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.

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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. Olivik, "Non-diffractive vector Bessel beams," J. Mod. Opt. 42, 1555-1566 (1995).
    [CrossRef]
  6. Z. Bouchal, R. Horak, and J. Wagner, "Propagation-invariant electromagnetic fields," J. Mod. Opt. 43, 1905-1920 (1996).
    [CrossRef]
  7. R. Horak, 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. Chavez-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. 57 772-778 (1967).
    [CrossRef]
  20. W. D. Montgomery, "Algebraic formulation of diffraction applied to self imaging," J. Opt. Soc. Am. 58 1112-1124 (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).

Other (23)

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

J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1987).
[CrossRef]

S. R. Mishra, "A vector wave analysis of a Bessel beam," Opt. Commun. 85, 159-161 (1991).
[CrossRef]

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

Z. Bouchal and M. Olivik, "Non-diffractive vector Bessel beams," J. Mod. Opt. 42, 1555-1566 (1995).
[CrossRef]

Z. Bouchal, R. Horak, and J. Wagner, "Propagation-invariant electromagnetic fields," J. Mod. Opt. 43, 1905-1920 (1996).
[CrossRef]

R. Horak, 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]

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

Y. Y. Schechner, R. Piestun, and J. Shamir, "Wave propagation with rotating intensity distributions," Phys. Rev. E 54, R50-R53 (1996).
[CrossRef]

S. Chavez-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]

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

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

S. Ruschin and A. Leizer, "Evanescent Bessel beams," J. Opt. Soc. Am. A 15, 1139-1143 (1998).
[CrossRef]

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

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

W. D. Montgomery, "Self-imaging objects of infinite aperture," J. Opt. Soc. Am. 57 772-778 (1967).
[CrossRef]

W. D. Montgomery, "Algebraic formulation of diffraction applied to self imaging," J. Opt. Soc. Am. 58 1112-1124 (1968).
[CrossRef]

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

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).

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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