Abstract

A theoretical procedure has been developed for the determination of internal and external electromagnetic fields that result from the interaction of a higher-order Gaussian beam with a homogeneous sphere. Specific calculations are performed for (1, 0), (0, 1), and (1, 1) mode Hermite–Gaussian beams and for doughnut mode beams of four different polarizations (radial, angular, arced, and helix). The effects of incident beam type on resonance excitation, and on the spatial distribution of the internal and near-surface electromagnetic fields, are examined.

© 1997 Optical Society of America

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References

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  1. J. P. Barton, D. R. Alexander, 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]
  2. J. P. Barton, D. R. Alexander, 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]
  3. J. P. Barton, D. R. Alexander, 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]
  4. J. P. Barton, D. R. Alexander, “Electromagnetic field calculations for a tightly-focused laser beam incident upon a microdroplet: applications to nonlinear optics,” in Nonlinear Optics and Materials, C. D. Cantrell, C. M. Bowden, eds., Proc. SPIE1497, 64–77 (1991).
  5. G. Gouesbet, B. Maheu, G. Grehan, “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]
  6. J. A. Lock, G. Gouesbet, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz–Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
    [Crossref]
  7. G. Gouesbet, J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz–Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516–2525 (1994).
    [Crossref]
  8. E. E. M. Khaled, S. C. Hill, P. W. Barber, D. Q. Chowdhury, “Near-resonance excitation of dielectric spheres with plane waves and off-axis Gaussian beams,” Appl. Opt. 31, 1166–1169 (1992).
    [Crossref] [PubMed]
  9. E. E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
    [Crossref]
  10. J. S. Kim, 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]
  11. S. Chang, S. S. Lee, “Optical torque exerted on a homogeneous sphere levitated in the circularly polarized fundamental-mode laser beam,” J. Opt. Soc. Am. B 2, 1853–1860 (1985).
    [Crossref]
  12. S. Chang, “Internal electromagnetic energy within a dielectric sphere in a plane-polarized TEM00 laser beam,” J. Opt. Soc. Am. B 6, 1332–1338 (1989).
    [Crossref]
  13. G. Roosen, C. Imbert, “The TEM01* mode laser beam—a powerful tool for optical levitation of various types of spheres,” Opt. Comm. 26, 432–436 (1978).
    [Crossref]
  14. S. Chang, S. S. Lee, “Radiation force and torque exerted on a stratified sphere in the circularly polarized TEM01*-mode beam,” J. Opt. Soc. Am. B 5, 61–66 (1988).
    [Crossref]
  15. J. P. Barton, W. Ma, S. A. Schaub, D. R. Alexander, “Electromagnetic field for a beam incident on two adjacent spherical particles,” Appl. Opt. 30, 4706–4715 (1991).
    [Crossref] [PubMed]
  16. E. Almaas, I. Brevik, “Radiation forces on a micrometer-sized sphere in an evanescent field,” J. Opt. Soc. Am. B 12, 2429–2438 (1995).
    [Crossref]
  17. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [Crossref]
  18. J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
    [Crossref]
  19. E. Zauderer, “Complex argument Hermite–Gaussian and Laguerre–Gaussian beams,” J. Opt. Soc. Am. A 3, 465–469 (1986).
    [Crossref]
  20. A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), pp. 330–331.
  21. H. Kogelnik, T. Li, “Laser beam resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [Crossref]
  22. C. Tamm, C. O. Weiss, “Bistability and optical switching of spatial patterns in a laser,” J. Opt. Soc. Am. B 7, 1034–1038 (1990).
    [Crossref]
  23. C. P. Smith, Y. Dihardja, C. O. Weiss, L. A. Lugiato, F. Prati, P. Vanotti, “Low energy switching of laser doughnut modes and pattern recognition,” Opt. Comm. 102, 505–514 (1993).
    [Crossref]
  24. L. D. Landau, E. M. Lifshitz, Electromagnetics of Continuous Media (Pergamon, Oxford, 1960), pp. 253–256.
  25. J. P. Barton, D. R. Alexander, “Electromagnetic fields for an irregularly shaped, near-spherical particle illuminated by a focused laser beam,” Appl. Opt. 69, 7973–7986 (1991).
  26. J. P. Barton, “Electromagnetic-field calculations for irregularly shaped, axisymmetric layered particles with focused illumination,” Appl. Opt. 35, 532–541 (1996).
    [Crossref] [PubMed]
  27. J. P. Barton, “Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,” Appl. Opt. 34, 5542–5551 (1995).
    [Crossref] [PubMed]

1996 (1)

1995 (2)

1994 (2)

1993 (2)

E. E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[Crossref]

C. P. Smith, Y. Dihardja, C. O. Weiss, L. A. Lugiato, F. Prati, P. Vanotti, “Low energy switching of laser doughnut modes and pattern recognition,” Opt. Comm. 102, 505–514 (1993).
[Crossref]

1992 (1)

1991 (2)

J. P. Barton, W. Ma, S. A. Schaub, D. R. Alexander, “Electromagnetic field for a beam incident on two adjacent spherical particles,” Appl. Opt. 30, 4706–4715 (1991).
[Crossref] [PubMed]

J. P. Barton, D. R. Alexander, “Electromagnetic fields for an irregularly shaped, near-spherical particle illuminated by a focused laser beam,” Appl. Opt. 69, 7973–7986 (1991).

1990 (1)

1989 (4)

S. Chang, “Internal electromagnetic energy within a dielectric sphere in a plane-polarized TEM00 laser beam,” J. Opt. Soc. Am. B 6, 1332–1338 (1989).
[Crossref]

J. P. Barton, 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, 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, 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 (3)

1986 (1)

1985 (1)

1983 (1)

1979 (1)

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

1978 (1)

G. Roosen, C. Imbert, “The TEM01* mode laser beam—a powerful tool for optical levitation of various types of spheres,” Opt. Comm. 26, 432–436 (1978).
[Crossref]

1966 (1)

H. Kogelnik, T. Li, “Laser beam resonators,” Proc. IEEE 54, 1312–1329 (1966).
[Crossref]

Alexander, D. R.

J. P. Barton, D. R. Alexander, “Electromagnetic fields for an irregularly shaped, near-spherical particle illuminated by a focused laser beam,” Appl. Opt. 69, 7973–7986 (1991).

J. P. Barton, W. Ma, S. A. Schaub, D. R. Alexander, “Electromagnetic field for a beam incident on two adjacent spherical particles,” Appl. Opt. 30, 4706–4715 (1991).
[Crossref] [PubMed]

J. P. Barton, D. R. Alexander, 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, 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, “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, 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. P. Barton, D. R. Alexander, “Electromagnetic field calculations for a tightly-focused laser beam incident upon a microdroplet: applications to nonlinear optics,” in Nonlinear Optics and Materials, C. D. Cantrell, C. M. Bowden, eds., Proc. SPIE1497, 64–77 (1991).

Almaas, E.

Barber, P. W.

E. E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[Crossref]

E. E. M. Khaled, S. C. Hill, P. W. Barber, D. Q. Chowdhury, “Near-resonance excitation of dielectric spheres with plane waves and off-axis Gaussian beams,” Appl. Opt. 31, 1166–1169 (1992).
[Crossref] [PubMed]

Barton, J. P.

J. P. Barton, “Electromagnetic-field calculations for irregularly shaped, axisymmetric layered particles with focused illumination,” Appl. Opt. 35, 532–541 (1996).
[Crossref] [PubMed]

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, W. Ma, S. A. Schaub, D. R. Alexander, “Electromagnetic field for a beam incident on two adjacent spherical particles,” Appl. Opt. 30, 4706–4715 (1991).
[Crossref] [PubMed]

J. P. Barton, D. R. Alexander, “Electromagnetic fields for an irregularly shaped, near-spherical particle illuminated by a focused laser beam,” Appl. Opt. 69, 7973–7986 (1991).

J. P. Barton, D. R. Alexander, 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, 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, “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, 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. P. Barton, D. R. Alexander, “Electromagnetic field calculations for a tightly-focused laser beam incident upon a microdroplet: applications to nonlinear optics,” in Nonlinear Optics and Materials, C. D. Cantrell, C. M. Bowden, eds., Proc. SPIE1497, 64–77 (1991).

Brevik, I.

Chang, S.

Chowdhury, D. Q.

Davis, L. W.

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

Dihardja, Y.

C. P. Smith, Y. Dihardja, C. O. Weiss, L. A. Lugiato, F. Prati, P. Vanotti, “Low energy switching of laser doughnut modes and pattern recognition,” Opt. Comm. 102, 505–514 (1993).
[Crossref]

Gouesbet, G.

Grehan, G.

Hill, S. C.

E. E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[Crossref]

E. E. M. Khaled, S. C. Hill, P. W. Barber, D. Q. Chowdhury, “Near-resonance excitation of dielectric spheres with plane waves and off-axis Gaussian beams,” Appl. Opt. 31, 1166–1169 (1992).
[Crossref] [PubMed]

Imbert, C.

G. Roosen, C. Imbert, “The TEM01* mode laser beam—a powerful tool for optical levitation of various types of spheres,” Opt. Comm. 26, 432–436 (1978).
[Crossref]

Khaled, E. E. M.

E. E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[Crossref]

E. E. M. Khaled, S. C. Hill, P. W. Barber, D. Q. Chowdhury, “Near-resonance excitation of dielectric spheres with plane waves and off-axis Gaussian beams,” Appl. Opt. 31, 1166–1169 (1992).
[Crossref] [PubMed]

Kim, J. S.

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beam resonators,” Proc. IEEE 54, 1312–1329 (1966).
[Crossref]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Electromagnetics of Continuous Media (Pergamon, Oxford, 1960), pp. 253–256.

Lee, S. S.

Li, T.

H. Kogelnik, T. Li, “Laser beam resonators,” Proc. IEEE 54, 1312–1329 (1966).
[Crossref]

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Electromagnetics of Continuous Media (Pergamon, Oxford, 1960), pp. 253–256.

Lock, J. A.

Lugiato, L. A.

C. P. Smith, Y. Dihardja, C. O. Weiss, L. A. Lugiato, F. Prati, P. Vanotti, “Low energy switching of laser doughnut modes and pattern recognition,” Opt. Comm. 102, 505–514 (1993).
[Crossref]

Ma, W.

Maheu, B.

Prati, F.

C. P. Smith, Y. Dihardja, C. O. Weiss, L. A. Lugiato, F. Prati, P. Vanotti, “Low energy switching of laser doughnut modes and pattern recognition,” Opt. Comm. 102, 505–514 (1993).
[Crossref]

Roosen, G.

G. Roosen, C. Imbert, “The TEM01* mode laser beam—a powerful tool for optical levitation of various types of spheres,” Opt. Comm. 26, 432–436 (1978).
[Crossref]

Schaub, S. A.

J. P. Barton, W. Ma, S. A. Schaub, D. R. Alexander, “Electromagnetic field for a beam incident on two adjacent spherical particles,” Appl. Opt. 30, 4706–4715 (1991).
[Crossref] [PubMed]

J. P. Barton, D. R. Alexander, 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, 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, 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]

Siegman, A. E.

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), pp. 330–331.

Smith, C. P.

C. P. Smith, Y. Dihardja, C. O. Weiss, L. A. Lugiato, F. Prati, P. Vanotti, “Low energy switching of laser doughnut modes and pattern recognition,” Opt. Comm. 102, 505–514 (1993).
[Crossref]

Tamm, C.

Vanotti, P.

C. P. Smith, Y. Dihardja, C. O. Weiss, L. A. Lugiato, F. Prati, P. Vanotti, “Low energy switching of laser doughnut modes and pattern recognition,” Opt. Comm. 102, 505–514 (1993).
[Crossref]

Weiss, C. O.

C. P. Smith, Y. Dihardja, C. O. Weiss, L. A. Lugiato, F. Prati, P. Vanotti, “Low energy switching of laser doughnut modes and pattern recognition,” Opt. Comm. 102, 505–514 (1993).
[Crossref]

C. Tamm, C. O. Weiss, “Bistability and optical switching of spatial patterns in a laser,” J. Opt. Soc. Am. B 7, 1034–1038 (1990).
[Crossref]

Zauderer, E.

Appl. Opt. (5)

IEEE Trans. Antennas Propag. (1)

E. E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[Crossref]

J. Appl. Phys. (4)

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

J. Opt. Soc. Am. (1)

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

J. Opt. Soc. Am. B (5)

Opt. Comm. (2)

C. P. Smith, Y. Dihardja, C. O. Weiss, L. A. Lugiato, F. Prati, P. Vanotti, “Low energy switching of laser doughnut modes and pattern recognition,” Opt. Comm. 102, 505–514 (1993).
[Crossref]

G. Roosen, C. Imbert, “The TEM01* mode laser beam—a powerful tool for optical levitation of various types of spheres,” Opt. Comm. 26, 432–436 (1978).
[Crossref]

Phys. Rev. A (1)

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

Proc. IEEE (1)

H. Kogelnik, T. Li, “Laser beam resonators,” Proc. IEEE 54, 1312–1329 (1966).
[Crossref]

Other (3)

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), pp. 330–331.

J. P. Barton, D. R. Alexander, “Electromagnetic field calculations for a tightly-focused laser beam incident upon a microdroplet: applications to nonlinear optics,” in Nonlinear Optics and Materials, C. D. Cantrell, C. M. Bowden, eds., Proc. SPIE1497, 64–77 (1991).

L. D. Landau, E. M. Lifshitz, Electromagnetics of Continuous Media (Pergamon, Oxford, 1960), pp. 253–256.

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Figures (13)

Fig. 1
Fig. 1

Intensity distribution at the focal point of a TEM00 (Gaussian) mode beam. The beam waist radius-to-wavelength ratio (w0/λ) is 6.10.

Fig. 2
Fig. 2

Intensity distribution at the focal point of a TEM01 mode beam. The beam waist radius-to-wavelength ratio (w0/λ) is 6.10.

Fig. 3
Fig. 3

Intensity distribution at the focal point of a TEM11 mode beam. The beam waist radius-to-wavelength ratio (w0/λ) is 6.10.

Fig. 4
Fig. 4

Intensity distribution at the focal point of a TEMdn (doughnut) mode beam. The beam waist radius-to-wavelength ratio (w0/λ) is 6.10.

Fig. 5
Fig. 5

Internal electromagnetic energy versus size parameter for the linearly polarized, plane-wave illumination of a sphere with relative index of refraction = 1.33.

Fig. 6
Fig. 6

Source function distribution in the x and y plane for a linearly polarized Gaussian-focused beam (TEM00x mode) incident upon a sphere at nonresonance. Size parameter α = 38.31549, relative index of refraction = 1.33, on-center focusing (0 = 0 = 0 = 0), and beam waist radius 0 = 1.414.

Fig. 7
Fig. 7

Source function distribution in the x and y plane for a TEM01x beam incident upon a sphere at nonresonance. α = 38.31549, ñ = 1.33 (0 = 0 = 0 = 0), and 0 = 1.414.

Fig. 8
Fig. 8

Source function distribution in the x and y plane for a TEM11x beam incident upon a sphere at nonresonance. α = 38.31549, = 1.33 (0 = 0 = 0 = 0), and 0 = 1.000.

Fig. 9
Fig. 9

Gray-level visualization (white is high, black is low, the circular sphere boundary is shown in gray) of source function distribution in the x and y plane for a TEM11x beam incident upon a sphere at nonresonance. α = 38.31549, = 1.33 (0 = 0 = 0 = 0), and 0 = 1.000.

Fig. 10
Fig. 10

Source function distribution in the x and y plane for a TEMdnlhel (helix doughnut) beam incident upon a sphere at nonresonance. α = 38.31549, = 1.33 (0 = 0 = 0 = 0), and 0 = 1.414.

Fig. 11
Fig. 11

Source function distribution in the x and y plane for a TEMdnlhel TEMdn (l hel) (helix doughnut) beam incident upon a sphere at electric wave resonance. α = 38.52481, = 1.33 (0 = 0 = 0 = 0), and 0 = 1.414.

Fig. 12
Fig. 12

Source function distribution in the x and y plane for a TEM11x beam incident upon a sphere at nonresonance with off-center focusing. α = 38.31549, = 1.33 (0 = 0.5, 0 = 0 = 0), and 0 = 1.000.

Fig. 13
Fig. 13

Gray-level visualization (white is high, black is low, the circular sphere boundary is shown in gray) of source function distribution in the x and y plane for a TEM11x beam incident upon a sphere at nonresonance with off-center focusing. α = 38.31549, = 1.33 (0 = 0.5, 0 = 0 = 0), and 0 = 1.000.

Tables (1)

Tables Icon

Table 1 Internal Electromagnetic Energy for Magnetic Wave Resonance, Nonresonance, and Electric Wave Resonance Conditions as a Function of Beam Typea

Equations (31)

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

TEMmnx=mnTEM00xξmηn,
TEM10x=TEM00xξ,
TEM01x=TEM00xη,
TEM11x=2TEM00xξη.
TEM00lcir=TEM00x+iTEM00y/2.
TEMdnrad=TEM10x+TEM01y/2
TEMdnang=TEM01x-TEM10y/2,
TEMdnarc=TEM10x-TEM01y/2,
TEMdnlhel=TEM10x+iTEM01x/2.
alm=ψln¯αψlα-n¯ψln¯αψlαn¯ψln¯αξl1α-ψln¯αξl1αAlm,
blm=n¯ψln¯αψlα-ψln¯αψlαψln¯αξl1α-n¯ψln¯αξl1αBlm,
clm=in¯2ψln¯αξl1α-n¯ψln¯αξl1α Alm,
dlm=iψln¯αξl1α-n¯ψln¯αξl1α Blm.
Alm=1ll+1ψlα02π0πEria, θ, ϕ×Ylm*θ, ϕsin θdθdϕ,
Blm=1ll+1ψlα02π0πHria, θ, ϕ×Ylm*θ, ϕsin θdθdϕ,
U=116πRe¯E2+H2.
U=116π010π02πReextn¯2E2+H2r˜2 sin θdϕdθdr˜.
U=116πα2nRnIl=1m=-llll+1extn¯22clm2+Ren¯2dlm2Imn¯ψl+1n¯αψl*n¯α+n¯22l+1Ren¯2extclm2+dlm2×Iml+1n¯ψln¯αψl-*n¯α+ln¯ψl+2n¯αψl+1*n¯α.
U=116πn2α22l=1m=-llll+1extn2clm2+dlm2×ψl2nα-ψl-1nα+12l+1×l+1ψl-12nα+ψl2nα-2l-1nα ψl-1nαψlnα+lψl+12-ψlnαψl+2nα.
Ex=E01+s2-ρ2Q2+iρ4Q3-2Q2ξ2ψ0 exp-iζ/s2,
Ey=E0s2-2Q2ξηψ0 exp-iζ/s2,
Ez=E0s-2Qξ+s3+6ρ2Q3-2iρ4Q4ξψ0 exp-iζ/s2,
Hx=E0s2-2Q2ξηψ0 exp-iζ/s2,
Hy=E01+s2-ρ2Q2+iρ4Q3-2Q2η2ψ0 exp-iζ/s2,
Hz=E0s-2Qη+s3+6ρ2Q3-2iρ4Q4ηψ0 exp-iζ/s2.
Ex=E0s2-2Q2ξηψ0 exp-iζ/s2,
Ey=E01+s2-ρ2Q2+iρ4Q3-2Q2η2ψ0 exp-iζ/s2,
Ez=E0s-2Qη+s3+6ρ2Q3-2iρ4Q4ηψ0 exp-iζ/s2,
Hy=-E0s2-2Q2ξηψ0 exp-iζ/s2,
Hx=-E01+s2-ρ2Q2+iρ4Q3-2Q2ξ2ψ0 exp-iζ/s2,
Hz=-E0s-2Qξ+s3+6ρ2Q3-2iρ4Q4ξψ0 exp-iζ/s2.

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