Abstract

This paper is an attempt to provide new insight into the behavior of near-field scalar diffraction phenomena by showing that the Rayleigh-Sommerfeld diffraction integral is equivalent to the Fourier transform integral of a generalized pupil function which includes a term that represents phase errors in the aperture. This term can be interpreted as describing a conventional wavefront aberration function. The resulting aberration coefficients are calculated and expressed in terms of the aperture diameter, observation distance, and appropriate field parameter for several different geometrical configurations of incident beam and observation space. These aberrations, which are inherently associated with the diffraction process, are precisely the effects ignored when making the usual Fresnel and Fraunhofer approximations.

© 1978 Optical Society of America

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  1. The title of this paper refers to aberrations that are inherently associated with the diffraction process and should not be confused with the diffraction theory of aberrations, which deals with the effects of propagation upon an aberrated wavefront resulting from some imperfect imaging system.
  2. H. G. Beutler, J. Opt. Soc. Am. 35, 311 (1945).
    [CrossRef]
  3. T. Namioka, J. Opt. Soc. Am. 49, 446 (1959).
    [CrossRef]
  4. W. T. Welford, “Aberration Theory of Gratings and Grating Mountings,” in Progress in Optics, Vol. 4, E. Wolf, Ed. (Interscience, New York, 1965).
    [CrossRef]
  5. W. Werner, Appl. Opt. 6, 1691 (1967).
    [CrossRef] [PubMed]
  6. J. Baumgardner, “Theory and Design of Unusual Concave Gratings,” Ph.D. Dissertation, U. Rochester (1969) (U. Microfilms 70-17864).
  7. H. Noda, T. Namioka, M. Seya, J. Opt. Soc. Am. 64, 1031 (1974).
    [CrossRef]
  8. R. W. Meier, J. Opt. Soc. Am. 55, 987 (1965).
    [CrossRef]
  9. E. B. Champagne, J. Opt. Soc. Am. 57, 51 (1967).
    [CrossRef]
  10. E. B. Champagne, “A Qualitative and Quantitative Study of Holographic Imaging,” Ph.D. Dissertation, Ohio State U. (July1967) (U. Microfilms, 67-10876).
  11. J. N. Latta, Appl. Opt. 10, 599 (1971).
    [CrossRef] [PubMed]
  12. A. Sommerfeld, Optics, Vol. 4 (Academic, New York, 1972), p. 199.
  13. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 42.
  14. M. Kline, I. W. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965). (This book also contains formulations of the diffraction problem in terms of direction cosines.)
  15. H. H. Hopkins, Wave Theory of Aberrations (Clarendon, Oxford, 1950).
  16. The presence of these aberrations may appear to contradict the well known fact that a plane wave incident upon a conventional grating results in diffracted orders made up of plane waves. The wavefront aberration function is given by the optical path difference between the actual diffracted wavefront (which is plane) and a spherical reference wavefront converging to the observation point. However, this optical path difference is then mapped onto the plane of the aperture. It therefore exhibits asymmetries which are described by the tabulated aberration coefficients.
  17. For small aberrations the term 1/(1 + ∊)2 in Eq. (8) is very small and can also be ignored.
  18. R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965).

1974 (1)

1971 (1)

1967 (2)

1965 (1)

1959 (1)

1945 (1)

Baumgardner, J.

J. Baumgardner, “Theory and Design of Unusual Concave Gratings,” Ph.D. Dissertation, U. Rochester (1969) (U. Microfilms 70-17864).

Beutler, H. G.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965).

Champagne, E. B.

E. B. Champagne, J. Opt. Soc. Am. 57, 51 (1967).
[CrossRef]

E. B. Champagne, “A Qualitative and Quantitative Study of Holographic Imaging,” Ph.D. Dissertation, Ohio State U. (July1967) (U. Microfilms, 67-10876).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 42.

Hopkins, H. H.

H. H. Hopkins, Wave Theory of Aberrations (Clarendon, Oxford, 1950).

Kay, I. W.

M. Kline, I. W. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965). (This book also contains formulations of the diffraction problem in terms of direction cosines.)

Kline, M.

M. Kline, I. W. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965). (This book also contains formulations of the diffraction problem in terms of direction cosines.)

Latta, J. N.

Meier, R. W.

Namioka, T.

Noda, H.

Seya, M.

Sommerfeld, A.

A. Sommerfeld, Optics, Vol. 4 (Academic, New York, 1972), p. 199.

Welford, W. T.

W. T. Welford, “Aberration Theory of Gratings and Grating Mountings,” in Progress in Optics, Vol. 4, E. Wolf, Ed. (Interscience, New York, 1965).
[CrossRef]

Werner, W.

Appl. Opt. (2)

J. Opt. Soc. Am. (5)

Other (11)

The title of this paper refers to aberrations that are inherently associated with the diffraction process and should not be confused with the diffraction theory of aberrations, which deals with the effects of propagation upon an aberrated wavefront resulting from some imperfect imaging system.

W. T. Welford, “Aberration Theory of Gratings and Grating Mountings,” in Progress in Optics, Vol. 4, E. Wolf, Ed. (Interscience, New York, 1965).
[CrossRef]

J. Baumgardner, “Theory and Design of Unusual Concave Gratings,” Ph.D. Dissertation, U. Rochester (1969) (U. Microfilms 70-17864).

A. Sommerfeld, Optics, Vol. 4 (Academic, New York, 1972), p. 199.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 42.

M. Kline, I. W. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965). (This book also contains formulations of the diffraction problem in terms of direction cosines.)

H. H. Hopkins, Wave Theory of Aberrations (Clarendon, Oxford, 1950).

The presence of these aberrations may appear to contradict the well known fact that a plane wave incident upon a conventional grating results in diffracted orders made up of plane waves. The wavefront aberration function is given by the optical path difference between the actual diffracted wavefront (which is plane) and a spherical reference wavefront converging to the observation point. However, this optical path difference is then mapped onto the plane of the aperture. It therefore exhibits asymmetries which are described by the tabulated aberration coefficients.

For small aberrations the term 1/(1 + ∊)2 in Eq. (8) is very small and can also be ignored.

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965).

E. B. Champagne, “A Qualitative and Quantitative Study of Holographic Imaging,” Ph.D. Dissertation, Ohio State U. (July1967) (U. Microfilms, 67-10876).

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Figures (8)

Fig. 1
Fig. 1

Rayleigh-Sommerfeld diffraction by a plane screen with a general illuminating wave.

Fig. 2
Fig. 2

Diffracted wave field on an observation hemisphere.

Fig. 3
Fig. 3

Diffraction pattern of a 10-line/mm Ronchi ruling placed in an f/6 cone of light with a 40-mm diam. Magnified images of diffracted orders at various field positions indicate that coma is predominant for small field angles with astigmatism also becoming significant at larger field angles.

Fig. 4
Fig. 4

Diffraction pattern of a 10-line/mm Ronchi ruling placed in an f/20 cone of light with a 12-mm diam. Magnified images of a diffracted order at different focal positions indicate that astigmatism is predominant. The relationship of the sagittal, medial, and tangential surface to the observation hemisphere is also shown.

Fig. 5
Fig. 5

Geometrical configuration when the incident beam strikes the diffracting aperture at an arbitrary angle.

Fig. 6
Fig. 6

The behavior of coma with field position for several angles of incidence.

Fig. 7
Fig. 7

The behavior of astigmatism with field position for several angles of incidence.

Fig. 8
Fig. 8

Illustration of the position of the diffracted orders in real space and direction cosine space.

Tables (1)

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Table I Tabulation of Expressions for the Aberration Coefficients for Several Different Geometrical Configurations of (a) Incident Wavefront and (b) Observation Space

Equations (18)

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U ( x ˆ , y ˆ ; z ˆ ) = 1 i U ( x ˆ , y ˆ ; 0 ) z ˆ l ˆ exp ( i 2 π l ˆ ) l ˆ d x ˆ d y ˆ ,
l ˆ = [ ( x ˆ x ˆ ) 2 + ( y ˆ y ˆ ) 2 + z ˆ 2 ] 1 / 2 .
α = x ˆ / r ˆ , β = y ˆ / r ˆ , γ = z ˆ / r ˆ ,
r ˆ 2 = x ˆ 2 + y ˆ 2 + z ˆ 2 .
l ˆ = r ˆ ( 1 + ) ; = ( l ˆ r ˆ ) / r ˆ ,
U ( α , β ; r ˆ ) = γ exp ( i 2 π r ˆ ) i r ˆ U 0 ( x ˆ , y ˆ ; 0 ) × 1 ( 1 + ) 2 exp [ i 2 π ( l ˆ r ˆ ) ] d x ˆ d y ˆ .
U ( α , β ; r ˆ ) = γ exp ( i 2 π r ˆ ) i r ˆ U 0 ( x ˆ , y ˆ ; α , β ) × exp [ i 2 π ( α x + β y ) ] d x ˆ d y ˆ ,
U 0 ( x ˆ , y ˆ ; α , β ) = T 0 ( x ˆ , y ˆ ; 0 ) 1 ( 1 + ) 2 exp ( i 2 π W ˆ )
W ˆ = W ˆ 200 ρ 2 + W ˆ 020 a ˆ 2 + W ˆ 111 ρ a ˆ cos ( ϕ ϕ ) + W ˆ 400 ρ 4 + W ˆ 040 a ˆ 4 + W ˆ 131 ρ a ˆ 3 cos ( ϕ ϕ ) + W ˆ 222 ρ 2 a ˆ 2 cos 2 ( ϕ ϕ ) + W ˆ 220 ρ 2 a ˆ 2 + W ˆ 311 ρ 3 a ˆ cos ( ϕ ϕ ) + higher - order terms ,
W ˆ 131 = 1.25 × 10 4 , W ˆ 222 = 2.50 × 10 1 .
W ˆ 131 = 0.545 , W ˆ 222 = 0.095.
W ˆ 131 = 0.625 , W ˆ 222 = 0.625.
W ˆ = ( l ˆ r ˆ ) ( l ˆ 0 r ˆ 0 ) + ( α x ˆ + β y ˆ ) .
W ˆ 31 = r ˆ 2 ( β β 0 ) ( d ˆ / 2 r ˆ ) 3 ,
W ˆ 22 = r ˆ 2 ( β 2 β 0 2 ) ( d ˆ / 2 r ˆ ) 2 .
U ( α , β ; r ˆ ) = γ exp ( i 2 π r ˆ ) i r ˆ T 0 ( x ˆ , y ˆ ; 0 ) × exp [ i 2 π ( α x ˆ + β y ˆ ) ] d x ˆ d y ˆ .
U ( α , β ; r ˆ ) = γ exp ( i 2 π r ˆ ) i r ˆ F [ T 0 ( x ˆ , y ˆ ; 0 ) ] .
U ( α , β β 0 ; r ˆ ) = γ exp ( i 2 π r ˆ ) i r ˆ F [ T ( x ˆ , y ˆ ; 0 ) exp ( i 2 π β 0 y ˆ ) ] .

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