Abstract

The oscillatory integrands of the Kirchhoff and the Rayleigh–Sommerfeld diffraction solutions mean that these two-dimensional integrals typically lead to challenging computations. By adoption of the Kirchhoff boundary conditions, the domain of the integrals is reduced to cover only the aperture. For perfect spherical (both diverging and focused) and plane incident fields, closed forms are derived for vector potentials that allow each of these solutions to be further simplified to just a one-dimensional, singularity-free integral around the aperture rim. The results offer easy numerical access to exact—although, given the approximate boundary conditions, not rigorous—solutions to important diffraction problems. They are derived by generalization of a standard theorem to extend previous results to the case of focused fields and the Rayleigh–Sommerfeld solutions.

© 1998 Optical Society of America

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  1. See the review given in A. Rubinowicz, “The Miyamoto–Wolf diffraction wave,” Prog. Opt. 4, 201–240 (1965), and the more historical account given in A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature 180, 160–162 (1957).
    [CrossRef]
  2. T. Gravelsaeter, J. J. Stamnes, “Diffraction by circular apertures. 1. Method of linear phase and amplitude approximation,” Appl. Opt. 21, 3644–3651 (1982).
    [CrossRef] [PubMed]
  3. W. B. Gordon, “Vector potentials and physical optics,” J. Math. Phys. 16, 448–454 (1975).
    [CrossRef]
  4. J. S. Asvestas, “Line integrals and physical optics. Part I. The transformation of the solid-angle surface integral to a line integral,” J. Opt. Soc. Am. A 2, 891–895 (1985), and Part II. The conversion of the Kirchhoff surface integral to a line integral,” 2, 896–902 (1985).
    [CrossRef]
  5. See W. Kaplan, Advanced Calculus (Addison-Wesley, Reading, Mass., 1973), p. 349. A variant of Green’s theorem leads to the more standard, but far less convenient, volume integral for this potential; see, e.g., W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, London, 1964), p. 2, and G. Arfken, Mathematical Methods for Physicists (Academic, Orlando, Fla., 1985), Sec. 1.15.
  6. E. W. Marchand, E. Wolf, “Boundary diffraction wave in the domain of the Rayleigh–Kirchhoff diffraction theory,” J. Opt. Soc. Am. 52, 761–767 (1962).
    [CrossRef]
  7. For example, the result deserves to be in books such as in J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 5; G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 165; and the second and third references given in Ref. 5.
  8. See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, Elmsford, New York, 1980), Sec. 8.3.
  9. This result is given by A. Rubinowicz, “Die Beugungswelle in der Kirchhoffschen Theorie der Beugungserscheinungen,” Ann. Phys. 53, 257–278 (1917). See the integrals presented after Fig. 1 in this reference.
    [CrossRef]
  10. By use of Eqs. (3.2) and (2.7) as described, it is found that D(r)=-(1-σ) δO×δS16π2×∫01 ik exp[ik(a2-bτ)/S(τ)]S3(τ)τdτ, where S(τ)=(a2-2bτ+c2τ2)1/2,a=|rO-rS|,b=(rO-rS)·(r-rS), and c=|r-rS|. Again, this integral can be done in closed form: The primitive is just (b2-a2c2)-1 exp[ik(a2-bτ)/S(τ)]. Equation (3.10) now follows.
  11. A good first step is to note that the apparent r dependence can be reduced by use of (r-rO)×(r-rS)=(r-rO)×(rO-rS).
  12. In fact, if this line of singularities cuts a surface of integration, the associated flux contribution is readily shown to mean that, instead of vanishing, the net flux is 2π/|rS-rO|. It follows that the last of the three terms inside the braces of each of Eqs. (3.8), (3.11), and (3.12) corresponds precisely to the discontinuous geometric field component discussed in Section 1. This observation clarifies the link to the results given in K. Miyamoto, E. Wolf, “Generalization of the Maggi–Rubinowicz theory of the boundary diffraction wave. Part I,” J. Opt. Soc. Am. 52, 615–625 (1962), and “Generalization of the Maggi–Rubinowicz theory of the boundary diffraction wave. Part II,” 52, 626–637 (1962). (As indicated previously in the text, the Miyamoto–Wolf potential becomes singular when the observation point falls on the shadow boundary.)
    [CrossRef]
  13. See, for example, L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Section 3.2.5.
  14. C. R. Schultheisz, “Numerical solution of the Huygens–Fresnel–Kirchhoff diffraction of spherical waves by a circu-lar aperture,” J. Opt. Soc. Am. A 11, 774–778 (1994).
    [CrossRef]
  15. In the case of the plane-wave results, recall that the incident field has already been scaled before taking the source point to infinity, so it is necessary to divide the potential given in Eq. (3.20), for example, by only exp(iku·rO), and this means that the phase factor now becomes exp(ik2L∞). The same is true of Eq. (4.4), and these results are in keeping with Eqs. (5.1) and (5.3). For the focused field, exp(-ik|rO-rS|)/(4π|rO-rS|) is to be divided out for points well before focus, but its complex conjugate is to be used far beyond the focal plane; the phase is small near the focus, so there is nothing to be gained there in this way.

1994

1985

1982

1975

W. B. Gordon, “Vector potentials and physical optics,” J. Math. Phys. 16, 448–454 (1975).
[CrossRef]

1965

See the review given in A. Rubinowicz, “The Miyamoto–Wolf diffraction wave,” Prog. Opt. 4, 201–240 (1965), and the more historical account given in A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature 180, 160–162 (1957).
[CrossRef]

1962

1917

This result is given by A. Rubinowicz, “Die Beugungswelle in der Kirchhoffschen Theorie der Beugungserscheinungen,” Ann. Phys. 53, 257–278 (1917). See the integrals presented after Fig. 1 in this reference.
[CrossRef]

Asvestas, J. S.

Born, M.

See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, Elmsford, New York, 1980), Sec. 8.3.

Gordon, W. B.

W. B. Gordon, “Vector potentials and physical optics,” J. Math. Phys. 16, 448–454 (1975).
[CrossRef]

Gravelsaeter, T.

Jackson, J. D.

For example, the result deserves to be in books such as in J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 5; G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 165; and the second and third references given in Ref. 5.

Kaplan, W.

See W. Kaplan, Advanced Calculus (Addison-Wesley, Reading, Mass., 1973), p. 349. A variant of Green’s theorem leads to the more standard, but far less convenient, volume integral for this potential; see, e.g., W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, London, 1964), p. 2, and G. Arfken, Mathematical Methods for Physicists (Academic, Orlando, Fla., 1985), Sec. 1.15.

Mandel, L.

See, for example, L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Section 3.2.5.

Marchand, E. W.

Miyamoto, K.

Rubinowicz, A.

See the review given in A. Rubinowicz, “The Miyamoto–Wolf diffraction wave,” Prog. Opt. 4, 201–240 (1965), and the more historical account given in A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature 180, 160–162 (1957).
[CrossRef]

This result is given by A. Rubinowicz, “Die Beugungswelle in der Kirchhoffschen Theorie der Beugungserscheinungen,” Ann. Phys. 53, 257–278 (1917). See the integrals presented after Fig. 1 in this reference.
[CrossRef]

Schultheisz, C. R.

Stamnes, J. J.

Wolf, E.

Ann. Phys.

This result is given by A. Rubinowicz, “Die Beugungswelle in der Kirchhoffschen Theorie der Beugungserscheinungen,” Ann. Phys. 53, 257–278 (1917). See the integrals presented after Fig. 1 in this reference.
[CrossRef]

Appl. Opt.

J. Math. Phys.

W. B. Gordon, “Vector potentials and physical optics,” J. Math. Phys. 16, 448–454 (1975).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Prog. Opt.

See the review given in A. Rubinowicz, “The Miyamoto–Wolf diffraction wave,” Prog. Opt. 4, 201–240 (1965), and the more historical account given in A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature 180, 160–162 (1957).
[CrossRef]

Other

For example, the result deserves to be in books such as in J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 5; G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 165; and the second and third references given in Ref. 5.

See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, Elmsford, New York, 1980), Sec. 8.3.

By use of Eqs. (3.2) and (2.7) as described, it is found that D(r)=-(1-σ) δO×δS16π2×∫01 ik exp[ik(a2-bτ)/S(τ)]S3(τ)τdτ, where S(τ)=(a2-2bτ+c2τ2)1/2,a=|rO-rS|,b=(rO-rS)·(r-rS), and c=|r-rS|. Again, this integral can be done in closed form: The primitive is just (b2-a2c2)-1 exp[ik(a2-bτ)/S(τ)]. Equation (3.10) now follows.

A good first step is to note that the apparent r dependence can be reduced by use of (r-rO)×(r-rS)=(r-rO)×(rO-rS).

See W. Kaplan, Advanced Calculus (Addison-Wesley, Reading, Mass., 1973), p. 349. A variant of Green’s theorem leads to the more standard, but far less convenient, volume integral for this potential; see, e.g., W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, London, 1964), p. 2, and G. Arfken, Mathematical Methods for Physicists (Academic, Orlando, Fla., 1985), Sec. 1.15.

See, for example, L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Section 3.2.5.

In the case of the plane-wave results, recall that the incident field has already been scaled before taking the source point to infinity, so it is necessary to divide the potential given in Eq. (3.20), for example, by only exp(iku·rO), and this means that the phase factor now becomes exp(ik2L∞). The same is true of Eq. (4.4), and these results are in keeping with Eqs. (5.1) and (5.3). For the focused field, exp(-ik|rO-rS|)/(4π|rO-rS|) is to be divided out for points well before focus, but its complex conjugate is to be used far beyond the focal plane; the phase is small near the focus, so there is nothing to be gained there in this way.

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Figures (2)

Fig. 1
Fig. 1

Solid angle (projected area on the unit sphere) of the surface S when viewed from rO. The result can be found from a line integral that associates a wedge with each element of the boundary. Note that the elemental contribution changes with the choice of the arbitrary reference point, r. A similar interpretation can be given to the boundary diffracted waves derived in the Sections 3 and 4.

Fig. 2
Fig. 2

Two surfaces S and S have the same boundary but do not subtend the same solid angle when viewed from rO. The flux used to compute the solid angle of S has a contribution associated with the line of singularities shown as a solid line starting at rO and directed away from r.

Equations (49)

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ϕ(r)=rrG(r)·dr.
ϕ(r)=(r-r)·01G[r+τ(r-r)]dτ.
A(r)=-(r-r)×01F[r+τ(r-r)]τdτ.
C(r)01F[r+τ(r-r)]τdτ.
·C(r)=01w[r+τ(r-r)]τ2dτ,
(r-r)·C(r)=01ddτF[r+τ(r-r)]τ2dτ=F(r)-limτ0{τ2F[r+τ(r-r)]}-2C(r),
D(r):=(r-r)×C(r)
=(r-r)×01F[r+τ(r-r)]τdτ,
×D(r)=-F(r)+limτ0{τ2F[r+τ(r-r)]}.
limτ{τ2F[r+τ(r-r)]},
×D¯(r)=F(r)-limτ{τ2F[r+τ(r-r)]},
Ω(rO)=S[(r-rO)/|r-rO|3]·dσ=S(-1/|r-rO|)·dσ,
D(r)=-(r-r)×(rO-r)01 τdτ(a2-2bτ+c2τ2)3/2,
a=|rO-r|,
b=(rO-r)·(r-r),
c=|r-r|.
01 τdτ(a2-2bτ+c2τ2)3/2=|rO-r||r-rO|+(rO-r)·(r-rO)|r-rO|{|rO-r|2|r-r|2-[(rO-r)·(r-r)]2}.
D(r)=-(r-rO)×(r-rO)|r-rO|[|r-rO||r-rO|+(r-rO)·(r-rO)].
uK(rO)=-S[u(r)v(r)-v(r)u(r)]·dσ,
limτ{τ2FK[r+τ(r-r)]}
=-(1-σ)ik4πexpik (rS-rO)·(r-r)|r-r|
×14π|r-r|.
limτ0{τ2FK[r+τ(r-r)]}=u(rO)14π|r-rO|,
limτ0{τ2FK[r+τ(r-r)]}=-v(rS)14π|r-rS|.
FK(r)=-×D(r)+u(rO)14π|r-rO|.
D(r)=(r-rO)×(rS-rO)16π2Cσ×01 exp{ik[Cτ+R(τ)]}R3(τ)[ikR(τ)-1]dτ,
D(r)=(r-rO)×(rS-rO)16π2|r-rO|exp[ik(|r-rO|+σ|r-rS|)]|r-rS|[σ|r-rO||r-rS|+(r-rO)·(r-rS)]
-exp(ikσ|rS-rO|)|rS-rO|[σ|r-rO||rS-rO|-(r-rO)·(rS-rO)].
DK(r)=-δO×δS16π2|δO||δS|exp[ik(|δO|+σ|δS|)]|δO×δS|2×{σ|δO||δS|-δO·δS-(1+σ)|δO||δS|×exp[ik(|δS-δO|-|δO|-|δS|)]}.
D¯(r)=δO×δS16π2|δO||δS|exp[ik(|δO|+σ|δS|)]|δO×δS|2×{σ|δO||δS|-δO·δS+(1-σ)|δO||δS|×exp[ik(|δS|-δO·δS/|δO|)]}.
D(r)=(1-σ) δO×δS16π2exp[ik(|δO|+σ|δS|)]|δO×δS|2×{exp[ik(|δS|-δO·δS/|δO|)]-exp[ik(|rS-rO|-|δO|+|δS|)]}.
D¯K0(r)=δO×δS16π2|δO||δS|exp[ik(|δO|+σ|δS|)]|δO×δS|2×{σ|δO||δS|-δO·δS+(1-σ)|δO||δS|×exp[ik(|rS-rO|-|δO|+|δS|)]}.
D¯KS(r)=δO×δS16π2|δO||δS|exp[ik(|δO|+σ|δS|)]|δO×δS|2×{σ|δO||δS|-δO·δS+(1-σ)|δO||δS|×exp[-ik(|δO|-|δS|+|rS-rO|)]}.
×(r-rO)×(r-rS)|(r-rO)×(r-rS)|2=0,
L½(|δO|+|δS|-|δO-δS|)=|δO×δS|2(|δO|+|δS|+|δO-δS|)(|δO||δS|-δO·δS).
DK+(r)=δO×δS16π2|δO||δS|exp[ik(|δO|+|δS|)](|δO||δS|-δO·δS)1-4ik|δO||δS|sinc(kL)exp(-ikL)(|δO|+|δS|+|δO-δS|),
M½(|δO|+|δO-δS|-|δS|)=|δO×δS|2(|δO|+|δS|+|δO-δS|)[|δO||δO-δS|-δO·(δO-δS)],
D¯K-S(r)=δO×δS16π2|δO||δS|exp[ik(|δO|-|δS|)](|δO||δS|+δO·δS)1-4ik|δO||δS|(|δO||δS|+δO·δS)sinc(kM)exp(-ikM)(|δO|+|δS|+|δO-δS|)[|δO||δO-δS|-δO·(δO-δS)].
D¯K-O(r)=δO×δS16π2|δO||δS|exp[ik(|δO|-|δS|)](|δO||δS|+δO·δS)1+4ik|δO||δS|(|δO||δS|+δO·δS)sinc(kN)exp(ikN)(|δO|+|δS|+|δO-δS|)[|δS||δO-δS|+δS·(δO-δS)],
N½(|δS|+|δO-δS|-|δO|)=|δO×δS|2(|δO|+|δS|+|δO-δS|)[|δS||δO-δS|+δS·(δO-δS)].
DK(r)=δO×u4π|δO|exp[ik(|δO|+u·r)](|δO|-δO·u)×[1-2ik|δO|sinc(kL)exp(-ikL)].
uRSα(r)=uK(rO)+(-1)αuK(rO*),
D¯K+(r)=δO*×δS16π2|δO*||δS|exp[ik(|δO*|+|δS|)](|δO*||δS|+δO*·δS),
DK-(r)=δO*×δS16π2|δO*||δS|exp[ik(|δO*|-|δS|)](|δO*||δS|-δO*·δS).
D¯K+(r)=δO*×u4π|δO*|exp[ik(|δO*|+u·r)](|δO*|+δO*·u).
PK+(r)=δO×δS4π|δO||δS||δO-δS|exp(ik2L)(|δO||δS|-δO·δS)×1-4ik|δO||δS|sinc(kL)exp(-ikL)(|δO|+|δS|+|δO-δS|).
uK(rO)=exp(ik|rO-rS|)4π|rO-rS|rimPK+(r)·dr.
P¯K+(r)=δO*×δS4π|δO*||δS||δO-δS|exp(ik2L)(|δO*||δS|+δO*·δS).
D(r)=-(1-σ) δO×δS16π2×01 ik exp[ik(a2-bτ)/S(τ)]S3(τ)τdτ,

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