Abstract

In Generalized Lorenz-Mie theories, (GLMTs), the most difficult task concerns the description of the illuminating beam. We provide an approach for expansions of the incident arbitrary shaped beam in spherical and spheroidal coordinates in the general case of oblique illumination. The representations for shaped beam coefficients are derived by using addition theorem for spherical vector wave functions under coordinate rotations. For

© 2007 Optical Society of America

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References

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  1. G. Gouesbet, B. Maheu, and G. Gréhan, "Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation," J. Opt. Soc. Am. A 5, 1427-1443 (1988).
    [CrossRef]
  2. G. Gouesbet, G. Gréhan, and B. Maheu, "Computations of the gn coefficients in the generalized Lorenz-Mie theory using three different methods," Appl. Opt. 27, 4874-4883 (1988).
    [CrossRef] [PubMed]
  3. G. Gouesbet, G. Gréhan, and B. Maheu, "Localized interpretation to compute all the coefficients gnm in the generalized Lorenz-Mie theory," J. Opt. Soc. Am. A 7, 998-1003 (1990).
    [CrossRef]
  4. K. F. Ren, G. Gréhan, and G. Gouesbet, "Scattering of a Gaussian beam by an infinite cylinder in the framework of generalized Lorenz-Mie theory formulation and numerical results," J. Opt. Soc. Am. A 14, 3014-3025 (1997).
    [CrossRef]
  5. Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Gréhan, "Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres," Appl. Opt. 36, 5188-5198 (1997).
    [CrossRef] [PubMed]
  6. A. Doicu and T. Wriedt, "Computation of the beam-shape coefficients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave functions," Appl. Opt. 36, 2971-2978 (1997).
    [CrossRef] [PubMed]
  7. J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys. 64, 1632-1639 (1988).
    [CrossRef]
  8. J. P. Barton, "Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination," Appl. Opt. 34, 5542-5551 (1995).
    [CrossRef] [PubMed]
  9. E. E. M. Khaled, S. C. Hill, and P. W. Barber, "Scattered and internal intensity of a sphere illuminated with a Gaussian beam," IEEE Trans. Antennas Propagat. 41, 259-303 (1993).
    [CrossRef]
  10. F. M. Schulz, K. Stamnes, and J. J. Stamnes, "Scattering of electromagnetic wave by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates," Appl. Opt. 37, 7875-7896 (1998).
    [CrossRef]
  11. H.-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea, "The imaging properties of scattering particles in laser beams," Meas. Sci. Technol. 10,564-574 (1999).
    [CrossRef]
  12. T. Evers, H. Dahl, and T. Wriedt, "Extension of the program 3D MMP with a fifth order Gaussian beam," Electron. Lett. 32, 1356-1357 (1996).
    [CrossRef]
  13. A. Doicu and T. Wriedt, "Formulations of the extended boundary condition method for incident Gaussian beams using multiple-multipole expansions," J. Modern. Opt. 44, 785-801 (1997).
    [CrossRef]
  14. Y. Han and Z. Wu, "Scattering of a spheroidal particle illuminated by a Gaussian beam," Appl. Opt. 40, 2501-2509 (2001).
    [CrossRef]
  15. Y. Han and Z. Wu, "The expansion coefficients of a Spheroidal Particle illuminated by Gaussian Beam," IEEE Trans. Antennas Propagat. 49, 615-620 (2001).
    [CrossRef]
  16. H. Y. Zhang and Y. P. Han, "Scattering by a confocal multilayered spheroidal particle illuminated by an axial Gaussian beam," IEEE Trans. Antennas Propagat. 53, 1514-1518 (2005).
    [CrossRef]
  17. Y. Han, H. Zhang, and X. Sun, "Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries," Appl. Phys. B. 84, 485-492 (2006).
    [CrossRef]
  18. B. Friedman and J. Russek, "Addition theorems for spherical waves," Q. Appl. Math. 12, 13-23 (1954).
  19. S. Stein, "Addition theorems for spherical wave functions," Q. Appl. Math. 19, 15-24 (1961).
  20. O. R. Cruzan, "Translational addition theorems for spherical vector wave functions," Q. Appl. Math. 20, 33-40 (1962).
  21. S. Asano and G. Yamamoto, "Light scattering by a spheroid particle," Appl. Opt. 14, 29-49 (1975).
    [PubMed]
  22. J. Dalmas and R. Deleuil, "Translational addition theorems for prolate spheroidal vector wave functions Mr and Nr ," Q. Appl. Math. 38,143-158 (1980).
  23. A. R. Edmonds, Angular momentum in quantum mechanics, (Princeton University Press, Princeton, N. J, 1957), Chap.4.
  24. B. P. Sinha and R. H. Macphie, "Translational addition theorems for spheroidal scalar and vector wave functions," Q. Appl. Math. 44,213-222 (1986).
  25. L. W. Davis, "Theory of electromagnetic beam," Phys. Rev. A 19, 1177-1179 (1979).
    [CrossRef]
  26. C. Flammer, Spheroidal wave functions, (Stanford University Press, Stanford, California, 1957).
  27. J. A. Stratton, Electromagnetic Theory, (New York: McGraw-Hill, 1941).

2006 (1)

Y. Han, H. Zhang, and X. Sun, "Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries," Appl. Phys. B. 84, 485-492 (2006).
[CrossRef]

2005 (1)

H. Y. Zhang and Y. P. Han, "Scattering by a confocal multilayered spheroidal particle illuminated by an axial Gaussian beam," IEEE Trans. Antennas Propagat. 53, 1514-1518 (2005).
[CrossRef]

2001 (2)

Y. Han and Z. Wu, "The expansion coefficients of a Spheroidal Particle illuminated by Gaussian Beam," IEEE Trans. Antennas Propagat. 49, 615-620 (2001).
[CrossRef]

Y. Han and Z. Wu, "Scattering of a spheroidal particle illuminated by a Gaussian beam," Appl. Opt. 40, 2501-2509 (2001).
[CrossRef]

1999 (1)

H.-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea, "The imaging properties of scattering particles in laser beams," Meas. Sci. Technol. 10,564-574 (1999).
[CrossRef]

1998 (1)

1997 (4)

1996 (1)

T. Evers, H. Dahl, and T. Wriedt, "Extension of the program 3D MMP with a fifth order Gaussian beam," Electron. Lett. 32, 1356-1357 (1996).
[CrossRef]

1995 (1)

1993 (1)

E. E. M. Khaled, S. C. Hill, and P. W. Barber, "Scattered and internal intensity of a sphere illuminated with a Gaussian beam," IEEE Trans. Antennas Propagat. 41, 259-303 (1993).
[CrossRef]

1990 (1)

1988 (3)

1986 (1)

B. P. Sinha and R. H. Macphie, "Translational addition theorems for spheroidal scalar and vector wave functions," Q. Appl. Math. 44,213-222 (1986).

1980 (1)

J. Dalmas and R. Deleuil, "Translational addition theorems for prolate spheroidal vector wave functions Mr and Nr ," Q. Appl. Math. 38,143-158 (1980).

1979 (1)

L. W. Davis, "Theory of electromagnetic beam," Phys. Rev. A 19, 1177-1179 (1979).
[CrossRef]

1975 (1)

1962 (1)

O. R. Cruzan, "Translational addition theorems for spherical vector wave functions," Q. Appl. Math. 20, 33-40 (1962).

1961 (1)

S. Stein, "Addition theorems for spherical wave functions," Q. Appl. Math. 19, 15-24 (1961).

1954 (1)

B. Friedman and J. Russek, "Addition theorems for spherical waves," Q. Appl. Math. 12, 13-23 (1954).

Albrecht, H.-E.

H.-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea, "The imaging properties of scattering particles in laser beams," Meas. Sci. Technol. 10,564-574 (1999).
[CrossRef]

Alexander, D. R.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys. 64, 1632-1639 (1988).
[CrossRef]

Asano, S.

Barber, P. W.

E. E. M. Khaled, S. C. Hill, and P. W. Barber, "Scattered and internal intensity of a sphere illuminated with a Gaussian beam," IEEE Trans. Antennas Propagat. 41, 259-303 (1993).
[CrossRef]

Barton, J. P.

J. P. Barton, "Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination," Appl. Opt. 34, 5542-5551 (1995).
[CrossRef] [PubMed]

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys. 64, 1632-1639 (1988).
[CrossRef]

Borys, M.

H.-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea, "The imaging properties of scattering particles in laser beams," Meas. Sci. Technol. 10,564-574 (1999).
[CrossRef]

Cruzan, O. R.

O. R. Cruzan, "Translational addition theorems for spherical vector wave functions," Q. Appl. Math. 20, 33-40 (1962).

Dahl, H.

T. Evers, H. Dahl, and T. Wriedt, "Extension of the program 3D MMP with a fifth order Gaussian beam," Electron. Lett. 32, 1356-1357 (1996).
[CrossRef]

Dalmas, J.

J. Dalmas and R. Deleuil, "Translational addition theorems for prolate spheroidal vector wave functions Mr and Nr ," Q. Appl. Math. 38,143-158 (1980).

Damaschke, N.

H.-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea, "The imaging properties of scattering particles in laser beams," Meas. Sci. Technol. 10,564-574 (1999).
[CrossRef]

Davis, L. W.

L. W. Davis, "Theory of electromagnetic beam," Phys. Rev. A 19, 1177-1179 (1979).
[CrossRef]

Deleuil, R.

J. Dalmas and R. Deleuil, "Translational addition theorems for prolate spheroidal vector wave functions Mr and Nr ," Q. Appl. Math. 38,143-158 (1980).

Doicu, A.

A. Doicu and T. Wriedt, "Formulations of the extended boundary condition method for incident Gaussian beams using multiple-multipole expansions," J. Modern. Opt. 44, 785-801 (1997).
[CrossRef]

A. Doicu and T. Wriedt, "Computation of the beam-shape coefficients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave functions," Appl. Opt. 36, 2971-2978 (1997).
[CrossRef] [PubMed]

Evers, T.

T. Evers, H. Dahl, and T. Wriedt, "Extension of the program 3D MMP with a fifth order Gaussian beam," Electron. Lett. 32, 1356-1357 (1996).
[CrossRef]

Friedman, B.

B. Friedman and J. Russek, "Addition theorems for spherical waves," Q. Appl. Math. 12, 13-23 (1954).

Gouesbet, G.

Gréhan, G.

Guo, L. X.

Han, Y.

Y. Han, H. Zhang, and X. Sun, "Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries," Appl. Phys. B. 84, 485-492 (2006).
[CrossRef]

Y. Han and Z. Wu, "Scattering of a spheroidal particle illuminated by a Gaussian beam," Appl. Opt. 40, 2501-2509 (2001).
[CrossRef]

Y. Han and Z. Wu, "The expansion coefficients of a Spheroidal Particle illuminated by Gaussian Beam," IEEE Trans. Antennas Propagat. 49, 615-620 (2001).
[CrossRef]

Han, Y. P.

H. Y. Zhang and Y. P. Han, "Scattering by a confocal multilayered spheroidal particle illuminated by an axial Gaussian beam," IEEE Trans. Antennas Propagat. 53, 1514-1518 (2005).
[CrossRef]

Hill, S. C.

E. E. M. Khaled, S. C. Hill, and P. W. Barber, "Scattered and internal intensity of a sphere illuminated with a Gaussian beam," IEEE Trans. Antennas Propagat. 41, 259-303 (1993).
[CrossRef]

Khaled, E. E. M.

E. E. M. Khaled, S. C. Hill, and P. W. Barber, "Scattered and internal intensity of a sphere illuminated with a Gaussian beam," IEEE Trans. Antennas Propagat. 41, 259-303 (1993).
[CrossRef]

Macphie, R. H.

B. P. Sinha and R. H. Macphie, "Translational addition theorems for spheroidal scalar and vector wave functions," Q. Appl. Math. 44,213-222 (1986).

Maheu, B.

Ren, K. F.

Russek, J.

B. Friedman and J. Russek, "Addition theorems for spherical waves," Q. Appl. Math. 12, 13-23 (1954).

Schaub, S. A.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys. 64, 1632-1639 (1988).
[CrossRef]

Schulz, F. M.

Sinha, B. P.

B. P. Sinha and R. H. Macphie, "Translational addition theorems for spheroidal scalar and vector wave functions," Q. Appl. Math. 44,213-222 (1986).

Stamnes, J. J.

Stamnes, K.

Stein, S.

S. Stein, "Addition theorems for spherical wave functions," Q. Appl. Math. 19, 15-24 (1961).

Sun, X.

Y. Han, H. Zhang, and X. Sun, "Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries," Appl. Phys. B. 84, 485-492 (2006).
[CrossRef]

Tropea, C.

H.-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea, "The imaging properties of scattering particles in laser beams," Meas. Sci. Technol. 10,564-574 (1999).
[CrossRef]

Wriedt, T.

A. Doicu and T. Wriedt, "Computation of the beam-shape coefficients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave functions," Appl. Opt. 36, 2971-2978 (1997).
[CrossRef] [PubMed]

A. Doicu and T. Wriedt, "Formulations of the extended boundary condition method for incident Gaussian beams using multiple-multipole expansions," J. Modern. Opt. 44, 785-801 (1997).
[CrossRef]

T. Evers, H. Dahl, and T. Wriedt, "Extension of the program 3D MMP with a fifth order Gaussian beam," Electron. Lett. 32, 1356-1357 (1996).
[CrossRef]

Wu, Z.

Y. Han and Z. Wu, "The expansion coefficients of a Spheroidal Particle illuminated by Gaussian Beam," IEEE Trans. Antennas Propagat. 49, 615-620 (2001).
[CrossRef]

Y. Han and Z. Wu, "Scattering of a spheroidal particle illuminated by a Gaussian beam," Appl. Opt. 40, 2501-2509 (2001).
[CrossRef]

Wu, Z. S.

Yamamoto, G.

Zhang, H.

Y. Han, H. Zhang, and X. Sun, "Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries," Appl. Phys. B. 84, 485-492 (2006).
[CrossRef]

Zhang, H. Y.

H. Y. Zhang and Y. P. Han, "Scattering by a confocal multilayered spheroidal particle illuminated by an axial Gaussian beam," IEEE Trans. Antennas Propagat. 53, 1514-1518 (2005).
[CrossRef]

Appl. Opt. (7)

Appl. Phys. B. (1)

Y. Han, H. Zhang, and X. Sun, "Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries," Appl. Phys. B. 84, 485-492 (2006).
[CrossRef]

Electron. Lett. (1)

T. Evers, H. Dahl, and T. Wriedt, "Extension of the program 3D MMP with a fifth order Gaussian beam," Electron. Lett. 32, 1356-1357 (1996).
[CrossRef]

IEEE Trans. Antennas Propagat. (3)

Y. Han and Z. Wu, "The expansion coefficients of a Spheroidal Particle illuminated by Gaussian Beam," IEEE Trans. Antennas Propagat. 49, 615-620 (2001).
[CrossRef]

H. Y. Zhang and Y. P. Han, "Scattering by a confocal multilayered spheroidal particle illuminated by an axial Gaussian beam," IEEE Trans. Antennas Propagat. 53, 1514-1518 (2005).
[CrossRef]

E. E. M. Khaled, S. C. Hill, and P. W. Barber, "Scattered and internal intensity of a sphere illuminated with a Gaussian beam," IEEE Trans. Antennas Propagat. 41, 259-303 (1993).
[CrossRef]

J. Appl. Phys. (1)

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys. 64, 1632-1639 (1988).
[CrossRef]

J. Modern. Opt. (1)

A. Doicu and T. Wriedt, "Formulations of the extended boundary condition method for incident Gaussian beams using multiple-multipole expansions," J. Modern. Opt. 44, 785-801 (1997).
[CrossRef]

J. Opt. Soc. Am. A (3)

Meas. Sci. Technol. (1)

H.-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea, "The imaging properties of scattering particles in laser beams," Meas. Sci. Technol. 10,564-574 (1999).
[CrossRef]

Phys. Rev. A (1)

L. W. Davis, "Theory of electromagnetic beam," Phys. Rev. A 19, 1177-1179 (1979).
[CrossRef]

Q. Appl. Math. (5)

B. Friedman and J. Russek, "Addition theorems for spherical waves," Q. Appl. Math. 12, 13-23 (1954).

S. Stein, "Addition theorems for spherical wave functions," Q. Appl. Math. 19, 15-24 (1961).

O. R. Cruzan, "Translational addition theorems for spherical vector wave functions," Q. Appl. Math. 20, 33-40 (1962).

J. Dalmas and R. Deleuil, "Translational addition theorems for prolate spheroidal vector wave functions Mr and Nr ," Q. Appl. Math. 38,143-158 (1980).

B. P. Sinha and R. H. Macphie, "Translational addition theorems for spheroidal scalar and vector wave functions," Q. Appl. Math. 44,213-222 (1986).

Other (3)

A. R. Edmonds, Angular momentum in quantum mechanics, (Princeton University Press, Princeton, N. J, 1957), Chap.4.

C. Flammer, Spheroidal wave functions, (Stanford University Press, Stanford, California, 1957).

J. A. Stratton, Electromagnetic Theory, (New York: McGraw-Hill, 1941).

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Figures (4)

Fig. 1.
Fig. 1.

The center of an arbitrarily oriented scatterer is located at origin O of the Cartesian coordinate systems Oxyz and Ox´´y´´z´´. (For spheroid, the major axe along the z axis of Oxyz ). The xyz axes are obtained by a rigid-body rotation of the x´´y´´z´´ axes through Euler angles α, β, γ. The scatterer is illuminated by a shaped beam propagating along the z´ axis with the middle of its waist located at origin O´. Theoz´´ is parallel to ´oz´ , and with similar conditions for the other axes. The Cartesian coordinates of O´ in the system Ox´´y´´z´´ are (x 0,y 0,z 0).

Fig. 2.
Fig. 2.

Geometry of the scattering description for a scatterer in the Cartesian coordinate system Oxyz . The rotation axis of the spheroid is the z axis and its orientation in space is specified by the Euler angles α,β,gamma; of the xyz axes with respect to the x´´y´´z´´ axes.

Fig. 3.
Fig. 3.

The convergence of the beam shape coefficients Gm n,TM

Fig. 4.
Fig. 4.

The convergence of the beam shape coefficients Gm n,TE

Equations (54)

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

E i = E 0 m = 0 n = m [ g n , T E m ¯ m emn r ( 1 ) ( kr , θ ´ ´ , ϕ ´ ´ ) + g ' n , TE m ¯ m omn r ( 1 ) ( kr , θ ´ ´ , ϕ´´ ) + i g ' n , TM m ¯ n emn r ( 1 ) ( kr , θ ´ ´ , ϕ ´ ´ ) + i g n , TM m ¯ n omn r ( 1 ) ( k r , θ ´ ´ , ϕ ´ ´ ) ]
P n m ( cos θ ´ ´ ) e imθ ´ ´ = s = n n ρ ( m , s , n ) P n s ( cos θ ) e i s ϕ
ρ ( m , s , n ) = ( 1 ) s + m e i s γ [ ( n + m ) ! ( n s ) ! ( n m ) ! ( n + s ) ! ] 1 2 u sm ( n ) ( β ) e i m α
u sm ( n ) ( β ) = [ ( n + s ) ! ( n s ) ! ( n + m ) ! ( n m ) ! ] 1 2 σ n + m n s σ n m σ ( 1 ) n s σ ( cos β 2 ) 2 σ + s + m ( sin β 2 ) 2 n 2 σ s m
P n m ( cos θ ' ' ) cos ( m ϕ ' ' ) sin ( m ϕ ' ' ) = s = n n [ ρ 1 ( m , s , n ) P n s ( cos θ ) cos ( s ϕ ) sin ( s ϕ )
+ 1 1 ρ 2 ( m , s , n ) P n s ( cos θ ) sin ( s ϕ ) cos ( s ϕ ) ]
w mn o e r ( 1 ) ( kr , θ ' ' , ϕ ' ' ) = s = n n ρ 1 ( m , s , n ) w sn o e r ( 1 ) ( kr , θ , ϕ ) s = n n ρ 2 ( m , s , n ) w sn e o r ( 1 ) ( k r , θ , ϕ )
P n m ( cos θ ) = ( 1 ) m ( n m ) ! ( n + m ) ! P n m ( cos θ ) m > 1
w -mn o e r ( i ) ( kr , θ , ϕ ) = ± ( 1 ) m ( n m ) ! ( n + m ) ! w mn o e r ( i ) ( kr , θ , ϕ ) m > 0
E i = E 0 m = 0 n = m s = 0 n [ g n , TE ms m esn r ( 1 ) ( kr , θ , ϕ ) g ' n , TE ms m osn r ( 1 ) ( kr , θ , ϕ )
+ ig ' n , TM ms n esn r ( 1 ) ( kr , θ , ϕ ) + ig n , TM ms n osn r ( 1 ) ( kr , θ , ϕ ) ]
g n , TE ms g ' n , TE ms g n , TM ms g ' n , TM ms = g n , T E m ¯ g n , T E m ¯ g n , T M m ¯ g n , T M m ¯ [ ρ 1 ( m , s , n ) ρ 2 ( m , s , n ) ρ 1 ( m , s , n ) ρ 2 ( m , s , n ) + 1 1 1 1 ( 1 δ 0 s ) ( 1 ) s ( n s ) ! ( n + s ) ! ρ 1 ( m , s , n ) ρ 2 ( m , s , n ) ρ 1 ( m , s , n ) ρ 2 ( m , s , n ) ]
+ 1 1 1 1 g' n , T E m ¯ g' n , T E m ¯ g' n , T M m ¯ g' n , T M m ¯ [ ρ 2 ( m , s , n ) ρ 1 ( m , s , n ) ρ 2 ( m , s , n ) ρ 1 ( m , s , n ) + 1 1 1 1 ( 1 δ 0 s ) ( 1 ) s ( n s ) ! ( n + s ) ! ρ 2 ( m , s , n ) ρ 1 ( m , s , n ) ρ 2 ( m , s , n ) ρ 1 ( m , s , n ) ]
c i c , ζ i ζ
P n m ( cos θ ) j n ( kr ) = 2 ( n + m ) ! ( 2 n + 1 ) ( n m ) ! ' l = m , m + 1 i l n N ml d n m ml ( c ) S ml ( c , η ) R ml ( 1 ) ( c , ζ )
w mn o e r ( 1 ) ( kr , θ , ϕ ) = l = m = m + l ' 2 ( n + m ) ! ( 2 n + 1 ) ( n m ) ! i l n N ml d n m ml ( c ) w ml o e r ( 1 ) ( c , ζ , η , ϕ )
E i = E 0 s = 0 l = s n = s , s + 1 m = 0 n 2 ( n + s ) ! ( 2 n + 1 ) ( n s ) ! i l n N sl d n s sl ( c ) [ g n , TE ms M esl r ( 1 ) ( c , ζ , η , ϕ )
g ' n , TE ms M osl r ( 1 ) ( c , ζ , η , ϕ ) + ig ' n , TM ms N esl r ( 1 ) ( c , ζ , η , ϕ ) + ig n , TM ms N osl r ( 1 ) ( c , ζ , η , ϕ ) ]
E i = E 0 m = 0 n = m [ G n , TE m M emn r ( 1 ) ( c , ζ , η , ϕ ) G ' n , TE m M o mn r ( 1 ) ( c , ζ , η , ϕ )
+ iG n , TM m N omn r ( 1 ) ( c , ζ , η , ϕ ) + iG ' n , TM m N emn r ( 1 ) c , ζ , η , ϕ ) ]
[ G n , TE m G ' n , TE m G n , TM m G ' n , TM m ] = ' r = 0,1 s = 0 r + m 2 ( r + 2 m ) ! ( 2 r + 2 m + 1 ) r ! i r m N mn d r mn ( c ) [ g r + m sm , TE g ' r + m sm , TE g r + m sm , TM g ' r + m sm , TM ]
E w = E 0 m = 0 n = m i n [ δ mn M emn r ( 1 ) ( c , ζ , η , ϕ ) + mn N omn r ( 1 ) ( c , ζ , η , ϕ )
+ δ ' mn M omn r ( 1 ) ( c , ζ , η , ϕ ) + i γ ' mn N emn r ( 1 ) ( c , ζ , η , ϕ ) ]
E s = E 0 m = 0 n = m i n [ β mn M emn r ( 3 ) ( c , ζ , η , ϕ β ' mn M omn r ( 3 ) ( c , ζ , η , ϕ )
+ i α mn N omn r ( 3 ) ( c , ζ , η , ϕ ) + i α ' mn N emn r ( 3 ) ( c , ζ , η , ϕ ) ]
H = 1 i w μ × E M mn o e = 1 k N mn o e N mn o e = 1 k M mn o e
g n , TE m ¯ = g n , TE m ¯ = g n , g ' n , TE m ¯ = g ' n , TM m ¯ = 0 m = 1 g n , TE m ¯ = g ' n , TE m ¯ = g n , TM m ¯ = g ' n , TM m ¯ = 0 m 1 }
g n , TE 1 m g ' n , TE 1 m g n , TM 1 m g ' n , TM 1 m = ( 1 ) m 1 ( n m ) ! ( n + m ) ! g n [ ( 2 δ m 0 ) dP n m ( cos β ) d β cos ( m γ ) cos α sin ( m γ ) cos α sin ( m γ ) sin α cos ( m γ ) sin α
+ 2 m P n m ( cos β ) sin β sin ( m γ ) sin α cos ( m γ ) sin α cos ( m γ ) cos α sin ( m γ ) cos α ]
G n , TE m G ' n , TE m G n , TM m G ' n , TM m = 2 N mn r = 0,1 / i r m 2 r + 2 m + 1 d r mn ( c ) g r + m ( 1 ) m 1
× [ ( 2 δ m 0 ) d P r + m m d β cos ( m γ ) cos α sin ( m γ ) cos α sin ( m γ ) sin α cos ( m γ ) sin α + 2 m P r + m m ( cos β ) sin β sin ( m γ ) sin α cos ( m γ ) sin α cos ( m γ ) cos α sin ( m γ ) cos α ]
ê y E 0 e ikr ( sin θ cos φ sin β + cos θ cos β ) = E 0 m = 0 n = m i n [ G n , TE m M emn r ( 1 ) ( c , ζ , η , ϕ ) + iG n , TM m N omn r ( 1 ) ( c , ζ , η , ϕ ]
G n , TE m G n , TM m = 2 N mn ( 1 ) m 1 r = 0,1 / d r mn ( c ) ( r + m ) ( r + m + 1 ) ( 2 δ m 0 ) dP r + m m ( cos β ) 2 m P r + m m ( cos β ) sin β
E r = E 0 ψ 0 [ cos ϕ′′ sin θ′′ ( 1 2 Q l r cos θ′′ ) + 2 Q l x 0 cos θ′′ ] exp ( K )
E θ = E 0 ψ 0 [ cos ϕ′′ ( cos θ′′ + 2 Q l r sin 2 θ′′ ) 2 Q l x 0 sin θ′′ ] exp ( K )
E ϕ = E 0 ψ 0 sin ϕ′′ exp ( K )
H r , TE = H 0 ψ 0 [ sin ϕ′′ sin θ′′ ( 1 2 Q l r cos θ′′ ) + 2 Q l y 0 cos θ′′ ] exp ( K )
H ϕ = H 0 ψ 0 [ sin ϕ′′ ( cos θ + 2 Q l r sin 2 θ′′ ) 2 Q l y 0 sin θ′′ ] exp ( K )
H ϕ = H 0 ψ 0 cos ϕ′′ exp ( K )
ψ 0 = iQ exp [ 2 iQ w 0 2 r sin θ′′ ( x 0 cos ϕ′′ + y sin ϕ′′ ) × exp ( iQ r 2 sin 2 θ′′ w 0 2 ) exp ( iQ x 0 2 + y 0 2 w 0 2 )
K = ik ( r cos θ z 0 )
Q = 1 i + ( z z 0 ) l
E 0 H 0 = ( μ ε ) 1 / 2
E r , TE = E 0 ψ 0 [ sin ϕ′′ sin θ′′ ( 1 2 Q l r cos θ′′ ) + 2 Q l y 0 cos θ′′ ] exp ( K )
E θ = E 0 ψ 0 [ sin ϕ′′ ( cos θ′′ + 2 Q l r sin 2 θ′′ ) 2 Q l y 0 sin θ′′ ] exp ( K )
E ϕ = E 0 ψ 0 cos ϕ′′ exp ( K )
H r = H 0 ψ 0 [ cos ϕ′′ sin θ′′ ( 1 2 Q l r cos θ′′ ) + 2 Q l x 0 cos θ′′ ] exp ( K )
H θ = H 0 ψ 0 [ cos ϕ′′ ( cos θ′′ + 2 Q l r sin 2 θ′′ ) 2 Q l x 0 sin θ′′ ] exp ( K )
H ϕ = H 0 ψ 0 sin ϕ′′ exp ( K )
E i = E 0 n = 1 m = n n C nm [ ig n , TE m m mn r ( 1 ) ( kr , θ′′ , ϕ′′ ) + g n , TM m n mn r ( 1 ) ( kr , θ′′ , ϕ′′ ) ]
C nm = { C n m 0 ( 1 ) m ( n + m ) ! ( n m ) ! C n m < 0
( C n = i n 1 2 n + 1 n ( n + 1 ) )
m mn r ( 1 ) ( kr , θ′′ , ϕ′′ ) n mn r ( 1 ) ( kr , θ′′ , ϕ′′ ) = m emn r ( 1 ) ( kr , θ′′ , ϕ′′ ) n emn r ( 1 ) ( kr , θ′′ , ϕ′′ ) + i m omn r ( 1 ) ( kr , θ′′ , ϕ′′ ) n omn r ( 1 ) ( kr , θ′′ , ϕ′′ )
( g n , T E m ¯ g n , T E ' m ¯ g n , T M m ¯ g n , T M ' m ¯ ) = i n 2 n + 1 n ( n + 1 ) 1 ( 1 + δ 0 m ) 1 i i 1 g n , TE m + g n , TE m g n , TE m g n , TE m g n , TM m g n , TM m g n , TM m + g n , TM m

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