Abstract

The vorticity of a monochromatic speckle beam is introduced as an expectation value of the local difference of densities of right and left vortices, or wave-front dislocations. Gaussian statistics allows for the complete description of a speckle beam on the basis of the correlation function E(r1)E*(r2) only, and this function depends on the coordinate R=(r1+r2)/2 explicitly for statistically inhomogeneous beams. An analytic expression is found both for the vorticity and for the sum of the right and the left vortex densities. The vorticity is shown to be nonzero for inhomogeneous beams only. The Poincaré–Cartan invariant of Hamilton’s classical mechanics or of geometrical optics is shown to be the topologically invariant integral of vorticity. An example is given of a beam with finite vorticity, which has Gaussian intensity profiles in both angular and spatial distributions. The conditions on the parameters that describe such a beam are found; these conditions follow from the positive character of probability.

© 1999 Optical Society of America

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References

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  1. J. F. Nye, R. G. Kyte, D. C. Threlfall, “Proposal for measuring the movement of a large ice sheet by observing radio echoes,” J. Glaciol. 2, 319–325 (1972).
  2. J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
    [CrossRef]
  3. N. B. Baranova, B. Ya. Zel’dovich, “Dislocations of the wave-front surface and zeroes of the amplitude,” JETP 53, 925–929 (1981).
  4. K. T. Gahagan, G. A. Swartzlander, “Trapping of low-index microparticles in an optical vortex,” J. Opt. Soc. Am. B 15, 524–534 (1998).
    [CrossRef]
  5. N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981).
  6. N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, “Experimental investigation of density of wavefront dislocations of speckle-inhomogeneous light fields,” JETP 56, 983–988 (1982).
  7. N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, “Wavefront dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. 73, 525–528 (1983).
    [CrossRef]
  8. B. Ya. Zel’dovich, N. F. Pilipetsky, V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, New York, 1985), Chap. 3.
  9. I. Freund, “Optical vortices in Gaussian random wave fields: statistical probability densities,” J. Opt. Soc. Am. A 11, 1644–1652 (1994).
    [CrossRef]
  10. N. R. Heckenberg, R. McDuff, C. P. Smith, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
    [CrossRef] [PubMed]
  11. I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, “Optics of light beams with screw dislocation,” Opt. Commun. 103, 422–428 (1993).
    [CrossRef]
  12. E. Wolf, L. Mandel, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  13. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  14. B. Ya. Zel’dovich, A. V. Mamaev, V. V. Shkunov, Speckle-Wave Interactions in Application to Holography and Nonlinear Optics (CRC Press, Boca Raton, 1995); V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, New York, 1989).
  15. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass.1980)
  16. V. S. Liberman, B. Ya. Zel’dovich, “Spin-orbit interaction of a photon in a inhomogeneous medium,” Phys. Rev. A 46, 5199–5207 (1992).
    [CrossRef] [PubMed]
  17. M. A. Bolshtyansky, A. Yu. Savchenko, B. Ya. Zel’dovich, “Use of skew rays in multimode fibers to generate speckle field with nonzero vorticity,” Opt. Lett. 24, 433–435 (1999).
    [CrossRef]

1999 (1)

1998 (1)

1994 (1)

1993 (1)

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, “Optics of light beams with screw dislocation,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

1992 (2)

V. S. Liberman, B. Ya. Zel’dovich, “Spin-orbit interaction of a photon in a inhomogeneous medium,” Phys. Rev. A 46, 5199–5207 (1992).
[CrossRef] [PubMed]

N. R. Heckenberg, R. McDuff, C. P. Smith, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[CrossRef] [PubMed]

1983 (1)

1982 (1)

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, “Experimental investigation of density of wavefront dislocations of speckle-inhomogeneous light fields,” JETP 56, 983–988 (1982).

1981 (2)

N. B. Baranova, B. Ya. Zel’dovich, “Dislocations of the wave-front surface and zeroes of the amplitude,” JETP 53, 925–929 (1981).

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981).

1974 (1)

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

1972 (1)

J. F. Nye, R. G. Kyte, D. C. Threlfall, “Proposal for measuring the movement of a large ice sheet by observing radio echoes,” J. Glaciol. 2, 319–325 (1972).

Baranova, N. B.

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, “Wavefront dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. 73, 525–528 (1983).
[CrossRef]

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, “Experimental investigation of density of wavefront dislocations of speckle-inhomogeneous light fields,” JETP 56, 983–988 (1982).

N. B. Baranova, B. Ya. Zel’dovich, “Dislocations of the wave-front surface and zeroes of the amplitude,” JETP 53, 925–929 (1981).

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981).

Basistiy, I. V.

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, “Optics of light beams with screw dislocation,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

Bazhenov, V. Yu.

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, “Optics of light beams with screw dislocation,” Opt. Commun. 103, 422–428 (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]

Bolshtyansky, M. A.

Freund, I.

Gahagan, K. T.

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass.1980)

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Heckenberg, N. R.

Kyte, R. G.

J. F. Nye, R. G. Kyte, D. C. Threlfall, “Proposal for measuring the movement of a large ice sheet by observing radio echoes,” J. Glaciol. 2, 319–325 (1972).

Liberman, V. S.

V. S. Liberman, B. Ya. Zel’dovich, “Spin-orbit interaction of a photon in a inhomogeneous medium,” Phys. Rev. A 46, 5199–5207 (1992).
[CrossRef] [PubMed]

Mamaev, A. V.

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, “Wavefront dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. 73, 525–528 (1983).
[CrossRef]

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, “Experimental investigation of density of wavefront dislocations of speckle-inhomogeneous light fields,” JETP 56, 983–988 (1982).

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981).

B. Ya. Zel’dovich, A. V. Mamaev, V. V. Shkunov, Speckle-Wave Interactions in Application to Holography and Nonlinear Optics (CRC Press, Boca Raton, 1995); V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, New York, 1989).

Mandel, L.

E. Wolf, L. Mandel, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

McDuff, R.

Nye, J. F.

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

J. F. Nye, R. G. Kyte, D. C. Threlfall, “Proposal for measuring the movement of a large ice sheet by observing radio echoes,” J. Glaciol. 2, 319–325 (1972).

Pilipetsky, N. F.

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, “Wavefront dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. 73, 525–528 (1983).
[CrossRef]

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, “Experimental investigation of density of wavefront dislocations of speckle-inhomogeneous light fields,” JETP 56, 983–988 (1982).

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981).

B. Ya. Zel’dovich, N. F. Pilipetsky, V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, New York, 1985), Chap. 3.

Savchenko, A. Yu.

Shkunov, V. V.

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, “Wavefront dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. 73, 525–528 (1983).
[CrossRef]

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, “Experimental investigation of density of wavefront dislocations of speckle-inhomogeneous light fields,” JETP 56, 983–988 (1982).

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981).

B. Ya. Zel’dovich, A. V. Mamaev, V. V. Shkunov, Speckle-Wave Interactions in Application to Holography and Nonlinear Optics (CRC Press, Boca Raton, 1995); V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, New York, 1989).

B. Ya. Zel’dovich, N. F. Pilipetsky, V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, New York, 1985), Chap. 3.

Smith, C. P.

Soskin, M. S.

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, “Optics of light beams with screw dislocation,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

Swartzlander, G. A.

Threlfall, D. C.

J. F. Nye, R. G. Kyte, D. C. Threlfall, “Proposal for measuring the movement of a large ice sheet by observing radio echoes,” J. Glaciol. 2, 319–325 (1972).

Wolf, E.

E. Wolf, L. Mandel, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

Zel’dovich, B. Ya.

M. A. Bolshtyansky, A. Yu. Savchenko, B. Ya. Zel’dovich, “Use of skew rays in multimode fibers to generate speckle field with nonzero vorticity,” Opt. Lett. 24, 433–435 (1999).
[CrossRef]

V. S. Liberman, B. Ya. Zel’dovich, “Spin-orbit interaction of a photon in a inhomogeneous medium,” Phys. Rev. A 46, 5199–5207 (1992).
[CrossRef] [PubMed]

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, “Wavefront dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. 73, 525–528 (1983).
[CrossRef]

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, “Experimental investigation of density of wavefront dislocations of speckle-inhomogeneous light fields,” JETP 56, 983–988 (1982).

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981).

N. B. Baranova, B. Ya. Zel’dovich, “Dislocations of the wave-front surface and zeroes of the amplitude,” JETP 53, 925–929 (1981).

B. Ya. Zel’dovich, N. F. Pilipetsky, V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, New York, 1985), Chap. 3.

B. Ya. Zel’dovich, A. V. Mamaev, V. V. Shkunov, Speckle-Wave Interactions in Application to Holography and Nonlinear Optics (CRC Press, Boca Raton, 1995); V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, New York, 1989).

J. Glaciol. (1)

J. F. Nye, R. G. Kyte, D. C. Threlfall, “Proposal for measuring the movement of a large ice sheet by observing radio echoes,” J. Glaciol. 2, 319–325 (1972).

J. Opt. Soc. Am. (1)

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

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

JETP (2)

N. B. Baranova, B. Ya. Zel’dovich, “Dislocations of the wave-front surface and zeroes of the amplitude,” JETP 53, 925–929 (1981).

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, “Experimental investigation of density of wavefront dislocations of speckle-inhomogeneous light fields,” JETP 56, 983–988 (1982).

JETP Lett. (1)

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981).

Opt. Commun. (1)

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, “Optics of light beams with screw dislocation,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (1)

V. S. Liberman, B. Ya. Zel’dovich, “Spin-orbit interaction of a photon in a inhomogeneous medium,” Phys. Rev. A 46, 5199–5207 (1992).
[CrossRef] [PubMed]

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]

Other (5)

B. Ya. Zel’dovich, N. F. Pilipetsky, V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, New York, 1985), Chap. 3.

E. Wolf, L. Mandel, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

B. Ya. Zel’dovich, A. V. Mamaev, V. V. Shkunov, Speckle-Wave Interactions in Application to Holography and Nonlinear Optics (CRC Press, Boca Raton, 1995); V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, New York, 1989).

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass.1980)

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Figures (1)

Fig. 1
Fig. 1

Contours of integration C0, at t0=0, and C1, at t1=t, in the phase space give the same values for the Poincaré–Cartan integral, and hence this integral is an invariant.

Equations (85)

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E(r, z)=r exp(-iωt+ik0z+iϕ-r2/2a2),
k0=ω/c,x=r cos ϕ,y=r sin ϕ.
r=x ex+y ey,r3D=z ez+r.
V(r, z)=N+(r, z)-N-(r, z).
E(r1, z)E*(r2, z)=G(r1, r2, z)exp{-H(r1, r2, z)},
H(r1, r2, z)=H*(r2, r1, z),
PCi=i=1Kdxi  dpii=1Kpidxi.
ω(2)=i=1Kdxi  dpi,
dxi/dt=H/pi,dpi/dt=-H/xi,
H=H(xi, pi, t).
ω(4)=ω(2)  ω(2),ω(6)=ω(4)  ω(2),
x1x,x2y,x3z,
p1px=Bkx,p2py=Bky,
p3pz=Bkz,
k=(ωn/c)s,HH(r, z)=(c/n){pp}1/2.
s=p/p=(c/ωn)k=s+ez/(1-s2)1/2;
dr3D/dt=(c/n)p/p(c/n)s,
dp/dt=(c/n)p ln(n).
dr/dz=H1/k,
dk/dz=-H1/r,
H1(r, k, z)={(ωn/c)2-kk}1/2.
E(r, z)|E(r, z)|exp[iψ(r, z)]
Cgradgrad ψ(r, z)dr=Ckdri=x,ydxi  dki;
PCii=x,ydxi dki=2π(n+-n-)=V(x, y)dxdy.
dP=const.×exp{-ΣTikZi*Zk}d Re(Z1)d Im(Z1)×d Re(Z2)d Im(Z2)d Re(Zk)d Im(Zk).
(T-1)ik=ZiZk*.
V(r, z)=N+(r, z)-N-(r, z),
N(r, z)=N+(r, z)+N-(r, z).
V(r, z)dxdy=δ {Re[E(r, z)]}×δ {Im[E(r, z)]}Ddxdy,
N(r, z)dxdy=δ {Re[E(r, z)]}×δ {Im[E(r, z)]}|D|dxdy.
D={Re[E(r, z)]/ x}{ Im[E(r, z)]/ y}-{ Re[E(r, z)]/ y}{ Im[E(r, z)]/ x}.
D=(i/2)(LM*-L*M).
LL1+iL2=E/x,L*=E*/x,
MM1+iM2=E/y,M*=E*/y.
l=L-ELE*/EE*,
m=M-EME*/EE*,
dPE=(πEE*)-1 exp(-EE*/EE*)×d Re(E1)d Im(E1)
D=(i/2)(lm*-l*m),
D=(i/2)[qm*-q*m+mm*×(lm*-ml*)/mm*].
uS exp(iα)=q/qq*1/2,
vT exp(iβ)=m/mm*1/2,
D=EE*[2πA(u*v-uv*)/(2i)+πBvv*]EE*[2πAST sin(β-α)+πBT2].
A=12π 2Hx1x2  2Hy1y2-2Hx1y2  2Hy1x21/2,
A>0,
B=i2π 2Hy1x2-2Hy2x1.
dP=π-2 exp[-(T2+S2)]STdSdTdαdβ.
V=2π2 02πdγ0π/2dτ cos τ sin τ[2πA cos τ sin τ sin γ+πB(cos τ)2],
N=2π2 02πdγ0π/2dτ cos τ sin τ|2πA cos τ sin τ sin γ+πB(cos τ)2|.
V=B.
N(A2+B2)1/2.
sx(R)=dkxdky(kx/k0)W(R, k)dkxdkyW(R, k).
V=k02syX-sxY.
|E(r, z=0)|2=exp(-r2/a2).
E(r1)E(r2)*=exp[-H(r1, r2, z=0)],
H(r1, r2, z=0)=14a2 {(r1+r2)2+μ2(r1-r2)2+4πia2V[r1×r2]ez}r122a2+r222a2+π(r1-r2)2×A2+V221/2+πiVez×(r1+r2)2(r1-r2),
W(R, k, z=0)
=const.×exp-R2a2-a2{k-πV[R×ez]}2μ2.
W(R, k, z)=W(R-kz/k0, k, z=0).
G(r1, r2, z)
=z02z2+z02 exp-(r1+r2)2+μ2(r1-r2)24a2(1+z2/z02)+iπ V[r1×r2]ez(1+z2/z02)+i k0z(r12-r22)2z2+2z02,
z02=a4k02[μ2+(πa2V)2]1/2,
G(r1, r2, z)
=n=0m=-In,mΨn,m(r1, z)Ψn,m*(r2, z),
In,m=exp[-α-β(2n+|m|)-γm],
α=lnμ2+1+2[μ2+(πa2V)2]1/24πa2,
β=lnμ2+1+2[μ2+(πa2V)2]1/2[(μ2-1)2-(2πa2V)2]1/2,
γ=lnμ2-1-2πa2Vμ2-1+2πa2V,
Ψm,n(r, z)
=Nm,n(z)exp-(1-iz/z0)r22b2(z)+imϕ rb(z)|m|×Ln|m|r2b2(z),
Nm,n(z)=1b(z) exp-i(2n+|m|+1)arctanzz0×n!π(n+|m|)!1/2,
b2(z)=a2(1+z2/z02)[μ2+(πa2V)2]1/2.
1+2πa2|V|μ2.
exp[-(En,m-LΩ)/kB T].
X =(Σn,m In,m)-1(Σn,mXn,m In,m).
K =(Σn,m In,m)2/[Σn,m(In,m)2].
K=μ2.
dxi(t+Δt)=dxi(t)+Δtd{H/pi}+O(Δt2),
dpi(t+Δt)=dpi(t)-Δtd{H/xi}+O(Δt2).
d{H/xi}=Σj {(2H/xixj)dxj+(2H/xipj)dpj},
d{H/pi}=Σj {(2H/pixj)dxj+(2H/pipj)dpj}.
dxi  dxj=-dxj  dxi,dxi  dpj=-dpj  dxi,
dpi  dpj=-dpj  dpi,
Σidxi(t+Δt)  dpi(t+Δt)
=Σidxi(t)  dpi(t)+O(Δt2).
Q.E.D.

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