Abstract

The design of a desired optical transfer function (OTF) is a common optical problem that has many possible applications. A well-known application for the OTF design is beam shaping for incoherent illumination. However, other applications, such as optical signal processing, can also be addressed with these systems. We derive a mathematical expression for an optimal phase-only filter that, when attached to an imaging lens, provides an optimal approximation in the sense of the minimal mean square error to the desired OTF function. Because of the fact that a phase-only filter is used, high efficiency is achieved.

© 1997 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).
  2. M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964).
  3. L. Mandel, E. Wolf “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
    [CrossRef]
  4. P. S. Considine, “Effects of coherence on imaging systems,” J. Opt. Soc. Am. 56, 1001–1009 (1966).
    [CrossRef]
  5. W. Lukosz, “Optical systems with resolving powers exceeding the classical limit,” J. Opt. Soc. Am. 56, 1463–1472 (1966).
    [CrossRef]
  6. A. Bachl, W. Lukosz, “Experiments on superresolution imaging of a reduced object field,” J. Opt. Soc. Am. 57, 163–168 (1967).
    [CrossRef]
  7. I. Glaser, “Holographic incoherent optical transfer function synthesis: analysis and optimization,” J. Opt. Soc. Am. A 3, 681–693 (1986).
    [CrossRef]
  8. E. Isaacson, H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966).
  9. R. W. Gerchberg, W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik (Stuttgart) 34, 275–284 (1971).
  10. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 227–246 (1972).
  11. Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Gerchberg–Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Let. 21, 842–844 (1996).
    [CrossRef]

1996

Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Gerchberg–Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Let. 21, 842–844 (1996).
[CrossRef]

1986

1972

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 227–246 (1972).

1971

R. W. Gerchberg, W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik (Stuttgart) 34, 275–284 (1971).

1967

1966

1965

L. Mandel, E. Wolf “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

Bachl, A.

Beran, M. J.

M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964).

Considine, P. S.

Dorsch, R. G.

Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Gerchberg–Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Let. 21, 842–844 (1996).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 227–246 (1972).

R. W. Gerchberg, W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik (Stuttgart) 34, 275–284 (1971).

Glaser, I.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

Isaacson, E.

E. Isaacson, H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966).

Keller, H. B.

E. Isaacson, H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966).

Lukosz, W.

Mandel, L.

L. Mandel, E. Wolf “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

Mendlovic, D.

Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Gerchberg–Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Let. 21, 842–844 (1996).
[CrossRef]

Parrent, G. B.

M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964).

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 227–246 (1972).

R. W. Gerchberg, W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik (Stuttgart) 34, 275–284 (1971).

Wolf, E.

L. Mandel, E. Wolf “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

Zalevsky, Z.

Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Gerchberg–Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Let. 21, 842–844 (1996).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Let.

Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Gerchberg–Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Let. 21, 842–844 (1996).
[CrossRef]

Optik (Stuttgart)

R. W. Gerchberg, W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik (Stuttgart) 34, 275–284 (1971).

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 227–246 (1972).

Rev. Mod. Phys.

L. Mandel, E. Wolf “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

Other

E. Isaacson, H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964).

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Figures (5)

Fig. 1
Fig. 1

Typical imaging system used for the OTF design.

Fig. 2
Fig. 2

Fourier transformer for the coherent beam-shaping application.

Fig. 3
Fig. 3

(a) Aperture chosen for the OTF design. (b) Desired and obtained OTF’s. (c) Display of the desired and obtained OTF’s within the aperture region.

Fig. 4
Fig. 4

(a) Input amplitude distribution. (b) Output intensity distribution.

Fig. 5
Fig. 5

Computer-simulation results for the beam-shaping application by use of the Fourier-transform configuration.

Equations (34)

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Pfx, fy=1k--hx, y2×exp-2πifxx+fyydxdy,
k=1--hx, y2dxdy
Idox, y=--hx-x0, y-y02Igix0, y0dx0dy0,
Hfx, fy=T-λUfx, -λUfy,
Hfx, fy=--hx, yexp-2πifxx+fyydxdy.
1U+1V=1F,
Pfx, fy=--Tξ+λUfx/2, η+λUfy/2T*ξ-ηUfx/2, η-λUfy/2dξdη--Tξ,η2dξdη.
P0, 0=1, P-fx, -fy=Pfx, fy,* Pfx, fyP0, 0.
Tu=Muexpiwu,
u=-λUfx.
px=-Pfxexp2πixfxdfx.
Ihx=-Muexpiwuexp+2πixλUudu×-Mu expiwuexp2πixλU udu*=--Mu1Mu2expiwu1-wu2+2πxλUu1-2πxλUu2du1du2.
=-px-Ihx2dx.
ψu1, u2, xwu1-wu2+2πxλUu1-2πxλUu2.
=-px2+Ihx2-2REIhxpx*dx,
wuwu+δwu.
expiδwu1-δwu21+iδwu1-δwu2.
Ihδx--Mu1Mu2expiψu1, u2, x×1+iδwu1-δwu2du1du2.
REIhδxpxIhxpx-px--Mu1Mu2×sinψu1, u2, x δwu1-δwu2×du1du2 Ihδx2Ihx2-2Ihx--Mu1Mu2×sinψu1, u2, xδwu1-δwu2×du1du2.
δ=-δ,
δ=-2-px-Ihx×--Mu1Mu2sinψu1, u2, x×δwu1du1du2---Mu1Mu2sinψu1, u2, x×δwu2du1du2dx,
ψu1, u2, x=-ψu2, u1, x,
δ=4-px-Ihx--Mu1Mu2×sinψu1, u2, xδwu2du1du2dx.
δ=4-Mu2δwu2 du2--px-Ihx×Mu1sinψu1, u2, xdu1dx.
--px-IhxMu1sinψu1, u2, xdu1dx=0.
sinwu1-2πxλUu2+2πxλUu1-wu2=sinwu1-2πxλUu2+2πxλUu1cos wu2-coswu1-2πxλUu2+2πxλUu1sin wu2
cos wu2--px-IhxMu1sin ϕdu1dx=sin wu2--px-IhxMu1cos ϕdu1dx,
tan wu2=--px-IhxMu1sin ϕdu1dx--px-IhxMu1cos ϕdu1dx
wu2=arctan--px-IhxMu1sin ϕdu1dx--px-IhxMu1cos ϕdu1 dx.
=-px2+c2Ihx2-2REcIhxpxdx.
c=0c=-REIhxpx dx-Ihx2 dx.
Pfx=FIdoxFIgix,
=-Idox-FEgiuexpiwu22 dx,
Ψx=α0+α1x+α2x2+α3x3+α4x4,

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