Abstract

In a number of techniques that measure weak fluxes of optical radiation, it is frequently necessary to keep a detector in a medium different from that of the radiating source by separating it from the source with a planar transparent window. However, sources such as systems of light-emitting diodes, large-fiber illuminators, and microscopic living objects that emit biological luminescence may sometimes be regarded as multiple-point sources. To estimate the fluxes of optical radiation illuminating a surface from a nonuniformly distributed multiple-point source, a method for calculating fluxes from a single off-axis point source is needed. A formula is derived to estimate a flux of temporally incoherent optical radiation incident on a circular disk from a single off-axis point source separated by a plane-parallel plate (PPP). This formula is expressed by a series of single integrals of some superposed elementary functions. These functions depend on the variables that characterize the point-source–plane-parallel-plate–circular-disk geometry and on the optical properties of the media that separate the source from the PPP and the PPP from the disk. The solution was obtained for isotropic media. For illustrative purposes some examples of the use of the formula are presented. The selected results are illustrated by three-dimensional surface plots and compared with the values of the fluxes calculated for radiation incident on the disk from a point source not separated by a PPP.

© 2001 Optical Society of America

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References

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  1. J. C. Maxwell, A Treatise on Electricity and Magnetism (Oxford U. Press, London, 1955), Vol. 2, pp. 331–346.
  2. A. G. Webster, Partial Differential Equations of Mathematical Physics (G. E. Stechert, New York, 1927), p. 368.
  3. Y. B. Kim, E. D. Platner, “Flux concentrator for high-intensity pulsed magnetic fields,” Rev. Sci. Instrum. 30, 524–533 (1959).
    [CrossRef]
  4. G. Rowlands, “Solid angle calculations,” Int. J. Appl. Radiat. Isot. 10, 86–93 (1961).
    [CrossRef]
  5. M. W. Garrett, “Solid angle subtended by a circular aperture,” Rev. Sci. Instrum. 25, 1208–1211 (1954).
    [CrossRef]
  6. A. H. Jaffey, “Solid angle subtended by a circular aperture at point and spread sources: formulas and some tables,” Rev. Sci. Instrum. 25, 349–354 (1954).
    [CrossRef]
  7. J. H. Smith, M. L. Storm, “Generalized off-axis distributions from disk sources of radiation,” J. Appl. Phys. 25, 519–527 (1954).
    [CrossRef]
  8. A. V. Masket, “Solid angle contour integrals, series, and tables,” Rev. Sci. Instrum. 28, 191–197 (1957).
    [CrossRef]
  9. P. A. Macklin, See solution presented as note in Ref. 8, p. 191.
  10. M. Naito, “A method of calculating the solid angle subtended by a circular aperture,” J. Phys. Soc. Jpn. 12, 1122–1129 (1957).
    [CrossRef]
  11. F. Paxton, “Solid angle calculation for a circular disk,” Rev. Sci. Instrum. 30, 254–258 (1959).
    [CrossRef]
  12. J. H. Hubbell, R. L. Bach, R. J. Herbold, “Radiation field from a circular disk source,” J. Res. Natl. Bur. Stand. 65C, 249–264 (1961).
  13. R. P. Gardner, K. Carnesale, “The solid angle subtended at a point by a circular disk,” Nucl. Instrum. Methods 73, 228–230 (1969).
    [CrossRef]
  14. R. P. Gardner, K. Verghese, “On the solid angle subtended by a circular disc,” Nucl. Instrum. Methods 93, 163–167 (1971).
    [CrossRef]
  15. C. J. Bland, “Tables of the geometrical factor for various source–detector configurations,” Nucl. Instrum. Methods Phys. Res. 223, 602–606 (1984).
    [CrossRef]
  16. P. Olivier, D. Gagnon, “Mathematical modelling of the solid angle function, part I: approximation in homogeneous medium,” Opt. Eng. 32, 2261–2265 (1993).
    [CrossRef]
  17. P. Olivier, S. Rioux, D. Gagnon, “Mathematical modelling of the solid angle function, part II: transmission through refractive media,” Opt. Eng. 32, 2266–2270 (1993).
    [CrossRef]
  18. S. Tryka, “Angular distribution of the solid angle at a point subtended by a circular disk,” Opt. Commun. 137, 317–333 (1997).
    [CrossRef]
  19. S. Tryka, “A method for calculating the average solid angle subtended by a circular disk from uniformly distributed points within a coaxial circular plane,” Rev. Sci. Instrum. 70, 3915–3920 (1999).
    [CrossRef]
  20. L. Colli, U. Facchini, “Light emission by germinating plants,” Nuovo Cimento 12, 150–153 (1954).
    [CrossRef]
  21. H. Inaba, A. Yamagishi, C. Tokyu, “Development of an ultra-high sensitive photon counting system and its application to biomedical measurements,” Opt. Lasers Eng. 3, 125–130 (1982).
    [CrossRef]
  22. S. Wollfram, Mathematica—A System for Doing Mathematics by Computer (Addison-Wesley, Reading, Mass., 1993), pp. 44–186.
  23. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), pp. 180–185.
  24. I. S. Sokolnikoff, R. M. Redheffer, Mathematics of Physics and Modern Engineering (McGraw-Hill, New York, 1958), pp. 684–686.
  25. J. I. Pankove, Optical Processes in Semiconductors (Prentice-Hall, Englewood Cliffs, N.J., 1971), Chap. 4.
  26. Ref. 23, pp. 38–40.

1999 (1)

S. Tryka, “A method for calculating the average solid angle subtended by a circular disk from uniformly distributed points within a coaxial circular plane,” Rev. Sci. Instrum. 70, 3915–3920 (1999).
[CrossRef]

1997 (1)

S. Tryka, “Angular distribution of the solid angle at a point subtended by a circular disk,” Opt. Commun. 137, 317–333 (1997).
[CrossRef]

1993 (2)

P. Olivier, D. Gagnon, “Mathematical modelling of the solid angle function, part I: approximation in homogeneous medium,” Opt. Eng. 32, 2261–2265 (1993).
[CrossRef]

P. Olivier, S. Rioux, D. Gagnon, “Mathematical modelling of the solid angle function, part II: transmission through refractive media,” Opt. Eng. 32, 2266–2270 (1993).
[CrossRef]

1984 (1)

C. J. Bland, “Tables of the geometrical factor for various source–detector configurations,” Nucl. Instrum. Methods Phys. Res. 223, 602–606 (1984).
[CrossRef]

1982 (1)

H. Inaba, A. Yamagishi, C. Tokyu, “Development of an ultra-high sensitive photon counting system and its application to biomedical measurements,” Opt. Lasers Eng. 3, 125–130 (1982).
[CrossRef]

1971 (1)

R. P. Gardner, K. Verghese, “On the solid angle subtended by a circular disc,” Nucl. Instrum. Methods 93, 163–167 (1971).
[CrossRef]

1969 (1)

R. P. Gardner, K. Carnesale, “The solid angle subtended at a point by a circular disk,” Nucl. Instrum. Methods 73, 228–230 (1969).
[CrossRef]

1961 (2)

J. H. Hubbell, R. L. Bach, R. J. Herbold, “Radiation field from a circular disk source,” J. Res. Natl. Bur. Stand. 65C, 249–264 (1961).

G. Rowlands, “Solid angle calculations,” Int. J. Appl. Radiat. Isot. 10, 86–93 (1961).
[CrossRef]

1959 (2)

Y. B. Kim, E. D. Platner, “Flux concentrator for high-intensity pulsed magnetic fields,” Rev. Sci. Instrum. 30, 524–533 (1959).
[CrossRef]

F. Paxton, “Solid angle calculation for a circular disk,” Rev. Sci. Instrum. 30, 254–258 (1959).
[CrossRef]

1957 (2)

A. V. Masket, “Solid angle contour integrals, series, and tables,” Rev. Sci. Instrum. 28, 191–197 (1957).
[CrossRef]

M. Naito, “A method of calculating the solid angle subtended by a circular aperture,” J. Phys. Soc. Jpn. 12, 1122–1129 (1957).
[CrossRef]

1954 (4)

M. W. Garrett, “Solid angle subtended by a circular aperture,” Rev. Sci. Instrum. 25, 1208–1211 (1954).
[CrossRef]

A. H. Jaffey, “Solid angle subtended by a circular aperture at point and spread sources: formulas and some tables,” Rev. Sci. Instrum. 25, 349–354 (1954).
[CrossRef]

J. H. Smith, M. L. Storm, “Generalized off-axis distributions from disk sources of radiation,” J. Appl. Phys. 25, 519–527 (1954).
[CrossRef]

L. Colli, U. Facchini, “Light emission by germinating plants,” Nuovo Cimento 12, 150–153 (1954).
[CrossRef]

Bach, R. L.

J. H. Hubbell, R. L. Bach, R. J. Herbold, “Radiation field from a circular disk source,” J. Res. Natl. Bur. Stand. 65C, 249–264 (1961).

Bland, C. J.

C. J. Bland, “Tables of the geometrical factor for various source–detector configurations,” Nucl. Instrum. Methods Phys. Res. 223, 602–606 (1984).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), pp. 180–185.

Carnesale, K.

R. P. Gardner, K. Carnesale, “The solid angle subtended at a point by a circular disk,” Nucl. Instrum. Methods 73, 228–230 (1969).
[CrossRef]

Colli, L.

L. Colli, U. Facchini, “Light emission by germinating plants,” Nuovo Cimento 12, 150–153 (1954).
[CrossRef]

Facchini, U.

L. Colli, U. Facchini, “Light emission by germinating plants,” Nuovo Cimento 12, 150–153 (1954).
[CrossRef]

Gagnon, D.

P. Olivier, D. Gagnon, “Mathematical modelling of the solid angle function, part I: approximation in homogeneous medium,” Opt. Eng. 32, 2261–2265 (1993).
[CrossRef]

P. Olivier, S. Rioux, D. Gagnon, “Mathematical modelling of the solid angle function, part II: transmission through refractive media,” Opt. Eng. 32, 2266–2270 (1993).
[CrossRef]

Gardner, R. P.

R. P. Gardner, K. Verghese, “On the solid angle subtended by a circular disc,” Nucl. Instrum. Methods 93, 163–167 (1971).
[CrossRef]

R. P. Gardner, K. Carnesale, “The solid angle subtended at a point by a circular disk,” Nucl. Instrum. Methods 73, 228–230 (1969).
[CrossRef]

Garrett, M. W.

M. W. Garrett, “Solid angle subtended by a circular aperture,” Rev. Sci. Instrum. 25, 1208–1211 (1954).
[CrossRef]

Herbold, R. J.

J. H. Hubbell, R. L. Bach, R. J. Herbold, “Radiation field from a circular disk source,” J. Res. Natl. Bur. Stand. 65C, 249–264 (1961).

Hubbell, J. H.

J. H. Hubbell, R. L. Bach, R. J. Herbold, “Radiation field from a circular disk source,” J. Res. Natl. Bur. Stand. 65C, 249–264 (1961).

Inaba, H.

H. Inaba, A. Yamagishi, C. Tokyu, “Development of an ultra-high sensitive photon counting system and its application to biomedical measurements,” Opt. Lasers Eng. 3, 125–130 (1982).
[CrossRef]

Jaffey, A. H.

A. H. Jaffey, “Solid angle subtended by a circular aperture at point and spread sources: formulas and some tables,” Rev. Sci. Instrum. 25, 349–354 (1954).
[CrossRef]

Kim, Y. B.

Y. B. Kim, E. D. Platner, “Flux concentrator for high-intensity pulsed magnetic fields,” Rev. Sci. Instrum. 30, 524–533 (1959).
[CrossRef]

Macklin, P. A.

P. A. Macklin, See solution presented as note in Ref. 8, p. 191.

Masket, A. V.

A. V. Masket, “Solid angle contour integrals, series, and tables,” Rev. Sci. Instrum. 28, 191–197 (1957).
[CrossRef]

Maxwell, J. C.

J. C. Maxwell, A Treatise on Electricity and Magnetism (Oxford U. Press, London, 1955), Vol. 2, pp. 331–346.

Naito, M.

M. Naito, “A method of calculating the solid angle subtended by a circular aperture,” J. Phys. Soc. Jpn. 12, 1122–1129 (1957).
[CrossRef]

Olivier, P.

P. Olivier, S. Rioux, D. Gagnon, “Mathematical modelling of the solid angle function, part II: transmission through refractive media,” Opt. Eng. 32, 2266–2270 (1993).
[CrossRef]

P. Olivier, D. Gagnon, “Mathematical modelling of the solid angle function, part I: approximation in homogeneous medium,” Opt. Eng. 32, 2261–2265 (1993).
[CrossRef]

Pankove, J. I.

J. I. Pankove, Optical Processes in Semiconductors (Prentice-Hall, Englewood Cliffs, N.J., 1971), Chap. 4.

Paxton, F.

F. Paxton, “Solid angle calculation for a circular disk,” Rev. Sci. Instrum. 30, 254–258 (1959).
[CrossRef]

Platner, E. D.

Y. B. Kim, E. D. Platner, “Flux concentrator for high-intensity pulsed magnetic fields,” Rev. Sci. Instrum. 30, 524–533 (1959).
[CrossRef]

Redheffer, R. M.

I. S. Sokolnikoff, R. M. Redheffer, Mathematics of Physics and Modern Engineering (McGraw-Hill, New York, 1958), pp. 684–686.

Rioux, S.

P. Olivier, S. Rioux, D. Gagnon, “Mathematical modelling of the solid angle function, part II: transmission through refractive media,” Opt. Eng. 32, 2266–2270 (1993).
[CrossRef]

Rowlands, G.

G. Rowlands, “Solid angle calculations,” Int. J. Appl. Radiat. Isot. 10, 86–93 (1961).
[CrossRef]

Smith, J. H.

J. H. Smith, M. L. Storm, “Generalized off-axis distributions from disk sources of radiation,” J. Appl. Phys. 25, 519–527 (1954).
[CrossRef]

Sokolnikoff, I. S.

I. S. Sokolnikoff, R. M. Redheffer, Mathematics of Physics and Modern Engineering (McGraw-Hill, New York, 1958), pp. 684–686.

Storm, M. L.

J. H. Smith, M. L. Storm, “Generalized off-axis distributions from disk sources of radiation,” J. Appl. Phys. 25, 519–527 (1954).
[CrossRef]

Tokyu, C.

H. Inaba, A. Yamagishi, C. Tokyu, “Development of an ultra-high sensitive photon counting system and its application to biomedical measurements,” Opt. Lasers Eng. 3, 125–130 (1982).
[CrossRef]

Tryka, S.

S. Tryka, “A method for calculating the average solid angle subtended by a circular disk from uniformly distributed points within a coaxial circular plane,” Rev. Sci. Instrum. 70, 3915–3920 (1999).
[CrossRef]

S. Tryka, “Angular distribution of the solid angle at a point subtended by a circular disk,” Opt. Commun. 137, 317–333 (1997).
[CrossRef]

Verghese, K.

R. P. Gardner, K. Verghese, “On the solid angle subtended by a circular disc,” Nucl. Instrum. Methods 93, 163–167 (1971).
[CrossRef]

Webster, A. G.

A. G. Webster, Partial Differential Equations of Mathematical Physics (G. E. Stechert, New York, 1927), p. 368.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), pp. 180–185.

Wollfram, S.

S. Wollfram, Mathematica—A System for Doing Mathematics by Computer (Addison-Wesley, Reading, Mass., 1993), pp. 44–186.

Yamagishi, A.

H. Inaba, A. Yamagishi, C. Tokyu, “Development of an ultra-high sensitive photon counting system and its application to biomedical measurements,” Opt. Lasers Eng. 3, 125–130 (1982).
[CrossRef]

Int. J. Appl. Radiat. Isot. (1)

G. Rowlands, “Solid angle calculations,” Int. J. Appl. Radiat. Isot. 10, 86–93 (1961).
[CrossRef]

J. Appl. Phys. (1)

J. H. Smith, M. L. Storm, “Generalized off-axis distributions from disk sources of radiation,” J. Appl. Phys. 25, 519–527 (1954).
[CrossRef]

J. Phys. Soc. Jpn. (1)

M. Naito, “A method of calculating the solid angle subtended by a circular aperture,” J. Phys. Soc. Jpn. 12, 1122–1129 (1957).
[CrossRef]

J. Res. Natl. Bur. Stand. (1)

J. H. Hubbell, R. L. Bach, R. J. Herbold, “Radiation field from a circular disk source,” J. Res. Natl. Bur. Stand. 65C, 249–264 (1961).

Nucl. Instrum. Methods (2)

R. P. Gardner, K. Carnesale, “The solid angle subtended at a point by a circular disk,” Nucl. Instrum. Methods 73, 228–230 (1969).
[CrossRef]

R. P. Gardner, K. Verghese, “On the solid angle subtended by a circular disc,” Nucl. Instrum. Methods 93, 163–167 (1971).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. (1)

C. J. Bland, “Tables of the geometrical factor for various source–detector configurations,” Nucl. Instrum. Methods Phys. Res. 223, 602–606 (1984).
[CrossRef]

Nuovo Cimento (1)

L. Colli, U. Facchini, “Light emission by germinating plants,” Nuovo Cimento 12, 150–153 (1954).
[CrossRef]

Opt. Commun. (1)

S. Tryka, “Angular distribution of the solid angle at a point subtended by a circular disk,” Opt. Commun. 137, 317–333 (1997).
[CrossRef]

Opt. Eng. (2)

P. Olivier, D. Gagnon, “Mathematical modelling of the solid angle function, part I: approximation in homogeneous medium,” Opt. Eng. 32, 2261–2265 (1993).
[CrossRef]

P. Olivier, S. Rioux, D. Gagnon, “Mathematical modelling of the solid angle function, part II: transmission through refractive media,” Opt. Eng. 32, 2266–2270 (1993).
[CrossRef]

Opt. Lasers Eng. (1)

H. Inaba, A. Yamagishi, C. Tokyu, “Development of an ultra-high sensitive photon counting system and its application to biomedical measurements,” Opt. Lasers Eng. 3, 125–130 (1982).
[CrossRef]

Rev. Sci. Instrum. (6)

F. Paxton, “Solid angle calculation for a circular disk,” Rev. Sci. Instrum. 30, 254–258 (1959).
[CrossRef]

A. V. Masket, “Solid angle contour integrals, series, and tables,” Rev. Sci. Instrum. 28, 191–197 (1957).
[CrossRef]

M. W. Garrett, “Solid angle subtended by a circular aperture,” Rev. Sci. Instrum. 25, 1208–1211 (1954).
[CrossRef]

A. H. Jaffey, “Solid angle subtended by a circular aperture at point and spread sources: formulas and some tables,” Rev. Sci. Instrum. 25, 349–354 (1954).
[CrossRef]

S. Tryka, “A method for calculating the average solid angle subtended by a circular disk from uniformly distributed points within a coaxial circular plane,” Rev. Sci. Instrum. 70, 3915–3920 (1999).
[CrossRef]

Y. B. Kim, E. D. Platner, “Flux concentrator for high-intensity pulsed magnetic fields,” Rev. Sci. Instrum. 30, 524–533 (1959).
[CrossRef]

Other (8)

J. C. Maxwell, A Treatise on Electricity and Magnetism (Oxford U. Press, London, 1955), Vol. 2, pp. 331–346.

A. G. Webster, Partial Differential Equations of Mathematical Physics (G. E. Stechert, New York, 1927), p. 368.

P. A. Macklin, See solution presented as note in Ref. 8, p. 191.

S. Wollfram, Mathematica—A System for Doing Mathematics by Computer (Addison-Wesley, Reading, Mass., 1993), pp. 44–186.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), pp. 180–185.

I. S. Sokolnikoff, R. M. Redheffer, Mathematics of Physics and Modern Engineering (McGraw-Hill, New York, 1958), pp. 684–686.

J. I. Pankove, Optical Processes in Semiconductors (Prentice-Hall, Englewood Cliffs, N.J., 1971), Chap. 4.

Ref. 23, pp. 38–40.

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Figures (8)

Fig. 1
Fig. 1

Definition of some variables and illustration of virtual images Pk obtained for the optical radiation from a point source P propagated through the PPP and incident on the disk surface S3.

Fig. 2
Fig. 2

Definition of the variables and the point-source–PPP–circular-disk-system geometry illustrating the approach used to obtain the first equality in Eq. (9) for the case of P lying under S3 at distance 0ρ<R3 and for 0η3,1R3-ρ.

Fig. 3
Fig. 3

Definition of the variables and the point-source–PPP–circular-disk-system geometry illustrating the approach used to obtain the second equality in Eq. (9) for the case of P lying under S3 at distance 0<ρ<R3 and for R3-ρ<η3,1R3+ρ.

Fig. 4
Fig. 4

Definition of the variables and the point-source–PPP–circular-disk-system geometry illustrating the approach used to obtain the fourth equality in Eq. (9) for the case of P lying outside S3 at distance 0<R3<ρ and for ρ-R3η3,1R3+ρ.

Fig. 5
Fig. 5

Dependencies of FP,S3 on R3 and ρ from formula (29) for rotationally symmetric optical radiation propagating through the PPP and described by the expression Iin=I0 cos Θ1, where I0=1. The dependencies were obtained for the relative values of H1=H3=0.4,H2=0.2, absolute indices of refraction n1=n3=1.0,n2=1.5, and absorption coefficients (a) aλ=0 and (b) aλ=0.05/H2.

Fig. 6
Fig. 6

Dependencies of FP,S3 on R3 and ρ from formula (31) for isotropic optical radiation propagating through the PPP for the relative values of H1=H3=0.4,H2=0.2, absolute indices of refraction n1=n3=1.0,n2=1.5, and absorption coefficients (a) aλ=0 and (b) aλ=0.05/H2.

Fig. 7
Fig. 7

Dependencies of FP,S3 on R3 and ρ for H1+H3=1.0,H2=0 and n1=n2=n3 (a) from formula (33) for the rotationally symmetric optical radiation incident directly on S3 from the source P and (b) from formula (34) for isotropic optical radiation incident directly on S3 from the source P.

Fig. 8
Fig. 8

Comparison of the results from Figs. 5(a), 6(a), 7(a), and 7(b) obtained for (a) R3=0.5 and (b) R3=5.0.

Equations (55)

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

dFPk,dS3=I(η3,k, α3, t)dΩPk,dS3,
η3,k=H1tan Θ1+(2k-1)H2tan Θ2+H3tan Θ3.
n1sin Θ1=n2sin Θ2=n3sin Θ3,
η3,k=2(k-1)H2n3sin Θ3(n22-n32sin2 Θ3)1/2+j=13Hjn3sin Θ3(nj2-n32sin2 Θ3)1/2,
nj>n3sin Θ3,
dΩPk,dS3=sin Θ3dΘ3dα3,
dFPk,dS3=I(η3,k, α3, t)sin Θ3dΘ3dα3.
dFPk,S3=α3,kα3,kdFPk,dS3=sin Θ3dΘ3α3,kα3,kI(η3,k, α3, t)dα3,
α3,k=0ρR3-η3,kπ-γ3,k,ρ>R3-η3,k,
α3,k=2πρR3-η3,kπ+γ3,k,ρ>R3-η3,k,
γ3,k=arccos[(ρ2+η3,k2-R32)/2ρη3,k].
dFPk,S3=fPk,S3dΘ3,
fPk,S3=dFPk,S3dΘ3α3,kα3,kI(η3,k,α3,,t)sin Θ3dα3
fPk,S3
=sin Θ302πI(η3,k ,α3, t)dα30ρ<R3,0η3,kR3-ρsin Θ3π-γ3,kπ+γ3,kI(η3,k, α3, t)dα30<ρ<R3,R3-ρη3,kR3+ρsin Θ3π-γR3,kπ+γR3,kI(η3,k, α3, t)dα30<ρ=R3,0η3,k2R3sin Θ3π-γ3,kπ+γ3,kI(η3,k, α3, t)dα30<R3<ρ,ρ-R3η3,kR3+ρ,
FPk,S3=0Θ3fPk,S3dΘ3.
FP,S3=k=10ΘR3,k02πI(η3,k, α3, t)sin Θ3dΘ3dα3,0=ρ<R3k=10ΘR3-ρ,k02πI(η3,k, α3, t)sin Θ3dΘ3dα3,+k=1ΘR3-ρ,kΘR3+ρ,kπ-γ3,kπ+γγ3,kI(η3,k, α3, t)sin Θ3dΘ3dα3,0<ρ<R3k=10Θ2R3,kπ-γR3,kπ+γR3,kI(η3,k, α3, t)sin Θ3dΘ3dα3,0<ρ=R3k=1kρ-R3,maxΘρ-R3,kΘR3+ρ,kπ-γ3,kπ+γ3,kI(η3,k, α3, t)sin Θ3dΘ3dα3,0<R3<ρ,
R3-2(k-1)H2n3sin ΘR3,k(n22-n32sin2 ΘR3,k)1/2
-j=13Hjn3sin ΘR3,k(nj2-n32sin2 ΘR3,k)1/2=0,
R3-ρ-2(k-1)H2n3sin ΘR3-ρ,k(n22-n32sin2 ΘR3-ρ,k)1/2
-j=13Hjn3sin ΘR3-ρ,k(nj2-n32sin2 ΘR3-ρ,k)1/2=0,
R3+ρ-2(k-1)H2n3sin ΘR3+ρ,k(n22-n32sin2 ΘR3+ρ,k)1/2
-j=13Hjn3sin ΘR3+ρ,k(nj2-n32sin2 ΘR3+ρ,k)1/2=0,
2R3-2(k-1)H2 n3sin Θ2R,3,k(n22-n32sin2 Θ2R3,k)1/2
-j=13Hjn3sin Θ2R3,k(nj2-n32sin2 Θ2R3,k)1/2=0,
ρ-R3-2(k-1)H2n3sin Θρ-R3,k(n22-n32sin2 Θp-R3,k)1/2
-j=13Hjn3sin Θρ-R3,k(nj2-n32sin2 Θρ-R3,k)1/2=0,
kρ-R3,max=R3(n22-n32sin2 Θρ-R3,min)1/2n3H2sin Θρ-R3,min+1,
 H2>0,n2>n3sin Θρ-R3,min,
A(η3,k, α3, t)=l=1mAl(η3,k, α3)exp[i(ωlt-ϕl)],
I(η3,k, α3, t)=A(η3,k, α3, t)A*(η3,k, α3, t)=l=1mp=1mAl(η3,k, α3)exp[i(ωlt-ϕl)]×Ap(η3,k, α3)exp[-i(ωpt-ϕp)]=l=1mIl(η3,k, α3)+2l=1,p>lmIl,p(η3,k, α3)×cos[(ωl-ωp)t+(ϕl-ϕp)],
I(η3,k, α3, t)=I(η3,k, α3)=l=1mIl(η3,k, α3).
I(η3,k, α3)=Iin(Θ1, α1)Bk,
Bk=R2(k-1)T2exp-2(2k-1)H2n2aλ(n22-n32sin2 Θ3)1/2,
ifn2>n3sin Θ3.
R2=r122r232,
T2=n3cos Θ3n1cos Θ1 t122t232=n3(1-sin2 Θ3)1/2(n12-n32sin2 Θ3)1/2 t122t232,
ifn2>n3sin Θ3,
R=(RP+RN)/2,T=(TP+TN)/2,
I(η3,k, α3)=Iin(θ1)Bk,
Iin(Θ1)=I0cos Θ1,
Iin(Θ1)=I0(n12-n32sin2 Θ3)1/2/n1,
FP,S3=2πI0n1k=10ΘR3,kBk(n12-n32sin2 Θ3)1/2sin Θ3dΘ3,0=ρ<R32I0n1k=10ΘR3-p,kπBk(n12-n32sin2 Θ3)1/2sin Θ3dΘ3+k=1ΘR3-p,kΘR3+p,kBk(n12-n32sin2Θ3)1/2sin Θ3arccosρ2+η3,k2-R322ρη3,kdΘ3,0<ρ<R3.2I0n1k=10Θ2R3,kBk(n12-n32sin2 Θ3)1/2sinΘ3arccosη3,k2R3dΘ3,0<ρ=R32I0n1k=1kp-R3,maxΘρ-R3,kΘR3+ρ,kBk(n12-n32sin2Θ3)1/2sin Θ3arccosρ2+η3,k2-R322ρη3,kdΘ3,0<R3<ρ
I(η3,k, α3)=IinBk=I0Bk,
FP,S3=2πI0k=10ΘR3,kBksin Θ3dΘ3,0=ρ<R32I0k=10ΘR3-ρ,kπBksin Θ3dΘ3+k=1ΘR3-ρ,kΘR3+ρ,kBksin Θ3arccosρ2+η3,k2-R322ρη3,kdΘ3,0<ρ<R3.2I0k=10Θ2R3,kBksin Θ3arccosη3,k2R3dΘ3,0<ρ=R32I0k=1kρ-R3,maxΘρ-R3,kΘR3+ρ,kBksin Θ3arccosρ2+η3,k2-R322ρη3,kdΘ3,0<R3<ρ
ΘR3,1=arctan(R3/H),
ΘR3-ρ,1=arctan[(R3-ρ)/H],
ΘR3+ρ,1=arctan[(R3+ρ)/H],
Θ2R3,1=arctan(2R3/H),
Θρ-R3,1=arctan[(ρ-R3)/H],
FP,S3
=πI00ΘR3,1sin(2Θ3)dΘ3,0=ρ<R3I00ΘR3-ρ,1π sin(2θ3)dΘ3+ΘR3-ρ,1ΘR3+ρ,1sin(2Θ3)×arccosρ2+η3,12-R322ρη3,1dΘ3,0<ρ<R3I00Θ2R3,1sin(2Θ3)arccosη3,12R3dΘ3,0<ρ=R3I0Θρ-R3,1ΘR3+ρ,1sin(2Θ3)×arccosρ2+η3,12-R322ρη3,1dΘ3,0<R3<ρ.
FP,S3=πI0R32R32+H2,ρ=0,R3>0πI021+R32-ρ2-H2[(R3-ρ)2+H2]1/2[(R3+ρ)2+H2]1/2,
ρ>0,R3>0.
FP,S3=I0ΩP,S3,

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