Abstract

This paper shows the requirements for the achromatization of an irradiance distribution which is a function of a space coordinate multiplied by wavelength raised to a power. The particular requirements for achromatic Fourier transformation are then presented. The theory is applied to the following problems: achromatization of Newton’s ring patterns and Fraunhofer diffraction patterns and frequency plane filtering of an object illuminated with a broadband light source. Experimental systems that perform these functions are presented.

© 1972 Optical Society of America

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References

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  1. Part 1 of this series. R. H. Katyl, Appl. Opt. 11, 1241 (1972).
    [Crossref] [PubMed]
  2. Part 2 of this series. R. H. Katyl, Appl. Opt. 11, 1248 (1972).
    [Crossref] [PubMed]
  3. R. S. Longhurst, Geometrical and Physical Optics (Longmans, Green and Co., London, 1960), p. 135.
  4. I. Newton, Opticks (Dover, New York, 1952), p. 222.
  5. J. W. Strutt, Philos. Mag. 28, 77, 189 (1889).
  6. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 86.
  7. A. Vander Lugt, Proc. IEEE 54, 1055 (1966).
    [Crossref]
  8. Schott SF-58 glass, Schott Optical Glass, Inc., York Avenue, Duryea, Pennsylvania 18642.
  9. S. Tolansky, Multiple-Beam Interferometry of Surfaces and Films (Dover, New York, 1970), p. 34.

1972 (2)

1966 (1)

A. Vander Lugt, Proc. IEEE 54, 1055 (1966).
[Crossref]

1889 (1)

J. W. Strutt, Philos. Mag. 28, 77, 189 (1889).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 86.

Katyl, R. H.

Longhurst, R. S.

R. S. Longhurst, Geometrical and Physical Optics (Longmans, Green and Co., London, 1960), p. 135.

Newton, I.

I. Newton, Opticks (Dover, New York, 1952), p. 222.

Strutt, J. W.

J. W. Strutt, Philos. Mag. 28, 77, 189 (1889).

Tolansky, S.

S. Tolansky, Multiple-Beam Interferometry of Surfaces and Films (Dover, New York, 1970), p. 34.

Vander Lugt, A.

A. Vander Lugt, Proc. IEEE 54, 1055 (1966).
[Crossref]

Appl. Opt. (2)

Philos. Mag. (1)

J. W. Strutt, Philos. Mag. 28, 77, 189 (1889).

Proc. IEEE (1)

A. Vander Lugt, Proc. IEEE 54, 1055 (1966).
[Crossref]

Other (5)

Schott SF-58 glass, Schott Optical Glass, Inc., York Avenue, Duryea, Pennsylvania 18642.

S. Tolansky, Multiple-Beam Interferometry of Surfaces and Films (Dover, New York, 1970), p. 34.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 86.

R. S. Longhurst, Geometrical and Physical Optics (Longmans, Green and Co., London, 1960), p. 135.

I. Newton, Opticks (Dover, New York, 1952), p. 222.

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Figures (7)

Fig. 1
Fig. 1

Example of a two-lens compensating system employing dispersive lenses.

Fig. 2
Fig. 2

Compensating system for production of achromatic Fourier transform followed by achromatic retransforming lens.

Fig. 3
Fig. 3

Second compensating system which follows the system of Fig. 2 for the production of an achromatic filtered image.

Fig. 4
Fig. 4

Experimental systems used for (a) Newton’s ring achromatization, (b) Fraunhofer diffraction pattern achromatization, and (c) frequency plane filtering with a broadband light source.

Fig. 5
Fig. 5

(a) Typical multiple reflection Newton’s ring pattern viewed with white light and no compensation, (b) pattern when compensating system is used.

Fig. 6
Fig. 6

Blurred and deblurred Fraunhofer diffraction patterns of following objects: (a), (b) halftone screen; (c), (d) three crossed Ronchi rulings; (e), (f) on-axis Fraunhofer hologram.

Fig. 7
Fig. 7

Images from frequency plane filtering experiment (a) without filter, (b) with 2-mm−1 horizontally oriented Ronchi ruling as filter, (c) with 4-mm−1 horizontally oriented Ronchi ruling as filter.

Tables (1)

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Table I Parameters of the Experimental Systemsa

Equations (33)

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1 f 1 = ( 1 z + 1 a ) + ( S z a ) · 1 M ,
1 f 2 = ( 1 S + 1 a ) + ( z S a ) · M ,
1 f 1 = ( 1 z + 1 a ) + ( S M 0 z a ) μ N ,
1 f 2 = ( 1 S + 1 a ) + ( M 0 z a S ) μ N ,
D 1 = f 1 μ ( 1 / f 1 ) ( 1 / μ ) = N ( f 1 a ) ( S M 0 z ) ,
D 2 = f 2 μ ( 1 / f 2 ) ( 1 / μ ) = N ( f 2 a ) ( M 0 z S ) .
E ( r 1 ) = ( 1 / i λ f ) exp [ i k r 1 2 2 f ( 1 d f ) ] T ˜ ( r 1 / λ f ) ,
k = 2 π / λ , T ˜ ( F ) = exp ( 2 π i F · r ) T ( r ) d r , f = focal length of the transforming lens .
1 f 1 = ( 1 z + 1 a ) + ( S M 0 z a ) μ ,
1 f 2 = ( 1 S + 1 a ) + ( M 0 z a S ) 1 μ .
( 1 / f 1 a ) = ( 1 / z ) + ( 1 / a ) ( achromat ) ,
( 1 / f 1 b ) = ( S / M 0 z a ) μ ( zone plate lens ) .
f 1 a = ,
f 1 b f 2 = D 2 a 2 ,
( 1 / S ) = [ ( 1 D 2 ) / f 2 ] ( 1 / a ) ,
M 0 = ( D 2 S / f 2 ) .
E ( r 2 ) = M exp ( i k l ) exp ( i k r 2 2 2 R ) T ˜ ( r 2 / M λ f 0 ) ,
E ( r 3 ) exp ( i k r 3 2 2 R 1 ) U ( r 3 / λ S 2 ) ,
1 R 1 = ( f d R ) ( d + R f ) 1 S 1 ,
S 2 = S 1 / [ 1 + ( d / R ) ] .
( 1 / f 4 ) = ( 1 / f 4 a ) ( μ / f 4 b ) ,
( 1 / f 5 ) = ( 1 / a 2 ) + ( 1 / S 3 ) ( M 20 / μ S 3 ) ,
( 1 / f 4 a ) = ( 1 / a 2 ) + ( 1 / S ) ( 1 / f 3 ) , ( 1 / f 4 b ) = ( 1 / M 10 S ) + ( S 3 / a 2 2 M 20 ) ,
a 2 2 = ( M 10 S S 3 / M 20 ) .
( 1 / f 3 ) = ( 1 / a 2 ) + ( 1 / S ) ,
a 2 2 = M 10 S ( f 5 / D 5 ) ,
M 20 = ( S 3 D 5 / f 5 ) ,
( 1 / S 3 ) = [ ( 1 D 5 ) / f 5 ] ( 1 / a 2 ) .
( 1 / s ) = ( 1 / f 1 ) ( 1 / z ) ,
( 1 / S ) = ( 1 / f 2 ) [ 1 / ( a s ) ] ,
M = ( s / z ) · [ S / ( a s ) ] .
( 1 / f 1 ) = ( 1 / a ) + ( 1 / z ) + ( S / z a ) ( 1 / M ) .
( 1 / f 2 ) = ( 1 / a ) + ( 1 / S ) + ( M z / S a ) ,

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