Abstract

Inhomogeneous optical vortex densities can be produced in stochastic optical fields by a combination of coherent and incoherent superposition of speckle fields. During subsequent propagation, the inhomogeneity in the vortex density decays away. However, the decay curves contain oscillatory features that are counterintuitive: for a short while, the inhomogeneity actually increases. We provide numerical simulations and analytic calculations to study the appearance of the anomalous features in the decay curves.

© 2011 Optical Society of America

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References

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  1. M. V. Berry, “Disruption of wavefronts: statistics of dislocations in incoherent Gaussian random waves,” J. Phys. A: Math. Gen. 11, 27–37 (1978).
    [CrossRef]
  2. I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
    [CrossRef]
  3. I. Freund, “Optical vortices in Gaussian random wave fields: statistical probability densities,” J. Opt. Soc. Am. A 11, 1644–1652 (1994).
    [CrossRef]
  4. N. Shvartsman and I. Freund, “Vortices in random wave fields: nearest neighbor anticorrelations,” Phys. Rev. Lett. 72, 1008–1011 (1994).
    [CrossRef] [PubMed]
  5. M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. London A 456, 2059–2079(2000).
    [CrossRef]
  6. M. Chen and F. S. Roux, “Evolution of the scintillation index and the optical vortex density in speckle fields after removal of the least-squares phase,” J. Opt. Soc. Am. A 27, 2138–2143(2010).
    [CrossRef]
  7. J. W. Goodman, Statistical Optics (Wiley, 1985).
  8. A. J. Devaney and G. C. Sherman, “Plane-wave representations for scalar wave fields,” SIAM Rev. 15, 765–786 (1973).
    [CrossRef]
  9. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

2010 (1)

2000 (1)

M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. London A 456, 2059–2079(2000).
[CrossRef]

1994 (2)

N. Shvartsman and I. Freund, “Vortices in random wave fields: nearest neighbor anticorrelations,” Phys. Rev. Lett. 72, 1008–1011 (1994).
[CrossRef] [PubMed]

I. Freund, “Optical vortices in Gaussian random wave fields: statistical probability densities,” J. Opt. Soc. Am. A 11, 1644–1652 (1994).
[CrossRef]

1993 (1)

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

1978 (1)

M. V. Berry, “Disruption of wavefronts: statistics of dislocations in incoherent Gaussian random waves,” J. Phys. A: Math. Gen. 11, 27–37 (1978).
[CrossRef]

1973 (1)

A. J. Devaney and G. C. Sherman, “Plane-wave representations for scalar wave fields,” SIAM Rev. 15, 765–786 (1973).
[CrossRef]

Berry, M. V.

M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. London A 456, 2059–2079(2000).
[CrossRef]

M. V. Berry, “Disruption of wavefronts: statistics of dislocations in incoherent Gaussian random waves,” J. Phys. A: Math. Gen. 11, 27–37 (1978).
[CrossRef]

Chen, M.

Dennis, M. R.

M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. London A 456, 2059–2079(2000).
[CrossRef]

Devaney, A. J.

A. J. Devaney and G. C. Sherman, “Plane-wave representations for scalar wave fields,” SIAM Rev. 15, 765–786 (1973).
[CrossRef]

Freilikher, V.

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

Freund, I.

I. Freund, “Optical vortices in Gaussian random wave fields: statistical probability densities,” J. Opt. Soc. Am. A 11, 1644–1652 (1994).
[CrossRef]

N. Shvartsman and I. Freund, “Vortices in random wave fields: nearest neighbor anticorrelations,” Phys. Rev. Lett. 72, 1008–1011 (1994).
[CrossRef] [PubMed]

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

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

Roux, F. S.

Sherman, G. C.

A. J. Devaney and G. C. Sherman, “Plane-wave representations for scalar wave fields,” SIAM Rev. 15, 765–786 (1973).
[CrossRef]

Shvartsman, N.

N. Shvartsman and I. Freund, “Vortices in random wave fields: nearest neighbor anticorrelations,” Phys. Rev. Lett. 72, 1008–1011 (1994).
[CrossRef] [PubMed]

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Phys. A: Math. Gen. (1)

M. V. Berry, “Disruption of wavefronts: statistics of dislocations in incoherent Gaussian random waves,” J. Phys. A: Math. Gen. 11, 27–37 (1978).
[CrossRef]

Opt. Commun. (1)

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

Phys. Rev. Lett. (1)

N. Shvartsman and I. Freund, “Vortices in random wave fields: nearest neighbor anticorrelations,” Phys. Rev. Lett. 72, 1008–1011 (1994).
[CrossRef] [PubMed]

Proc. R. Soc. London A (1)

M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. London A 456, 2059–2079(2000).
[CrossRef]

SIAM Rev. (1)

A. J. Devaney and G. C. Sherman, “Plane-wave representations for scalar wave fields,” SIAM Rev. 15, 765–786 (1973).
[CrossRef]

Other (2)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

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

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Figures (6)

Fig. 1
Fig. 1

Setup to produce an inhomogeneous initial vortex density. It consists of two Mach–Zehnder interferometers that are used to produce interference patterns in two mutually incoherent speckle fields with different speckle sizes, which are then combined to give a sinusoidal variation in the vortex density.

Fig. 2
Fig. 2

Example of the (a) intensity and (b) phase of the initial optical field used in the numerical simulation, showing the variations of the speckle sizes and vortex density. The parameter values are a 0 = 2 , W 1 = 64 , and W 2 = 16 .

Fig. 3
Fig. 3

Numerically simulated evolution of an inhomogeneous vortex density, shown in terms of the Fourier coefficients V ( 0 ) , V ( k x ) , and V ( 2 k x ) , where k x = 4 π a 0 , plotted as a function of logarithmic propagation distance. The curves are obtained from the average over hundreds of simulations with the error bars indicating the standard deviations.

Fig. 4
Fig. 4

Comparison of the analytically calculated and numerically simulated evolution of an inhomogeneous optical vortex density, shown in terms of three Fourier coefficients of the optical vortex density function: the global average vortex density V ( 0 ) , the fundamental spatial frequency V ( k x ) , and the first harmonic V ( 2 k x ) , where k x = 4 π a 0 . The numerical results are shown as discreet points representing the average values over several simulations with the error bars indicating the standard deviations. The solid curves represent the analytical results.

Fig. 5
Fig. 5

Curves of the normalized amplitude variation Δ V ( z ) for different values of W 1 , with W 2 = 16 and a 0 = 1 .

Fig. 6
Fig. 6

Curves of the normalized amplitude variation Δ V ( z ) for different values of W 2 , with W 1 = 64 and a 0 = 1 .

Equations (31)

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ψ in ( x , y , z = 0 ) = ψ 1 ( x , y ) sin ( 2 π a 0 x ) + ψ 2 ( x , y ) cos ( 2 π a 0 x ) ,
ψ n ( x , y ) = F 1 { χ ˜ n ( a ) A n ( a ) } n = 1 , 2 ,
A n ( a ) = 1 W n exp ( | a | 2 W n 2 ) n = 1 , 2 ,
χ ˜ n ( a ) = χ ˜ m ( a 1 ) χ ˜ n ( a 2 ) = χ ˜ m * ( a 1 ) χ ˜ n * ( a 2 ) = 0 ,
χ ˜ m ( a 1 ) χ ˜ n * ( a 2 ) = Δ a 2 δ m n δ ( a 1 a 2 ) ,
G in ( a ) = i χ ˜ 1 ( a a 0 , b ) W 1 exp [ ( a a 0 ) 2 b 2 W 1 2 ] i χ ˜ 1 ( a + a 0 , b ) W 1 exp [ ( a + a 0 ) 2 b 2 W 1 2 ] + χ ˜ 2 ( a a 0 , b ) W 2 exp [ ( a a 0 ) 2 b 2 W 2 2 ] + χ ˜ 2 ( a + a 0 , b ) W 2 exp [ ( a + a 0 ) 2 b 2 W 2 2 ]
g in ( x , y , z ) = G in ( a ) exp [ i 2 π ( a x + b y ) + i π z λ ( a 2 + b 2 ) ] d a d b ,
V A = 1 A A δ ( ψ r ) δ ( ψ i ) | x ψ r y ψ i x ψ i y ψ r | d x d y ,
V ( x ) = exp [ i 2 π p · ( q U ˜ ) ] | q 2 q 5 q 3 q 4 | d 6 p d 4 q | q 1 = q 2 = 0 ,
U ˜ = [ ψ r , ψ i , x ψ r , x ψ i , y ψ r , y ψ i ] ,
p = [ p 1 , p 2 , p 3 , p 4 , p 5 , p 6 ] ,
q = [ q 1 , q 2 , q 3 , q 4 , q 5 , q 6 ] ,
i 2 π p · U ˜ = i π ( P W ˜ + W ˜ P ) ,
P = [ p 1 + i p 2 p 3 + i p 4 p 5 + i p 6 ] and W ˜ = [ ψ ψ x ψ y ] ,
exp ( i 2 π p · U ˜ ) = exp ( π 2 P W ˜ W ˜ P ) ,
W ˜ W ˜ = M = [ ψ ψ * ψ x ψ * ψ y ψ * ψ ψ x * ψ x ψ x * ψ y ψ x * ψ ψ y * ψ x ψ y * ψ y ψ y * ] .
F ( Q , Q ) = exp ( i 2 π p · q π 2 P MP ) d 6 p = exp ( Q M 1 Q ) π 3 det ( M ) ,
Q = [ q 1 + i q 2 q 3 + i q 4 q 5 + i q 6 ] .
V ( x ) = F ( Q , Q ) | q 2 q 5 q 3 q 4 | d 4 q | q 1 = q 2 = 0 .
ψ ( x ) ψ * ( x ) = 2 π + π C ( x ) [ h 1 ( z ) h 2 ( z ) ] ,
ψ x ( x ) ψ * ( x ) = 2 π 2 a 0 S ( x ) [ ( 1 + i π z λ W 2 2 ) h 2 ( z ) ( 1 + i π z λ W 1 2 ) h 1 ( z ) ] ,
ψ x ( x ) ψ x * ( x ) = π 3 [ C ( x ) ( 4 a 0 2 T ( z ) W 2 2 ) h 2 ( z ) C ( x ) ( 4 a 0 2 T ( z ) W 1 2 ) h 1 ( z ) + 8 a 0 2 + W 1 2 + W 2 2 ] ,
ψ y ( x ) ψ y * ( x ) = π 3 [ W 1 2 + W 2 2 + C ( x ) W 1 2 h 1 ( z ) C ( x ) W 2 2 h 2 ( z ) ] ,
T ( z ) = 1 4 π 2 λ 2 a 0 2 z 2 ,
C ( x ) = cos ( 4 π a 0 x ) ,
S ( x ) = sin ( 4 π a 0 x ) ,
h n ( z ) = exp ( 2 π 2 λ 2 a 0 2 z 2 W n 2 ) .
V ( x ) = π { W 1 2 + W 2 2 + C ( x ) [ W 1 2 h 1 ( z ) W 2 2 h 2 ( z ) ] } 1 / 2 2 { 2 + C ( x ) [ h 1 ( z ) h 2 ( z ) ] } 3 / 2 { 4 [ 4 [ h 1 ( z ) h 2 ( z ) ] 2 π 2 λ 2 ( C ( x ) { 2 + C ( x ) [ h 1 ( z ) h 2 ( z ) ] } [ W 1 4 h 1 ( z ) W 2 4 h 2 ( z ) ] + S ( x ) 2 [ W 1 2 h 1 ( z ) W 2 2 h 2 ( z ) ] 2 ) z 2 ] a 0 2 + { 2 + C ( x ) [ h 1 ( z ) h 2 ( z ) ] } × { W 1 2 + W 2 2 + C ( x ) [ W 1 2 h 1 ( z ) W 2 2 h 2 ( z ) ] } } 1 / 2 .
V ( x ) = π 4 { 1 + 8 a 0 2 [ W 1 2 + W 2 2 + C ( x ) ( W 1 2 W 2 2 ) ] } 1 / 2 × [ W 1 2 + W 2 2 + C ( x ) ( W 1 2 W 2 2 ) ] ,
V ( x ) = π 4 ( W 1 2 + W 2 2 ) ( 1 + 8 a 0 2 W 1 2 + W 2 2 ) 1 / 2 ,
Δ V ( z ) = 2 [ h 1 ( z ) + h 2 ( z ) ] 4 [ h 1 ( z ) h 2 ( z ) ] 2 8 a 0 2 ( π 2 λ 2 z 2 [ W 1 4 h 1 ( z ) W 2 4 h 2 ( z ) ] + 2 [ h 1 ( z ) h 2 ( z ) ] { 4 [ h 1 ( z ) h 2 ( z ) ] 2 } ( W 1 2 W 2 2 ) ) .

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