Abstract

A method for the Fourier transformation of one-dimensional functions, especially interferograms, with coherent light is described. The functions to be transformed are represented in the form of transparent masks with amplitude transmittance T=±1. Negative values of amplitude transmittance are realized with a phase plate. As the essential result, the Fourier transform of the interferogram is displayed by a lens along the abscissa of the frequency plane.

© 1971 Optical Society of America

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References

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  1. G. W. Stroke and A. T. Funkhouser, Phys. Letters 16, 272 (1965).
    [Crossref]
  2. S. Lowenthal, C. Froehly, and T. Serres, Compt. Rend. 268, 1481 (1969).
  3. H. C. Montgomery, Bell System Tech. J. 17, 406 (1938).
    [Crossref]
  4. A. Lohman, Opt. Acta 6, 37 (1959).
    [Crossref]
  5. R. F. Edgar, B. Lawrenson, and J. Ring, J. Phys. 28, Colloq. C2, Suppl. 3–4, 73 (1967).
  6. R. L. Lamberts and G. C. Higgins, Phot, Sci. Eng. 10, 209, 213 (1966).

1969 (1)

S. Lowenthal, C. Froehly, and T. Serres, Compt. Rend. 268, 1481 (1969).

1967 (1)

R. F. Edgar, B. Lawrenson, and J. Ring, J. Phys. 28, Colloq. C2, Suppl. 3–4, 73 (1967).

1966 (1)

R. L. Lamberts and G. C. Higgins, Phot, Sci. Eng. 10, 209, 213 (1966).

1965 (1)

G. W. Stroke and A. T. Funkhouser, Phys. Letters 16, 272 (1965).
[Crossref]

1959 (1)

A. Lohman, Opt. Acta 6, 37 (1959).
[Crossref]

1938 (1)

H. C. Montgomery, Bell System Tech. J. 17, 406 (1938).
[Crossref]

Edgar, R. F.

R. F. Edgar, B. Lawrenson, and J. Ring, J. Phys. 28, Colloq. C2, Suppl. 3–4, 73 (1967).

Froehly, C.

S. Lowenthal, C. Froehly, and T. Serres, Compt. Rend. 268, 1481 (1969).

Funkhouser, A. T.

G. W. Stroke and A. T. Funkhouser, Phys. Letters 16, 272 (1965).
[Crossref]

Higgins, G. C.

R. L. Lamberts and G. C. Higgins, Phot, Sci. Eng. 10, 209, 213 (1966).

Lamberts, R. L.

R. L. Lamberts and G. C. Higgins, Phot, Sci. Eng. 10, 209, 213 (1966).

Lawrenson, B.

R. F. Edgar, B. Lawrenson, and J. Ring, J. Phys. 28, Colloq. C2, Suppl. 3–4, 73 (1967).

Lohman, A.

A. Lohman, Opt. Acta 6, 37 (1959).
[Crossref]

Lowenthal, S.

S. Lowenthal, C. Froehly, and T. Serres, Compt. Rend. 268, 1481 (1969).

Montgomery, H. C.

H. C. Montgomery, Bell System Tech. J. 17, 406 (1938).
[Crossref]

Ring, J.

R. F. Edgar, B. Lawrenson, and J. Ring, J. Phys. 28, Colloq. C2, Suppl. 3–4, 73 (1967).

Serres, T.

S. Lowenthal, C. Froehly, and T. Serres, Compt. Rend. 268, 1481 (1969).

Stroke, G. W.

G. W. Stroke and A. T. Funkhouser, Phys. Letters 16, 272 (1965).
[Crossref]

Bell System Tech. J. (1)

H. C. Montgomery, Bell System Tech. J. 17, 406 (1938).
[Crossref]

Compt. Rend. (1)

S. Lowenthal, C. Froehly, and T. Serres, Compt. Rend. 268, 1481 (1969).

J. Phys. (1)

R. F. Edgar, B. Lawrenson, and J. Ring, J. Phys. 28, Colloq. C2, Suppl. 3–4, 73 (1967).

Opt. Acta (1)

A. Lohman, Opt. Acta 6, 37 (1959).
[Crossref]

Phot, Sci. Eng. (1)

R. L. Lamberts and G. C. Higgins, Phot, Sci. Eng. 10, 209, 213 (1966).

Phys. Letters (1)

G. W. Stroke and A. T. Funkhouser, Phys. Letters 16, 272 (1965).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1

Interferogram of infrared absorption of CsBr.

Fig. 2
Fig. 2

Mask of the interferogram, input of the coherent-optical Fourier-transformation setup.

Fig. 3
Fig. 3

Coherent-optical setup for Fourier transformation of one-dimensional functions.

Fig. 4
Fig. 4

Photograph of the irradiance distribution in the frequency plane.

Fig. 5
Fig. 5

Spectrum of the interferogram of Fig. 1 obtained with the digital computer (above) and with the coherent-optical setup (below).

Equations (17)

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f ( x ) = f p ( x ) + f n ( x ) .
T ( x , y ) = T p ( x , y ) + T n ( x , y ) ,
T p ( x , y ) = { + 1 for 0 y f p ( x ) 0 elsewhere
T n ( x , y ) = { - 1 for 0 y f n ( x ) 0 elsewhere .
F ( ν , μ ) = - + T ( x , y ) e - 2 π i ( x ν + y μ ) d x d y .
F ( ν , μ ) = F p ( ν , μ ) + F n ( ν , μ ) ,
F p ( ν , μ ) = - + T p ( x , y ) e - 2 π i ( x ν + y μ ) d x d y ,
F n ( ν , μ ) = - + T n ( x , y ) e - 2 π i ( x ν + y μ ) d x d y .
F ( ν , 0 ) = - + e - 2 π i x ν [ - + T ( x , y ) d y ] d x .
F ( ν , 0 ) = - + [ f p ( x ) + f n ( x ) ] e - 2 π i x ν d x = - + f ( x ) e - 2 π i x ν d x .
μ 0 = d / ( λ · f ) ,
μ 0 ( 1 / y max ) ,
F ( ν , μ ) = - + [ T p ( x , y - b 2 ) + T n ( x , y + b 2 ) ] × e - 2 π i ( x ν + y μ ) d x d y .
F ( ν , μ ) = e - 2 π i ( b / 2 ) μ - + T p ( x , y ) e - 2 π i ( x ν + y μ ) d x d y + e + 2 n i ( b / 2 ) μ - + T n ( x , y ) e - 2 π i ( x ν + y μ ) d x d y = e - 2 π i ( b / 2 ) μ F p ( ν , μ ) + e + 2 π i ( b / 2 ) μ F n ( ν , μ ) = ( cos 2 π b 2 μ ) [ F p ( ν , μ ) + F n ( ν , μ ) ] - i ( sin 2 π b 2 μ ) [ F p ( ν , μ ) - F n ( ν , μ ) ] .
F ( ν , μ ) = [ 1 + 1 2 ( 2 π b 2 μ ) 2 ] [ F p ( ν , μ ) + F n ( ν , μ ) ] - i ( 2 π b 2 μ ) [ F p ( ν , μ ) - F n ( ν , μ ) ] .
μ 0 ( 1 / b ) ,
d 2 mm ,