Abstract

The relation for the angular spectra of rotated planes is evaluated, starting from the knowledge of the monochromatic scalar field on a given plane. Diffracted light on a tilted plane can then be calculated in the frequency domain by fast-Fourier-transform algorithms. Unlike the Fresnel and Fraunhofer approaches, this analysis does not require approximations; as a consequence, it permits any positions in space for the planes under investigation. Digital images are generated, which show the effects of rotation.

© 1992 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  2. S. Ganci, Eur. J. Phys. 2, 158 (1981).
    [CrossRef]
  3. K. Patorski, Opt. Acta 30, 673 (1982).
    [CrossRef]
  4. H. J. Rabal, N. Bolognini, E. E. Sicre, Opt. Acta 32, 1309 (1985).
    [CrossRef]
  5. D. Leseberg, C. Frere, Appl. Opt. 27, 3020 (1988).
    [CrossRef] [PubMed]
  6. C. Frere, D. Leseberg, Appl. Opt. 28, 2422 (1989).
    [CrossRef] [PubMed]

1989

1988

1985

H. J. Rabal, N. Bolognini, E. E. Sicre, Opt. Acta 32, 1309 (1985).
[CrossRef]

1982

K. Patorski, Opt. Acta 30, 673 (1982).
[CrossRef]

1981

S. Ganci, Eur. J. Phys. 2, 158 (1981).
[CrossRef]

Bolognini, N.

H. J. Rabal, N. Bolognini, E. E. Sicre, Opt. Acta 32, 1309 (1985).
[CrossRef]

Frere, C.

Ganci, S.

S. Ganci, Eur. J. Phys. 2, 158 (1981).
[CrossRef]

Goodman, J. W.

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

Leseberg, D.

Patorski, K.

K. Patorski, Opt. Acta 30, 673 (1982).
[CrossRef]

Rabal, H. J.

H. J. Rabal, N. Bolognini, E. E. Sicre, Opt. Acta 32, 1309 (1985).
[CrossRef]

Sicre, E. E.

H. J. Rabal, N. Bolognini, E. E. Sicre, Opt. Acta 32, 1309 (1985).
[CrossRef]

Appl. Opt.

Eur. J. Phys.

S. Ganci, Eur. J. Phys. 2, 158 (1981).
[CrossRef]

Opt. Acta

K. Patorski, Opt. Acta 30, 673 (1982).
[CrossRef]

H. J. Rabal, N. Bolognini, E. E. Sicre, Opt. Acta 32, 1309 (1985).
[CrossRef]

Other

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

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Figures (3)

Fig. 1
Fig. 1

Example of progressive waves for the system (x, y, z) but regressive waves for the system (ξ, η, ζ).

Fig. 2
Fig. 2

Light intensities on three rotated planes compared with that in the original transparency.

Fig. 3
Fig. 3

Moduli of the Fourier transforms for the images in Fig. 2.

Equations (16)

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A ( f x , f y , z ) = A ( f x , f y , 0 ) × exp { 2 π j z [ ( 1 / λ ) 2 - f x 2 - f y 2 ] 1 / 2 } ,
( x , y , z ) t = M ( ξ , η , ζ ) t .
U ( x , y , z ) = A ( f x , f y , 0 ) × exp [ 2 π j ( x f x + y f y + z f z ) ] d f x d f y ,
f z = [ ( 1 / λ ) 2 - f x 2 - f y 2 ] 1 / 2 .
U ( ξ , η , ζ ) = A ( f x , f y , 0 ) × exp [ 2 π j ( ξ f ξ + η f η + ζ f ζ ) ] d f x d f y ,
( f ξ , f η , f ζ ) t = M t ( f x , f y , f z ) t .
U ( ξ , η , 0 ) = C A ( f x , f y , 0 ) × exp [ 2 π j ( ξ f ξ + η f η ) ] | ( f x , f y ) ( f ξ , f η ) | d f ξ d f η ,
f x / f ξ = m 11 + m 13 f ζ / f ξ = m 11 - m 13 f ξ / f ζ ,
f x / f η = m 12 + m 13 f ζ / f η = m 12 - m 13 f η / f ζ ,
f y / f ξ = m 21 - m 23 f ξ / f ζ ,
f y / f η = m 22 - m 23 f η / f ζ .
( f x , f y ) / ( f ξ , f η ) = m 11 m 22 - m 12 m 21 + ( - m 13 m 22 + m 12 m 23 ) f ξ / f ζ + ( m 13 m 21 - m 11 m 23 ) f η / f ζ .
m 11 m 22 - m 12 m 21 = m 33 ,
( f x , f y ) / ( f ξ , f η ) = f z / f ζ .
f z > 0 ,             f ζ > 0.
A ( f ξ , f η , 0 ) = { 0 outside C A ( f x , f y , 0 ) f z / f ζ inside C .

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