Abstract

Light scattering from particles or structures on a surface is calculated by using a modification of the coupled-dipole method. The interaction matrix is modified for the presence of the surface by inclusion of the dipole reflection terms, calculated with the Sommerfeld integral formulation. The incident plane wave is replaced by the incident plus the surface reflected waves. In some cases the dipole reflections may be approximated by a Fresnel image dipole. Light scattering for structures up to a wavelength in size can be calculated in a reasonable amount of time.

© 1993 Optical Society of America

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References

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  1. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  2. G. L. Wojcik, D. K. Vaughn, L. K. Galbraith, “Calculation of light scatter from structures on silicon surfaces,” in Lasers in Microlithography, J. S. Batchelder, O. J. Ehrlich, J. Y. Tsao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.774, 21–31 (1987).
    [Crossref]
  3. P. A. Bobbert, J. Vleiger, “Light scattering by a sphere on a substrate,” Physica 137A, 209–242 (1986).
  4. F. I. Assi, “Electromagnetic wave scattering by a sphere on a layered substrate,” master’s thesis (Center for Micro-contamination Control, University of Arizona, Tucson, Ariz., 1990).
  5. G. Videen, “Light scattering from a sphere on or near a surface,”J. Opt. Soc. Am. 8, 483–489 (1991).
    [Crossref]
  6. B. R. Johnson, “Light scattering from a spherical particle on a conducting plane: I. Normal incidence,” J. Opt. Soc. Am. A 9, 1341–1351 (1992).
    [Crossref]
  7. D. L. Lager, R. J. Little, Numerical Evaluation of Sommerfeld Integrals, (Lawrence Livermore Laboratory, Livermore, Calif., 1974).
  8. D. L. Lager, R. J. Little, fortran Subroutines for the Numerical Evaluation of Sommerfeld Integrals unter Anterem, (Lawrence Livermore Laboratory, Livermore, Calif., 1975).
  9. G. J. Burke, A. J. Pogio, Numerical Electromagnetics Code (NEC)—Method of Moments, (Lawrence Livermore Laboratory, Livermore, Calif., 1981).
  10. S. B. Singham, C. R. Bohren, “Light scattering by an arbitrary particle: a physical reformulation of the coupled-dipole method,” Opt. Lett. 12, 10–12 (1987).
    [Crossref] [PubMed]
  11. S. B. Singham, C. F. Bohren, “Light scattering by an arbitrary particle: the scattering-order formulation of the coupled-dipole method,” J. Opt. Soc. Am. A 5, 1867–1872 (1988).
    [Crossref] [PubMed]
  12. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
    [Crossref]
  13. M. A. Taubenblatt, “Light scattering from cylindrical structures on surfaces,” Opt. Lett. 15, 255–257 (1989).
    [Crossref]
  14. M. A. Taubenblatt, T. K. Tran, “Calculation of light scattering from particles and structures on surfaces by using the coupled-dipole method,” in Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 32.
  15. J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross-section shape,”IEEE Trans. Antennas Propag. AP-13, 334–343 (1965).
    [Crossref]
  16. E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
    [Crossref]
  17. A. Sommerfeld, “Über die Ausbreitung der Wellen in der drahtlosen Telegraphie,” Ann. Physik 28, 665–737 (1909).
    [Crossref]
  18. H. Weyl, “Ausbreitung elektromagnetischer Wellen über einem ebenen Leiter,” Ann. Physik 60, 481–500 (1919).
    [Crossref]
  19. A. Banos, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, Oxford, 1966).
  20. A. Mohsen, “On the evaluation Sommerfeld integrals,”IEE Conf. Publ. 129 H, 177–182 (1982).
  21. K. A. Michalski, “On the efficient evaluation of integrals arising in the Sommerfeld halfspace problem,”IEE Conf. Publ. 132 H, 312–318 (1985).
  22. K. A. Michalski, C. M. Butler, “Evaluation of Sommerfeld integrals arising in the ground stake antenna problem,”IEE Conf. Publ. 134 H, 93–97 (1987).
  23. B. Drachman, M. Cloud, D. P. Nyquist, “Accurate evaluation of Sommerfeld integrals using the fast Fourier transform,”IEEE Trans. Antennas Propag. 37, 403–406 (1989).
    [Crossref]
  24. C. E. Dungey, C. F. Bohren, “Scattering by nonspherical particles: a refinement to the coupled-dipole method,” J. Opt. Soc. Am. A 8, 81–87 (1991).
    [Crossref]
  25. This relation may be derived from Ref. 24 by substitution of Eq. (1) into Eq. (3) and converting to Gaussian units by dropping the factor 4π.
  26. J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986).
  27. J. R. Wait, Electromagnetic Wave Theory (Harper ∊ Row, New York, 1985).
  28. K. A. Norton, “The propagation of radio waves over the surface of the earth and in the upper atmosphere,” Proc. Inst. Radio Eng. 25, 1203–1236 (1937).
  29. P. J. Flatau, G. L. Stephens, B. T. Draine, “Light scattering by rectangular solids in the discrete-dipole approximation: a new algorithm exploiting the Block–Toeplitz structure,” J. Opt. Soc. Am. A 7, 593–600 (1990).
    [Crossref]
  30. J. J. Goodman, B. T. Draine, P. J. Flatau, “Application of fast-Fourier-transform techniques to the discrete-dipole approximation,” Opt. Lett. 16, 1198–1200 (1991).
    [Crossref] [PubMed]

1992 (1)

1991 (3)

1990 (1)

1989 (2)

B. Drachman, M. Cloud, D. P. Nyquist, “Accurate evaluation of Sommerfeld integrals using the fast Fourier transform,”IEEE Trans. Antennas Propag. 37, 403–406 (1989).
[Crossref]

M. A. Taubenblatt, “Light scattering from cylindrical structures on surfaces,” Opt. Lett. 15, 255–257 (1989).
[Crossref]

1988 (2)

S. B. Singham, C. F. Bohren, “Light scattering by an arbitrary particle: the scattering-order formulation of the coupled-dipole method,” J. Opt. Soc. Am. A 5, 1867–1872 (1988).
[Crossref] [PubMed]

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[Crossref]

1987 (2)

S. B. Singham, C. R. Bohren, “Light scattering by an arbitrary particle: a physical reformulation of the coupled-dipole method,” Opt. Lett. 12, 10–12 (1987).
[Crossref] [PubMed]

K. A. Michalski, C. M. Butler, “Evaluation of Sommerfeld integrals arising in the ground stake antenna problem,”IEE Conf. Publ. 134 H, 93–97 (1987).

1986 (1)

P. A. Bobbert, J. Vleiger, “Light scattering by a sphere on a substrate,” Physica 137A, 209–242 (1986).

1985 (1)

K. A. Michalski, “On the efficient evaluation of integrals arising in the Sommerfeld halfspace problem,”IEE Conf. Publ. 132 H, 312–318 (1985).

1982 (1)

A. Mohsen, “On the evaluation Sommerfeld integrals,”IEE Conf. Publ. 129 H, 177–182 (1982).

1973 (1)

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

1965 (1)

J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross-section shape,”IEEE Trans. Antennas Propag. AP-13, 334–343 (1965).
[Crossref]

1937 (1)

K. A. Norton, “The propagation of radio waves over the surface of the earth and in the upper atmosphere,” Proc. Inst. Radio Eng. 25, 1203–1236 (1937).

1919 (1)

H. Weyl, “Ausbreitung elektromagnetischer Wellen über einem ebenen Leiter,” Ann. Physik 60, 481–500 (1919).
[Crossref]

1909 (1)

A. Sommerfeld, “Über die Ausbreitung der Wellen in der drahtlosen Telegraphie,” Ann. Physik 28, 665–737 (1909).
[Crossref]

Assi, F. I.

F. I. Assi, “Electromagnetic wave scattering by a sphere on a layered substrate,” master’s thesis (Center for Micro-contamination Control, University of Arizona, Tucson, Ariz., 1990).

Banos, A.

A. Banos, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, Oxford, 1966).

Bobbert, P. A.

P. A. Bobbert, J. Vleiger, “Light scattering by a sphere on a substrate,” Physica 137A, 209–242 (1986).

Bohren, C. F.

Bohren, C. R.

Burke, G. J.

G. J. Burke, A. J. Pogio, Numerical Electromagnetics Code (NEC)—Method of Moments, (Lawrence Livermore Laboratory, Livermore, Calif., 1981).

Butler, C. M.

K. A. Michalski, C. M. Butler, “Evaluation of Sommerfeld integrals arising in the ground stake antenna problem,”IEE Conf. Publ. 134 H, 93–97 (1987).

Cloud, M.

B. Drachman, M. Cloud, D. P. Nyquist, “Accurate evaluation of Sommerfeld integrals using the fast Fourier transform,”IEEE Trans. Antennas Propag. 37, 403–406 (1989).
[Crossref]

Drachman, B.

B. Drachman, M. Cloud, D. P. Nyquist, “Accurate evaluation of Sommerfeld integrals using the fast Fourier transform,”IEEE Trans. Antennas Propag. 37, 403–406 (1989).
[Crossref]

Draine, B. T.

Dungey, C. E.

Flatau, P. J.

Galbraith, L. K.

G. L. Wojcik, D. K. Vaughn, L. K. Galbraith, “Calculation of light scatter from structures on silicon surfaces,” in Lasers in Microlithography, J. S. Batchelder, O. J. Ehrlich, J. Y. Tsao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.774, 21–31 (1987).
[Crossref]

Goodman, J. J.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Johnson, B. R.

Kong, J. A.

J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986).

Lager, D. L.

D. L. Lager, R. J. Little, Numerical Evaluation of Sommerfeld Integrals, (Lawrence Livermore Laboratory, Livermore, Calif., 1974).

D. L. Lager, R. J. Little, fortran Subroutines for the Numerical Evaluation of Sommerfeld Integrals unter Anterem, (Lawrence Livermore Laboratory, Livermore, Calif., 1975).

Little, R. J.

D. L. Lager, R. J. Little, fortran Subroutines for the Numerical Evaluation of Sommerfeld Integrals unter Anterem, (Lawrence Livermore Laboratory, Livermore, Calif., 1975).

D. L. Lager, R. J. Little, Numerical Evaluation of Sommerfeld Integrals, (Lawrence Livermore Laboratory, Livermore, Calif., 1974).

Michalski, K. A.

K. A. Michalski, C. M. Butler, “Evaluation of Sommerfeld integrals arising in the ground stake antenna problem,”IEE Conf. Publ. 134 H, 93–97 (1987).

K. A. Michalski, “On the efficient evaluation of integrals arising in the Sommerfeld halfspace problem,”IEE Conf. Publ. 132 H, 312–318 (1985).

Mohsen, A.

A. Mohsen, “On the evaluation Sommerfeld integrals,”IEE Conf. Publ. 129 H, 177–182 (1982).

Norton, K. A.

K. A. Norton, “The propagation of radio waves over the surface of the earth and in the upper atmosphere,” Proc. Inst. Radio Eng. 25, 1203–1236 (1937).

Nyquist, D. P.

B. Drachman, M. Cloud, D. P. Nyquist, “Accurate evaluation of Sommerfeld integrals using the fast Fourier transform,”IEEE Trans. Antennas Propag. 37, 403–406 (1989).
[Crossref]

Pennypacker, C. R.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

Pogio, A. J.

G. J. Burke, A. J. Pogio, Numerical Electromagnetics Code (NEC)—Method of Moments, (Lawrence Livermore Laboratory, Livermore, Calif., 1981).

Purcell, E. M.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

Richmond, J. H.

J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross-section shape,”IEEE Trans. Antennas Propag. AP-13, 334–343 (1965).
[Crossref]

Singham, S. B.

Sommerfeld, A.

A. Sommerfeld, “Über die Ausbreitung der Wellen in der drahtlosen Telegraphie,” Ann. Physik 28, 665–737 (1909).
[Crossref]

Stephens, G. L.

Taubenblatt, M. A.

M. A. Taubenblatt, “Light scattering from cylindrical structures on surfaces,” Opt. Lett. 15, 255–257 (1989).
[Crossref]

M. A. Taubenblatt, T. K. Tran, “Calculation of light scattering from particles and structures on surfaces by using the coupled-dipole method,” in Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 32.

Tran, T. K.

M. A. Taubenblatt, T. K. Tran, “Calculation of light scattering from particles and structures on surfaces by using the coupled-dipole method,” in Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 32.

Vaughn, D. K.

G. L. Wojcik, D. K. Vaughn, L. K. Galbraith, “Calculation of light scatter from structures on silicon surfaces,” in Lasers in Microlithography, J. S. Batchelder, O. J. Ehrlich, J. Y. Tsao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.774, 21–31 (1987).
[Crossref]

Videen, G.

G. Videen, “Light scattering from a sphere on or near a surface,”J. Opt. Soc. Am. 8, 483–489 (1991).
[Crossref]

Vleiger, J.

P. A. Bobbert, J. Vleiger, “Light scattering by a sphere on a substrate,” Physica 137A, 209–242 (1986).

Wait, J. R.

J. R. Wait, Electromagnetic Wave Theory (Harper ∊ Row, New York, 1985).

Weyl, H.

H. Weyl, “Ausbreitung elektromagnetischer Wellen über einem ebenen Leiter,” Ann. Physik 60, 481–500 (1919).
[Crossref]

Wojcik, G. L.

G. L. Wojcik, D. K. Vaughn, L. K. Galbraith, “Calculation of light scatter from structures on silicon surfaces,” in Lasers in Microlithography, J. S. Batchelder, O. J. Ehrlich, J. Y. Tsao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.774, 21–31 (1987).
[Crossref]

Ann. Physik (2)

A. Sommerfeld, “Über die Ausbreitung der Wellen in der drahtlosen Telegraphie,” Ann. Physik 28, 665–737 (1909).
[Crossref]

H. Weyl, “Ausbreitung elektromagnetischer Wellen über einem ebenen Leiter,” Ann. Physik 60, 481–500 (1919).
[Crossref]

Astrophys. J. (2)

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[Crossref]

IEE Conf. Publ. (3)

A. Mohsen, “On the evaluation Sommerfeld integrals,”IEE Conf. Publ. 129 H, 177–182 (1982).

K. A. Michalski, “On the efficient evaluation of integrals arising in the Sommerfeld halfspace problem,”IEE Conf. Publ. 132 H, 312–318 (1985).

K. A. Michalski, C. M. Butler, “Evaluation of Sommerfeld integrals arising in the ground stake antenna problem,”IEE Conf. Publ. 134 H, 93–97 (1987).

IEEE Trans. Antennas Propag. (2)

B. Drachman, M. Cloud, D. P. Nyquist, “Accurate evaluation of Sommerfeld integrals using the fast Fourier transform,”IEEE Trans. Antennas Propag. 37, 403–406 (1989).
[Crossref]

J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross-section shape,”IEEE Trans. Antennas Propag. AP-13, 334–343 (1965).
[Crossref]

J. Opt. Soc. Am. (1)

G. Videen, “Light scattering from a sphere on or near a surface,”J. Opt. Soc. Am. 8, 483–489 (1991).
[Crossref]

J. Opt. Soc. Am. A (4)

Opt. Lett. (3)

Physica (1)

P. A. Bobbert, J. Vleiger, “Light scattering by a sphere on a substrate,” Physica 137A, 209–242 (1986).

Proc. Inst. Radio Eng. (1)

K. A. Norton, “The propagation of radio waves over the surface of the earth and in the upper atmosphere,” Proc. Inst. Radio Eng. 25, 1203–1236 (1937).

Other (11)

This relation may be derived from Ref. 24 by substitution of Eq. (1) into Eq. (3) and converting to Gaussian units by dropping the factor 4π.

J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986).

J. R. Wait, Electromagnetic Wave Theory (Harper ∊ Row, New York, 1985).

F. I. Assi, “Electromagnetic wave scattering by a sphere on a layered substrate,” master’s thesis (Center for Micro-contamination Control, University of Arizona, Tucson, Ariz., 1990).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

G. L. Wojcik, D. K. Vaughn, L. K. Galbraith, “Calculation of light scatter from structures on silicon surfaces,” in Lasers in Microlithography, J. S. Batchelder, O. J. Ehrlich, J. Y. Tsao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.774, 21–31 (1987).
[Crossref]

D. L. Lager, R. J. Little, Numerical Evaluation of Sommerfeld Integrals, (Lawrence Livermore Laboratory, Livermore, Calif., 1974).

D. L. Lager, R. J. Little, fortran Subroutines for the Numerical Evaluation of Sommerfeld Integrals unter Anterem, (Lawrence Livermore Laboratory, Livermore, Calif., 1975).

G. J. Burke, A. J. Pogio, Numerical Electromagnetics Code (NEC)—Method of Moments, (Lawrence Livermore Laboratory, Livermore, Calif., 1981).

M. A. Taubenblatt, T. K. Tran, “Calculation of light scattering from particles and structures on surfaces by using the coupled-dipole method,” in Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 32.

A. Banos, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, Oxford, 1966).

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Figures (9)

Fig. 1
Fig. 1

(a) Ratio of the observed field of a vertical dipole (Sommerfeld calculation) to a pure image dipole compared with the Fresnel reflection coefficients, (b) ratio of the observed field of a horizontal dipole (Sommerfeld calculation) to a pure image dipole. The dipole is situated at the surface, and the observation point is at a fixed radius of 0.05 μm, shown as a function of the angle from the surface normal; the wavelength is 632.8 nm.

Fig. 2
Fig. 2

(a) Ratio of the observed field of a vertical dipole (Sommerfeld calculation) to a pure image dipole compared with the Fresnel reflection coefficients, (b) ratio of the observed field of a horizontal dipole (Sommerfeld calculation) to a pure image dipole. The dipole is situated at the surface, and the observation point is at a fixed radius of 0.5 μm, shown as a function of the angle from the surface normal; the wavelength is 632.8 nm.

Fig. 3
Fig. 3

Scattering geometry used in the calculations. The surface is at the z = 0 plane, while the light is incident from the ϕ = 180° direction along the y = 0 plane at angle γ from the surface normal. The scattered light is observed at the scattering plane defined by azimuthal angle ϕ and at polar angle θ from the surface normal.

Fig. 4
Fig. 4

Comparison of coupled-dipole calculation (present study) to a finite-element solution of Maxwell’s equations (Wojcik) for a 0.54-μm polystyrene latex sphere on Si. Light scattering for s-polarized 632.8-nm radiation incident at γ = 0 is shown as a function of the scattering angle measured from the surface normal. The scattering plane is the x axis, and s-polarized scattered radiation is shown.

Fig. 5
Fig. 5

Comparison of exact (Sommerfeld), Fresnel-approximation, and no-surface-interaction terms in the computation of light scattering from a 0.3-μm polystyrene sphere on Si. The angle of incidence is −65°, and the scattering plane is the x axis; the wavelength is 632.8 nm.

Fig. 6
Fig. 6

Comparison of the exact (Sommerfeld) terms and the Fresnel approximation for a 0.45-μm-diameter Si3N4 particle on Si. The angle of incidence is −65°, and the scattering plane is the x axis; the wavelength is 632.8 nm.

Fig. 7
Fig. 7

Comparison of the exact (Sommerfeld) terms and the Fresnel approximation for a square tungsten plate of dimension 0.16 × 0.16 × 0.08 μm3 on Si. the angle of incidence is 0°, and the scattering plane is the x axis; the wavelength is 632.8 nm.

Fig. 8
Fig. 8

(a) Light scattered from an SiO2 plate of dimensions 0.54 × 0.54 × 0.09 μm3 on Si as a function of azimuthal angle (ϕ) into a cone defined by θ = 30°, (b) light scattered into a cone defined by θ = 60°; the wavelength is 632.8 nm.

Fig. 9
Fig. 9

Light scattered from a cubical stud of SiO2 on Si, with 0.3-μm sides parallel to the coordinate axes. Comparison of light scattering from this stud with no material missing and with one of three octants missing: top left (−x, + y, + z); top right (+x, + y, + z); or bottom right (+ x, + y, −z). The angle of incidence is 65° and s polarized. The scattering plane is the x axis; the wavelength is 632.8 nm.

Equations (19)

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AP = E 0 ,
P = α E .
α = a 3 f ( m 2 - 1 m 2 + 2 ) ,
A i i = 1 / α .
A i j · P j = - E i j ,
E i j = k 2 ( n i j × P j ) × n i j exp ( i k r ) r + [ 3 n i j ( n i j · P j ) - P j ] ( 1 r 3 - i k r 2 ) exp ( i k r ) ,
C abs = 4 π k E 0 2 j Im ( α ) P j 2 α 2 ,
C ext = 4 π k E 0 2 j Im ( E 0 * ) j P j .
C sca = C ext - C abs .
A i i = 1 / α - R i i ,
A i j · P j = - ( E i j + R i j ) ,
E ρ V = C 2 ρ z ( - G 21 + k 1 2 V 22 ) ,
E z V = C ( 2 z 2 + k 2 2 ) ( - G 21 + k 1 2 V 22 ) ,
E ρ H = C cos ϕ [ 2 ρ 2 ( - G 21 + k 2 2 V 22 ) + k 2 2 ( - G 21 + U 22 ) ] ,
E ϕ H = - C sin ϕ × [ 1 ρ ρ ( - G 21 + k 2 2 V 22 ) + k 2 2 ( - G 21 + U 22 ) ] ,
E z H = C cos ϕ 2 z ρ ( G 21 - k 1 2 V 22 ) ,
G 21 = exp ( i k 2 R 1 ) / R 1 ,
U 22 = 2 0 exp [ - γ 2 ( z + h ) ] J 0 ( λ ρ ) λ d λ γ 1 + γ 2 ,
V 22 = 2 0 exp [ - γ 2 ( z + h ) ] J 0 ( λ ρ ) λ d λ k 1 2 γ 1 + k 2 2 γ 2 ,

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