Abstract

We compare the radiative transfer equation for media with constant refractive index with the radiative transfer equation for media with spatially varying refractive indices [J. Opt. A Pure App. Opt. 1, L1 (1999)] and obtain approximate conditions under which the former equation is accurate for modeling light propagation in scattering media with spatially varying refractive indices. These conditions impose restrictions on the variations of the refractive index, the gradient of the refractive index, the divergence of the rays, the frequency of modulation, and the widths of light pulses, which are related to the mean refractive index, the absorption coefficient, and the reduced scattering coefficient of the medium. Each condition is geometrically interpreted. Some implications of the results are discussed.

© 2003 Optical Society of America

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  27. J. B. Fishkin, O. Coquoz, E. R. Anderson, M. Brenner, B. J. Tromberg, “Frequency-domain photon migration measurements of normal and malignant tissue optical properties in a human subject,” App. Opt. 36, 10–20 (1997).
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    [Crossref]
  29. M. Firbank, M. Oda, D. T. Delpy, “An improved design for a stable and reproducible phantom material for use in near-infrared spectroscopy and imaging,” Phys. Med. Biol. 40, 955–961 (1995).
    [Crossref] [PubMed]

2003 (1)

T. Khan, H. Jiang, “A new diffusion approximation to the radiative transfer equation for scattering media with spatially varying refractive indices,” J. Opt. A Pure Appl. Opt. 5, 137–141 (2003).
[Crossref]

2001 (1)

B. Chen, K. Stamnes, J. Stamnes, “Validity of the diffusion approximation in bio-optical imaging,” App. Opt. 40, 6356–6366 (2001).
[Crossref]

2000 (5)

F. Martelli, M. Bassani, L. Alianelli, L. Zanghaeri, G. Zaccanti, “Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation,” Phys. Med. Biol. 45, 1359–1373 (2000).
[Crossref] [PubMed]

D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W.-F. Cheong, D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20, 469–477 (2000).
[Crossref] [PubMed]

F. E. W. Schmidt, M. E. Fry, E. M. C. Hillman, J. C. Hebden, D. T. Delpy, “A 32-channel time-resolved instrument for medical optical tomography,” Rev. Sci. Instrum. 71, 256–265 (2000).
[Crossref]

M. A. Franceschini, V. Toronov, M. E. Filiaci, E. Gratton, S. Fantini, “On-line optical imaging of the human brain with 160-ms temporal resolution,” Opt. Express 6, 49–57 (2000).
[Crossref] [PubMed]

H. Dehghani, S. R. Arridge, M. Schweiger, D. T. Delpy, “Optical tomography in the presence of void regions,” J. Opt. Soc. Am. A 17, 1659–1670 (2000).
[Crossref]

1999 (4)

A. Roggan, D. Schädel, U. Netz, J.-P. Ritz, C.-T. Germer, G. Müller, “The effect of preparation technique on the optical parameters of biological tissue,” Appl. Phys. B 69, 445–453 (1999).
[Crossref]

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, M. Tamura, “Multichannel time-resolved optical tomographic system,” Rev. Sci. Instrum. 70, 3595–3602 (1999).
[Crossref]

H. A. Ferwerda, “The radiative transfer equation for scattering media with spatially varying refractive index,” J. Opt. A Pure Appl. Opt. 1, L1–L2 (1999).
[Crossref]

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

1998 (1)

1997 (4)

B. W. Pogue, M. Testorf, T. McBride, U. Osterberg, K. Paulsen, “Instrumentation and design of a frequency-domain diffuse optical tomography imager for breast cancer detection,” Opt. Express 1, 391–403 (1997).
[Crossref] [PubMed]

J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[Crossref] [PubMed]

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[Crossref] [PubMed]

J. B. Fishkin, O. Coquoz, E. R. Anderson, M. Brenner, B. J. Tromberg, “Frequency-domain photon migration measurements of normal and malignant tissue optical properties in a human subject,” App. Opt. 36, 10–20 (1997).
[Crossref]

1996 (1)

J. Beuthan, O. Minet, J. Helfmann, M. Herrig, G. Müller, “The spatial variation of refractive index in biological cells,” Phys. Med. Biol. 41, 369–382 (1996).
[Crossref] [PubMed]

1995 (3)

A. Duncan, J. H. Meek, M. Clemence, C. E. Elwell, L. Tyszczuk, M. Cope, D. T. Delpy, “Optical pathlength measurements on adult head, calf and forearm and the head of the newborn infant using phase resolved optical spectroscopy,” Phys. Med. Biol. 40, 295–304 (1995).
[Crossref] [PubMed]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography,” Opt. Lett. 20, 426–428 (1995).
[Crossref] [PubMed]

M. Firbank, M. Oda, D. T. Delpy, “An improved design for a stable and reproducible phantom material for use in near-infrared spectroscopy and imaging,” Phys. Med. Biol. 40, 955–961 (1995).
[Crossref] [PubMed]

1991 (1)

M. S. Patterson, B. S. C. Wilson, D. R. Wyman, “The propagation of optical radiation in tissue I. Models of radiation transport and their application,” Lasers Med. Sci. 6, 155–168 (1991).
[Crossref]

1990 (1)

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990). Available also at http://omlc.ogi.edu/pubs/pdf/cheong90a.pdf .
[Crossref]

1989 (1)

F. P. Bolin, L. E. Preuss, R. C. Taylor, R. J. Ference, “Refractive index of some mammalian tissues using a fiber optic cladding method,” App. Opt. 28, 2297–2303 (1989).
[Crossref]

1988 (1)

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, J. S. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurements,” Phys. Med. Biol. 33, 1433–1442 (1988).
[Crossref] [PubMed]

Alianelli, L.

F. Martelli, M. Bassani, L. Alianelli, L. Zanghaeri, G. Zaccanti, “Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation,” Phys. Med. Biol. 45, 1359–1373 (2000).
[Crossref] [PubMed]

Anderson, E. R.

J. B. Fishkin, O. Coquoz, E. R. Anderson, M. Brenner, B. J. Tromberg, “Frequency-domain photon migration measurements of normal and malignant tissue optical properties in a human subject,” App. Opt. 36, 10–20 (1997).
[Crossref]

Arridge, S. R.

H. Dehghani, S. R. Arridge, M. Schweiger, D. T. Delpy, “Optical tomography in the presence of void regions,” J. Opt. Soc. Am. A 17, 1659–1670 (2000).
[Crossref]

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

J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[Crossref] [PubMed]

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[Crossref] [PubMed]

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, J. S. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurements,” Phys. Med. Biol. 33, 1433–1442 (1988).
[Crossref] [PubMed]

S. R. Arridge, “Diffusion tomography in dense media,” in Scattering and Inverse Scattering in Pure and Applied Science, R. Pike, P. Sabatier, eds. (Academic, San Diego, Calif., 2002), pp. 920–936.

Bassani, M.

F. Martelli, M. Bassani, L. Alianelli, L. Zanghaeri, G. Zaccanti, “Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation,” Phys. Med. Biol. 45, 1359–1373 (2000).
[Crossref] [PubMed]

Benaron, D. A.

D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W.-F. Cheong, D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20, 469–477 (2000).
[Crossref] [PubMed]

Beuthan, J.

J. Beuthan, O. Minet, J. Helfmann, M. Herrig, G. Müller, “The spatial variation of refractive index in biological cells,” Phys. Med. Biol. 41, 369–382 (1996).
[Crossref] [PubMed]

Boas, D.

D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W.-F. Cheong, D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20, 469–477 (2000).
[Crossref] [PubMed]

Boas, D. A.

Bolin, F. P.

F. P. Bolin, L. E. Preuss, R. C. Taylor, R. J. Ference, “Refractive index of some mammalian tissues using a fiber optic cladding method,” App. Opt. 28, 2297–2303 (1989).
[Crossref]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1975).

Brenner, M.

J. B. Fishkin, O. Coquoz, E. R. Anderson, M. Brenner, B. J. Tromberg, “Frequency-domain photon migration measurements of normal and malignant tissue optical properties in a human subject,” App. Opt. 36, 10–20 (1997).
[Crossref]

Chance, B.

Chen, B.

B. Chen, K. Stamnes, J. Stamnes, “Validity of the diffusion approximation in bio-optical imaging,” App. Opt. 40, 6356–6366 (2001).
[Crossref]

Cheong, W. F.

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990). Available also at http://omlc.ogi.edu/pubs/pdf/cheong90a.pdf .
[Crossref]

Cheong, W.-F.

D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W.-F. Cheong, D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20, 469–477 (2000).
[Crossref] [PubMed]

Clemence, M.

A. Duncan, J. H. Meek, M. Clemence, C. E. Elwell, L. Tyszczuk, M. Cope, D. T. Delpy, “Optical pathlength measurements on adult head, calf and forearm and the head of the newborn infant using phase resolved optical spectroscopy,” Phys. Med. Biol. 40, 295–304 (1995).
[Crossref] [PubMed]

Cope, M.

A. Duncan, J. H. Meek, M. Clemence, C. E. Elwell, L. Tyszczuk, M. Cope, D. T. Delpy, “Optical pathlength measurements on adult head, calf and forearm and the head of the newborn infant using phase resolved optical spectroscopy,” Phys. Med. Biol. 40, 295–304 (1995).
[Crossref] [PubMed]

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, J. S. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurements,” Phys. Med. Biol. 33, 1433–1442 (1988).
[Crossref] [PubMed]

Coquoz, O.

J. B. Fishkin, O. Coquoz, E. R. Anderson, M. Brenner, B. J. Tromberg, “Frequency-domain photon migration measurements of normal and malignant tissue optical properties in a human subject,” App. Opt. 36, 10–20 (1997).
[Crossref]

Dehghani, H.

Delpy, D. T.

H. Dehghani, S. R. Arridge, M. Schweiger, D. T. Delpy, “Optical tomography in the presence of void regions,” J. Opt. Soc. Am. A 17, 1659–1670 (2000).
[Crossref]

F. E. W. Schmidt, M. E. Fry, E. M. C. Hillman, J. C. Hebden, D. T. Delpy, “A 32-channel time-resolved instrument for medical optical tomography,” Rev. Sci. Instrum. 71, 256–265 (2000).
[Crossref]

J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[Crossref] [PubMed]

A. Duncan, J. H. Meek, M. Clemence, C. E. Elwell, L. Tyszczuk, M. Cope, D. T. Delpy, “Optical pathlength measurements on adult head, calf and forearm and the head of the newborn infant using phase resolved optical spectroscopy,” Phys. Med. Biol. 40, 295–304 (1995).
[Crossref] [PubMed]

M. Firbank, M. Oda, D. T. Delpy, “An improved design for a stable and reproducible phantom material for use in near-infrared spectroscopy and imaging,” Phys. Med. Biol. 40, 955–961 (1995).
[Crossref] [PubMed]

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, J. S. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurements,” Phys. Med. Biol. 33, 1433–1442 (1988).
[Crossref] [PubMed]

Duncan, A.

A. Duncan, J. H. Meek, M. Clemence, C. E. Elwell, L. Tyszczuk, M. Cope, D. T. Delpy, “Optical pathlength measurements on adult head, calf and forearm and the head of the newborn infant using phase resolved optical spectroscopy,” Phys. Med. Biol. 40, 295–304 (1995).
[Crossref] [PubMed]

Eda, H.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, M. Tamura, “Multichannel time-resolved optical tomographic system,” Rev. Sci. Instrum. 70, 3595–3602 (1999).
[Crossref]

Elwell, C. E.

A. Duncan, J. H. Meek, M. Clemence, C. E. Elwell, L. Tyszczuk, M. Cope, D. T. Delpy, “Optical pathlength measurements on adult head, calf and forearm and the head of the newborn infant using phase resolved optical spectroscopy,” Phys. Med. Biol. 40, 295–304 (1995).
[Crossref] [PubMed]

Fantini, S.

Ference, R. J.

F. P. Bolin, L. E. Preuss, R. C. Taylor, R. J. Ference, “Refractive index of some mammalian tissues using a fiber optic cladding method,” App. Opt. 28, 2297–2303 (1989).
[Crossref]

Ferwerda, H. A.

H. A. Ferwerda, “The radiative transfer equation for scattering media with spatially varying refractive index,” J. Opt. A Pure Appl. Opt. 1, L1–L2 (1999).
[Crossref]

Filiaci, M. E.

Firbank, M.

M. Firbank, M. Oda, D. T. Delpy, “An improved design for a stable and reproducible phantom material for use in near-infrared spectroscopy and imaging,” Phys. Med. Biol. 40, 955–961 (1995).
[Crossref] [PubMed]

Fishkin, J. B.

J. B. Fishkin, O. Coquoz, E. R. Anderson, M. Brenner, B. J. Tromberg, “Frequency-domain photon migration measurements of normal and malignant tissue optical properties in a human subject,” App. Opt. 36, 10–20 (1997).
[Crossref]

Frahm, J.

D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W.-F. Cheong, D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20, 469–477 (2000).
[Crossref] [PubMed]

Franceschini, M. A.

Fry, M. E.

F. E. W. Schmidt, M. E. Fry, E. M. C. Hillman, J. C. Hebden, D. T. Delpy, “A 32-channel time-resolved instrument for medical optical tomography,” Rev. Sci. Instrum. 71, 256–265 (2000).
[Crossref]

Germer, C.-T.

A. Roggan, D. Schädel, U. Netz, J.-P. Ritz, C.-T. Germer, G. Müller, “The effect of preparation technique on the optical parameters of biological tissue,” Appl. Phys. B 69, 445–453 (1999).
[Crossref]

Gratton, E.

Hebden, J. C.

F. E. W. Schmidt, M. E. Fry, E. M. C. Hillman, J. C. Hebden, D. T. Delpy, “A 32-channel time-resolved instrument for medical optical tomography,” Rev. Sci. Instrum. 71, 256–265 (2000).
[Crossref]

J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[Crossref] [PubMed]

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[Crossref] [PubMed]

Helfmann, J.

J. Beuthan, O. Minet, J. Helfmann, M. Herrig, G. Müller, “The spatial variation of refractive index in biological cells,” Phys. Med. Biol. 41, 369–382 (1996).
[Crossref] [PubMed]

Herrig, M.

J. Beuthan, O. Minet, J. Helfmann, M. Herrig, G. Müller, “The spatial variation of refractive index in biological cells,” Phys. Med. Biol. 41, 369–382 (1996).
[Crossref] [PubMed]

Hillman, E. M. C.

F. E. W. Schmidt, M. E. Fry, E. M. C. Hillman, J. C. Hebden, D. T. Delpy, “A 32-channel time-resolved instrument for medical optical tomography,” Rev. Sci. Instrum. 71, 256–265 (2000).
[Crossref]

Hintz, S. R.

D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W.-F. Cheong, D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20, 469–477 (2000).
[Crossref] [PubMed]

Hirth, C.

D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W.-F. Cheong, D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20, 469–477 (2000).
[Crossref] [PubMed]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

Ito, Y.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, M. Tamura, “Multichannel time-resolved optical tomographic system,” Rev. Sci. Instrum. 70, 3595–3602 (1999).
[Crossref]

Jiang, H.

T. Khan, H. Jiang, “A new diffusion approximation to the radiative transfer equation for scattering media with spatially varying refractive indices,” J. Opt. A Pure Appl. Opt. 5, 137–141 (2003).
[Crossref]

Kermit, E. L.

D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W.-F. Cheong, D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20, 469–477 (2000).
[Crossref] [PubMed]

Khan, T.

T. Khan, H. Jiang, “A new diffusion approximation to the radiative transfer equation for scattering media with spatially varying refractive indices,” J. Opt. A Pure Appl. Opt. 5, 137–141 (2003).
[Crossref]

Kleinschmidt, A.

D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W.-F. Cheong, D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20, 469–477 (2000).
[Crossref] [PubMed]

Levin, G. G.

G. G. Levin, G. N. Vishnyakov, Optical Tomography (Radio i Svyaz’, Moscow, 1989) (in Russian).

Martelli, F.

F. Martelli, M. Bassani, L. Alianelli, L. Zanghaeri, G. Zaccanti, “Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation,” Phys. Med. Biol. 45, 1359–1373 (2000).
[Crossref] [PubMed]

McBride, T.

Meek, J. H.

A. Duncan, J. H. Meek, M. Clemence, C. E. Elwell, L. Tyszczuk, M. Cope, D. T. Delpy, “Optical pathlength measurements on adult head, calf and forearm and the head of the newborn infant using phase resolved optical spectroscopy,” Phys. Med. Biol. 40, 295–304 (1995).
[Crossref] [PubMed]

Minet, O.

J. Beuthan, O. Minet, J. Helfmann, M. Herrig, G. Müller, “The spatial variation of refractive index in biological cells,” Phys. Med. Biol. 41, 369–382 (1996).
[Crossref] [PubMed]

Müller, G.

A. Roggan, D. Schädel, U. Netz, J.-P. Ritz, C.-T. Germer, G. Müller, “The effect of preparation technique on the optical parameters of biological tissue,” Appl. Phys. B 69, 445–453 (1999).
[Crossref]

J. Beuthan, O. Minet, J. Helfmann, M. Herrig, G. Müller, “The spatial variation of refractive index in biological cells,” Phys. Med. Biol. 41, 369–382 (1996).
[Crossref] [PubMed]

Murrer, L. H. P.

Netz, U.

A. Roggan, D. Schädel, U. Netz, J.-P. Ritz, C.-T. Germer, G. Müller, “The effect of preparation technique on the optical parameters of biological tissue,” Appl. Phys. B 69, 445–453 (1999).
[Crossref]

O’Leary, M. A.

Obrig, H.

D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W.-F. Cheong, D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20, 469–477 (2000).
[Crossref] [PubMed]

Oda, I.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, M. Tamura, “Multichannel time-resolved optical tomographic system,” Rev. Sci. Instrum. 70, 3595–3602 (1999).
[Crossref]

Oda, M.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, M. Tamura, “Multichannel time-resolved optical tomographic system,” Rev. Sci. Instrum. 70, 3595–3602 (1999).
[Crossref]

M. Firbank, M. Oda, D. T. Delpy, “An improved design for a stable and reproducible phantom material for use in near-infrared spectroscopy and imaging,” Phys. Med. Biol. 40, 955–961 (1995).
[Crossref] [PubMed]

Oikawa, Y.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, M. Tamura, “Multichannel time-resolved optical tomographic system,” Rev. Sci. Instrum. 70, 3595–3602 (1999).
[Crossref]

Osterberg, U.

Patterson, M. S.

M. S. Patterson, B. S. C. Wilson, D. R. Wyman, “The propagation of optical radiation in tissue I. Models of radiation transport and their application,” Lasers Med. Sci. 6, 155–168 (1991).
[Crossref]

Paulsen, K.

Pogue, B. W.

Prahl, S. A.

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990). Available also at http://omlc.ogi.edu/pubs/pdf/cheong90a.pdf .
[Crossref]

Preuss, L. E.

F. P. Bolin, L. E. Preuss, R. C. Taylor, R. J. Ference, “Refractive index of some mammalian tissues using a fiber optic cladding method,” App. Opt. 28, 2297–2303 (1989).
[Crossref]

Rinzema, K.

Ritz, J.-P.

A. Roggan, D. Schädel, U. Netz, J.-P. Ritz, C.-T. Germer, G. Müller, “The effect of preparation technique on the optical parameters of biological tissue,” Appl. Phys. B 69, 445–453 (1999).
[Crossref]

Roggan, A.

A. Roggan, D. Schädel, U. Netz, J.-P. Ritz, C.-T. Germer, G. Müller, “The effect of preparation technique on the optical parameters of biological tissue,” Appl. Phys. B 69, 445–453 (1999).
[Crossref]

Sassaroli, A.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, M. Tamura, “Multichannel time-resolved optical tomographic system,” Rev. Sci. Instrum. 70, 3595–3602 (1999).
[Crossref]

Schädel, D.

A. Roggan, D. Schädel, U. Netz, J.-P. Ritz, C.-T. Germer, G. Müller, “The effect of preparation technique on the optical parameters of biological tissue,” Appl. Phys. B 69, 445–453 (1999).
[Crossref]

Schmidt, F. E. W.

F. E. W. Schmidt, M. E. Fry, E. M. C. Hillman, J. C. Hebden, D. T. Delpy, “A 32-channel time-resolved instrument for medical optical tomography,” Rev. Sci. Instrum. 71, 256–265 (2000).
[Crossref]

Schweiger, M.

Stamnes, J.

B. Chen, K. Stamnes, J. Stamnes, “Validity of the diffusion approximation in bio-optical imaging,” App. Opt. 40, 6356–6366 (2001).
[Crossref]

Stamnes, K.

B. Chen, K. Stamnes, J. Stamnes, “Validity of the diffusion approximation in bio-optical imaging,” App. Opt. 40, 6356–6366 (2001).
[Crossref]

Star, W. M.

Stevenson, D. K.

D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W.-F. Cheong, D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20, 469–477 (2000).
[Crossref] [PubMed]

Takada, M.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, M. Tamura, “Multichannel time-resolved optical tomographic system,” Rev. Sci. Instrum. 70, 3595–3602 (1999).
[Crossref]

Tamura, M.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, M. Tamura, “Multichannel time-resolved optical tomographic system,” Rev. Sci. Instrum. 70, 3595–3602 (1999).
[Crossref]

Taylor, R. C.

F. P. Bolin, L. E. Preuss, R. C. Taylor, R. J. Ference, “Refractive index of some mammalian tissues using a fiber optic cladding method,” App. Opt. 28, 2297–2303 (1989).
[Crossref]

Testorf, M.

Toronov, V.

Tromberg, B. J.

J. B. Fishkin, O. Coquoz, E. R. Anderson, M. Brenner, B. J. Tromberg, “Frequency-domain photon migration measurements of normal and malignant tissue optical properties in a human subject,” App. Opt. 36, 10–20 (1997).
[Crossref]

Tsuchiya, Y.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, M. Tamura, “Multichannel time-resolved optical tomographic system,” Rev. Sci. Instrum. 70, 3595–3602 (1999).
[Crossref]

Tsunazawa, Y.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, M. Tamura, “Multichannel time-resolved optical tomographic system,” Rev. Sci. Instrum. 70, 3595–3602 (1999).
[Crossref]

Tuchin, V. V.

V. V. Tuchin, “Light interaction with biological tissues (overview),” in Static and Dynamic Light Scattering in Medicine and Biology, R. J. Nossal, R. Pecora, A. V. Priezzhev, eds., Proc. SPIE1884, 234–272 (1994).
[Crossref]

Tyszczuk, L.

A. Duncan, J. H. Meek, M. Clemence, C. E. Elwell, L. Tyszczuk, M. Cope, D. T. Delpy, “Optical pathlength measurements on adult head, calf and forearm and the head of the newborn infant using phase resolved optical spectroscopy,” Phys. Med. Biol. 40, 295–304 (1995).
[Crossref] [PubMed]

van der Zee, P.

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, J. S. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurements,” Phys. Med. Biol. 33, 1433–1442 (1988).
[Crossref] [PubMed]

van Houten, J. C.

D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W.-F. Cheong, D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20, 469–477 (2000).
[Crossref] [PubMed]

Villringer, A.

D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W.-F. Cheong, D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20, 469–477 (2000).
[Crossref] [PubMed]

Vishnyakov, G. N.

G. G. Levin, G. N. Vishnyakov, Optical Tomography (Radio i Svyaz’, Moscow, 1989) (in Russian).

Wada, Y.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, M. Tamura, “Multichannel time-resolved optical tomographic system,” Rev. Sci. Instrum. 70, 3595–3602 (1999).
[Crossref]

Welch, A. J.

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990). Available also at http://omlc.ogi.edu/pubs/pdf/cheong90a.pdf .
[Crossref]

Wilson, B. S. C.

M. S. Patterson, B. S. C. Wilson, D. R. Wyman, “The propagation of optical radiation in tissue I. Models of radiation transport and their application,” Lasers Med. Sci. 6, 155–168 (1991).
[Crossref]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1975).

Wray, S.

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, J. S. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurements,” Phys. Med. Biol. 33, 1433–1442 (1988).
[Crossref] [PubMed]

Wyatt, J. S.

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, J. S. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurements,” Phys. Med. Biol. 33, 1433–1442 (1988).
[Crossref] [PubMed]

Wyman, D. R.

M. S. Patterson, B. S. C. Wilson, D. R. Wyman, “The propagation of optical radiation in tissue I. Models of radiation transport and their application,” Lasers Med. Sci. 6, 155–168 (1991).
[Crossref]

Yamada, Y.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, M. Tamura, “Multichannel time-resolved optical tomographic system,” Rev. Sci. Instrum. 70, 3595–3602 (1999).
[Crossref]

Yamashita, Y.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, M. Tamura, “Multichannel time-resolved optical tomographic system,” Rev. Sci. Instrum. 70, 3595–3602 (1999).
[Crossref]

Yodh, A. G.

Zaccanti, G.

F. Martelli, M. Bassani, L. Alianelli, L. Zanghaeri, G. Zaccanti, “Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation,” Phys. Med. Biol. 45, 1359–1373 (2000).
[Crossref] [PubMed]

Zanghaeri, L.

F. Martelli, M. Bassani, L. Alianelli, L. Zanghaeri, G. Zaccanti, “Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation,” Phys. Med. Biol. 45, 1359–1373 (2000).
[Crossref] [PubMed]

App. Opt. (3)

B. Chen, K. Stamnes, J. Stamnes, “Validity of the diffusion approximation in bio-optical imaging,” App. Opt. 40, 6356–6366 (2001).
[Crossref]

F. P. Bolin, L. E. Preuss, R. C. Taylor, R. J. Ference, “Refractive index of some mammalian tissues using a fiber optic cladding method,” App. Opt. 28, 2297–2303 (1989).
[Crossref]

J. B. Fishkin, O. Coquoz, E. R. Anderson, M. Brenner, B. J. Tromberg, “Frequency-domain photon migration measurements of normal and malignant tissue optical properties in a human subject,” App. Opt. 36, 10–20 (1997).
[Crossref]

Appl. Phys. B (1)

A. Roggan, D. Schädel, U. Netz, J.-P. Ritz, C.-T. Germer, G. Müller, “The effect of preparation technique on the optical parameters of biological tissue,” Appl. Phys. B 69, 445–453 (1999).
[Crossref]

IEEE J. Quantum Electron. (1)

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990). Available also at http://omlc.ogi.edu/pubs/pdf/cheong90a.pdf .
[Crossref]

Inverse Probl. (1)

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

J. Cereb. Blood Flow Metab. (1)

D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W.-F. Cheong, D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20, 469–477 (2000).
[Crossref] [PubMed]

J. Opt. A Pure Appl. Opt. (2)

H. A. Ferwerda, “The radiative transfer equation for scattering media with spatially varying refractive index,” J. Opt. A Pure Appl. Opt. 1, L1–L2 (1999).
[Crossref]

T. Khan, H. Jiang, “A new diffusion approximation to the radiative transfer equation for scattering media with spatially varying refractive indices,” J. Opt. A Pure Appl. Opt. 5, 137–141 (2003).
[Crossref]

J. Opt. Soc. Am. A (2)

Lasers Med. Sci. (1)

M. S. Patterson, B. S. C. Wilson, D. R. Wyman, “The propagation of optical radiation in tissue I. Models of radiation transport and their application,” Lasers Med. Sci. 6, 155–168 (1991).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Phys. Med. Biol. (7)

M. Firbank, M. Oda, D. T. Delpy, “An improved design for a stable and reproducible phantom material for use in near-infrared spectroscopy and imaging,” Phys. Med. Biol. 40, 955–961 (1995).
[Crossref] [PubMed]

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[Crossref] [PubMed]

J. Beuthan, O. Minet, J. Helfmann, M. Herrig, G. Müller, “The spatial variation of refractive index in biological cells,” Phys. Med. Biol. 41, 369–382 (1996).
[Crossref] [PubMed]

J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[Crossref] [PubMed]

F. Martelli, M. Bassani, L. Alianelli, L. Zanghaeri, G. Zaccanti, “Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation,” Phys. Med. Biol. 45, 1359–1373 (2000).
[Crossref] [PubMed]

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, J. S. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurements,” Phys. Med. Biol. 33, 1433–1442 (1988).
[Crossref] [PubMed]

A. Duncan, J. H. Meek, M. Clemence, C. E. Elwell, L. Tyszczuk, M. Cope, D. T. Delpy, “Optical pathlength measurements on adult head, calf and forearm and the head of the newborn infant using phase resolved optical spectroscopy,” Phys. Med. Biol. 40, 295–304 (1995).
[Crossref] [PubMed]

Rev. Sci. Instrum. (2)

F. E. W. Schmidt, M. E. Fry, E. M. C. Hillman, J. C. Hebden, D. T. Delpy, “A 32-channel time-resolved instrument for medical optical tomography,” Rev. Sci. Instrum. 71, 256–265 (2000).
[Crossref]

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, M. Tamura, “Multichannel time-resolved optical tomographic system,” Rev. Sci. Instrum. 70, 3595–3602 (1999).
[Crossref]

Other (5)

G. G. Levin, G. N. Vishnyakov, Optical Tomography (Radio i Svyaz’, Moscow, 1989) (in Russian).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1975).

V. V. Tuchin, “Light interaction with biological tissues (overview),” in Static and Dynamic Light Scattering in Medicine and Biology, R. J. Nossal, R. Pecora, A. V. Priezzhev, eds., Proc. SPIE1884, 234–272 (1994).
[Crossref]

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

S. R. Arridge, “Diffusion tomography in dense media,” in Scattering and Inverse Scattering in Pure and Applied Science, R. Pike, P. Sabatier, eds. (Academic, San Diego, Calif., 2002), pp. 920–936.

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Figures (1)

Fig. 1
Fig. 1

Elementary scattering volumes. ·Ω0, n0: elementary scattering volume for the RTEvri. ·Ω=0, n=0: elementary scattering volume for the RTE.

Equations (130)

Equations on this page are rendered with MathJax. Learn more.

n(r)ctL(r, Ω, t)+Ω(r)·rL(r, Ω, t)+[μa(r)+μs(r)]L(r, Ω, t)+[r·Ω(r)]L(r, Ω, t)+r ln n(r)·ΩL(r, Ω, t)=μs(r)4πL(r, Ω, t)θ(Ω, Ω)dω+(r, Ω, t),
Ω(r)=rL(r)|rL(r)|=rL(r)n(r),
rL(r)·rL(r)=n2(r).
L(r)=const.
Ω[R(s)]=ddsR(s).
ddsn[R(s)]ddsR(s)=rn[R(s)].
r·Ω(r)=1n(r)1Ωxn(r)x+1Ωyn(r)y+1Ωzn(r)z-3n(r)rn(r)·Ω(r),
L(r)=nr,
Ω(r)=rL(r)n(r)=rr,
r·Ω(r)=2r0.
r·Ω(r)=r2L(r)n(r)-Ω(r)·r ln n(r).
n¯=1VVn(r)dV,
n˜(r)=n(r)-n¯,
L(r, Ω, t)=-r ln n(r)·ΩL(r, Ω, t)-n˜(r)ctL(r, Ω, t).
n¯ctL(r, Ω, t)+Ω·rL(r, Ω, t)+[μa(r)+μs(r)+r·Ω(r)]L(r, Ω, t)
=μs(r)4πL(r, Ω, t)θ(Ω, Ω)dω+(r, Ω, t)+L(r, Ω, t).
n¯ctL(r, Ω, t)+Ω·rL(r, Ω, t)+[μa(r)+μs(r)]L(r, Ω, t)
=μs(r)4πL(r, Ω, t)θ(Ω, Ω)dω+(r, Ω, t).
|n˜(r)|n¯,
n¯ctLvri(r, Ω, t)+Ω(r)·rLvri(r, Ω, t)+[μa(r)+μs(r)+r·Ω(r)]Lvri(r, Ω, t)=μs(r)4πLvri(r, Ω, t)θ(r, Ω, Ω)dω+(r, Ω, t)+L(r, Ω, t),
n¯ctLvri(r, Ω, t)+Ω·rLvri(r, Ω, t)
+[μar(r)+μsr(r)]Lvri(r, Ω, t)
=μsr(r)4πLvri(r, Ω, t)θ(r, Ω, Ω)dω
+(r, Ω, t).
r·Ω(r)=L(r, Ω, t)Lvri(r, Ω, t),
n¯ctL(r, Ω, t)+Ω·rL(r, Ω, t)
+[μamod(r)+μs(r)]L(r, Ω, t)
=μs(r)4πL(r, Ω, t)θ(r, Ω, Ω)dω
+mod(r, Ω, t),
Lvri(r, Ω, t)=14πIvri(r, t)+34πΩ·Jvri(r, t),
(r, Ω, t)=14π0(r, t)+34πΩ·1(r, t),
Ivri(r, t)=4πLvri(r, Ω, t)dω,
Jvri(r, t)=4πLvri(r, Ω, t)Ωdω.
0(r, t)=4π(r, Ω, t)dω,
1(r, t)=4π(r, Ω, t)Ωdω.
θ(r, Ω·Ω)14π+3g(r)4πΩ·Ω,
g(r)=4πΩ·Ωθ(r, Ω, Ω)dω.
n¯ctIvri(r, t)+r·Jvri(r, t)+μa(r)Ivri(r, t)
+2r ln n(r)·Jvri(r, t)
=0(r, t)-n˜(r)ctIvri(r, t).
n¯ctJvri(r, t)-13[r ln n(r)]Ivri(r, t)+13rIvri(r, t)
+[μa(r)+μs(r)+μd(r)]Jvri(r, t)
=1n¯1(r, t)-n˜(r)ctJvri(r, t),
μs(r)=[1-g(r)]μs(r),
μd(r)=r2L(r)n(r).
r·Jvri(r)+μa(r)Ivri(r)+2r ln n(r)·Jvri(r)=0(r),
-13[r ln n(r)]Ivri(r)+13rIvri(r)+[μa(r)+μs(r)+μd(r)]Jvri(r)=1(r).
|r ln n(r)·Jvri(r)|12|μa(r)Ivri(r)|,
|[r ln n(r)]Ivri(r)|3|min[μa(r), μs(r)]Jvri(r)|,
|μd(r)Jvri(r)||min[μa(r), μs(r)]Jvri(r)|,
|r ln n(r)·Jvri(r)||r ln n(r)||Jvri(r)||r ln n(r)|Ivri(r),
|r ln n(r)|μa(r)2,
|r ln n(r)|3η min[μa(r), μs(r)],
|μd(r)||μd max|=min[μa(r), μs(r)],
|r ln n(r)|minμa(r)2, 3η min[μa(r), μs(r)].
RRmin=max2μa(r),13ηmax1μa(r),1μs(r),
RRmin=max2la,13ηmax(la, ls).
|r·Ω(r)||μd(r)|+|r ln n(r)|.
|r·Ω(r)||r·Ω(r)|max=min[μa(r), μs(r)]+minμa(r)2, 3η min[μa(r), μs(r)],
1cn˜(r)tIvri(r, t)n¯ctIvri(r, t),
n˜(r)ctIvri(r, t)μa(r)Ivri(r, t),
n˜(r)ctJvri(r, t)1cn¯tJvri(r, t),
n˜(r)ctJvri(r, t)[μa(r)+μs(r)]Jvri(r, t)|.
Lvri(r, Ω, t)=Li(r, Ω)exp-(t-t0)2w02(r),
Ivri(r, t)=Ii(r)exp-(t-t0)2w02(r),
Jvri(r, t)=Ji(r)exp-(t-t0)2w02(r).
2(t-t0)n˜(r)cw02Ivri(r)exp-(t-t0)2w02
μa(r)Ivri(r)exp-(t-t0)2w02,
2(t-t0)n˜(r)cw02Jvri(r)exp-(t-t0)2w02
[μa(r)+μs(r)]Ivri(r)exp-(t-t0)2w02,
|n˜(r)|w0(r)2|n˜(r)|wmin=c min[μa(r), μa(r)],
|n˜(r)|n¯lwidth2 max[la, ls].
fmax<1wmin=c2|n˜(r)|min[μa(r), μs(r)],
0(r, t)=0i(r)[1+γ cos(2πft)],
1(r, t)=1i(r)[1+γ cos(2πft)],
Ivri(r, t)=Ii(r)+Id(r, t),
Jvri(r, t)=Ji(r)+Jd(r, t),
r·Ji(r)+μa(r)Ii(r)+2r ln n(r)·Ji(r)=0i(r),
-13[r ln n(r)]Ii(r)+13rIi(r)+[μa(r)+μs(r)]Ji(r)=1i(r)
n¯ctId(r, t)+n˜(r)ctId(r, t)+r·Jd(r, t)
+μa(r)Id(r, t)+2r ln n(r)·Jd(r, t)
=γ0i(r)cos(2πft),
n¯ctJd(r, t)-13[r ln n(r)]Id(r, t)+13rId(r, t)
+[μa(r)+μs(r)+μd(r)]Jd(r, t)+n˜(r)ctJd(r, t)
=γ1i(r)cos(2πft).
Ivri(r, t)¯=Ii(r),Id(r, t)¯=0,
Jvri(r, t)¯=Ji(r),Jd(r, t)¯=0,
n˜(r)c2Id(r, t)t2¯μa2(r)Id2(r, t)¯,
n˜(r)c2Jd(r, t)t2¯[μa(r)+μs(r)]2|Jd(r, t)|2¯.
Id(r, t)=αIi(r)cos[2πft+ϕI(r)],
Jd(r, t)=βJi(r)cos[2πft+ϕJ(r)],
ffmax=c2π|n˜(r)|min[μa(r), μs(r)],
|n˜(r)|n¯c min[μa(r), μs(r)]2πfn¯=lperiod2π max[la, ls].
|r ln n(r)|3ημa(r)=0.03ηmm-1,
RRmin=33ηmm.
|r ln n(r)|μa(r)2=0.005mm-1,
RRmin=2μa(r)=200mm.
|μd(r)||μd max|=0.01mm-1,
|r·Ω(r)||μd(r)|max+|r ln n(r)|max=μa(1+3η)=0.01(1+3η)mm-1,
|r·Ω(r)||μd(r)|max+|r ln n(r)|max=0.015mm-1,
2w0(r)wmin=2|n˜(r)|c min[μa(r), μs(r)]33.3ps,
ffmax=c2π|n˜(r)|min[μa(r), μs(r)]9.5GHz.
Rinac0.25/μt,2/μt,
Rinac2/μs.
rRinac2/min[μa(r), μs(r)].
r·[ΩL(r, Ω, t)]=Ω·rL(r, Ω, t)+(r·Ω)L(r, Ω, t),
n(r)ctL(r, Ω, t)+r[ΩL(r, Ω, t)]
+[μa(r)+μs(r)]L(r, Ω, t)
+r ln n(r)·ΩL(r, Ω, t)
=μs(r)4πL(r, Ω, t)θ(Ω, Ω)dω+(r, Ω, t).
n(r)c4πtL(r, Ω, t)dω=n(r)ct4πL(r, Ω, t)dω=n(r)ctI(r, t),
4πr·[ΩL(r, Ω, t)]dω=r·4πΩL(r, Ω, t)dω=r·J(r, t),
4π[μa(r)+μs(r)]L(r, Ω, t)dω=[μa(r)+μs(r)]I(r, t),
4πr ln n(r)·ΩL(r, Ω, t)dω=2r ln n(r)·4πΩL(r, Ω, t)dω=2r ln n(r)·J(r, t),
4πμs(r)4πL(r, Ω, t)θ(Ω, Ω)dωdω=μs(r)I(r, t),
n¯4πc4πtIvri(r, t)Ωdω+3n¯4πc4πtΩ·Jvri(r, t)Ωdω+14πΩ·rIvri(r, t)Ωdω+34π4πΩ·r[Ω·Jvri(r, t)]Ωdω+14π4π[μd(r)-r ln n(r)·Ω]Ivri(r, t)Ωdω+34π4π[μd(r)-r ln n(r)·Ω][Ω·Jvri(r, t)]Ωdω+[μa(r)+μs(r)]4π4πIvri(r, t)Ωdω+3[μa(r)+μs(r)]4π4πΩ·Jvri(r, t)Ωdω+34π4π[r ln n(r)]·Ω[Ω·Jvri(r, t)]Ωdω=μs(r)4π4π4πIvri(r, t)θ(Ω, Ω)Ωdωdω+3μs(r)4π4π4πΩ·Jvri(r, t)θ(Ω, Ω)Ωdωdω+1(r, t).
n¯4πc4πtIvri(r, t)Ωdω=0,
3n¯4πc4πtΩ·Jvri(r, t)Ωdω
=3n¯4πct4πΩ·Jvri(r, t)Ωdω
=n(r)ctJ(r, t),
14π4πΩ·rIvri(r, t)Ωdω=13rIvri(r, t),
34π4πΩ·r[Ω·Jvri(r, t)]Ωdω=0,
4πμd(r)I(r, t)Ωdω=0,
34π4πμd(r)Ω·J(r, t)Ωdω=μd(r)J(r, t)
14π4πΩ·[r ln n(r)]I(r, t)Ωdω=13[r ln n(r)]I(r, t),
34π4πΩ·[r ln n(r)]Ω·J(r, t)Ωdω=0,
34π4π[μa(r)+μs(r)]Ω·J(r, t)Ωdω=[μa(r)+μs(r)]J(r, t),
34π4π[r ln n(r)]·Ω[Ω·J(r, t)]Ωdω=0,
4π4πIvri(r, t)θ(Ω, Ω)Ωdωdω=0,
3μs(r)4π4π4πΩ·Jvri(r, t)θ(Ω, Ω)Ωdωdω=μs(r)Jvri(r, t)

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