Abstract

Necessary and sufficient conditions are presented that determine a generalized class of propagation-invariant wave fields. The existence of wave fields with transverse distributions that are periodically reproduced with different azimuthal orientations is demonstrated. These fields are conveniently described in the longitudinal-azimuthal frequency representation. An interesting subclass is characterized by aperiodic rotated self-images, in the sense that they never return to their original orientation along the propagation. Other subclasses include the conventional self-imaging wave fields, the so-called nondiffracting beams, and the rotating wave fields.

© 1998 Optical Society of America

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References

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  1. J. Durnin, “Exact solutions for nondiffracting beams,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  2. G. Indebetouw, “Nondiffracting optical fields: some remarks on their analysis and synthesis,” J. Opt. Soc. Am. A 6, 150–152 (1989).
    [CrossRef]
  3. A. B. Valyaev, S. G. Krivoshlykov, “Mode properties of Bessel beams,” Sov. J. Quantum Electron. 19, 679–680 (1989).
    [CrossRef]
  4. P. Szwaykowski, J. Ojeda-Castañeda, “Nondiffracting beams and the self-imaging phenomenon,” Opt. Commun. 83, 1–4 (1991).
    [CrossRef]
  5. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
    [CrossRef]
  6. E. Abramochkin, V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993).
    [CrossRef]
  7. Y. Y. Schechner, R. Piestun, J. Shamir, “Wave propagation with rotating intensity distributions,” Phys. Rev. E 54, R50–R53 (1996).
    [CrossRef]
  8. S. Chavez-Cerda, G. S. McDonald, G. H. C. New, “Nondiffracting beams: travelling, standing, rotating and spiral waves,” Opt. Commun. 123, 225–233 (1996).
    [CrossRef]
  9. C. Paterson, R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. 124, 131–140 (1996).
    [CrossRef]
  10. V. V. Kotlyar, V. A. Soifer, S. N. Khonina, “An algorithm for the generation of laser beams with longitudinal periodicity: rotating images,” J. Mod. Opt. 44, 1409–1416 (1997).
    [CrossRef]
  11. G. Indebetouw, “Polychromatic self-imaging,” J. Mod. Opt. 35, 243–252 (1988).
    [CrossRef]
  12. J. Turunen, A. Vasara, A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. 8, 282–289 (1991).
    [CrossRef]
  13. Z. Bouchal, R. Horak, J. Wagner, “Propagation-invariant electromagnetic fields: theory and experiment,” J. Mod. Opt. 43, 1905–1920 (1996).
    [CrossRef]
  14. R. Piestun, Y. Y. Schechner, J. Shamir, “Self-imaging with finite energy,” Opt. Lett. 22, 200–202 (1997).
    [CrossRef] [PubMed]
  15. R. Piestun, Y. Y. Schechner, J. Shamir, “Generalized self-imaging in free space,” Diffractive Optics ’97, Vol. 12 of European Optical Society Topical Meetings Digest Series (European Optical Society, Orsay, France, 1997), pp. 128–129.
  16. W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57, 772–778 (1967).
    [CrossRef]
  17. W. D. Montgomery, “Algebraic formulation of diffraction applied to self-imaging,” J. Opt. Soc. Am. 58, 1112–1124 (1968).
    [CrossRef]
  18. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).
  19. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).
  20. J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
    [CrossRef]
  21. G. Indebetouw, “Quasi-self-imaging using aperiodic sequences,” J. Opt. Soc. Am. A 9, 549–558 (1992).
    [CrossRef]

1997 (2)

V. V. Kotlyar, V. A. Soifer, S. N. Khonina, “An algorithm for the generation of laser beams with longitudinal periodicity: rotating images,” J. Mod. Opt. 44, 1409–1416 (1997).
[CrossRef]

R. Piestun, Y. Y. Schechner, J. Shamir, “Self-imaging with finite energy,” Opt. Lett. 22, 200–202 (1997).
[CrossRef] [PubMed]

1996 (4)

Z. Bouchal, R. Horak, J. Wagner, “Propagation-invariant electromagnetic fields: theory and experiment,” J. Mod. Opt. 43, 1905–1920 (1996).
[CrossRef]

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

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

C. Paterson, R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. 124, 131–140 (1996).
[CrossRef]

1993 (2)

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

E. Abramochkin, V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993).
[CrossRef]

1992 (1)

1991 (2)

J. Turunen, A. Vasara, A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. 8, 282–289 (1991).
[CrossRef]

P. Szwaykowski, J. Ojeda-Castañeda, “Nondiffracting beams and the self-imaging phenomenon,” Opt. Commun. 83, 1–4 (1991).
[CrossRef]

1989 (2)

A. B. Valyaev, S. G. Krivoshlykov, “Mode properties of Bessel beams,” Sov. J. Quantum Electron. 19, 679–680 (1989).
[CrossRef]

G. Indebetouw, “Nondiffracting optical fields: some remarks on their analysis and synthesis,” J. Opt. Soc. Am. A 6, 150–152 (1989).
[CrossRef]

1988 (1)

G. Indebetouw, “Polychromatic self-imaging,” J. Mod. Opt. 35, 243–252 (1988).
[CrossRef]

1987 (1)

1974 (1)

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

1968 (1)

1967 (1)

Abramochkin, E.

E. Abramochkin, V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993).
[CrossRef]

Berry, M. V.

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Bouchal, Z.

Z. Bouchal, R. Horak, J. Wagner, “Propagation-invariant electromagnetic fields: theory and experiment,” J. Mod. Opt. 43, 1905–1920 (1996).
[CrossRef]

Chavez-Cerda, S.

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

Durnin, J.

Friberg, A. T.

J. Turunen, A. Vasara, A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. 8, 282–289 (1991).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

Horak, R.

Z. Bouchal, R. Horak, J. Wagner, “Propagation-invariant electromagnetic fields: theory and experiment,” J. Mod. Opt. 43, 1905–1920 (1996).
[CrossRef]

Indebetouw, G.

Khonina, S. N.

V. V. Kotlyar, V. A. Soifer, S. N. Khonina, “An algorithm for the generation of laser beams with longitudinal periodicity: rotating images,” J. Mod. Opt. 44, 1409–1416 (1997).
[CrossRef]

Kotlyar, V. V.

V. V. Kotlyar, V. A. Soifer, S. N. Khonina, “An algorithm for the generation of laser beams with longitudinal periodicity: rotating images,” J. Mod. Opt. 44, 1409–1416 (1997).
[CrossRef]

Krivoshlykov, S. G.

A. B. Valyaev, S. G. Krivoshlykov, “Mode properties of Bessel beams,” Sov. J. Quantum Electron. 19, 679–680 (1989).
[CrossRef]

McDonald, G. S.

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

Montgomery, W. D.

New, G. H. C.

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

Nye, J. F.

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Ojeda-Castañeda, J.

P. Szwaykowski, J. Ojeda-Castañeda, “Nondiffracting beams and the self-imaging phenomenon,” Opt. Commun. 83, 1–4 (1991).
[CrossRef]

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Paterson, C.

C. Paterson, R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. 124, 131–140 (1996).
[CrossRef]

Piestun, R.

R. Piestun, Y. Y. Schechner, J. Shamir, “Self-imaging with finite energy,” Opt. Lett. 22, 200–202 (1997).
[CrossRef] [PubMed]

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

R. Piestun, Y. Y. Schechner, J. Shamir, “Generalized self-imaging in free space,” Diffractive Optics ’97, Vol. 12 of European Optical Society Topical Meetings Digest Series (European Optical Society, Orsay, France, 1997), pp. 128–129.

Schechner, Y. Y.

R. Piestun, Y. Y. Schechner, J. Shamir, “Self-imaging with finite energy,” Opt. Lett. 22, 200–202 (1997).
[CrossRef] [PubMed]

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

R. Piestun, Y. Y. Schechner, J. Shamir, “Generalized self-imaging in free space,” Diffractive Optics ’97, Vol. 12 of European Optical Society Topical Meetings Digest Series (European Optical Society, Orsay, France, 1997), pp. 128–129.

Shamir, J.

R. Piestun, Y. Y. Schechner, J. Shamir, “Self-imaging with finite energy,” Opt. Lett. 22, 200–202 (1997).
[CrossRef] [PubMed]

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

R. Piestun, Y. Y. Schechner, J. Shamir, “Generalized self-imaging in free space,” Diffractive Optics ’97, Vol. 12 of European Optical Society Topical Meetings Digest Series (European Optical Society, Orsay, France, 1997), pp. 128–129.

Smith, R.

C. Paterson, R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. 124, 131–140 (1996).
[CrossRef]

Soifer, V. A.

V. V. Kotlyar, V. A. Soifer, S. N. Khonina, “An algorithm for the generation of laser beams with longitudinal periodicity: rotating images,” J. Mod. Opt. 44, 1409–1416 (1997).
[CrossRef]

Szwaykowski, P.

P. Szwaykowski, J. Ojeda-Castañeda, “Nondiffracting beams and the self-imaging phenomenon,” Opt. Commun. 83, 1–4 (1991).
[CrossRef]

Turunen, J.

J. Turunen, A. Vasara, A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. 8, 282–289 (1991).
[CrossRef]

Valyaev, A. B.

A. B. Valyaev, S. G. Krivoshlykov, “Mode properties of Bessel beams,” Sov. J. Quantum Electron. 19, 679–680 (1989).
[CrossRef]

Vasara, A.

J. Turunen, A. Vasara, A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. 8, 282–289 (1991).
[CrossRef]

Volostnikov, V.

E. Abramochkin, V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993).
[CrossRef]

Wagner, J.

Z. Bouchal, R. Horak, J. Wagner, “Propagation-invariant electromagnetic fields: theory and experiment,” J. Mod. Opt. 43, 1905–1920 (1996).
[CrossRef]

J. Mod. Opt. (4)

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

V. V. Kotlyar, V. A. Soifer, S. N. Khonina, “An algorithm for the generation of laser beams with longitudinal periodicity: rotating images,” J. Mod. Opt. 44, 1409–1416 (1997).
[CrossRef]

G. Indebetouw, “Polychromatic self-imaging,” J. Mod. Opt. 35, 243–252 (1988).
[CrossRef]

Z. Bouchal, R. Horak, J. Wagner, “Propagation-invariant electromagnetic fields: theory and experiment,” J. Mod. Opt. 43, 1905–1920 (1996).
[CrossRef]

J. Opt. Soc. Am. (3)

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

Opt. Commun. (4)

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

C. Paterson, R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. 124, 131–140 (1996).
[CrossRef]

E. Abramochkin, V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993).
[CrossRef]

P. Szwaykowski, J. Ojeda-Castañeda, “Nondiffracting beams and the self-imaging phenomenon,” Opt. Commun. 83, 1–4 (1991).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. E (1)

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

Proc. R. Soc. London Ser. A (1)

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Sov. J. Quantum Electron. (1)

A. B. Valyaev, S. G. Krivoshlykov, “Mode properties of Bessel beams,” Sov. J. Quantum Electron. 19, 679–680 (1989).
[CrossRef]

Other (3)

R. Piestun, Y. Y. Schechner, J. Shamir, “Generalized self-imaging in free space,” Diffractive Optics ’97, Vol. 12 of European Optical Society Topical Meetings Digest Series (European Optical Society, Orsay, France, 1997), pp. 128–129.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

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Figures (6)

Fig. 1
Fig. 1

Longitudinal-azimuthal frequency domain (β-m plane) and the representation of PI WF’s. Bessel modes composing PI WF’s lie on equidistant parallel lines. Δz is the distance between self-images, and γ is their relative rotation.

Fig. 2
Fig. 2

Longitudinal spatial frequencies (β) of PI WF’s are restricted to specific periodic sequences of values. These values are obtained from the projection onto the β axis of the intersection of specific sets of equidistant parallel lines (see text). For clarity, only two lines are shown. Note that classical SI WF’s (a special case) possess uniformly spaced longitudinal spatial frequencies.

Fig. 3
Fig. 3

Longitudinal-azimuthal frequency domain of (a) self-imaging WF’s, (b) nondiffracting beams, (c) rotating WF’s, and (d) rotated self-imaging WF’s, periodic (γ rational) and aperiodic (γ irrational).

Fig. 4
Fig. 4

Transverse intensity of a rotating WF. The field distribution continuously rotates along and about the propagation axis. From left to right, γ(0)=0, γ(Δz¯/3)=120°, γ(2Δz¯/3)=240°. All the units are micrometers.

Fig. 5
Fig. 5

Aperiodic self-imaging WF: The transverse field is reproduced at different orientations but never returns to its original state. Note the change in the field at intermediate planes. From left to right, z=0, Δz/2, Δz; γ52°. All the units are micrometers.

Fig. 6
Fig. 6

Graphical representation of the relation among the different PI WF’s. Previously described PI WF’s are the SI with γ=0, the nondiffracting, and part of the rotating WF’s.

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

U(ρ, ϕ, z, t)=U(ρ, ϕ, z)exp(-iwt)
R[γ]g(r, ϕ)=g(r, ϕ-γ),
D[Δz]=F-1 exp(ik2-α2Δz)F.
R[γ]D[Δz]U(ρ, ϕ, z1)=μU(ρ, ϕ, z1),
U(ρ, ϕ, 0)=m=-am(ρ, 0)exp(imϕ),
am(ρ, 0)=12π -ππU(ρ,ϕ, 0)exp(-imϕ)dϕ.
FU(ρ, ϕ, 0)=2πm=-Am(α)exp(-imϕ),
Am(α)=0ρam(ρ, 0)Jm(αρ)dρ.
U(r)=m=-am(ρ, z)exp(imϕ),
am(ρ, z)=0kαAm(α)exp(ik2-α2z)Jm(αρ)dα.
umα(r)exp(-iwt)Jm(αρ)exp(iβz)×exp(imϕ)exp(-iwt),
U(ρ, ϕ-γ, z2)=μU(ρ, ϕ, z1)forallρ,ϕ.
U(ρ, ϕ-γ, z2)-μU(ρ, ϕ, z1)=m=- exp(imϕ)0αAm(α)×{exp[iβ(α)z2-imγ]-μ exp[iβ(α)z1]}Jm(αρ)dα=0
exp[iβ(α)z2-imγ]-μ exp[iβ(α)z1]=0.
β=βmn=β0+mγ+2πnΔz,0βk,
Am(α)=ncmnδ(α-αmn);
UPI(r)=m,nCmnJm(αmnρ)exp[i(βmnz+mϕ)],
αn2=k2-k2-α02+2πnΔz2,
UND(ρ, ϕ, z)=exp[iβ0z]m=-CmJm(α0ρ)exp(imϕ),
βm=β0+m γ1Δz1;0βmk,
dγdz=γ1Δz1.
ρ=ρ0,
ϕ=ϕ0+γ1Δz1z.
α2=k2-β0+mγ+2πnΔz2=k2-β0+2πn¯Δz¯2,

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