Abstract

The electromagnetic theory of self-imaging fields is considered. Several features are presented, which have no counterparts within the scalar theory of self-imaging. For example, the electromagnetic field self-images at one half of the classical self-imaging distance for scalar fields, the electric and magnetic energy densities can self-image while the scalar field components do not, and the self-imaging distances of the electric and magnetic energy densities can be different. In addition, general expressions for TE and TM polarized fields are presented by using the concept of the angular spectrum of the field.

© 2001 Optical Society of America

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References

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  1. K. Patorski, “The self-imaging phenomenon and its applications,” in Progr. Opt., Vol. XXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1989), Chap. 1.
    [Crossref]
  2. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [Crossref]
  3. S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85, 159–161 (1991).
    [Crossref]
  4. J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
    [Crossref]
  5. Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
    [Crossref]
  6. Z. Bouchal, R. Horák, and J. Wagner, “Propagation-invariant electromagnetic fields,” J. Mod. Opt. 43, 1905–1920 (1996).
    [Crossref]
  7. R. Horák, Z. Bouchal, and J. Bajer, “Nondiffracting stationary electromagnetic field,” Opt. Commun. 133, 315–327 (1997).
    [Crossref]
  8. J. Tervo and J. Turunen, “Generation of vectorial propagation-invariant propagation-invariant fields with polarization-grating axicons,” Opt. Commun. 192, 13–18 (2001).
    [Crossref]
  9. J. Tervo, P. Vahimaa, and J. Turunen, “On propagation-invariance and self-imaging of intensity distributions of electromganetic fields,” J. Mod. Opt. (In press).
  10. Y. Y. Schechner, R. Piestun, and J. Shamir, “Wave propagation with rotating intensity distributions,” Phys. Rev. E 54, R50–R53 (1996).
    [Crossref]
  11. 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]
  12. C. Paterson and R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124, 121–130 (1996).
    [Crossref]
  13. R. Piestun and J. Shamir, “Generalized propagation-invariant fields,” J. Opt. Soc. Am. A 15, 3039–3044 (1998).
    [Crossref]
  14. J. Tervo and J. Turunen, “Rotating scale-invariant electromagnetic fields,” Opt. Express 9, 9–15 (2001), http://www.opticsexpress.org/oearchive/source/33955.htm.
    [Crossref] [PubMed]
  15. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995), sect. 3.2.
  16. S. Ruschin and A. Leizer, “Evanescent Bessel beams,” J. Opt. Soc. Am. A 15, 1139–1143 (1998).
    [Crossref]
  17. H. F. Talbot, “Facts relating to optical science. No. IV,” Philos. Mag. 9, 401–407 (1836).
  18. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999).
  19. W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57772–778 (1967).
    [Crossref]
  20. W. D. Montgomery, “Algebraic formulation of diffraction applied to self imaging,” J. Opt. Soc. Am. 581112–1124 (1968).
    [Crossref]
  21. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic Press, New York, 2001), p. 681.
  22. J. Tervo and J. Turunen, “Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings,” Opt. Lett. 25, 785–786 (2000).
    [Crossref]
  23. M. Honkanen, V. Kettunen, J. Tervo, and J. Turunen, “Fourier array illuminators with 100% efficiency: analytical Jones-matrix construction,” J. Mod. Opt. 47, 2351–2359 (2000).

2001 (2)

J. Tervo and J. Turunen, “Generation of vectorial propagation-invariant propagation-invariant fields with 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/oearchive/source/33955.htm.
[Crossref] [PubMed]

2000 (2)

M. Honkanen, V. Kettunen, J. Tervo, and J. Turunen, “Fourier array illuminators with 100% efficiency: analytical Jones-matrix construction,” J. Mod. Opt. 47, 2351–2359 (2000).

J. Tervo and J. Turunen, “Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings,” Opt. Lett. 25, 785–786 (2000).
[Crossref]

1998 (2)

1997 (1)

R. Horák, Z. Bouchal, and J. Bajer, “Nondiffracting stationary electromagnetic field,” Opt. Commun. 133, 315–327 (1997).
[Crossref]

1996 (4)

Y. Y. Schechner, R. Piestun, and J. Shamir, “Wave propagation 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]

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

1995 (1)

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

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 (1)

1968 (1)

1967 (1)

1836 (1)

H. F. Talbot, “Facts relating to optical science. No. IV,” Philos. Mag. 9, 401–407 (1836).

Arfken, G. B.

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

Bajer, J.

R. Horák, Z. Bouchal, and J. Bajer, “Nondiffracting stationary electromagnetic field,” Opt. Commun. 133, 315–327 (1997).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999).

Bouchal, Z.

R. Horák, Z. Bouchal, and J. Bajer, “Nondiffracting stationary electromagnetic field,” Opt. Commun. 133, 315–327 (1997).
[Crossref]

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]

Durnin, J.

Friberg, A. T.

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

Honkanen, M.

M. Honkanen, V. Kettunen, J. Tervo, and J. Turunen, “Fourier array illuminators with 100% efficiency: analytical Jones-matrix construction,” J. Mod. Opt. 47, 2351–2359 (2000).

Horák, R.

R. Horák, Z. Bouchal, and J. Bajer, “Nondiffracting stationary electromagnetic field,” Opt. Commun. 133, 315–327 (1997).
[Crossref]

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

Kettunen, V.

M. Honkanen, V. Kettunen, J. Tervo, and J. Turunen, “Fourier array illuminators with 100% efficiency: analytical Jones-matrix construction,” J. Mod. Opt. 47, 2351–2359 (2000).

Leizer, A.

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]

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]

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 Progr. Opt., Vol. XXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1989), Chap. 1.
[Crossref]

Piestun, R.

R. Piestun and J. Shamir, “Generalized propagation-invariant fields,” J. Opt. Soc. Am. A 15, 3039–3044 (1998).
[Crossref]

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

Ruschin, S.

Schechner, Y. Y.

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

Shamir, J.

R. Piestun and J. Shamir, “Generalized propagation-invariant fields,” J. Opt. Soc. Am. A 15, 3039–3044 (1998).
[Crossref]

Y. Y. Schechner, R. Piestun, and J. Shamir, “Wave propagation 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]

Talbot, H. F.

H. F. Talbot, “Facts relating to optical science. No. IV,” Philos. Mag. 9, 401–407 (1836).

Tervo, J.

J. Tervo and J. Turunen, “Generation of vectorial propagation-invariant propagation-invariant fields with 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/oearchive/source/33955.htm.
[Crossref] [PubMed]

J. Tervo and J. Turunen, “Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings,” Opt. Lett. 25, 785–786 (2000).
[Crossref]

M. Honkanen, V. Kettunen, J. Tervo, and J. Turunen, “Fourier array illuminators with 100% efficiency: analytical Jones-matrix construction,” J. Mod. Opt. 47, 2351–2359 (2000).

J. Tervo, P. Vahimaa, and J. Turunen, “On propagation-invariance and self-imaging of intensity distributions of electromganetic fields,” J. Mod. Opt. (In press).

Turunen, J.

J. Tervo and J. Turunen, “Generation of vectorial propagation-invariant propagation-invariant fields with 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/oearchive/source/33955.htm.
[Crossref] [PubMed]

J. Tervo and J. Turunen, “Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings,” Opt. Lett. 25, 785–786 (2000).
[Crossref]

M. Honkanen, V. Kettunen, J. Tervo, and J. Turunen, “Fourier array illuminators with 100% efficiency: analytical Jones-matrix construction,” J. Mod. Opt. 47, 2351–2359 (2000).

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

J. Tervo, P. Vahimaa, and J. Turunen, “On propagation-invariance and self-imaging of intensity distributions of electromganetic fields,” J. Mod. Opt. (In press).

Vahimaa, P.

J. Tervo, P. Vahimaa, and J. Turunen, “On propagation-invariance and self-imaging of intensity distributions of electromganetic fields,” J. Mod. Opt. (In press).

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.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999).

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

J. Mod. Opt. (3)

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]

M. Honkanen, V. Kettunen, J. Tervo, and J. Turunen, “Fourier array illuminators with 100% efficiency: analytical Jones-matrix construction,” J. Mod. Opt. 47, 2351–2359 (2000).

J. Opt. Soc. Am. (2)

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

Opt. Commun. (5)

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

R. Horák, Z. Bouchal, and J. Bajer, “Nondiffracting stationary electromagnetic field,” Opt. Commun. 133, 315–327 (1997).
[Crossref]

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

Opt. Express (1)

Opt. Lett. (1)

Philos. Mag. (1)

H. F. Talbot, “Facts relating to optical science. No. IV,” Philos. Mag. 9, 401–407 (1836).

Phys. Rev. E (1)

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

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 (5)

K. Patorski, “The self-imaging phenomenon and its applications,” in Progr. Opt., Vol. XXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1989), Chap. 1.
[Crossref]

J. Tervo, P. Vahimaa, and J. Turunen, “On propagation-invariance and self-imaging of intensity distributions of electromganetic fields,” J. Mod. Opt. (In press).

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999).

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

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

Supplementary Material (3)

» Media 1: MOV (1390 KB)     
» Media 2: MOV (740 KB)     
» Media 3: MOV (1921 KB)     

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Figures (4)

Fig. 1.
Fig. 1.

The x/y decompositions of the radial (left) and azimuthal (right) components of the angular spectrum vector in two different points of the Fourier-plane.

Fig. 2.
Fig. 2.

(1.35 MB) Movie of the squared absolute value of the z-component of the electric field and the time-averaged electric energy density within one self-imaging distance. The field is calculated by using the parameters given in Table 1.

Fig. 3.
Fig. 3.

(740 kB) Movie of the squared absolute values of the cartesian components of the electric field and the electric density within one self-imaging distance. The field is calculated by using the values given in Table 2.

Fig. 4.
Fig. 4.

(1.87 MB) Movie of the electric and the magnetic energy densities given by Eqs. (34) and (35) within one self-imaging distance.

Tables (2)

Tables Icon

Table 1. The parameters assumed in Fig. 2.

Tables Icon

Table 2. The parameters assumed in Fig. 3.

Equations (52)

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E ( ρ , ϕ , z ) = 0 0 2 π A ( α , ψ ) exp [ i α ρ cos ( ϕ ψ ) + i β z ] α d α d ψ ,
β = { k 2 α 2 when α < k i α 2 k 2 otherwise
A ( α , ψ ) = 1 ( 2 π ) 2 0 0 2 π E ( ρ , ϕ , 0 ) exp [ i α ρ cos ( ϕ ψ ) ] ρ d ρ d ϕ .
· E ( r ) = 0
A z ( α , ψ ) = α β [ A x ( α , ψ ) cos ψ + A y ( α , ψ ) sin ψ ] ,
× E ( r ) = i ω μ 0 H ( r ) ,
B x ( α , ψ ) = 1 k β ε 0 μ 0 [ A x ( α , ψ ) α 2 sin ψ cos ψ + A y ( α , ψ ) ( α 2 sin 2 ψ + β 2 ) ] ,
B y ( α , ψ ) = 1 k β ε 0 μ 0 [ A x ( α , ψ ) ( α 2 cos 2 ψ + β 2 ) + A y ( α , ψ ) α 2 sin ψ cos ψ ] ,
B z ( α , ψ ) = α k ε 0 μ 0 [ A x ( α , ψ ) sin ψ + A y ( α , ψ ) cos ψ ] ,
{ A α ( α , ψ ) = A x ( α , ψ ) cos ψ + A y ( α , ψ ) sin ψ A ψ ( α , ψ ) = A x ( α , ψ ) sin ψ + A y ( α , ψ ) cos ψ .
E z ( ρ , ϕ , z ) = 0 A α ( α , ψ ) = 0 ,
H z ( ρ , ϕ , z ) 0 A ψ ( α , ψ ) 0 ,
w ( ρ , ϕ , z + z T , t ) = w ( ρ , ϕ , z , t ) ,
w ( r , t ) = w e ( r , t ) + w h ( r , t ) = 1 4 ε 0 E ( r ) · E * ( r ) + 1 4 μ 0 H ( r ) · H * ( r ) ,
E j ( ρ , ϕ , z + z T ) = E j ( ρ , ϕ , z ) ,
β q = ξ q 2 π z T ,
A ( α , ψ ) = q = 0 Q δ ( α α q ) A q ( ψ ) ,
exp ( i ϱ cos σ ) = m = i m J m ( ϱ ) exp ( i m σ )
E ( ρ , ϕ , z ) = q = 0 Q exp ( i β q z ) m = a m , q J m ( α q ρ ) exp ( i m ϕ ) ,
a m , q = i m 0 2 π A ( α q , ψ ) exp ( i m ψ ) d ψ .
H ( ρ , ϕ , z ) = q = 0 Q exp ( i β q z ) m = b m , q J m ( α q ρ ) exp ( i m ϕ ) ,
E q ( r ) = exp ( i β q z ) m = a m , q { [ J m 1 ( α q ρ ) exp ( i ϕ ) + J m + 1 ( α q ρ ) exp ( i ϕ ) ] x ̂
+ i [ J m 1 ( α q ρ ) exp ( i ϕ ) J m + 1 exp ( i ϕ ) ( α q ρ ) ] y ̂ } exp ( i m ϕ )
H q ( r ) = i k ε 0 μ 0 exp ( i β q z ) m = a m , q exp ( i m ϕ )
× { β q [ J m 1 ( α q ρ ) exp ( i ϕ ) J m + 1 ( α q ρ ) exp ( i ϕ ) ] x ̂
+ i β q [ J m 1 ( α q ρ ) exp ( i ϕ ) + J m + 1 ( α q ρ ) exp ( i ϕ ) ] y ̂
+ 2 α q J m ( α q ρ ) z ̂ } ,
E q ( r ) = 1 β q exp ( i β q z ) m = a m , q exp ( i m ϕ ) { 2 i α q J m ( α q ρ ) } z ̂
+ β q [ J m 1 ( α q ρ ) exp ( i ϕ ) J m + 1 ( α q ρ ) exp ( i ϕ ) ] ρ ̂
+ i β q [ J m 1 ( α q ρ ) exp ( i ϕ ) + J m + 1 ( α q ρ ) exp ( i ϕ ) ϕ ̂ ]
H q ( r ) = i k β q ε 0 μ 0 exp ( i β q z ) m = a m , q exp ( i m ϕ )
× { [ J m 1 ( α q ρ ) exp ( i ϕ ) + J m + 1 ( α q ρ ) exp ( i ϕ ) ] x ̂
+ i [ J m 1 ( α q ρ ) exp ( i ϕ ) J m + 1 ( α q ρ ) exp ( i ϕ ) ] y ̂ } .
β q = β s = ξ ( 2 s + 1 ) 2 π z T = ξ ' s 2 π z T 2 ,
a m , q = a m , q x x ̂ + a m , q y y ̂
b m , q = ε 0 μ 0 [ a m , y y x ̂ + a m , y x y ̂ ] .
a m , y y = exp ( i β q z T 2 + i φ ) a m , q x ,
w e ( r , t ) = q = 0 Q exp ( i β q z ) m = a m , q x J m ( α q ρ ) exp ( i m ϕ ) 2
+ q = 0 Q exp [ i β q ( z + z T 2 ) ] m = a m x J m ( α q ρ ) exp ( i m ϕ ) 2 .
w e ( ρ , ϕ , z + z T 2 , t ) = w e ( ρ , ϕ , z , t )
E ( ρ , ϕ , z ) = a 2 J 0 ( α 1 ρ ) exp ( i β 1 z ) x ̂ + a 2 { exp ( i k z ) + exp [ i ( α 2 ρ cos ϕ + β 2 z ) ] } y ̂
i a a 1 β 1 cos ϕ J 1 ( α 1 ρ ) exp ( i β 1 z ) z ̂ ,
H ( ρ , ϕ , z ) = a ε 0 μ 0 { 1 2 exp ( i k z ) β 2 2 k exp [ i ( α 2 ρ cos ϕ + β 2 z ) ]
+ α 1 2 2 k β 1 J 2 ( α 1 ρ ) sin ( 2 ϕ ) exp ( i β 1 z ) } x ̂
+ a 2 k β 1 ε 0 μ 0 [ ( β 1 2 + k 2 ) J 0 ( α 1 ρ ) α 1 2 J 2 ( α 1 ρ ) cos ( 2 ϕ ) ] exp ( i β 1 z ) y ̂
+ a ε 0 μ 0 { α 2 2 k exp [ i ( α 2 ρ cos ϕ + β 2 z ) ] i α 1 k J 1 ( α 1 ρ ) sin ϕ exp ( i β 1 z ) } z ̂ .
w e ( r , t ) = a 2 ε 0 4 [ 1 2 + 1 2 cos ( 4 π z z T α 2 ρ cos ϕ ) + J 0 2 ( α 1 ρ ) + α 1 2 β 1 2 cos 2 ϕ J 1 2 ( α 1 ρ ) ]
w h ( r , t ) = a 2 ε 0 { 1 8 + α 1 4 16 k 2 β 1 2 J 2 2 ( α 1 ρ ) + α 1 2 4 k 2 J 1 2 ( α 1 ρ ) sin 2 ϕ
+ β 1 2 + k 2 16 k 2 β 1 2 J 0 ( α 1 ρ ) [ ( β 1 2 + k 2 ) J 0 ( α 1 ρ ) 2 α 1 2 J 2 ( α 1 ρ ) cos ( 2 ϕ ) ]
α 1 2 8 k β 1 J 2 ( α 1 ρ ) sin ( 2 ϕ ) cos ( 2 π z z T ) + β 2 8 k cos ( 4 π z z T α 2 ρ cos ϕ )
α 1 2 β 2 8 k 2 β 1 J 2 ( α 1 ρ ) sin ( 2 ϕ ) cos ( 2 π z z T α 2 ρ cos ϕ )
+ α 1 α 2 4 k 2 J 1 ( α 1 ρ ) sin ϕ sin ( 2 π z z T α 2 ρ cos ϕ ) } .

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