Abstract

A novel decomposition of the transversal part of the electric field vector of a general non-paraxial electromagnetic field is presented, which is an extension of the radial/aximuthal decomposition and is known as γζ decomposition. Purely γ and ζ polarized fields are examined and the decomposition is applied to propagation-invariant, rotating, and self-imaging electromagnetic fields. An experimental example on the effect of state of polarization in the propagation characteristics of the field: its is shown that a simple modification of the polarization conditions of the angular spectrum converts a self-imaging field into a propagation-invariant field.

© 2002 Optical Society of America

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References

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  1. W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57, 772–778 (1967).
    [Crossref]
  2. K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in OpticsVol. XXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1989), Chap. 1.
  3. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [Crossref]
  4. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [Crossref] [PubMed]
  5. Y. Y. Schechner, R. Piestun, and J. Shamir, “Wave propagating with rotating intensity distributions,” Phys. Rev. E 54, R50–R53 (1996).
    [Crossref]
  6. S. Chávez-Cerda, G. S. McDonald, and G. H. S. New, “Nondiffracting Beams: travelling, standing, rotating and spiral waves,” Opt. Commun. 123, 225–233 (1996).
    [Crossref]
  7. C. Paterson and R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124, 121–130 (1996).
    [Crossref]
  8. S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85, 159–161 (1991).
    [Crossref]
  9. J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
    [Crossref]
  10. Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
    [Crossref]
  11. Z. Bouchal, R. Horák, and J. Wagner, “Propagation-invariant electromagnetic fields,” J. Mod. Opt. 43, 1905–1920 (1996).
    [Crossref]
  12. J. Tervo and J. Turunen, “Rotating scale-invariant electromagnetic fields,” Opt. Express 9, 9–15 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-1-9.
    [Crossref] [PubMed]
  13. P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotl-yar, V. A. Soifer, and A. T. Friberg, “Rotating optical fields: experimental demonstration with diffractive optics,” J. Mod. Opt. 46, 2355–2369 (1998).
    [Crossref]
  14. F. Gori, “Polarization basis for vortex beams,” J. Opt. Soc. Am. A 18, 1612–1617 (2001).
    [Crossref]
  15. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995), Sect. 3.2.
  16. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic Press, New York, 2001), p. 681.
  17. R. H. Jordan and D. G. Hall, “Highly directional surface emission from concentric-circle gratings on planar optical waveguides: the field-expansion method,” J. Opt. Soc. Am. A 12, 84–94 (1995).
    [Crossref]
  18. J. Tervo, P. Vahimaa, and J. Turunen, “On propagation-invariant and self-imaging intensity distributions of electromagnetic fields,” J. Mod. Opt. 49, 1537–1543 (2002).
    [Crossref]
  19. A. Lapucci and M. Ciofini, “Polarization state modifications in the propagation of high azimuthal order annular beams,” Opt. Express 9, 603–609 (2001),http: //www. opticsexpress.org/abstract.cfm?URI=OPEX-9-12-603.
    [Crossref] [PubMed]
  20. J. Tervo and J. Turunen, “Self-imaging of electromagnetic fields,” Opt. Express 9, 622–630 (2001), http: //www. opticsexpress.org/abstract.cfm?URI=OPEX-9-12-622.
    [Crossref] [PubMed]
  21. J. Tervo and J. Turunen, “Generation of vectorial propagation-invariant fields by polarization-grating axicons,” Opt. Commun. 192, 13–18 (2001).
    [Crossref]

2002 (1)

J. Tervo, P. Vahimaa, and J. Turunen, “On propagation-invariant and self-imaging intensity distributions of electromagnetic fields,” J. Mod. Opt. 49, 1537–1543 (2002).
[Crossref]

2001 (5)

1998 (1)

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotl-yar, V. A. Soifer, and A. T. Friberg, “Rotating optical fields: experimental demonstration with diffractive optics,” J. Mod. Opt. 46, 2355–2369 (1998).
[Crossref]

1996 (4)

Z. Bouchal, R. Horák, and J. Wagner, “Propagation-invariant electromagnetic fields,” J. Mod. Opt. 43, 1905–1920 (1996).
[Crossref]

Y. Y. Schechner, R. Piestun, and J. Shamir, “Wave propagating with rotating intensity distributions,” Phys. Rev. E 54, R50–R53 (1996).
[Crossref]

S. Chávez-Cerda, G. S. McDonald, and G. H. S. New, “Nondiffracting Beams: travelling, standing, rotating and spiral waves,” Opt. Commun. 123, 225–233 (1996).
[Crossref]

C. Paterson and R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124, 121–130 (1996).
[Crossref]

1995 (2)

1993 (1)

J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
[Crossref]

1991 (1)

S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85, 159–161 (1991).
[Crossref]

1987 (2)

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[Crossref]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

1967 (1)

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. 43, 1905–1920 (1996).
[Crossref]

Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (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. 123, 225–233 (1996).
[Crossref]

Ciofini, M.

Durnin, J.

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[Crossref]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Friberg, A. T.

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotl-yar, V. A. Soifer, and A. T. Friberg, “Rotating optical fields: experimental demonstration with diffractive optics,” J. Mod. Opt. 46, 2355–2369 (1998).
[Crossref]

J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
[Crossref]

Gori, F.

Hall, D. G.

Honkanen, M.

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotl-yar, V. A. Soifer, and A. T. Friberg, “Rotating optical fields: experimental demonstration with diffractive optics,” J. Mod. Opt. 46, 2355–2369 (1998).
[Crossref]

Horák, R.

Z. Bouchal, R. Horák, and J. Wagner, “Propagation-invariant electromagnetic fields,” J. Mod. Opt. 43, 1905–1920 (1996).
[Crossref]

Jordan, R. H.

Khonina, S. N.

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotl-yar, V. A. Soifer, and A. T. Friberg, “Rotating optical fields: experimental demonstration with diffractive optics,” J. Mod. Opt. 46, 2355–2369 (1998).
[Crossref]

Kotl-yar, V. V.

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotl-yar, V. A. Soifer, and A. T. Friberg, “Rotating optical fields: experimental demonstration with diffractive optics,” J. Mod. Opt. 46, 2355–2369 (1998).
[Crossref]

Kuittinen, M.

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotl-yar, V. A. Soifer, and A. T. Friberg, “Rotating optical fields: experimental demonstration with diffractive optics,” J. Mod. Opt. 46, 2355–2369 (1998).
[Crossref]

Lapucci, A.

Lautanen, J.

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotl-yar, V. A. Soifer, and A. T. Friberg, “Rotating optical fields: experimental demonstration with diffractive optics,” J. Mod. Opt. 46, 2355–2369 (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. 123, 225–233 (1996).
[Crossref]

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Mishra, S. R.

S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85, 159–161 (1991).
[Crossref]

Montgomery, W. D.

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. 123, 225–233 (1996).
[Crossref]

Olivík, M.

Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[Crossref]

Pääkkönen, P.

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotl-yar, V. A. Soifer, and A. T. Friberg, “Rotating optical fields: experimental demonstration with diffractive optics,” J. Mod. Opt. 46, 2355–2369 (1998).
[Crossref]

Paterson, C.

C. Paterson and R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124, 121–130 (1996).
[Crossref]

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in OpticsVol. XXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1989), Chap. 1.

Piestun, R.

Y. Y. Schechner, R. Piestun, and J. Shamir, “Wave propagating with rotating intensity distributions,” Phys. Rev. E 54, R50–R53 (1996).
[Crossref]

Schechner, Y. Y.

Y. Y. Schechner, R. Piestun, and J. Shamir, “Wave propagating with rotating intensity distributions,” Phys. Rev. E 54, R50–R53 (1996).
[Crossref]

Shamir, J.

Y. Y. Schechner, R. Piestun, and J. Shamir, “Wave propagating with rotating intensity distributions,” Phys. Rev. E 54, R50–R53 (1996).
[Crossref]

Smith, R.

C. Paterson and R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124, 121–130 (1996).
[Crossref]

Soifer, V. A.

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotl-yar, V. A. Soifer, and A. T. Friberg, “Rotating optical fields: experimental demonstration with diffractive optics,” J. Mod. Opt. 46, 2355–2369 (1998).
[Crossref]

Tervo, J.

J. Tervo, P. Vahimaa, and J. Turunen, “On propagation-invariant and self-imaging intensity distributions of electromagnetic fields,” J. Mod. Opt. 49, 1537–1543 (2002).
[Crossref]

J. Tervo and J. Turunen, “Generation of vectorial propagation-invariant fields by polarization-grating axicons,” Opt. Commun. 192, 13–18 (2001).
[Crossref]

J. Tervo and J. Turunen, “Self-imaging of electromagnetic fields,” Opt. Express 9, 622–630 (2001), http: //www. opticsexpress.org/abstract.cfm?URI=OPEX-9-12-622.
[Crossref] [PubMed]

J. Tervo and J. Turunen, “Rotating scale-invariant electromagnetic fields,” Opt. Express 9, 9–15 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-1-9.
[Crossref] [PubMed]

Turunen, J.

J. Tervo, P. Vahimaa, and J. Turunen, “On propagation-invariant and self-imaging intensity distributions of electromagnetic fields,” J. Mod. Opt. 49, 1537–1543 (2002).
[Crossref]

J. Tervo and J. Turunen, “Generation of vectorial propagation-invariant fields by polarization-grating axicons,” Opt. Commun. 192, 13–18 (2001).
[Crossref]

J. Tervo and J. Turunen, “Rotating scale-invariant electromagnetic fields,” Opt. Express 9, 9–15 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-1-9.
[Crossref] [PubMed]

J. Tervo and J. Turunen, “Self-imaging of electromagnetic fields,” Opt. Express 9, 622–630 (2001), http: //www. opticsexpress.org/abstract.cfm?URI=OPEX-9-12-622.
[Crossref] [PubMed]

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotl-yar, V. A. Soifer, and A. T. Friberg, “Rotating optical fields: experimental demonstration with diffractive optics,” J. Mod. Opt. 46, 2355–2369 (1998).
[Crossref]

J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
[Crossref]

Vahimaa, P.

J. Tervo, P. Vahimaa, and J. Turunen, “On propagation-invariant and self-imaging intensity distributions of electromagnetic fields,” J. Mod. Opt. 49, 1537–1543 (2002).
[Crossref]

Wagner, J.

Z. Bouchal, R. Horák, and J. Wagner, “Propagation-invariant electromagnetic fields,” J. Mod. Opt. 43, 1905–1920 (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)

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotl-yar, V. A. Soifer, and A. T. Friberg, “Rotating optical fields: experimental demonstration with diffractive optics,” J. Mod. Opt. 46, 2355–2369 (1998).
[Crossref]

Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[Crossref]

Z. Bouchal, R. Horák, and J. Wagner, “Propagation-invariant electromagnetic fields,” J. Mod. Opt. 43, 1905–1920 (1996).
[Crossref]

J. Tervo, P. Vahimaa, and J. Turunen, “On propagation-invariant and self-imaging intensity distributions of electromagnetic fields,” J. Mod. Opt. 49, 1537–1543 (2002).
[Crossref]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (4)

S. Chávez-Cerda, G. S. McDonald, and G. H. S. New, “Nondiffracting Beams: travelling, standing, rotating and spiral waves,” Opt. Commun. 123, 225–233 (1996).
[Crossref]

C. Paterson and R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124, 121–130 (1996).
[Crossref]

S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85, 159–161 (1991).
[Crossref]

J. Tervo and J. Turunen, “Generation of vectorial propagation-invariant fields by polarization-grating axicons,” Opt. Commun. 192, 13–18 (2001).
[Crossref]

Opt. Express (3)

Phys. Rev. E (1)

Y. Y. Schechner, R. Piestun, and J. Shamir, “Wave propagating with rotating intensity distributions,” Phys. Rev. E 54, R50–R53 (1996).
[Crossref]

Phys. Rev. Lett. (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Pure Appl. Opt. (1)

J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
[Crossref]

Other (3)

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in OpticsVol. XXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1989), Chap. 1.

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 (2)

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» Media 2: MOV (1415 KB)     

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Figures (4)

Fig. 1.
Fig. 1.

Directions of the transversal basis vectors in the cases q = 1 and q = - 2.

Fig. 2.
Fig. 2.

The principle of the experimental setup. Here a and b denote an annular half-wave plate and a ring aperture, respectively, and c is a thin lens.

Fig. 3.
Fig. 3.

(1.21 MB) An animation of the experimental fields within one self-imaging distance zT ≈ 20 mm. The field in the left-hand side is obtained with parallel polarization states and the field in the right-hand side with orthogonal polarization states.

Fig. 4.
Fig. 4.

(1.38 MB) An animation of the intensity distributions of the central parts of the x- and γ-components and the total electric energy density as functions of the z-coordinate, calculated with the parameters given in Table 1. The movie is best appreciated if viewed repeatedly.

Tables (1)

Tables Icon

Table 1. The parameters assumed in Fig. 4. The constant βC must be less than k but is otherwise arbitrary.

Equations (40)

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{ x = ρ cos ϕ y = ρ sin ϕ
{ k x = α cos ψ k y = α sin ψ k z = β
E ( ρ , ϕ , z ) = 0 2 π 0 A α ψ exp [ i αρ cos ( ϕ ψ ) + i βz ] α d α d ψ ,
β = { k 2 α 2 if α k i α 2 k 2 otherwise
A α ψ = 1 ( 2 π ) 2 0 2 π 0 E ( ρ , ϕ , 0 ) exp [ i αρ cos ( ϕ ψ ) ] ρ d ρ d ϕ .
· E ( ρ , ϕ , z ) = 0 .
{ E γ ( q ) ( ρ , ϕ , z ) = cos ( ) E x ( ρ , ϕ , z ) + sin ( ) E y ( ρ , ϕ , z ) E ζ ( q ) ( ρ , ϕ , z ) = sin ( ) E x ( ρ , ϕ , z ) + cos ( ) E y ( ρ , ϕ , z ) ,
E γ ( q ) ( ρ , ϕ , z ) = m = exp ( i ) 0 J m ( αρ ) exp ( i βz )
× [ a m x ( α ) cos ( ) + a m y ( α ) sin ( ) ] ,
E ζ ( q ) ( ρ , ϕ , z ) = m = exp ( i ) 0 J m ( αρ ) exp ( i βz )
× [ a m x ( α ) sin ( ) + a m y ( α ) cos ( ) ] .
exp ( cos τ ) = m = i m J m ( ϑ ) exp ( i ) ,
a m j ( α ) = i m α 0 2 π A j α ψ exp ( i ) d ψ ,
m = exp ( i ) 0 { J m q ( αρ ) [ i a m q x ( α ) a m q y ( α ) ]
+ J m + q ( αρ ) [ i a m + q x ( α ) + a m + q y ( α ) ] } exp ( i βz ) d α = 0 .
J m q ( αρ ) [ i a m q x ( α ) + a m q y ( α ) ] + J m + q ( αρ ) [ i a m + q x ( α ) + a m + q y ( α ) ] 0 ,
{ a q y ( α ) = i ( 1 ) q a q x ( α ) a q y ( α ) = i a q x ( α ) . a q x ( α ) = ( 1 ) q a q x ( α )
E γ ( q ) ( ρ , ϕ , z ) = E γ ( q ) ρ z = 0 f q ( α ) J q ( αρ ) exp ( i βz ) d α ,
f q ( α ) = 2 a q x ( α ) .
E z ( q ) ρ ϕ z = i cos [ ( q 1 ) ϕ ] 0 α β f q ( α ) J q 1 ( αρ ) exp ( i βz ) d α .
E ζ ( q ) ρ ϕ z = E ζ ( q ) ρ z = 0 g q ( α ) J q ( αρ ) exp ( i βz ) d α
E z ( q ) ρ ϕ z = i sin [ ( q 1 ) ϕ ] 0 α β g q ( α ) J q 1 ( αρ ) exp ( i βz ) d α ,
g q ( α ) = i f q ( α ) .
w ( ρ , ϕ , z + Δ z , t ) = w ( ρ , ϕ , z , t )
z = w ( ρ , ϕ , z , t ) 0 ,
E γ ( q ) ( ρ , ϕ , z + Δ z ) = exp [ ( ρ , ϕ , Δ z ) ] E γ ( q ) ( ρ , ϕ , z )
E ζ ( q ) ( ρ , ϕ , z + Δ z ) = exp [ ( ρ , ϕ , Δ z ) ] E ζ ( q ) ( ρ , ϕ , z )
E γ ( q ) ( ρ , z ) = c γ J q ( α γ ρ ) exp ( i β γ z )
E ζ ( q ) ( ρ , z ) = c ζ J q ( α ζ ρ ) exp ( i β ζ z ) ,
E z ( ρ , ϕ , z ) = i c γ α γ β γ cos [ ( q 1 ) ϕ ] J q 1 ( α γ ρ ) exp ( i β γ z )
+ i c ζ α ζ β ζ sin [ ( q 1 ) ϕ ] J q 1 ( α ζ ρ ) exp ( i β ζ z ) .
w ( ρ , ϕ + η Δ z , z + Δ z , t ) = w ( ρ , ϕ , z , t )
{ E γ ( q ) ( ρ , ϕ + η Δ z , z + Δ z ) = exp [ i ξ ( ρ , ϕ , Δ z ) ] E γ ( q ) ( ρ , ϕ , z ) E ζ ( q ) ( ρ , ϕ + η Δ z , z + Δ z ) = exp [ i ξ ( ρ , ϕ , Δ z ) ] E ζ ( q ) ( ρ , ϕ , z ) ,
a m y ( α ) = i s a m x ( α )
β ms = β 0 ( m + sq ) η ,
E γ ( q ) ( ρ , ϕ , z ) = m 𝓜 s = 1,1 a ms J m ( α ms ρ ) exp { i [ ( m + sq ) ϕ + β ms z ] }
E ζ ( q ) ( ρ , ϕ , z ) = m 𝓜 s = 1,1 i s a ms J m ( α ms ρ ) exp { i [ ( m + sq ) ϕ + β ms z ] } ,
E z ( ρ , ϕ , z ) = m 𝓜 s = 1,1 s a ms α ms i β ms J m + s ( α ms ρ ) exp { i [ ( m + s ) ϕ + β ms z ] } .
E ( ρ , ϕ , z ) = exp ( i β m , 1 z ) a m , 1 J m ( α m , 1 ρ ) exp ( i m ϕ ) [ 1 i ]
+ exp ( i β m , 1 z ) a m , 1 J m ( α m , 1 ρ ) exp ( i m ϕ ) [ 1 i ] ,

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