Abstract

In coherent optical filtering systems, the paraxial description of the imaging process represents the basis for the functional representation of the optical filter. With this approximation, the light amplitude in the secondary focal plane of an aberration-free lens appears as the Fourier transform of the light amplitude in the primary focal plane, as long as the field angles of all chief rays remain within the limits of Gaussian optics. The introduction of Abbe’s sine condition into the chief ray path allows the paraxial restriction to be dropped, and the Fourier transform relationship becomes valid for large as well as small field angles. The resulting Fourier transform lens designs are remarkably different from conventional imaging systems.

© 1971 Optical Society of America

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References

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  1. E. Abbe, Gesammelte Abhandlungen (G. Fischer, Jena, 1904), Band 1, p. 137.
  2. F. Zernike, Physica 1, 689 (1934).
    [CrossRef]
  3. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1950).
  4. P. Elias, D. Grey, D. Robinson, J. Opt. Soc. Am. 42, 127 (1952).
    [CrossRef]
  5. J. E. Rhodes, Am. J. Phys. 21, 337 (1953).
    [CrossRef]
  6. E. L. O’Neil, IRE Trans. Inform. Theory IT-2, 56 (1965).
  7. L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inform. Theory IT-6, 386 (1960).
    [CrossRef]

1965 (1)

E. L. O’Neil, IRE Trans. Inform. Theory IT-2, 56 (1965).

1960 (1)

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inform. Theory IT-6, 386 (1960).
[CrossRef]

1953 (1)

J. E. Rhodes, Am. J. Phys. 21, 337 (1953).
[CrossRef]

1952 (1)

1934 (1)

F. Zernike, Physica 1, 689 (1934).
[CrossRef]

Abbe, E.

E. Abbe, Gesammelte Abhandlungen (G. Fischer, Jena, 1904), Band 1, p. 137.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1950).

Cutrona, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inform. Theory IT-6, 386 (1960).
[CrossRef]

Elias, P.

Grey, D.

Leith, E. N.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inform. Theory IT-6, 386 (1960).
[CrossRef]

O’Neil, E. L.

E. L. O’Neil, IRE Trans. Inform. Theory IT-2, 56 (1965).

Palermo, C. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inform. Theory IT-6, 386 (1960).
[CrossRef]

Porcello, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inform. Theory IT-6, 386 (1960).
[CrossRef]

Rhodes, J. E.

J. E. Rhodes, Am. J. Phys. 21, 337 (1953).
[CrossRef]

Robinson, D.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1950).

Zernike, F.

F. Zernike, Physica 1, 689 (1934).
[CrossRef]

Am. J. Phys. (1)

J. E. Rhodes, Am. J. Phys. 21, 337 (1953).
[CrossRef]

IRE Trans. Inform. Theory (2)

E. L. O’Neil, IRE Trans. Inform. Theory IT-2, 56 (1965).

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inform. Theory IT-6, 386 (1960).
[CrossRef]

J. Opt. Soc. Am. (1)

Physica (1)

F. Zernike, Physica 1, 689 (1934).
[CrossRef]

Other (2)

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1950).

E. Abbe, Gesammelte Abhandlungen (G. Fischer, Jena, 1904), Band 1, p. 137.

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Figures (4)

Fig. 1
Fig. 1

Fourier transform geometry.

Fig. 2
Fig. 2

Symmetrical Fourier transform lens design.

Fig. 3
Fig. 3

Asymmetrical Fourier transform lens.

Fig. 4
Fig. 4

Asymmetrical Fourier transform lens predesign.

Tables (1)

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Table I Design Requirements

Equations (11)

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z z + y y = γ f z + β f y ,
Φ = ( 2 π / λ ) ( γ z + β y ) .
d E = f ( z , y ) exp { j [ ω t - ( 2 π / λ ) ( z z / f + y y / f ) ] } d z d y .
ω y = ( 2 π / λ f ) y , ω z = ( 2 π / λ f ) z
E ( ω z , ω y ) = A f ( z , y ) exp [ - j ( ω z z + ω y y ) ] d z d y .
s = λ · f / number ,
l 1 = f · tan θ .
l 2 = f · sin θ .
l 2 - l 1 = s
tan θ ¯ - sin θ ¯ = λ / D .
θ ¯ = ( 2 λ / D ) 1 3 .

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