Abstract

The generalized optical transfer function and the spectral correlation function are investigated for nonparaxial two-dimensional wave fields. The angle-impact marginal of the four-dimensional Wigner function is derived directly. For focused wave fields of semiangle greater than 90°, the spectral correlation function exhibits overlapping and interference. For focused wave fields for which the semiangle is known to be less than 180°, the magnitude and phase can be recovered directly from knowledge of the intensity in the focal region.

© 2001 Optical Society of America

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References

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  1. K. B. Wolf, M. A. Alonso, G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999).
    [CrossRef]
  2. K. G. Larkin, C. J. R. Sheppard, “Direct method for phase retrieval from the intensity of cylindrical wavefronts,” J. Opt. Soc. Am. A 16, 1838–1844 (1999).
    [CrossRef]
  3. L. Mertz, Transformations in Optics (Wiley, New York, 1965).
  4. C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Optik (Stuttgart) 72, 131–133 (1986).
  5. C. J. R. Sheppard, T. J. Connolly, M. Gu, “Scattering by a one-dimensional rough surface and surface reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
    [CrossRef] [PubMed]
  6. C. J. R. Sheppard, M. Gu, Y. Kawata, S. Kawata, “Three-dimensional transfer functions for high aperture systems obeying the sine condition,” J. Opt. Soc. Am. A 11, 593–598 (1994).
    [CrossRef]
  7. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [CrossRef]
  8. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964).
    [CrossRef]
  9. B. R. Frieden, “Optical transfer of the three-dimensional object,” J. Opt. Soc. Am. 57, 56–66 (1967).
    [CrossRef]
  10. A. Papoulis, “Ambiguity function in Fourier optics,” J. Opt. Soc. Am. 64, 779–788 (1974).
    [CrossRef]
  11. A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).
  12. M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1144 (1994).
    [CrossRef] [PubMed]
  13. D. F. McAlister, M. Beck, L. Clarke, A. Mayer, M. G. Raymes, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995).
    [CrossRef] [PubMed]

1999

1995

1994

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1144 (1994).
[CrossRef] [PubMed]

C. J. R. Sheppard, M. Gu, Y. Kawata, S. Kawata, “Three-dimensional transfer functions for high aperture systems obeying the sine condition,” J. Opt. Soc. Am. A 11, 593–598 (1994).
[CrossRef]

1993

C. J. R. Sheppard, T. J. Connolly, M. Gu, “Scattering by a one-dimensional rough surface and surface reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
[CrossRef] [PubMed]

1986

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Optik (Stuttgart) 72, 131–133 (1986).

1974

1967

1964

1959

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Alonso, M. A.

Beck, M.

Clarke, L.

Connolly, T. J.

C. J. R. Sheppard, T. J. Connolly, M. Gu, “Scattering by a one-dimensional rough surface and surface reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
[CrossRef] [PubMed]

Forbes, G. W.

Frieden, B. R.

Gu, M.

C. J. R. Sheppard, M. Gu, Y. Kawata, S. Kawata, “Three-dimensional transfer functions for high aperture systems obeying the sine condition,” J. Opt. Soc. Am. A 11, 593–598 (1994).
[CrossRef]

C. J. R. Sheppard, T. J. Connolly, M. Gu, “Scattering by a one-dimensional rough surface and surface reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
[CrossRef] [PubMed]

Kawata, S.

Kawata, Y.

Larkin, K. G.

Mayer, A.

McAlister, D. F.

McCutchen, C. W.

Mertz, L.

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

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Papoulis, A.

Raymer, M. G.

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1144 (1994).
[CrossRef] [PubMed]

Raymes, M. G.

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Sheppard, C. J. R.

K. G. Larkin, C. J. R. Sheppard, “Direct method for phase retrieval from the intensity of cylindrical wavefronts,” J. Opt. Soc. Am. A 16, 1838–1844 (1999).
[CrossRef]

C. J. R. Sheppard, M. Gu, Y. Kawata, S. Kawata, “Three-dimensional transfer functions for high aperture systems obeying the sine condition,” J. Opt. Soc. Am. A 11, 593–598 (1994).
[CrossRef]

C. J. R. Sheppard, T. J. Connolly, M. Gu, “Scattering by a one-dimensional rough surface and surface reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
[CrossRef] [PubMed]

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Optik (Stuttgart) 72, 131–133 (1986).

Wolf, E.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Wolf, K. B.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Lett.

Optik (Stuttgart)

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Optik (Stuttgart) 72, 131–133 (1986).

Phys. Rev. Lett.

C. J. R. Sheppard, T. J. Connolly, M. Gu, “Scattering by a one-dimensional rough surface and surface reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
[CrossRef] [PubMed]

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1144 (1994).
[CrossRef] [PubMed]

Proc. R. Soc. London Ser. A

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Other

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

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

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Figures (6)

Fig. 1
Fig. 1

Field at a point in x, z space produced from an angular spectrum (pupil function) P(θ).

Fig. 2
Fig. 2

Generalized OTF calculated from the overlap of two circles and centered at m/2, s/2 and -m/2, -s/2.

Fig. 3
Fig. 3

Transformation of the integration variables from θ1, θ2 to θ, α. For an optical field of small semiangle, the support repeats periodically.

Fig. 4
Fig. 4

The rectangular region inclined at 45° contains the same information as the square θ1, θ2 region but rearranged.

Fig. 5
Fig. 5

(a) Example of an intensity plot for a field with a chirped phase that has a semiangle equal to π/2. This corresponds to the limiting case for spectral correlation function overlap. (b) Magnitude (left) and phase (right) of the spectral correlation function extracted from the intensity map by using the algorithm developed in a previous paper.2 The magnitude component clearly shows that the functions do not overlap but just touch at the corners.

Fig. 6
Fig. 6

(a) Intensity plot for a field with a chirped phase that has a semiangle almost equal to π, in other words, almost the full circle. (b) Magnitude (left) and phase (right) of the spectral correlation function again extracted from the intensity map by using the algorithm developed in a previous paper.2 The magnitude component clearly shows almost complete overlap and interference, except in an xy cross pattern at the center. The actual position of the cross depends on the location of the gap in the radiation direction.

Equations (33)

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U(x, z)=-ππP(θ)exp[ik(x sin θ-z cos θ)]dθ,
U(x, z)=k2πΠ(m, s)exp[ik(mx+sz)]dmds,
m=K sin θ,
s=-K cos θ,
Π(m, s)=2πkP(θ)δ(K-1).
Π(m, s)=k2πU(x, z)exp[-ik(mx+sz)]dxdz.
G(m, s)=k2π|U(x, z)|2 exp[-ik(mx+sz)]dxdz,
G(m, s)=k2πΠ(m+m/2, s+s/2)Π*(m-m/2, s-s/2)dmds.
G(m, s)=2πkP(θ1)P*(θ2)|sin(θ1-θ2)|=P(θ+α/2)P*(θ-α/2)|sin α|,
K=2 sin(α/2),
m=2 sinα2cos θ,
s=2 sinα2sin θ,
|sin α|=|K|(1-K2/4)1/2.
s/m=tan θ,
γ(α, θ)=P(θ+α/2)P*(θ-α/2),
G(m, s)=2πkγ2 arcsinm2+s22,arctansm+γ2π-2 arcsinm2+s22,arctansm(m2+s2)1/21-m2+s241/2,
γ(α, θ)=k2π|sin α|G2 sinα2cos θ, 2 sinα2sin θ.
m=1-K241/2 sin θ,
m=sK1-K241/2,
|U(x, z)|2=k2πG(m, s)exp[ik(mx+sz)]dmds,
l=x cos θ+z sin θ.
|U(x, z)|2=P(θ1)exp[ik(x sin θ1-z cos θ1)]×P*(θ2)exp[-ik(x sin θ2-z cos θ2)]dθ1dθ2.
|U(x, z)|2=-ππ-ππγ(α, θ )×exp2ik sinα2(x cos θ+z sin θ)dθdα.
M(θ, l)=k2π1/2-ππγ(α, θ)exp2ikl sinα2dα,
|U(x, z)|2=2πk1/2M(θ, x cos θ+z sin θ)dθ.
γ(α, θ)=k2π1/2cosα2-M(θ, l)exp-2ikl sinα2dl.
|P(θ)|2=k2π1/2-M(θ, l)dl.
|P(θ)|2dθ=k2π|U(x, z)|2dl=k2π1/2M(θ, l)dldθ=E,
G(m, s)=2πk1/21(m2+s2)1/2-Marctansm, l×exp(-iklK)dl.
C(m, z)=k2π1/2G(m, s)exp(iksz)ds,
C(m, z)=2πk1/2γ(α, θ)(cos2 θ-m2/4)1/2exp(ikmz tan θ)dθ
=2πk1/2γ(α, θ)(1-K2/4)1/2(K2-m2)1/2×exp[ikz(K2-m2)1/2]dK.
C(0, z)=2πk1/2-ππ|P(θ)|2cos θdθ

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