Abstract

A new method is proposed for modeling wave propagation in two-dimensional, smoothly varying inhomogeneous media. This method is valid in the limit of small wavelength and constructs the wave propagator by combining contributions from all relevant rays. The form of each contribution was reported in J. Opt. Soc. Am. A 14, 1279 (1997). The resulting estimate for the total field takes a strikingly simple form, which associates a Gaussian field with each ray. This form is asymptotically correct and free of the usual difficulties associated with caustics.

© 1998 Optical Society of America

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References

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  1. The assumption that the field is forward propagating is essential to the definition of the propagator. A discussion of forward-propagating fields is given in M. A. Alonso, G. W. Forbes, “Semigeometrical estimation of Green’s functions and wave propagators in optics,” J. Opt. Soc. Am. A 14, 1076–1086 (1997).
    [CrossRef]
  2. M. A. Alonso, G. W. Forbes, “Uniform asymptotic expansions for wave propagators via fractional transformations,” J. Opt. Soc. Am. A 14, 1279–1292 (1997).
    [CrossRef]
  3. A. W. Conway, J. L. Synge, eds., The Mathematical Papers of Sir William Rowan Hamilton (Cambridge U. Press, Cambridge, 1931), Vol. 1.
  4. H. A. Buchdahl, An Introduction to Hamiltonian Optics (Dover, New York, 1993), p. 8.
  5. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1989), p. 133.
  6. A. Walther, “Lenses, wave optics and eikonal functions,” J. Opt. Soc. Am. 59, 1325–1333 (1969).
    [CrossRef]
  7. A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, Cambridge, 1995), pp. 169–187.
  8. 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]
  9. Y. A. Kravtsov, Y. A. Orlov, Caustics, Catastrophes and Wave Fields (Springer-Verlag, Berlin, 1993), p. 18.
  10. The definition of the mixed characteristic varies from one author to the next. The one used here is consistent with the one given by R. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), pp. 111–115.
  11. R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1962), Vol. II, pp. 32–39.
  12. An analogous expression for three-dimensional propagation is given in Refs. 6 and 7. A rigorous derivation of a more general result is given in Eq. (5.21) of Ref. 1.
  13. A momentum caustic occurs when the rays in the bundle are locally parallel. Momentum caustics carry a sign: They are positive when the bundle goes from being divergent to convergent and negative when the converse is true. The net sum of momentum caustics is given by the difference between the number of positive ones and the number of negative ones.
  14. V. P. Maslov, M. V. Fedoriuk, Semiclassical Approximation in Quantum Mechanics (Reidel, Boston, Mass., 1981).
  15. J. B. Delos, “Semiclassical calculation of quantum mechanical wavefunctions,” in Advances in Chemical Physics, Vol. 65 of Advances in Chemical Physics, I. Prigogine, S. A. Rice, eds. (Wiley/Interscience, New York, 1986), pp. 161–213.
  16. See Chap. 6 of Ref. 9.
  17. M. A. Alonso, G. W. Forbes, “Fractional Legendre transformation,” J. Phys. A 28, 5509–5527 (1995).
    [CrossRef]
  18. E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).
    [CrossRef] [PubMed]
  19. For the purposes of the asymptotics considered here, a factor of k is included in the exponent of the kernel of the FrFT (with the corresponding factor of k in the amplitude for normalization). Comprehensive bibliographies on this subject are given in A. W. Lohmann, D. Mendlovic, Z. Zalevsky, eds., The Fractional Fourier Transform–Status Report 1995 (Tel Aviv U. Press, Tel Aviv, 1995); “Fractional Fourier transform and its applications in optics and signal processing—a bibliography,” http://www.ee.bilkent.edu.tr/∼haldun/ffbiblio/ffbiblio.html .
  20. See Eq. (5.21) of Ref. 1. In that reference the initial representation was also allowed to be fractional, but the initial parameter ϕ is set to zero here. The fractional propagator and characteristic, referred to here as Kθ(x, y; x′, ρ′) and Vθ(x, y; x′, ρ′), correspond, respectively, to Q(x, ρ, 0;x′, ρ′, θ) and F(x, ρ, 0; x′, ρ′, θ) of Ref. 1.
  21. Let y′ and p′ be the parametric position and momentum at x′ of a ray. The relative change of y′ as x′ increases is given by the ratio of the components of the normal vector, that is, ∂y′/∂x′=αy′/αx′=p′/[n2(x′, y′)-p′2]1/2. Notice that for a vertical phase-space curve segment, y′ is constant. Because ∂y′/∂x′ is evidently a monotonically increasing function of p′, the points with higher momentum have greater displacement as x′ increases, and this corresponds to a clockwise rotation through the vertical in phase space.
  22. A Gaussian of different width or even a different window function could be chosen here. However, it turns out that this particular choice greatly simplifies the results that follow.
  23. Given that, from Eq. (4.3), ϑ is the angle from the y′ axis to the local tangent of the phase-space curve, Eq. (3.10) states that |ρs′-ρc′|max=2|R|.
  24. This theorem states that phase space area (or étendue) is conserved during propagation; i.e., ∂y′/∂y ∂p′/∂p-∂p′/∂y ∂y′/∂p=1. See, for example, W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, San Diego, Calif., 1989), pp. 225–226.
  25. M. Alonso, G. Forbes, “Asymptotic estimation of the optical wave propagator. II. Relative validity,” J. Opt. Soc. Am. A 15, 1341–1354 (1998).
    [CrossRef]
  26. A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956), p. 51.

1998 (1)

1997 (2)

1995 (1)

M. A. Alonso, G. W. Forbes, “Fractional Legendre transformation,” J. Phys. A 28, 5509–5527 (1995).
[CrossRef]

1969 (1)

1937 (1)

E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).
[CrossRef] [PubMed]

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]

Alonso, M.

Alonso, M. A.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1989), p. 133.

Buchdahl, H. A.

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Dover, New York, 1993), p. 8.

Condon, E. U.

E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).
[CrossRef] [PubMed]

Courant, R.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1962), Vol. II, pp. 32–39.

Delos, J. B.

J. B. Delos, “Semiclassical calculation of quantum mechanical wavefunctions,” in Advances in Chemical Physics, Vol. 65 of Advances in Chemical Physics, I. Prigogine, S. A. Rice, eds. (Wiley/Interscience, New York, 1986), pp. 161–213.

Erdélyi, A.

A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956), p. 51.

Fedoriuk, M. V.

V. P. Maslov, M. V. Fedoriuk, Semiclassical Approximation in Quantum Mechanics (Reidel, Boston, Mass., 1981).

Forbes, G.

Forbes, G. W.

Hilbert, D.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1962), Vol. II, pp. 32–39.

Kravtsov, Y. A.

Y. A. Kravtsov, Y. A. Orlov, Caustics, Catastrophes and Wave Fields (Springer-Verlag, Berlin, 1993), p. 18.

Luneburg, R.

The definition of the mixed characteristic varies from one author to the next. The one used here is consistent with the one given by R. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), pp. 111–115.

Maslov, V. P.

V. P. Maslov, M. V. Fedoriuk, Semiclassical Approximation in Quantum Mechanics (Reidel, Boston, Mass., 1981).

Orlov, Y. A.

Y. A. Kravtsov, Y. A. Orlov, Caustics, Catastrophes and Wave Fields (Springer-Verlag, Berlin, 1993), p. 18.

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.

Welford, W. T.

This theorem states that phase space area (or étendue) is conserved during propagation; i.e., ∂y′/∂y ∂p′/∂p-∂p′/∂y ∂y′/∂p=1. See, for example, W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, San Diego, Calif., 1989), pp. 225–226.

Winston, R.

This theorem states that phase space area (or étendue) is conserved during propagation; i.e., ∂y′/∂y ∂p′/∂p-∂p′/∂y ∂y′/∂p=1. See, for example, W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, San Diego, Calif., 1989), pp. 225–226.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1989), p. 133.

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

J. Phys. A (1)

M. A. Alonso, G. W. Forbes, “Fractional Legendre transformation,” J. Phys. A 28, 5509–5527 (1995).
[CrossRef]

Proc. Natl. Acad. Sci. USA (2)

E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).
[CrossRef] [PubMed]

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 (19)

Y. A. Kravtsov, Y. A. Orlov, Caustics, Catastrophes and Wave Fields (Springer-Verlag, Berlin, 1993), p. 18.

The definition of the mixed characteristic varies from one author to the next. The one used here is consistent with the one given by R. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), pp. 111–115.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1962), Vol. II, pp. 32–39.

An analogous expression for three-dimensional propagation is given in Refs. 6 and 7. A rigorous derivation of a more general result is given in Eq. (5.21) of Ref. 1.

A momentum caustic occurs when the rays in the bundle are locally parallel. Momentum caustics carry a sign: They are positive when the bundle goes from being divergent to convergent and negative when the converse is true. The net sum of momentum caustics is given by the difference between the number of positive ones and the number of negative ones.

V. P. Maslov, M. V. Fedoriuk, Semiclassical Approximation in Quantum Mechanics (Reidel, Boston, Mass., 1981).

J. B. Delos, “Semiclassical calculation of quantum mechanical wavefunctions,” in Advances in Chemical Physics, Vol. 65 of Advances in Chemical Physics, I. Prigogine, S. A. Rice, eds. (Wiley/Interscience, New York, 1986), pp. 161–213.

See Chap. 6 of Ref. 9.

For the purposes of the asymptotics considered here, a factor of k is included in the exponent of the kernel of the FrFT (with the corresponding factor of k in the amplitude for normalization). Comprehensive bibliographies on this subject are given in A. W. Lohmann, D. Mendlovic, Z. Zalevsky, eds., The Fractional Fourier Transform–Status Report 1995 (Tel Aviv U. Press, Tel Aviv, 1995); “Fractional Fourier transform and its applications in optics and signal processing—a bibliography,” http://www.ee.bilkent.edu.tr/∼haldun/ffbiblio/ffbiblio.html .

See Eq. (5.21) of Ref. 1. In that reference the initial representation was also allowed to be fractional, but the initial parameter ϕ is set to zero here. The fractional propagator and characteristic, referred to here as Kθ(x, y; x′, ρ′) and Vθ(x, y; x′, ρ′), correspond, respectively, to Q(x, ρ, 0;x′, ρ′, θ) and F(x, ρ, 0; x′, ρ′, θ) of Ref. 1.

Let y′ and p′ be the parametric position and momentum at x′ of a ray. The relative change of y′ as x′ increases is given by the ratio of the components of the normal vector, that is, ∂y′/∂x′=αy′/αx′=p′/[n2(x′, y′)-p′2]1/2. Notice that for a vertical phase-space curve segment, y′ is constant. Because ∂y′/∂x′ is evidently a monotonically increasing function of p′, the points with higher momentum have greater displacement as x′ increases, and this corresponds to a clockwise rotation through the vertical in phase space.

A Gaussian of different width or even a different window function could be chosen here. However, it turns out that this particular choice greatly simplifies the results that follow.

Given that, from Eq. (4.3), ϑ is the angle from the y′ axis to the local tangent of the phase-space curve, Eq. (3.10) states that |ρs′-ρc′|max=2|R|.

This theorem states that phase space area (or étendue) is conserved during propagation; i.e., ∂y′/∂y ∂p′/∂p-∂p′/∂y ∂y′/∂p=1. See, for example, W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, San Diego, Calif., 1989), pp. 225–226.

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

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Dover, New York, 1993), p. 8.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1989), p. 133.

A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956), p. 51.

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

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Figures (6)

Fig. 1
Fig. 1

Definition of V(r; r) as the optical length of a ray that joins points r and r. The momenta p and p are defined as the products of the y components of the unit tangents at r and r (namely, α and α) and the local refractive indices.

Fig. 2
Fig. 2

Phase-space diagram at x of the rays from (x, y), where each ray is characterized by its position and momentum. Also shown are the point and angle caustics, indicated by filled and open circles, respectively, and the cuts (including overlaps) proposed by Maslov.

Fig. 3
Fig. 3

Phase-space diagram described by the plot of lVθ/ρ versus ρ in a rotated reference frame. The filled circles indicate the caustics for the θ representation.

Fig. 4
Fig. 4

Phase-space diagram at x for the rays associated with parameter values t to t+dt. Φ is the (necessarily negative) angle that this infinitesimal segment (or, more precisely, its tangent at t) has rotated away from its initially vertical position during propagation. Notice that, since the variation of γ is due in part to the variation of ϑ, the coordinate for the ray labeled t+dt in the ϑ(t) representation takes the form given in the figure.

Fig. 5
Fig. 5

A ray and its associated field, as assigned by Eq. (5.10). This ray is propagating in a waveguidelike index distribution. Here l is assumed to be real and independent of the distance of propagation. Notice that, at any given value of x, the transverse spacing of the wave fronts—curves of constant L+p(y-y)—is uniform and is determined solely by the ray slope and the value of the refractive index at the ray intercept. The resulting wave fronts are shown with uniformly spaced contours starting at L+p(y-y)=0. As l is varied, only the width of the Gaussian envelope varies; the wave fronts do not change.

Fig. 6
Fig. 6

Saddle points occur at tj and tj+1 when the propagator is evaluated at ρ in the θ representation. (Although Φ mod π is shown as a positive angle, recall that Φ is always negative and typically larger than π in modulus.) At tj, |arctan{tan[θ-Φ(tj)]}|R(tj)/s(tj) is an estimate for |tj+1-tj|/2, where arctan lies inside the interval (-π/2, π/2] and s(t) is the local speed of the parameterization.

Equations (123)

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[2+n2(r)k2]U(r)=0,
U(x, y)=U(x, y)K(x, y; x, y)dy,
K(r; r)=k2π φ2Vyy(r; r)1/2+O(k-1)×exp-i π4exp[ikV(r; r)],
φ=-Vx(r; r)Vx(r; r)
V(r; r)=-n(r)α,V(r; r)=n(r)α,
p=-Vy(r; r),
p=Vy(r; r).
K(r; r)=jk2π φ(j)2V(j)yy(r; r)1/2+O(k-1)×exp-i π212+M(j)exp[ikV(j)(r; r)],
y=-Wp(x, y; x, p),
W(x, y; x, p)=V(x, y; x, Y)-Yp,
Vy[x, y; x,Y(x, y; x, p)]-p=0.
K(x, y; x, y)=k2π K˜(x, y; x, p)×exp(ikyp)dp,
K˜(x, y; x, p)
=jk2π φ(j)2W(j)yp(x, y; x, p)1/2+O(k-1)×exp-i π2M˜(j)exp[ikW(j)(x, y; x, p)],
φ(j)=-W(j)x(x, y; x, p)W(j)x(x, y; x, p).
Vθ(x, y; x, ρ)=V(x, y; x, Y)-Yρl sin θ+(Y2+ρ2)2l tan θ,
Vy(x, y; x, Y)-ρl sin θ+Yl tan θ=0.
Vθρ=-Yl sin θ+ρl tan θ.
ρ=l Vysin θ+Y cos θ,
l Vθρ=l Vycos θ-Y sin θ.
V0(x, y; x, ρ)V(x, y; x, ρ),
Vπ/2(x, y; x, ρ)W(x, y; x, ρ/l).
Kθ(x, y; x, ρ)K(x, y; x, y)Γθ(y, ρ)dy,
Γθ(y, ρ)k2πl|sin θ|1/2 exp-iπ4-θ mod π2
×exp-ikyρl sin θ-y2+ρ22l tan θ.
K0(x, y; x, ρ)K(x, y; x, ρ),
Kπ/2(x, y; x, ρ)1lK˜(x, y; x, ρ/l).
Γθ(y, ρ)Γϕ(ρ, σ)dρ=Γθ+ϕ(y, σ).
K(x, y; x, y)=Kθ(x, y; x, ρ)Γ-θ(ρ, y)dρ.
Kθ(x, y; x, ρ)
=jk2π φ(j)2Vθ(j)yρ(x, y; x, ρ)1/2+O(k-1)×expi2θ-π2-πMθ(j)×exp[ikVθ(j)(x, y; x, ρ)],
φ(j)=-Vθ(j)x(x, y; x, ρ)Vθ(j)x(x, y; x, ρ)
Mθθ-Φπ,
tan θtan Φ.
Kϑ(t)A(x, y; x, ρ)
=kφ2π2Vϑ(t)yρ(x, y; x, ρ)1/2 expiϑ(t)2-π4×exp[ikVϑ(t)(x, y; x, ρ)],
φ=-Vϑ(t)x(x, y; x, ρ)Vϑ(t)x(x, y; x, ρ),
Φ(t)<ϑ(t)<Φ(t)+π.
γ(t)=y(t)cos ϑ(t)+lp(t)sin ϑ(t),
ρ=y(t)cos θ+lp(t)sin θ=γ(t)cos[θ-ϑ(t)]+l Vϑ(t)ρ[x, y; x,γ(t)]×sin[θ-ϑ(t)].
CθI(t, ρ)dtKϑ(t)A[x, y; x,γ(t)]Γθ-ϑ(t)×[γ(t), ρ]w(t)dt,
w(t)=γ˙(t)-lϑ˙(t) Vϑ(t)ρ[x, y; x,γ(t)],
KθI(x, y; x, ρ)=CθI(t, ρ)dt=Kϑ(t)A[x, y; x,γ(t)]×Γθ-ϑ(t)[γ(t), ρ]w(t)dt.
ρc(t, θ)=γ(t)cos[θ-ϑ(t)]+l Vϑ(t)ρ[x, y; x,γ(t)]×sin[θ-ϑ(t)],
ρs(t, θ)=γ(t)cos[θ-ϑ(t)]+2 γ˙(t)ϑ˙(t)-l Vϑ(t)ρ[x, y; x,γ(t)]sin[θ-ϑ(t)].
|ρs-ρc|max=2γ˙(t)ϑ˙(t)-l Vϑ(t)ρ[x, y; x,γ(t)]=2 w(t)|ϑ˙(t)|.
Cϑ(t)+π/2II(t, ρ)
Cϑ(t)+π/2I(t, ρ)exp-k2l{ρ-ρc[t, ϑ(t)+π/2]}2.
CθII(t, ρ)=Cϑ(t)+π/2II(t, σ)Γθ-ϑ(t)-π/2(σ, ρ)dσ=Kϑ(t)A[x, y; x,γ(t)]Gθ-ϑ(t){γ(t),ρc[t, ϑ(t)+π/2], ρ}w(t),
Gϕ(γ, ρc, ρ)
Γπ/2(γ, σ)exp-k(σ-ρc)22lΓϕ-π/2(σ, ρ)dσ.
Gϕ(γ, ρc, ρ)
=k2πl exp-k2l(ρ-γ cos ϕ-ρc sin ϕ)2×exp-ik2l[(γ2+ρc2)sin ϕ cos ϕ+2(ρc cos ϕ-γ sin ϕ)(γ cos ϕ-ρ)].
KθII(x, y; x, ρ)
=CθII(t, ρ)dt=Kϑ(t)A[x, y; x, γ(t)]×Gθ-ϑ(t)γ(t), l Vϑ(t)ρ[x, y; x, γ(t)], ρw(t)dt,
KθII(x, y; x, ρ)Γϕ(ρ, σ)dρ
=KϑA(x, y; x, γ)Gθ-ϑγ, l Vϑρ, ρΓϕ(ρ, σ)×w(t)dρdt
=KϑA(x, y; x, γ)Gθ+ϕ-ϑγ, l Vϑρ, σw(t)dt=Kθ+ϕII(x, y; x, σ).
KθII(x, y; x, ρ)
=jk2π φ(j)2Vθ(j)yρ(x, y; x, ρ)1/2+O(k-1)×1+l 2Vϑ(tj)ρ2[x, y; x,γ(tj)]2-1/4×expi2ϑ(tj)-Ω(j)-π2×exp[ikVθ(j)(x, y; x, ρ)],
Ω(j)arctanl 2Vϑ(tj)ρ2[x, y; x, γ(tj)]cos[θ-ϑ(tj)]-sin[θ-ϑ(tj)]l 2Vϑ(tj)ρ2[x, y; x, γ(tj)]sin[θ-ϑ(tj)]+cos[θ-ϑ(tj)].
2Vϑ(t)ρ2[x, y; x,γ(t)]0.
ϑ(t)=Φ(t)+(π/2).
ϑ-Ω-π2=Φ+arctantanθ-Φ-π2=θ-π2-πθ-Φπ=θ-π2-πMθ.
[θ-ϑ(t)]mod π-π2R(t)lk |cos[θ-ϑ(t)]|-1/2,
R(t)l/kforallt.
KII(x, y; x, y)=k2πγ˙ϑ˙lVϑρφl2Vϑyρ1/2expiϑ2π4×expk2lyγ cos ϑ+lVϑρsin ϑ2×expikVϑ+12lγ2+lVϑρ2×sin ϑ cos ϑ2lVϑρcos ϑ+γ sin ϑ×(γ cos ϑy)dt, 
p(x, y, p; x)=Lp(x, y, p; x)yp(x, y, p; x).
tan ϑ=l 2Vy2(x, y; x, y)=l ppyp,
ϑ=arctanl ppyp-M(p)π,
γ=y cos ϑ+lp sin ϑ,
l Vϑρ=lp cos ϑ-y sin ϑ.
γ˙-ϑ˙ Vϑρ=ypcos ϑ+l ppsin ϑ=yp2+l pp21/2.
2Vϑyρ=cos ϑpy-1lyytan ϑ=cos ϑyp-1yypp-pyyp=cos ϑyp-1=yp2+l pp2-1/2.
Vϑ(x, y; x, γ)
=V(x, y; x, y)-yγl sin ϑ+y2+γ22l tan ϑ=L-py sin2 ϑ+(l2p2-y2)2lsin ϑ cos ϑ.
KII(x, y; x, y)
=k2πexp-i π4φ1/2pp2+1lyp21/4×expiϑ2exp-k2l(y-y)2×exp{ik[L+p(y-y)]}dp,
φ=n2(x, y)-p2n2(x, y)-p21/2.
KII(x, y; x, y)
=k2πexp-i π4φ1/21lyp+i pp1/2×exp-k2l(y-y)2×exp{ik[L+p(y-y)]}dp.
l3yp2+l pp2-3yp2pp2-pp2yp22k
forallp.
K˜(x, y; x, p)
=k2π K(x, y; x, y)exp(-ikyp)dy=k2πexp-i π4φ1/2yp+il pp1/2×exp-kl2(p-p)2exp[ik(L-yp)]dp.
A(t)exp[ikΩ(t)]dt
2πk mA(t˜m)|Ω¨(t˜m)|expiπ4sgn[Ω¨(t˜m)]×exp[ikΩ(t˜m)],
Ω(t)=Vϑ(t)[x, y; x, γ(t)]-ργ(t)l sin[θ-ϑ(t)]+ρ2+γ2(t)2l tan[θ-ϑ(t)],
Ω˙(t)=Vϑρ+γ cos(θ-ϑ)-ρl sin(θ-ϑ)γ˙+Vϑθ+ρ2+γ2-2ργ cos(θ-ϑ)2l sin2(θ-ϑ)ϑ˙.
Vθθ(x, y; x, ρ)
=-12ll Vθρ(x, y; x, ρ)2+ρ2,
Ω˙(t)=Vϑρ+γ cos(θ-ϑ)-ρl sin(θ-ϑ)γ˙-lϑ˙2Vϑρ-γ cos(θ-ϑ)-ρl sin(θ-ϑ).
Vθ(x, y; x, ρ)=Vϑ(x, y; x, P)-Pρl sin(θ-ϑ)+P2+ρ22l tan(θ-ϑ),
Vϑρ(x, y; x, P)=ρ-P cos(θ-ϑ)l sin(θ-ϑ).
Vθx(x, y; x, ρ)=Vϑx(x, y; x, P),
Vθy(x, y; x, ρ)=Vϑy(x, y; x, P),
Vθx(x, y; x, ρ)=Vϑx(x, y; x, P),
Vθρ(x, y; x, ρ)=ρ cos(θ-ϑ)-Pl sin(θ-ϑ).
Vϑρ(x, y; x, P)
=ρlsin(θ-ϑ)+Vθρ(x, y; x, ρ)cos(θ-ϑ).
2Vϑyρ(x, y; x, P)
=2Vθyρ(x, y; x, ρ)2Vϑρ2(x, y; x, P)
×l sin(θ-ϑ)+cos(θ-ϑ)],
KθII(x, y; x, ρ)=k2πlA(t)exp-klΨ(t)dt,
Aφ2Vϑyρ(x, y; x, γ)1/2 expiϑ2-π4w,
Ψ12ρ-γ cos(θ-ϑ)-l Vϑρ(x, y; x, γ)×sin(θ-ϑ)2-ilVϑ(x, y; x, γ)-12γ2+l Vϑρ(x, y; x, γ)2×sin(θ-ϑ)cos(θ-ϑ)-l Vϑρ(x, y; x, γ)cos(θ-ϑ)-γ sin(θ-ϑ)[γ cos(θ-ϑ)-ρ].
KθII(x, y; x, ρ)=k2π j{A(tj)[Ψ¨(tj)]-1/2+O(k-1)}exp-klΨ(tj),
|tj+1-tj|kl|Ψ¨(tj)|-1/2+kl|Ψ¨(tj+1)|-1/2
forallj.
Vϑθ(x, y; x, γ)
=-12ll Vϑρ(x, y; x, γ)2+γ2,
2Vϑρθ(x, y; x, γ)
=-l Vϑρ(x, y; x, γ) 2Vϑρ2(x, y; x, γ)-γl.
Ψ˙=-ρ-γ cos(θ-ϑ)-l Vϑρ(x, y; x, γ)×sin(θ-ϑ)γ˙-lϑ˙ Vϑρ(x, y; x, γ)×1+il 2Vϑρ2(x, y; x, γ)exp[-i(θ-ϑ)]=0.
Vϑ(tj)ρ[x, y; x, γ(tj)]-ρ-γ(tj)cos[θ-ϑ(tj)]l sin[θ-ϑ(tj)]=0.
Ψ(tj)=-ilVθ(j)(x, y; x, ρ),
A(tj)=w(tj)expiϑ2-π4φ(j)2Vθ(j)yρ(x, y; x, ρ)×2Vϑρ2(x, y; x, γ)l sin(θ-ϑ)+cos(θ-ϑ)1/2,
φ(j)=-Vθ(j)x(x, y; x, ρ)Vθ(j)x(x, y; x, ρ),
Ψ¨(tj)=γ˙-lϑ˙ Vϑρ(x, y; x, γ)2l 2Vϑρ2(x, y; x, γ)sin(θ-ϑ)+cos(θ-ϑ)2×1+i l 2Vϑρ2(x, y; x, γ)cos(θ-ϑ)-sin(θ-ϑ)l 2Vϑρ2(x, y; x, γ)sin(θ-ϑ)+cos(θ-ϑ)=γ˙-lϑ˙ Vϑρ(x, y; x, γ)2×l 2Vϑρ2(x, y; x, γ)sin(θ-ϑ)+cos(θ-ϑ)1+l 2Vϑρ2(x, y; x, γ)21/2 exp[iΩ(j)],
Ω(j)arctanl 2Vϑ(tj)ρ2[x, y; x,γ(tj)]cos[θ-ϑ(tj)]-sin[θ-ϑ(tj)]l 2Vϑ(tj)ρ2[x, y; x,γ(tj)]sin[θ-ϑ(tj)]+cos[θ-ϑ(tj)].
|arctan{tan[θ-Φ(tj)]}|R(tj)s(tj)kl|Ψ¨(tj)|-1/2
forallj,
s(t)w(t)1+l 2Vϑρ2(x, y; x, γ)21/2,

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