Abstract

It is shown that branch points present in a turbulence-distorted optical field can be visualized as peaks and valleys of a certain potential function. Peaks correspond to positive branch points and valleys correspond to negative ones, thus allowing one to study the formation, movements, and merging of branch points. A closed-form formula is given for the potential in terms of wave-front-sensor measurements; branch points appear as logarithmic singularities that are easy to detect visually through computer-generated images. In fact, the branch-point potential is obtained by means of a single matrix multiplication. An electrostatic analogy is given, as well as a proof that the continuous part of the wave front does not change the location of the potential singularities. Applications can be found in adaptive optics, in the airborne laser system, in speckle or coherent imaging, and in high-bandwidth laser communication.

© 1999 Optical Society of America

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References

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  1. D. L. Fried, J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992).
    [CrossRef] [PubMed]
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    [CrossRef]
  3. D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759–2768 (1998).
    [CrossRef]
  4. E.-O. Le Bigot, W. J. Wild, E. J. Kibblewhite, “Branch point reconstructors for discontinuous light phase functions,” in Airborne Laser Advanced Technology, T. D. Steiner, P. H. Merritt, eds., Proc. SPIE3381, 76–87 (1998).
    [CrossRef]
  5. E.-O. Le Bigot, W. J. Wild, E. J. Kibblewhite, “Reconstruction of discontinuous light phase functions,” Opt. Lett. 23, 10–12 (1998).
    [CrossRef]
  6. V. Aksenov, V. Banakh, O. Tikhomirova, “Potential and vortex features of optical speckle fields and visualization of wave-front singularities,” Appl. Opt. 37, 4536–4540 (1998).
    [CrossRef]
  7. M. C. Roggemann, D. J. Lee, “Two-deformable-mirror concept for correcting scintillation effects in laser beam propagation through the turbulent atmosphere,” Appl. Opt. 37, 4577–4578 (1998).
    [CrossRef]
  8. J. Matthew, R. Walker, Mathematical Methods of Physics, 2nd ed. (Benjamin, New York, 1970).
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    [CrossRef]
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    [CrossRef]
  11. The Mathematica code that we wrote for this purpose can be obtained by sending a request to Eric.Le.Bigot@ens.fr.

1998

1992

1979

1977

Aksenov, V.

Banakh, V.

Fried, D. L.

Hudgin, R. H.

Hunt, B. R.

Kibblewhite, E. J.

E.-O. Le Bigot, W. J. Wild, E. J. Kibblewhite, “Reconstruction of discontinuous light phase functions,” Opt. Lett. 23, 10–12 (1998).
[CrossRef]

E.-O. Le Bigot, W. J. Wild, E. J. Kibblewhite, “Branch point reconstructors for discontinuous light phase functions,” in Airborne Laser Advanced Technology, T. D. Steiner, P. H. Merritt, eds., Proc. SPIE3381, 76–87 (1998).
[CrossRef]

Kouznetsov, D.

Le Bigot, E.-O.

E.-O. Le Bigot, W. J. Wild, E. J. Kibblewhite, “Reconstruction of discontinuous light phase functions,” Opt. Lett. 23, 10–12 (1998).
[CrossRef]

E.-O. Le Bigot, W. J. Wild, E. J. Kibblewhite, “Branch point reconstructors for discontinuous light phase functions,” in Airborne Laser Advanced Technology, T. D. Steiner, P. H. Merritt, eds., Proc. SPIE3381, 76–87 (1998).
[CrossRef]

Lee, D. J.

Matthew, J.

J. Matthew, R. Walker, Mathematical Methods of Physics, 2nd ed. (Benjamin, New York, 1970).

Morozov, D. K.

Roggemann, M. C.

Tikhomirova, O.

Vaughn, J. L.

Voitsekhovich, V. V.

Walker, R.

J. Matthew, R. Walker, Mathematical Methods of Physics, 2nd ed. (Benjamin, New York, 1970).

Wild, W. J.

E.-O. Le Bigot, W. J. Wild, E. J. Kibblewhite, “Reconstruction of discontinuous light phase functions,” Opt. Lett. 23, 10–12 (1998).
[CrossRef]

E.-O. Le Bigot, W. J. Wild, E. J. Kibblewhite, “Branch point reconstructors for discontinuous light phase functions,” in Airborne Laser Advanced Technology, T. D. Steiner, P. H. Merritt, eds., Proc. SPIE3381, 76–87 (1998).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Lett.

Other

E.-O. Le Bigot, W. J. Wild, E. J. Kibblewhite, “Branch point reconstructors for discontinuous light phase functions,” in Airborne Laser Advanced Technology, T. D. Steiner, P. H. Merritt, eds., Proc. SPIE3381, 76–87 (1998).
[CrossRef]

J. Matthew, R. Walker, Mathematical Methods of Physics, 2nd ed. (Benjamin, New York, 1970).

The Mathematica code that we wrote for this purpose can be obtained by sending a request to Eric.Le.Bigot@ens.fr.

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Figures (3)

Fig. 1
Fig. 1

(a) Phase gradients, showing simulated wave-front measurements. The curl is zero except in the middle, where there is a branch point (since the contour integral around the central point is non-null). (b) Rotated phase gradients, showing that the measurement of a given contour integral is equivalent to the measurement of the divergence (flux) of the rotated wave-front measurements.

Fig. 2
Fig. 2

Example of branch-point detection by (a) the branch-point-potential method presented in this paper and (b) the gradient-circulation method. The calculations are based on a set of 9×9 simulated, noisy phase gradients. The circles indicate the theoretical positions of the three branch points along with their signs. No spurious extrema appear with the potential method, and the global structure of the branch points is more visible than with the gradient-circulation method.

Fig. 3
Fig. 3

(a) Branch point potential, (b) gradient circulation. (a) The equipotential lines of the branch-point potential defined in this paper converge toward branch points, which makes their localization more effective. Since the branch-point potential is calculated by means of all the phase gradient measurements, equipotential lines are much smoother and more legible than lines in the contour plot of the phase-gradient circulation (b).

Equations (15)

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curl g(x)=2πi=1N+δ(x-xi+)-j=1N-δ(x-xj-)zˆ,
div(Rot-π/2 g)=(curl g)zˆ.
grad V=-Rot-π/2 g=Rotπ/2 g.
Sd2 x grad V(x)-R(x)2,
kikV˜k-R˜k2,
Vksol˜-i kR˜kk2.
ikR˜k=2πi=1N+ exp(-ikxi+)-j=1N- exp(-ikxj-),
Vksol˜=-2πi=1N+ exp(-ikxi+)k2-j=1N- exp(-ikxj-)k2.
R(x)=- 2πx rˆ=-2π grad logx.
Vsol(x)=-2π logx.
Vksol˜=-2πk2.
Vsol(x)grad+(Rotπ/2 g)(x)=-2πi=1N+ logx-xi+-j=1N- logx-xj-,
A+=limm0 (AA+m2)-1A,
VsolA+Rπ/2g,
Rπ / 2-1AVsol.

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