Abstract

The complete Rayleigh–Sommerfeld scalar diffraction formula contains (1-ikR) in the integrand. Usually the wavelength is small compared with the distance of the observation point from the aperture and (1-ikR) is approximated by -ikR alone. Other approximations usually made in the Rayleigh–Sommerfeld formula are addressed as well. Closed-form solutions, without approximations, are possible wherein interesting consequences of these approximations become apparent.

© 2004 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 6th ed (Pergamon, Oxford, UK, 1980), pp. 1–104, 370–455, 556–591.
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 30–76.
  3. H. Osterberg, L. Smith, “Closed solutions of Rayleigh’s diffraction integral for axial points,” J. Opt. Soc. Am. 51, 1050–1054 (1961).
    [CrossRef]
  4. J. C. Heurtley, “Scalar Rayleigh–Sommerfeld and Kirchhoff diffraction integrals: a comparison of exact evaluations for axial points,” J. Opt. Soc. Am. 63, 1003–1008 (1973).
    [CrossRef]
  5. E. Marchand, E. Wolf, “Boundary diffraction wave in the domain of the Rayleigh–Kirchhoff diffraction theory,” J. Opt. Soc. Am. 52, 761–767 (1962).
    [CrossRef]
  6. A. Rubinowicz, “Boundary wave diffraction,” in Progress in Optics, Vol. IV, E. Wolf, ed. (North-Holland, Amsterdam, 1965), p. 199.
  7. T. Young, ”Bakerian lecture on the mechanism of the theory of light and colour,” Philos. Trans.12–48 (1802).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed (Pergamon, Oxford, UK, 1980), pp. 1–104, 370–455, 556–591.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 30–76.

Heurtley, J. C.

Marchand, E.

Osterberg, H.

Rubinowicz, A.

A. Rubinowicz, “Boundary wave diffraction,” in Progress in Optics, Vol. IV, E. Wolf, ed. (North-Holland, Amsterdam, 1965), p. 199.

Smith, L.

Wolf, E.

E. Marchand, E. Wolf, “Boundary diffraction wave in the domain of the Rayleigh–Kirchhoff diffraction theory,” J. Opt. Soc. Am. 52, 761–767 (1962).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed (Pergamon, Oxford, UK, 1980), pp. 1–104, 370–455, 556–591.

Young, T.

T. Young, ”Bakerian lecture on the mechanism of the theory of light and colour,” Philos. Trans.12–48 (1802).

J. Opt. Soc. Am.

Philos. Trans.

T. Young, ”Bakerian lecture on the mechanism of the theory of light and colour,” Philos. Trans.12–48 (1802).

Other

M. Born, E. Wolf, Principles of Optics, 6th ed (Pergamon, Oxford, UK, 1980), pp. 1–104, 370–455, 556–591.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 30–76.

A. Rubinowicz, “Boundary wave diffraction,” in Progress in Optics, Vol. IV, E. Wolf, ed. (North-Holland, Amsterdam, 1965), p. 199.

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Figures (9)

Fig. 1
Fig. 1

Circular aperture of radius a in a thin, nonconducting opaque screen at z=0. The origin of the coordinate system is at the center of the aperture. The aperture is illuminated by a normally incident plane wave of amplitude A propagating along the z axis in the +z direction.

Fig. 2
Fig. 2

(a) Plot of the on-axis irradiance for the case of a circular aperture of radius a=8λ for z=0 to z=20λ. (b) Plot of the on-axis irradiance for the case of a circular aperture of radius a=8λ for z=20λ to z=140λ.

Fig. 3
Fig. 3

Plot of the on-axis irradiance for the case of a circular disk of radius a=8λ.

Fig. 4
Fig. 4

(a) Vector representation of Babinet’s principle. (b) Vector diagram representation of Babinet’s principle in the region of Fresnel diffraction associated with a circular aperture of radius 10λ and its complementary aperture, an opaque disk of radius 10λ. (c) Vector diagram representation of Babinet’s principle in the region of Fraunhofer diffraction associated with a circular aperture of radius 10λ and its complementary aperture, an opaque disk of radius 10λ.

Fig. 5
Fig. 5

Pair of complementary zone plates. (a) Odd Fresnel zone plate. (b) Even Fresnel zone plate.

Fig. 6
Fig. 6

On-axis irradiance as a function of the distance along the z axis for an even zone plate illuminated by a normally incident plane wave (a) about the primary focal point and (b) about a subsidiary focal point.

Fig. 7
Fig. 7

Converging-wave illumination of a circular aperture of radius a.

Fig. 8
Fig. 8

On-axis irradiance along the z axis from a circular diffracting aperture of radius 7λ. The aperture is illuminated by a converging wave with a 40λ radius of curvature.

Fig. 9
Fig. 9

On-axis irradiance I(z) due to the diffracted field of a circular aperture. (a) Solid curve, complete solution; dotted curve, irradiance calculated by ignoring the 1 in the (1-ikr) term of the Rayleigh–Sommerfeld diffraction formula. (b) Solid curve, complete solution; dotted curve, irradiance calculated by setting the cosine term equal to unity in the Rayleigh–Sommerfeld diffraction formula. (c) Solid curve, complete solution; dotted curve, irradiance calculated by making the paraxial approximation in the Rayleigh–Sommerfeld diffraction formula. (d) Solid curve is the complete solution and the dotted curve is the irradiance calculated by ignoring the 1 in the (1-ikr) term, setting the cosine term equal to unity, and making the paraxial approximation in the Rayleigh–Sommerfeld diffraction formula.

Equations (18)

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V(x, y, z)=apertureV(x, y, 0)12πzR(1-ikR)exp(ikR)R2dxdy.
Vcir(0, 0, z)=A12πzR×(1-ikR)exp(ikR)R2RdRdf,
Vcir(z)=AzzRaexp(ikR)R2-ikexp(ikR)RdR.
Vcir(z)=Az-exp(ikR)RzRa+ikzRaexp(ikR)RdR-ikzRaexp(ikR)RdR.
Vcir(z)=A exp(ikz)-Aza2+z2exp(ika2+z2).
Icir(z)=|Vcir|2=A21+z2Ra2-2zRacos[k(Ra-z)].
Vdisk(z)=Azexp(ika2+z2)a2+z2.
Idisk(z)=A2z2a2+z2.
Vcir+Vdisk=Vun,
F(dn)Azdnexp(ikdn).
Veven=1N[F(d2n-1)-F(d2n)],
Vodd=0N[F(d2n)-F(d2n+1)].
Iodd(2N)2Azf2sin(γ2N)(2N)sin(γ)2,
γπΔz2f.
Iodd4N2Azf2.
V(x, y, 0)=ARcexp(-ikRc),
Rc=(x2+y2+Rc2)1/2=(ρ2+Rc2)1/2.
V(0, 0, z)=Az0aexp[ik(R-Rc)]RcR2-ikexp[ik(R-Rc)]RcRρdρ.

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