Abstract

A generalization of the Fresnel approximation in diffraction theory is proposed. The phase term in the diffraction integral is approximated by a paraboloidal variation, not by a binomial expansion but rather by a matching at the critical points in asymptotic evaluation of the integral. The method provides a correction to the optical coordinates of the Fresnel diffraction theory that extends its region of validity. It is applied to diffraction by a circular aperture of a plane wave or focused beam, including effects caused by a large numerical aperture, finite Fresnel number, off-axis illumination, and the presence of aberrations. The method may also be used with other geometries: It is readily applied to cylindrical focusing.

© 1992 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).
  2. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, 1986).
  3. J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
    [CrossRef]
  4. S. F. Gibson, F. Lanni, “Diffraction by a circular aperture as a model for three-dimensional optical microscopy,” J. Opt. Soc. Am. A 6, 1357–1367 (1989).
    [CrossRef] [PubMed]
  5. C. J. R. Sheppard, H. J. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
    [CrossRef]
  6. W. H. Southwell, “Validity of the Fresnel approximation in the near field,”J. Opt. Soc. Am. 71, 7–41 (1981).
    [CrossRef]
  7. H. Osterberg, L. W. Smith, “Closed solutions of Rayleigh’s diffraction integral for axial points,”J. Opt. Soc. Am. 51, 1050–1054 (1961).
    [CrossRef]
  8. A. M. Steane, H. N. Rutt, “Diffraction calculations in the near field and the validity of the Fresnel approximation,” J. Opt. Soc. Am. A 6, 1809–1814 (1989).
    [CrossRef]
  9. 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]
  10. C. J. R. Sheppard, “Imaging in systems of finite Fresnel number,” J. Opt. Soc. Am. A 3, 1428–1432 (1986).
    [CrossRef]
  11. M. K. Hu, “Modified Zernike polynomials and their application to the analysis of Fresnel region fields of circular apertures with nonuniform and nonsymmetric illumination,”J. Opt. Soc. Am. 53, 261–266 (1963).
    [CrossRef]
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    [CrossRef] [PubMed]

1989 (2)

1987 (1)

1986 (1)

1984 (1)

1981 (1)

1979 (1)

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

1978 (1)

1963 (1)

1961 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

Gibson, S. F.

Harvey, J. E.

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

Hu, M. K.

Lanni, F.

Li, Y.

Matthews, H. J.

Osterberg, H.

Rutt, H. N.

Sheppard, C. J. R.

Smith, L. W.

Southwell, W. H.

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, 1986).

Steane, A. M.

Wolf, E.

Am. J. Phys. (1)

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

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (5)

Opt. Lett. (1)

Other (2)

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, 1986).

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Figures (8)

Fig. 1
Fig. 1

Geometry of diffraction of a plane wave by a circular aperture.

Fig. 2
Fig. 2

(a) Axial intensity predicted by the Fresnel approximation (solid curve) and the Rayleigh–Sommerfeld theory (dashed curve) for a circular aperture (ka = 100) illuminated by a plane wave. (b) Axial intensity predicted by the generalized Fresnel approximation (solid curve).

Fig. 3
Fig. 3

Contours of constants u and v as functions of the coordinates ρ and z for a circular aperture illuminated by a plane wave.

Fig. 4
Fig. 4

Geometry of diffraction for a wave convergent on point F by a circular aperture for high aperture systems.

Fig. 5
Fig. 5

Contours of constants u and v in the focal region of a focused wave diffracted by a circular aperture for various values of semiangular aperture α: (a) α = π/6, (b) α = π/3, (c) α = π/2. The shadow edge is also shown.

Fig. 6
Fig. 6

Geometry of diffraction for a wave convergent on point F by a circular aperture for finite values of Fresnel number.

Fig. 7
Fig. 7

Geometry of diffraction for a wave convergent on an off-axis point F by a circular aperture.

Fig. 8
Fig. 8

Contours of constants u and v in the focal region of a focused wave diffracted by a circular aperture. The wave is focused on the points (a) r1 = a, d1 = 50a, (b) r1 = a, d1 = 50a.

Equations (69)

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U ( x , y , z ) = 1 2 π A U ( x , u , 0 ) exp ( i k r ) r ( z r ) ( i k + 1 r ) d x d y ,
r = [ z 2 + ( x - x ) 2 + ( y - y ) 2 ] 1 / 2 .
U ( ρ , z ) = 1 2 π 0 1 0 2 π a 2 exp ( i k r ) r ( z r ) ( i k + 1 r ) d ϕ ρ d ρ ,
r = ( z 2 + a 2 ρ 2 + a 2 ρ 2 - 2 a 2 ρ ρ cos ϕ ) 1 / 2
r = z ( 1 + a 2 ρ 2 2 z 2 + a 2 ρ 2 2 z 2 - a 2 ρ ρ z 2 cos ϕ ) .
R 2 = z 2 + a 2 ρ 2 ,
r = R ( 1 + a 2 ρ 2 2 R 2 - a 2 ρ ρ R 2 cos ϕ ) .
k r = 1 2 u ρ 2 - v ρ cos ϕ + w ,
u = k a 2 R , v = k a 2 ρ R , w = k R .
U ( ρ , R ) = a 2 ( z R ) ( i k + 1 R ) exp ( i w ) R 0 1 exp ( 1 2 i u ρ 2 ) × J 0 ( v ρ ) ρ d ρ .
u 3 ( k a ) 2 ,
R 3 k a 4 .
R 2 a 2 ,
k z [ ( 1 + a 2 ρ 2 z 2 ) 1 / 2 - 1 ] 1 2 u ρ 2 .
u = 2 k z [ ( 1 + a 2 z 2 ) 1 / 2 - 1 ]
u 2 + 4 k z u = 4 k 2 a 2 .
u = 2 k a 2 z / [ ( 1 + a 2 z 2 ) 1 / 2 + 1 ] ,
a 2 z 2
z = k a 2 u ( 1 - u 2 4 k 2 a 2 ) ,
u 2 4 k 2 a 2 .
I = k 2 a 4 z 2 sin 2 ( u / 4 ) u 2
I ( z ) = 1 4 [ 1 + ( 1 + a 2 z 2 ) 1 / 2 ] 2 × sin 2 { k a 2 2 z / [ ( 1 + a 2 z 2 ) 1 / 2 + 1 ] }
= 1 8 [ 1 + ( 1 + a 2 z 2 ) 1 / 2 ] 2 × ( 1 - cos { k a 2 z / [ ( 1 + a 2 z 2 ) 1 / 2 + 1 ] } ) .
I ( z ) = 1 4 ( 1 + [ 1 / ( 1 + a 2 z 2 ) ] - [ 2 / ( 1 + a 2 z 2 ) 1 / 2 ] × cos { k a 2 z / [ ( 1 + a 2 z 2 ) 1 / 2 + 1 ] } ) .
I ( z ) = sin 2 ( k a 2 2 z ) = 1 2 [ 1 - cos ( k a 2 z ) ] .
u = 2 k a .
U ( ρ , R ) = a 2 [ z / ( R 2 + a 2 2 ) ] { i k + [ 1 / ( R 2 + a 2 2 ) 1 / 2 ] } × exp ( i w ) 0 1 exp ( 1 2 i u ρ 1 / 2 ) J 0 ( v ρ ) ρ d ρ .
I ( z ) = 1 4 { [ 1 + ( 1 + a 2 z 2 ) 1 / 2 ] / ( 1 + a 2 2 z 2 ) } × sin 2 { k a 2 2 z / [ ( 1 + a 2 z 2 ) 1 / 2 + 1 ] } .
k ( z 2 + a 2 ρ 2 + a 2 ρ 2 - 2 a 2 ρ ρ cos ϕ ) 1 / 2 ( 1 / 2 ) u ρ 2 - v ρ cos ϕ + w .
ρ = ρ ,             ϕ = 0
ρ = v / u ,             ϕ = 0
k [ z 2 + a 2 ( 1 - ρ ) 2 ] 1 / 2 = ( 1 / 2 ) u - v + w , k [ z 2 + a 2 ( 1 + ρ ) 2 ] 1 / 2 = ( 1 / 2 ) u + v + w , k z = - ( v 2 / 2 u ) + w ,
p = { [ z 2 + a 2 ( 1 + ρ ) 2 ] 1 / 2 - z } 1 / 2 ,
q = { + { [ z 2 + a 2 ( 1 - ρ ) 2 ] 1 / 2 - z } 1 / 2 ρ < 1 - { [ z 2 + a 2 ( ρ - 1 ) 2 ] 1 / 2 - z } 1 / 2 ρ > 1 ,
k p 2 = 1 2 u + v + v 2 2 u = ( u + v ) 2 2 u , k q 2 = 1 2 u - v + v 2 2 u = ( u - v ) 2 2 u ,
u = ( k / 2 ) ( p + q ) 2 , v = ( k / 2 ) ( p 2 - q 2 ) , w = ( k / 4 ) ( p - q ) 2 + k z ,
v u = p - q p + q .
ρ 2 - ( u 2 + v 2 ) u v ρ + 1 + 1 4 k 2 a 2 ( u 2 - v 2 ) u 2 = 0 ,
( k z ) 2 - ( u 2 + v 2 ) u v 2 ( k 2 a 2 - v 2 ) ( k z ) - [ ( u 2 + v 2 ) 2 4 u v 2 - k 2 a 2 v 2 ] × ( k 2 a 2 - v 2 ) = 0 ,
ρ = u 2 v [ 1 - ( 1 - v 2 k 2 a 2 ) 1 / 2 ] + v 2 u [ 1 + ( 1 - v 2 k 2 a 2 ) 1 / 2 ] ,
z = k a 2 2 v { u v [ ( 1 - v 2 k 2 a 2 ) - ( 1 - v 2 k 2 a 2 ) 1 / 2 ] + v u [ ( 1 - v 2 k 2 a 2 ) + ( 1 - v 2 k 2 a 2 ) 1 / 2 ] } .
u = k a 2 z [ 1 - a 2 4 z 2 ( 1 + 3 ρ 2 ) ] , v = k a 2 ρ z [ 1 - a 2 2 z 2 ( 1 + ρ 2 ) ] ,
ρ = v u ( 1 + u 2 - v 2 4 k 2 a 2 ) , z = k a 2 u ( 1 - u 2 + 3 v 2 4 k 2 a 2 ) .
U ( r , z ) = 0 α 0 2 π exp [ i k ( z cos θ - r sin θ cos ϕ ) ] × d ϕ sin ( θ ) d θ ,
ρ = sin θ / sin α ,
U ( r , z ) = 0 1 0 2 π exp { i k [ z ( 1 - ρ 2 sin 2 α ) 1 / 2 - r ρ sin α cos ϕ ] } d ϕ ρ d ρ .
k z ( 1 - ρ 2 sin 2 α ) 1 / 2 - k r ρ sin α cos ϕ ( 1 / 2 ) u ρ 2 - v ρ cos ϕ + w
k z cos α - k r sin α = u / 2 - v + w , k z cos α + k r sin α = u / 2 + v + w ,
v = k r sin α ,
k z cos α = u / 2 + w .
ρ = ± ( r sin α ) / ( z 2 + r 2 ) 1 / 2
ρ = v / u
k ( z 2 + r ) 1 / 2 = - v 2 2 u + w
= - k 2 r 2 sin 2 α 2 u + w .
u = - k [ z + ( z 2 + r 2 ) 1 / 2 ] ( 1 - cos α ) ,
w = k 2 { [ z + ( z 2 + r 2 ) 1 / 2 ] + cos α [ z - ( z 2 + r 2 ) } .
k z = - u 4 sin 2 α / 2 ( 1 - v 2 u 2 tan 2 α 2 ) .
r = ± z tan α
v = u .
k [ ( z + d ) 2 + r 2 + a 2 ρ 2 - 2 a r ρ cos ϕ ] 1 / 2 - k ( a 2 ρ 2 + d 2 ) 1 / 2 ( 1 / 2 ) u ρ 2 - v ρ cos ϕ + w ,
u = - ( k / 2 ) ( p + q ) 2 , v = ( k / 2 ) ( q 2 - p 2 ) , w = ( k / 4 ) [ 4 n - ( p - q ) 2 ] ,
p = { { ( a 2 + + d 2 ) 1 / 2 - [ ( a + r ) 2 + ( z + d ) 2 ] 1 / 2 + n } 1 / 2 r 2 > a z / d - { ( a 2 + d 2 ) 1 / 2 - [ ( a + r ) 2 + ( z + d ) 2 ] 1 / 2 + n } 1 / 2 r 2 < a z / d , q = { { ( a 2 + d 2 ) 1 / 2 - [ ( a - r ) 2 + ( z + d ) 2 ] 1 / 2 + n } 1 / 2 r 2 < - a z / d - { ( a 2 + d 2 ) 1 / 2 - [ ( a - r ) 2 + ( z + d ) 2 ] 1 / 2 + n } 1 / 2 r 2 > - a z / d , n = { ( r 2 + z 2 ) 1 / 2 z > 0 - ( r 2 + z 2 ) 1 / 2 z < 0 .
a d , r d ,
u = - k a 2 z d ( z + d ) , v = k a r z + d ,
k ( r 2 2 + r 2 - 2 r 2 r cos ϕ + d 2 2 ) 1 / 2 - k ( r 1 2 + r 2 - 2 r 1 r cos ϕ + d 1 2 ) 1 / 2 ( 1 / 2 ) u ρ 2 - v ρ cos ϕ + w ,
p = { ( l 1 - l 2 + n ) 1 / 2 r 2 d 1 > a ( d 2 - d 1 ) + r 1 d 2 - ( l 1 - l 2 + n ) 1 / 2 r 2 d 1 < a ( d 2 - d 1 ) + r 1 d 2 , q = { ( m 1 - m 2 + n ) 1 / 2 r 2 d 1 < a ( d 1 - d 2 ) + r 1 d 2 - ( m 1 - m 2 + n ) 1 / 2 r 2 d 1 > a ( d 1 - d 2 ) + r 1 d 2 , l 1 , 2 = ( r 1 , 2 2 + a 2 + 2 a r 1 , 2 + d 1 , 2 2 ) 1 / 2 , n = { [ ( d 2 - d 1 ) 2 + ( r 2 - r 1 ) 2 ] 1 / 2 d 2 > d 1 - [ ( d 2 - d 1 ) 2 + ( r 2 - r 1 ) 2 ] 1 / 2 d 1 > d 2 .
v = u , q = 0 , v = - u , p = 0.
( 1 / 2 ) u ρ 2 + A ρ 4 ( 1 / 2 ) u eff ρ 2 .
u eff = u + 2 A ,

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