Abstract

We report on an analytical formulation for evaluating the amplitude distribution along any line directed toward the geometrical focus of a spherical wave front that passes through a rotationally nonsymmetric diffracting screen. Our formula consists of two factors. The first factor involves the one-dimensional Fourier transform of the projection of the screen function onto the off-axis line. The second factor depends on the inverse distance to the screen and permits us to recognize the existence of focal shift along off-axis lines.

© 1993 Optical Society of America

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References

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  1. D. G. Stavenga, J. H. van Hateren, J. Opt. Soc. Am. A 8, 14 (1991).
    [CrossRef]
  2. M. A. Gusinow, M. E. Riley, M. A. Palmer, Opt. Quantum Electron. 9, 465 (1977).
    [CrossRef]
  3. Y. Li, E. Wolf, Opt. Commun. 39, 211 (1981).
    [CrossRef]
  4. C. W. McCutchen, J. Opt. Soc. Am. 54, 240 (1964).
    [CrossRef]
  5. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), p. 436.

1991 (1)

1981 (1)

Y. Li, E. Wolf, Opt. Commun. 39, 211 (1981).
[CrossRef]

1977 (1)

M. A. Gusinow, M. E. Riley, M. A. Palmer, Opt. Quantum Electron. 9, 465 (1977).
[CrossRef]

1964 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), p. 436.

Gusinow, M. A.

M. A. Gusinow, M. E. Riley, M. A. Palmer, Opt. Quantum Electron. 9, 465 (1977).
[CrossRef]

Li, Y.

Y. Li, E. Wolf, Opt. Commun. 39, 211 (1981).
[CrossRef]

McCutchen, C. W.

Palmer, M. A.

M. A. Gusinow, M. E. Riley, M. A. Palmer, Opt. Quantum Electron. 9, 465 (1977).
[CrossRef]

Riley, M. E.

M. A. Gusinow, M. E. Riley, M. A. Palmer, Opt. Quantum Electron. 9, 465 (1977).
[CrossRef]

Stavenga, D. G.

van Hateren, J. H.

Wolf, E.

Y. Li, E. Wolf, Opt. Commun. 39, 211 (1981).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), p. 436.

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

Y. Li, E. Wolf, Opt. Commun. 39, 211 (1981).
[CrossRef]

Opt. Quantum Electron. (1)

M. A. Gusinow, M. E. Riley, M. A. Palmer, Opt. Quantum Electron. 9, 465 (1977).
[CrossRef]

Other (1)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), p. 436.

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Figures (3)

Fig. 1
Fig. 1

Geometry used for diffraction investigation under converging spherical-wave illumination. The origin of the cylindrical coordinates, A, coincides with the point where the off-axis line intersects the spherical wave front.

Fig. 2
Fig. 2

Profile of the azimuthally averaged transmittance A o ( ζ ) of a circular aperture versus the normalized axial coordinate, ζ/ζM = (r/R)2, for four different positions of the origin of the polar coordinates.

Fig. 3
Fig. 3

Normalized irradiance distribution for a clear circular aperture, with N = 2, along four different lines directed toward the geometrical focus.

Equations (7)

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U ( P ) = - i exp ( - i k f ) f W A ¯ ( S ) exp ( i k s ) s d S ,
s 2 = f 2 + z 2 - 2 f z cos ( s ^ ) = f 2 + z 2 + 2 f z [ 1 - ( r / f ) 2 ] 1 / 2 .
s ( f + z ) - z r 2 2 f ( f + z ) .
U A ( z ) = - i λ exp ( i k z ) f ( f + z ) 0 2 π r min r max A ( r , ϕ ) × exp [ - i k 2 z f ( f + z ) r 2 ] r d r d ϕ ,
ζ = r 2 2 f ,             A ( ζ , ϕ ) = A ( r , ϕ ) .
U A ( z ) = - i λ ( f + z ) exp ( i k z ) 0 2 π ζ min ζ max A ( ζ , ϕ ) × exp [ - i k z ( f + z ) ζ ] d ζ d ϕ .
U A ( z ) = - i k f + z exp ( i k z ) ζ min ζ max A ( ζ ) × exp [ - i k z ( f + z ) ζ ] d ζ .

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