Abstract

Asymptotic expressions for both the Green's function and the wave propagator of scalar diffraction theory are derived in terms of the point characteristic of geometrical optics. These expressions are valid when the refractive index varies slowly on the scale of a wavelength. They are derived directly from the wave equation by using coincidence limits to select unique forms. The relationship between the Green's function and the propagator is discussed along with a careful treatment of their interpretation and validity.

© 1997 Optical Society of America

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References

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  1. A. Walther, “Lenses, wave optics and eikonal functions,” J. Opt. Soc. Am. 59, 1325–1333 (1969).
    [CrossRef]
  2. A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, Cambridge, 1995), pp. 169–187.
  3. J. H. Van Vleck, “The correspondence principle in the statistical interpretation of quantum mechanics,” Proc. Natl. Acad. Sci. USA 14, 178–188 (1928).
    [CrossRef] [PubMed]
  4. C. Morette, “On the definition and approximation of Feynman’s path integrals,” Phys. Rev. 81, 848–852 (1951).
    [CrossRef]
  5. J. B. Delos, “Semiclassical calculation of quantum mechanical wavefunctions,” in Advances in Chemical Physics, I. Prigogine, S. A. Rice, eds. (John Wiley Interscience, 1986), Vol. 67, pp. 161–213.
  6. M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer-Verlag, New York, 1990), pp. 184, 283.
  7. A. W. Conway, J. L. Synge, eds., The Mathematical Papers of Sir William Rowan Hamilton, (Cambridge, 1931), Vol. 1.
  8. H. A. Buchdahl, An Introduction to Hamiltonian Optics (Dover, New York, 1993).
  9. M. Born, E. Wolf, Principles of Optics, 6th (corrected) ed. (Pergamon, Oxford, 1989), pp. 128–130.
  10. This equation is also known as the Hamilton–Jacobi equation. See Ref. 8, p. 112.
  11. Consider, for example, f(r; r′)=1-a2(x-x′)+a[(y-y′)2+(z-z′)2]1/2, which satisfies the eikonal equation for free space in both the primed and the unprimed variables, but it does not correspond to the free-space point characteristic.
  12. An analogous condition is required for world functions in general relativity. See, for example, J. L. Synge, Relativity: The General Theory (North-Holland, Amsterdam, 1964), pp. 51–57.
  13. G. W. Forbes, “On variational problems in parametric form,” Am. J. Phys. 59, 1130–1140 (1991). See Eq. (72).
    [CrossRef]
  14. See J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1988), pp. 38–39.
  15. See Ref. 14, p. 36. Notice that this coincidence limit becomes invalid in the unlikely event that the field emanating from r is focused back onto this same point.
  16. See, for example, Y. A. Kravtsov, Y. A. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990), p. 7.
  17. See Ref. 16, pp. 22–26.
  18. If U is square integrable over any plane of constant x, then the operator Hˆ2 can be shown to be Hermitian, with use of a two-dimensional version of Green’s theorem. The operator Hˆ2 then has real eigenvalues hl2 (which may be discrete or continuous) associated with a complete orthogonal set of eigenfunctions. The operator Hˆ is then defined as an operator with the same eigenfunctions as Hˆ2, and with eigenvalues hl=hl2 for hl2>0 (associated with homogeneous eigenfunctions) and hl=i|hl2| for hl2<0 (associated with evanescent eigenfunctions). In the special case when n is independent of y and z, this means of determining HˆU reduces to a simple application of Fourier methods.
  19. Hˆ-1 is an operator with the same eigenfunctions as Ĥ and with eigenvalues hl-1. Any component of the eigenvector for hl=0 (which amounts to a field that propagates perpendicularly to the x axis at x=x0) is dropped in the calculation of Hˆ-1∂U/∂χ and can be propagated independently of the forward and backward components.
  20. The specification of U+ at x=x0 naturally complements the Sommerfeld radiation condition for U, as both prescribe the value of a linear combination of U and its normal derivative at the boundary of the half-space and serve to single out the component of the field that is propagating across the boundary with a particular sense. Such conditions represent a variation of the more standard Dirichlet and Neumann boundary conditions, in which a form is prescribed for either the field or its normal derivative, respectively.
  21. With an angular spectrum method, relation (4.8) can be shown to be the propagation kernel for a forward-propagating field in a homogeneous medium. This Rayleigh–Sommerfield diffraction formula can also be derived with Green’s theorem and can be found, for example, in J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 431. The standard expression is valid only for δx>0: even when |δ| is understood to mean δx[1+(δy2+δz2)/δx2]1/2, the evanescent component is not correctly accounted for when it comes to inverse propagation.
  22. See Ref. 16, p. 41.
  23. See Ref. 1, Eq. (23d) or Ref. 2, Eq. (18.21). Notice that a factor of -i is suppressed in these references.
  24. See, for example, Secs. 2.5.3 and 2.5.4 of Ref. 16.
  25. G. W. Forbes, Bryan D. Stone, “Restricted characteristic functions for general optical configurations,” J. Opt. Soc. Am. A 10, 1263–1269 (1993).
    [CrossRef]

1993 (1)

1991 (1)

G. W. Forbes, “On variational problems in parametric form,” Am. J. Phys. 59, 1130–1140 (1991). See Eq. (72).
[CrossRef]

1969 (1)

1951 (1)

C. Morette, “On the definition and approximation of Feynman’s path integrals,” Phys. Rev. 81, 848–852 (1951).
[CrossRef]

1928 (1)

J. H. Van Vleck, “The correspondence principle in the statistical interpretation of quantum mechanics,” Proc. Natl. Acad. Sci. USA 14, 178–188 (1928).
[CrossRef] [PubMed]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th (corrected) ed. (Pergamon, Oxford, 1989), pp. 128–130.

Buchdahl, H. A.

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Dover, New York, 1993).

Delos, J. B.

J. B. Delos, “Semiclassical calculation of quantum mechanical wavefunctions,” in Advances in Chemical Physics, I. Prigogine, S. A. Rice, eds. (John Wiley Interscience, 1986), Vol. 67, pp. 161–213.

Forbes, G. W.

G. W. Forbes, Bryan D. Stone, “Restricted characteristic functions for general optical configurations,” J. Opt. Soc. Am. A 10, 1263–1269 (1993).
[CrossRef]

G. W. Forbes, “On variational problems in parametric form,” Am. J. Phys. 59, 1130–1140 (1991). See Eq. (72).
[CrossRef]

Goodman, J. W.

See J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1988), pp. 38–39.

Gutzwiller, M. C.

M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer-Verlag, New York, 1990), pp. 184, 283.

Jackson, J. D.

With an angular spectrum method, relation (4.8) can be shown to be the propagation kernel for a forward-propagating field in a homogeneous medium. This Rayleigh–Sommerfield diffraction formula can also be derived with Green’s theorem and can be found, for example, in J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 431. The standard expression is valid only for δx>0: even when |δ| is understood to mean δx[1+(δy2+δz2)/δx2]1/2, the evanescent component is not correctly accounted for when it comes to inverse propagation.

Kravtsov, Y. A.

See, for example, Y. A. Kravtsov, Y. A. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990), p. 7.

Morette, C.

C. Morette, “On the definition and approximation of Feynman’s path integrals,” Phys. Rev. 81, 848–852 (1951).
[CrossRef]

Orlov, Y. A.

See, for example, Y. A. Kravtsov, Y. A. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990), p. 7.

Stone, Bryan D.

Synge, J. L.

An analogous condition is required for world functions in general relativity. See, for example, J. L. Synge, Relativity: The General Theory (North-Holland, Amsterdam, 1964), pp. 51–57.

Van Vleck, J. H.

J. H. Van Vleck, “The correspondence principle in the statistical interpretation of quantum mechanics,” Proc. Natl. Acad. Sci. USA 14, 178–188 (1928).
[CrossRef] [PubMed]

Walther, A.

A. Walther, “Lenses, wave optics and eikonal functions,” J. Opt. Soc. Am. 59, 1325–1333 (1969).
[CrossRef]

A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, Cambridge, 1995), pp. 169–187.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th (corrected) ed. (Pergamon, Oxford, 1989), pp. 128–130.

Am. J. Phys. (1)

G. W. Forbes, “On variational problems in parametric form,” Am. J. Phys. 59, 1130–1140 (1991). See Eq. (72).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Phys. Rev. (1)

C. Morette, “On the definition and approximation of Feynman’s path integrals,” Phys. Rev. 81, 848–852 (1951).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

J. H. Van Vleck, “The correspondence principle in the statistical interpretation of quantum mechanics,” Proc. Natl. Acad. Sci. USA 14, 178–188 (1928).
[CrossRef] [PubMed]

Other (20)

A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, Cambridge, 1995), pp. 169–187.

J. B. Delos, “Semiclassical calculation of quantum mechanical wavefunctions,” in Advances in Chemical Physics, I. Prigogine, S. A. Rice, eds. (John Wiley Interscience, 1986), Vol. 67, pp. 161–213.

M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer-Verlag, New York, 1990), pp. 184, 283.

A. W. Conway, J. L. Synge, eds., The Mathematical Papers of Sir William Rowan Hamilton, (Cambridge, 1931), Vol. 1.

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Dover, New York, 1993).

M. Born, E. Wolf, Principles of Optics, 6th (corrected) ed. (Pergamon, Oxford, 1989), pp. 128–130.

This equation is also known as the Hamilton–Jacobi equation. See Ref. 8, p. 112.

Consider, for example, f(r; r′)=1-a2(x-x′)+a[(y-y′)2+(z-z′)2]1/2, which satisfies the eikonal equation for free space in both the primed and the unprimed variables, but it does not correspond to the free-space point characteristic.

An analogous condition is required for world functions in general relativity. See, for example, J. L. Synge, Relativity: The General Theory (North-Holland, Amsterdam, 1964), pp. 51–57.

See J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1988), pp. 38–39.

See Ref. 14, p. 36. Notice that this coincidence limit becomes invalid in the unlikely event that the field emanating from r is focused back onto this same point.

See, for example, Y. A. Kravtsov, Y. A. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990), p. 7.

See Ref. 16, pp. 22–26.

If U is square integrable over any plane of constant x, then the operator Hˆ2 can be shown to be Hermitian, with use of a two-dimensional version of Green’s theorem. The operator Hˆ2 then has real eigenvalues hl2 (which may be discrete or continuous) associated with a complete orthogonal set of eigenfunctions. The operator Hˆ is then defined as an operator with the same eigenfunctions as Hˆ2, and with eigenvalues hl=hl2 for hl2>0 (associated with homogeneous eigenfunctions) and hl=i|hl2| for hl2<0 (associated with evanescent eigenfunctions). In the special case when n is independent of y and z, this means of determining HˆU reduces to a simple application of Fourier methods.

Hˆ-1 is an operator with the same eigenfunctions as Ĥ and with eigenvalues hl-1. Any component of the eigenvector for hl=0 (which amounts to a field that propagates perpendicularly to the x axis at x=x0) is dropped in the calculation of Hˆ-1∂U/∂χ and can be propagated independently of the forward and backward components.

The specification of U+ at x=x0 naturally complements the Sommerfeld radiation condition for U, as both prescribe the value of a linear combination of U and its normal derivative at the boundary of the half-space and serve to single out the component of the field that is propagating across the boundary with a particular sense. Such conditions represent a variation of the more standard Dirichlet and Neumann boundary conditions, in which a form is prescribed for either the field or its normal derivative, respectively.

With an angular spectrum method, relation (4.8) can be shown to be the propagation kernel for a forward-propagating field in a homogeneous medium. This Rayleigh–Sommerfield diffraction formula can also be derived with Green’s theorem and can be found, for example, in J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 431. The standard expression is valid only for δx>0: even when |δ| is understood to mean δx[1+(δy2+δz2)/δx2]1/2, the evanescent component is not correctly accounted for when it comes to inverse propagation.

See Ref. 16, p. 41.

See Ref. 1, Eq. (23d) or Ref. 2, Eq. (18.21). Notice that a factor of -i is suppressed in these references.

See, for example, Secs. 2.5.3 and 2.5.4 of Ref. 16.

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Figures (5)

Fig. 1
Fig. 1

When the rays from r intersect and form a caustic, there may be more than one stationary path connecting r and r, as shown.

Fig. 2
Fig. 2

Unit tangents α and α at the end points r and r of a stationary path of optical length V(r; r).

Fig. 3
Fig. 3

Volume R, defined as the region of space occupied by a thin bundle of rays emanating from r, between the planes (r -r-δ)·u=0 and (r-r)·u=0. The areas of the intersections of these planes with the bundle are denoted da and da, respectively.

Fig. 4
Fig. 4

In the presence of a caustic, the ray tube shown in Fig. 3 becomes flattened as represented here.

Fig. 5
Fig. 5

Definition of the function Vr(r; r) as the optical length of a ray between r and r, being specularly reflected by the plane x=x0.

Equations (92)

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V(r; r)=-n(r)α,
V(r; r)=n(r)α,
[V(r; r)]2=n2(r),
[V(r; r)]2=n2(r).
f(r; r+δ)n(r)|δ|.
|r˙(t)|=n[r(t)],
r˙(t)=f[r; r(t)].
r¨(t)=[r˙(t)·]f[r; r(t)]={f[r; r(t)]·}f[r; r(t)]=12{f[r; r(t)]·f[r; r(t)]}=n[r(t)]n[r(t)],
f(r;r+δ)=n(r)|δ|[1+O(δ)],
[2+n2(r)k2]U(r)=-ρ(r),
[2+n2(r)k2]G(r; r)=-δ(r-r).
limr rGr(r; r)-ikn(r)G(r; r)=0.
U(r)=ρ(r)G(r; r)d3r,
[2+n2(r)k2]G(r; r)=0.
G(r; r+δ)14π|δ|exp[ikn(r)|δ|].
G(r; r)=A(r; r)exp[ikΘ(r; r)],
k2{n2(r)-[Θ(r; r)]2}A(r; r)+ik{2A(r; r)·Θ(r; r)+A(r; r)2Θ(r; r)}
+2A(r; r)=0.
A(r; r)=l=01(ik)lAl(r; r).
[Θ(r; r)]2-n2(r)=0,
2A0(r; r)·Θ(r; r)+A0(r; r)2Θ(r; r)=0,
2Al(r; r)·Θ(r; r)+Al(r; r)2Θ(r; r)
+2Al-1(r; r)=0,l1.
Θ(r;r+δ)=n(r)|δ|[1+O(δ)].
A0(r; r+δ)=1+O(δ2)4π|δ|.
·[A02(r; r)V(r; r)]=0,
0=R·[A02(r; r)V(r; r)]d3r=daA02(r; r)[u·V(r; r)]+daA02(r; r+δ)[u·V(r; r+δ)].
G(r; r)=14πu·F·u(u·V)(u·V)+O(k-1)exp[ikV(r; r)],
G(r; r)=14πn(r)n(r)[V·F·V+O(k-1)]exp[ikV(r; r)].
G(r; r)
=14π-VxVx-12Vyy2Vzz-2Vyz2Vzy+O(k-1)exp[ikV(r; r)].
[2+n2(r)k2]U(r)=0.
xikHˆU±(r)=0,
x±ikHˆU(r)=x±ikHˆU±(r)=±2ikHˆU±(r).
U±(r)=12U(r)±ikHˆ-1Ux.
U+(r)= U+(r)K(r; r)dydz,
U(r)= U+(r)K(r; r)dydz,
[2+n2(r)k2]K(r; r)=0.
K(r; r+δ)-ikn(r)2πδx|δ|21+ikn(r)|δ|exp[ikn(r)|δ|],δx>00,otherwise,
K(r; r)=ikl=01(ik)lCl(r; r)exp[ikV(r; r)],
2C0(r; r)·V(r; r)+C0(r; r)2V(r; r)=0,
2Cl(r; r)·V(r; r)+Cl(r; r)2V(r; r)
+2Cl-1(r; r)=0,l1.
C0(r; r+δ)=-n(r)2π|δ|2[δx+O(δ2)],δx>00,otherwise.
K(r; r)=ik2πVxu·F·u(u·V)(u·V)+O(k-1)exp[ikV(r; r)],Vx(r; r)<00,otherwise.
K(r; r)=-ik2π-Vx(r; r)Vx(r; r)2Vyy2Vzz-2Vyz2Vzy1/2+O(k-1)exp[ikV(r; r)].
D [G˜(r; r)2U(r)-U(r)2G˜(r; r)]d3r
=surface of D[G˜(r;r)U(r)-U(r)G˜(r; r)]·nda,
U(r)= U(x0, y, z)G˜x(x0, y, z; r)dydz.
Gr(r; r)
=14πVrxVrx-12Vryy2Vrzz-2Vryz2Vrzy+O(k-1)exp[ikVr(r; r)],
G˜(r; r)=[G(r; r)-Gr(r; r)][1+O(k-1)],
Vr[x0+(x-x0), y, z; r]=V[x0-(x-x0), y, z; r]+O[(x-x0)2].
Grx(x0, y, z; r)=-[1+O(k-1)]Gx(x0, y, z; r).
U(r)2U(x0, y, z)Gx(x0, y, z; r)dydz.
N(r, δ)=n(r+δ).
f(r; r+δ)=|δ|F(r, δ),
(2δ·δF+F)F+|δ|2|δF|2=N2,
Fi1im(m)=1m+1Ni1im(m)+fi1im(m)(N(1),,N(m-1)),
F(0)=N(0)=n(r),
Fi(1)=12Ni(1),
Fij(2)=13Nij(2)+124N(0)(Ni(1)Nj(1)-Nk(1)Nk(1)δij),
F(r, δ)=n+12N·δ+13δ··δ+124n[(N·δ)2-|N|2|δ|2]+O(δ3).
A0(r; r+δ)=B(r, δ)4π|δ|,
Fδ·δB+|δ|2(δF)·(δB)+12Bδ2F=0.
Bi(1)=0,
Bij(2)=-12Fkk(2)δij=1614n(r)NkNk-Nkkδij,
Bi1im(m)=-δi1i2m-12Fjji3im(m)+fi3im(m-2)(F(2),,F(m-1))=-δi1i2m-12(m+1)Njji3im(m)+gi3im(m-2)(N(2),,N(m-1)),
m3,
B(r, δ)=1+16|δ|214n|N|2-tr{}+O(δ),
A0(r; r+δ)=14π|δ|+|δ|E(r, δ),
A2l-1(r; r+δ)=B2l-1(r, δ),
A2l(r; r+δ)=|δ|B2l(r, δ),l1,
V˜(ρ; ρ)=V(A-1·ρ; A-1·ρ).
η=(η, μ, ν)=-1n(A-1·ρ)V˜ρ(ρ; ρ).
dΩ=dμdνη=daηδ(μ, ν)δ(σ, τ)=daη1n2(A-1·ρ)2V˜σσ2V˜ττ-2V˜στ2V˜τσ
=da2η1n2(A-1·ρ)1jk1mn2V˜ρjρm2V˜ρkρn,
dΩ=daη1n2(r)uiul12ijklmn2Vrjrm2Vrkrn=daη1n2(r)u·F·u,
da=1η|δ|2dΩ=da(u·V)2|δ|2u·F·u.
u·V(r; r+δ)=-u·V(r; r)+O(δ).
A0(r; r)=14πu·F·u(u·V)(u·V).
C0(r; r+δ)=-n(r)2π|δ|2D(r, δ),δx>00,otherwise,
F(δ·δD-D)+D12|δ|2δ2F-δ·δF
+|δ|2(δD)·(δF)=0.
D(0)=0,
Dij(2)=Di(1)Fj(1)-Dk(1)Fk(1)δij
=12(Di(1)Nj(1)-Dk(1)Nk(1)δij).
Di1im(m)=δ1im[Fi1im-1(m-1)-(m-2)δi1i2Fjji3im-1(m-1)]+δi1imF1i2im-1(m-1)+fi1im(m)(F(2),,F(m-2))=1mδ1im[Ni1im-1(m-1)-(m-2)δi1i2Njji3im-1(m-1)]+1mδi1imN1i2im-1(m-1)+gi1im(m)(N(2),,N(m-2)),m3,
D(r, δ)=δx+12[δx(N·δ)-Nx|δ|2]+O(δ3),
C0(r; r+δ)=-n(r)2π|δ|2[δx+O(δ2)],δx>00,otherwise.
Cl(r; r+δ)=|δ|-(l+2)Dl(r, δ),δx>00,otherwise,
C1(r; r+δ)=12π|δ|3[δx+O(δ2)],δx>00,otherwise.

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