas, M. Ishii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
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M. Ishii, J. S. Leigh, J. C. Schotland, “Photon diffusion imaging of model and biological systems,” in Optical Tomography, Photon Migration and Spectroscopy of Tissue and Model Media: Theory, Human Studies, B. Chance, ed., Proc. SPIE2389, 312–317 (1995).

Singer, J. R.

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
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Spinehirne, D.

A. B. Davis, R. F. Cahalan, D. Spinehirne, M. J. McGill, S. P. Love, “Off-beam lidar: an emerging technique in cloud remote sensing based on radiative Green-function theory in the diffusion domain,” Phys. Chem. Earth B 24, 177–185 (1999).
[CrossRef]

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D. A. Benaron, D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science 259, 1463–1466 (1993).
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I. Ersh, L. S. Muratov, S. Y. Novozhilov, B. M. Stockman, M. I. Stockman, “Kinetics of the immunological reaction of agglutination and rapid determination of bacteria using an automated laser photon-correlation spectrometer,” Dok. Biochem. 287, 125–129 (1986).

Stockman, M. I.

I. Ersh, L. S. Muratov, S. Y. Novozhilov, B. M. Stockman, M. I. Stockman, “Kinetics of the immunological reaction of agglutination and rapid determination of bacteria using an automated laser photon-correlation spectrometer,” Dok. Biochem. 287, 125–129 (1986).

Swanson, E. A.

Valdmanis, J.

van der Zee, P.

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “New results for the development of infrared absorption imaging,” in Biomedical Image Processing, A. C. Bovik, W. E. Higgins, eds., Proc. SPIE1245, 92–103 (1991).
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M. C. W. van Rossum, T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
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P. K. Venkatesh, R. W. Carr, M. H. Cohen, A. M. Dean, “Microcanonical transition state theory rate coefficients from thermal rate constants via inverse Laplace transformation,” J. Phys. Chem. A 102, 8104–8115 (1998).
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V. B. Volkonskii, O. V. Kravtsenyuk, V. V. Lyubimov, E. P. Mironov, A. G. Murzin, “The use of statistical characteristics of photon trajectories for the tomographic studies of optical macroheterogeneities in strongly scattering objects,” Opt. Spectrosc. 86, 253–260 (1999).

Wang, L.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
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Yodh, A. G.

Yoo, K. M.

Zhang, G.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Zubelli, J. P.

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Appl. Opt. (2)

Biophys. J. (1)

J. Schotland, J. S. Leigh, “Photon diffusion imaging,” Biophys. J. 61, 446 (1992).

Dok. Biochem. (1)

I. Ersh, L. S. Muratov, S. Y. Novozhilov, B. M. Stockman, M. I. Stockman, “Kinetics of the immunological reaction of agglutination and rapid determination of bacteria using an automated laser photon-correlation spectrometer,” Dok. Biochem. 287, 125–129 (1986).

Inverse Probl. (1)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

J. Chem. Phys. (1)

R. Bauer, M. Hansen, S. Hansen, L. Ogendal, S. Lomholt, K. Qvist, “The structure of casein aggregates during renneting studied by indirect Fourier transformation and inverse Laplace transformation of static and dynamic light scattering data, respectively,” J. Chem. Phys. 103, 2725–2737 (1995).
[CrossRef]

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

J. Phys. Chem. A (1)

P. K. Venkatesh, R. W. Carr, M. H. Cohen, A. M. Dean, “Microcanonical transition state theory rate coefficients from thermal rate constants via inverse Laplace transformation,” J. Phys. Chem. A 102, 8104–8115 (1998).
[CrossRef]

Nature (1)

A. Rebane, J. Feinberg, “Time-resolved holography,” Nature 351, 378–380 (1991).
[CrossRef]

Opt. Express (2)

Opt. Lett. (5)

Opt. Spectrosc. (6)

V. V. Lyubimov, “Image transfer in a plane layer of a scattering medium and estimation of the resolving power of optical tomography using 1st transmitted photons of ultrashort pulses,” Opt. Spectrosc. 76, 725–726 (1994).

V. V. Lyubimov, “Spatial resolution in probing a strongly scattering medium with a short optical pulse,” Opt. Spectrosc. 78, 259–260 (1995).

V. V. Lyubimov, “On the spatial resolution of optical tomography of strongly scattering media with the use of the directly passing photons,” Opt. Spectrosc. 86, 251–252 (1999).

V. V. Lyubimov, “Optics of photon density waves in strongly scattering media and spatial resolution in tomography,” Opt. Spectrosc. 81, 299–301 (1996).

V. B. Volkonskii, O. V. Kravtsenyuk, V. V. Lyubimov, E. P. Mironov, A. G. Murzin, “The use of statistical characteristics of photon trajectories for the tomographic studies of optical macroheterogeneities in strongly scattering objects,” Opt. Spectrosc. 86, 253–260 (1999).

O. V. Kravtsenyuk, V. V. Lyubimov, “Specific features of statistical characteristics of photon trajectories in a strongly scattering medium near an object surface,” Opt. Spectrosc. 88, 608–614 (2000).
[CrossRef]

Phys. Chem. Earth B (1)

A. B. Davis, R. F. Cahalan, D. Spinehirne, M. J. McGill, S. P. Love, “Off-beam lidar: an emerging technique in cloud remote sensing based on radiative Green-function theory in the diffusion domain,” Phys. Chem. Earth B 24, 177–185 (1999).
[CrossRef]

Phys. Rev. E (2)

C. P. Gonatas, M. Ishii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
[CrossRef]

X. Li, D. N. Pattanayak, T. Durduran, J. P. Culver, B. Chance, A. G. Yodh, “Near-field diffraction tomography with diffuse photon-density waves,” Phys. Rev. E 61, 4295–4309 (2000).
[CrossRef]

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[CrossRef]

Rev. Mod. Phys. (1)

M. C. W. van Rossum, T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[CrossRef]

Science (3)

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

D. A. Benaron, D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science 259, 1463–1466 (1993).
[CrossRef] [PubMed]

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Z. Phys. C (1)

E. Schnedermann, “The inverse Laplace transform as the ultimate tool for transverse mass spectroscopy,” Z. Phys. C 64, 85–90 (1994).
[CrossRef]

Other (9)

The right-hand side of this equation is the same in different versions of perturbation theory, such as the first Born approximation or the first Rytov approximation, but the definition of the data function ϕ is different. Definition (10) is obtained in the first Born approximation; reexponentiation according to the Rytov expansion leads to ϕ(r1, r2)=-G0(r1, r2)ln[G(r1, r2)/G0(r1, r2)].

M. Ishii, J. S. Leigh, J. C. Schotland, “Photon diffusion imaging of model and biological systems,” in Optical Tomography, Photon Migration and Spectroscopy of Tissue and Model Media: Theory, Human Studies, B. Chance, ed., Proc. SPIE2389, 312–317 (1995).

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “New results for the development of infrared absorption imaging,” in Biomedical Image Processing, A. C. Bovik, W. E. Higgins, eds., Proc. SPIE1245, 92–103 (1991).
[CrossRef]

R. L. Barbour, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. SPIE1431, 192–203 (1991).
[CrossRef]

G. Mueller, ed., Medical Optical Tomography: Functional Imaging and Monitoring (SPIE Press, Bellingham, Wash., 1993).

R. Alfano, ed., Advances in Optical Imaging and Photon Migration (Optical Society of America, Washington, D.C., 1994).

B. Chance, R. Alfano, eds., Proceedings of Optical Tomography and Spectroscopy of Tissue (SPIE Press, Bellingham, Wash., 1995).

B. Chance, R. Alfano, eds., Proceedings of Optical Tomography and Spectroscopy of Tissue II (SPIE Press, Bellingham, Wash., 1997).

B. Chance, R. Alfano, B. Tromberg, eds., Proceedings of Optical Tomography and Spectroscopy of Tissue III (SPIE Press, Bellingham, Wash., 1999).

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Figures (1)

Fig. 1
Fig. 1

Different geometrical arrangements of source-detector pairs. The gray areas indicate the objects to be imaged. In the case of (a) planar geometry, the sources and detectors can be placed either on the same plane or on different planes. For the cylindrical geometry (b), the sources and detectors are placed on the boundary of the cylinder, which is assumed to be infinitely long (much longer than the characteristic size of the sample). For the spherical geometry (c) the sources and detectors are placed on the surface of the sphere.

Equations (95)

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u(r, t)t=·[D(r)u(r, t)]-α(r)u(r, t)+S(r, t),
(2-k2)u(r)=1D0[δα(r)-·δD(r)]u(r)-1D0S(r),
γ(A)=Aifω>01+Aifω=0,
k2=α0-iωD0,
u(r)=γ(A) G0(r, r)S(r)d3r+ G0(r, r)×[r·δD(r)r-δα(r)]u(r)d3r,
(r2-k2)G0(r, r)=-1D0δ(r-r).
G0(r, r)=exp(-k|r-r|)4πD0|r-r|.
G(r1, r2)=G0(r1, r2)+ G0(r1, r)×[r·δD(r)D(r)r-δα(r)]G(r, r2)d3r.
G(r1, r2)=G0(r1, r2)- G0(r1, r)G0(r, r2)δα(r)d3r- rG0(r1, r)·rG0(r, r2)δD(r)d3r.
ϕ(r1, r2)=G0(r1, r2)-G(r1, r2).
ϕ(r1, r2)= G0(r1, r)G0(r, r2)δα(r)d3r+ rG0(r1, r)·rG0(r, r2)δD(r)d3r.
G0(r1, r2)=1(2π)3D0  exp[iq·(r1-r2)]q2+k2 d3q.
G0(ρ1, φ1, z1; ρ2, φ2, z2)=1(2π)2D0 m=- exp[im(φ1-φ2)]×- exp[iq(z1-z2)]Km[(q2+k2)1/2ρ>]×Im[(q2+k2)1/2ρ<]dq,
G0(r1, r2)=k2π2D0 l=0(2l+1)il(kr<)kl(kr>)Pl(rˆ1·rˆ2).
ϕα(ρ1, zs; ρ2, zd)=1(2π)6D02  d2ρdzδα(ρ, z)× d3p1d3p2 exp{i[p1·(ρ1-ρ)+p2·(ρ-ρ2)+p1z(zd-z)+p2z(z-zs)]}[(p1)2+(p1z)2+k2][(p2)2+(p2z)2+k2],
ϕα(q1, q2)= d2ρ1d2ρ2ϕα(ρ1, zs; ρ2, zd)×exp[i(q1·ρ1+q2·ρ2)],
ϕα(q1, q2)= d2ρdzδα(p, z) exp[i(q1+q2)·ρ-Q(q1)|z-zs|-Q(q2)|z-zd|](2D0)2Q(q1)Q(q2),
Q(q)(q2+k2)1/2
- exp(izt)dtt2+q2+k2=π exp[-Q(q)|z|]Q(q).
ϕα(q1, q2)= d2ρdzδα(ρ, z)×exp [i(q1+q2)·ρ-(Q(q1)+Q(q2))z](2D0)2Q(q1)Q(q2).
ζ=q1+q2,
η1=2Q(q1),
η2=2Q(q2).
J=(ζ, η1, η2)(q1, q2)=4(q2×q1)·eˆzQ(q1)Q(q2),
q1,2=ζ2±ez×ζζ η22-ζ22-k21/2.
ψα(ζ, η)= d2ρdzδα(ρ, z)exp(iζ·ρ+ηz),
ψα(ζ, η)=(D0η)2ϕα[q1(ζ, η), q2(ζ, η)].
δα(ρ, z)=1i(2π)3  d2ζ dηψα(ζ, η)exp(ηz-iζ·ρ),
ϕα(ρ1, 0; ρ2, 0)=α0(4πD0)2exp(-k{[(ρ1-ρ0)2+z02]1/2+[(ρ2-ρ0)2+z02]1/2}){[(ρ1-ρ0)2+z02][(ρ2-ρ0)2+z02]}1/2.
ϕα(q1, q2)=α0(2D0)2 exp[i(q1+q2)·ρ0-(Q(q1)+Q(q2))z0]Q(q1)Q(q2).
ψα(ζ, η)=α0 exp(iζ·ρ0-ηz0).
δα(ρ, z)=1(2π)3  d2ζ-dσψα(ζ, iσ)×exp[i(σz-ζ·ρ)].
Kα(ρ, z; ρ1, ρ2)=-D02(2π)3 -dσσ2 exp(iσz)× d2ζ expiζ·ρ1+ρ22-ρ-eˆ×ζζ·(ρ1-ρ2)×[(σ/2)2+(ζ/2)2+k2]1/2.
 d2ρ1d2ρ2Kα(ρ, z; ρ1, ρ2)ϕα(ρ1, ρ2)=δα(ρ, z),
 d2ρ1d2ρ2Kα(ρ, z; ρ1, ρ2)G0(ρ1, 0; ρ, z)×G0(ρ2, 0; ρ, z)=δ(ρ-ρ)δ(z-z).
ϕD(ρ1, ρ2)=1(2π)6D02  d3rδD(ρ, z)×r d3p1 exp[ip1·(ρ1+zdeˆz-r)]p12+k2·r d3p2 exp[ip2·(r-ρ2-zseˆs)]p22+k2,
ϕD(q1, q2)=1(2D0)2Q(q1)Q(q2)  d2ρdzδD(ρ, z)×[r exp(iq1·ρ-Q(q1)|z-zs|)]·[r exp(iq2·ρ-Q(q2)|z-zd)].
ϕD(q1, q2)=Q(q1)Q(q2)-q1·q2(2D0)2Q(q1)Q(q2)  d2ρdzδD(ρ, z)×exp{i(q1+q2)·ρ-[Q(q1)+Q(q2)]z}.
ψD(ζ, η)= d2ρdzδD(ρ, z)exp(iζ·ρ+ηz)
ψD(ζ, η)=(D0η)2η2-ζ2-2k2ϕD[q1(ζ, η), q2(ζ, η)].
KD(ρ, z; ρ1, ρ2)=2D02(2π)3 -dσσ2 exp(iσz) d2ζσ2+ζ2+2k2×expiζ·ρ1+ρ22-ρ-eˆz×ζζ·(ρ1-ρ2)×[(σ/2)2+(ζ/2)2+k2]1/2.
ϕtot(q1, q2)=1(2D0)2Q(q1)Q(q2) d3rδα(ρ, z)×exp[i(q1+q2)·ρ-(Q(q1)+Q(q2))z]+Q(q1)Q(q2)-q1·q2(2D0)2Q(q1)Q(q2)  d3rδD(ρ, z)×exp[i(q1+q2)·ρ-(Q(q1)+Q(q2))z].
ϕtot(ζ, η, k)= d2ρdz[cα(ζ, η)δα(ρ, z)+cD(ζ, η, k)δD(ρ, z)]exp(iζ·ρ-ηz),
cα(ζ, η)=1/(D0η)2,
cD(ζ, η, k)=(η2/2-ζ2/2-k2)/(D0η)2.
ϕtot(ζ, η, k1)=cα(ζ, η)ψα(ζ, η)+cD(ζ, η, k1)ψD(ζ, η),
ϕtot(ζ, η, k2)=cα(ζ, η)ψα(ζ, η)+cD(ζ, η, k2)ψD(ζ, η),
ψα(ζ, η)=(D0η)2k12-k22{k12ϕtot(ζ, η, k2)-k22ϕtot(ζ, η, k1)+(η2/2-ζ2/2)[ϕtot(ζ, η, k1)-ϕtot(ζ, η, k2)]},
ψD(ζ, η)=(D0η)2 ϕtot(ζ, η, k2)-ϕtot(ζ, η, k1)k12-k22.
ϕα(q1, q2)=exp[-LQ(q2)](2D0)2Q(q1)Q(q2)  d2ρdzδα(ρ, z)×exp{i(q1+q2)·ρ-[Q(q1)-Q(q2)]z}.
ψα(ζ, η)=-(D0η)2 exp(-ηL/2)ϕα[q1(ζ, η),q2(ζ, η)].
ϕD(q1, q2)=-exp[-LQ(q1)][Q(q1)Q(q2)+q1·q2](2D0)2Q(q1)Q(q2)× d2ρdzδα(ρ, z)exp[i(q1+q2)·ρ-(Q(q1)-Q(q2))z].
ψD(ζ, η)=-(D0η)2 exp(-ηL/2)ϕD[q1(ζ, η), q2(ζ, η)](η2-ζ2-2k2).
ϕα(ρ2, k)=1(2π)3D0  d3rδα(r)G0(r, 0)× d3p exp[ip·(ρ1-r)]p2+k2,
ϕα(q, k)= ϕα(ρ1, k)exp(iq·ρ1)d2ρ1
ϕα(q, k)=12D0  d3ra(r) exp[iq·ρ-zQ(q)]Q(q).
ζ=q,
η=Q(q).
ψα(ζ, η)= d2ρdza(ρ, z)exp(iζ·ρ-ηz),
ϕα(φ1, z1; φ2, z2)=1(2π)4D02 m1,m2=- exp[-i(m1φ1-m2φ2)]×-dq1-dq2 exp[i(q1z1-q2z2)]×Km1(RQ(q1))Km2(RQ(q2))× d3rδα(r)exp{i[(m2-m1)φ+(q2-q1)z]}×Im1(ρQ(q1))Im2(ρQ(q2)),
ϕα(m1, q1; m2, q2; φ˜)=-dz1-dz202πdφ102πdφ2ϕα(φ1, z1; φ2, z2)×exp{i[q1z1+q2z2+m1(φ1-φ˜)+m2(φ2-φ˜)]},
ϕα(m1, q1; m2, q2; φ˜)=exp[-i(m1+m2)φ˜](2π)4D02Km1(RQ(q1))Km2(RQ(q2))× d3rδα(r)exp[i(q1+q2)z+i(m1+m2)φ]×Im1(ρQ(q1))Im2(ρQ(q2)).
ψα(q1, q2; φ˜)=m1,m2=- (2π)4D02ϕα(m1, q1; m2, q2; φ˜)Km1(RQ(q1))Km2(RQ(q2))
m=-Im(w)exp(imθ)=exp(w cos θ)
ψα(q1, q2; φ˜)= d3rδα(r)exp[i(q1+q2)z+ρ·eˆ(Q(q1)+Q(q2))],
η=q1+q2,
ζ=(Q(q1)+Q(q2))eˆ.
J=(η, ζ)(q1, q2, eˆ)=q2Q(q2)-q1Q(q1)[Q(q1)+Q(q2)]
q1,2=η4+4k2ζ2-ζ44(η2-ζ2)±ηζ2 4k2η2-ζ2+11/21/2,
ψα(η, ζ)= d3rδα(r)exp(iηz+ζ·ρ),
Kα(ρ, z; φ1, z1, φ2, z2)=-2πD02m1,m2=- exp[i(m1φ1+m2φ2)]-dη exp(-iηz)× d2ζ exp[-ζ·ρ-i(m1+m2)φ˜+i(q1z1+q2z2)]Km1(RQ(q1))Km2(RQ(q2)),
ϕD(φ1, z1; φ2, z2)=1(2π)4D02 m1,m2=- exp[i(m1φ1-m2φ2)]×-dq1-dq2 exp[i(q1z1-q2z2)]×Km1(RQ(q1))Km2(RQ(q2)) d3rδD(r)×[r exp[-i(q1z+m1φ)]Im1(ρQ(q1))]·[r exp[i(q2z+m2φ)]Im2(ρQ(q2))].
ϕD(m1, q1; m2, q2; φ˜)=(2π)4D02 exp[-i(m1+m2)φ˜]×Km1(RQ(q1))Km2(RQ(q2))× d3rδα(r){r exp[i(q1z+m1φ)]Im1(ρQ(q1))}·{r exp[i(q2z+m2φ)]Im2(ρQ(q2))}.
ψD(q1, q2; φ˜)= d3rδD(r)[r exp(iq1z+ρ·eˆQ(q1))]·[r exp(iq2z+ρ·eˆQ(q2))].
ψD(q1, q2; φ˜)=[Q(q1)Q(q2)-q1q2]× d3rδD(r)exp{i(q1+q2)z+[Q(q1)+Q(q2)]eˆ·ρ}.
ψD(η, ζ)=ζ2-η2-2k22  d3rδD(r)exp(iηz+ζ·ρ),
Q(q1)Q(q2)-q1q2=ζ2-η2-2k22,
ψtot(q1, q2; φ˜)=m1,m2=- (2π)4D02ϕtot(m1, q1; m2, q2; φ˜)Km1(RQ(q1))Km2(RQ(q2)).
ψtot(η, ζ, k)= d3r[δα(r)+c(η, ζ, k)δD(r)]×exp(iηz+ζ·ρ),
ϕα(m1, q1; m2, q2; φ˜1, φ˜2)=-dz1-dz202πdφ102πdφ2ϕα(φ1, z1; φ2, z2)×exp{i[q1z1+q2z2+m1(φ1-φ˜1)+m2(φ2-φ˜2)]}.
ψα(q1, q2; φ˜1, φ˜2)= d3rδα(r)exp[i(q1+q2)z+ρ·(eˆ1Q(q1)+eˆ2Q(q2))].
η=q1+q2,
ζ=eˆ1Q(q1)+eˆ2Q(q2).
η=2q,
ζ=Q(q)(eˆ1+eˆ2).
ϕα(rˆ1, rˆ2)=k2π2D02l1,l2=0(2l1+1)(2l2+1)kl1(kR)kl2(kR)× d3rδα(r)il1(kr)il2(kr)Pl1(rˆ·rˆ1)Pl2(rˆ·rˆ1).
ϕα(l1, l2; eˆ1, eˆ2)= ϕα(rˆ1, rˆ2)Pl1(eˆ1·rˆ1)×Pl2(eˆ2·rˆ2)dΩ1dΩ2,
 Pl1(aˆ·xˆ)Pl2(bˆ·xˆ)dΩxˆ=4πδl1l22l1+1Pl1(aˆ·bˆ),
ϕα(l1, l2; eˆ1, eˆ2)=2kπD02kl1(kR)kl2(kR)× d3rδα(r)il1(kr)il2(kr)Pl1(rˆ·eˆ1)Pl2(rˆ·eˆ2).
exp(a·b)=l=0(2l+1)il(ab)Pl(aˆ·bˆ)
ψα(eˆ1, eˆ2)=l1,l2=0πD02k2×(2l1+1)(2l2+1)ϕα(l1, l2; eˆ1, eˆ2)kl1(kR)kl2(kR).
ψa(eˆ1, eˆ2)= d3rδα(r)exp[k(eˆ1+eˆ2)·r].
ψD(eˆ1, eˆ2)= d3rδD(r)[r exp(keˆ1·r)]·[r exp(keˆ2·r)],
ψD(eˆ1, eˆ2)=k2(eˆ1·eˆ2) d3rδD(r)exp[k(eˆ1+eˆ2)·r].
ψtot(eˆ1, eˆ2)= d3r[δα(r)+k2(eˆ1·eˆ2)δD(r)]×exp[k(eˆ1+eˆ2)·r)].

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