Abstract

Various formulations have been proposed for the Fresnel approximation for diffraction by a circular aperture. These formulations can describe the effects of convergent illumination, finite value of Fresnel number, and off-axis illumination. The retention of a further term, which is dependent on the coordinates of the focus and the observation point, is proposed. This results in a redefined axial optical coordinate, giving improved prediction of the diffracted field for off-axis points.

© 1993 Optical Society of America

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References

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  1. S. F. Gibson, F. Lanni, “Diffraction by a circular aperture as a model for three-dimensional microscopy,” J. Opt. Soc. Am. A 6, 1357–1367 (1989).
    [CrossRef] [PubMed]
  2. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).
  3. Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984).
    [CrossRef]
  4. C. J. R. Sheppard, “Imaging in optical systems of finite Fresnel number,” J. Opt. Soc. Am. A 3, 1428–1432 (1986).
    [CrossRef]
  5. J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
    [CrossRef]

1989 (1)

1986 (1)

1984 (1)

1979 (1)

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Geometry of the diffraction problem.

Fig. 2
Fig. 2

Intensity along the axis of illumination of a convergent beam incident off axis upon a circular aperture. Curve (i) according to the approximation of Gibson and Lanni,1 curve (ii) retaining an extra defocus term dependent on the transverse coordinates of the focus and the observation point, and curve (iii) retaining the astigmatism term. The angle of illumination is 60°.

Equations (15)

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U ( P d ) = - i A 0 λ A exp [ - i k ( r - s ) ] s d S ,
U ( P d ) = - i k a 2 A o r d exp [ i k ( r d - r f ) ] × 0 1 J 0 ( v ρ ) exp ( 1 2 i u ρ 2 ) ρ d ρ ,
v = k a [ ( x d r d - x f r f ) 2 + ( y d r d - y f r f ) 2 ] 1 / 2 ,
u = k a 2 ( 1 r d - 1 r f ) .
r - s = - ( r d - r f ) + a ρ [ ( x d r d - x f r f ) cos θ + ( y d r d - y f r f ) sin θ ] - a 2 ρ 2 2 ( 1 r d - 1 r f ) - a 2 ρ 2 4 [ ( x d r d cos θ + y d r d sin θ ) 2 × 1 r d - ( x f r f cos θ + y f r f sin θ ) 2 1 r f ] + a 4 ρ 4 8 ( 1 r d 3 - 1 r f 3 ) - a 3 ρ 3 4 × [ ( x d r d 3 - x f r f 3 ) cos θ + ( y d r f 3 - y f r f 3 ) sin θ ] + a 3 ρ 3 8 [ ( x d r d cos θ + y d r d sin θ ) 3 1 r d 2 - ( x f r f cos θ + y f r f sin θ ) 3 1 r d 2 ] .
ξ = x r ,             η = y r ,             ζ = 1 r ,
r - s = - ( r d - r f ) + a ρ [ ( ξ d - ξ f ) cos θ + ( η d - η f ) sin θ ] - a 2 ρ 2 2 [ ζ d ( 1 + ξ d 2 + η d 2 4 ) - ζ f ( 1 + ξ f 2 + η f 2 4 ) ] - a 2 ρ 2 8 { [ ζ d ( ξ d 2 - η d 2 ) - ζ f ( ξ f 2 - η f 2 ) ] cos 2 θ + 2 ( ξ d η d ζ d - ξ f η f ζ f ) sin 2 θ } .
N A 2 ( k a ) - 1 / 2 .
u = k a 2 × { 1 r d [ 1 + 1 4 ( x d 2 + y d 2 r d 2 ) ] - 1 r f [ 1 + 1 4 ( x f 2 + y f 2 r f 2 ) ] }
u = k a 2 × { ζ d [ 1 + 1 4 ( ξ d 2 + η d 2 ) ] - ζ f [ 1 + 1 4 ( ξ f 2 + η f 2 ) ] } .
ξ d = ξ f = ξ , η d = η f = 0.
U ( P d ) = - i A 0 k a 2 ζ d 0 1 0 2 π exp { 1 2 i k a 2 ρ 2 ( ζ d - ζ f ) × [ 1 + ξ 2 4 ( 1 + cos 2 θ ) ] } d θ ρ d ρ .
I = | 0 1 J 0 [ k a 2 ξ 2 ( ζ d - ζ f ) t 8 ] × exp { 1 2 i k a 2 ( ζ d - ζ f ) t ( 1 + ξ 2 4 ) } d t | 2 ,
ζ d - ζ f ζ d .
I = { sin [ k a 2 4 ( ζ d - ζ f ) ( 1 + ξ 2 4 ) ] k a 2 4 ( ζ d - ζ f ) ( 1 + ξ 2 4 ) } 2 .

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