Abstract

While adaptive optical systems are able to remove moderate wavefront distortions in scintillated optical beams, phase singularities that appear in strongly scintillated beams can severely degrade the performance of such an adaptive optical system. Therefore the detection of these phase singularities is an important aspect of strong-scintillation adaptive optics. We investigate the detection of phase singularities with the aid of a Shack–Hartmann wavefront sensor and show that, in spite of some systematic deficiencies inherent to the Shack–Hartmann wavefront sensor, it can be used for the reliable detection of phase singularities, irrespective of their morphologies. We provide full analytical results, together with numerical simulations of the detection process.

© 2007 Optical Society of America

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References

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2006 (2)

F. S. Roux, "Fluid dynamical entrophy and the number of optical vortices in a paraxial beam," Opt. Commun. 268, 15-22 (2006).
[CrossRef]

A. Talmi and E. N. Ribak, "Wavefront reconstruction from its gradients," J. Opt. Soc. Am. A 23, 1-10 (2006).
[CrossRef]

2003 (1)

M. Hattori and S. Komatsu, "An exact formulation of a filter for rotation in phase gradients and its applications to wavefront reconstruction problems," J. Mod. Opt. 50, 1705-1723 (2003).

2002 (3)

2000 (2)

1999 (1)

1998 (1)

1989 (1)

P. Coullet, L. Gil, and F. Rocca, "Optical vortices," Opt. Commun. 73, 403-408 (1989).
[CrossRef]

1981 (1)

1980 (2)

1977 (2)

1974 (1)

J. Nye and M. Berry, "Dislocations in wave trains," Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
[CrossRef]

Aksenov, V. P.

Arrasmith, W. W.

Barchers, J. D.

Berry, M.

J. Nye and M. Berry, "Dislocations in wave trains," Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
[CrossRef]

Coullet, P.

P. Coullet, L. Gil, and F. Rocca, "Optical vortices," Opt. Commun. 73, 403-408 (1989).
[CrossRef]

Fried, D. L.

Gil, L.

P. Coullet, L. Gil, and F. Rocca, "Optical vortices," Opt. Commun. 73, 403-408 (1989).
[CrossRef]

Hattori, M.

M. Hattori and S. Komatsu, "An exact formulation of a filter for rotation in phase gradients and its applications to wavefront reconstruction problems," J. Mod. Opt. 50, 1705-1723 (2003).

Herrmann, J.

Hudgin, R. H.

Koivunen, A. C.

Komatsu, S.

M. Hattori and S. Komatsu, "An exact formulation of a filter for rotation in phase gradients and its applications to wavefront reconstruction problems," J. Mod. Opt. 50, 1705-1723 (2003).

Lane, R. G.

Link, D. J.

Nye, J.

J. Nye and M. Berry, "Dislocations in wave trains," Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
[CrossRef]

Ribak, E. N.

Rocca, F.

P. Coullet, L. Gil, and F. Rocca, "Optical vortices," Opt. Commun. 73, 403-408 (1989).
[CrossRef]

Roggemann, M. C.

Roux, F. S.

F. S. Roux, "Fluid dynamical entrophy and the number of optical vortices in a paraxial beam," Opt. Commun. 268, 15-22 (2006).
[CrossRef]

Southwell, W.

Talmi, A.

Tikhomirova, O. V.

Tyler, G. A.

van Dam, M. A.

Appl. Opt. (2)

J. Mod. Opt. (1)

M. Hattori and S. Komatsu, "An exact formulation of a filter for rotation in phase gradients and its applications to wavefront reconstruction problems," J. Mod. Opt. 50, 1705-1723 (2003).

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (6)

Opt. Commun. (2)

F. S. Roux, "Fluid dynamical entrophy and the number of optical vortices in a paraxial beam," Opt. Commun. 268, 15-22 (2006).
[CrossRef]

P. Coullet, L. Gil, and F. Rocca, "Optical vortices," Opt. Commun. 73, 403-408 (1989).
[CrossRef]

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

J. Nye and M. Berry, "Dislocations in wave trains," Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

One-dimensional representation of a Shack–Hartmann wavefront sensor.

Fig. 2
Fig. 2

Array of subapertures (small squares) within the system aperture of the SHWS. The average phase slope values are associated with the dots inside the small squares.

Fig. 3
Fig. 3

One-dimensional representation of one lenslet in the SHWS, showing the shift of the focal point due to the average tilt of the incident wavefront.

Fig. 4
Fig. 4

Circulation D m , n over four subapertures with a singularity located either at the center (assumed to be the origin), denoted A, or at some arbitrary location ( x 0 , y 0 ), denoted B. The four subapertures are represented by the four squares. The dot at the center of each subaperture is the position with which the average phase slope value G of that subaperture is associated. The arrows represent the components of G. The dashed lines represent the contour used for calculation of the circulation.

Fig. 5
Fig. 5

Circulation D for a canonical singularity. (a) Topview of D shown as a function of the relative position of the singularity inside the four subaperture area, shown in Fig. 4, for 2 < μ < 2 and 2 < ν < 2 . (b) One-dimensional functions of D plotted as functions of r = μ 2 + ν 2 along the diagonal line and middle line, respectively, as indicated in (a).

Fig. 6
Fig. 6

Peak value of the circulation D as a function of the morphology angles 0 < α < π and 0 < β < 2 π . The jump at α = π 2 is due to the change of the topological charge of the singularity.

Fig. 7
Fig. 7

Circulation D for a noncanonical singularity, with α = π 4 and β = π . (a) Topview of D is shown as a function of the relative position of the singularity inside the four subaperture area, shown in Fig. 4, for 2 < μ < 2 and 2 < ν < 2 . (b) One-dimensional functions of D plotted as functions of r = μ 2 + ν 2 along the diagonal line, μ line and ν line, respectively, as indicated in (a).

Fig. 8
Fig. 8

Circulation D for a noncanonical singularity, with α = 4 π 9 and β = π 2 . (a) Topview of D shown as a function of the relative position of the singularity inside the four subaperture area, shown in Fig. 4, for 2 < μ < 2 and 2 < μ < 2 . (b) One-dimensional functions of D plotted as functions of r = μ 2 + ν 2 along diagonal line I (perpendicular to the orientation of the singularity), diagonal line II (along the orientation of the singularity), and the middle line, respectively, as indicated in (a).

Fig. 9
Fig. 9

Numerical simulation results for a Gaussian beam that propagated over a distance of 100 km through a turbulent atmosphere. (a) Resulting phase of the beam inside the system aperture. There are two pairs of oppositely charged phase singularities. The pairs are, respectively, located at the lower left and the upper right of the system aperture, (b) Circulation D, numerically calculated from the output of the Shack-Hartmann wavefront sensor.

Equations (22)

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θ ( x , y ) = θ C ( x , y ) + n ϕ ( x x n , y y n ; α n , β n ) ,
ϕ ( x , y ; α , β ) = i 2 ln [ ξ ( x + i y ) + ζ ( x i y ) ξ * ( x i y ) + ζ * ( x + i y ) ] ,
ξ = cos ( α 2 ) exp ( i β 2 ) ,
ζ = sin ( α 2 ) exp ( i β 2 ) ,
C θ d l = τ 2 π ,
G x ( x , y ) y = G y ( x , y ) x .
D ( x , y ) = T × G ( x , y ) .
× ϕ ( x , y , α , β ) = τ 2 π δ ( x ) δ ( y ) ,
× θ ( x , y ) = 2 π n τ n δ ( x x n ) δ ( y y n ) .
u m , n = H I ( u ) u d 2 u H I ( u ) d 2 u u 0 m , n ,
G m , n = Ω θ ( x ) d 2 x Ω d 2 x k f u m , n ,
D m , n = w 2 ( G x m , n + G x m , n + 1 + G y m , n + 1 + G y m + 1 , n + 1 G x m + 1 , n + 1 G x m + 1 , n G y m + 1 , n G y m , n ) ,
ϕ ( x , y ) = x y ̂ y x ̂ x 2 + y 2 .
G m , n = 1 w 2 w 0 w 0 x y ̂ y x ̂ x 2 + y 2 d x d y = ( π 4 w + ln 2 2 w ) ( x ̂ y ̂ ) .
D m , n = 4 w G x m , n = π + 2 ln ( 2 ) = 4.527887 .
ϕ ( x x 0 , y y 0 ) = ( x x 0 ) y ̂ ( y y 0 ) x ̂ ( x x 0 ) 2 + ( y y 0 ) 2 .
D = π 2 2 ln ( 5 ) 2 arctan ( 2 ) + 5 ln ( 2 ) = 0.396641 .
θ ( x , y ) = ( x y ̂ y x ̂ ) C x 2 ( 1 + A ) 2 y x B + y 2 ( 1 A ) ,
A = sin ( α ) cos ( β ) ,
B = sin ( α ) sin ( β ) ,
C = cos ( α ) ,
D m , n = ( μ + 1 ) ( A m B ) 2 A m arctan [ ( μ + 1 ) B ( ν 1 ) A m ( μ + 1 ) C ] ( μ + 1 ) ( A m + B ) 2 A m arctan [ ( μ + 1 ) B ( ν + 1 ) A m ( μ + 1 ) C ] + ( μ 1 ) ( A m + B ) 2 A m arctan [ ( μ 1 ) B ( ν 1 ) A m ( μ 1 ) C ] ( μ 1 ) ( A m B ) 2 A m arctan [ ( μ 1 ) B ( ν + 1 ) A m ( μ 1 ) C ] + ( ν + 1 ) ( A p B ) 2 A p arctan [ ( ν + 1 ) B ( μ 1 ) A p ( ν + 1 ) C ] ( ν + 1 ) ( A p + B ) 2 A p arctan [ ( ν + 1 ) B ( μ + 1 ) A p ( ν + 1 ) C ] + ( ν 1 ) ( A p + B ) 2 A p arctan [ ( ν 1 ) B ( μ 1 ) A p ( ν 1 ) C ] ( ν 1 ) ( A p B ) 2 A p arctan [ ( ν 1 ) B ( μ + 1 ) A p ( ν 1 ) C ] ( μ + 1 ) B A m arctan [ ν A m ( μ + 1 ) B ( μ + 1 ) C ] + ( μ 1 ) B A m arctan [ ν A m ( μ 1 ) B ( μ 1 ) C ] ( ν + 1 ) B A p arctan [ μ A p ( ν + 1 ) B ( ν + 1 ) C ] + ( ν 1 ) B A p arctan [ μ A p ( ν 1 ) B ( ν 1 ) C ] + μ arctan [ μ B ( ν + 1 ) A m μ C ] μ arctan [ μ B ( ν 1 ) A m μ C ] + ν arctan [ ν B ( μ + 1 ) A p ν C ] ν arctan [ ν B ( μ 1 ) A p ν C ] + ( μ + 1 ) C 4 A m { ln [ 2 ( μ + 1 ) ( ν 1 ) B ( μ + 1 ) 2 A p ( ν 1 ) 2 A m ] + ln [ 2 ( μ + 1 ) ( ν + 1 ) B ( μ + 1 ) 2 A p ( ν + 1 ) 2 A m ] 2 ln [ 2 ( μ + 1 ) ν B ( μ + 1 ) 2 A p ν 2 A m ] } ( μ 1 ) C 4 A m { [ 2 ( μ 1 ) ( ν + 1 ) B ( μ 1 ) 2 A p ( ν + 1 ) 2 A m ] + ln [ 2 ( μ 1 ) ( ν 1 ) B ( μ 1 ) 2 A p ( ν 1 ) 2 A m ] 2 ln [ 2 ( μ 1 ) ν B ( μ 1 ) 2 A p ν 2 A m ] } + ( ν + 1 ) C 4 A p { ln [ 2 ( μ + 1 ) ( ν + 1 ) B ( μ + 1 ) 2 A p ( ν + 1 ) 2 A m ] + ln [ 2 ( μ 1 ) ( ν + 1 ) B ( μ 1 ) 2 A p ( ν + 1 ) 2 A m ] 2 ln [ 2 μ ( ν + 1 ) B μ 2 A p ( ν + 1 ) 2 A m ] } ( ν 1 ) C 4 A p { ln [ 2 ( μ 1 ) ( ν 1 ) B ( μ 1 ) 2 A p ( ν 1 ) 2 A m ] + ln [ 2 ( μ + 1 ) ( ν 1 ) B ( μ + 1 ) 2 A p ( ν 1 ) 2 A m ] 2 ln [ 2 μ ( ν 1 ) B μ 2 A p ( ν 1 ) 2 A m ] } ,

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