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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    [CrossRef]

J. Mod. Opt.

Z. Bouchal and M. Olivýk, ???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]

P. Paakkonen, 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. 46, 2355???2369 (1998).
[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.

J. Opt. Soc. Am. A

Opt. Commun.

J. Tervo and J. Turunen, ???Generation of vectorial propagation-invariant fields by polarizationgrating axicons,??? Opt. Commun. 192, 13???18 (2001).
[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 computergenerated 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]

Opt. Express

Phys. Rev. E

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.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, ???Diffraction-free beams,??? Phys. Rev. Lett. 58, 1499???1501 (1987).
[CrossRef] [PubMed]

Pure Appl. Opt.

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

Other

K. Patorski, ???The self-imaging phenomenon and its applications,??? in Progress in Optics Vol. 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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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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