Abstract

In terms of the Fourier spectrum, a simple but general analytical expression is given for the evanescent field associated to a certain kind of non-paraxial exact solutions of the Maxwell equations. This expression enables one to compare the relative weight of the evanescent wave with regard to the propagating field. In addition, in those cases in which the evanescent term is significant, the magnitude of the field components across the transverse profile (including the evanescent features) can be determined. These results are applied to some illustrative examples.

© 2008 Optical Society of America

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References

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  1. G. P. Agrawal and M. Lax, "Free-space wave propagation beyond the paraxial approximation," Phys. Rev. A 27, 1693-1695 (1983).
    [CrossRef]
  2. D. G. Hall, "Vector-beam solutions of Maxwell’s wave equations," Opt. Lett. 21, 9-11 (1996).
    [CrossRef] [PubMed]
  3. P. Varga and P. Török, "Exact and approximate solutions of Maxwell’s equations for a confocal cavity," Opt. Lett. 21, 1523-1525 (1996).
    [CrossRef] [PubMed]
  4. A. Doicu and T. Wriedt, "Plane wave spectrum of electromagnetic beams," Opt. Commun. 136, 114-124 (1997)
    [CrossRef]
  5. S. R. Seshadri, "Electromagnetic Gaussian beam," J. Opt. Soc. Am. A 15, 2712-2719 (1998).
    [CrossRef]
  6. P. Varga and P. Török, "The Gaussian wave solution of Maxvell’s equations and the validity of the scalar wave approximation," Opt. Commun. 152, 108-118 (1998).
    [CrossRef]
  7. S. R. Seshadri, "Partially coherent Gaussian Schell-model electromagnetic beam," J. Opt. Soc. Am. A 16, 1373-1380 (1999).
    [CrossRef]
  8. T. Setala, A. T. Friberg, and M. Kaivola, "Decomposition of the point-dipole field into homogeneous and evanescent parts," Phys. Rev. E 59, 1200-1206 (1999).
    [CrossRef]
  9. A. V. Shchegrov and P. S. Carney, "Far-field contribution to the electromagnetic Green’s tensor from evanescent modes," J. Opt. Soc. Am. A,  16, 2583-2584 (1999).
    [CrossRef]
  10. C. J. R. Sheppard and S. Saghafi, "Electromagnetic Gaussian beams beyond the paraxial approximation," J. Opt. Soc. Am. A 16, 1381-1386 (1999).
    [CrossRef]
  11. S. R. Seshadri, "Average characteristics of partially coherent electromagnetic beams," J. Opt. Soc. Am. A 17, 780-789 (2000).
    [CrossRef]
  12. C. J. R. Sheppard, "Polarization of almost-planes waves," J. Opt. Soc. Am. A 17, 335-341 (2000).
    [CrossRef]
  13. R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, "Vectorial structure of nonparaxial electromagnetic beams," J. Opt. Soc. Am. A 18, 1678-1680 (2001).
    [CrossRef]
  14. J. Tervo and J. Turunen, "Self-imaging of electromagnetic fields," Opt. Express 9, 622-630 (2001).
    [CrossRef] [PubMed]
  15. P. Pääkkönen, J. Tervo, P. Vahimaa, J. Turunen and F. Gori, General vectorial decomposition of electromagnetic fields with application to propagation-invariant and rotating fields," Opt. Express 10, 949-959 (2002).
    [PubMed]
  16. C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilman and M. L. Schattenburg, "Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximation," J. Opt. Soc. Am. A 19, 404-412 (2002).
    [CrossRef]
  17. R. Borghi, A. Ciattoni and M. Santarsiero, "Exact axial electromagnetic field for vectorial Gaussian and flattened Gaussian boundary distributions," J. Opt. Soc. Am. A 19, 1207-1211 (2002).
    [CrossRef]
  18. A Ciattoni, B. Crosignani and P. Di Porto, "Vectorial analytical description of propagation of a highly nonparaxial beam," Opt. Commun. 202, 17-20 (2002).
    [CrossRef]
  19. P. M. Mejías, R. Martínez-Herrero, G. Piquero and J. Movilla, "Parametric characterization of the spatial structure of non-uniformly polarized laser beams," Prog. Quantum Electron. 26, 65-130 (2002).
    [CrossRef]
  20. H. F. Arnoldus and J. T. Foley, "Traveling and evanescent fields of an electric point dipole," J. Opt. Soc. Am. A 19, 1701-1711 (2002).
    [CrossRef]
  21. N. I. Petrov, "Evanescent and propagating fields of a strongly focused beam, " J. Opt. Soc. Am. A 20, 2385-2389 (2003).
    [CrossRef]
  22. K. Duan and B. Lü, "Polarization properties of vectorial nonparaxial Gaussian beam in the far field," Opt. Lett. 30, 308-310 (2005).
    [CrossRef] [PubMed]
  23. G. Zhou, X. Chu and L. Zhao, "Propagation characteristics of TM Gaussian beam," Opt. Laser Technol. 37, 470-474 (2005).
    [CrossRef]
  24. K. Belkebir, P. C. Chaumet and A. Sentetac, "Influence of multiple scatering on three-dimensional imaging with optical diffraction tomography," J. Opt. Soc. Am. A 23, 586-595 (2006).
    [CrossRef]
  25. R. Martínez-Herrero, P. M. Mejías, S. Bosch and A. Carnicer, "Structure of the transverse profile of Gaussian-model non-paraxial electromagnetic beams," J. Opt. A: Pure Appl. Opt. 8, 524-530 (2006).
    [CrossRef]
  26. P. C. Chaumet, "Fully vectorial highly nonparaxial beam close to the waist," J. Opt. Soc. Am. A 23, 3197-3202 (2006).
    [CrossRef]
  27. H. Guo, J. Chen and S. Zhuang, "Vector plane spectrum of an arbitrary polarized electromagnetic wave," Opt. Express 14, 2095-2100 (2006).
    [CrossRef] [PubMed]
  28. G. C. Sherman, J. J. Stamnes and E. Lalor, "Asymptotic approximations to angular-spectrum representations," J. Math. Phys. 17, 760-776 (1976).
    [CrossRef]

2006 (4)

2005 (2)

K. Duan and B. Lü, "Polarization properties of vectorial nonparaxial Gaussian beam in the far field," Opt. Lett. 30, 308-310 (2005).
[CrossRef] [PubMed]

G. Zhou, X. Chu and L. Zhao, "Propagation characteristics of TM Gaussian beam," Opt. Laser Technol. 37, 470-474 (2005).
[CrossRef]

2003 (1)

2002 (6)

2001 (2)

2000 (2)

1999 (4)

1998 (2)

S. R. Seshadri, "Electromagnetic Gaussian beam," J. Opt. Soc. Am. A 15, 2712-2719 (1998).
[CrossRef]

P. Varga and P. Török, "The Gaussian wave solution of Maxvell’s equations and the validity of the scalar wave approximation," Opt. Commun. 152, 108-118 (1998).
[CrossRef]

1997 (1)

A. Doicu and T. Wriedt, "Plane wave spectrum of electromagnetic beams," Opt. Commun. 136, 114-124 (1997)
[CrossRef]

1996 (2)

1983 (1)

G. P. Agrawal and M. Lax, "Free-space wave propagation beyond the paraxial approximation," Phys. Rev. A 27, 1693-1695 (1983).
[CrossRef]

1976 (1)

G. C. Sherman, J. J. Stamnes and E. Lalor, "Asymptotic approximations to angular-spectrum representations," J. Math. Phys. 17, 760-776 (1976).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal and M. Lax, "Free-space wave propagation beyond the paraxial approximation," Phys. Rev. A 27, 1693-1695 (1983).
[CrossRef]

Arnoldus, H. F.

Belkebir, K.

Borghi, R.

Bosch, S.

R. Martínez-Herrero, P. M. Mejías, S. Bosch and A. Carnicer, "Structure of the transverse profile of Gaussian-model non-paraxial electromagnetic beams," J. Opt. A: Pure Appl. Opt. 8, 524-530 (2006).
[CrossRef]

R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, "Vectorial structure of nonparaxial electromagnetic beams," J. Opt. Soc. Am. A 18, 1678-1680 (2001).
[CrossRef]

Carney, P. S.

Carnicer, A.

R. Martínez-Herrero, P. M. Mejías, S. Bosch and A. Carnicer, "Structure of the transverse profile of Gaussian-model non-paraxial electromagnetic beams," J. Opt. A: Pure Appl. Opt. 8, 524-530 (2006).
[CrossRef]

R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, "Vectorial structure of nonparaxial electromagnetic beams," J. Opt. Soc. Am. A 18, 1678-1680 (2001).
[CrossRef]

Chaumet, P. C.

Chen, C. G.

Chen, J.

Chu, X.

G. Zhou, X. Chu and L. Zhao, "Propagation characteristics of TM Gaussian beam," Opt. Laser Technol. 37, 470-474 (2005).
[CrossRef]

Ciattoni, A

A Ciattoni, B. Crosignani and P. Di Porto, "Vectorial analytical description of propagation of a highly nonparaxial beam," Opt. Commun. 202, 17-20 (2002).
[CrossRef]

Ciattoni, A.

Crosignani, B.

A Ciattoni, B. Crosignani and P. Di Porto, "Vectorial analytical description of propagation of a highly nonparaxial beam," Opt. Commun. 202, 17-20 (2002).
[CrossRef]

Di Porto, P.

A Ciattoni, B. Crosignani and P. Di Porto, "Vectorial analytical description of propagation of a highly nonparaxial beam," Opt. Commun. 202, 17-20 (2002).
[CrossRef]

Doicu, A.

A. Doicu and T. Wriedt, "Plane wave spectrum of electromagnetic beams," Opt. Commun. 136, 114-124 (1997)
[CrossRef]

Duan, K.

Ferrera, J.

Foley, J. T.

Friberg, A. T.

T. Setala, A. T. Friberg, and M. Kaivola, "Decomposition of the point-dipole field into homogeneous and evanescent parts," Phys. Rev. E 59, 1200-1206 (1999).
[CrossRef]

Gori, F.

Guo, H.

Hall, D. G.

Heilman, R. K.

Kaivola, M.

T. Setala, A. T. Friberg, and M. Kaivola, "Decomposition of the point-dipole field into homogeneous and evanescent parts," Phys. Rev. E 59, 1200-1206 (1999).
[CrossRef]

Konkola, P. T.

Lalor, E.

G. C. Sherman, J. J. Stamnes and E. Lalor, "Asymptotic approximations to angular-spectrum representations," J. Math. Phys. 17, 760-776 (1976).
[CrossRef]

Lax, M.

G. P. Agrawal and M. Lax, "Free-space wave propagation beyond the paraxial approximation," Phys. Rev. A 27, 1693-1695 (1983).
[CrossRef]

Lü, B.

Martínez-Herrero, R.

R. Martínez-Herrero, P. M. Mejías, S. Bosch and A. Carnicer, "Structure of the transverse profile of Gaussian-model non-paraxial electromagnetic beams," J. Opt. A: Pure Appl. Opt. 8, 524-530 (2006).
[CrossRef]

P. M. Mejías, R. Martínez-Herrero, G. Piquero and J. Movilla, "Parametric characterization of the spatial structure of non-uniformly polarized laser beams," Prog. Quantum Electron. 26, 65-130 (2002).
[CrossRef]

R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, "Vectorial structure of nonparaxial electromagnetic beams," J. Opt. Soc. Am. A 18, 1678-1680 (2001).
[CrossRef]

Mejías, P. M.

R. Martínez-Herrero, P. M. Mejías, S. Bosch and A. Carnicer, "Structure of the transverse profile of Gaussian-model non-paraxial electromagnetic beams," J. Opt. A: Pure Appl. Opt. 8, 524-530 (2006).
[CrossRef]

P. M. Mejías, R. Martínez-Herrero, G. Piquero and J. Movilla, "Parametric characterization of the spatial structure of non-uniformly polarized laser beams," Prog. Quantum Electron. 26, 65-130 (2002).
[CrossRef]

R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, "Vectorial structure of nonparaxial electromagnetic beams," J. Opt. Soc. Am. A 18, 1678-1680 (2001).
[CrossRef]

Movilla, J.

P. M. Mejías, R. Martínez-Herrero, G. Piquero and J. Movilla, "Parametric characterization of the spatial structure of non-uniformly polarized laser beams," Prog. Quantum Electron. 26, 65-130 (2002).
[CrossRef]

Pääkkönen, P.

Petrov, N. I.

Piquero, G.

P. M. Mejías, R. Martínez-Herrero, G. Piquero and J. Movilla, "Parametric characterization of the spatial structure of non-uniformly polarized laser beams," Prog. Quantum Electron. 26, 65-130 (2002).
[CrossRef]

Saghafi, S.

Santarsiero, M.

Schattenburg, M. L.

Sentetac, A.

Seshadri, S. R.

Setala, T.

T. Setala, A. T. Friberg, and M. Kaivola, "Decomposition of the point-dipole field into homogeneous and evanescent parts," Phys. Rev. E 59, 1200-1206 (1999).
[CrossRef]

Shchegrov, A. V.

Sheppard, C. J. R.

Sherman, G. C.

G. C. Sherman, J. J. Stamnes and E. Lalor, "Asymptotic approximations to angular-spectrum representations," J. Math. Phys. 17, 760-776 (1976).
[CrossRef]

Stamnes, J. J.

G. C. Sherman, J. J. Stamnes and E. Lalor, "Asymptotic approximations to angular-spectrum representations," J. Math. Phys. 17, 760-776 (1976).
[CrossRef]

Tervo, J.

Török, P.

Turunen, J.

Vahimaa, P.

Varga, P.

P. Varga and P. Török, "The Gaussian wave solution of Maxvell’s equations and the validity of the scalar wave approximation," Opt. Commun. 152, 108-118 (1998).
[CrossRef]

P. Varga and P. Török, "Exact and approximate solutions of Maxwell’s equations for a confocal cavity," Opt. Lett. 21, 1523-1525 (1996).
[CrossRef] [PubMed]

Wriedt, T.

A. Doicu and T. Wriedt, "Plane wave spectrum of electromagnetic beams," Opt. Commun. 136, 114-124 (1997)
[CrossRef]

Zhao, L.

G. Zhou, X. Chu and L. Zhao, "Propagation characteristics of TM Gaussian beam," Opt. Laser Technol. 37, 470-474 (2005).
[CrossRef]

Zhou, G.

G. Zhou, X. Chu and L. Zhao, "Propagation characteristics of TM Gaussian beam," Opt. Laser Technol. 37, 470-474 (2005).
[CrossRef]

Zhuang, S.

J. Math. Phys. (1)

G. C. Sherman, J. J. Stamnes and E. Lalor, "Asymptotic approximations to angular-spectrum representations," J. Math. Phys. 17, 760-776 (1976).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

R. Martínez-Herrero, P. M. Mejías, S. Bosch and A. Carnicer, "Structure of the transverse profile of Gaussian-model non-paraxial electromagnetic beams," J. Opt. A: Pure Appl. Opt. 8, 524-530 (2006).
[CrossRef]

J. Opt. Soc. Am. A (13)

P. C. Chaumet, "Fully vectorial highly nonparaxial beam close to the waist," J. Opt. Soc. Am. A 23, 3197-3202 (2006).
[CrossRef]

H. F. Arnoldus and J. T. Foley, "Traveling and evanescent fields of an electric point dipole," J. Opt. Soc. Am. A 19, 1701-1711 (2002).
[CrossRef]

N. I. Petrov, "Evanescent and propagating fields of a strongly focused beam, " J. Opt. Soc. Am. A 20, 2385-2389 (2003).
[CrossRef]

K. Belkebir, P. C. Chaumet and A. Sentetac, "Influence of multiple scatering on three-dimensional imaging with optical diffraction tomography," J. Opt. Soc. Am. A 23, 586-595 (2006).
[CrossRef]

S. R. Seshadri, "Electromagnetic Gaussian beam," J. Opt. Soc. Am. A 15, 2712-2719 (1998).
[CrossRef]

S. R. Seshadri, "Partially coherent Gaussian Schell-model electromagnetic beam," J. Opt. Soc. Am. A 16, 1373-1380 (1999).
[CrossRef]

A. V. Shchegrov and P. S. Carney, "Far-field contribution to the electromagnetic Green’s tensor from evanescent modes," J. Opt. Soc. Am. A,  16, 2583-2584 (1999).
[CrossRef]

C. J. R. Sheppard and S. Saghafi, "Electromagnetic Gaussian beams beyond the paraxial approximation," J. Opt. Soc. Am. A 16, 1381-1386 (1999).
[CrossRef]

S. R. Seshadri, "Average characteristics of partially coherent electromagnetic beams," J. Opt. Soc. Am. A 17, 780-789 (2000).
[CrossRef]

C. J. R. Sheppard, "Polarization of almost-planes waves," J. Opt. Soc. Am. A 17, 335-341 (2000).
[CrossRef]

R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, "Vectorial structure of nonparaxial electromagnetic beams," J. Opt. Soc. Am. A 18, 1678-1680 (2001).
[CrossRef]

C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilman and M. L. Schattenburg, "Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximation," J. Opt. Soc. Am. A 19, 404-412 (2002).
[CrossRef]

R. Borghi, A. Ciattoni and M. Santarsiero, "Exact axial electromagnetic field for vectorial Gaussian and flattened Gaussian boundary distributions," J. Opt. Soc. Am. A 19, 1207-1211 (2002).
[CrossRef]

Opt. Commun. (3)

A Ciattoni, B. Crosignani and P. Di Porto, "Vectorial analytical description of propagation of a highly nonparaxial beam," Opt. Commun. 202, 17-20 (2002).
[CrossRef]

P. Varga and P. Török, "The Gaussian wave solution of Maxvell’s equations and the validity of the scalar wave approximation," Opt. Commun. 152, 108-118 (1998).
[CrossRef]

A. Doicu and T. Wriedt, "Plane wave spectrum of electromagnetic beams," Opt. Commun. 136, 114-124 (1997)
[CrossRef]

Opt. Express (3)

Opt. Laser Technol. (1)

G. Zhou, X. Chu and L. Zhao, "Propagation characteristics of TM Gaussian beam," Opt. Laser Technol. 37, 470-474 (2005).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (1)

G. P. Agrawal and M. Lax, "Free-space wave propagation beyond the paraxial approximation," Phys. Rev. A 27, 1693-1695 (1983).
[CrossRef]

Phys. Rev. E (1)

T. Setala, A. T. Friberg, and M. Kaivola, "Decomposition of the point-dipole field into homogeneous and evanescent parts," Phys. Rev. E 59, 1200-1206 (1999).
[CrossRef]

Progr. Quantum Electron. (1)

P. M. Mejías, R. Martínez-Herrero, G. Piquero and J. Movilla, "Parametric characterization of the spatial structure of non-uniformly polarized laser beams," Prog. Quantum Electron. 26, 65-130 (2002).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Ratio Δ in terms of the propagation distance z for the field defined by Eqs. (18), with ω 0=0.1λ (highly non-paraxial case).

Fig. 2.
Fig. 2.

Transverse distributions (pseudocolour) of the squared modulus of the propagating waves for the field given by Eqs. (18). In this figure, ω 0=0.1λ and the field magnitude is computed after propagating a distance z=0.1λ. Abcises and ordinates follow the conventional Cartesian representation, namely, abcises are parallel to the lines of writing and ordinates are orthogonal. (a) plot of (|Ex |2)pr; (b) (|Ey |2)pr; (c) (|Ez |2)pr ; (d) (|Ex |2+|Ey |2+|Ez |2)pr. Integration throughout the transverse plane gives the values (a) 0.69; (b) 0.035; (c) 0.085; (d) 0.81. These values have been normalised with respect to that of Fig. 4.d (see below).

Fig. 3.
Fig. 3.

The same as in Fig. 2 but now referred to the evanescent waves: (a) plot of (|Ex |2)ev; (b) (|Ey |2)ev; (c) (|Ez |2)ev ; (d) (|Ex |2+|Ey |2+|Ez |2)ev. Integration throughout the transverse plane gives the values (a) 0.14; (b) 0.025; (c) 0.025; (d) 0.19.

Fig. 4.
Fig. 4.

The same as in Fig. 2 but now referred to the combined field (propagating + evanescent waves): (a) plot of (|Ex |2)pr+ev; (b) (|Ey |2)pr+ev; (c) (|Ez |2)pr+ev; (d) (|Ex |2+|Ey |2+|Ez |2)pr+ev. Integration throughout the transverse plane gives the values (a) 0.83; (b) 0.06; (c) 0.11; (d) 1.

Fig. 5.
Fig. 5.

The same as in Fig. 2 but now for the closest field associated to a circularly polarized Gaussian beam (see Eq. (20)). Integration over the entire transverse plane has now been normalised with respect to the value of Fig. 7.d (see below). (a) plot of (|Ex |2)pr; (b) (|Ey |2)pr; (c) (|Ez |2)pr; (d) (|Ex |2+|Ey |2+|Ez |2)pr. Integration throughout the transverse plane gives the values (a) 0.362; (b) 0.362; (c) 0.085; (d) 0.81.

Fig. 6.
Fig. 6.

The same as in Fig. 5 but now referred to the evanescent waves: (a) plot of (|Ex |2)ev; (b) (|Ey |2)ev; (c) (|Ez |2)ev; (d) (|Ex |2+|Ey |2+|Ez |2)ev. Integration throughout the transverse plane gives the values (a) 0.082; (b) 0.082; (c) 0.025; (d) 0.19.

Fig. 7.
Fig. 7.

The same as in Fig. 5 but now referred to the combined field (propagating + evanescent waves): (a) plot of (|Ex |2)pr+ev; (b) (|Ey |2)pr+ev; (c) (|Ez |2)pr+ev; (d) (|Ex |2+|Ey |2+|Ez |2)pr+ev. Integration throughout the transverse plane gives the values (a) 0.445; (b) 0.445; (c) 0.11; (d) 1.

Equations (37)

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

× H + ik E = 0 ,
× E ik H = 0 ,
· E = · H = 0 ,
E ( x , y , z ) = E ˜ ( u , ν , z ) exp [ i k ( xu + y ν ) ] dudν ,
L × H ˜ + ik E ˜ = 0
L × E ˜ ik H ˜ = 0 ,
L · E ˜ = L · H ˜ = 0 ,
x = R cos θ y = R sin θ
u = ρ cos ϕ ν = ρ sin ϕ .
E ˜ ( ρ , ϕ , z ) = E ˜ 0 ( ρ , ϕ ) exp ( ikz ξ ) ,
E ˜ 0 x σ x + E ˜ 0 y σ y + E ˜ 0 z σ z = 0 ,
H ˜ ( ρ , ϕ , z ) = ( σ × E ˜ 0 ) exp ( ikz ξ ) ,
ξ = 1 ρ 2 ρ 1 ,
ξ = i ρ 2 1 ρ > 1 ,
E ( R , θ , z ) = 0 0 2 π E ˜ 0 ( ρ , ϕ ) exp [ i k ρ R cos ( θ ϕ ) ] exp ( i kz ξ ) ρ d ρ d ϕ ,
E ( R , θ , z ) = E pr ( R , θ , z ) + E ev ( R , θ , z ) ,
E pr ( R , θ , z ) = 0 1 0 2 π E ˜ 0 ( ρ , ϕ ) exp [ i k ρ R cos ( θ ϕ ) ] exp ( i kz ξ ) ρ d ρ d ϕ ,
E ev ( R , θ , z ) = 1 0 2 π E ˜ 0 ( ρ , ϕ ) exp [ i k ρ R cos ( θ ϕ ) ] exp ( i kz ξ ) ρ d ρ d ϕ .
E ( R , θ , z ) = 0 1 0 2 π a ( ρ , ϕ ) e 1 ( ϕ ) exp [ i k s · r ) ] ρ d ρ d ϕ
+ 0 1 0 2 π b ( ρ , ϕ ) e 2 ( ρ , ϕ ) exp [ i k s · r ) ] ρ d ρ d ϕ
+ 1 0 2 π [ a ( ρ , ϕ ) e 1 ( ϕ ) + b ev ( ρ , ϕ ) e ev ( ρ , ϕ ) ] exp [ i k s ev · r ) ] ρ d ρ d ϕ
e 1 = ( sin ϕ , cos ϕ , 0 ) ,
e 2 = ( 1 ρ 2 cos ϕ , 1 ρ 2 sin ϕ , ρ )
s ( ρ , ϕ ) = ( ρ cos ϕ , ρ sin ϕ , 1 ρ 2 ) , with ρ [ 0 , 1 ]
e ev = 1 2 ρ 2 1 ( i ( ρ 2 1 ) 1 2 cos ϕ , i ( ρ 2 1 ) 1 2 sin ϕ , ρ )
S ev = ( ρ cos ϕ , ρ sin ϕ , i ( ρ 2 1 ) 1 2 ) , with ρ [ 1 , ] ,
a ( ρ , ϕ ) = E ˜ 0 · e 1 ,
b ( ρ , ϕ ) = E ˜ 0 · e 2 ,
b ev ( ρ , ϕ ) = E ˜ 0 · e ev
S 0 = ( ρ cos ϕ , ρ sin ϕ , 0 ) ,
f ˜ ( ρ , ϕ ) = C exp ( ρ 2 D 2 ) ( 1 , 0 , 0 ) ,
a ( ρ , ϕ ) = C sin ϕ exp ( ρ 2 D 2 ) ,
b ( ρ , ϕ ) = C cos ϕ 1 ρ 2 exp ( ρ 2 D 2 ) ,
b ev ( ρ , ϕ ) = i 2 ρ 2 1 C cos ϕ ( ρ 2 1 ) 1 2 exp ( ρ 2 D 2 ) .
I pr = 0 1 0 2 π [ a ( ρ , ϕ ) 2 + b ( ρ , ϕ ) 2 ] ρ d ρ d ϕ ,
I ev = 1 0 2 π [ a ( ρ , ϕ ) 2 + b ev ( ρ , ϕ ) 2 ] exp ( 2 kz ρ 2 1 ) ρ d ρ d ϕ .
f ˜ ( ρ , ϕ ) = C exp ( ρ 2 D 2 ) ( 1 , i , 0 ) ,

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