Abstract

When a small object is placed in a multiple-scattering medium the stationary diffusion equation can be used to derive the disturbance in the transmitted and backscattered light intensity. The diffusion equation will describe the intensity outside and inside the object. The object is characterized by a size, a diffusion constant, and an absorption length. In this way absorbing objects as well as nonabsorbing objects can be treated. The results are derived for two and three dimensions. Experiments are performed on suspended titanium dioxide particles in glycerine, wherein objects could be placed. There is good agreement between theory and experiment. This work shows that with the use of continuous light sources, it may be possible to recover the location of objects accurately inside a diffusive scattering medium.

© 1993 Optical Society of America

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References

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  1. S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infra-red absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 204–351 (1991).
    [Crossref]
  2. D. T. Delpy, M. Cope, P. van der Zee, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
    [Crossref] [PubMed]
  3. H. Chen, Y. Chen, D. Dilworth, E. Leith, J. Lopez, J. Valdmanis, “Two-dimensional imaging through diffusing media using 150-fs gated electronic holography techniques,” Opt. Lett. 16, 487–489 (1991).
    [Crossref] [PubMed]
  4. K. M. Yoo, Q. Xing, R. R. Alfano, “Imaging objects hidden in highly scattering media using femtosecond second-harmonic-generation cross-correlation time gating,” Opt. Lett. 16, 1019–1021 (1991).
    [Crossref] [PubMed]
  5. R. W. Waynant, presider, session on biomedical imaging, in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 101–104.
  6. E. N. Leith, C. Chen, H. Chen, Y. Chen, J. Lopez, P. C. Sun, “Imaging through scattering media using spatial incoherence techniques,” Opt. Lett. 16, 1820–1822 (1991).
    [Crossref] [PubMed]
  7. R. Berkovits, S. Feng, “Theory of speckle-pattern tomography in multiple-scattering media,” Phys. Rev. Lett. 65, 3120–3123 (1990).
    [Crossref] [PubMed]
  8. I. Freund, “Image reconstruction through multiple scattering media,” Opt. Commun. 86, 216–227 (1991).
    [Crossref]
  9. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. I and II.
  10. H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vols. I and II.
  11. J. Fishkin, E. Gratton, M. J. van de Ven, W. W. Mantulin, “Diffusion of intensity modulated near-infrared light in turbid media,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 122–135 (1991).
    [Crossref]
  12. S. R. Arridge, M. Cope, P. van der Zee, P. J. Hilson, D. T. Delpy, “Visualization of the oxygenation state of brain and muscle in newborn infants by near infra-red transillumination,” in Information Processing in Medical Imaging, S. L. Bacharach, ed. (Nijhoff, Dordrecht, 1985), pp. 155–177.
  13. E. M. Sevick, B. Chance, J. Leigh, S. Nioka, M. Maris “Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxygenation,” Ann. Biochem. Exp. Med. 195, 330–351 (1991).
    [Crossref]
  14. M. B. van der Mark, M. P. van Albada, A. Lagendijk, “Light scattering in strongly media: multiple scattering and weak localization,” Phys. Rev. B 37, 3575–3592 (1988).
    [Crossref]
  15. See, for instance, J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  16. E. Akkermans, P. E. Wolf, R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
    [Crossref] [PubMed]
  17. See, for instance, J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

1991 (5)

1990 (1)

R. Berkovits, S. Feng, “Theory of speckle-pattern tomography in multiple-scattering media,” Phys. Rev. Lett. 65, 3120–3123 (1990).
[Crossref] [PubMed]

1988 (2)

M. B. van der Mark, M. P. van Albada, A. Lagendijk, “Light scattering in strongly media: multiple scattering and weak localization,” Phys. Rev. B 37, 3575–3592 (1988).
[Crossref]

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

1986 (1)

E. Akkermans, P. E. Wolf, R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[Crossref] [PubMed]

Akkermans, E.

E. Akkermans, P. E. Wolf, R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[Crossref] [PubMed]

Alfano, R. R.

Arridge, S. R.

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infra-red absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 204–351 (1991).
[Crossref]

S. R. Arridge, M. Cope, P. van der Zee, P. J. Hilson, D. T. Delpy, “Visualization of the oxygenation state of brain and muscle in newborn infants by near infra-red transillumination,” in Information Processing in Medical Imaging, S. L. Bacharach, ed. (Nijhoff, Dordrecht, 1985), pp. 155–177.

Berkovits, R.

R. Berkovits, S. Feng, “Theory of speckle-pattern tomography in multiple-scattering media,” Phys. Rev. Lett. 65, 3120–3123 (1990).
[Crossref] [PubMed]

Chance, B.

E. M. Sevick, B. Chance, J. Leigh, S. Nioka, M. Maris “Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxygenation,” Ann. Biochem. Exp. Med. 195, 330–351 (1991).
[Crossref]

Chen, C.

Chen, H.

Chen, Y.

Cope, M.

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

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infra-red absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 204–351 (1991).
[Crossref]

S. R. Arridge, M. Cope, P. van der Zee, P. J. Hilson, D. T. Delpy, “Visualization of the oxygenation state of brain and muscle in newborn infants by near infra-red transillumination,” in Information Processing in Medical Imaging, S. L. Bacharach, ed. (Nijhoff, Dordrecht, 1985), pp. 155–177.

Delpy, D. T.

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

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infra-red absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 204–351 (1991).
[Crossref]

S. R. Arridge, M. Cope, P. van der Zee, P. J. Hilson, D. T. Delpy, “Visualization of the oxygenation state of brain and muscle in newborn infants by near infra-red transillumination,” in Information Processing in Medical Imaging, S. L. Bacharach, ed. (Nijhoff, Dordrecht, 1985), pp. 155–177.

Dilworth, D.

Feng, S.

R. Berkovits, S. Feng, “Theory of speckle-pattern tomography in multiple-scattering media,” Phys. Rev. Lett. 65, 3120–3123 (1990).
[Crossref] [PubMed]

Fishkin, J.

J. Fishkin, E. Gratton, M. J. van de Ven, W. W. Mantulin, “Diffusion of intensity modulated near-infrared light in turbid media,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 122–135 (1991).
[Crossref]

Freund, I.

I. Freund, “Image reconstruction through multiple scattering media,” Opt. Commun. 86, 216–227 (1991).
[Crossref]

Goodman, J. W.

See, for instance, J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Gratton, E.

J. Fishkin, E. Gratton, M. J. van de Ven, W. W. Mantulin, “Diffusion of intensity modulated near-infrared light in turbid media,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 122–135 (1991).
[Crossref]

Hilson, P. J.

S. R. Arridge, M. Cope, P. van der Zee, P. J. Hilson, D. T. Delpy, “Visualization of the oxygenation state of brain and muscle in newborn infants by near infra-red transillumination,” in Information Processing in Medical Imaging, S. L. Bacharach, ed. (Nijhoff, Dordrecht, 1985), pp. 155–177.

Ishimaru, A.

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

Jackson, J. D.

See, for instance, J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

Lagendijk, A.

M. B. van der Mark, M. P. van Albada, A. Lagendijk, “Light scattering in strongly media: multiple scattering and weak localization,” Phys. Rev. B 37, 3575–3592 (1988).
[Crossref]

Leigh, J.

E. M. Sevick, B. Chance, J. Leigh, S. Nioka, M. Maris “Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxygenation,” Ann. Biochem. Exp. Med. 195, 330–351 (1991).
[Crossref]

Leith, E.

Leith, E. N.

Lopez, J.

Mantulin, W. W.

J. Fishkin, E. Gratton, M. J. van de Ven, W. W. Mantulin, “Diffusion of intensity modulated near-infrared light in turbid media,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 122–135 (1991).
[Crossref]

Maris, M.

E. M. Sevick, B. Chance, J. Leigh, S. Nioka, M. Maris “Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxygenation,” Ann. Biochem. Exp. Med. 195, 330–351 (1991).
[Crossref]

Maynard, R.

E. Akkermans, P. E. Wolf, R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[Crossref] [PubMed]

Nioka, S.

E. M. Sevick, B. Chance, J. Leigh, S. Nioka, M. Maris “Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxygenation,” Ann. Biochem. Exp. Med. 195, 330–351 (1991).
[Crossref]

Sevick, E. M.

E. M. Sevick, B. Chance, J. Leigh, S. Nioka, M. Maris “Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxygenation,” Ann. Biochem. Exp. Med. 195, 330–351 (1991).
[Crossref]

Sun, P. C.

Valdmanis, J.

van Albada, M. P.

M. B. van der Mark, M. P. van Albada, A. Lagendijk, “Light scattering in strongly media: multiple scattering and weak localization,” Phys. Rev. B 37, 3575–3592 (1988).
[Crossref]

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vols. I and II.

van de Ven, M. J.

J. Fishkin, E. Gratton, M. J. van de Ven, W. W. Mantulin, “Diffusion of intensity modulated near-infrared light in turbid media,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 122–135 (1991).
[Crossref]

van der Mark, M. B.

M. B. van der Mark, M. P. van Albada, A. Lagendijk, “Light scattering in strongly media: multiple scattering and weak localization,” Phys. Rev. B 37, 3575–3592 (1988).
[Crossref]

van der Zee, P.

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

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infra-red absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 204–351 (1991).
[Crossref]

S. R. Arridge, M. Cope, P. van der Zee, P. J. Hilson, D. T. Delpy, “Visualization of the oxygenation state of brain and muscle in newborn infants by near infra-red transillumination,” in Information Processing in Medical Imaging, S. L. Bacharach, ed. (Nijhoff, Dordrecht, 1985), pp. 155–177.

Waynant, R. W.

R. W. Waynant, presider, session on biomedical imaging, in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 101–104.

Wolf, P. E.

E. Akkermans, P. E. Wolf, R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[Crossref] [PubMed]

Wray, S.

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

Wyatt, J.

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

Xing, Q.

Yoo, K. M.

Ann. Biochem. Exp. Med. (1)

E. M. Sevick, B. Chance, J. Leigh, S. Nioka, M. Maris “Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxygenation,” Ann. Biochem. Exp. Med. 195, 330–351 (1991).
[Crossref]

Opt. Commun. (1)

I. Freund, “Image reconstruction through multiple scattering media,” Opt. Commun. 86, 216–227 (1991).
[Crossref]

Opt. Lett. (3)

Phys. Med. Biol. (1)

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

Phys. Rev. B (1)

M. B. van der Mark, M. P. van Albada, A. Lagendijk, “Light scattering in strongly media: multiple scattering and weak localization,” Phys. Rev. B 37, 3575–3592 (1988).
[Crossref]

Phys. Rev. Lett. (2)

R. Berkovits, S. Feng, “Theory of speckle-pattern tomography in multiple-scattering media,” Phys. Rev. Lett. 65, 3120–3123 (1990).
[Crossref] [PubMed]

E. Akkermans, P. E. Wolf, R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[Crossref] [PubMed]

Other (8)

See, for instance, J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

See, for instance, J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infra-red absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 204–351 (1991).
[Crossref]

R. W. Waynant, presider, session on biomedical imaging, in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 101–104.

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

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vols. I and II.

J. Fishkin, E. Gratton, M. J. van de Ven, W. W. Mantulin, “Diffusion of intensity modulated near-infrared light in turbid media,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 122–135 (1991).
[Crossref]

S. R. Arridge, M. Cope, P. van der Zee, P. J. Hilson, D. T. Delpy, “Visualization of the oxygenation state of brain and muscle in newborn infants by near infra-red transillumination,” in Information Processing in Medical Imaging, S. L. Bacharach, ed. (Nijhoff, Dordrecht, 1985), pp. 155–177.

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Figures (9)

Fig. 1
Fig. 1

Schematic view of a multiple-scattering system with an embedded spherical object. The scattering system is characterized by a diffusion constant D1 and the object by radius a, diffusion constant D2, and absorbing parameter α. A plane wave is incident at x = L. The intensity is derived at x = 0 and x = L.

Fig. 2
Fig. 2

Calculated diffuse intensities are shown for an object embedded in a two-dimensional multiple scattering system, as a function of the transversal distance to the object. The intensity is plotted relative to the undisturbed intensity (no object present), according to Eq. (26). (a) Transmission for absorbing rod, a = l, α = ∞, κ = 0. Solid curve, x0 = 20 l; dotted curve, x0 = 80 l. (b) Transmission for scattering rod, a = l, α = κ = 0, D2 = ∞. Solid curve, x0 = 20 l; dotted curve, x0 = 30 l; dashed curve, x0 = 40 l. In all cases L = 100 l. L is the thickness of the slab, and l is the mean free path.

Fig. 3
Fig. 3

Calculated transmitted diffuse intensities as a function of the transversal distance y for a slab containing an absorbing spherical object at various positions, following Eq. (16). The light source is pointlike and placed at r = (l,0,0). Slab thickness L = 50 l, absorbing object radius a = l, α = ∞, scattering term neglected. Solid curve, no object; long-dashed curve, x0 = 45 l, y0 = 0; dashed-dotted curve, x0 = 45 l, y0 = 10 l; short-dashed curve, x0 = 5 l, y0 = 0; dotted curve x0 = 5 l, y0 = 10 l.

Fig. 4
Fig. 4

Experimental setup: L, laser, Sp, spatial filter; S, screen; C, cell with the scattering medium and the object; Le, lens for imaging the diffuse intensity on D, the one-dimensional diode array.

Fig. 5
Fig. 5

Transmitted intensity profiles from a cell containing a diffusive scattering suspension, for various positions of the embedded absorbing wire ϕ = 350 μm: a, x0 = 1.00 mm; b, x0 =2.60 mm, and c, x0 = 4.10 mm. The experimental data are shown together with the best numerical fits for Eq. (25): a, x0 = 1.41 ± 0.05 mm, D1/αa = 0.95 ± 0.03; b, x0 = 2.86 ± 0.05 mm, D1/αa = 1.04 ± 0.03; c, x0 = 4.33 ± 0.05 mm, D1/αa = 1.11 ± 0.03. In all cases a = 0.175, p = 0, l = 20 ± 3 and cell thickness L = 10.0 mm.

Fig. 6
Fig. 6

Minima of the intensity profiles for different positions of the absorbing wire. The experimental points of Table 1 are plotted. The solid curve is obtained from Eq. (25) with L = 10.0 mm, a = 0.175 mm, D1/αa = 1.00, p = 0, and D2 = 0.

Fig. 7
Fig. 7

Transmitted intensity profiles from a cell containing a diffusive scattering suspension for a transparent fiber with ϕ = 350 μm at different positions inside the cell: a, x0 = 0.74 ± 0.05; b, x0 = 2.5 mm; l = 20 ± 3 μm, cell thickness L = 5.0 mm. Experimental data are shown together with the best numerical fits. The fits are obtained from Eq. (25): a, x0 = 0.60 ± 0.02 mm, D2 = ∞, D1/αa = 1 × 102; b, x0 = 2.4 ± 0.1 mm, D2 = ∞, D1/αa = 1 × 102.

Fig. 8
Fig. 8

Backscattered intensity profiles for an absorbing wire ϕ = 350 μm inside a scattering suspension at different positions: a, x0 = 1.68 mm; b, x0 = 1.38 mm; c, x0 = 0.78 mm. Transport mean-free path l = 47 ± 3 μm, cell thickness L = 4.0 mm. Experimental data are shown together with numerical fits for the experimental data from Eq. (26); see also Table 3.

Fig. 9
Fig. 9

Measured transmitted intensity from a slablike scattering medium containing an absorbing wire with Ø = 350 μm. A line source is used instead of a plane wave. Normal position of wire x0 = 1.05 ± 0.05 mm. Transport mean-free path l = 20 ± 3 μm, cell thickness L = 5.0 mm. The smooth curve shows the nunerical fit to the experimental data, x0 = 1.03 ± 0.02 mm. The fit is derived from Eq. (29).

Tables (3)

Tables Icon

Table 1 Fits of the Calculated Line Shape to Experimental Data Obtained from a Transmission Experiment with an Absorbing Roda

Tables Icon

Table 2 Fits of the Calculated Line Shape to Transmission Experiment Dataa

Tables Icon

Table 3 Fits of the Calculated Line Shape to Experimental Data Obtained from a Backscatter Experiment with an Absorbing Roda

Equations (46)

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Δ I ( r ) = 0.
I ( L , y , z ) = S ( y , z ) ,             I ( 0 , y , z ) = 0
I ( R , θ , ϕ ) = S ( θ , ϕ ) ,
Δ I ( r ) = κ 2 I ( r )
I out ( a + ) = I in ( a - ) ,
D 1 I out n | a + = D 2 I in n | a - .
I out ( a + ) = I in ( a - ) ,
D 1 I out n | a + = D 2 I in n | a - + α I in ( a ) .
α = κ = 0 , D 2 D 1 : pure scatterer , no absorption , α = κ = 0 , D 2 = : glass object or air bubble , α 0 , or κ 0 : absorber , absorbing behavior dominates scattering .
I ( r ) = I 0 x L + q n = - 1 r - r 0 + 2 L n x ^ - 1 r + r 0 + 2 L n x ^ + p n = - x - x 0 + 2 n L [ ( x - x 0 + 2 n L ) 2 + ρ 2 ] 3 / 2 + x + x 0 + 2 n L [ ( x + x 0 + 2 n L ) 2 + ρ 2 ] 3 / 2 ,
I in ( r ) = A sinh κ r κ r - 3 B κ ( sinh κ r κ 2 r 2 - cosh κ r κ r ) Y 10 ( θ , ϕ ) ,
I in ( r ) = A + B ( x - x 0 ) .
lim κ 0 I in ( r ) = A + ( - 3 B κ ) ( - κ r 3 ) ( x - x 0 r ) .
I ( r ) I 0 x L + q [ 1 r - r 0 + O ( a L ) ] + p [ x - x 0 r - r 0 3 + O ( a L ) ] .
q = - I 0 a x 0 L D 2 ( cosh κ a - sinhc κ a ) / sinhc κ a + α a D 1 + D 2 ( cosh κ a - sinhc κ a ) / sinhc κ a + α a , p = I 0 a 3 L D 1 - D 2 ( 2 cosh κ a - 2 sinhc κ a - κ a sinh κ a ) / ( sinhc κ a - cosh κ a ) - α a 2 D 1 + D 2 ( 2 cosh κ a - 2 sinhc κ a - κ a sinh κ a ) / ( sinhc κ a - cosh κ a ) + α a , A = D 1 q α a 2 sinhc κ a + A d 2 ( cosh κ a - sinhc κ a ) , B = 1 3 κ 2 a 2 sinhc κ a - cosh κ a ( I 0 L + p a 3 ) ,
q = - I 0 x 0 a L α a D 1 + α a , p = I 0 a 3 L D 1 - D 2 - α a 2 D 1 + D 2 + α a , A = D 1 q α a 2 , B = I 0 L + p 1 a 3 .
T ( ρ ) l I 0 I x | x = 0 = l L + 2 q l n = - x 0 + 2 n L [ ( x 0 + 2 n L ) 2 + ρ 2 ] 3 / 2 + 2 p l n = - ρ 2 - 2 ( x 0 + 2 n L ) 2 [ ( x 0 + 2 n L ) 2 + ρ 2 ] 5 / 2
B ( ρ ) 1 - l I 0 I x | x = L = 1 - ( l L + 2 q l n = - x 0 + L ( 2 n + 1 ) { [ x 0 + L ( 2 n + 1 ) ] 2 + ρ 2 } 3 / 2 + 2 p l n = - ρ 2 - 2 [ x 0 + ( 2 n + 1 ) L ] 2 { [ x 0 + ( 2 n + 1 ) L ] 2 + ρ 2 } 5 / 2 ) .
I out ( ζ ) = I 0 x L + q Re { ln sin [ ( π / 2 L ) ( ζ - x 0 ) ] sin [ ( π / 2 L ) ( ζ + x 0 ) ] } + p Re { cot [ π 2 L ( ζ - x 0 ) ] + cot [ π 2 L ( ζ + x 0 ) ] } ,
I in ( ζ ) = A + B ( x - x 0 ) .
I in ( ζ ) = A I 0 ( κ ρ ) + 2 B κ I 1 ( κ ρ ) cos ϕ .
I out ( ζ ) = I 0 x 0 + ( x - x 0 ) L + q { ln π 2 L [ ( x - x 0 ) 2 + y 2 ] 1 / 2 sin ( π / L ) x 0 + O ( a 2 L 2 ) } + p { 2 L ( x - x 0 ) π [ ( x - x 0 ) 2 + y 2 ] 1 / 2 + cot π L x 0 + O ( a L ) } .
q = I 0 [ x 0 L + p cot ( π x 0 / L ) ] × α a + D 2 κ a I 1 ( κ a ) / I 0 ( κ a ) D 1 + [ α a + D 2 κ a I 1 ( κ a ) / I 0 ( κ a ) ] ln ( L / a * ) , p = I 0 π a 2 2 L 2 D 1 - D 2 [ κ a I 0 ( κ a ) - I 1 ( κ a ) ] / I 1 ( κ a ) - α a D 1 + D 2 [ κ a I 0 ( κ a ) - I 1 ( κ a ) ] / I 1 ( κ a ) + α a , A = I 0 ( κ a ) κ a I 1 ( κ a ) D 2 + I 0 ( κ a ) α a D 1 q , B = κ a 2 I 1 ( κ a ) ( I 0 1 L + p 2 L π a 2 ) ,
a * = π a 2 sin ( π x 0 / L ) .
q = I 0 [ x 0 L + cot ( π x 0 / L ) D 1 - D 2 - α a D 1 + D 2 + α a π a 2 2 L 2 ] × α a D 1 + α a ln ( L / a * ) , p = I 0 D 1 - D 2 - α a D 1 + D 2 + α a π a 2 2 L 2 , A = D 1 q α a , B = I 0 1 L + p 2 L π a 2 .
T ( y ) l I 0 I ( r ) x | x = 0 , = l L [ 1 - q π sin ( π x 0 / L ) cosh ( π y / L ) - cos ( π x 0 / L ) - 2 π p 1 - cosh ( π y / L ) cos ( π x 0 / L ) [ cosh ( π y / L ) - cos ( π x 0 / L ) ] 2 ]
B ( y ) 1 - l I 0 I ( r ) x | x = L , = 1 - l L [ 1 + q π sin ( π x 0 / L ) cosh ( π y / L ) + cos ( π x 0 / L ) - 2 π p 1 + cosh ( π y / L ) cos ( π x 0 / L ) [ cosh ( π y / L ) + cos ( π x 0 / L ) ] 2 ] .
G ( l , ζ ) = Re { ln sin [ ( π / 2 L ) ( ζ - l ) ] sin [ ( π / 2 L ) ( ζ + l ) ] }
G ( l , r ) = n = - 1 [ ( l - x + 2 L n ) 2 + y 2 + z 2 ] 1 / 2 - 1 [ ( l + x + 2 L n ) 2 + y 2 + z 2 ] 1 / 2
I out ( ζ ) = I 0 G ( I , ζ ) + q Re { ln sin [ ( π / 2 L ) ( ζ - x 0 - i y 0 ) ] sin [ ( π / 2 L ) ( ζ + x 0 + i y 0 ) ] } + Re { p cot [ π 2 L ( ζ - x 0 + i y 0 ) ] + p * cot [ π 2 L ( ζ + x 0 - i y 0 ) ] } ,
p p ^ s 1 ( x 0 , y 0 ) + i p ^ s 2 ( x 0 , y 0 ) .
s 1 = sin [ ( π / L ) ( x 0 - l ) ] cosh [ ( π / L ) y 0 ] - cos [ ( π / L ) ( x 0 - l ) ] - sin [ ( π / L ) ( x 0 + l ) ] cosh [ ( π / L ) y 0 ] - cos [ ( π / L ) ( x 0 + l ) ] , s 2 = sinh [ ( π / L ) y 0 ] cosh [ ( π / L ) y 0 ] - cos [ ( π / L ) ( x 0 - l ) ] - sinh [ ( π / L ) x 0 ] cosh [ ( π / L ) y 0 ] - cos [ ( π / L ) ( x 0 - l ) ] .
I in = A + B [ s 1 ( x 0 , y 0 ) ( x - x 0 ) + s 2 ( x 0 , y 0 ) ( y - y 0 ) ] .
q = [ I 0 G ( l , ζ 0 ) + p ^ s 1 ( x 0 , y 0 ) cot ( π x 0 / L ) ] α a D 1 + α a ln ( L / a * ) , p ^ = I 0 ( π a 2 L ) 2 D 1 - D 2 - α a D 1 + D 2 + α a , A = D 1 q / α a , B = I 0 π 2 L 2 D 1 D 1 + D 2 + α a .
I out ( r ) = I 0 G ( l , r ) + q I q ( r ) + p r | r 0 G ( l , r ) · r 0 I q ( r ) ,
I in ( r ) = A + B ( r - r 0 ) · r | r 0 G ( l , r ) ,
I q ( r ) = n = - 1 [ ( x - x 0 + 2 L n ) 2 + ( y - y 0 ) 2 + ( z - z 0 ) 2 ] 1 / 2 - 1 [ ( x + x 0 + 2 L n ) 2 + ( y - y 0 ) 2 + ( z - z 0 ) 2 ] 1 / 2 .
q = - a I 0 G ( l , r 0 ) α a D 1 + α a , p = a 3 I 0 D 1 - D 2 - α a 2 D 1 + D 2 + α a , A = D 1 a α a 2 , B = I 0 3 D 1 2 D 1 + D 2 + α a .
I ( r ) = I 0 G R ( r , r s ) .
G R ( r 1 , r 2 ) = [ 1 r 1 - r 2 - R r 2 1 | r 1 - R 2 r 2 2 r 2 | ] .
I out ( r ) = I ( r ) + q G R ( r , r 0 ) - p · r 0 G R ( r , r 0 ) .
I in = A + I 0 B · ( r - r 0 ) .
q = - a [ I 0 G R ( r s , r 0 ) - p · r 0 R ( r 0 2 - R 2 ) 2 ] α a D 1 + α a [ 1 - a R / ( r b 2 - R 2 ) ] ,
p = a 3 I 0 r | r 0 G R ( r , r s ) D 1 - D 2 - α a 2 D 1 [ 1 - ½ a 3 R 3 / ( r 0 2 - R 2 ) 3 ] + ( D 2 + α a ) [ 1 + a 3 R 3 / ( r 0 2 - R 2 ) 3 ] + q r 0 R r 0 2 - R 2 D 1 + D 2 + α a 2 D 1 [ 1 - ½ a 3 R 3 / ( r 0 2 - R 2 ) 3 ] + ( D 2 + α a ) [ 1 + a 3 R 3 / ( r 0 2 - R 2 ) 3 ] ,
A = - D 1 q α a 2 ,
B = I 0 r | r 0 G R ( r , r s ) q r 0 R ( r 0 2 - R 2 ) 2 + p [ 1 a 3 + R 3 ( r 0 2 - R 2 ) 3 ] .

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