Abstract

We present a general analysis of the sensitivity of a diffuse photon density wave in a homogeneous multiple-scattering medium to the presence of a small spherical object. From our calculations in both infinite and semi-infinite geometry we derive the charge and dipole coefficients that typify the object’s most significant response.

© 1996 Optical Society of America

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References

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  1. A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Physics Today 48, 34–40 (1995).
    [CrossRef]
  2. B. Chance, R. R. Alfano, eds., Photon Migration and Imaging in Random Media and Tissues, Proc. SPIE1888(1993).
    [CrossRef]
  3. R. R. Alfano, ed., Advances in Optical Imaging and Photon Migration, Vol. 21 of 1994 OSA Technical Digest Series (Opt. Soc. of Am., Washington, D.C., 1994).
  4. B. Chance, R. R. Alfano, eds., Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, Proc. SPIE2389(1995).
    [CrossRef]
  5. S. Feng, F. Zeng, B. Chance, “Monte Carlo simulations of photon migration path distribution in multiple scattering media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 78–89 (1993).
    [CrossRef]
  6. Shechao Feng, Fan-An Zeng, B. Chance, “Photon migration in the presence of a single defect—a perturbation analysis,” Appl. Opt. 34, 3826–3837 (1995).
    [CrossRef] [PubMed]
  7. S. R. Arridge, M. Schweiger, D. T. Delpy, “Iterative reconstruction of new infra-red absorption images,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE1767, 372–383 (1992).
    [CrossRef]
  8. P. N. den Outer, Th. M. Nieuwenhuizen, A. Lagendijk, “Location of objects in multiple-scattering media,” J. Opt. Soc. Am. A 10, 1209–1218 (1993).
    [CrossRef]
  9. J. M. Schmitt, A. Knüttel, J. R. Knutson, “Interference of diffusive light waves,” J. Opt. Soc. Am. A 9, 1832–1843 (1992); A. Knüttel, J. M. Schmitt, J. R. Knutson, “Spatial localization of absorbing bodies by interfering diffusive photon-density waves,” Appl. Opt. 32, 381–389 (1993); A. Knüttel, J. M. Schmitt, R. Barnes, J. R. Knutson, “Acoustic-optic scanning and interfering photon density waves for precise localization of an absorbing (or fluorescent) body in a turbid medium,” Rev. Sci. Instrum. 64, 638–644 (1993); B. Chance, K. Kang, L. He, J. Weng, E. Sevick, “Highly sensitive object location in tissue models with linear in-phase and anti-phase multi-element optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. USA (Medical Sciences) 90, 3423–3427 (1993).
    [CrossRef] [PubMed]
  10. D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
    [CrossRef] [PubMed]
  11. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  12. H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vols. I and II; M. B. van der Mark, M. P. van Albada, A. Lagendijk, “Light scattering in strongly scattering media; Multiple scattering and weak localization,” Phys. Rev. B 37, 3575–3592 (1988).
    [CrossRef]
  13. G. Arfken, Mathematical Methods for Physicists, 2nd ed. (Academic, New York, 1970), pp. 531–532.
  14. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  15. Th. M. Nieuwenhuizen, M. C. W. van Rossum, “Role of a single scatterer in a multiple scattering medium,” Phys. Lett. A 177, 102–106 (1993).
    [CrossRef]
  16. R. Berkovits, Shechao Feng, “Theory of speckle-pattern tomography in multiple-scattering media,” Phys. Rev. Lett. 65, 3120–3123 (1990).
    [CrossRef] [PubMed]
  17. R. Aronson, “Extrapolation distance for diffusion of light,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 297–305 (1993).
    [CrossRef]
  18. I. S. Grashteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed., A. Jeffrey, ed. (Academic, New York, 1994).

1995 (2)

1994 (1)

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef] [PubMed]

1993 (2)

Th. M. Nieuwenhuizen, M. C. W. van Rossum, “Role of a single scatterer in a multiple scattering medium,” Phys. Lett. A 177, 102–106 (1993).
[CrossRef]

P. N. den Outer, Th. M. Nieuwenhuizen, A. Lagendijk, “Location of objects in multiple-scattering media,” J. Opt. Soc. Am. A 10, 1209–1218 (1993).
[CrossRef]

1992 (1)

1990 (1)

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

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 2nd ed. (Academic, New York, 1970), pp. 531–532.

Aronson, R.

R. Aronson, “Extrapolation distance for diffusion of light,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 297–305 (1993).
[CrossRef]

Arridge, S. R.

S. R. Arridge, M. Schweiger, D. T. Delpy, “Iterative reconstruction of new infra-red absorption images,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE1767, 372–383 (1992).
[CrossRef]

Berkovits, R.

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

Boas, D. A.

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef] [PubMed]

Chance, B.

Shechao Feng, Fan-An Zeng, B. Chance, “Photon migration in the presence of a single defect—a perturbation analysis,” Appl. Opt. 34, 3826–3837 (1995).
[CrossRef] [PubMed]

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Physics Today 48, 34–40 (1995).
[CrossRef]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef] [PubMed]

S. Feng, F. Zeng, B. Chance, “Monte Carlo simulations of photon migration path distribution in multiple scattering media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 78–89 (1993).
[CrossRef]

Delpy, D. T.

S. R. Arridge, M. Schweiger, D. T. Delpy, “Iterative reconstruction of new infra-red absorption images,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE1767, 372–383 (1992).
[CrossRef]

den Outer, P. N.

Feng, S.

S. Feng, F. Zeng, B. Chance, “Monte Carlo simulations of photon migration path distribution in multiple scattering media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 78–89 (1993).
[CrossRef]

Feng, Shechao

Shechao Feng, Fan-An Zeng, B. Chance, “Photon migration in the presence of a single defect—a perturbation analysis,” Appl. Opt. 34, 3826–3837 (1995).
[CrossRef] [PubMed]

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

Grashteyn, I. S.

I. S. Grashteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed., A. Jeffrey, ed. (Academic, New York, 1994).

Ishimaru, A.

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

Jackson, J. D.

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

Knutson, J. R.

Knüttel, A.

Lagendijk, A.

Nieuwenhuizen, Th. M.

P. N. den Outer, Th. M. Nieuwenhuizen, A. Lagendijk, “Location of objects in multiple-scattering media,” J. Opt. Soc. Am. A 10, 1209–1218 (1993).
[CrossRef]

Th. M. Nieuwenhuizen, M. C. W. van Rossum, “Role of a single scatterer in a multiple scattering medium,” Phys. Lett. A 177, 102–106 (1993).
[CrossRef]

O’Leary, M. A.

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef] [PubMed]

Ryzhik, I. M.

I. S. Grashteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed., A. Jeffrey, ed. (Academic, New York, 1994).

Schmitt, J. M.

Schweiger, M.

S. R. Arridge, M. Schweiger, D. T. Delpy, “Iterative reconstruction of new infra-red absorption images,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE1767, 372–383 (1992).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vols. I and II; M. B. van der Mark, M. P. van Albada, A. Lagendijk, “Light scattering in strongly scattering media; Multiple scattering and weak localization,” Phys. Rev. B 37, 3575–3592 (1988).
[CrossRef]

van Rossum, M. C. W.

Th. M. Nieuwenhuizen, M. C. W. van Rossum, “Role of a single scatterer in a multiple scattering medium,” Phys. Lett. A 177, 102–106 (1993).
[CrossRef]

Yodh, A.

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Physics Today 48, 34–40 (1995).
[CrossRef]

Yodh, A. G.

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef] [PubMed]

Zeng, F.

S. Feng, F. Zeng, B. Chance, “Monte Carlo simulations of photon migration path distribution in multiple scattering media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 78–89 (1993).
[CrossRef]

Zeng, Fan-An

Appl. Opt. (1)

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

Phys. Lett. A (1)

Th. M. Nieuwenhuizen, M. C. W. van Rossum, “Role of a single scatterer in a multiple scattering medium,” Phys. Lett. A 177, 102–106 (1993).
[CrossRef]

Phys. Rev. Lett. (1)

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

Physics Today (1)

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Physics Today 48, 34–40 (1995).
[CrossRef]

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

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef] [PubMed]

Other (11)

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

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vols. I and II; M. B. van der Mark, M. P. van Albada, A. Lagendijk, “Light scattering in strongly scattering media; Multiple scattering and weak localization,” Phys. Rev. B 37, 3575–3592 (1988).
[CrossRef]

G. Arfken, Mathematical Methods for Physicists, 2nd ed. (Academic, New York, 1970), pp. 531–532.

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

R. Aronson, “Extrapolation distance for diffusion of light,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 297–305 (1993).
[CrossRef]

I. S. Grashteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed., A. Jeffrey, ed. (Academic, New York, 1994).

B. Chance, R. R. Alfano, eds., Photon Migration and Imaging in Random Media and Tissues, Proc. SPIE1888(1993).
[CrossRef]

R. R. Alfano, ed., Advances in Optical Imaging and Photon Migration, Vol. 21 of 1994 OSA Technical Digest Series (Opt. Soc. of Am., Washington, D.C., 1994).

B. Chance, R. R. Alfano, eds., Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, Proc. SPIE2389(1995).
[CrossRef]

S. Feng, F. Zeng, B. Chance, “Monte Carlo simulations of photon migration path distribution in multiple scattering media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 78–89 (1993).
[CrossRef]

S. R. Arridge, M. Schweiger, D. T. Delpy, “Iterative reconstruction of new infra-red absorption images,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE1767, 372–383 (1992).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Small spherical defect object with radius of curvature a embedded in an infinite multiple-scattering medium. The origin of the coordinate system overlaps the center of the object. A point light source is placed at r0.

Fig. 2
Fig. 2

Small spherical defect object with radius of curvature a embedded in a semi-infinite multiple-scattering medium (the sample region) that occupies the half-space with z > 0. The scattering-free region occupies the half space with z < 0. A point light source is placed inside the sample region at rs = (d, 0, z0). Special attention should be paid to the definitions of θ s ( + ) and θ s ( - ) that are used in Appendix B.

Fig. 3
Fig. 3

Same as Fig. 2 except for the definitions of θ image ( + ) and θ image ( - ) that are used in Appendix B.

Equations (45)

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1 c Φ ( r , t ) t - D 2 Φ ( r , t ) + μ a Φ ( r , t ) = S 0 ( t ) δ ( r - r 0 ) .
- 2 Φ ( r , Ω ) + ( μ a D - i Ω D c ) Φ ( r , Ω ) = S 0 D δ ( r - r 0 ) .
Φ 0 ( r , Ω ) = S 0 4 π D exp ( - κ r - r 0 ) r - r 0 ,
J 0 = - D Φ 0 ( r , Ω ) .
Φ out ( r , Ω ) = Φ 0 ( r , Ω ) + m = 0 B m k m ( κ r ) P m ( cos θ ) ,
B m = ( - ) S 0 κ ( 2 m + 1 ) k m ( κ r 0 ) 4 π D × { κ D i m ( κ a ) [ i m ( κ a ) ] ( 1 ) - κ D i m ( κ a ) [ i m ( κ a ) ] ( 1 ) κ D i m ( κ a ) [ k m ( κ a ) ] ( 1 ) - κ D k m ( κ a ) [ i m ( κ a ) ] ( 1 ) } ,
q = B 0 / κ
p = B 1 / κ 2 .
Φ 1 ( r ) Φ out ( r ) - Φ 0 ( r ) = [ q 0 r + p 0 · r r 3 ( 1 + κ r ) ] exp ( - κ r ) .
q 0 = - Φ 0 ( 0 ) a
p 0 = - Φ 0 ( 0 ) a 3 ,
Φ 1 ( r ) [ q r + p · r r 3 ( 1 + κ r ) ] exp ( - κ r ) .
q = - Φ 0 ( 0 ) a ( μ a a 2 3 D ) = q 0 μ a a 2 3 D
p = - Φ 0 ( 0 ) a 3 D - D D + 2 D = p 0 D - D D + 2 D .
B 0 = - Φ 0 ( 0 ) { ( δ κ / κ ) [ sinh ( κ a ) cosh ( κ a ) - κ a ] + ( δ D / D ) [ sinh ( κ a ) / κ a ] × [ κ a cosh ( κ a ) - sinh ( κ a ) ] }
B 1 = - 3 Φ 0 ( 0 ) κ 4 a 3 { ( δ κ / κ ) [ κ a sinh ( κ a ) cosh ( κ a ) + κ 2 a 2 - 2 sinh 2 ( κ a ) ] κ 2 a 2 + ( δ D / D ) [ κ a cosh ( κ a ) - sinh ( κ a ) ] [ κ 2 a 2 sinh ( κ a ) - 2 κ a cosh ( κ a ) + 2 sinh ( κ a ) ] } .
q = - Φ 0 ( 0 ) κ { ( δ κ / κ ) [ sinh ( κ a ) cosh ( κ a ) - κ a ] + ( δ D / D ) [ sinh ( κ a ) / κ a ] × [ κ a cosh ( κ a ) - sinh ( κ a ) ] } ,
p = - 3 Φ 0 ( 0 ) κ 6 a 3 { ( δ κ / κ ) [ κ a sinh ( κ a ) cosh ( κ a ) + κ 2 a 2 - 2 sinh 2 ( κ a ) ] κ 2 a 2 + ( δ D / D ) [ κ a cosh ( κ a ) - sinh ( κ a ) ] [ κ 2 a 2 sinh ( κ a ) - 2 κ a cosh ( κ a ) + 2 sinh ( κ a ) ] } .
q = q 0 a 2 ( μ a - μ a ) 3 D
p p 0 3 [ ( δ D / D ) + 2 5 ( δ κ / κ ) κ 2 a 2 ] ,
Φ 0 ( semi ) ( r ) = Φ 0 , orig ( r ) + Φ 0 , image ( r ) = S 0 4 π D exp ( - κ r - d x ^ - z 0 z ^ ) r - d x ^ - z 0 z ^ + ( - ) S 0 4 π D exp ( - κ r - d x ^ + z 0 z ^ ) r - d x ^ + z 0 z ^ 2 z z 0 S 0 ( 1 + κ r - d x ^ ) exp ( - κ r - d x ^ ) 4 π D r - d x ^ 3 .
Φ 1 ( semi ) ( r ) q ( semi ) [ exp ( - κ r - r ) r - r - exp ( - κ r - r image ) r - r image ] + p x ( semi ) ( x - x ) [ ( 1 + κ r - r ) exp ( - κ r - r ) r - r 3 - ( 1 + κ r - r image ) exp ( - κ r - r image ) r - r image 3 ] + p z ( semi ) [ ( z - z ) ( 1 + κ r - r ) exp ( - κ r - r ) r - r 3 + ( z + z ) ( 1 + κ r - r image ) exp ( - κ r - r image ) r - r image 3 ] .
q 0 ( semi ) = - Φ 0 ( semi ) ( r ) a
p 0 ( semi ) = - Φ 0 ( semi ) ( r ) a 3 ,
q ( semi ) - Φ 0 ( semi ) ( r ) a 3 ( μ a - μ a ) 3 D
p ( semi ) - Φ 0 ( semi ) ( r ) a 3 3 [ ( D - D D ) + 2 5 ( κ - κ κ ) κ 2 a 2 ] .
J 0 = - D Φ 0 ( semi ) ( r ) r = 0 - S 0 z 0 ( 1 + κ d ) exp ( - κ d ) 2 π d 3 z ^ ,
J q = - 2 D q ( semi ) z ( 1 + κ r ) exp ( - κ r ) r 3 z ^ ,
J p = 2 D z ( p ( semi ) · r ) ( 3 + 3 κ r + κ 2 r 2 ) exp ( - κ r ) r 5 z ^ - 2 D p z ( semi ) ( 1 + κ r ) exp ( - κ r ) r 3 z ^ .
J q J p r 2 a 2 1 ,
J q 2 Φ 0 ( semi ) ( r ) a 3 ( μ a - μ a ) z ( 1 + κ r ) exp ( - κ r ) 3 r 3 z ^ .
Φ in ( r ) = m = 0 A m i m ( κ r ) P m ( cos θ ) .
Φ out ( r ) = Φ 0 ( r ) + m = 0 B m k m ( κ r ) P m ( cos θ ) .
Φ 0 ( r ) = S 0 κ 4 π D m = 0 ( 2 m + 1 ) i m ( κ r ) k m ( κ r 0 ) P m ( cos θ )
Φ out ( r ) = m = 0 [ B m k m ( κ r ) + ( 2 m + 1 ) κ S 0 k m ( κ r 0 ) 4 π D i m ( κ r ) ] × P m ( cos θ ) .
Φ in ( r ) r = a = Φ out ( r ) r = a ,
D Φ in ( r ) r | r = a = D Φ out ( r ) r | r = a .
A m = ( - ) S 0 κ ( 2 m + 1 ) k m ( κ r 0 ) 4 π D × { κ D k m ( κ a ) [ i m ( κ a ) ] ( 1 ) - κ D i m ( κ a ) [ k m ( κ a ) ] ( 1 ) κ D i m ( κ a ) [ k m ( κ a ) ] ( 1 ) - κ D k m ( κ a ) [ i m ( κ a ) ] ( 1 ) } ,
B m = ( - ) S 0 κ ( 2 m + 1 ) k m ( κ r 0 ) 4 π D × { κ D i m ( κ a ) [ i m ( κ a ) ] ( 1 ) - κ D i m ( κ a ) [ i m ( κ a ) ] ( 1 ) κ D i m ( κ a ) [ k m ( κ a ) ] ( 1 ) - κ D k m ( κ a ) [ i m ( κ a ) ] ( 1 ) } .
Φ 1 ( infinite ) ( r ) = m = 0 B m ( s ) k m ( κ r - r ) P m [ cos θ s ( + ) ]
B m ( s ) = ( - ) S 0 κ ( 2 m + 1 ) k m ( κ r s - r ) 4 π D × { κ D i m ( κ a ) [ i m ( κ a ) ] ( 1 ) - κ D i m ( κ a ) [ i m ( κ a ) ] ( 1 ) κ D i m ( κ a ) [ k m ( κ a ) ] ( 1 ) - κ D k m ( κ a ) [ i m ( κ a ) ] ( 1 ) } .
Φ 1 , s ( semi ) ( r ) = m = 0 B m ( s ) { k m ( κ r - r ) P m [ cos θ s ( + ) ] - k m ( κ r - r image ) P m [ cos θ s ( - ) ] } .
Φ 1 , image ( semi ) ( r ) = m = 0 B m ( image ) { k m ( κ r - r ) P m [ cos θ image ( + ) ] - k m ( κ r - r image ) P m [ cos θ image ( - ) ] } .
B m ( image ) = ( + ) S 0 κ ( 2 m + 1 ) k m ( κ r s , image - r ) 4 π D × { κ D i m ( κ a ) [ i m ( κ a ) ] ( 1 ) - κ D i m ( κ a ) [ i m ( κ a ) ] ( 1 ) κ D i m ( κ a ) [ k m ( κ a ) ] ( 1 ) - κ D k m ( κ a ) [ i m ( κ a ) ] ( 1 ) } .
Φ 1 ( semi ) ( r ) = Φ 1 , s ( semi ) ( r ) + Φ 1 , image ( semi ) ( r ) = m = 0 B m ( s ) { k m ( κ r - r ) P m [ cos θ s ( + ) ] - k m ( κ r - r image ) P m [ cos θ s ( - ) ] } + m = 0 B m ( image ) { k m ( κ r - r ) P m [ cos θ image ( + ) ] - k m ( κ r - r image ) P m [ cos θ image ( - ) ] } .

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