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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  1. W. D. Montgomery, "Self-imaging objects of infinite aperture," J. Opt. Soc. Am. 57 772-778 (1967).
    [CrossRef]
  2. W. D. Montgomery, "Algebraic formulation of diffraction applied to self imaging," J. Opt. Soc. Am. 58 1112-1124 (1968).
    [CrossRef]
  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, Jr., and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58 1499-1501 (1987).
    [CrossRef] [PubMed]
  5. S. Ch�vez-Cerda, G. S. McDonald, and G. H. S. New, "Nondiffracting Beams: travelling, standing, rotating and spiral waves," Opt. Commun. 123 225-233 (1996).
    [CrossRef]
  6. C. Paterson and R. Smith, "Higher-order Bessel waves produced by axicon-type computer-generated holograms," Opt.Commun. 124 121-130 (1996).
    [CrossRef]
  7. V. V. Kotlyar, V. A. Soifer and S. N. Khonina, "An algorithm for the generation of laser-beams with longitudinal periodicity," J. Mod. Opt. 44 1409-1416 (1997).
    [CrossRef]
  8. P. P��kk�nen, 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]
  9. 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 110 137-144 (1999).
  10. S. R. Mishra, "A vector wave analysis of a Bessel beam," Opt. Commun. 85 159-161 (1991).
    [CrossRef]
  11. J. Turunen and A. T. Friberg, "Self-imaging and propagation-invariance in electromagnetic fields," Pure Appl. Opt. 2 51-60 (1993).
    [CrossRef]
  12. Z. Bouchal and M. Oliv�k, "Non-diffractive vector Bessel beams," J. Mod. Opt. 8 1555-1566 (1995).
    [CrossRef]
  13. Z. Bouchal, R. Hor�k and J. Wagner, "Propagation-invariant electromagnetic fields," J. Mod. Opt. 9 1905-1920 (1996).
    [CrossRef]
  14. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995), sect.3.2.
  15. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic Press, New York, 2001), p. 681.
  16. H. F. Talbot, "Facts relating to optical science, No. IV," Philos. Mag. 9 401-407 (1836).

Other

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]

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

S. Ch�vez-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]

V. V. Kotlyar, V. A. Soifer and S. N. Khonina, "An algorithm for the generation of laser-beams with longitudinal periodicity," J. Mod. Opt. 44 1409-1416 (1997).
[CrossRef]

P. P��kk�nen, 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]

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 110 137-144 (1999).

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. Oliv�k, "Non-diffractive vector Bessel beams," J. Mod. Opt. 8 1555-1566 (1995).
[CrossRef]

Z. Bouchal, R. Hor�k and J. Wagner, "Propagation-invariant electromagnetic fields," J. Mod. Opt. 9 1905-1920 (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. 9 401-407 (1836).

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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Figures (4)

Fig. 1.
Fig. 1.

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

Fig. 2.
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.

Fig. 3.
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.

Fig. 4.
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.

Tables (2)

Tables Icon

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

Tables Icon

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

Equations (33)

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U ( ρ , ϕ , z ) = 0 0 2 π α A ( α , ψ ) exp [ i α ρ cos ( ϕ ψ ) + i β z ] d α d ψ ,
A ( α , ψ ) = 1 2 π 0 0 2 π ρ U ( ρ , ϕ , 0 ) exp [ i α ρ cos ( ϕ ψ ) ] d ρ d ϕ ,
β = { ( k 2 α 2 ) 1 2 if α k i ( α 2 k 2 ) 1 2 if k < α .
U ( ρ , ϕ + η Δ z , z + Δ z ) = exp [ i ξ ( ρ , ϕ , Δ z ) ] U ( ρ , ϕ , z ) .
0 = m = 0 a m ( α ) J m ( αρ ) exp ( i m ϕ )
× { exp [ i ξ ( ρ , ϕ , Δ z ) ] exp [ i ( + β ) Δ z ] } d α ,
a m ( α ) = i m α 0 2 π A ( α , ψ ) exp ( i ) d ψ .
exp ( i ϑ cos φ ) = m =- i m J m ( ϑ ) exp ( i ) .
ξ ( ρ , ϕ , Δ z ) = ξ ( Δ z ) = ( + β ) Δ z + 2 π q ,
β m = β 0 ,
U ( ρ , ϕ , z ) = m M a m J m ( α m ρ ) exp [ i ( + 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
{ 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 ) ,
E ρ ( ρ , ϕ , z ) = m = 0 [ a m ( α ) cos ϕ + b m ( α ) sin ϕ ]
× J m ( αρ ) exp [ i ( + βz ) ] d α ,
E ϕ ( ρ , ϕ , z ) m = 0 [ a m ( α ) sin ϕ + b m ( α ) cos ϕ ]
× J m ( αρ ) exp [ i ( + β z ) ] d α ,
{ 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 ,
{ 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 ) ] .
h m ( ρ , ϕ , Δ z ) = ± i g m ( ρ , ϕ , Δ z )
b m ( α ) = ± i a m ( α ) ,
[ ( m ± 1 ) γ + k z ] Δ z = ξ ( ρ , ϕ , Δ z ) .
β mn = β 0 n ( m + n ) γ ,
E ρ ( ρ , ϕ , z ) = m M n = 1,1 a mn J m ( α mn ρ ) exp { i [ ( m + n ) ϕ + β mn z ] }
E ϕ ( ρ , ϕ , z ) = m M n = 1,1 i na mn J m ( α mn ρ ) exp { i [ ( m + n ) ϕ + β mn z ] } ,
· E ( ρ , ϕ , z ) = 1 ρ ρ [ ρ E ρ ( ρ , ϕ , z ) ] + 1 ρ ϕ E ϕ ( ρ , ϕ , z ) + z E z ( ρ , ϕ , z ) = 0 ,
E z ( ρ , ϕ , z ) = m M n = 1,1 na mn α mn i β mn J m + n ( α mn ρ ) exp { i [ ( m + n ) ϕ + β mn z ] } .
{ β j = β 0 = β 0 n γ ( m + n j ) , β m = β 0 = β 0 n γ ( m + n m ) .
η γ = P Q = m j + n m n j m j .

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