Abstract

General solutions and conditions are presented for paraxial waves that image themselves with different scales through free propagation. These waves, represented as superpositions of Gauss–Laguerre modes, have finite energy and thus finite effective width. The self-imaging wave fields described by Montgomery [J.  Opt.  Soc.  Am.    57, 772 (1967)], which possess a Fourier transform that is confined to a ring structure, are obtained as a specific limiting case of an infinite aperture.

© 1997 Optical Society of America

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References

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  1. G. Indebetouw, J. Opt. Soc. Am. A 6, 150 (1989).
    [CrossRef]
  2. G. Indebetouw, J. Mod. Opt. 40, 73 (1993).
    [CrossRef]
  3. Y. Y. Schechner, R. Piestun, and J. Shamir, Phys. Rev. E 54, R50 (1996).
    [CrossRef]
  4. W. D. Montgomery, J. Opt. Soc. Am. 57, 772 (1967).
    [CrossRef]
  5. W. D. Montgomery, J. Opt. Soc. Am. 58, 1112 (1968).
    [CrossRef]
  6. J. M. Cowley and A. F. Moodie, Proc. Phys. Soc. B 70, 505 (1957).
    [CrossRef]
  7. A. Kalestynski and B. Smolinska, Opt. Acta 25, 125 (1978).
    [CrossRef]
  8. A. P. Smirnov, Opt. Spectrosc. 44, 208 (1978).
  9. K. Patorski, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, The Netherlands, 1989), Vol. XXVII, pp. 3–108.
  10. P. Szwaykowski and J. Ojeda-Castaneda, Opt. Commun. 83, 1 (1991).
    [CrossRef]
  11. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).
  12. G. Nienhuis and L. Allen, Phys. Rev. A 48, 656 (1993).
    [CrossRef] [PubMed]

1996 (1)

Y. Y. Schechner, R. Piestun, and J. Shamir, Phys. Rev. E 54, R50 (1996).
[CrossRef]

1993 (2)

G. Nienhuis and L. Allen, Phys. Rev. A 48, 656 (1993).
[CrossRef] [PubMed]

G. Indebetouw, J. Mod. Opt. 40, 73 (1993).
[CrossRef]

1991 (1)

P. Szwaykowski and J. Ojeda-Castaneda, Opt. Commun. 83, 1 (1991).
[CrossRef]

1989 (1)

1978 (2)

A. Kalestynski and B. Smolinska, Opt. Acta 25, 125 (1978).
[CrossRef]

A. P. Smirnov, Opt. Spectrosc. 44, 208 (1978).

1968 (1)

1967 (1)

1957 (1)

J. M. Cowley and A. F. Moodie, Proc. Phys. Soc. B 70, 505 (1957).
[CrossRef]

Allen, L.

G. Nienhuis and L. Allen, Phys. Rev. A 48, 656 (1993).
[CrossRef] [PubMed]

Cowley, J. M.

J. M. Cowley and A. F. Moodie, Proc. Phys. Soc. B 70, 505 (1957).
[CrossRef]

Indebetouw, G.

Kalestynski, A.

A. Kalestynski and B. Smolinska, Opt. Acta 25, 125 (1978).
[CrossRef]

Montgomery, W. D.

Moodie, A. F.

J. M. Cowley and A. F. Moodie, Proc. Phys. Soc. B 70, 505 (1957).
[CrossRef]

Nienhuis, G.

G. Nienhuis and L. Allen, Phys. Rev. A 48, 656 (1993).
[CrossRef] [PubMed]

Ojeda-Castaneda, J.

P. Szwaykowski and J. Ojeda-Castaneda, Opt. Commun. 83, 1 (1991).
[CrossRef]

Patorski, K.

K. Patorski, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, The Netherlands, 1989), Vol. XXVII, pp. 3–108.

Piestun, R.

Y. Y. Schechner, R. Piestun, and J. Shamir, Phys. Rev. E 54, R50 (1996).
[CrossRef]

Schechner, Y. Y.

Y. Y. Schechner, R. Piestun, and J. Shamir, Phys. Rev. E 54, R50 (1996).
[CrossRef]

Shamir, J.

Y. Y. Schechner, R. Piestun, and J. Shamir, Phys. Rev. E 54, R50 (1996).
[CrossRef]

Smirnov, A. P.

A. P. Smirnov, Opt. Spectrosc. 44, 208 (1978).

Smolinska, B.

A. Kalestynski and B. Smolinska, Opt. Acta 25, 125 (1978).
[CrossRef]

Szwaykowski, P.

P. Szwaykowski and J. Ojeda-Castaneda, Opt. Commun. 83, 1 (1991).
[CrossRef]

J. Mod. Opt. (1)

G. Indebetouw, J. Mod. Opt. 40, 73 (1993).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

A. Kalestynski and B. Smolinska, Opt. Acta 25, 125 (1978).
[CrossRef]

Opt. Commun. (1)

P. Szwaykowski and J. Ojeda-Castaneda, Opt. Commun. 83, 1 (1991).
[CrossRef]

Opt. Spectrosc. (1)

A. P. Smirnov, Opt. Spectrosc. 44, 208 (1978).

Phys. Rev. A (1)

G. Nienhuis and L. Allen, Phys. Rev. A 48, 656 (1993).
[CrossRef] [PubMed]

Phys. Rev. E (1)

Y. Y. Schechner, R. Piestun, and J. Shamir, Phys. Rev. E 54, R50 (1996).
[CrossRef]

Proc. Phys. Soc. B (1)

J. M. Cowley and A. F. Moodie, Proc. Phys. Soc. B 70, 505 (1957).
[CrossRef]

Other (2)

K. Patorski, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, The Netherlands, 1989), Vol. XXVII, pp. 3–108.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).

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Equations (28)

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f(r,t)=u(r)exp[i(kz-ωt)],
un,m(r)=G(ρ˜,z˜)Rn,m(ρ˜)Φm(ϕ)Zn(z˜),
G(ρ˜,z˜)=w0w(z˜)exp(-ρ˜2)exp(iρ˜2z˜)exp[-iψ(z˜)],
Rn,m(ρ˜)=2ρ˜|m|L(n-|m|)/2|m|(2ρ˜2),
Φm(ϕ)=exp(imϕ),
Zn(z˜)=exp[-inψ(z˜)],
n=|m|,|m|+2,|m|+4,.
u(r)=j=1SAjunj,mj(r),
I(r)=|G(ρ˜)|2j=1S|Aj|2Rnj,mj2(ρ˜)+j=1Sp=j+1S2|Aj||Ap|Rnj,mj(ρ˜)×Rnp,mp(ρ˜)cos[Δmjpϕ-Δnjpψ(z˜)-ϑjp]
I(ρ˜,ϕ,z1)=I(ρ˜,ϕ+2πN,z2)for all ρ˜,ϕ.
cos[Δmjpϕ-Δnjpψ(z˜1)-ϑjp]=cos[Δmjp(ϕ+2πN)-Δnjpψ(z˜2)-ϑjp].
Δmjpϕ-Δnjpψ(z˜1)-ϑjp=[Δmjp(ϕ+2πN)-Δnjpψ(z˜2)-ϑjp]+2πN
Δnjp=NjpΩ(z˜2,z˜1)for all modes j,p,
Ω(z˜2,z˜1)2πψ(z˜2)-ψ(z˜1)=2πΔψ
nj=n1+NjΩ(z˜2,z˜1)for all j,
M=w(z˜2)/w(z˜1)={[1+(z˜2)2]/[1+(z˜1)2]}1/2.
u(r)=w0w(z˜)exp[-i(1+n1)ψ(z˜)]×exp(iρ˜2z˜)j=1MAjKnj,mj(ρ˜)Φmj(ϕ)×exp[-i(nj-n1)ψ(z˜)],
phase(z̃2)=phase(z̃1)-(1+n1)[ψ(z̃2)-ψ(z̃1)]+(z̃2-z̃1)ρ̃2.
z02=[Δz2+2z1Δz-(M2-1)z12]/(M2-1),
ξ2=ξ12+(2/λΔz)N,
nj-n12πNj(z0/Δz)z˜0.
(π/λ)tan(θbeam)=n+1/w0n1n/w0.
λnj/w0=constant(j)>0for all j
(Δx)2=(1/4)(n+1)w2(z˜)z˜1,n1nw02/4.
η-aLηa(χ/η)ηχ-a/2Ja(2χ),
un,mlimit(r)=c(n,m)exp-ρ2w02×J|m|2πρξ12+2λΔzN1/2×exp(imϕ)exp-inz0z
Un,mlimit(ξ,Θ)exp[-π(w0ξ)2]* ringN,m(ξ,Θ),
ringN,m(ξ,Θ)δ{ξ-[ξ12+(2/λΔz)N]1/2}exp(imΘ).

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