Abstract

Recently exact and approximate solutions were given for focusing of two-dimensional scalar and electromagnetic waves through a slit aperture in a perfectly reflecting screen. We present numerical comparisons, based on these solutions, between exact results and approximate Rayleigh–Sommerfeld results for a variety of different focusing geometries. These comparisons show that, at sufficiently low Fresnel numbers and large angular apertures, approximate solutions based on the first and the second Rayleigh–Sommerfeld diffraction integrals agree well with the corresponding exact solutions for hard and soft screens, respectively. Inasmuch as these results are contrary to what one would expect from considerations of the boundary conditions, we give an analytical explanation of them through comparisons between exact and approximate near-field solutions for the corresponding half-plane problems.

© 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. H. A. Eide, J. J. Stamnes, “Exact and approximate solutions for focusing of two-dimensional waves. II. Numerical comparisons among exact, Debye, and Kirchhoff theories,” J. Opt. Soc. Am. A 15, 1292–1307 (1998).
    [CrossRef]
  3. C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
    [CrossRef]
  4. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), p. 447.
  5. Ref. 4, Eqs. (8.19).
  6. Ref. 1, Eqs. (3.1a) and (3.1c).
  7. Ref. 4, lowest-order terms of Eqs. (8.8a) and (8.14a).
  8. Ref. 4, Sec. 4.4.

1998 (2)

1954 (1)

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
[CrossRef]

J. Opt. Soc. Am. A (2)

Rep. Prog. Phys. (1)

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
[CrossRef]

Other (5)

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), p. 447.

Ref. 4, Eqs. (8.19).

Ref. 1, Eqs. (3.1a) and (3.1c).

Ref. 4, lowest-order terms of Eqs. (8.8a) and (8.14a).

Ref. 4, Sec. 4.4.

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Figures (10)

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, θ1=45°. (b) Contour map of the exact total electric-field amplitude |Ey| in the xz plane. (c) |Ey| along the z axis obtained with the exact solution (solid curve), the first Rayleigh–Sommerfeld solution (dashed curve), and the second Rayleigh–Sommerfeld solution (dotted curve). (d)–(h) |Ey| obtained as in (c) along the lines z=constant shown in (b) and (c). All the distances in (b)–(h) are in wavelengths.

Fig. 3
Fig. 3

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

Fig. 4
Fig. 4

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

Fig. 5
Fig. 5

Two-dimensional electromagnetic focusing with H polarization. (a) Focusing geometry, 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 with the exact solution (solid curves), the first Rayleigh–Sommerfeld solution (dashed curves), and the second Rayleigh–Sommerfeld solution (dotted curves). All the distances in (b)–(h) are in wavelengths.

Fig. 6
Fig. 6

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

Fig. 7
Fig. 7

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

Fig. 8
Fig. 8

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

Fig. 9
Fig. 9

Two-dimensional electromagnetic focusing with H polarization. (a) Focusing geometry, 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 with the exact solution (solid curves), the first Rayleigh–Sommerfeld solution (dashed curves), and the second Rayleigh–Sommerfeld solution (dotted curves). All the distances in (b)–(h) are in wavelengths.

Fig. 10
Fig. 10

Plane wave normally incident upon the edge of a half-plane at an angle θ0 and the diffracted field observed at an angle θ and a distance s.

Equations (39)

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Ey=us,Ex=Ez=0,
Hx=-ciωμEyz,Hz=ciωμEyx,Hy=0.
Hy=uh,Hx=Hz=0,
Ex=-icωHyz,Ez=icωHyx,Ey=0.
ui=exp[iks cos(θ-θ0)],
ur=exp[iks cos(θ+θ0)].
usuh=F(ξi)uiF(ξr)ur,
F(x)=½{1-2 exp(-iπ/4)[C(x)+iS(x)]},
C(x)=2π 0x cos(t2)dt,
S(x)=2π 0x sin(t2)dt.
F(x)-F(-x)=-2 exp(-iπ/4)[C(x)+iS(x)],
F(x)+F(-x)=1.
ξi=-2ks sin ½(θ-θ0),
ξr=2ks sin ½(θ+θ0).
ξi=-ξr=-2ks cos ½θ0<0,
ui=ur=exp(-iks cos θ0).
uh(x0, z=0)=ui,
us(x0, z=0)=-ui2 exp(-iπ/4)[C(ξi)+iS(ξi)].
us(x0, z=0+)=ui-exp[i(ks+π/4)]2πks cos ½θ0.
ξi=ξr=2ks sin ½θ0>0,
ui=ur=exp(iks cos θ0),
us(x>0, z=0+)uh(x>0, z=0+)=02F(ξi)ui.
2F(ξi)uiexp[i(ks+π/4)]2πks sin ½θ0.
us(x, 0+)=-ui2 exp-iπ4[C(ξi)+iS(ξi)]ui-1cos ½θ0expiks+π42πksx<00x>0,
uh(x, 0+)=uix<02uiF(ξi)1sin ½θ0exp[i(ks+π/4)]2πksx>0.
uI=12i-0ui(x, 0) z2H0(1)(kR2)dx=ik2-0ui(x, 0) z2R2H1(1)(kR2)dx,
uII=12i-0 ui(x, z)zz=0H0(1)(kR2)dx,
R2=[(x-x2)2+z22]1/2,
ui(x, 0)=exp(ikx cos θ0),
ui(x, z)xz=0=ikui sin θ0.
uj=-0gj(x)exp[ikf(x)]dx(j=I, II),
gI(x)=exp(-iπ/4)λz2z2R23/2,
gII(x)=exp(-iπ/4)λz2z2R21/2 sin θ0,
f(x)=R2+x cos θ0.
uI(s, θ)uiH(θ-θ0)-sin θcos θ0-cos θexp[i(ks+π/4)]2πks,
uII(s, θ)uiH(θ-θ0)-sin θ0cos θ0-cos θexp[i(ks+π/4)]2πks,
s=x22+z22,θ=arctan(x2/z2).
uI(x, 0+)=uix<00x>0,
uII(x, 0+)ui-sin ½θ0cos ½θ0expiks+π42πksx<0cos ½θ0sin ½θ0expiks+π42πksx>0.

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