Abstract

The Richards–Wolf theory and the complex point-source method are both used to express the phasor of the electric field of tightly focused beams, but the connection between these two approaches is not straightforward. In this paper, the Richards–Wolf vector field equations are used to find the electromagnetic field of a TM01 beam in the neighborhood of the focus of a 4π focusing system, such as a parabolic mirror with infinite transverse dimensions. Closed-form solutions are found for the distribution of the fields at any point in the vicinity of the focus; these solutions are identical to the electromagnetic field obtained with the complex source-point method in which sources are accompanied by sinks. This work thus establishes a connection between the Richards–Wolf theory and the complex sink/source model. The vector magnetic potential is introduced to simplify the computation of the six electromagnetic field components. The method is then used to find analytical expressions for the electromagnetic field of strongly focused TM01 beams affected by primary aberrations such as curvature of field, coma, astigmatism and spherical aberration.

© 2010 OSA

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References

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  1. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000).
    [Crossref]
  2. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
    [Crossref] [PubMed]
  3. H. Dehez, M. Piché, and Y. De Koninck, “Enhanced resolution in two-photon imaging using a TM(01) laser beam at a dielectric interface,” Opt. Lett. 34(23), 3601–3603 (2009).
    [Crossref] [PubMed]
  4. C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(2), 026603 (2005).
    [Crossref] [PubMed]
  5. B. Richards and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
    [Crossref]
  6. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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  13. G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7(23), 684–685 (1971).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  24. N. Bokor and N. Davidson, “Toward a spherical spot distribution with 4π focusing of radially polarized light,” Opt. Lett. 29(17), 1968–1970 (2004).
    [Crossref] [PubMed]
  25. N. Bokor and N. Davidson, “4π Focusing with single paraboloid mirror,” Opt. Commun. 281(22), 5499–5503 (2008).
    [Crossref]
  26. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19(3), 1177–1179 (1979).
    [Crossref]
  27. C. J. R. Sheppard, “Orthogonal aberration functions for high-aperture optical systems,” J. Opt. Soc. Am. A 21(5), 832–838 (2004).
    [Crossref]
  28. R. Kant, “An analytical method of vector diffraction for focusing optical systems with Seidel aberrations II: Astigmatism and coma,” J. Mod. Opt. 42(2), 299–320 (1995).
    [Crossref]
  29. C. J. R. Sheppard, “Vector diffraction in paraboloidal mirrors with Seidel aberrations: effects of small object displacements,” Opt. Commun. 138(4-6), 262–264 (1997).
    [Crossref]
  30. R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary coma on the focusing of a Laguerre–Gaussian beam by a high numerical aperture system; vectorial diffraction theory,” J. Opt. A, Pure Appl. Opt. 10(9pp), (2008).
    [Crossref]
  31. D. P. Biss and T. G. Brown, “Primary aberrations in focused radially polarized vortex beams,” Opt. Express 12(3), 384–393 (2004).
    [Crossref] [PubMed]

2009 (1)

2008 (5)

J. Stadler, C. Stanciu, C. Stupperich, and A. J. Meixner, “Tighter focusing with a parabolic mirror,” Opt. Lett. 33(7), 681–683 (2008).
[Crossref] [PubMed]

A. April, “Nonparaxial TM and TE beams in free space,” Opt. Lett. 33(14), 1563–1565 (2008).
[Crossref] [PubMed]

A. April, “Nonparaxial elegant Laguerre-Gaussian beams,” Opt. Lett. 33(12), 1392–1394 (2008).
[Crossref] [PubMed]

N. Bokor and N. Davidson, “4π Focusing with single paraboloid mirror,” Opt. Commun. 281(22), 5499–5503 (2008).
[Crossref]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary coma on the focusing of a Laguerre–Gaussian beam by a high numerical aperture system; vectorial diffraction theory,” J. Opt. A, Pure Appl. Opt. 10(9pp), (2008).
[Crossref]

2005 (1)

C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(2), 026603 (2005).
[Crossref] [PubMed]

2004 (4)

2003 (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

2001 (2)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light – theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

M. A. Lieb and A. J. Meixner, “A high numerical aperture parabolic mirror as imaging device for confocal microscopy,” Opt. Express 8(7), 458–474 (2001).
[Crossref] [PubMed]

2000 (4)

1999 (2)

1998 (1)

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: A scalar treatment,” Phys. Rev. A 57(4), 2971–2979 (1998).
[Crossref]

1997 (1)

C. J. R. Sheppard, “Vector diffraction in paraboloidal mirrors with Seidel aberrations: effects of small object displacements,” Opt. Commun. 138(4-6), 262–264 (1997).
[Crossref]

1995 (1)

R. Kant, “An analytical method of vector diffraction for focusing optical systems with Seidel aberrations II: Astigmatism and coma,” J. Mod. Opt. 42(2), 299–320 (1995).
[Crossref]

1994 (1)

C. J. R. Sheppard and K. G. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41(7), 1495–1505 (1994).
[Crossref]

1981 (2)

L. W. Davis and G. Patsakos, “TM and TE electromagnetic beams in free space,” Opt. Lett. 6(1), 22–23 (1981).
[Crossref] [PubMed]

M. Couture and P.-A. Bélanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24(1), 355–359 (1981).
[Crossref]

1979 (1)

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19(3), 1177–1179 (1979).
[Crossref]

1971 (1)

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7(23), 684–685 (1971).
[Crossref]

1959 (1)

B. Richards and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

April, A.

Bélanger, P.-A.

M. Couture and P.-A. Bélanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24(1), 355–359 (1981).
[Crossref]

Biss, D. P.

Bokor, N.

Brown, T. G.

Couture, M.

M. Couture and P.-A. Bélanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24(1), 355–359 (1981).
[Crossref]

Davidson, N.

Davis, L. W.

De Koninck, Y.

Dehez, H.

Deschamps, G. A.

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7(23), 684–685 (1971).
[Crossref]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light – theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000).
[Crossref]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light – theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000).
[Crossref]

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light – theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000).
[Crossref]

Kant, R.

R. Kant, “An analytical method of vector diffraction for focusing optical systems with Seidel aberrations II: Astigmatism and coma,” J. Mod. Opt. 42(2), 299–320 (1995).
[Crossref]

Larkin, K. G.

C. J. R. Sheppard and K. G. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41(7), 1495–1505 (1994).
[Crossref]

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light – theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000).
[Crossref]

Lieb, M. A.

Ludlow, I. K.

Meixner, A. J.

Patsakos, G.

Piché, M.

H. Dehez, M. Piché, and Y. De Koninck, “Enhanced resolution in two-photon imaging using a TM(01) laser beam at a dielectric interface,” Opt. Lett. 34(23), 3601–3603 (2009).
[Crossref] [PubMed]

C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(2), 026603 (2005).
[Crossref] [PubMed]

Porras, M. A.

C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(2), 026603 (2005).
[Crossref] [PubMed]

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light – theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000).
[Crossref]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

Saghafi, S.

Senthilkumaran, P.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary coma on the focusing of a Laguerre–Gaussian beam by a high numerical aperture system; vectorial diffraction theory,” J. Opt. A, Pure Appl. Opt. 10(9pp), (2008).
[Crossref]

Sheppard, C. J. R.

C. J. R. Sheppard, “Orthogonal aberration functions for high-aperture optical systems,” J. Opt. Soc. Am. A 21(5), 832–838 (2004).
[Crossref]

C. J. R. Sheppard and S. Saghafi, “Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,” Opt. Lett. 24(22), 1543–1545 (1999).
[Crossref]

C. J. R. Sheppard and S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 16(6), 1381–1386 (1999).
[Crossref]

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: A scalar treatment,” Phys. Rev. A 57(4), 2971–2979 (1998).
[Crossref]

C. J. R. Sheppard, “Vector diffraction in paraboloidal mirrors with Seidel aberrations: effects of small object displacements,” Opt. Commun. 138(4-6), 262–264 (1997).
[Crossref]

C. J. R. Sheppard and K. G. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41(7), 1495–1505 (1994).
[Crossref]

Singh, K.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary coma on the focusing of a Laguerre–Gaussian beam by a high numerical aperture system; vectorial diffraction theory,” J. Opt. A, Pure Appl. Opt. 10(9pp), (2008).
[Crossref]

Singh, R. K.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary coma on the focusing of a Laguerre–Gaussian beam by a high numerical aperture system; vectorial diffraction theory,” J. Opt. A, Pure Appl. Opt. 10(9pp), (2008).
[Crossref]

Stadler, J.

Stanciu, C.

Stupperich, C.

Török, P.

Ulanowski, Z.

Varga, P.

Varin, C.

C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(2), 026603 (2005).
[Crossref] [PubMed]

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

Youngworth, K. S.

Appl. Phys. B (1)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light – theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

Electron. Lett. (1)

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7(23), 684–685 (1971).
[Crossref]

J. Mod. Opt. (2)

C. J. R. Sheppard and K. G. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41(7), 1495–1505 (1994).
[Crossref]

R. Kant, “An analytical method of vector diffraction for focusing optical systems with Seidel aberrations II: Astigmatism and coma,” J. Mod. Opt. 42(2), 299–320 (1995).
[Crossref]

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

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary coma on the focusing of a Laguerre–Gaussian beam by a high numerical aperture system; vectorial diffraction theory,” J. Opt. A, Pure Appl. Opt. 10(9pp), (2008).
[Crossref]

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

Opt. Commun. (3)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000).
[Crossref]

C. J. R. Sheppard, “Vector diffraction in paraboloidal mirrors with Seidel aberrations: effects of small object displacements,” Opt. Commun. 138(4-6), 262–264 (1997).
[Crossref]

N. Bokor and N. Davidson, “4π Focusing with single paraboloid mirror,” Opt. Commun. 281(22), 5499–5503 (2008).
[Crossref]

Opt. Express (3)

Opt. Lett. (9)

Phys. Rev. A (3)

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19(3), 1177–1179 (1979).
[Crossref]

M. Couture and P.-A. Bélanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24(1), 355–359 (1981).
[Crossref]

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: A scalar treatment,” Phys. Rev. A 57(4), 2971–2979 (1998).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(2), 026603 (2005).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Proc. R. Soc. Lond. A Math. Phys. Sci. (1)

B. Richards and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

Other (2)

L. Novotny, and B. Hecht, Principles of nano-optics, (Cambridge University Press, 2006, Chap. 3).

G. Arfken, Mathematical Methods for Physicists, 3rd ed., (Oxford, Ohio, Academic Press, Inc., 1985).

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Figures (4)

Fig. 1
Fig. 1

The parabolic mirror of very short focal length f is an example of a 4π focusing system that can be used to generate tightly focused beams.

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Fig. 2
Fig. 2

The angular amplitude distribution given by Eq. (6) characterizes a doughnut shape beam in the far-field. The amplitude profile has a significant value on a broader range of α as k a = 2 f 2 / W o 2 decreases.

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Fig. 3
Fig. 3

The point P is located with the coordinates ( r , ϕ ) or ( r ′ , ϕ ′ ) . The second reference system is translated horizontally with respect to the first by an amount C 1 , 1 .

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Fig. 4
Fig. 4

Electric energy density distributions of a TM01 beam for ka = 1 (a) without aberration, affected (b) by curvature of field with C 2,0 = 2λ, (c) by spherical aberration with C 4,0 = 3 λ, (d) by primary coma with C 3,1 = 2 λ, (e) and by astigmatism with C 2,2 = λ.

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Tables (1)

Tables Icon

Table 1 Aberration function for each primary aberration.

View Table

Equations (33)

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E = E o [ j a ^ x V ˜ 0 , 1 e + j a ^ y V ˜ 0 , 1 o + a ^ z U ˜ 1 , 0 e ] ,
H = j H o [ − a ^ x U ˜ 0 , 1 o + a ^ y U ˜ 0 , 1 e ] ,
U ˜ p , m e = ( 2 p ) ! ! ∑ s = 0 p ( p + m s + m ) ( 4 s + 2 m + 1 ) ( 2 s − 1 ) ! ! ( 2 p + 2 s + 2 m + 1 ) ! ! ψ ˜ 2 s + m , m e ,
V ˜ p , m e = − j ( 2 p ) ! ! ∑ s = 0 p ( p + m s + m ) ( 4 s + 2 m + 3 ) ( 2 s + 1 ) ! ! ( 2 p + 2 s + 2 m + 3 ) ! ! ψ ˜ 2 s + m + 1 , m e ,
ψ ˜ n , m e = exp ( − k a ) j n ( k R ˜ ) P n m ( cos θ ˜ ) cos ( m ϕ ) ,
[ E x E y E z ] = E o 4 π ∫ 0 2 π ∫ 0 α max q ( α ) l 0 ( α ) [ cos α cos β cos α sin β sin α ] exp [ − j k Γ ( α , β ) ] sin α d α d β ,
Γ ( α , β ) ≡ Φ ( α , β ) − r sin α cos ( ϕ − β ) + z cos α ,
∫ 0 2 π exp [ j k r sin α cos ( ϕ − β ) ] cos ( m β ) d β = 2 π j m J m ( k r sin α ) cos ( m ϕ ) ,
[ E x E y E z ] = 1 2 E o ∫ 0 α max q ( α ) l 0 ( α ) exp ( − j k z cos α ) [ j cos α cos ϕ J 1 ( k r sin α ) j cos α sin ϕ J 1 ( k r sin α ) sin α J 0 ( k r sin α ) ] sin α d α .
q ( α ) l 0 ( α ) = sin α exp [ − 4 f 2 W o 2 sin 2 ( 1 2 α ) ] = sin α exp [ − 2 f 2 W o 2 ( 1 − cos α ) ] ,
[ E x E y E z ] = 1 2 E o exp ( − k a ) ∫ 0 π sin 2 α exp ( − j k z ˜ cos α ) [ j cos α cos ϕ J 1 ( k r sin α ) j cos α sin ϕ J 1 ( k r sin α ) sin α J 0 ( k r sin α ) ] d α ,
U ˜ p , m e = 1 2 exp ( − k a ) cos ( m ϕ ) ∫ 0 π sin 2 p + m + 1 α exp ( − j k z ˜ cos α ) J m ( k r sin α ) d α ,
V ˜ p , m e = 1 2 exp ( − k a ) cos ( m ϕ ) ∫ 0 π sin 2 p + m + 1 α cos α exp ( − j k z ˜ cos α ) J m ( k r sin α ) d α ,
H = 1 μ 0 ∇ × A .
E = − j ω k 2 ∇ × ∇ × A .
A = 1 4 π ∫ 0 2 π ∫ 0 α max A o ( α ) exp [ − j k Γ ( α , β ) ] sin α d α d β ,
∇ × ∇ × [ a ^ z exp ( − j k Γ ) ] = k 2 sin α ( a ^ x cos α cos β + a ^ y cos α sin β + a ^ z sin α ) exp ( − j k Γ ) ,
A o ( α ) = a z A o exp [ − k a ( 1 − cos α ) ] ,
A = 1 2 a ^ z A o exp ( − k a ) ∫ 0 π exp ( − j k z ˜ cos α ) J 0 ( k r sin α ) sin α d α ,
∇ × ( a ^ z U ˜ 0 , 0 e ) = − k a ^ x U ˜ 0 , 1 o + k a ^ y U ˜ 0 , 1 e ,
∇ × ∇ × (   a ^ z U ˜ 0 , 0 e ) = j k 2 a ^ x V ˜ 0 , 1 e + j k 2 a ^ y V ˜ 0 , 1 o + k 2 a ^ z U ˜ 1 , 0 e ,
Φ ( α , β ) = ∑ n = 0 ∞ ∑ m = 0 ∞ C n , m sin n α cos ( m β ) ,
Γ ( α , β ) = C 1 , 1 sin α cos β − r sin α cos ( ϕ − β ) + z cos α = − ( r cos ϕ − C 1 , 1 ) sin α cos β − r sin ϕ sin α sin β + z cos α = − r ′ sin α cos ( ϕ ′ − β ) + z cos α ,
exp ( − j k C n , 0 sin n α ) = ∑ s = 0 ∞ ( − j k C n , 0 ) s s ! sin n s α .
A = a ^ z A o 4 π exp ( − k a ) ∑ s = 0 ∞ ( − j k C n , 0 ) s s ! × ∫ 0 2 π ∫ 0 π sin n s + 1 α exp [ j k r sin α cos ( ϕ − β ) − j k z ˜ cos α ] d α d β ,
A = 1 2 a ^ z A o exp ( − k a ) ∑ s = 0 ∞ ( − j k C n , 0 ) s s ! ∫ 0 π sin n s + 1 α exp ( − j k z ˜ cos α ) J 0 ( k r sin α ) d α .
A = A o ∑ s = 0 ∞ ( − j k C n , 0 ) s s ! a ^ z U ˜ n s / 2 , 0 e .
exp [ − j ζ cos ( m β ) ] = J 0 ( ζ ) + 2 ∑ q = 1 ∞ ( − j ) q J q ( ζ ) cos ( q m β ) ,
J q ( ζ ) = ∑ s = 0 ∞ ( − 1 ) s ( k C n , m sin n α ) 2 s + q ( 2 s ) ! ! ( 2 s + 2 q ) ! ! .
A = a ^ z A o 4 π exp ( − k a ) ∑ s = 0 ∞ ( − j k C n , m ) 2 s ( 2 s ) ! ! ∫ 0 2 π ∫ 0 π sin α exp ( − j k z ˜ cos α ) [ sin 2 n s α ( 2 s ) ! ! + 2 ∑ q = 1 ∞ ( − j k C n , m ) q sin 2 n s + n q α ( 2 s + 2 q ) ! ! cos ( q m β ) ] exp [ j k r sin α cos ( θ − β ) ] d α d β .
A = 2 A o ∑ s = 0 ∞ ∑ q = 0 ∞ j m q ( − j k C n , m ) 2 s + q ( 2 s ) ! ! ( 2 s + 2 q ) ! ! ( 1 + δ q , 0 ) a ^ z U ˜ n s + ( n − m ) q / 2 ,   q m e ,
∇ × ( a ^ z U ˜ p , m e   ) = − 1 2 k a ^ x ( U ˜ p , m + 1 o + U ˜ p + 1 , m − 1 o ) + 1 2 k a ^ y ( U ˜ p , m + 1 e − U ˜ p + 1 , m − 1 e )
∇ × ∇ × (   a ^ z U ˜ p , m e ) = j 1 2 k 2 a ^ x ( V ˜ p , m + 1 e − V ˜ p + 1 , m − 1 e ) + j 1 2 k 2 a ^ y ( V ˜ p , m + 1 o + V ˜ p + 1 , m − 1 o ) + k 2 a ^ z U ˜ p + 1 , m e

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