Abstract

Recently exact and approximate solutions were presented for focusing of two-dimensional scalar and electromagnetic waves through a slit aperture in a perfectly conducting screen. We present, on the basis of these solutions, numerical comparisons between exact results and approximate Debye and Kirchhoff results for a variety of different focusing geometries. At moderately low Fresnel numbers large discrepancies are found between Debye and Kirchhoff results, and the latter are found to show greater agreement with the exact results. However, at low Fresnel numbers and large angular apertures significant discrepancies are also found between exact and Kirchhoff results, particularly at observation points inside the geometrical focus.

© 1998 Optical Society of America

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References

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  1. J. J. Stamnes, H. A. Eide, “Exact and approximate solutions for focusing of two-dimensional waves. I. Theory,” J. Opt. Soc. Am. A 15, 1285–1291 (1998).
    [CrossRef]
  2. J. J. Stamnes, “Exact two-dimensional scattering by perfectly reflecting elliptical cylinders, strips, and slits,” Pure Appl. Opt. 4, 841–855 (1995).
    [CrossRef]
  3. J. J. Stamnes, B. Spjelkavik, “New method for computing eigenfunctions (Mathieu functions) for scattering by elliptical cylinders,” Pure Appl. Opt. 4, 251–262 (1995).
    [CrossRef]
  4. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).
  5. 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]
  6. Section 7.2 of Ref. 4.
  7. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, 5th ed. (Dover, New York, 1968).
  8. J. J. Stamnes, “Focusing of two-dimensional waves,” J. Opt. Soc. Am. A 71, 15–31 (1981).
    [CrossRef]
  9. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [CrossRef]
  10. Y. Li, H. Platzer, “An experimental investigation of the diffraction pattern in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
    [CrossRef]
  11. V. Dhayalan, J. J. Stamnes, “Focusing of electric-dipole waves in the Debye and Kirchhoff approximations,” Pure Appl. Opt. 6, 347–372 (1997).
    [CrossRef]
  12. D. Jiang, J. J. Stamnes, “Theoretical and experimental results for focusing of two-dimensional scalar waves,” Pure Appl. Opt. 6, 211–224 (1997).
    [CrossRef]
  13. H. A. Eide, J. J. Stamnes, “Exact and approximate solutions for focusing of two-dimensional waves. III. Numerical comparisons between exact and Rayleigh–Sommerfeld theories,” J. Opt. Soc. Am. A 15, 1308–1319 (1998).
    [CrossRef]

1998

1997

V. Dhayalan, J. J. Stamnes, “Focusing of electric-dipole waves in the Debye and Kirchhoff approximations,” Pure Appl. Opt. 6, 347–372 (1997).
[CrossRef]

D. Jiang, J. J. Stamnes, “Theoretical and experimental results for focusing of two-dimensional scalar waves,” Pure Appl. Opt. 6, 211–224 (1997).
[CrossRef]

1995

J. J. Stamnes, “Exact two-dimensional scattering by perfectly reflecting elliptical cylinders, strips, and slits,” Pure Appl. Opt. 4, 841–855 (1995).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, “New method for computing eigenfunctions (Mathieu functions) for scattering by elliptical cylinders,” Pure Appl. Opt. 4, 251–262 (1995).
[CrossRef]

1983

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]

Y. Li, H. Platzer, “An experimental investigation of the diffraction pattern in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
[CrossRef]

1981

J. J. Stamnes, “Focusing of two-dimensional waves,” J. Opt. Soc. Am. A 71, 15–31 (1981).
[CrossRef]

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, 5th ed. (Dover, New York, 1968).

Dhayalan, V.

V. Dhayalan, J. J. Stamnes, “Focusing of electric-dipole waves in the Debye and Kirchhoff approximations,” Pure Appl. Opt. 6, 347–372 (1997).
[CrossRef]

Eide, H. A.

Jiang, D.

D. Jiang, J. J. Stamnes, “Theoretical and experimental results for focusing of two-dimensional scalar waves,” Pure Appl. Opt. 6, 211–224 (1997).
[CrossRef]

Li, Y.

Y. Li, H. Platzer, “An experimental investigation of the diffraction pattern in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
[CrossRef]

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

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]

Platzer, H.

Y. Li, H. Platzer, “An experimental investigation of the diffraction pattern in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
[CrossRef]

Spjelkavik, B.

J. J. Stamnes, B. Spjelkavik, “New method for computing eigenfunctions (Mathieu functions) for scattering by elliptical cylinders,” Pure Appl. Opt. 4, 251–262 (1995).
[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]

Stamnes, J. J.

J. J. Stamnes, H. A. Eide, “Exact and approximate solutions for focusing of two-dimensional waves. I. Theory,” J. Opt. Soc. Am. A 15, 1285–1291 (1998).
[CrossRef]

H. A. Eide, J. J. Stamnes, “Exact and approximate solutions for focusing of two-dimensional waves. III. Numerical comparisons between exact and Rayleigh–Sommerfeld theories,” J. Opt. Soc. Am. A 15, 1308–1319 (1998).
[CrossRef]

D. Jiang, J. J. Stamnes, “Theoretical and experimental results for focusing of two-dimensional scalar waves,” Pure Appl. Opt. 6, 211–224 (1997).
[CrossRef]

V. Dhayalan, J. J. Stamnes, “Focusing of electric-dipole waves in the Debye and Kirchhoff approximations,” Pure Appl. Opt. 6, 347–372 (1997).
[CrossRef]

J. J. Stamnes, “Exact two-dimensional scattering by perfectly reflecting elliptical cylinders, strips, and slits,” Pure Appl. Opt. 4, 841–855 (1995).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, “New method for computing eigenfunctions (Mathieu functions) for scattering by elliptical cylinders,” Pure Appl. Opt. 4, 251–262 (1995).
[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]

J. J. Stamnes, “Focusing of two-dimensional waves,” J. Opt. Soc. Am. A 71, 15–31 (1981).
[CrossRef]

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

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, 5th ed. (Dover, New York, 1968).

Wolf, E.

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Acta

Y. Li, H. Platzer, “An experimental investigation of the diffraction pattern in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
[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. Commun.

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

Pure Appl. Opt.

J. J. Stamnes, “Exact two-dimensional scattering by perfectly reflecting elliptical cylinders, strips, and slits,” Pure Appl. Opt. 4, 841–855 (1995).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, “New method for computing eigenfunctions (Mathieu functions) for scattering by elliptical cylinders,” Pure Appl. Opt. 4, 251–262 (1995).
[CrossRef]

V. Dhayalan, J. J. Stamnes, “Focusing of electric-dipole waves in the Debye and Kirchhoff approximations,” Pure Appl. Opt. 6, 347–372 (1997).
[CrossRef]

D. Jiang, J. J. Stamnes, “Theoretical and experimental results for focusing of two-dimensional scalar waves,” Pure Appl. Opt. 6, 211–224 (1997).
[CrossRef]

Other

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

Section 7.2 of Ref. 4.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, 5th ed. (Dover, New York, 1968).

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Figures (13)

Fig. 1
Fig. 1

Converging wave with focus at (0, z1) incident upon a slit aperture of width 2a in a perfectly reflecting screen in the plane z=0.

Fig. 2
Fig. 2

Two-dimensional electromagnetic focusing with E polarization. (a) Focusing geometry: N=10, θ1=45°. (b) Contour map of the exact total electric-field amplitude |Ey| in the xz plane. (c) |Ey| along the z axis obtained by use of the exact solution (solid curve), the Kirchhoff solution (dashed curve), and the Debye solution (dotted curve). (d)–(h) |Ey| obtained as in (c) along the lines z=constant shown in (b) and (c). All the distances given in (b)–(h) are in wavelengths.

Fig. 3
Fig. 3

Two-dimensional electromagnetic focusing with H polarization. (a) Focusing geometry: N=10, θ1=45°. (b) Contour map of the exact total magnetic-field amplitude |Hy| in the xz plane. (c) |Hy| along the z axis obtained by use of the exact solution (solid curve), the Kirchhoff solution (dashed curve), and the Debye solution (dotted curve). (d)–(h) |Hy| obtained as in (c) along the lines z=constant shown in (b) and (c). All the distances given in (b)–(h) are in wavelengths.

Fig. 4
Fig. 4

Two-dimensional electromagnetic focusing with H polarization. (a) Focusing geometry: N=10, θ1=45°. (b) Contour map of the exact transverse electric-field amplitude |Ex| in the xz plane. (c) |Ex| along the z axis obtained by use of the exact solution (solid curve), the Kirchhoff solution (dashed curve), and the Debye solution (dotted curve). (d)–(h) |Ex| obtained as in (c) along the lines z=constant shown in (b) and (c). All the distances given in (b)–(h) are in wavelengths.

Fig. 5
Fig. 5

Two-dimensional electromagnetic focusing with H polarization. (a) Focusing geometry: N=10, θ1=45°. (b) Contour map of the exact longitudinal electric-field amplitude |Ez| in the xz plane. (c) |Ez| is zero along the z axis because of symmetry. (d)–(h) |Ez| along the lines z=constant shown in (b) obtained by use of the exact solution (solid curves), the Kirchhoff solution (dashed curves), and the Debye solution (dotted curves). All the distances given in (b)–(h) are in wavelengths.

Fig. 6
Fig. 6

Two-dimensional electromagnetic focusing with E polarization. (a) Focusing geometry: N=2.5, θ1=26.56°. (b) Contour map of the exact total electric-field amplitude |Ey| in the xz plane. (c) |Ey| along the z axis obtained by use of the exact solution (solid curve), the Kirchhoff solution (dashed curve), and the Debye solution (dotted curve). (d)–(h) |Ey| obtained as in (c) along the lines z=constant shown in (b) and (c). All the distances given in (b)–(h) are in wavelengths.

Fig. 7
Fig. 7

Two-dimensional electromagnetic focusing with H polarization. (a) Focusing geometry: N=2.5, θ1=26.56°. (b) Contour map of the exact total magnetic-field amplitude |Hy| in the xz plane. (c) |Hy| along the z axis obtained by use of the exact solution (solid curve), the Kirchhoff solution (dashed curve), and the Debye solution (dotted curve). (d)–(h) |Hy| obtained as in (c) along the lines z=constant shown in (b) and (c). All the distances given in (b)–(h) are in wavelengths.

Fig. 8
Fig. 8

Two-dimensional electromagnetic focusing with H polarization. (a) Focusing geometry: N=2.5, θ1=26.56°. (b) Contour map of the exact transverse electric-field amplitude |Ex| in the xz plane. (c) |Ex| along the z axis obtained by use of the exact solution (solid curve), the Kirchhoff solution (dashed curve), and the Debye solution (dotted curve). (d)–(h) |Ex| obtained as in (c) along the lines z=constant shown in (b) and (c). All the distances given in (b)–(h) are in wavelengths.

Fig. 9
Fig. 9

Two-dimensional electromagnetic focusing with H polarization. (a) Focusing geometry: N=2.5, θ1=26.56°. (b) Contour map of the exact longitudinal electric-field amplitude |Ez| in the xz plane. (c) |Ez| is zero along the z axis owing to symmetry. (d)–(h) |Ez| along the lines z=constant shown in (b) obtained by use of the exact solution (solid curves), the Kirchhoff solution (dashed curves), and the Debye solution (dotted curves). All the distances given in (b)–(h) are in wavelengths.

Fig. 10
Fig. 10

Two-dimensional electromagnetic focusing with E polarization. (a) Focusing geometry: N=2.5, θ1=45°. (b) Contour map of the total electric-field amplitude |Ey| in the xz plane. (c) |Ey| along the z axis obtained by use of the exact solution (solid curve), the Kirchhoff solution (dashed curve), and the Debye solution (dotted curve). (d)–(h) |Ey| obtained as in (c) along the lines z=constant shown in (b) and (c). All the distances given in (b)–(h) are in wavelengths.

Fig. 11
Fig. 11

Two-dimensional electromagnetic focusing with H polarization. (a) Focusing geometry: N=2.5, θ1=45°. (b) Contour map of the exact total magnetic-field amplitude |Hy| in the xz plane. (c) |Hy| along the z axis obtained by use of the exact solution (solid curve), the Kirchhoff solution (dashed curve), and the Debye solution (dotted curve). (d)–(h) |Hy| obtained as in (c) along the lines z=constant shown in (b) and (c). All the distances given in (b)–(h) are in wavelengths.

Fig. 12
Fig. 12

Two-dimensional electromagnetic focusing with H polarization. (a) Focusing geometry: N=2.5, θ1=45°. (b) Contour map of the exact transverse electric-field amplitude |Ex| in the xz plane. (c) |Ex| along the z axis obtained by use of the exact solution (solid curve), the Kirchhoff solution (dashed curve), and the Debye solution (dotted curve). (d)–(h) |Ex| obtained as in (c) along the lines z=constant shown in (b) and (c). All the distances given in (b)–(h) are in wavelengths.

Fig. 13
Fig. 13

Two-dimensional electromagnetic focusing with H polarization. (a) Focusing geometry: N=2.5, θ1=45°. (b) Contour map of the exact longitudinal electric-field amplitude |Ez| in the xz plane. (c) |Ez| is zero along the z axis owing to symmetry. (d)–(h) |Ez| along the lines z=constant shown in (b) obtained by use of the exact solution (solid curves), the Kirchhoff solution (dashed curves), and the Debye solution (dotted curves). All the distances given in (b)–(h) are in wavelengths.

Equations (56)

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R1=(x12+z12)1/2,R2=[(x-x2)2+z22]1/2.
uc=½kH0(2)(kR1)(λz1)-1/2z1R11/2×exp[i(π/4-kR1)],
ud=i2z1H0(2)(kR1)=kz12iR1H1(2)(kR1)z1R1uc.
us,q=m=0Bom(s)Comq(s)Hom(1)(s, u)Som(s, v),
uh,q=m=0Bem(s)Cemq(s)Hem(1)(s, u)Sem(s, v),
Bpm(s)=(8π)1/2 imNpmJpm(s, 0)Hpm(1)(s, 0),
Bpm(s)=(8π)1/2 imNpmJpm(s, 0)Hpm(1)(s, 0),
Se2r+γ(s, v)=k=0De2k+γ(2r+γ)(s)cos(2k+γ)v
(γ=0, 1),
So2r+γ(s, v)=k=0Do2k+γ(2r+γ)(s)sin(2k+γ)v
(γ=0, 1),
Cp2r+γq(s)=k=0Dp2k+γ(2r+γ)(s)Lp2k+γq(kz1)
(γ=0, 1),
Lenq(w)=½[Lnq+(w)+Lnq-(w)],
Lonq(w)=12i[Lnq+(w)-Lnq-(w)],
Lnd±=k2π0π sin t exp[-i(w sin tnt)]dt,
Lnc±=k2π0π exp[-i(w sin tnt)]dt,
s=(ka)2,x2=a cosh u cos v,z2=a sinh u cos v.
Eyq=us,q,Hxq=-ciωμus,qz2,
Hzq=ciωμus,qx2,
us,qx2=m=0Bom(s)Comq(s)Xom,
us,qz2=m=0Bom(s)Comq(s)Zom,
Xpm=ux2Hpm(1)(s, u)Spm(s, v)+vx2Hpm(1)(s, u)Spm(s, v),
Zpm=uz2Hpm(1)(s, u)Spm(s, v)+vz2Hpm(1)(s, u)Spm(s, v),
ux2=vz2=1asinh u cos vcosh2 u-cos2 v,
uz2=-vx2=1acosh u sin vcosh2 u-cos2 v.
Hyq=uh,q,Exq=-icωεuh,qz2,Ezq=icωεuh,qx2,
uh,qx2=m=0Bem(s)Cemq(s)Xem,
uh,qz2=m=0Bem(s)Cemq(s)Zem,
uIc=ik24-aa z2R2H0(2)(kR1)H1(1)(kR2)dx,
uIIc=k24i-aa z1R1H1(2)(kR1)H0(1)(kR2)dx,
uId=k24-aa z1z2R1R2H1(2)(kR1)H1(1)(kR2)dx,
uIId=k24-aa z12R12H0(2)(kR1)A(x)H0(1)(kR2)dx,
A(x)=1+1kR1R1z12-2 H1(2)(kR1)H0(2)(kR1).
uK=½(uI+uII).
Hn(2)(kR1)in2πkR11/2 exp[i(π/4-kR1)].
ujq=-aaGjq(x)exp[iF(x)]dx(j=I,II,q=c,d),
GIc(x)=ik242πkR11/2 z2R2H1(1)(kR2),
GId(x)=z1R1GIc(x),
GIIc(x)=k242πkR11/2 z1R1H0(1)(kR2),
GIId(x)=z1R1GIIc(x),
F(x)=π/4-kR1.
Hn(1)(-i)n2πkR21/2Bn(x)exp{i[kR2-π/4+ψn(x)]},
Bn(x)=[Pn2(kR2)+Qn2(kR2)]1/2,
ψn(x)=arctanQn(kR2)Pn(kR2).
ujq=-aaGjq(x)exp[iFj(x)]dx(j=I,II,q=c, d),
GIc=z2R2B1(x)λ(R1R2)1/2,GId(x)=z1R1GIc(x),
GIIc=z1R1B0(x)λ(R1R2)1/2,GIId(x)=z1R1GId(x),
FI(x)=k(R2-R1)+ψ1(x),
FII(x)=k(R2-R1)+ψ0(x).
uDq=-θθgq(θ)exp[if(θ)]dθ,
gc=1/λ,gd(θ)=cos θ gc(θ),
f(θ)=kr cos(θ-ϕ),r=[x22+(z2-z1)2]1/2,
ϕ=arctanx2z2-z1.
Ejyq=ujq,Hjxq=-ciωμujqz2,Hjzq=ciωμujqx2.
Hjyq=ujq,Ejxq=-icωεujqz2,Ejzq=icωεujqx2.

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