Abstract

An algorithm for fast numerical integration of near-field scalar diffraction formulas is presented, based on the local approximation of the integrand of the diffraction equation by a variant of the Fresnel kernel. The two-dimensional local propagation integral is solved analytically for an integration domain enclosed between two mutually perpendicular line segments and a parabolic arc. We show that, by combining rectangular and arched elements, one can achieve accurate computation of the field diffracted at complicated aperture shapes without having to resort to time-consuming numerical quadrature techniques. The numerical accuracy and the computational speed of the algorithm are assessed and compared with the performance of the linear-phase approximation method developed by Hopkins and Yzuel [ Opt. Acta 17, 157 ( 1970)].

© 1994 Optical Society of America

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References

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  1. H. H. Hopkins, M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
    [Crossref]
  2. T. Gravelsaeter, J. J. Stamnes, “Diffraction by circular apertures. 1: Method of linear phase and amplitude approximations,” Appl. Opt. 21, 3644–3651 (1982).
    [Crossref] [PubMed]
  3. J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
    [Crossref]
  4. H. G. Kraus, “Finite element area and line integral transforms for generalization of aperture function and geometry in Kirchhoff scalar diffraction theory,” Opt. Eng. 32, 368–383 (1993).
    [Crossref]
  5. J. J. Stamnes, “Hybrid integration technique for efficient and accurate computation of diffraction integrals,” J. Opt. Soc. Am. A 6, 1330–1342 (1989).
    [Crossref]
  6. H. Osterberg, L. W. Smith, “Closed solutions of Rayleigh’s diffraction integral for axial points,” J. Opt. Soc. Am. 51, 1050–1054 (1961).
    [Crossref]
  7. J. E. Harvey, R. V. Shack, “Aberrations of diffracted wave fields,” Appl. Opt. 17, 3003–3009 (1978).
    [Crossref] [PubMed]
  8. C. J. R. Sheppard, P. P. Roberts, M. Gu, “Fresnel approximation for off-axis illumination of a circular aperture,” J. Opt. Soc. Am. A 10, 984–986 (1993).
    [Crossref]
  9. 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]

1993 (2)

H. G. Kraus, “Finite element area and line integral transforms for generalization of aperture function and geometry in Kirchhoff scalar diffraction theory,” Opt. Eng. 32, 368–383 (1993).
[Crossref]

C. J. R. Sheppard, P. P. Roberts, M. Gu, “Fresnel approximation for off-axis illumination of a circular aperture,” J. Opt. Soc. Am. A 10, 984–986 (1993).
[Crossref]

1989 (2)

1983 (1)

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[Crossref]

1982 (1)

1978 (1)

1970 (1)

H. H. Hopkins, M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[Crossref]

1961 (1)

Gibson, S. F.

Gravelsaeter, T.

Gu, M.

Harvey, J. E.

Hopkins, H. H.

H. H. Hopkins, M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[Crossref]

Kraus, H. G.

H. G. Kraus, “Finite element area and line integral transforms for generalization of aperture function and geometry in Kirchhoff scalar diffraction theory,” Opt. Eng. 32, 368–383 (1993).
[Crossref]

Lanni, F.

Osterberg, H.

Pedersen, H. M.

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[Crossref]

Roberts, P. P.

Shack, R. V.

Sheppard, C. J. R.

Smith, L. W.

Spjelkavik, B.

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[Crossref]

Stamnes, J. J.

Yzuel, M. J.

H. H. Hopkins, M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[Crossref]

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Acta (2)

H. H. Hopkins, M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[Crossref]

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[Crossref]

Opt. Eng. (1)

H. G. Kraus, “Finite element area and line integral transforms for generalization of aperture function and geometry in Kirchhoff scalar diffraction theory,” Opt. Eng. 32, 368–383 (1993).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1

Geometry of the arched surface elements. The actual boundary η(bound)(ξ) (thick curve) is approximated by a parabolic arc segment η(parab)(ξ) = C + + 2 (thin curve).

Fig. 2
Fig. 2

Normalized theoretical intensity along the axis of symmetry of the circular aperture, around the focal region z = zF.

Fig. 3
Fig. 3

Relative errors for intensity I(z) (upper row) and phase Φ(z) (lower row) in the cases of segmentation performed on the full aperture (left-hand column) and on the aperture divided into 18 adjacent sectors (right-hand column) at N = 60. We have obtained each curve by computing the error for 120 values of the axial coordinate z. The errors around the focal point z = zF are too small to be visible at the scale used for the plots.

Fig. 4
Fig. 4

Average and peak values for the relative errors in intensity I and phase Φ as a function of the number of grid divisions N.

Fig. 5
Fig. 5

CPU times as obtained by running the program on a Sun SPARC Station 10 Model 40 workstation.

Equations (63)

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U ( x , y , z ) = S g ( x , y ; x , y , z ) × exp [ i f ( x , y ; x , y , z ) ] d x d y ,
U m ( x , y , z ) = S m g ( x m + ξ , y m + η ; x , y , z ) × exp [ i f ( x m + ξ , y m + η ; x , y , z ) ] d ξ d η .
U m P ( x , y , z ) = A h = 0 N x l = 0 N y B h , l × S m ξ h η l exp [ i ( β x ξ + γ x ξ 2 + β y η + γ y η 2 ) ] d ξ d η .
U m P ( x , y , z ) = A p = 0 N B p H ˜ p ( β , γ ) ( L ) .
H p ( β , γ ) ( u ) = u p exp [ i ( β u + γ u 2 ) ] d u
H ˜ p ( β , γ ) ( L ) = H p ( β , γ ) ( L / 2 ) - H p ( β , γ ) ( - L / 2 ) = - L / 2 L / 2 u p exp [ i ( β u + γ u 2 ) ] d u
H p ( β , γ ) ( v - β 2 γ ) = exp ( - i β 2 4 γ ) × j = 0 p ( p j ) ( - β 2 γ ) p - j R j ( γ ) ( v ) ,
R j ( γ ) ( v ) = v j exp ( i γ v 2 ) d v
R 0 ( γ ) ( v ) = ( π 2 γ ) 1 / 2 F [ ( 2 γ π ) 1 / 2 v ] ,
R 1 ( γ ) ( v ) = - i 2 γ exp ( i γ v 2 ) ,
R j ( γ ) ( v ) = - i 2 γ [ v j - 1 exp ( i γ v 2 ) - ( j - 1 ) R 1 - 2 ( γ ) ( v ) ]             ( j 2 ) .
F ( v ) = 0 v exp ( i π 2 χ 2 ) d χ .
H p ( β , γ ) ( u ) = n = 0 + ( i γ ) n n ! Q 2 n + p ( β ) ( u ) .
Q l ( β ) ( u ) = u l exp ( i β u ) d u
Q 0 ( β ) ( u ) = - i β exp ( i β u ) ,
Q l ( β ) ( u ) = - i β [ u l exp ( i β u ) - l Q l = 1 ( β ) ( u ) ]             ( l 1 ) ,
Q l ( β ) ( u ) = - i β exp ( i β u ) j = 0 l l ! j ! ( i β ) l - j u j .
Q ˜ l ( β ) ( L ) = - 2 i β ( L 2 ) l { cos ( β L / 2 ) i sin ( β L / 2 ) } + i l β Q ˜ l - 1 ( β ) ( L )             ( l odd ) ( l even ) .
H p ( β , γ ) ( u ) = n = 0 + i n n ! j = 0 n ( n j ) β n - j γ j p + n + j + 1 u p + n + j + 1 .
H ˜ p ( β , γ ) ( L ) = 2 ( L 2 ) p + 1 n = 0 + i n n ! j ( n j ) β n - j γ j p + n + j + 1 × ( L 2 ) n + j ,
U m P ( x , y , z ) = A h = 0 N x l = 0 N y B h , l H ˜ h ( β x , γ x ) ( L x , m ) H ˜ l ( β y , γ y ) ( L y , m ) .
U m P ( x , y , z ) = A h = 0 N x l = 0 N y B h , l - L x , m / 2 L x , m / 2 ξ h - L y , m / 2 η ( bound ) ( ξ ) η l × exp [ ( i β x ξ + γ x ξ 2 + β y η + γ y η 2 ) ] d ξ d η .
U m P ( x , y ; z ) = A h = 0 N x l = 0 N y B h , l × [ T ˜ h , l ( L x , m ) - H ˜ h ( β x , γ x ) ( L x , m ) H l ( β y , γ y ) ( - L y , m 2 ) ] ,
T ˜ h , l ( L x , m ) = - L x , m / 2 L x , m / 2 ξ h H l ( β y , γ y ) [ η ( bound ) ( ξ ) ] × exp [ i ( β x ξ + γ x ξ 2 ) ] d ξ .
η ( parab ) ( ξ ) = C + D ξ + E ξ 2 .
D - L y , m / L x , m ,             E - 4 C / L x , m 2 .
T ˜ h , l ( L x , m ) = exp ( - i β y 2 4 γ y ) j = 0 l ( l j ) ( - β y 2 γ y ) l - j G ˜ h , j ( L x , m ) ,
G ˜ h , j ( L x , m ) = - L x , m / 2 L x , m / 2 ξ h R j ( γ y ) ( β y 2 γ y + C + D ξ + E ξ 2 ) × exp [ i ( β x ξ + γ x ξ 2 ) ] d ξ .
G ˜ h , 0 ( L x , m ) = ( π 2 γ y ) 1 / 2 - L x , m / 2 L x , m / 2 ξ h × F [ ( 2 γ y π ) 1 / 2 ( β y 2 γ y + C + D ξ + E ξ 2 ) ] × exp [ i ( β x ξ + γ x ξ 2 ) ] d ξ ,
G ˜ h , j ( L x , m ) = - i 2 γ y [ V ˜ h , j - 1 ( L x , m ) - ( j - 1 ) G ˜ h , j - 2 ( L x , m ) ]             ( j 1 ) ,
V ˜ h , n ( L x , m ) = - L x , m / 2 L x , m / 2 ξ h ( β y 2 γ y + C + D ξ + E ξ 2 ) n × exp [ i γ y ( β y 2 γ y + C + D ξ + E ξ 2 ) 2 ] × exp [ i ( β x ξ + γ x ξ 2 ) ] d ξ .
V ˜ h , n ( L x , m ) = exp ( i α ) t = 0 + ( i γ y ) t t ! w = 0 t ( t w ) ( 2 D ) t - w E t + w × P ˜ h + 3 t + w , n ( β , γ ) ( L x , m ) ,
α = γ y ( β y 2 γ y + C ) 2 , β = β x + 2 γ y D ( β y 2 γ y + C ) , γ = γ x + γ y [ 2 E ( β y 2 γ y + C ) + D 2 ] ,
P ˜ k , n ( β , γ ) ( L x , m ) = - L x , m / 2 L x , m / 2 ξ k ( β y 2 γ y + C + D ξ + E ξ 2 ) 2 × exp [ i ( β ξ + γ ξ 2 ) ] d ξ .
P ˜ k , n ( β , γ ) ( L x , m ) = q = 0 2 n r = r min ( q ) r max ( q ) ( n r ) ( r q - r ) ( β y 2 γ y + C ) n - r × D 2 r - q E q - r H ˜ k + q ( β , γ ) ( L x , m ) ,
F [ ( 2 γ y π ) 1 / 2 ( β y 2 γ y + C + D ξ + E ξ 2 ) ] = F [ ( 2 γ y π ) 1 / 2 ( β y 2 γ y + C ) ] + ( 2 γ y π ) 1 / 2 exp ( i α ) × 0 D ξ + E ξ 2 exp ( i γ y χ 2 ) exp ( i β χ ) d χ ,
F [ ( 2 γ y π ) 1 / 2 ( β y 2 γ y + C + D ξ + E ξ 2 ) ] = F [ ( 2 γ y π ) 1 / 2 ( β y 2 γ y + C ) ] + ( 2 γ y π ) 1 / 2 exp ( i α ) × t = 0 + ( i γ y ) t t ! [ Q 2 t ( β ) ( D ξ + E ξ 2 ) - Q 2 t ( β ) ( 0 ) ] .
G ˜ h , 0 ( 1 ) ( L x , m ) = ( π 2 γ y ) 1 / 2 F [ ( 2 γ y π ) 1 / 2 ( β y 2 γ y + C ) ] × H ˜ h ( β x , γ x ) ( L x , m ) ,
G ˜ h , 0 ( 2 ) ( L x , m ) = exp ( i α ) t = 0 + ( i γ y ) t ( 2 t ) ! t ! ( i β ) 2 t + 1 × [ H ˜ h ( β x , γ x ) ( L x , m ) - q = 0 2 t 1 q ! ( i β ) - q × w = 0 q ( q w ) D q - w E w H ˜ h + q + w ( β , γ ) ( L x , m ) ] ,
γ = γ x + E β .
G ˜ h , 0 ( 2 ) ( L x , m ) = exp ( i α ) t = 0 + i t t ! q = 0 t ( t q ) ( β ) t - q γ y q t + q + 1 × w = 0 t + q + 1 ( t + q + 1 w ) D t + q - w + 1 E w H ˜ h + t + q + w + 1 ( β x , γ x ) ( L x , m ) .
γ y L y , m 2 1
γ y L y , m 2 + β L y , m 1
T ˜ h , l ( L x , m ) = - i β y exp ( i β y C ) n = 0 + ( i γ y ) n n ! j = 0 2 n + l ( 2 n + l ) ! j ! × ( i β y ) 2 n + l - j P ˜ h , j ( β , γ ) ( L x , m ) .
T ˜ h , l ( L x , m ) = n = 0 + i n n ! j = 0 n ( n j ) β y n - j γ y j p + n + j + 1 × P ˜ h , l + n + j + 1 ( β x , γ x ) ( L x , m ) ,
η ( bound ) ( ξ ) = [ R 2 - ( x m + ξ ) 2 ] 1 / 2 - y m .
η ( bound ) ( ξ ) = ( R 2 - x m 2 ) 1 / 2 ( 1 - ξ 2 + 2 x m ξ R 2 - x m 2 ) 1 / 2 - y m
η ( bound ) ( ξ ) = η ( parab ) ( ξ ) + ( bound ) ( ξ ) ,
η ( parab ) ( ξ ) = [ ( R 2 - x m 2 ) 1 / 2 - y m ] - x m ( R 2 - x m 2 ) 1 / 2 ξ - R 2 2 ( R 2 - x m 2 ) 3 / 2 ξ 2 ,
( bound ) ( ξ ) ξ 3 8 [ ξ + 4 x m ( R 2 - x m 2 ) 3 / 2 + ξ + 2 x m 3 2 ( R 2 - x m 2 ) 5 / 2 ] .
g ( x , y ; x , y , z ; x F , y F , z F ) = - i z λ 1 d ( x , y ; x F , y F , z F ) s 2 ( x , y ; x , y , z ) × [ 1 + i k s ( x , y ; x , y , z ) ] ,
f ( x , y ; x , y , z ; x F , y F , z F ) = k [ s ( x , y ; x , y , z ) - d ( x , y ; x F , y F , z F ) ] ,
s ( x , y ; x , y , z ) = [ z 2 + ( x - x ) 2 + ( y - y ) 2 ] 1 / 2 ,
d ( x , y ; x F , y F , z F ) = [ z F 2 + ( x - x F ) 2 + ( y - y F ) 2 ] 1 / 2 .
s a ( x m + ξ , y m + η ; x , y , z ) = ( s m ) a ( 1 + ξ 2 + 2 X m ξ + η 2 + 2 Y m η s m 2 ) a / 2 ,
s ( x m + ξ , y m + η ; x , y , z ) s m + X m s m ξ + Y m s m η + 1 2 s m ( 1 - X m 2 s m 2 ) ξ 2 + 1 2 s m ( 1 - Y m 2 s m 2 ) η 2 - X m Y m s m 3 ξ η .
d ( x m + ξ , y m + η ; x , y , z ) d m + X ˜ m d m ξ + Y ˜ m d m η + 1 2 d m ( 1 - X ˜ m 2 d m 2 ) ξ 2 + 1 2 d m ( 1 - Y ˜ m 2 d m 2 ) η 2 - X ˜ m Y ˜ m d m ξ η .
β x = k ( X m s m - X ˜ m d m ) , γ x = k 2 s m ( 1 - X m 2 s m 2 ) - k 2 d m ( 1 - X ˜ m 2 d m 2 ) , β y = k ( Y m s m - Y ˜ m d m ) , γ y = k 2 s m ( 1 - Y m 2 s m 2 ) - k 2 d m ( 1 - Y ˜ m 2 d m 2 ) .
W x y ( ξ , η ) = exp [ - i k ( X m Y m s m 3 - X ˜ m Y ˜ m d m 3 ) ξ η ] = t = 0 + 1 t ! [ - i k ( X m Y m s m 3 - X ˜ m Y ˜ m d m 3 ) ξ η ] t ,
k | X m Y m s m 3 - X ˜ m Y ˜ m d m 3 | L x , m L y , m 4 1.
A = - i z λ s m 2 d m exp [ i k ( s m - d m ) ] ,
B 0 , 0 = ( 1 + i k s m ) , B 1 , 0 = - ( 2 X m s m 2 + X ˜ m d m 2 ) - i k s m ( 3 X m s m 2 + X ˜ m d m 2 ) , B 0 , 1 = - ( 2 Y m s m 2 + Y ˜ m d m 2 ) - i k s m ( 3 Y m s m 2 + Y ˜ m d m 2 ) , B 1 , 1 = ( 8 X m Y m s m 4 + 3 X ˜ m Y ˜ m d m 4 + 2 X m Y ˜ m + Y m X ˜ m d m 2 s m 2 ) + i k s m ( 15 X m Y m s m 4 + 3 X ˜ m Y ˜ m d m 4 + 3 X m Y ˜ m + Y m X ˜ m d m 2 s m 2 ) , B 2 , 0 = - ( 1 s m 2 + 1 2 d m 2 - 4 X m 2 s m 4 - 3 X ˜ m 2 2 d m 4 - 2 X m X ˜ m s m 2 d m 2 ) - i k s m ( 3 2 s m 2 + 1 2 d m 2 - 15 X m 2 2 s m 4 - 3 X ˜ m 2 2 d m 4 - 3 X m X ˜ m s m 2 d m 2 ) , B 0 , 2 = - ( 1 s m 2 + 1 2 d m 2 - 4 Y m 2 s m 4 - 3 Y ˜ m 2 2 d m 4 - 2 Y m Y ˜ m s m 2 d m 2 ) - i k s m ( 3 2 s m 2 + 1 2 d m 2 - 15 Y m 2 2 s m 4 - 3 Y ˜ m 2 2 d m 4 - 3 Y m Y ˜ m s m 2 d m 2 ) .
B h + j , l + j = 1 j ! [ - i k ( X m Y m s m 3 - X ˜ m Y ˜ m d m 3 ) ] j B h , l .

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