Abstract

We derive a straightforward theoretical method to determine the electromagnetic fields for the incidence of a monochromatic laser beam on a near-spherical dielectric particle. The beam-shape coefficients are obtained from the radial laser fields and expressed as a finite series in a form that has, to our knowledge, not been published before. Our perturbation approach to solve Maxwell’s equations in spherical coordinates employs two alternative techniques to match the boundary conditions: an analytic approach for small particles with low eccentricity and an adapted point-matching method for larger spheroids with higher aspect ratios. We present results for the internal and external fields, scattering intensities, and stresses exerted on the particle. While similarly accurate as others, our approach is easily implemented numerically and thus particularly useful in praxis, e.g., for analyzing optical traps, such as the optical stretcher.

© 2009 Optical Society of America

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References

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  1. T. Oguchi, “Attenuation and phase rotation of radio waves due to rain: calculations at 19.3 and 34.8 GHz,” Radio Sci. 8, 31-38 (1973).
    [CrossRef]
  2. J. Guck, R. Ananthakrishnan, and H. Mahmood, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81, 767-784 (2001).
    [CrossRef]
  3. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1991).
  4. H. Van De Hulst, Light Scattering by Small Particles (Dover, 1982).
  5. J. W. Strutt (Lord Rayleigh), “On the dispersal of light by a dielectric cylinder,” Philos. Mag. 36, 365-376 (1918).
  6. C. Yeh, “Backscattering cross section of a dielectric elliptical cylinder,” J. Opt. Soc. Am. 55, 309-314 (1965).
    [CrossRef]
  7. S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29-49 (1975).
  8. Y. Han, G. Gréhan, and G. Gouesbet, “Generalized Lorenz-Mie theory for a spheroidal particle with off-axis Gaussian-beam illumination,” Appl. Opt. 42, 6621-6629 (2003).
    [CrossRef]
  9. M. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A 8, 871-882 (1991).
    [CrossRef]
  10. K. J. Chalut, M. G. Giacomelli, and A. Wax, “Application of Mie theory to assess structure of spheroidal scattering in backscattering geometries,”J. Opt. Soc. Am. A 25, 1866-1874 (2008).
  11. A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).
  12. J. P. Barton and D. R. Alexander, “Electromagnetic fields for an irregularly shaped, near-spherical particle illuminated by a focused laser beam,” Appl. Phys. 69, 7972-7986 (1991).
  13. C. Yeh, “Perturbation approach to the diffraction of electromagnetic waves by arbitrarily shaped dielectric obstacles,” Phys. Rev. A 135, 1193-1201 (1964).
    [CrossRef]
  14. 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,”Appl. Phys. 64, 1632-1639 (1988).
    [CrossRef]
  15. 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]
  16. F. Xu, K. Ren, G. Gouesbet, X. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E 75, 026613-1-026613-14 (2007).
  17. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177-1179 (1979).
    [CrossRef]
  18. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1965).
  19. A. P. Prudnikov, Y. A. Brychkow, and O. I. Marichev, Integrals and Series, Vol. 2 (Overseas Publisher Association, 1998).
  20. J. P. Barton, “Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,” Appl. Opt. 34, 5542-5551 (1995).
  21. J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800-2802 (1989).
    [CrossRef]
  22. J. A. Stratton, Electromagnetic Theory, 1st ed. (McGraw-Hill, 1941).
  23. S. Asano, “Light scattering properties of spheroidal particles,” Appl. Opt. 18, 712-723 (1979).
    [CrossRef]
  24. J. Guck, R. Ananthakrishnan, and T. J. Moon, “Optical deformability of soft biological dielectrics,” Phys. Rev. Lett. 84, 5451-5454 (2000).
    [CrossRef]

2008

2007

F. Xu, K. Ren, G. Gouesbet, X. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E 75, 026613-1-026613-14 (2007).

2003

2001

J. Guck, R. Ananthakrishnan, and H. Mahmood, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81, 767-784 (2001).
[CrossRef]

2000

J. Guck, R. Ananthakrishnan, and T. J. Moon, “Optical deformability of soft biological dielectrics,” Phys. Rev. Lett. 84, 5451-5454 (2000).
[CrossRef]

1995

1991

M. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A 8, 871-882 (1991).
[CrossRef]

J. P. Barton and D. R. Alexander, “Electromagnetic fields for an irregularly shaped, near-spherical particle illuminated by a focused laser beam,” Appl. Phys. 69, 7972-7986 (1991).

1989

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800-2802 (1989).
[CrossRef]

1988

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]

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,”Appl. Phys. 64, 1632-1639 (1988).
[CrossRef]

1979

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

S. Asano, “Light scattering properties of spheroidal particles,” Appl. Opt. 18, 712-723 (1979).
[CrossRef]

1975

1973

T. Oguchi, “Attenuation and phase rotation of radio waves due to rain: calculations at 19.3 and 34.8 GHz,” Radio Sci. 8, 31-38 (1973).
[CrossRef]

1965

1964

C. Yeh, “Perturbation approach to the diffraction of electromagnetic waves by arbitrarily shaped dielectric obstacles,” Phys. Rev. A 135, 1193-1201 (1964).
[CrossRef]

1918

J. W. Strutt (Lord Rayleigh), “On the dispersal of light by a dielectric cylinder,” Philos. Mag. 36, 365-376 (1918).

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

Alexander, D. R.

J. P. Barton and D. R. Alexander, “Electromagnetic fields for an irregularly shaped, near-spherical particle illuminated by a focused laser beam,” Appl. Phys. 69, 7972-7986 (1991).

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800-2802 (1989).
[CrossRef]

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,”Appl. Phys. 64, 1632-1639 (1988).
[CrossRef]

Ananthakrishnan, R.

J. Guck, R. Ananthakrishnan, and H. Mahmood, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81, 767-784 (2001).
[CrossRef]

J. Guck, R. Ananthakrishnan, and T. J. Moon, “Optical deformability of soft biological dielectrics,” Phys. Rev. Lett. 84, 5451-5454 (2000).
[CrossRef]

Asano, S.

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).

J. P. Barton and D. R. Alexander, “Electromagnetic fields for an irregularly shaped, near-spherical particle illuminated by a focused laser beam,” Appl. Phys. 69, 7972-7986 (1991).

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800-2802 (1989).
[CrossRef]

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,”Appl. Phys. 64, 1632-1639 (1988).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1991).

Brychkow, Y. A.

A. P. Prudnikov, Y. A. Brychkow, and O. I. Marichev, Integrals and Series, Vol. 2 (Overseas Publisher Association, 1998).

Cai, X.

F. Xu, K. Ren, G. Gouesbet, X. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E 75, 026613-1-026613-14 (2007).

Chalut, K. J.

Davis, L. W.

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

Giacomelli, M. G.

Gouesbet, G.

Gréhan, G.

Guck, J.

J. Guck, R. Ananthakrishnan, and H. Mahmood, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81, 767-784 (2001).
[CrossRef]

J. Guck, R. Ananthakrishnan, and T. J. Moon, “Optical deformability of soft biological dielectrics,” Phys. Rev. Lett. 84, 5451-5454 (2000).
[CrossRef]

Hagness, S.

A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Han, Y.

Maheu, B.

Mahmood, H.

J. Guck, R. Ananthakrishnan, and H. Mahmood, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81, 767-784 (2001).
[CrossRef]

Marichev, O. I.

A. P. Prudnikov, Y. A. Brychkow, and O. I. Marichev, Integrals and Series, Vol. 2 (Overseas Publisher Association, 1998).

Mishchenko, M.

Moon, T. J.

J. Guck, R. Ananthakrishnan, and T. J. Moon, “Optical deformability of soft biological dielectrics,” Phys. Rev. Lett. 84, 5451-5454 (2000).
[CrossRef]

Oguchi, T.

T. Oguchi, “Attenuation and phase rotation of radio waves due to rain: calculations at 19.3 and 34.8 GHz,” Radio Sci. 8, 31-38 (1973).
[CrossRef]

Prudnikov, A. P.

A. P. Prudnikov, Y. A. Brychkow, and O. I. Marichev, Integrals and Series, Vol. 2 (Overseas Publisher Association, 1998).

Ren, K.

F. Xu, K. Ren, G. Gouesbet, X. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E 75, 026613-1-026613-14 (2007).

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,”Appl. Phys. 64, 1632-1639 (1988).
[CrossRef]

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory, 1st ed. (McGraw-Hill, 1941).

Strutt (Lord Rayleigh), J. W.

J. W. Strutt (Lord Rayleigh), “On the dispersal of light by a dielectric cylinder,” Philos. Mag. 36, 365-376 (1918).

Taflove, A.

A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Van De Hulst, H.

H. Van De Hulst, Light Scattering by Small Particles (Dover, 1982).

Wax, A.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1991).

Xu, F.

F. Xu, K. Ren, G. Gouesbet, X. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E 75, 026613-1-026613-14 (2007).

Yamamoto, G.

Yeh, C.

C. Yeh, “Backscattering cross section of a dielectric elliptical cylinder,” J. Opt. Soc. Am. 55, 309-314 (1965).
[CrossRef]

C. Yeh, “Perturbation approach to the diffraction of electromagnetic waves by arbitrarily shaped dielectric obstacles,” Phys. Rev. A 135, 1193-1201 (1964).
[CrossRef]

Appl. Opt.

Appl. Phys.

J. P. Barton and D. R. Alexander, “Electromagnetic fields for an irregularly shaped, near-spherical particle illuminated by a focused laser beam,” Appl. Phys. 69, 7972-7986 (1991).

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,”Appl. Phys. 64, 1632-1639 (1988).
[CrossRef]

Biophys. J.

J. Guck, R. Ananthakrishnan, and H. Mahmood, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81, 767-784 (2001).
[CrossRef]

J. Appl. Phys.

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800-2802 (1989).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Philos. Mag.

J. W. Strutt (Lord Rayleigh), “On the dispersal of light by a dielectric cylinder,” Philos. Mag. 36, 365-376 (1918).

Phys. Rev. A

C. Yeh, “Perturbation approach to the diffraction of electromagnetic waves by arbitrarily shaped dielectric obstacles,” Phys. Rev. A 135, 1193-1201 (1964).
[CrossRef]

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

Phys. Rev. E

F. Xu, K. Ren, G. Gouesbet, X. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E 75, 026613-1-026613-14 (2007).

Phys. Rev. Lett.

J. Guck, R. Ananthakrishnan, and T. J. Moon, “Optical deformability of soft biological dielectrics,” Phys. Rev. Lett. 84, 5451-5454 (2000).
[CrossRef]

Radio Sci.

T. Oguchi, “Attenuation and phase rotation of radio waves due to rain: calculations at 19.3 and 34.8 GHz,” Radio Sci. 8, 31-38 (1973).
[CrossRef]

Other

A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1991).

H. Van De Hulst, Light Scattering by Small Particles (Dover, 1982).

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

A. P. Prudnikov, Y. A. Brychkow, and O. I. Marichev, Integrals and Series, Vol. 2 (Overseas Publisher Association, 1998).

J. A. Stratton, Electromagnetic Theory, 1st ed. (McGraw-Hill, 1941).

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Figures (10)

Fig. 1
Fig. 1

Definition of fields and coordinates. The center of the scattering object (here, a spheroid with semiaxes a , b , and c ) defines the origin of the spherical coordinates. The radiation source (e.g., a laser with fields E ( L ) E ( i ) , etc.) is placed a distance d away from the coordinate center and is collinear with the z-axis. Regions I and II have different permittivity and conductivity that is reflected in the two parameters k I and k II that enter Maxwell’s equations.

Fig. 2
Fig. 2

Logarithmic plots of the far-field scattering intensity computed with approach A, approach B (both Matlab), and a T-matrix code [9] for plane-wave incidence ( w 0 , λ = 628.3 nm ). The scattering particles are prolate spheroids ( a b = 1.05 ) of different sizes ( α = 1 , 3 , 5 ) . The refractive indices of the medium and object are n I = 1 and n II = 1.33 , respectively. Relative intensities are plotted to scale.

Fig. 3
Fig. 3

Logarithmic plots of the far-field scattering intensity computed with approach B (Matlab) and a T-matrix code [9] for plane-wave incidence ( w 0 , λ = 628.3 nm ). The scattering particles are prolate spheroids ( a b = 1.10 ) of different sizes ( α = 5 , 10 ) . The refractive indices of medium and object are n I = 1 and n II = 1.33 , respectively. For clarity, the intensity of the lower curve ( α = 5 ) is scaled down by a factor of 10 3 .

Fig. 4
Fig. 4

Plot of scattering efficiency factors for different size parameters and plane wave incidence ( w 0 , λ = 628.3 nm ) on oblate spheroids with aspect ratios b a = 1.1 and b a = 1.5 . The results obtained by [23] and those computed with the Matlab code based on approach B are shown. Refractive indices are n I = 1 and n II = 1.33 .

Fig. 5
Fig. 5

Logarithmic plot of the tangential and normal convergence parameters Δ t and Δ n for equal-volume prolate spheroids with aspect ratios a b = 1.2 and a b = 1.4 in dependence on l max . The size parameter is set to α = 5 . The conditions under which the point-matching method is carried out are described in the text.

Fig. 6
Fig. 6

Logarithmic plot of the tangential and normal convergence parameters Δ t and Δ n for different values of m max l max . The prolate spheroids ( α = 5 ) are of equal volume and have aspect ratios a b = 1.2 and a b = 1.4 . The m max points are spread linearly over three unequal size intervals (see text).

Fig. 7
Fig. 7

3D plot and contour plot of the normalized internal source function S in the x - z plane for incidence of a Gaussian laser beam ( λ = 628 nm , w 0 = 3 λ , z 0 = 60 μ m ) on an oblate spheroid with b a = 1.2 . The size parameter is α = 10 . The coordinates are normalized with respect to the spherical radius: X = x a and Z = z a .

Fig. 8
Fig. 8

3D plot and contour plot of the normalized internal source function S for a prolate spheroid ( a b = 1.2 ) . All parameters are identical to those of Fig. 7.

Fig. 9
Fig. 9

Time-averaged radial stresses for large spherical and spheroidal objects ( α = 47 ) placed in a single Gaussian laser beam with λ = 1064 nm , w 0 = 3 λ , and z 0 = 90 μ m . Left: sphere with ϵ = 0 ( a b = 1 ) . Middle: prolate spheroid with ϵ = 0.05 ( a b = 1.05 ) . Right: prolate spheroid with ϵ = 0.10 ( a b = 1.10 ) . The refractive indices are common for a cell ( n II = 1.375 ) in an aqueous solution ( n I = 1.335 ) .

Fig. 10
Fig. 10

Time-averaged radial stresses for a prolate spheroid ( α = 47 , a b = 1.1 , and n II = 1.375 ) in aqueous solution ( n I = 1.335 ) trapped in a double-beam laser stretcher ( λ = 1064 nm , w 0 = 3 λ ). The laser cell distances are z 0 = ± 60 μ m (left), z 0 = ± 120 μ m (middle), and z 0 = ± 200 μ m (right).

Equations (50)

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r Π ( r , θ , ϕ ) = l = 0 m = l + l [ a m l ψ l ( k r ) + b m l χ l ( k r ) ] P l ( m ) ( cos θ ) e i m ϕ
[ r Π ( i ) e r Π ( i ) m ] = 1 k I 2 l = 1 [ B l e cos ϕ B l m sin ϕ ] ψ l ( k I r ) P l ( 1 ) ( cos θ ) ,
[ E r ( i ) H r ( i ) ] = l = 1 [ A l e cos ϕ A l m sin ϕ ] ψ l ( k I r ) r 2 P l ( 1 ) ( cos θ ) ,
B ( θ ) = a ( 1 + ϵ ) [ 1 + ϵ ( 2 + ϵ ) sin 2 θ ] 1 2 .
sin θ | θ ( k 2 I r Π ( i ) m + k 2 I r Π ( s ) m k 2 II r Π ( w ) m ) | r = B ( θ ) = | 2 r ϕ ( r Π ( i ) e + r Π ( s ) e r Π ( w ) e ) | r = B ( θ ) ,
| ϕ ( k 2 I r Π ( i ) m + k 2 I r Π ( s ) m k 2 II r Π ( w ) m ) | r = B ( θ ) = sin θ | 2 r θ ( r Π ( i ) e + r Π ( s ) e r Π ( w ) e ) | r = B ( θ ) ,
[ r Π 0 ( w ) e r Π 1 ( w ) e ] = 1 k II 2 l = 1 [ C l e c l e ] ψ l ( k II r ) P l ( 1 ) ( cos θ ) cos ϕ ,
[ r Π 0 ( s ) e r Π 1 ( s ) e ] = 1 k I 2 l = 1 [ D l e d l e ] ξ l ( k I r ) P l ( 1 ) ( cos θ ) cos ϕ .
C l e = k II k 2 I [ ξ l ( k I a ) ψ l ( k I a ) ξ l ( k I a ) ψ l ( k I a ) ] k 1 II k II ξ l ( k I a ) ψ l ( k II a ) k 1 I k I ξ l ( k I a ) ψ l ( k II a ) B l e ,
C l m = k II k 1 I [ ξ l ( k I a ) ψ l ( k I a ) ξ l ( k I a ) ψ l ( k I a ) ] k 2 II k II ξ l ( k I a ) ψ l ( k II a ) k 2 I k I ξ l ( k I a ) ψ l ( k II a ) B l m ,
D l e = k 1 I k I ψ l ( k I a ) ψ l ( k II a ) k 1 II k II ψ l ( k I a ) ψ l ( k II a ) k 1 II k II ξ l ( k I a ) ψ l ( k II a ) k 1 I k I ξ l ( k I a ) ψ l ( k II a ) B l e ,
D l m = k 2 I k I ψ l ( k I a ) ψ l ( k II a ) k 2 II k II ψ l ( k I a ) ψ l ( k II a ) k 2 II k II ξ l ( k I a ) ψ l ( k II a ) k 2 I k I ξ l ( k I a ) ψ l ( k II a ) B l m ,
l = 1 Λ l ( 1 ) e P l ( 1 ) ( x ) B 1 ( x ) l = 1 Λ l ( 2 ) m ( 1 x 2 ) P l ( 1 ) ( x ) B 1 ( x ) = l = 1 α l e P l ( 1 ) ( x ) l = 1 β l m ( 1 x 2 ) P l ( 1 ) ( x ) ,
l = 1 Λ l ( 1 ) e ( 1 x 2 ) P l ( 1 ) ( x ) B 1 ( x ) l = 1 Λ l ( 2 ) m P l ( 1 ) ( x ) B 1 ( x ) = l = 1 α l e ( 1 x 2 ) P l ( 1 ) ( x ) l = 1 β l m P l ( 1 ) ( x ) ,
Λ l ( 1 ) e = B l e ψ l ( k I a ) C l e ψ l ( k II a ) + D l e ξ l ( k I a ) ,
Λ l ( 2 ) m = k 2 I k I [ B l m ψ l ( k I a ) + D l m ξ l ( k I a ) ] k 2 II k II C l m ψ l ( k II a ) ;
α l e = + c l e k II ψ l ( k II a ) d l e k I ξ l ( k I a ) ,
β l m = c l m k 1 II ψ l ( k II a ) + d l m k 1 I ξ l ( k I a ) .
g m ( l ) = α l e ( l + 2 ) 2 2 l + 3 β l + 1 m + ( l 1 ) 2 2 l 1 β l 1 m ,
f e ( l ) = β l m ( l + 2 ) 2 2 l + 3 α l + 1 e + ( l 1 ) 2 2 l 1 α l 1 e ,
f e ( l ) = ( l 1 ) 2 2 l 1 Λ l 1 ( 1 ) e + Λ l ( 2 ) m ( l + 2 ) 2 2 l + 3 Λ l + 1 ( 1 ) e + n = 1 [ Λ n ( 2 ) m I l , n + Λ n ( 1 ) e 2 n + 1 ( n 2 I l , n + 1 ( n + 1 ) 2 I l , n 1 ) ] ,
g m ( l ) = ( l 1 ) 2 2 l 1 Λ l 1 ( 2 ) m + Λ l ( 1 ) e ( l + 2 ) 2 2 l + 3 Λ l + 1 ( 2 ) m + n = 1 [ Λ n ( 1 ) e I l , n + Λ n ( 2 ) m 2 n + 1 ( n 2 I l , n + 1 ( n + 1 ) 2 I l , n 1 ) ] .
I l , n = n ( n + 1 ) ( 2 l + 1 ) Γ ( n + l 2 1 2 ) 4 π Γ ( n l 2 + 2 ) Γ ( n l 2 + 1 2 ) Γ ( l n 2 + 1 2 ) Γ ( l n 2 + 2 ) Γ ( n + l 2 + 5 2 ) .
γ 1 ( l ) α l 2 e + γ 2 ( l ) α l e + γ 3 ( l ) α l + 2 e = ( l + 2 ) 2 2 l + 3 f e ( l + 1 ) ( l 1 ) 2 2 l 1 f e ( l 1 ) + g m ( l ) ,
γ 1 ( l ) β l 2 m + γ 2 ( l ) β l m + γ 3 ( l ) β l + 2 m = ( l + 2 ) 2 2 l + 3 g m ( l + 1 ) ( l 1 ) 2 2 l 1 g m ( l 1 ) + f e ( l ) .
γ 1 ( l ) = ( l 1 ) 2 ( l 2 ) 2 ( 2 l 3 ) ( 2 l 1 ) ,
γ 2 ( l ) = 1 + ( l 1 ) 2 ( l + 1 ) 2 ( 2 l 1 ) ( 2 l + 1 ) + l 2 ( l + 2 ) 2 ( 2 l + 1 ) ( 2 l + 3 ) ,
γ 3 ( l ) = ( l + 2 ) 2 ( l + 3 ) 2 ( 2 l + 3 ) ( 2 l + 5 ) .
M γ = [ γ 2 ( 1 ) 0 γ 3 ( 1 ) 0 0 0 γ 2 ( 2 ) 0 γ 3 ( 2 ) 0 γ 1 ( 3 ) 0 γ 2 ( 3 ) 0 γ 3 ( 3 ) ] ,
v α = [ 9 5 f e ( 2 ) + g m ( 1 ) 16 7 f e ( 3 ) 1 3 f e ( 1 ) + g m ( 2 ) 25 9 f e ( 4 ) 4 5 f e ( 2 ) + g m ( 3 ) ] .
c l e = k 2 II k II [ α l e k I ξ l ( k I a ) + β e k 2 I ξ l ( k I a ) ] k 2 II k I ψ l ( k II a ) ξ l ( k I a ) k 2 I k II ψ l ( k II a ) ξ l ( k I a ) ,
c l m = k 1 II k II [ α l m k I ξ l ( k I a ) + β m k 1 I ξ l ( k I a ) ] k 1 II k I ψ l ( k II a ) ξ l ( k I a ) k 1 I k II ψ l ( k II a ) ξ l ( k I a ) ,
d l e = k 2 I k I [ α l e k II ψ l ( k II a ) + β e k 2 II ψ l ( k II a ) ] k 2 II k I ψ l ( k II a ) ξ l ( k I a ) k 2 I k II ψ l ( k II a ) ξ l ( k I a ) ,
d l m = k 1 I k I [ α l m k II ψ l ( k II a ) + β m k 1 II ψ l ( k II a ) ] k 1 II k I ψ l ( k II a ) ξ l ( k I a ) k 1 I k II ψ l ( k II a ) ξ l ( k I a ) .
l = 1 [ B l e k I ψ l ( k I B ) C l e k II ψ l ( k II B ) + D l e k I ξ l ( k I B ) ] P l ( 1 ) ( x ) 1 x 2 + l = 1 [ B l m k I ψ l ( k I B ) C l m k II ψ l ( k II B ) + D l m k I ξ l ( k I B ) ] P l ( 1 ) ( x ) = l = 1 [ c ̃ l e k II ψ l ( k II B ) d ̃ l e k I ξ l ( k I B ) ] P l ( 1 ) ( x ) 1 x 2 + l = 1 [ c ̃ l m k 1 II ψ l ( k II B ) d ̃ l m k 1 I ξ l ( k I B ) ] P l ( 1 ) ( x ) .
l = 1 { ϵ I B l e [ ψ l ( k I B ) + ψ l ( k I B ) ] ϵ II C l e [ ψ l ( k II B ) + ψ l ( k II B ) ] + ϵ I D l e [ ξ l ( k I B ) + ξ l ( k I B ) ] } P l ( 1 ) ( x ) = l = 1 { c ̃ l e [ ψ l ( k II B ) + ψ l ( k II B ) ] ϵ II } P l ( 1 ) ( x ) l = 1 { d ̃ l e [ ξ l ( k I B ) + ξ l ( k I B ) ] ϵ I } P l ( 1 ) ( x ) .
j = 1 4 M ( i j ) v ( j ) = N ( i ) , for i = 1 , , 6 .
[ M ( 11 ) ] m l = 1 k II ψ l [ k II B ( x m ) ] P l ( 1 ) ( x m ) 1 x m 2 ,
[ M ( 21 ) ] m l = 1 k II ψ l [ k II B ( x m ) ] P l ( 1 ) ( x m ) .
[ M ( 11 ) M ( 14 ) M ( 61 ) M ( 64 ) ] [ v ( 1 ) v ( 4 ) ] = [ N ( 1 ) N ( 6 ) ] .
[ E x ( L ) H y ( L ) ] = [ E 0 H 0 ] i Q e i k I ( z + z 0 ) exp { i Q ( x 2 + y 2 ) w 0 2 } ,
[ E z ( L ) H z ( L ) ] = [ x E x ( L ) y H y ( L ) ] Q z R ,
[ E r ( L ) H r ( L ) ] = [ E 0 cos ϕ H 0 sin ϕ ] ( 1 + 2 Q 0 r cos θ z R ) sin θ exp ( i k I r cos θ i Q 0 r 2 sin 2 θ w 0 2 i Q 0 2 r 3 sin 2 θ cos θ z R w 0 2 ) .
f ( l ) = 2 l + 1 l ( l + 1 ) 2 l + 1 l ! ( l 1 ) ! ( 2 l + 1 ) ! k I 2 .
I = 1 1 d x x l 2 m 2 n 1 ( 1 x 2 ) m + n + 1 2 P l ( 1 ) ( x ) .
[ B l e B l m ] = [ E 0 H 0 ] e i k I z 0 2 Q 0 ( l + 1 2 ) 2 ( 2 l 1 ) ! ( l + 1 ) ! ( i 2 ) l m = 0 n = 0 2 m + 3 n l 1 Γ ( l 2 m n ) Γ ( m + n + 2 ) Γ ( l 2 m 3 n ) Γ ( l 2 + 2 ) m ! n ! ( 1 ) m 2 + 1 Q 0 m ( k I z R ) n F 2 3 ( l 2 + 1 , 1 l 2 , m + n + 2 l 2 + 2 , 2 ; 1 ) ( Q 0 k I w 0 ) 2 m + 2 n [ 1 ( l 2 m 3 n 1 ) Δ m , n 2 i Q 0 k I z R ] .
[ B ̃ l e B ̃ l m ] = 1 2 π 2 l + 1 2 l ( l + 1 ) ( a k I ) 2 l ( l + 1 ) 1 ψ l ( k I a ) 0 π 0 2 π d θ d ϕ [ E r ( L ) ( a , θ , ϕ ) H r ( L ) ( a , θ , ϕ ) ] P l ( 1 ) ( cos θ ) sin θ .
C scat = lim r r 2 0 π 0 2 π d θ d ϕ sin θ ( | E ϕ ( s ) | 2 + | E θ ( s ) | 2 ) ,
Δ r e = 0 π 0 2 π d θ d ϕ sin θ | E r ext n II n I E r int | ,
σ r r t = 1 16 π Re [ ϵ I ( E r E r * E θ E θ * E ϕ E ϕ * ) + μ I ( H r H r * H θ H θ * H ϕ H ϕ * ) ] .

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