Abstract

Analytical expressions are derived and computational algorithms are constructed for retrieving optical-field phase distribution under strong scintillation. The input data for the phase reconstruction are the wave-front slopes registered by a Hartmann sensor or shearing interferometer. The theory is based on representing the slope-vector field as the sum of its potential and solenoid components; it introduces the concept of phase-source and phase-vortex density and uses strict integral expressions relating these quantities to the wave-front slopes. To overcome the difficulties arising from the singular character of phase distribution, use is made of regularization of the wave-front slopes. The slopes can be measured with an ideal point wave-front sensor. It is shown that the slopes measured at the output of a nonideal sensor can be treated as regularized values of these slopes. Numerical simulation of phase unwrapping from the reference values of the wave-front slopes has shown that the algorithm designed for visualization of local phase singularities and those for phase reconstruction are very helpful in eliminating the measurement noise.

© 2002 Optical Society of America

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References

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  32. B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, S. A. Werner, “Imaging: phase radiography with neutrons,” Nature 408, 158–159 (2000).
    [CrossRef] [PubMed]

2000 (2)

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, S. A. Werner, “Imaging: phase radiography with neutrons,” Nature 408, 158–159 (2000).
[CrossRef] [PubMed]

C. Paterson, J. C. Dainty, “Hybrid curvature and gradient wave-front sensor,” Opt. Lett. 25, 1687–1689 (2000).
[CrossRef]

1999 (4)

E.-O. Le Bigot, W. J. Wild, “Theory of branch-point detection and its implementation,” J. Opt. Soc. Am. A 16, 1724–1729 (1999).
[CrossRef]

W. W. Arrasmitth, “Branch-point-tolerant least-squares phase reconstructor,” J. Opt. Soc. Am. A 16, 1864–1872 (1999).
[CrossRef]

V. P. Aksenov, O. V. Tikhomirova, “Reconstruction of optical field phase from the wave front slopes,” Proc. IEEE 246, 30–32 (1999).

B. V. Fortes, “Phase correction for turbulent blurring of an image under conditions of strong intensity fluctuations,” Atmos. Oceanic Opt. 12, 406–411 (1999).

1998 (5)

1995 (1)

1992 (2)

1989 (1)

E. G. Abramochkin, V. G. Volostnikov, “Relationship between two-dimensional intensity and phase in Fresnel diffraction zone,” Opt. Commun. 74, 144–148 (1989).
[CrossRef]

1988 (1)

1980 (1)

Abramochkin, E. G.

E. G. Abramochkin, V. G. Volostnikov, “Relationship between two-dimensional intensity and phase in Fresnel diffraction zone,” Opt. Commun. 74, 144–148 (1989).
[CrossRef]

Aksenov, V. P.

V. P. Aksenov, O. V. Tikhomirova, “Reconstruction of optical field phase from the wave front slopes,” Proc. IEEE 246, 30–32 (1999).

V. P. Aksenov, V. A. Banakh, O. V. Tikhomirova, “Potential and vortex features of optical speckle field and visualization of wave-front singularities,” Appl. Opt. 37, 4536–4540 (1998).
[CrossRef]

V. P. Aksenov, Yu. N. Isaev, “Analytical representation of the phase and its mode components reconstructed according to the wave-front slopes,” Opt. Lett. 17, 1180–1182 (1992).
[CrossRef] [PubMed]

V. P. Aksenov, “Hydrodynamic model of wave front dislocations and equation for the vortex of phase gradient,” in Fourth International Symposium on Atmospheric and Ocean Optics, Abstracts (Institute of Atmosphere Optics SB RAS, Tomsk, Russia, 1997), p. 43–44.

V. P. Aksenov, “Hydrodynamic description of wave front dislocations and equations for the rotor of phase gradient,” in International Conference on Singular Optics, M. S. Soskin ed., Proc. SPIE3487, 42–45 (1998).
[CrossRef]

Allman, B. E.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, S. A. Werner, “Imaging: phase radiography with neutrons,” Nature 408, 158–159 (2000).
[CrossRef] [PubMed]

Arif, M.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, S. A. Werner, “Imaging: phase radiography with neutrons,” Nature 408, 158–159 (2000).
[CrossRef] [PubMed]

Arrasmitth, W. W.

Banakh, V. A.

Barclay, H. T.

Dainty, J. C.

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Fortes, B. V.

B. V. Fortes, “Phase correction for turbulent blurring of an image under conditions of strong intensity fluctuations,” Atmos. Oceanic Opt. 12, 406–411 (1999).

Fried, D. L.

D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 13, 2759–2768 (1998).
[CrossRef]

D. L. Fried, J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992).
[CrossRef] [PubMed]

Gakhov, F. D.

F. D. Gakhov, Boundary Value Problems (Nauka, Moscow, 1977).

Ghiglia, D. C.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley Interscience, New York, 1998).

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, Oxford, UK, 1998).

Herrmann, J.

Humphreys, R. A.

Isaev, Yu. N.

Jacobson, D. L.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, S. A. Werner, “Imaging: phase radiography with neutrons,” Nature 408, 158–159 (2000).
[CrossRef] [PubMed]

Kibblewhite, E. J.

Kravtsov, Yu. A.

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, Berlin, Heidelberg, 1989).

Lavrent’ev, M. A.

M. A. Lavrent’ev, B. V. Shabat, Methods for Theory of Complex Variable Function (Nauka, Moscow, 1973).

Le Bigot, E.-O.

Loitsyanskii, L. G.

L. G. Loitsyanskii, Fluid Mechanics (Nauka, Moscow, 1987).

McMahon, P. J.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, S. A. Werner, “Imaging: phase radiography with neutrons,” Nature 408, 158–159 (2000).
[CrossRef] [PubMed]

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Nugent, K. A.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, S. A. Werner, “Imaging: phase radiography with neutrons,” Nature 408, 158–159 (2000).
[CrossRef] [PubMed]

D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[CrossRef]

Paganin, D.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, S. A. Werner, “Imaging: phase radiography with neutrons,” Nature 408, 158–159 (2000).
[CrossRef] [PubMed]

D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[CrossRef]

Paterson, C.

Pries, R.

Primmerman, A.

Pritt, M. D.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley Interscience, New York, 1998).

Roddier, F.

Roggemann, M. C.

M. C. Roggemann, B. M. Welsh, Imaging Through Turbulence (CRC Press, Boca Raton, Fla., 1996).

Rytov, S. M.

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, Berlin, Heidelberg, 1989).

Shabat, B. V.

M. A. Lavrent’ev, B. V. Shabat, Methods for Theory of Complex Variable Function (Nauka, Moscow, 1973).

Takahashi, T.

Takijo, H.

Tatarskii, V. I.

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, Berlin, Heidelberg, 1989).

Tikhomirova, O. V.

V. P. Aksenov, O. V. Tikhomirova, “Reconstruction of optical field phase from the wave front slopes,” Proc. IEEE 246, 30–32 (1999).

V. P. Aksenov, V. A. Banakh, O. V. Tikhomirova, “Potential and vortex features of optical speckle field and visualization of wave-front singularities,” Appl. Opt. 37, 4536–4540 (1998).
[CrossRef]

Vaughn, J. L.

Vekua, I. N.

I. N. Vekua, Generalized Analytic Functions (Pergamon-Addison-Wesley, Reading, Mass., 1962).

Vladimirov, V. S.

V. S. Vladimirov, Generalized Function in Mathematical Physics (Nauka, Moscow, 1979).

Volostnikov, V. G.

E. G. Abramochkin, V. G. Volostnikov, “Relationship between two-dimensional intensity and phase in Fresnel diffraction zone,” Opt. Commun. 74, 144–148 (1989).
[CrossRef]

Welsh, B. M.

M. C. Roggemann, B. M. Welsh, Imaging Through Turbulence (CRC Press, Boca Raton, Fla., 1996).

Werner, S. A.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, S. A. Werner, “Imaging: phase radiography with neutrons,” Nature 408, 158–159 (2000).
[CrossRef] [PubMed]

Weyl, H.

H. Weyl, The Concept of a Riemann Surface (Addison-Wesley, Reading, Mass., 1964).

Wild, W. J.

Zollars, B. G.

Appl. Opt. (4)

Atmos. Oceanic Opt. (1)

B. V. Fortes, “Phase correction for turbulent blurring of an image under conditions of strong intensity fluctuations,” Atmos. Oceanic Opt. 12, 406–411 (1999).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Nature (1)

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, S. A. Werner, “Imaging: phase radiography with neutrons,” Nature 408, 158–159 (2000).
[CrossRef] [PubMed]

Opt. Commun. (1)

E. G. Abramochkin, V. G. Volostnikov, “Relationship between two-dimensional intensity and phase in Fresnel diffraction zone,” Opt. Commun. 74, 144–148 (1989).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. Lett. (1)

D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[CrossRef]

Proc. IEEE (1)

V. P. Aksenov, O. V. Tikhomirova, “Reconstruction of optical field phase from the wave front slopes,” Proc. IEEE 246, 30–32 (1999).

Other (15)

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley Interscience, New York, 1998).

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, Oxford, UK, 1998).

M. C. Roggemann, B. M. Welsh, Imaging Through Turbulence (CRC Press, Boca Raton, Fla., 1996).

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, Berlin, Heidelberg, 1989).

V. P. Aksenov, “Hydrodynamic model of wave front dislocations and equation for the vortex of phase gradient,” in Fourth International Symposium on Atmospheric and Ocean Optics, Abstracts (Institute of Atmosphere Optics SB RAS, Tomsk, Russia, 1997), p. 43–44.

V. P. Aksenov, “Hydrodynamic description of wave front dislocations and equations for the rotor of phase gradient,” in International Conference on Singular Optics, M. S. Soskin ed., Proc. SPIE3487, 42–45 (1998).
[CrossRef]

I. N. Vekua, Generalized Analytic Functions (Pergamon-Addison-Wesley, Reading, Mass., 1962).

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

M. A. Lavrent’ev, B. V. Shabat, Methods for Theory of Complex Variable Function (Nauka, Moscow, 1973).

L. G. Loitsyanskii, Fluid Mechanics (Nauka, Moscow, 1987).

F. D. Gakhov, Boundary Value Problems (Nauka, Moscow, 1977).

http://documents.wolfram.com/v4/MainBook/3DGraphics/G.2.19.html

http://www.mbay.net/∼cgd/flt/flt04.htm

H. Weyl, The Concept of a Riemann Surface (Addison-Wesley, Reading, Mass., 1964).

V. S. Vladimirov, Generalized Function in Mathematical Physics (Nauka, Moscow, 1979).

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Figures (10)

Fig. 1
Fig. 1

Model phase.

Fig. 2
Fig. 2

(a) Model x-slope μ(x, y, z)=(/x)S(x, y, z), (b) with fragment.

Fig. 3
Fig. 3

(a) Model and (b) regularized phase-source density.

Fig. 4
Fig. 4

Regularized phase-vortex density.

Fig. 5
Fig. 5

(a) Model noisy x slope μ(x, y, z)=(/x)S(x, y, z), (b) with fragment.

Fig. 6
Fig. 6

Regularized source density reconstructed from the noisy slopes.

Fig. 7
Fig. 7

(a) Averaged slope (b) slope reconstructed by using Eq. (15) for the initial noisy data with averaging, (c) without averaging.

Fig. 8
Fig. 8

Regularized phase-vortex density reconstructed from the noisy slopes.

Fig. 9
Fig. 9

(a) Phase reconstructed from the ideal model slopes and (b) from the noisy model slopes.

Fig. 10
Fig. 10

Instrumental regularized phase-vortex density.

Equations (55)

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γS(ρ, z)dρ=2πm,
DΩ(ρ, z)dρ=2πm,
Ω(ρ, z)=2πmδ(x-xd, y-yd).
S(ρ, z)=S(ρ, z)+Ss(ρ, z),
rot{S(ρ, z)}=0
div{Ss(ρ, z)}=0.
div{S(ρ, z)}=Q(ρ, z).
F(ζ, ζ*)=-μ+iν,
Fζ*=12x+i y(-μ+iν)=-12μx+νy+12 iνx-μy.
μx+νy=div{S(x, y, z)},
νx-μyn=rot{S(x, y, z)},
Fζ*=-12 Q+i2 Ω.
F(ζ, ζ*)=12πiDF(ζ, ζ*)ζ-ζdζ-DF(ζ, ζ*)ζ*dξdηζ-ζ,
F(ζ, ζ*)=12πiDF(ζ, ζ*)ζ-ζdζ+12πDQ(ξ, η)ζ-ζdξdη- i2πDΩ(ξ, η)ζ-ζdξdη,
S(x, y)=-D{[ν(ξ, η)h1(x-ξ, y-η)+μ(ξ, η)h2(x-ξ, y-η)]dξ}+{[ν(ξ, η)h2(x-ξ, y-η)-μ(ξ, η)h1(x-ξ, y-η)]dη}-DQ(ξ, η)×h1(x-ξ, y-η)dξdη+DΩ(ξ, η)×h2(x-ξ, y-η)dξdη,
h1(x-ξ, y-η)=-12πln[(x-ξ)2+(y-η)2]1/2,
h2(x-ξ, y-η)=12πarg[(x-ξ)+i(y-η)].
S(x, y)=S(x0, y0)+ΓS(x, y)dρ
=S(x0, y0)+μ(x, y)dx+ν(x, y)dy-ReΓF(ζ)dζ,
S(x, y)=S(x0, y0)+Γ0μ(x, y)dx+ν(x, y)dy+N1M1+N2M2++NjMj,
S(x, y)=S(x0, y0)+Γ0μ(x, y)dx+ν(x, y)dy+2πjNjmj,
SP(x, y)=arg{exp[iS(x, y)]},
Ω(x, y)=2πj=1Jmjδ(x-xdj, y-ydj).
S(x, y)=S(x0, y0)-Re12πiΓdζDF(ζ, ζ¯*)ζ-ζdζ+12πΓdζDQ(ξ, η)ζ-ζdξdη-ij=1JmjΓdζζdj-ζ.
Φ(ζ)=-12πiΓdζζ-ζ.
Γz0zdζζ-ζ=lnz-ζz0-ζ+iαπ=ln|z-ζ|-ln|z0-ζ|+i arg(z-ζ)-i arg(z0-ζ)+iπα,
S(x, y)=S(x0, y0)-D{[ν(ξ, η)h1(x-ξ, y-η)+μ(ξ, η)h2(x-ξ, y-η)] dξ}+{[ν(ξ, η)h2(x-ξ, y-η)-μ(ξ, η)h1(x-ξ, y-η)] dη}-DQ(ξ, η)h1(x-ξ, y-η)dξdη+2πj=1Jmjh2(x-ξj, y-ηj),
h1(x-ξ, y-η)=-12πln(x-ξ)2+(y-η)2(x0-η)2+(y0-η)21/2,
h2(x-ξ, y-η)=12πarg(x-ξ)+i(y-η)(x0-ξ)+i(y0-η).
U(x, y, z)=q(1+q2)3exp-q2a2(x2+y2)(1+q2)+i q(x2+y2)a2(1+q2)(Ur+iUi)(1+iq)3,
Ur=-3+2q-q2+2 q2a2 (x2+3y2),
Ui=3+2q+q2-2 q2a2 (3x2+y2).
Sp(x, y, z)=arg(U),
μ(x, y, z)=q(x+y)1+q2+2[q1x-q2y+q3(x-y)]q4,
ν(x, y, z)=q(x-y)1+q2+2[q1x+q2y-q3(x+y)]q4,
q1=3-4q+q2,q2=3+4q+q2,
q3=8q2xy,
q4=9+2q2[5-12(x2+y2)]-8q3(x2-y2)+q4[1-8(x2+y2)+20(x4+y4)+24x2y2].
U(x, y, z)=U1(x, y, z)+iU2(x, y, z)=U1x(xd, yd, z)x+U1y(xd, yd, z)y+i[U2x(xd, yd, z)x+U2y(xd, yd, z)y],
S(x, y, z)=μ(x, y, z)I+ν(x, y, z)m=arctanU2xx+U2yyU1xx+U1yy
x=U1x+U1yy;
y=U2xx+U2yy,
μ(x, y, z)=U2xx-U1xyx2+y2,
ν(x, y, z)=U2yx-U1yyx2+y2.
μ(ρ, ϕ, z)=U2xcos ϕ-U1xsin ϕρ,
ν(ρ, ϕ, z)=U2ycos ϕ-U1ysin ϕρ,
{S(ρ, z)}=μ(ρ, z)l+ν(ρ, z)m=lμ(ρ-ρ, z)ω(ρ)d2ρ+mν(ρ-ρ, z)ω(ρ)d2ρ.
ω(x, y)
=-Cexp22-r2,r=(x2+y2)1/2<0,|r|.
Dαμ(ρ, ζ)=μ(ρ)Dραω(|ρ-ρ|)d2ρ,
Ω(ρ, z)=ν(ρ, z)x-μ(ρ, z)y.
mj=12πDjΩ(ρ)d2ρ.
ρdj=DjΩ(ρ)ρd2ρDjΩ(ρ)d2ρ
{S(ρ, z)}I=W(ρ-ρ0)I(ρ0)S(ρ0)d2ρ0W(ρ-ρ0)I(ρ0)d2ρ0,
{I(ρ, z)S}=-k Iz

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