Abstract

The interaction of a highly focused beam of light with spherical nanoparticles is investigated for linear and radial polarizations. An analytical solution is obtained to calculate this interaction. The Richards-Wolf theory is used to express the incident electric field near the focus of an aplanatic lens. The incident beam is expressed as an integral where the integrand is separated into transverse-electric (TE) and transverse-magnetic (TM) waves. The interaction of each TE and TM wave with a spherical nanoparticle is calculated using the Mie theory. The resulting analytical solution is then obtained by integrating the scattered waves over the entire angular spectrum. A finite element method solution is also obtained for comparison.

© 2008 Optical Society of America

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References

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  1. A. Ashkin, "Acceleration and trapping of particles by radiation pressure," Phys. Rev. Lett. 24, 156-159 (1970).
    [CrossRef]
  2. C. Godefroy and M. Adjouadi, "Particle sizing in a flow environment using light scattering patterns," Part. Part. Syst. Charact. 17, 47-55 (2000).
    [CrossRef]
  3. A. C. Eckbreth, "Effects of laser-modulated particulate incandescence on Raman scattering diagnostics," J. Appl. Phys. 48, 4473-4479 (1977).
    [CrossRef]
  4. N. Morita, T. Tanaka, T. Yamasaki, and Y. Yakanishi, "Scattering of a beam by a spherical object," IEEE Trans. Antennas Propag. 16, 724-727 (1968).
    [CrossRef]
  5. W.-C. Tsai and R. J. Pogorzelski, "Eigenfunction solution of the scattering of beam radiation fields by spherical objects," J. Opt. Soc. Am. A 65, 1457-1463 (1975).
    [CrossRef]
  6. W. G. Tam and R. Corriveau, "Scattering of electromagnetic beams by spherical objects," J. Opt. Soc. Am. 68, 763-767 (1978).
    [CrossRef]
  7. J. S. Kim and S. S. Lee, "Scattering of laser beams and the optical potential well for a homogeneous sphere," J. Opt. Soc. Am. 73, 303-312 (1983).
    [CrossRef]
  8. 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]
  9. L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979).
    [CrossRef]
  10. J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal fields of a spherical particle illuminated by a tightly focused laser beam: Focal point positioning effects at resonance," J. Appl. Phys. 65, 2900-2906 (1989).
    [CrossRef]
  11. 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]
  12. J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989).
    [CrossRef]
  13. J. P. Barton, "Electromagnetic-field calculations for a sphere illuminated by a higher-order Gaussian beam. I. Internal and near-field effects," Appl. Opt. 36, 1303-1311 (1997).
    [CrossRef] [PubMed]
  14. J. P. Barton, "Electromagnetic-field calculations for a sphere illuminated by a higher-order Gaussian beam. II. Far-field scattering," Appl. Opt. 37, 3339-3344 (1998).
    [CrossRef]
  15. E. Wolf, "Electromagnetic diffraction in optical systems I. An integral representation of the image field," Proc. Roy. Soc. London Ser. A 253, 349-357 (1959).
    [CrossRef]
  16. B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. Roy. Soc. London Ser. A 253, 358-379 (1959).
    [CrossRef]
  17. A. Hartschuh, E. J. S’anchez, X. S. Xie, and L. Novotny, "High-resolution near-field Raman microscopy of singlewalled carbon nanotubes," Phys. Rev. Lett. 90, 095503 (2003).
    [CrossRef] [PubMed]
  18. W. A. Challener, I. K. Sendur, and C. Peng, "Scattered field formulation of finite difference time domain for a focused light beam in dense media with lossy materials," Opt. Express 11, 3160-3170 (2003).
    [CrossRef] [PubMed]
  19. K. Sendur, W. Challener, and C. Peng, "Ridge waveguide as a near field aperture for high density data storage," J. Appl. Phys. 96, 2743-2752 (2004).
    [CrossRef]
  20. G. Mie, "Beitr¨age zur optik truber medien, speziell kolloida ler metallosungen," Ann. d. Physik 25, 377 (1908).
    [CrossRef]
  21. M. Born and E. Wolf, Principles of Optics 5th ed., section 13.5 (Pergamon Press, Oxford, 1975).
    [PubMed]
  22. J. M. Jin, The Finite Element Method in Electomagnetics (John Wiley & Sons, New York, NY, 2000).
  23. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, San Diego, CA, 1998).

2004 (1)

K. Sendur, W. Challener, and C. Peng, "Ridge waveguide as a near field aperture for high density data storage," J. Appl. Phys. 96, 2743-2752 (2004).
[CrossRef]

2003 (2)

A. Hartschuh, E. J. S’anchez, X. S. Xie, and L. Novotny, "High-resolution near-field Raman microscopy of singlewalled carbon nanotubes," Phys. Rev. Lett. 90, 095503 (2003).
[CrossRef] [PubMed]

W. A. Challener, I. K. Sendur, and C. Peng, "Scattered field formulation of finite difference time domain for a focused light beam in dense media with lossy materials," Opt. Express 11, 3160-3170 (2003).
[CrossRef] [PubMed]

2000 (1)

C. Godefroy and M. Adjouadi, "Particle sizing in a flow environment using light scattering patterns," Part. Part. Syst. Charact. 17, 47-55 (2000).
[CrossRef]

1998 (1)

1997 (1)

1989 (3)

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal fields of a spherical particle illuminated by a tightly focused laser beam: Focal point positioning effects at resonance," J. Appl. Phys. 65, 2900-2906 (1989).
[CrossRef]

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, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989).
[CrossRef]

1988 (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]

1983 (1)

1979 (1)

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

1978 (1)

1977 (1)

A. C. Eckbreth, "Effects of laser-modulated particulate incandescence on Raman scattering diagnostics," J. Appl. Phys. 48, 4473-4479 (1977).
[CrossRef]

1975 (1)

W.-C. Tsai and R. J. Pogorzelski, "Eigenfunction solution of the scattering of beam radiation fields by spherical objects," J. Opt. Soc. Am. A 65, 1457-1463 (1975).
[CrossRef]

1970 (1)

A. Ashkin, "Acceleration and trapping of particles by radiation pressure," Phys. Rev. Lett. 24, 156-159 (1970).
[CrossRef]

1968 (1)

N. Morita, T. Tanaka, T. Yamasaki, and Y. Yakanishi, "Scattering of a beam by a spherical object," IEEE Trans. Antennas Propag. 16, 724-727 (1968).
[CrossRef]

1959 (2)

E. Wolf, "Electromagnetic diffraction in optical systems I. An integral representation of the image field," Proc. Roy. Soc. London Ser. A 253, 349-357 (1959).
[CrossRef]

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. Roy. Soc. London Ser. A 253, 358-379 (1959).
[CrossRef]

Adjouadi, M.

C. Godefroy and M. Adjouadi, "Particle sizing in a flow environment using light scattering patterns," Part. Part. Syst. Charact. 17, 47-55 (2000).
[CrossRef]

Alexander, D. R.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal fields of a spherical particle illuminated by a tightly focused laser beam: Focal point positioning effects at resonance," J. Appl. Phys. 65, 2900-2906 (1989).
[CrossRef]

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, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (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," J. Appl. Phys. 64, 1632-1639 (1988).
[CrossRef]

Ashkin, A.

A. Ashkin, "Acceleration and trapping of particles by radiation pressure," Phys. Rev. Lett. 24, 156-159 (1970).
[CrossRef]

Barton, J. P.

J. P. Barton, "Electromagnetic-field calculations for a sphere illuminated by a higher-order Gaussian beam. II. Far-field scattering," Appl. Opt. 37, 3339-3344 (1998).
[CrossRef]

J. P. Barton, "Electromagnetic-field calculations for a sphere illuminated by a higher-order Gaussian beam. I. Internal and near-field effects," Appl. Opt. 36, 1303-1311 (1997).
[CrossRef] [PubMed]

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal fields of a spherical particle illuminated by a tightly focused laser beam: Focal point positioning effects at resonance," J. Appl. Phys. 65, 2900-2906 (1989).
[CrossRef]

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

Challener, W.

K. Sendur, W. Challener, and C. Peng, "Ridge waveguide as a near field aperture for high density data storage," J. Appl. Phys. 96, 2743-2752 (2004).
[CrossRef]

Challener, W. A.

Corriveau, R.

Davis, L. W.

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

Eckbreth, A. C.

A. C. Eckbreth, "Effects of laser-modulated particulate incandescence on Raman scattering diagnostics," J. Appl. Phys. 48, 4473-4479 (1977).
[CrossRef]

Godefroy, C.

C. Godefroy and M. Adjouadi, "Particle sizing in a flow environment using light scattering patterns," Part. Part. Syst. Charact. 17, 47-55 (2000).
[CrossRef]

Hartschuh, A.

A. Hartschuh, E. J. S’anchez, X. S. Xie, and L. Novotny, "High-resolution near-field Raman microscopy of singlewalled carbon nanotubes," Phys. Rev. Lett. 90, 095503 (2003).
[CrossRef] [PubMed]

Kim, J. S.

Lee, S. S.

Mie, G.

G. Mie, "Beitr¨age zur optik truber medien, speziell kolloida ler metallosungen," Ann. d. Physik 25, 377 (1908).
[CrossRef]

Morita, N.

N. Morita, T. Tanaka, T. Yamasaki, and Y. Yakanishi, "Scattering of a beam by a spherical object," IEEE Trans. Antennas Propag. 16, 724-727 (1968).
[CrossRef]

Peng, C.

Pogorzelski, R. J.

W.-C. Tsai and R. J. Pogorzelski, "Eigenfunction solution of the scattering of beam radiation fields by spherical objects," J. Opt. Soc. Am. A 65, 1457-1463 (1975).
[CrossRef]

Richards, B.

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. Roy. Soc. London Ser. A 253, 358-379 (1959).
[CrossRef]

Schaub, S. A.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal fields of a spherical particle illuminated by a tightly focused laser beam: Focal point positioning effects at resonance," J. Appl. Phys. 65, 2900-2906 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (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," J. Appl. Phys. 64, 1632-1639 (1988).
[CrossRef]

Sendur, I. K.

Sendur, K.

K. Sendur, W. Challener, and C. Peng, "Ridge waveguide as a near field aperture for high density data storage," J. Appl. Phys. 96, 2743-2752 (2004).
[CrossRef]

Tam, W. G.

Tanaka, T.

N. Morita, T. Tanaka, T. Yamasaki, and Y. Yakanishi, "Scattering of a beam by a spherical object," IEEE Trans. Antennas Propag. 16, 724-727 (1968).
[CrossRef]

Tsai, W.-C.

W.-C. Tsai and R. J. Pogorzelski, "Eigenfunction solution of the scattering of beam radiation fields by spherical objects," J. Opt. Soc. Am. A 65, 1457-1463 (1975).
[CrossRef]

Wolf, E.

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. Roy. Soc. London Ser. A 253, 358-379 (1959).
[CrossRef]

E. Wolf, "Electromagnetic diffraction in optical systems I. An integral representation of the image field," Proc. Roy. Soc. London Ser. A 253, 349-357 (1959).
[CrossRef]

Yakanishi, Y.

N. Morita, T. Tanaka, T. Yamasaki, and Y. Yakanishi, "Scattering of a beam by a spherical object," IEEE Trans. Antennas Propag. 16, 724-727 (1968).
[CrossRef]

Yamasaki, T.

N. Morita, T. Tanaka, T. Yamasaki, and Y. Yakanishi, "Scattering of a beam by a spherical object," IEEE Trans. Antennas Propag. 16, 724-727 (1968).
[CrossRef]

Appl. Opt. (2)

IEEE Trans. Antennas Propag. (1)

N. Morita, T. Tanaka, T. Yamasaki, and Y. Yakanishi, "Scattering of a beam by a spherical object," IEEE Trans. Antennas Propag. 16, 724-727 (1968).
[CrossRef]

J. Appl. Phys. (6)

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal fields of a spherical particle illuminated by a tightly focused laser beam: Focal point positioning effects at resonance," J. Appl. Phys. 65, 2900-2906 (1989).
[CrossRef]

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, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (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," J. Appl. Phys. 64, 1632-1639 (1988).
[CrossRef]

A. C. Eckbreth, "Effects of laser-modulated particulate incandescence on Raman scattering diagnostics," J. Appl. Phys. 48, 4473-4479 (1977).
[CrossRef]

K. Sendur, W. Challener, and C. Peng, "Ridge waveguide as a near field aperture for high density data storage," J. Appl. Phys. 96, 2743-2752 (2004).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

W.-C. Tsai and R. J. Pogorzelski, "Eigenfunction solution of the scattering of beam radiation fields by spherical objects," J. Opt. Soc. Am. A 65, 1457-1463 (1975).
[CrossRef]

Opt. Express (1)

Part. Part. Syst. Charact. (1)

C. Godefroy and M. Adjouadi, "Particle sizing in a flow environment using light scattering patterns," Part. Part. Syst. Charact. 17, 47-55 (2000).
[CrossRef]

Phys. Rev. A (1)

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

Phys. Rev. Lett. (2)

A. Hartschuh, E. J. S’anchez, X. S. Xie, and L. Novotny, "High-resolution near-field Raman microscopy of singlewalled carbon nanotubes," Phys. Rev. Lett. 90, 095503 (2003).
[CrossRef] [PubMed]

A. Ashkin, "Acceleration and trapping of particles by radiation pressure," Phys. Rev. Lett. 24, 156-159 (1970).
[CrossRef]

Proc. Roy. Soc. London Ser. A (2)

E. Wolf, "Electromagnetic diffraction in optical systems I. An integral representation of the image field," Proc. Roy. Soc. London Ser. A 253, 349-357 (1959).
[CrossRef]

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. Roy. Soc. London Ser. A 253, 358-379 (1959).
[CrossRef]

Other (4)

G. Mie, "Beitr¨age zur optik truber medien, speziell kolloida ler metallosungen," Ann. d. Physik 25, 377 (1908).
[CrossRef]

M. Born and E. Wolf, Principles of Optics 5th ed., section 13.5 (Pergamon Press, Oxford, 1975).
[PubMed]

J. M. Jin, The Finite Element Method in Electomagnetics (John Wiley & Sons, New York, NY, 2000).

E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, San Diego, CA, 1998).

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Figures (9)

Fig. 1.
Fig. 1.

Various electric fields components for the linearly polarized focused beam at the focal plane. The results are normalized with the maximum value of the total electric field. (a) Ex (x,y), (b) Ey (x,y), (c) Ez (x,y), and (d) Et (x,y).

Fig. 2.
Fig. 2.

Various electric fields components for the radially polarized focused beam at the focal plane. The results are normalized with the maximum value of the total electric field. (a) Ex (x,y), (b) Ey (x,y), (c) Ez (x,y), and (d) Et (x,y).

Fig. 3.
Fig. 3.

Spherical particle illuminated by (a) TM polarized plane wave, and (a) TE polarized plane wave.

Fig. 4.
Fig. 4.

Interaction of a radially polarized focused beam with a silver sphere with a 50 nm radius. The total electric field is plotted on the x̂-ẑ plane. (a) Solution using Mie series for |Ex (x,y)|2, (b) FEM solution for |Ex (x,y)|2, (c) Solution using Mie series for |Ez (x,y)|2, (d) FEM solution for |Ez (x,y)|2. |Ey (x,y)|2 components for both solutions are negligible.

Fig. 5.
Fig. 5.

Interaction of a linearly polarized focused beam with a silver sphere with a 50 nm radius. The total electric field is plotted on the x̂-ẑ plane. (a) Solution using Mie series for |Ex (x,y)|2, (b) FEM solution for |Ex (x,y)|2, (c) Solution using Mie series for |Ez (x,y)|2, (d) FEM solution for |Ez (x,y)|2. |Ey (x,y)|2 components for both solutions are negligible.

Fig. 6.
Fig. 6.

Interaction of a radially polarized focused beam with a silver sphere with a 250 nm radius. The total electric field is plotted on the x̂-ẑ plane. (a) Solution using Mie series for |Ex (x,y)|2, (b) FEM solution for |Ex (x,y)|2, (c) Solution using Mie series for |Ez (x,y)|2, (d) FEM solution for |Ez (x,y)|2. |Ey (x,y)|2 components for both solutions are negligible.

Fig. 7.
Fig. 7.

Interaction of a linearly polarized focused beam with a silver sphere with a 250 nm radius. The total electric field is plotted on the x̂-zĮ plane. (a) Solution using Mie series for |Ex (x,y)|2, (b) FEM solution for |Ex (x,y)|2, (c) Solution using Mie series for |Ez (x,y)|2, (d) FEM solution for |Ez (x,y)|2. |Ey (x,y)|2 components for both solutions are negligible.

Fig. 8.
Fig. 8.

Interaction of a radially polarized focused beam with a dielectric sphere with a 250 nm radius. Dielectic index of the sphere is 2. The total electric field is plotted on the x̂-ẑ plane. (a) Solution using Mie series for |Ex (x,y)|2, (b) FEM solution for |Ex (x,y)|2, (c) Solution using Mie series for |Ez (x,y)|2, (d) FEM solution for |Ez (x,y)|2. |Ey (x,y)|2 components for both solutions are negligible.

Fig. 9.
Fig. 9.

Interaction of a linearly polarized focused beam with a dielectic sphere with a 250 nm radius. Dielectic index of the sphere is 2. The total electric field is plotted on the x̂-ẑ plane. (a) Solution using Mie series for |Ex (x,y)|2, (b) FEM solution for |Ex (x,y)|2, (c) Solution using Mie series for |Ez (x,y)|2, (d) FEM solution for |Ez (x,y)|2. |Ey (x,y)|2 components for both solutions are negligible.

Equations (20)

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E ( r p ) = i λ 0 α d θ sin θ 0 2 π d ϕ a ( θ , ϕ ) exp ( i k · r p )
r p = x p x ̂ + y p y ̂ + z p z ̂ = r p cos ϕ p x ̂ + r p sin ϕ p y ̂ + z p z ̂
k = 2 π λ ( sin θ cos ϕ x ̂ + sin θ sin ϕ y ̂ cos θ z ̂ ) .
a ( θ , ϕ ) = [ cos θ cos 2 ϕ + sin 2 ϕ cos θ cos ϕ sin ϕ cos ϕ sin ϕ sin θ cos ϕ ] cos θ ,
a ( θ , ϕ ) = [ cos θ cos ϕ cos θ sin ϕ sin θ ] cos θ ,
E ( r p ) = i λ i = 1 N θ + 1 j = 1 N ϕ + 1 ω i j sin θ i cos θ i a ( θ i , ϕ j ) exp ( i k ij · r p )
θ i = ( i 1 ) θ max N θ ,
ϕ j = ( j 1 ) 2 π N ϕ ,
k ij = 2 π λ ( sin θ i cos ϕ j x ̂ + sin θ i sin ϕ j y ̂ cos θ i z ̂ ) .
E inc x ( r ) = x ̂ exp ( i k · r ) .
E inc TE ( r ) = ( sin ϕ inc x ̂ + cos ϕ inc y ̂ ) exp ( i k · r ) ,
E inc TM ( r ) = ( cos θ inc cos ϕ inc x ̂ + cos θ inc sin ϕ inc y ̂ + sin θ inc z ̂ ) exp ( i k · r ) ,
E tot TM ( r ) = [ cos θ cos ϕ sin ϕ sin θ cos ϕ cos θ sin ϕ cos ϕ sin θ sin ϕ sin θ 0 cos θ ] E tot x ( r )
E tot TE ( r ) = [ sin ϕ cos θ sin ϕ sin θ cos ϕ cos ϕ cos θ sin ϕ sin θ sin ϕ 0 sin θ cos θ ] E tot x ( r )
E inc lin ( r ) = iA π 0 α d θ 0 2 π d ϕ sin θ cos θ exp ( i k · r ) [ cos θ cos 2 ϕ + sin 2 ϕ cos θ cos ϕ sin ϕ cos ϕ sin ϕ sin θ cos ϕ ]
E inc lin ( r ) = iA π 0 α d θ 0 2 π d ϕ sin θ cos θ [ cos ϕ exp ( i k · r ) [ cos θ cos ϕ cos θ sin ϕ sin θ ]
sin ϕ exp ( i k · r ) [ sin ϕ cos ϕ 0 ] ]
E tot lin ( r ) = iA π 0 α d θ 0 2 π d ϕ sin θ cos θ [ cos ϕ E tot TM ( r ) sin ϕ E tot TE ( r ) ]
E inc rad ( r ) = iA π 0 α d θ 0 2 π d ϕ sin θ cos θ exp ( i k · r ) [ cos θ cos ϕ cos θ sin ϕ sin θ ] .
E tot rad ( r ) = iA π 0 α d θ 0 2 π d ϕ sin θ cos θ E tot TM ( r )

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