Abstract

A numerical study of single-source and dual-interfering-source modalities for optical imaging through biological media with a view to practical implications is given. Our comparisons are based on multigrid finite-difference frequency-domain solutions of the diffusion equation, for example, inhomogeneous imaging problems. Detector noise levels, the influence of boundaries, and ease of measurement are considered. This investigation indicates that a dual source, or dual-source data synthesized from single-source measurements, offers significant imaging opportunities when compared with single-source data.

© 1997 Optical Society of America

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References

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  1. A. Ishimaru, “Wave propagation and scattering in random media and rough surfaces,” Proc. IEEE 17, 1359–1366 (1991).
    [CrossRef]
  2. B. Wilson, E. Sevick, M. Patterson, B. Chance, “Time-dependent optical spectroscopy and imaging for biomedical applications,” Proc. IEEE 80, 918–930 (1992).
    [CrossRef]
  3. C. A. Thompson, J. S. Reynolds, F. P. LaPlant, D. Ben-Amotz, K. J. Webb, “Raman spectroscopic studies of diamond in intralipid,” Opt. Lett. 20, 1195–1197 (1995).
    [CrossRef] [PubMed]
  4. J. S. Reynolds, A. Przadka, S. P. Yeung, K. J. Webb, “Optical diffusion imaging: a comparative numerical and experimentalstudy,” Appl. Opt. 35, 3671–3679 (1996).
    [CrossRef] [PubMed]
  5. A. Knüttel, J. M. Schmitt, J. R. Knutson, “Improvement of spatial resolution in reflectance near-infrared imagingby laser-beam interference,” in Time-Resolved Laser Spectroscopy in Biochemistry III, J. R. Lakowicz, ed., Proc. SPIE1640, 405–416 (1992).
    [CrossRef]
  6. A. Knüttel, J. M. Schmitt, J. R. Knutson, “Spatial localization of absorbing bodies by interfering diffusive photon-densitywaves,” Appl. Opt. 32, 381–389 (1993).
    [CrossRef]
  7. A. Knüttel, J. M. Schmitt, R. Barnes, J. R. Knutson, “Acousto-optic scanning and interfering photon density waves for preciselocalization of an absorbing (or fluorescent) body in a turbid medium,” Rev. Sci. Instrum. 64, 636–644 (1993).
  8. A. Knüttel, J. M. Schmitt, R. Barnes, J. R. Knutson, “Spatial localization using interfering photon-density waves: contrastenhancement and limitations,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. Alfano, eds., Proc. SPIE1888, 322–333 (1993).
    [CrossRef]
  9. B. Chance, K. Kang, L. He, J. Weng, E. Sevick, “Highly sensitive object localization in tissue models with linear in-phaseand anti-phase multi-element optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. (USA) 90, 3423–3427 (1993).
    [CrossRef]
  10. B. Chance, “Multi-element arrays for phase modulation imaging,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. Alfano, eds., Proc. SPIE1888, 354–358 (1993).
    [CrossRef]
  11. D. A. Boas, M. A. O'Leary, B. Chance, A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffusephoton density waves: a signal-to-noise analysis,” Appl. Opt. 36, 75–92 (1997).
    [CrossRef] [PubMed]
  12. M. S. Patterson, B. Chance, B. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurementof tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
    [CrossRef] [PubMed]
  13. J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).
  14. J. C. Adams, Multigrid Software for Elliptic Partial Differential Equations (National Center for Atmospheric Research, Boulder, Colo., February1991).
  15. J. C. Adams, “MUDPACK: multigrid portable fortran softwarefor the efficient solution of linear elliptic partial differential equations,” Appl. Math. Comput. 34, 113–146 (1989).
    [CrossRef]
  16. B. W. Pogue, M. S. Patterson, “Forward and inverse calculations for near-infrared imaging using multigrid finite difference method,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of 1994 OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 176–180.
  17. V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissues in thevisible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
    [CrossRef] [PubMed]
  18. R. C. Haskell, L. O. Svaasand, T. Tsay, T. Feng, M. S. McAdams, B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
    [CrossRef]
  19. E. Dereniak, D. Crowe, Optical Radiation Detectors (Wiley, New York, 1984).
  20. H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, M. S. Patterson, “Optical image reconstruction using frequency-domain data: simulationsand experiments,” J. Opt. Soc. Am. A 13, 253–266 (1996).
    [CrossRef]

1997 (1)

1996 (2)

1995 (1)

1994 (1)

1993 (3)

A. Knüttel, J. M. Schmitt, J. R. Knutson, “Spatial localization of absorbing bodies by interfering diffusive photon-densitywaves,” Appl. Opt. 32, 381–389 (1993).
[CrossRef]

A. Knüttel, J. M. Schmitt, R. Barnes, J. R. Knutson, “Acousto-optic scanning and interfering photon density waves for preciselocalization of an absorbing (or fluorescent) body in a turbid medium,” Rev. Sci. Instrum. 64, 636–644 (1993).

B. Chance, K. Kang, L. He, J. Weng, E. Sevick, “Highly sensitive object localization in tissue models with linear in-phaseand anti-phase multi-element optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. (USA) 90, 3423–3427 (1993).
[CrossRef]

1992 (1)

B. Wilson, E. Sevick, M. Patterson, B. Chance, “Time-dependent optical spectroscopy and imaging for biomedical applications,” Proc. IEEE 80, 918–930 (1992).
[CrossRef]

1991 (1)

A. Ishimaru, “Wave propagation and scattering in random media and rough surfaces,” Proc. IEEE 17, 1359–1366 (1991).
[CrossRef]

1990 (1)

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissues in thevisible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef] [PubMed]

1989 (2)

M. S. Patterson, B. Chance, B. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurementof tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

J. C. Adams, “MUDPACK: multigrid portable fortran softwarefor the efficient solution of linear elliptic partial differential equations,” Appl. Math. Comput. 34, 113–146 (1989).
[CrossRef]

Adams, J. C.

J. C. Adams, “MUDPACK: multigrid portable fortran softwarefor the efficient solution of linear elliptic partial differential equations,” Appl. Math. Comput. 34, 113–146 (1989).
[CrossRef]

J. C. Adams, Multigrid Software for Elliptic Partial Differential Equations (National Center for Atmospheric Research, Boulder, Colo., February1991).

Barnes, R.

A. Knüttel, J. M. Schmitt, R. Barnes, J. R. Knutson, “Acousto-optic scanning and interfering photon density waves for preciselocalization of an absorbing (or fluorescent) body in a turbid medium,” Rev. Sci. Instrum. 64, 636–644 (1993).

A. Knüttel, J. M. Schmitt, R. Barnes, J. R. Knutson, “Spatial localization using interfering photon-density waves: contrastenhancement and limitations,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. Alfano, eds., Proc. SPIE1888, 322–333 (1993).
[CrossRef]

Ben-Amotz, D.

Boas, D. A.

Chance, B.

D. A. Boas, M. A. O'Leary, B. Chance, A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffusephoton density waves: a signal-to-noise analysis,” Appl. Opt. 36, 75–92 (1997).
[CrossRef] [PubMed]

B. Chance, K. Kang, L. He, J. Weng, E. Sevick, “Highly sensitive object localization in tissue models with linear in-phaseand anti-phase multi-element optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. (USA) 90, 3423–3427 (1993).
[CrossRef]

B. Wilson, E. Sevick, M. Patterson, B. Chance, “Time-dependent optical spectroscopy and imaging for biomedical applications,” Proc. IEEE 80, 918–930 (1992).
[CrossRef]

M. S. Patterson, B. Chance, B. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurementof tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

B. Chance, “Multi-element arrays for phase modulation imaging,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. Alfano, eds., Proc. SPIE1888, 354–358 (1993).
[CrossRef]

Crowe, D.

E. Dereniak, D. Crowe, Optical Radiation Detectors (Wiley, New York, 1984).

Dereniak, E.

E. Dereniak, D. Crowe, Optical Radiation Detectors (Wiley, New York, 1984).

Duderstadt, J. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).

Feng, T.

Frank, G. L.

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissues in thevisible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef] [PubMed]

Hamilton, L. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).

Haskell, R. C.

He, L.

B. Chance, K. Kang, L. He, J. Weng, E. Sevick, “Highly sensitive object localization in tissue models with linear in-phaseand anti-phase multi-element optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. (USA) 90, 3423–3427 (1993).
[CrossRef]

Ishimaru, A.

A. Ishimaru, “Wave propagation and scattering in random media and rough surfaces,” Proc. IEEE 17, 1359–1366 (1991).
[CrossRef]

Jiang, H.

Kang, K.

B. Chance, K. Kang, L. He, J. Weng, E. Sevick, “Highly sensitive object localization in tissue models with linear in-phaseand anti-phase multi-element optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. (USA) 90, 3423–3427 (1993).
[CrossRef]

Knutson, J. R.

A. Knüttel, J. M. Schmitt, R. Barnes, J. R. Knutson, “Acousto-optic scanning and interfering photon density waves for preciselocalization of an absorbing (or fluorescent) body in a turbid medium,” Rev. Sci. Instrum. 64, 636–644 (1993).

A. Knüttel, J. M. Schmitt, J. R. Knutson, “Spatial localization of absorbing bodies by interfering diffusive photon-densitywaves,” Appl. Opt. 32, 381–389 (1993).
[CrossRef]

A. Knüttel, J. M. Schmitt, J. R. Knutson, “Improvement of spatial resolution in reflectance near-infrared imagingby laser-beam interference,” in Time-Resolved Laser Spectroscopy in Biochemistry III, J. R. Lakowicz, ed., Proc. SPIE1640, 405–416 (1992).
[CrossRef]

A. Knüttel, J. M. Schmitt, R. Barnes, J. R. Knutson, “Spatial localization using interfering photon-density waves: contrastenhancement and limitations,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. Alfano, eds., Proc. SPIE1888, 322–333 (1993).
[CrossRef]

Knüttel, A.

A. Knüttel, J. M. Schmitt, J. R. Knutson, “Spatial localization of absorbing bodies by interfering diffusive photon-densitywaves,” Appl. Opt. 32, 381–389 (1993).
[CrossRef]

A. Knüttel, J. M. Schmitt, R. Barnes, J. R. Knutson, “Acousto-optic scanning and interfering photon density waves for preciselocalization of an absorbing (or fluorescent) body in a turbid medium,” Rev. Sci. Instrum. 64, 636–644 (1993).

A. Knüttel, J. M. Schmitt, R. Barnes, J. R. Knutson, “Spatial localization using interfering photon-density waves: contrastenhancement and limitations,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. Alfano, eds., Proc. SPIE1888, 322–333 (1993).
[CrossRef]

A. Knüttel, J. M. Schmitt, J. R. Knutson, “Improvement of spatial resolution in reflectance near-infrared imagingby laser-beam interference,” in Time-Resolved Laser Spectroscopy in Biochemistry III, J. R. Lakowicz, ed., Proc. SPIE1640, 405–416 (1992).
[CrossRef]

LaPlant, F. P.

McAdams, M. S.

O'Leary, M. A.

Osterberg, U. L.

Patterson, M.

B. Wilson, E. Sevick, M. Patterson, B. Chance, “Time-dependent optical spectroscopy and imaging for biomedical applications,” Proc. IEEE 80, 918–930 (1992).
[CrossRef]

Patterson, M. S.

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, M. S. Patterson, “Optical image reconstruction using frequency-domain data: simulationsand experiments,” J. Opt. Soc. Am. A 13, 253–266 (1996).
[CrossRef]

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissues in thevisible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef] [PubMed]

M. S. Patterson, B. Chance, B. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurementof tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

B. W. Pogue, M. S. Patterson, “Forward and inverse calculations for near-infrared imaging using multigrid finite difference method,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of 1994 OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 176–180.

Paulsen, K. D.

Peters, V. G.

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissues in thevisible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef] [PubMed]

Pogue, B. W.

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, M. S. Patterson, “Optical image reconstruction using frequency-domain data: simulationsand experiments,” J. Opt. Soc. Am. A 13, 253–266 (1996).
[CrossRef]

B. W. Pogue, M. S. Patterson, “Forward and inverse calculations for near-infrared imaging using multigrid finite difference method,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of 1994 OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 176–180.

Przadka, A.

Reynolds, J. S.

Schmitt, J. M.

A. Knüttel, J. M. Schmitt, R. Barnes, J. R. Knutson, “Acousto-optic scanning and interfering photon density waves for preciselocalization of an absorbing (or fluorescent) body in a turbid medium,” Rev. Sci. Instrum. 64, 636–644 (1993).

A. Knüttel, J. M. Schmitt, J. R. Knutson, “Spatial localization of absorbing bodies by interfering diffusive photon-densitywaves,” Appl. Opt. 32, 381–389 (1993).
[CrossRef]

A. Knüttel, J. M. Schmitt, R. Barnes, J. R. Knutson, “Spatial localization using interfering photon-density waves: contrastenhancement and limitations,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. Alfano, eds., Proc. SPIE1888, 322–333 (1993).
[CrossRef]

A. Knüttel, J. M. Schmitt, J. R. Knutson, “Improvement of spatial resolution in reflectance near-infrared imagingby laser-beam interference,” in Time-Resolved Laser Spectroscopy in Biochemistry III, J. R. Lakowicz, ed., Proc. SPIE1640, 405–416 (1992).
[CrossRef]

Sevick, E.

B. Chance, K. Kang, L. He, J. Weng, E. Sevick, “Highly sensitive object localization in tissue models with linear in-phaseand anti-phase multi-element optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. (USA) 90, 3423–3427 (1993).
[CrossRef]

B. Wilson, E. Sevick, M. Patterson, B. Chance, “Time-dependent optical spectroscopy and imaging for biomedical applications,” Proc. IEEE 80, 918–930 (1992).
[CrossRef]

Svaasand, L. O.

Thompson, C. A.

Tromberg, B. J.

Tsay, T.

Webb, K. J.

Weng, J.

B. Chance, K. Kang, L. He, J. Weng, E. Sevick, “Highly sensitive object localization in tissue models with linear in-phaseand anti-phase multi-element optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. (USA) 90, 3423–3427 (1993).
[CrossRef]

Wilson, B.

B. Wilson, E. Sevick, M. Patterson, B. Chance, “Time-dependent optical spectroscopy and imaging for biomedical applications,” Proc. IEEE 80, 918–930 (1992).
[CrossRef]

M. S. Patterson, B. Chance, B. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurementof tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

Wyman, D. R.

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissues in thevisible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef] [PubMed]

Yeung, S. P.

Yodh, A. G.

Appl. Math. Comput. (1)

J. C. Adams, “MUDPACK: multigrid portable fortran softwarefor the efficient solution of linear elliptic partial differential equations,” Appl. Math. Comput. 34, 113–146 (1989).
[CrossRef]

Appl. Opt. (4)

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

Opt. Lett. (1)

Phys. Med. Biol. (1)

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissues in thevisible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef] [PubMed]

Proc. IEEE (2)

A. Ishimaru, “Wave propagation and scattering in random media and rough surfaces,” Proc. IEEE 17, 1359–1366 (1991).
[CrossRef]

B. Wilson, E. Sevick, M. Patterson, B. Chance, “Time-dependent optical spectroscopy and imaging for biomedical applications,” Proc. IEEE 80, 918–930 (1992).
[CrossRef]

Proc. Natl. Acad. Sci. (USA) (1)

B. Chance, K. Kang, L. He, J. Weng, E. Sevick, “Highly sensitive object localization in tissue models with linear in-phaseand anti-phase multi-element optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. (USA) 90, 3423–3427 (1993).
[CrossRef]

Rev. Sci. Instrum. (1)

A. Knüttel, J. M. Schmitt, R. Barnes, J. R. Knutson, “Acousto-optic scanning and interfering photon density waves for preciselocalization of an absorbing (or fluorescent) body in a turbid medium,” Rev. Sci. Instrum. 64, 636–644 (1993).

Other (7)

A. Knüttel, J. M. Schmitt, R. Barnes, J. R. Knutson, “Spatial localization using interfering photon-density waves: contrastenhancement and limitations,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. Alfano, eds., Proc. SPIE1888, 322–333 (1993).
[CrossRef]

B. Chance, “Multi-element arrays for phase modulation imaging,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. Alfano, eds., Proc. SPIE1888, 354–358 (1993).
[CrossRef]

A. Knüttel, J. M. Schmitt, J. R. Knutson, “Improvement of spatial resolution in reflectance near-infrared imagingby laser-beam interference,” in Time-Resolved Laser Spectroscopy in Biochemistry III, J. R. Lakowicz, ed., Proc. SPIE1640, 405–416 (1992).
[CrossRef]

B. W. Pogue, M. S. Patterson, “Forward and inverse calculations for near-infrared imaging using multigrid finite difference method,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of 1994 OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 176–180.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).

J. C. Adams, Multigrid Software for Elliptic Partial Differential Equations (National Center for Atmospheric Research, Boulder, Colo., February1991).

E. Dereniak, D. Crowe, Optical Radiation Detectors (Wiley, New York, 1984).

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Figures (10)

Fig. 1
Fig. 1

2-D and 3-D simulation geometries. Strong absorbers are embedded in a scattering, tissuelike domain. The dual source and detector are scanned in tandem along opposite sides of the phantom, and the detector is located at the midpoint of the sources, which are modulated at 100 MHz. Throughout, for the background, μ a = 0.035   cm - 1 , μ s = 15   cm - 1 , and for the phantom absorber, μ a = 35   cm - 1 , μ s = 15   cm - 1 . For the dual source, one source is modulated 180° with respect to the other. The phase data presented is relative to 0° at the source. The arbitrary magnitude data is normalized for a 0-dB maximum.

Fig. 2
Fig. 2

Calculated detected power level through a 1-mm-square aperture for a fixed dual source with a homogeneous 3-D simulation domain of side 9.6 cm. The noise floor is calculated for an example PMT detector to illustrate the measurable features.

Fig. 3
Fig. 3

Dual-source data for a central absorber as a function of source–detector scan position: (a) intensity, (b) phase. The magnitude scale is in decibels and is normalized for this and all subsequent figures (which present 2-D data).

Fig. 4
Fig. 4

Dual-source data with absorber located at x = 4.8   cm , y = 3.2   cm .

Fig. 5
Fig. 5

Magnitude of fluence, Φ, as a function of cross-sectional position for a dual source and offset absorber of 1-cm diameter, located at x = 4.8   cm , y = 3.2   cm . The amplitude scale is in decibels with an arbitrary reference. Note how the interference line bends toward the absorber.

Fig. 6
Fig. 6

Magnitude and phase contour plots, where sources 1 and 2 are in the positions indicated and the detected quantity is midway between the two sources: (a) and (b) are relative decibel magnitude and phase for a homogeneous domain; (c) and (d) are for a central absorber of 1-cm diameter; (e) and (f) are for an offset 1-cm absorber located at x = 4.8   cm , y = 3.2   cm .

Fig. 7
Fig. 7

Single-source data with central absorber.

Fig. 8
Fig. 8

Single-source data with absorber at x = 4.8   cm , y = 3.2   cm .

Fig. 9
Fig. 9

Difference of magnitude in decibels and phase, with and without absorber, for single- and dual-source cases for a central absorber in (a) and (b) and an offset absorber in (c) and (d).

Fig. 10
Fig. 10

Illustration of reciprocity in measurements.

Equations (5)

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

- 2 Φ ( r ,   ω ) + k 2 Φ ( r ,   ω ) = S ( r ,   ω ) D ,
k = i ω / v + μ a D 1 / 2 ,
J ( r ,   ω ) = - D ( r ) Φ ( r ,   ω ) .
J in ( r b ,   ω ) = 1 4 Φ ( r b ,   ω ) + 1 2 [ D ( r ) n ˆ Φ ( r ,   ω ) ] | r = r b = 0 ,
i nrms = 2 q ( i d + i s ) B n ,

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