Abstract

The design of a desired optical transfer function (OTF) is a common problem that has many possible applications. A well-known application for OTF design is beam shaping for incoherent illumination. However, other applications such as optical signal processing can also be addressed with this system. We design and realize an optimal phase only filter that, when attached to the imaging lens, enables an optimization (based on the minimal mean square error criterion) to a desired OTF. By combining several OTF design goal requirements, each represents a different plane along the beam propagation direction, an imaging system with an increased depth of focus is obtained. Because a phase only filter is used, high energetic efficiency is achieved.

© 2003 Optical Society of America

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References

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1999 (1)

1998 (1)

1997 (2)

1996 (1)

1995 (1)

1989 (1)

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

1971 (1)

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

Bradurn, S.

Cathey, W. T.

Dorsch, R. G.

Dowski, E. R.

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, 237–246 (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).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd. edition (McGraw-Hill, San Francisco, 1996).

Herzig, H. P.

H. P. Herzig, Micro-optics: Elements, systems and applications (Taylor Francis, London, 1997).

Longhurst, R. S.

R. S. Longhurst, Geometrical and Physical Optics2nd edition (Wiley, New York, 1967).

Mait, J. N.

Mendlovic, D.

Rhodes, W. T.

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, 237–246 (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).

Shabtay, G.

Tucker, S.

Turunen, J.

J. Turunen, F. Wyrowski, “Diffractive optics for industrial and commercial applications” (Akademie, Berlin, 1997).

Wach, H. B.

Wyrowski, F.

J. Turunen, F. Wyrowski, “Diffractive optics for industrial and commercial applications” (Akademie, Berlin, 1997).

Zalevsky, Z.

Appl. Opt. (5)

Opt. Express (1)

Opt. Lett. (1)

Optik (Stuttgart) (2)

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, 237–246 (1972).

Other (4)

J. W. Goodman, Introduction to Fourier Optics, 2nd. edition (McGraw-Hill, San Francisco, 1996).

R. S. Longhurst, Geometrical and Physical Optics2nd edition (Wiley, New York, 1967).

H. P. Herzig, Micro-optics: Elements, systems and applications (Taylor Francis, London, 1997).

J. Turunen, F. Wyrowski, “Diffractive optics for industrial and commercial applications” (Akademie, Berlin, 1997).

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Figures (4)

Fig. 1
Fig. 1

(a) Schematic drawing of phase of the Gerchberg-Saxton algorithm, (b) the specific algorithm applied for the presented iterative approach.

Fig. 2
Fig. 2

(a) Three OTF for comparison at the left-hand side edge of the DOF range, (b) three OTF for comparison at the right-hand side edge of the DOF range, (c) three OTF for comparison at the focal plane.

Fig. 3
Fig. 3

The experimental setup.

Fig. 4
Fig. 4

(a) Perfectly focused image (without the filter), (b) misfocused image without the filter, (c) with the filter.

Equations (9)

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ϕx, y=πλZax2+y2,
kWx, y=πλZax2+y2-πλZix2+y2.
Wx, y=121Za-1Zix2+y2.
Wm=121Za-1Ziw2.
Wx, y=Wmx2+y2w2.
OTFfx, fy= Px+λZifx2, y+λZify2Px-λZifx2, y-λZify2dxdy Px, ydxdy,
Pgx, y=Px, yexpjkWx, y.
OTFfx, fy=Afx,fyexpjkWx+λZifx2, y+λZify2-Wx-λZifx2, y-λZify2dxdyA00dxdy,
OTFfx, fy=Λfx2f0Λfy2f0×sinc8Wmλfx2f01-|fx|2f0×sinc8Wmλfy2f01-|fy|2f0,

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