Abstract

The angle-impact Wigner function for highly convergent three-dimensional scalar wave fields is derived directly by use of the three-dimensional generalized optical transfer function rather than from a six-dimensional Wigner function. The angle-impact Wigner function is a real four-dimensional function from which the intensity at any point in space is readily determined.

© 2001 Optical Society of America

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References

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2000 (1)

1999 (2)

1994 (1)

1987 (1)

1974 (1)

1967 (1)

1964 (1)

Alieva, T.

Alonso, M. A.

Bastiaans, M. J.

Bracewell, R. N.

R. N. Bracewell, Two-Dimensional Imaging (Prentice-Hall, Englewood Cliffs, N.J., 1995).

Forbes, G. W.

Frieden, B. R.

Gu, M.

Kawata, S.

Kawata, Y.

Larkin, K. G.

Matthews, H. J.

McCutchen, C. W.

Mertz, L.

L. Mertz, Transformations in Optics (Wiley, New York, 1965).

Papoulis, A.

Sheppard, C. J. R.

Wolf, K. B.

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Figures (1)

Fig. 1
Fig. 1

Geometry of two intersecting generalized pupil functions, shown in the meridional plane containing vector m.

Equations (30)

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Ur=k/2π3/2-+mexpikm·rd3m,
m=2π/kPθ,ϕδm-1,
Ir=k/2π3∭∭-+m1*m2×exp-ikm2-m1·rd3m1d3m2,
m=m2-m1,p=12m2+m1,p=1-m241/2,
Ir=k2π3∭∭-+p-m2*p+m2×exp-ikm·rd3md3p.
Ir=k2π3-Cmexp-ikm·rd3m,
Cm=-p-m2*p+m2d3p,
Cm=2πk2-ππPθ1,ϕ1P*θ2,ϕ2mdψ,
Ir=k2π--ππPθ1,ϕ1P*θ2,ϕ2m×exp-ikm·rdψd3m=k2πΩ02-ππPθ1,ϕ1P*θ2,ϕ2×exp-ikm·rmdψdmd2Ω.
l=m·rm=m^·r,
γα,θ,ϕ,ψ=Pθ1,ϕ1P*θ2,ϕ2,
m=2sinα/2,
Ir=k2πΩ02-ππγα,θ,ϕ,ψexp-ikml×mdψdmd2Ω.
Ir=k4πΩ-22-ππγα,θ,ϕ,ψexp-ikml×mdψdmd2Ω.
Ir=12Ω-ππMθ,ϕ,m^·r,ψdψd2Ω,
Mθ,ϕ,l,ψ=k2π-22γα,θ,ϕ,ψexp-ikmlmdm,
Mθ,ϕ,l,ψ=k2π-ππγα,θ,ϕ,ψexp-2iklsinα2×sinαdα,
-ππMθ,ϕ,l,ψdψ=k/2π3-22Cm×exp-ikmlm2dm=Nθ,ϕ,l.
Cm=2πk21m2--ππMθ,ϕ,l,ψexpikmldψdl.
γα,θ,ϕ,ψ=12sinα/2-Mθ,ϕ,l,ψ×exp2iklsinα2dl.
Pπ-θ,π-ϕP*θ,ϕ=12-Mθ,ϕ,l,ψ×exp2ikldl
m=mcosθcosϕi+cosθsinϕj+sinθk,
m1,2=cosα2sinψsinϕ-sinθcosψcosϕsinα2cosθcosϕi-cosα2sinψcosϕ+sinθcosψsinϕ±sinα2cosθsinϕj+cosα2cosθcosψsinα2sinθk,
m1,2=sinθ1,2cosϕ1,2i+sinθ1,2sinϕ1,2j+cosθ1,2k.
sinψ=p·m×km1-m2/41/2=k·m1×m2.
cosθ1,2=cosα2cosθcosψsinα2sinθ
d2Ω=cosθdθdϕ.
Mθ,ϕ,l,ψ=2sin2kl2kl-sinklkl2,
Ir=2πsinkrkr2,
Ir=mx2+my2<2-π-ππMθ,ϕ,m^·r,ψmzdψdmxdmy.

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