Rotating scale-invariant electromagnetic fields

Jani Tervo; Jari Turunen

doi:10.1364/OE.9.000009

Abstract

The concept of scalar fields with uniformly rotating intensity distributions and propagation-invariant radial scales is extended to the case of electromagnetic fields with rotating but otherwise propagation-invariant states of polarization. It is shown that the conditions for field rotation are different for scalar and electromagnetic fields and that the electromagentic analysis brings in new aspects such as the possibility that different components of a rotating electromagnetic field can rotate in opposite directions.

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References

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|
Year
|
Author
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Publication

W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57772–778 (1967).
[Crossref]
W. D. Montgomery, “Algebraic formulation of diffraction applied to self imaging,” J. Opt. Soc. Am. 581112–1124 (1968).
[Crossref]
J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4651–654 (1987).
[Crossref]
J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 581499–1501 (1987).
[Crossref] [PubMed]
S. Chávez-Cerda, G. S. McDonald, and G. H. S. New, “Nondiffracting Beams: travelling, standing, rotating and spiral waves,” Opt. Commun. 123225–233 (1996).
[Crossref]
C. Paterson and R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124121–130 (1996).
[Crossref]
V. V. Kotlyar, V. A. Soifer, and S. N. Khonina, “An algorithm for the generation of laser-beams with longitudinal periodicity,” J. Mod. Opt. 441409–1416 (1997).
[Crossref]
P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soifer, and A. T. Friberg, “Rotating optical fields: experimental demonstration with diffractive optics,” J. Mod. Opt. 462355–2369 (1998).
[Crossref]
S. N. Khonina, S. N., V. V. Kotlyar, V. A. Soifer, J. Lautanen, M. Honkanen, and J. Turunen, “Generating a couple of rotating nondiffracting beams using a binary-phase DOE,” Optik 110137–144 (1999).
S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85159–161 (1991).
[Crossref]
J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 251–60 (1993).
[Crossref]
Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 81555–1566 (1995).
[Crossref]
Z. Bouchal, R. Horák, and J. Wagner, “Propagation-invariant electromagnetic fields,” J. Mod. Opt. 91905–1920 (1996).
[Crossref]
L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995), sect. 3.2.
G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic Press, New York, 2001), p. 681.
H. F. Talbot, “Facts relating to optical science, No. IV,” Philos. Mag. 9401–407 (1836).

1999 (1)

S. N. Khonina, S. N., V. V. Kotlyar, V. A. Soifer, J. Lautanen, M. Honkanen, and J. Turunen, “Generating a couple of rotating nondiffracting beams using a binary-phase DOE,” Optik 110137–144 (1999).

1998 (1)

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soifer, and A. T. Friberg, “Rotating optical fields: experimental demonstration with diffractive optics,” J. Mod. Opt. 462355–2369 (1998).
[Crossref]

1997 (1)

V. V. Kotlyar, V. A. Soifer, and S. N. Khonina, “An algorithm for the generation of laser-beams with longitudinal periodicity,” J. Mod. Opt. 441409–1416 (1997).
[Crossref]

1996 (3)

S. Chávez-Cerda, G. S. McDonald, and G. H. S. New, “Nondiffracting Beams: travelling, standing, rotating and spiral waves,” Opt. Commun. 123225–233 (1996).
[Crossref]

C. Paterson and R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124121–130 (1996).
[Crossref]

Z. Bouchal, R. Horák, and J. Wagner, “Propagation-invariant electromagnetic fields,” J. Mod. Opt. 91905–1920 (1996).
[Crossref]

1995 (1)

Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 81555–1566 (1995).
[Crossref]

1993 (1)

J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 251–60 (1993).
[Crossref]

1991 (1)

S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85159–161 (1991).
[Crossref]

1987 (2)

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4651–654 (1987).
[Crossref]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 581499–1501 (1987).
[Crossref] [PubMed]

1968 (1)

W. D. Montgomery, “Algebraic formulation of diffraction applied to self imaging,” J. Opt. Soc. Am. 581112–1124 (1968).
[Crossref]

1967 (1)

W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57772–778 (1967).
[Crossref]

1836 (1)

H. F. Talbot, “Facts relating to optical science, No. IV,” Philos. Mag. 9401–407 (1836).

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic Press, New York, 2001), p. 681.

Bouchal, Z.

Z. Bouchal, R. Horák, and J. Wagner, “Propagation-invariant electromagnetic fields,” J. Mod. Opt. 91905–1920 (1996).
[Crossref]

Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 81555–1566 (1995).
[Crossref]

Chávez-Cerda, S.

S. Chávez-Cerda, G. S. McDonald, and G. H. S. New, “Nondiffracting Beams: travelling, standing, rotating and spiral waves,” Opt. Commun. 123225–233 (1996).
[Crossref]

Durnin, J.

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4651–654 (1987).
[Crossref]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 581499–1501 (1987).
[Crossref] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 581499–1501 (1987).
[Crossref] [PubMed]

Friberg, A. T.

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soifer, and A. T. Friberg, “Rotating optical fields: experimental demonstration with diffractive optics,” J. Mod. Opt. 462355–2369 (1998).
[Crossref]

J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 251–60 (1993).
[Crossref]

Honkanen, M.

S. N. Khonina, S. N., V. V. Kotlyar, V. A. Soifer, J. Lautanen, M. Honkanen, and J. Turunen, “Generating a couple of rotating nondiffracting beams using a binary-phase DOE,” Optik 110137–144 (1999).

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soifer, and A. T. Friberg, “Rotating optical fields: experimental demonstration with diffractive optics,” J. Mod. Opt. 462355–2369 (1998).
[Crossref]

Horák, R.

Z. Bouchal, R. Horák, and J. Wagner, “Propagation-invariant electromagnetic fields,” J. Mod. Opt. 91905–1920 (1996).
[Crossref]

Khonina, S. N.

S. N. Khonina, S. N., V. V. Kotlyar, V. A. Soifer, J. Lautanen, M. Honkanen, and J. Turunen, “Generating a couple of rotating nondiffracting beams using a binary-phase DOE,” Optik 110137–144 (1999).

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soifer, and A. T. Friberg, “Rotating optical fields: experimental demonstration with diffractive optics,” J. Mod. Opt. 462355–2369 (1998).
[Crossref]

V. V. Kotlyar, V. A. Soifer, and S. N. Khonina, “An algorithm for the generation of laser-beams with longitudinal periodicity,” J. Mod. Opt. 441409–1416 (1997).
[Crossref]

Kotlyar, V. V.

S. N. Khonina, S. N., V. V. Kotlyar, V. A. Soifer, J. Lautanen, M. Honkanen, and J. Turunen, “Generating a couple of rotating nondiffracting beams using a binary-phase DOE,” Optik 110137–144 (1999).

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soifer, and A. T. Friberg, “Rotating optical fields: experimental demonstration with diffractive optics,” J. Mod. Opt. 462355–2369 (1998).
[Crossref]

V. V. Kotlyar, V. A. Soifer, and S. N. Khonina, “An algorithm for the generation of laser-beams with longitudinal periodicity,” J. Mod. Opt. 441409–1416 (1997).
[Crossref]

Kuittinen, M.

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soifer, and A. T. Friberg, “Rotating optical fields: experimental demonstration with diffractive optics,” J. Mod. Opt. 462355–2369 (1998).
[Crossref]

Lautanen, J.

S. N. Khonina, S. N., V. V. Kotlyar, V. A. Soifer, J. Lautanen, M. Honkanen, and J. Turunen, “Generating a couple of rotating nondiffracting beams using a binary-phase DOE,” Optik 110137–144 (1999).

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soifer, and A. T. Friberg, “Rotating optical fields: experimental demonstration with diffractive optics,” J. Mod. Opt. 462355–2369 (1998).
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995), sect. 3.2.

McDonald, G. S.

S. Chávez-Cerda, G. S. McDonald, and G. H. S. New, “Nondiffracting Beams: travelling, standing, rotating and spiral waves,” Opt. Commun. 123225–233 (1996).
[Crossref]

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 581499–1501 (1987).
[Crossref] [PubMed]

Mishra, S. R.

S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85159–161 (1991).
[Crossref]

Montgomery, W. D.

W. D. Montgomery, “Algebraic formulation of diffraction applied to self imaging,” J. Opt. Soc. Am. 581112–1124 (1968).
[Crossref]

W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57772–778 (1967).
[Crossref]

New, G. H. S.

S. Chávez-Cerda, G. S. McDonald, and G. H. S. New, “Nondiffracting Beams: travelling, standing, rotating and spiral waves,” Opt. Commun. 123225–233 (1996).
[Crossref]

Olivík, M.

Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 81555–1566 (1995).
[Crossref]

Pääkkönen, P.

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soifer, and A. T. Friberg, “Rotating optical fields: experimental demonstration with diffractive optics,” J. Mod. Opt. 462355–2369 (1998).
[Crossref]

Paterson, C.

C. Paterson and R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124121–130 (1996).
[Crossref]

S. N.,

S. N. Khonina, S. N., V. V. Kotlyar, V. A. Soifer, J. Lautanen, M. Honkanen, and J. Turunen, “Generating a couple of rotating nondiffracting beams using a binary-phase DOE,” Optik 110137–144 (1999).

Smith, R.

C. Paterson and R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124121–130 (1996).
[Crossref]

Soifer, V. A.

S. N. Khonina, S. N., V. V. Kotlyar, V. A. Soifer, J. Lautanen, M. Honkanen, and J. Turunen, “Generating a couple of rotating nondiffracting beams using a binary-phase DOE,” Optik 110137–144 (1999).

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soifer, and A. T. Friberg, “Rotating optical fields: experimental demonstration with diffractive optics,” J. Mod. Opt. 462355–2369 (1998).
[Crossref]

V. V. Kotlyar, V. A. Soifer, and S. N. Khonina, “An algorithm for the generation of laser-beams with longitudinal periodicity,” J. Mod. Opt. 441409–1416 (1997).
[Crossref]

Talbot, H. F.

H. F. Talbot, “Facts relating to optical science, No. IV,” Philos. Mag. 9401–407 (1836).

Turunen, J.

S. N. Khonina, S. N., V. V. Kotlyar, V. A. Soifer, J. Lautanen, M. Honkanen, and J. Turunen, “Generating a couple of rotating nondiffracting beams using a binary-phase DOE,” Optik 110137–144 (1999).

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soifer, and A. T. Friberg, “Rotating optical fields: experimental demonstration with diffractive optics,” J. Mod. Opt. 462355–2369 (1998).
[Crossref]

J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 251–60 (1993).
[Crossref]

Wagner, J.

Z. Bouchal, R. Horák, and J. Wagner, “Propagation-invariant electromagnetic fields,” J. Mod. Opt. 91905–1920 (1996).
[Crossref]

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic Press, New York, 2001), p. 681.

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995), sect. 3.2.

J. Mod. Opt. (4)

V. V. Kotlyar, V. A. Soifer, and S. N. Khonina, “An algorithm for the generation of laser-beams with longitudinal periodicity,” J. Mod. Opt. 441409–1416 (1997).
[Crossref]

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soifer, and A. T. Friberg, “Rotating optical fields: experimental demonstration with diffractive optics,” J. Mod. Opt. 462355–2369 (1998).
[Crossref]

Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 81555–1566 (1995).
[Crossref]

Z. Bouchal, R. Horák, and J. Wagner, “Propagation-invariant electromagnetic fields,” J. Mod. Opt. 91905–1920 (1996).
[Crossref]

J. Opt. Soc. Am. (2)

W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57772–778 (1967).
[Crossref]

W. D. Montgomery, “Algebraic formulation of diffraction applied to self imaging,” J. Opt. Soc. Am. 581112–1124 (1968).
[Crossref]

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

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4651–654 (1987).
[Crossref]

Opt. Commun. (3)

S. Chávez-Cerda, G. S. McDonald, and G. H. S. New, “Nondiffracting Beams: travelling, standing, rotating and spiral waves,” Opt. Commun. 123225–233 (1996).
[Crossref]

C. Paterson and R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124121–130 (1996).
[Crossref]

S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85159–161 (1991).
[Crossref]

Optik (1)

S. N. Khonina, S. N., V. V. Kotlyar, V. A. Soifer, J. Lautanen, M. Honkanen, and J. Turunen, “Generating a couple of rotating nondiffracting beams using a binary-phase DOE,” Optik 110137–144 (1999).

Philos. Mag. (1)

H. F. Talbot, “Facts relating to optical science, No. IV,” Philos. Mag. 9401–407 (1836).

Phys. Rev. Lett. (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 581499–1501 (1987).
[Crossref] [PubMed]

Pure Appl. Opt. (1)

J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 251–60 (1993).
[Crossref]

Other (2)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995), sect. 3.2.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic Press, New York, 2001), p. 681.

Supplementary Material (4)

» Media 1: MOV (1568 KB)

» Media 2: MOV (1388 KB)

» Media 3: MOV (1249 KB)

» Media 4: MOV (1404 KB)

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Fig. 1. (1.53 MB) The squared absolute values of the central parts of E_ρ , E_z , and E_x as a function of z-coordinate, calculated with the parameters given in Table 1 (left). The movie should be viewed repeatedly.

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Fig. 2. (1.35 MB) Same as Fig. 1, except that the parameters used in the calculations given in Table 1 (right). The movie should be viewed repeatedly.

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Fig. 3. (1.21 MB) Same as Fig. 1, except that the parameters used in the calculations given in Table 2 (left). The movie should be viewed repeatedly.

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Fig. 4. (1.37 MB) Same as Fig. 1, except that the parameters used in the calculations given in Table 2 (right). The movie should be viewed repeatedly.

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Tables (2)

Table 1. The parameters assumed in Figs. 1 (left) and 2 (right). The constant β_c must be less than k but is otherwise arbitrary.

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Table 2. The parameters assumed in Figs. 3 (left) and 4 (right). The constant β_c must be less than k but is otherwise arbitrary.

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Equations (33)

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

U (ρ, ϕ, z) = \int_{0}^{\infty} \int_{0}^{2 π} α A (α, ψ) \exp [i α ρ \cos (ϕ - ψ) + i β z] d α d ψ,

A (α, ψ) = \frac{1}{2 π} \int_{0}^{\infty} \int_{0}^{2 π} ρ^{U} (ρ, ϕ, 0) \exp [- i α ρ \cos (ϕ - ψ)] d ρ d ϕ,

β = {\begin{matrix} {(k^{2} - α^{2})}^{\frac{1}{2}} if α \leq k \\ i {(α^{2} - k^{2})}^{\frac{1}{2}} if k < α \end{matrix} .

U (ρ, ϕ + η Δ z, z + Δ z) = \exp {[i ξ (ρ, ϕ, Δ z)]}^{U} (ρ, ϕ, z) .

0 = \sum_{m = - \infty}^{\infty} \int_{0}^{\infty} a_{m} (α) J_{m} (αρ) \exp (i m ϕ)

\times {\exp [i ξ (ρ, ϕ, Δ z)] - \exp [i (mη + β) Δ z]} d α,

a_{m} (α) = i^{m} α \int_{0}^{2 π} A (α, ψ) \exp (- i mψ) d ψ .

\exp (i ϑ \cos φ) = \sum_{m =- \infty}^{\infty} i^{m} J_{m} (ϑ) \exp (i mφ) .

ξ (ρ, ϕ, Δ z) = ξ (Δ z) = (mη + β) Δ z + 2 π q,

β_{m} = β_{0} - mη,

U (ρ, ϕ, z) = \sum_{m \in M} a_{m} J_{m} (α_{m} ρ) \exp [i (mϕ + β_{m} z)],

z_{T} = 2 π / η .

z_{T} / Q = 2 π / η Q,

I (ρ, ϕ + 2 π q / Q, z) = I (ρ, ϕ, z),

E_{ρ} (ρ, ϕ, z) = E_{x} (ρ, ϕ, z) \cos ϕ + E_{y} (ρ, ϕ, z) \sin ϕ

E_{ϕ} (ρ, ϕ, z) = - E_{x} (ρ, ϕ, z) \sin ϕ + E_{y} (ρ, ϕ, z) \cos ϕ

{\begin{matrix} E_{ρ} (ρ, ϕ + γ Δ z, z + Δ z) = \exp [i ξ (ρ, ϕ, Δ z)] E_{ρ} (ρ, ϕ, z) \\ E_{ϕ} (ρ, ϕ + γ Δ z, z + Δ z) = \exp [i ξ (ρ, ϕ, Δ z)] E_{ϕ} (ρ, ϕ, z) \end{matrix},

E_{ρ} (ρ, ϕ, z) = \sum_{m = - \infty}^{\infty} \int_{0}^{\infty} [a_{m} (α) \cos ϕ + b_{m} (α) \sin ϕ]

\times J_{m} (αρ) \exp [i (mϕ + βz)] d α,

E_{ϕ} (ρ, ϕ, z) \sum_{m = - \infty}^{\infty} \int_{0}^{\infty} [- a_{m} (α) \sin ϕ + b_{m} (α) \cos ϕ]

\times J_{m} (αρ) \exp [i (mϕ + β z)] d α,

{\begin{matrix} g_{m} (ρ, ϕ, Δ z) a_{m} (α) + h_{m} (ρ, ϕ, Δ z) b_{m} (α) = 0 \\ - h_{m} (ρ, ϕ, Δ z) a_{m} (α) + g_{m} (ρ, ϕ, Δ z) b_{m} (α) = 0 \end{matrix},

{\begin{matrix} g_{m} (ρ, ϕ, Δ z) = \cos (ϕ + γ Δ z) \exp [i (m γ + β) Δ z] - \cos ϕ \exp [i ξ (ρ, ϕ, Δ z)] \\ h_{m} (ρ, ϕ, Δ z) = \sin (ϕ + γ Δ z) \exp [i (m γ + β) Δ z] - \sin ϕ \exp [i ξ (ρ, ϕ, Δ z)] \end{matrix} .

h_{m} (ρ, ϕ, Δ z) = \pm i g_{m} (ρ, ϕ, Δ z)

b_{m} (α) = \pm i a_{m} (α),

[(m \pm 1) γ + k_{z}] Δ z = ξ (ρ, ϕ, Δ z) .

β_{mn} = β_{0_{n}} - (m + n) γ,

E_{ρ} (ρ, ϕ, z) = \sum_{m \in M} \sum_{n = - 1,1} a_{mn} J_{m} (α_{mn} ρ) \exp {i [(m + n) ϕ + β_{mn} z]}

E_{ϕ} (ρ, ϕ, z) = \sum_{m \in M} \sum_{n = - 1,1} i {na}_{mn} J_{m} (α_{mn} ρ) \exp {i [(m + n) ϕ + β_{mn} z]},

\nabla \cdot E (ρ, ϕ, z) = \frac{1}{ρ} \frac{\partial}{\partial ρ} [ρ E_{ρ} (ρ, ϕ, z)] + \frac{1}{ρ} \frac{\partial}{\partial ϕ} E_{ϕ} (ρ, ϕ, z) + \frac{\partial}{\partial z} E_{z} (ρ, ϕ, z) = 0,

E_{z} (ρ, ϕ, z) = \sum_{m \in M} \sum_{n = - 1,1} {na}_{mn} \frac{α_{mn}}{i β_{mn}} J_{m + n} (α_{mn} ρ) \exp {i [(m + n) ϕ + β_{mn} z]} .

{\begin{matrix} β_{j} = β_{0} - jη = β_{0_{n}} - γ (m + n_{j}), \\ β_{m} = β_{0} - mη = β_{0_{n}} - γ (m + n_{m}) . \end{matrix}

\frac{η}{γ} = \frac{P}{Q} = \frac{m - j + n_{m} - n_{j}}{m - j} .

m	β_m	n	a_mn
0	β_c /2	1	1
2	β_c /2	-1	i
-3	β_c	1	1
m	β_m	n	a_mn
0	β_c /2	1	1
-3	β_c	-1	i

m	β_m	n	a_mn
2	β_c /2	-1	1
1	β_c	1	i
m	β_m	n	a_mn
2	β_c /3	-1	1
0	2β_c /3	-1	-1
-2	3	β_c	-i

m	β_m	n	a_mn
0	β_c /2	1	1
2	β_c /2	-1	i
-3	β_c	1	1
m	β_m	n	a_mn
0	β_c /2	1	1
-3	β_c	-1	i

m	β_m	n	a_mn
2	β_c /2	-1	1
1	β_c	1	i
m	β_m	n	a_mn
2	β_c /3	-1	1
0	2β_c /3	-1	-1
-2	3	β_c	-i

Abstract

References

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Supplementary Material (4)

Cited By

Figures (4)

Tables (2)

Equations (33)

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