Abstract

An achromatic Fourier transformer working with spatially coherent but temporally incoherent light is described both theoretically and experimentally. The achromatic Fraunhofer diffraction pattern results from the incoherent superposition of the monochromatic diffraction patterns generated by every spectral component of light. Wavelength compensation is achieved by a specific configuration combining airspaced zone plates and lenses. Several parameters, such as the spectral resolution of the setup or the spectral broadness of the source, govern the accuracy of coincidence of the various monochromatic patterns. Experimental results demonstrate the broadband operation of the system; the error on the achromatic Fourier transformation is 2% over the visible spectrum. Applications in colored image processing are also reported.

© 1983 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. Leith, J. Upatnieks, J. Opt. Soc. Am. 57, 975 (1967).
    [CrossRef]
  2. O. Bryngdahl, A. Lohmann, J. Opt. Soc. Am. 60, 281 (1970).
    [CrossRef]
  3. G. Morris, N. George, Opt. Lett. 5, 202 (1980); Appl. Opt. 19, 3843 (1980).
    [CrossRef] [PubMed]
  4. H. Bartelt, J. Optics 12, 169 (1981).
    [CrossRef]
  5. J. P. Goedgebuer, R. Gazeu, Opt. Commun. 27, 53 (1978).
    [CrossRef]
  6. F. T. Yu, Opt. Commun. 27, 23 (1978).
    [CrossRef]
  7. E. N. Leith, J. A. Roth, Appl. Opt. 16, 2565 (1977).
    [CrossRef] [PubMed]
  8. R. H. Katyl, Appl. Opt. 11, 1255 (1972).
    [CrossRef] [PubMed]
  9. G. M. Morris, Appl. Opt. 20, 2017 (1981).
    [CrossRef] [PubMed]
  10. R. Ferriere, J. P. Goedgebuer, J. C. Vienot, Opt. Commun. 31, 285 (1979).
    [CrossRef]
  11. G. M. Morris, N. George, Opt. Lett. 5, 446 (1980).
    [CrossRef] [PubMed]
  12. J. Upatnieks, J. G. Duthie, P. R. Ashley, “Matched Filtering with Achromatic Optical Correlator,” U.S. Army Missile Command, Tech. Rep. RR-82-5 (1982).
  13. R. Ferriere, J. P. Goedgebuer, J. C. Vienot, Conférence Européenne d’Optique 1980, “Horizons de l’Optique,” 22–25 avril 1980.
  14. R. Ferriere, J. P. Goedgebuer, Opt. Commun. 42, 223 (1982).
    [CrossRef]
  15. P. Chavel, J. Opt. Soc. Am. 70, 935 (1980).
    [CrossRef]

1982 (1)

R. Ferriere, J. P. Goedgebuer, Opt. Commun. 42, 223 (1982).
[CrossRef]

1981 (2)

1980 (3)

1979 (1)

R. Ferriere, J. P. Goedgebuer, J. C. Vienot, Opt. Commun. 31, 285 (1979).
[CrossRef]

1978 (2)

J. P. Goedgebuer, R. Gazeu, Opt. Commun. 27, 53 (1978).
[CrossRef]

F. T. Yu, Opt. Commun. 27, 23 (1978).
[CrossRef]

1977 (1)

1972 (1)

1970 (1)

1967 (1)

Ashley, P. R.

J. Upatnieks, J. G. Duthie, P. R. Ashley, “Matched Filtering with Achromatic Optical Correlator,” U.S. Army Missile Command, Tech. Rep. RR-82-5 (1982).

Bartelt, H.

H. Bartelt, J. Optics 12, 169 (1981).
[CrossRef]

Bryngdahl, O.

Chavel, P.

Duthie, J. G.

J. Upatnieks, J. G. Duthie, P. R. Ashley, “Matched Filtering with Achromatic Optical Correlator,” U.S. Army Missile Command, Tech. Rep. RR-82-5 (1982).

Ferriere, R.

R. Ferriere, J. P. Goedgebuer, Opt. Commun. 42, 223 (1982).
[CrossRef]

R. Ferriere, J. P. Goedgebuer, J. C. Vienot, Opt. Commun. 31, 285 (1979).
[CrossRef]

R. Ferriere, J. P. Goedgebuer, J. C. Vienot, Conférence Européenne d’Optique 1980, “Horizons de l’Optique,” 22–25 avril 1980.

Gazeu, R.

J. P. Goedgebuer, R. Gazeu, Opt. Commun. 27, 53 (1978).
[CrossRef]

George, N.

Goedgebuer, J. P.

R. Ferriere, J. P. Goedgebuer, Opt. Commun. 42, 223 (1982).
[CrossRef]

R. Ferriere, J. P. Goedgebuer, J. C. Vienot, Opt. Commun. 31, 285 (1979).
[CrossRef]

J. P. Goedgebuer, R. Gazeu, Opt. Commun. 27, 53 (1978).
[CrossRef]

R. Ferriere, J. P. Goedgebuer, J. C. Vienot, Conférence Européenne d’Optique 1980, “Horizons de l’Optique,” 22–25 avril 1980.

Katyl, R. H.

Leith, E.

Leith, E. N.

Lohmann, A.

Morris, G.

Morris, G. M.

Roth, J. A.

Upatnieks, J.

E. Leith, J. Upatnieks, J. Opt. Soc. Am. 57, 975 (1967).
[CrossRef]

J. Upatnieks, J. G. Duthie, P. R. Ashley, “Matched Filtering with Achromatic Optical Correlator,” U.S. Army Missile Command, Tech. Rep. RR-82-5 (1982).

Vienot, J. C.

R. Ferriere, J. P. Goedgebuer, J. C. Vienot, Opt. Commun. 31, 285 (1979).
[CrossRef]

R. Ferriere, J. P. Goedgebuer, J. C. Vienot, Conférence Européenne d’Optique 1980, “Horizons de l’Optique,” 22–25 avril 1980.

Yu, F. T.

F. T. Yu, Opt. Commun. 27, 23 (1978).
[CrossRef]

Appl. Opt. (3)

J. Opt. Soc. Am. (3)

J. Optics (1)

H. Bartelt, J. Optics 12, 169 (1981).
[CrossRef]

Opt. Commun. (4)

J. P. Goedgebuer, R. Gazeu, Opt. Commun. 27, 53 (1978).
[CrossRef]

F. T. Yu, Opt. Commun. 27, 23 (1978).
[CrossRef]

R. Ferriere, J. P. Goedgebuer, J. C. Vienot, Opt. Commun. 31, 285 (1979).
[CrossRef]

R. Ferriere, J. P. Goedgebuer, Opt. Commun. 42, 223 (1982).
[CrossRef]

Opt. Lett. (2)

Other (2)

J. Upatnieks, J. G. Duthie, P. R. Ashley, “Matched Filtering with Achromatic Optical Correlator,” U.S. Army Missile Command, Tech. Rep. RR-82-5 (1982).

R. Ferriere, J. P. Goedgebuer, J. C. Vienot, Conférence Européenne d’Optique 1980, “Horizons de l’Optique,” 22–25 avril 1980.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Experimental setup: ZP1 and ZP2, holographic on-axis zone plates; V, diffraction patterns of the pupil P chromatically dispersed along the optical axis; O1 and O2, achromatic lenses.

Fig. 2
Fig. 2

Imaging and Fourier transforming properties of a zone plate illuminated in white light. In (a), ZP2 Fourier transforms the input pupil g(x1,y1). At its back focal plane, the diffraction patterns are displayed in the region V. In (b), ZP2 acts as a negative lens imaging the plane object G(x4,y4) set at infinity. At its front focal plane, the images are displayed in V′ with wavelength-dependent sizes (e.g., GR and GB are the images of G generated by the red and blue components of light). In (c) are well-known conjugation properties of a conventional lens O1 imaging a 3-D pattern V as a 3-D distorted pattern V′. By suitably inserting the latter system between the setups (a) and (b), the volume of diffraction V is first transformed into the volume V′ and then is compressed through ZP2 at a single plane (x4,y4); in that plane one observes an achromatic diffraction pattern of the pupil g(x1,y1).

Fig. 3
Fig. 3

Evaluation of the distortion introduced by the imaging lens O1 [see Fig. 2(c)]: (a) variations of p′(σ) vs p(σ) in the visible region; (b) relative deviation from linearity of p′(σ) vs σ in the visible region around σ0 = 2 μm−1.

Fig. 4
Fig. 4

Notations used for the description of the propagation of light in the system of Fig. 1.

Fig. 5
Fig. 5

Variations of the curvature factor F(σ) altering the exact Fourier transform at the achromatic plane for different values of the wave number σ0 used for recording the holographic zone plates.

Fig. 6
Fig. 6

Influence of the wave number of illumination on the size S(σ) of an achromatic diffraction pattern: theoretical (solid line) and experimental (dashed line) curves.

Fig. 7
Fig. 7

Fraunhofer diffraction patterns of pupils illuminated in white light. On the left, the achromatic arrangement (ZP1,O1,ZP2), is removed from the setup, yielding a chromatic blurring. On the right, the patterns are recorded at the output plane of the setup: (a) diffraction pattern of two slits; (b), of a grid; (c) speckle pattern. (Experimental conditions: R = R′ = 0.3 m, D = 0.9 m, zone plates recorded with the 5145-Å argon radiation; σ0 = 1.94 μm−1.)

Fig. 8
Fig. 8

(a) Achromatic Fourier hologram of the letter T; (b) achromatic reconstruction in white light.

Fig. 9
Fig. 9

Achromatic diffraction pattern of a colored object.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

Z K ( σ ) = 1 K R σ σ 0 ,
G 0 ( x , y ) | j σ z ( σ ) - + g ( x 1 , y 1 ) × exp [ - j 2 π σ Z + 1 ( σ ) ( x 1 x + y 1 y ) ] d x 1 d y 1 | 2 σ 0 2 R 2 | - + g ( x 1 , y 1 ) × exp [ - j 2 π σ 0 R ( x 1 x + y 1 y ) ] d x 1 d y 1 | 2 ,
G 0 [ x , y , z ( σ ) ] = G 0 ( x , y ) · δ ( z - R σ σ 0 ) .
p ( σ ) = f 1 p ( σ ) / [ f 1 + p ( σ ) ] ,
m ( σ ) = f 1 / [ f 1 + p ( σ ) ] .
m = 1 - p f 1 ,
p ( σ ) = f 1 2 [ f 1 + p ( σ 0 ) ] 2 p ( σ ) + f 1 p 2 ( σ 0 ) [ f 1 + p ( σ 0 ) ] 2 .
τ 1 ( x 1 , y 1 ) = exp ( - j π x 1 2 + y 1 2 R σ 0 ) g ( x 1 , y 1 ) ,
τ 2 ( x 2 , y 2 ) = exp ( - j π x 2 2 + y 2 2 f 1 σ ) ,
τ 3 ( x 3 , y 3 ) = exp [ - j π ( x 3 2 + y 3 2 ) ( R σ - f 2 σ 0 ) f 2 R ]
E 2 ( x 2 , y 2 , σ ) = - j σ 2 R + f 1 exp [ j π σ 2 R + f 1 ( x 2 2 + y 2 2 ) ] · τ 2 ( x 2 , y 2 ) × τ 1 ( x 1 , y 1 ) · exp [ j π σ 2 R + f 1 ( x 1 2 + y 1 2 ) ] × exp [ - j 2 π σ 2 R + f 1 ( x 1 x 2 + y 1 y 2 ) ] d x 1 d y 1 ,
E 3 ( x 3 , y 3 , σ ) = - j σ f 1 exp [ j π σ f 1 ( x 3 2 + y 3 2 ) ] × τ 3 ( x 3 , y 3 ) E 2 ( x 2 , y 2 , σ ) × exp [ j π σ f 1 ( x 2 2 + y 2 2 ) ] × exp [ - j 2 π σ f 1 ( x 2 x 3 + y 2 y 3 ) ] d x 2 d y 2 .
E 4 ( x 4 y 4 , σ ) = - j f 1 σ 2 f 2 R ( σ 0 - 2 σ ) · exp { j π σ f 2 [ 1 - f 1 2 σ f 2 R ( σ 0 - 2 σ ) ] × ( x 4 2 + y 4 2 ) } · g ( x 1 , y 1 ) × exp [ - j π ( x 1 2 + y 1 2 ) ( σ 0 - σ ) 2 R ( σ 0 - 2 σ ) ] × exp [ - j 2 π σ 2 f 1 f 2 R ( σ 0 - 2 σ ) ( x 1 x 4 + y 1 y 4 ) ] d x 1 d y 1 .
R [ ( x 1 2 + y 1 2 ) · F ( σ ) ] max ,
F ( σ ) = π ( σ 0 - σ ) 2 / σ 0 - 2 σ .
S ( σ ) = σ 2 f 1 / R f 2 ( σ 0 - 2 σ ) ,

Metrics