Abstract

Two methods are presented for efficient computation of the wave field that results when a spherical wave is diffracted by circular apertures. One method is based on the Kirchhoff diffraction integral, the other on the boundary-diffraction-wave (BDW) integral. In each method the integration domain is divided into subdomains, and the amplitude and phase within each subdomain are linearized to make an analytical integration possible. Explicit and simple formulas are derived that specify the number of subdomains needed to obtain a desired accuracy for a given geometry and wavelength. Also we determine the number of subdomains needed in the BDW integral to obtain a sufficient accuracy in the vicinity of the shadow boundary. The speed of computation of each method is compared with that using direct numerical integration. As an illustration, the BDW method is used to compute the image field of a holographic lens.

© 1982 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), Sec. 8.8.
  2. B. Richards, E. Wolf, Phys. Rev. B 138, 1561 (1965); A. Boivin, E. Wolf, J. Opt. Soc. Am. 57, 1171 (1967); A. Boivin, N. Brousseau, S. C. Biswas, Opt. Acta 25, 415 (1978).
    [Crossref]
  3. J. J. Stamnes, J. Opt. Soc. Am. 71, 15 (1981).
    [Crossref]
  4. J. J. Stamnes, New Methods and Results in Focusing,” in Optics in Four Dimensions—1980, M. A. Machado, L. M. Narducci, Eds., AIP Conf. Proc. 65 (American Institute of Physics, New York, 1981).
  5. J. J. Stamnes, T. Gravelsaeter, “Methods for Efficient Computation of the Image Field of Holographic Lenses for Sound Waves,” in Acoustical Imaging, Vol. 10, P. Alais, A. F. Metherell, Eds. (Plenum, New York, 1982).
  6. J. J. Stamnes, T. Gravelsaeter, “Diffraction by Circular Apertures, 2. Asymptotic Methods (in preparation).
  7. E. W. Marchand, E. Wolf, J. Opt. Soc. Am. 56, 1712 (1966).
    [Crossref]
  8. H. H. Hopkins, Proc. Phys. Soc., London Sect. B 70, 1002 (1957).
    [Crossref]
  9. H. H. Hopkins, M. J. Yzuel, Opt. Acta 17, 157 (1970).
    [Crossref]
  10. Ref. 1, Sec. 8.3.2.
  11. Ref. 1, Sec. 8.9.
  12. J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of Diffraction Integrals using Local Phase and Amplitude Approximations” (in preparation).
  13. J. J. Stamnes, T. Gravelsaeter, O. Bentsen, “Image Quality and Diffraction Efficiency of a Holographic Lens for Sound Waves,” in Acoustical Imaging, Vol. 10, P. Alais, A. F. Metherell, Eds. (Plenum, New York, 1982).

1981 (1)

1970 (1)

H. H. Hopkins, M. J. Yzuel, Opt. Acta 17, 157 (1970).
[Crossref]

1966 (1)

1965 (1)

B. Richards, E. Wolf, Phys. Rev. B 138, 1561 (1965); A. Boivin, E. Wolf, J. Opt. Soc. Am. 57, 1171 (1967); A. Boivin, N. Brousseau, S. C. Biswas, Opt. Acta 25, 415 (1978).
[Crossref]

1957 (1)

H. H. Hopkins, Proc. Phys. Soc., London Sect. B 70, 1002 (1957).
[Crossref]

Bentsen, O.

J. J. Stamnes, T. Gravelsaeter, O. Bentsen, “Image Quality and Diffraction Efficiency of a Holographic Lens for Sound Waves,” in Acoustical Imaging, Vol. 10, P. Alais, A. F. Metherell, Eds. (Plenum, New York, 1982).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), Sec. 8.8.

Gravelsaeter, T.

J. J. Stamnes, T. Gravelsaeter, “Methods for Efficient Computation of the Image Field of Holographic Lenses for Sound Waves,” in Acoustical Imaging, Vol. 10, P. Alais, A. F. Metherell, Eds. (Plenum, New York, 1982).

J. J. Stamnes, T. Gravelsaeter, “Diffraction by Circular Apertures, 2. Asymptotic Methods (in preparation).

J. J. Stamnes, T. Gravelsaeter, O. Bentsen, “Image Quality and Diffraction Efficiency of a Holographic Lens for Sound Waves,” in Acoustical Imaging, Vol. 10, P. Alais, A. F. Metherell, Eds. (Plenum, New York, 1982).

Hopkins, H. H.

H. H. Hopkins, M. J. Yzuel, Opt. Acta 17, 157 (1970).
[Crossref]

H. H. Hopkins, Proc. Phys. Soc., London Sect. B 70, 1002 (1957).
[Crossref]

Marchand, E. W.

Pedersen, H. M.

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of Diffraction Integrals using Local Phase and Amplitude Approximations” (in preparation).

Richards, B.

B. Richards, E. Wolf, Phys. Rev. B 138, 1561 (1965); A. Boivin, E. Wolf, J. Opt. Soc. Am. 57, 1171 (1967); A. Boivin, N. Brousseau, S. C. Biswas, Opt. Acta 25, 415 (1978).
[Crossref]

Spjelkavik, B.

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of Diffraction Integrals using Local Phase and Amplitude Approximations” (in preparation).

Stamnes, J. J.

J. J. Stamnes, J. Opt. Soc. Am. 71, 15 (1981).
[Crossref]

J. J. Stamnes, New Methods and Results in Focusing,” in Optics in Four Dimensions—1980, M. A. Machado, L. M. Narducci, Eds., AIP Conf. Proc. 65 (American Institute of Physics, New York, 1981).

J. J. Stamnes, T. Gravelsaeter, “Methods for Efficient Computation of the Image Field of Holographic Lenses for Sound Waves,” in Acoustical Imaging, Vol. 10, P. Alais, A. F. Metherell, Eds. (Plenum, New York, 1982).

J. J. Stamnes, T. Gravelsaeter, “Diffraction by Circular Apertures, 2. Asymptotic Methods (in preparation).

J. J. Stamnes, T. Gravelsaeter, O. Bentsen, “Image Quality and Diffraction Efficiency of a Holographic Lens for Sound Waves,” in Acoustical Imaging, Vol. 10, P. Alais, A. F. Metherell, Eds. (Plenum, New York, 1982).

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of Diffraction Integrals using Local Phase and Amplitude Approximations” (in preparation).

Wolf, E.

E. W. Marchand, E. Wolf, J. Opt. Soc. Am. 56, 1712 (1966).
[Crossref]

B. Richards, E. Wolf, Phys. Rev. B 138, 1561 (1965); A. Boivin, E. Wolf, J. Opt. Soc. Am. 57, 1171 (1967); A. Boivin, N. Brousseau, S. C. Biswas, Opt. Acta 25, 415 (1978).
[Crossref]

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), Sec. 8.8.

Yzuel, M. J.

H. H. Hopkins, M. J. Yzuel, Opt. Acta 17, 157 (1970).
[Crossref]

J. Opt. Soc. Am. (2)

Opt. Acta (1)

H. H. Hopkins, M. J. Yzuel, Opt. Acta 17, 157 (1970).
[Crossref]

Phys. Rev. B (1)

B. Richards, E. Wolf, Phys. Rev. B 138, 1561 (1965); A. Boivin, E. Wolf, J. Opt. Soc. Am. 57, 1171 (1967); A. Boivin, N. Brousseau, S. C. Biswas, Opt. Acta 25, 415 (1978).
[Crossref]

Proc. Phys. Soc., London Sect. B (1)

H. H. Hopkins, Proc. Phys. Soc., London Sect. B 70, 1002 (1957).
[Crossref]

Other (8)

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), Sec. 8.8.

J. J. Stamnes, New Methods and Results in Focusing,” in Optics in Four Dimensions—1980, M. A. Machado, L. M. Narducci, Eds., AIP Conf. Proc. 65 (American Institute of Physics, New York, 1981).

J. J. Stamnes, T. Gravelsaeter, “Methods for Efficient Computation of the Image Field of Holographic Lenses for Sound Waves,” in Acoustical Imaging, Vol. 10, P. Alais, A. F. Metherell, Eds. (Plenum, New York, 1982).

J. J. Stamnes, T. Gravelsaeter, “Diffraction by Circular Apertures, 2. Asymptotic Methods (in preparation).

Ref. 1, Sec. 8.3.2.

Ref. 1, Sec. 8.9.

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of Diffraction Integrals using Local Phase and Amplitude Approximations” (in preparation).

J. J. Stamnes, T. Gravelsaeter, O. Bentsen, “Image Quality and Diffraction Efficiency of a Holographic Lens for Sound Waves,” in Acoustical Imaging, Vol. 10, P. Alais, A. F. Metherell, Eds. (Plenum, New York, 1982).

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Figures (9)

Fig. 1
Fig. 1

Geometry of the diffraction problem.

Fig. 2
Fig. 2

Illustration of the difference between our phase linearization procedure and the one based on a Taylor series expansion (Hopkins procedure).

Fig. 3
Fig. 3

Number of subintervals M that are needed in the radial direction in Eq. (1) to obtain an accuracy better than 1%. The quantity NF on the horizontal axis is the number of Fresnel zones computed by Eq. (33). The geometry is a1 = a2 = 0, z1 = 2000, ρU = 105, ρL = 95. λ was varied between 0.05 and 1, and z2 was in each case adjusted so as to give a maximum value of the integral.

Fig. 4
Fig. 4

Percentage error in the computed amplitude of the BDW integral as a function of the maximum allowable phase error (ΔF)max in the angular direction. The variable L along the horizontal axis is given by L = 2π/(ΔF)max.

Fig. 5
Fig. 5

Geometry concerning use of the BDW integral in the vicinity of the shadow boundary. As the angular distance β tends to zero, the integrand of the BDW integral becomes singular.

Fig. 6
Fig. 6

Number of samples N that are needed to obtain 1% accuracy in the amplitude of the BDW integral in the vicinity of the shadow boundary. β is the angular distance between the observation point and the shadow boundary as seen from the aperture edge (see Fig. 5). The geometry is a = 100λ, z1 = z2 = 2000λ, and a1 varies between zero (on-axis) and 800λ (22° off-axis).

Fig. 7
Fig. 7

Integrand of the BDW integral is singular at the shadow boundary: (a), (b) Typical field and amplitude distributions across the shadow boundary computed by the Kirchhoff integral; (c), (d) corresponding distributions computed by the BDW integral, using the same number of samples N in the angular direction as in the Kirchhoff integral; (e) difference between (a) and (c); (f) difference between (b) and (d). The geometry is a1 = 0, ρL = 97.31 mm, ρU = 100.73 mm, z1 = z2 = 2000 mm, and λ = 0.7 mm [see Eq. (1)].

Fig. 8
Fig. 8

Plots of the percentage error as a function of the number of subdomains N in the BDW integral obtained when using our method and the Simpson integration formula. By the percentage error we mean [1 − (R/M) ± 2σ] · 100, where R is a reference value, obtained by employing a large number of subdomains. M and σ are, respectively, the mean value and the standard deviation obtained as explained in the text. In those cases in which only one value is plotted, the standard deviation is so small as to be negligible.

Fig. 9
Fig. 9

Plots of intensity distributions in the vicinity of focus of a holographic lens for IR (solid curves): (a) and (b), amplitude along the optical axis and in the focal plane, respectively. The dashed curve in (a) shows the corresponding results for an ordinary aberration free lens of the same f/No. In (b) the difference between the results of the holographic and the ordinary lens is too small to be noticed in the figure.

Equations (45)

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u ( P 2 ) = - i C 2 λ 0 2 π ρ L ρ U G ( ρ , ϕ ) exp [ i F ( ρ , ϕ ) ] d ρ d ϕ ,
G ( ρ , ϕ ) = ρ r 1 r 2 [ z 1 r 1 ( 1 + i k r 1 ) + z 2 r 2 ( 1 + i k r 2 ) ] ,
F ( ρ , ϕ ) = k ( r 1 + r 2 ) ,
r l = [ z l 2 + a l 2 + ρ 2 - 2 ρ a l cos ( ϕ - θ 1 ) ] 1 / 2 ;             l = 1 , 2.
C = A exp ( i ψ ) .
u ( P 2 ) = u ( g ) ( P 2 ) + u ( d ) ( P 2 ) ,
u ( d ) ( P 2 ) = C [ I ( ρ U ) - I ( ρ L ) ] ,
I ( ρ ) = 0 2 π g ( ϕ ) exp [ i f ( ϕ ) ] d ϕ ,
f ( ϕ ) = k ( r 1 + r 2 ) ,
g ( ϕ ) = ρ 4 π r 1 r 2 · g 1 ( ϕ ) g 2 ( ϕ ) ,
g 1 ( ϕ ) = - z 2 [ ρ - a 1 cos ( ϕ - θ 1 ) ] - z 1 [ ρ - a 2 cos ( ϕ - θ 2 ) ] ,
g 2 ( ϕ ) = r 1 r 2 + [ ρ - a 1 cos ( ϕ - θ 1 ) ] [ ρ - a 2 cos ( ϕ - θ 2 ) ] + a 1 a 2 sin ( ϕ - θ 1 ) sin ( ϕ - θ 2 ) - z 1 z 2 .
I = y L y U x L x U g ( x , y ) exp [ i f ( x , y ) ] d x d y .
I = n = 1 N m = 1 M I m , n
I m , n = y n L y n U x m L x m U g ( x , y ) exp [ i f ( x , y ) ] d x d y .
Δ x m = ½ ( x m U - x m L ) ;             Δ y n = ½ ( y n U - y n L ) ,
x m A = ½ ( x m U + x m L ) ;             y n A = ½ ( y n U + y n L ) .
f ( x , y ) f m ( x , y ) = a m ( y ) + b m ( y ) x - x m A Δ x m ;             x m L < x < x m U ,
a m = f ( x m A , y ) + [ f ( x m U , y ) + f ( x m L , y ) ] ,
b m = ½ [ f ( x m U , y ) - f ( x m L , y ) ] .
I m , n y n L y n U ( I m + - I m - ) d y ,
I m ± = g ( x m A , y ) Δ x m i b m ( y ) exp { i [ a m ( y ) ± b m ( y ) ] } .
I m + - I m - = 2 Δ x m g ( x m A , y ) sinc [ b m ( y ) ] exp [ i a m ( y ) ] ,
I m , n = y n L y n U G m ( y ) exp [ i F m ( y ) ] d y .
G m ( y ) G m , n ( y ) = c m , n + d m , n y - y n A Δ y n ,
F m ( y ) F m , n ( y ) = e m , n + f m , n y - y n A Δ y n ,
I m , n = A m , n + i B m , n ,
A m , n = 2 Δ y n [ c m , n w 1 ( f m , n ) cos ( e m , n ) + d m , n w 2 ( f m , n ) sin ( e m , n ) ] ,
B m , n = 2 Δ y n [ c m , n w 1 ( f m , n ) sin ( e m , n ) - d m , n w 2 ( f m , n ) cos ( e m , n ) ] ,
w 2 ( x ) = x cos ( x ) - sin ( x ) x 2 - 1 3 x + 1 30 x 3 - 1 840 x 5 .
I = y L Y U g ( y ) exp [ i f ( y ) ] d y .
I = n = 1 N y n L y n U g ( y ) exp [ i f ( y ) ] d y .
N F ρ A ( ρ U - ρ L ) ( z 1 + z 2 ) / z 1 z 2 λ .
Δ = k l l = 1 2 Δ r l = k l l = 1 2 2 [ r ¯ l ( ρ , ϕ ) - r l ( ρ , ϕ ) ] ,
Δ r l = [ r ( ρ , ϕ U ) + r l ( ρ , ϕ L ) - 2 r l ( ρ , ϕ A ) ] ,
Δ r l 1 3 β 2 2 r l ( ρ , ϕ ) ϕ 2 = 1 3 β 2 a l ρ r l 3 ( ρ , ϕ ) { ( z l 2 + a l 2 + ρ 2 ) cos ( ϕ - θ l ) - a l ρ [ 1 + cos 2 ( ϕ - θ l ) ] } 1 3 β 2 a l ρ / r l ( ρ , θ l ) ,
β max = [ ( 3 λ / L ρ ) / ( a 1 r 2 + a 2 r 2 ) ] 1 / 2 ,
N = 2 π β max = 2 π [ L 3 ρ λ ( a 1 r 1 + a 2 r 2 ) ] 1 / 2 .
g 2 ( ϕ ) = r 1 r 2 ( 1 - r ^ 1 · r ^ 2 ) ,
d = R 0 - R ,
R 0 = a ( 1 + z 2 / z 1 ) ;             R = [ ( x 2 + x 1 z 2 / z 1 ) 2 + ( y 1 + y 1 z 2 / z 1 ) 2 ] 1 / 2 ,
( 1 - r ^ 1 · r ^ 2 ) min = 1 - cos β β 2 / 2 ,
D = Q P 2 = [ r 02 2 + ( R 0 - a ) 2 ] 1 / 2 = z 2 z 1 ( a 1 2 + z 1 2 + a 2 ) .
( 1 - r ^ 1 · r ^ 2 ) min ½ β 2 ½ ( d D ) 2 ,
ρ n = ( 2 n λ f ) 1 / 2 ( 1 + n λ 2 f ) 1 / 2 .

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