Abstract

The validity of a new asymptotic method for propagating waves in two-dimensional, smoothly varying inhomogeneous media is compared with that of other standard methods. Simple geometric validity conditions are derived along with expressions for the maximum wavelength at which the modulus of the complex-valued field error can be expected to be no more than approximately a few percent, and then 20%, of the local peak field amplitude. It is shown that the limiting error in the Maslov method is generated by the process of switching between representations. The new method is predicted to be accurate to within a few percent for wavelengths that are one to two orders of magnitude larger than the corresponding cutoff for the Maslov method. This ratio exceeds two orders of magnitude for accuracy of approximately 20%. These predictions are confirmed by numerical investigation of a simple example.

© 1998 Optical Society of America

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References

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  1. M. A. Alonso, G. W. Forbes, “Asymptotic estimation of the optical wave propagator. I. Derivation of a new method,” J. Opt. Soc. Am. A 15, 1329–1340 (1998).
    [CrossRef]
  2. The assumption that the field is forward propagating is essential to the definition of a 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]
  3. See Sec. 5 of Ref. 1.
  4. A. W. Conway, J. L. Synge, eds., The Mathematical Papers of Sir William Rowan Hamilton (Cambridge U. Press, Cambridge, UK, 1931), Vol. 1.
  5. H. A. Buchdahl, An Introduction to Hamiltonian Optics (Dover, New York, 1993), p. 8.
  6. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 133.
  7. Y. A. Kravtsov, Y. A. Orlov, Caustics, Catastrophes and Wave Fields, (Springer-Verlag, Berlin, 1993), p. 18.
  8. 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. Calif. Press, Berkeley, Calif., 1964), pp. 111–115.
  9. R. Courant, D. Hilbert, Methods of Mathematical Physics, (Interscience, New York, 1962), Vol. 2, pp. 32–39.
  10. A differential ray bundle has a momentum caustic when the rays in the bundle become parallel. Momentum caustics carry a sign: They are positive when the rays go from being divergent to convergent and negative otherwise.
  11. 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]
  12. M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer-Verlag, New York, 1990), pp. 184, 283.
  13. A. Walther, “Lenses, wave optics and eikonal functions,” J. Opt. Soc. Am. 59, 1325–1333 (1969).
    [CrossRef]
  14. A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, Cambridge, UK1995), pp. 169–187.
  15. A rigorous derivation for the three-dimensional analog of Eq. (3.2) in the absence of caustics is given in the reference cited in Ref. 2. For the two-dimensional case, the form of each of the contributions given in Eqs. (3.2) and (3.8) are seen to be particular cases of a more general result derived in 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]
  16. It may be surprising that ∂2V/∂y∂y′ depends on b, which can be seen from Eq. (3.11) to be ∂3W/∂p′3 evaluated at the caustic. One can alternatively derive Eq. (3.15) by including the cross term A(y-y0)(p′-p0′) in Eq. (3.11) and performing a Legendre transformation to derive the expansion for V.
  17. The Fourier transform can be done asymptotically, and the associated saddle points stay within our allowed region for p when |y′-y0′|<αk-1/2; this means that we can use the fact that the Fourier transform of a cubic phase is given in terms of an Airy function by ∫ exp[i(xt+at3/3)]dt=2π|a|-1/3Ai(xa-1/3), where a and x are real and a-1/3 is the real root when a<0.
  18. Since we are to consider the Fourier transform of this difference only in an asymptotically small neighborhood of y1′, it can be shown that it is only necessary to fit the component of the error that is so evident in Fig. 4.
  19. V. P. Maslov, M. V. Fedoriuk, Semiclassical Approximation in Quantum Mechanics (Reidel, Boston, Mass., 1981).
  20. J. B. Delos, “Semiclassical calculation of quantum mechanical wavefunctions,” Adv. Chem. Phys., I. Prigogine, S. A. Rice, eds. (Wiley Interscience, 1986), Vol. 67, pp. 161–213.
  21. If the Taylor expansion for g(x) about the origin is written as g(x)=Σjgjxj, then ∫g(x)exp(ikx2/2)dx=2πi/k[g0+ig2/k+O(k-2)].
  22. Kl may work in such a case because, in place of just the two options employed by KM, it makes up the field by use of a continuum of asymptotic representations.
  23. The curvature of a three-dimensional parametric curve r(t) is given by c=|r˙×r¨|/|r˙|3, and, for the plane curve in the scaled phase space considered here, this becomes l(y˙p¨-y¨p˙)[y˙2+l2p˙2]-3/2, where the dot denotes (d/dp).
  24. To be consistent with the caustic phase shifts of Section 3, the phase-space curve shown in Fig. 3 must be regarded as if it originated as a vertical line segment. [As it rotates away from the vertical in a clockwise sense, one half of the regular phase shift associated with rotating fully through the vertical is acquired, and this is the exp(-iπ/4) factor that is present in the Rayleigh–Sommerfeld solution and therefore in both Eqs. (2.3) and (3.2).] Under propagation, the curve rotates further: first through the horizontal, and then the upper branch is again passed through the vertical. [In terms of Eq. (3.16), this means that the upper branch of Fig. 3 has M=1, while M=0 for the lower branch, and M˜=1 for the whole parabola used in deriving Eq. (3.17).] In terms of Eq. (2.3), this means that ∂p/∂p′<0 and, in fact, that ∂p/∂p′ has phase exp(-iπ/2).
  25. It is important not to confuse the differences between various asymptotic estimates and their errors—i.e., their differences from an exact solution. Nevertheless, we estimate the errors here by considering no more than the differences between asymptotic estimates. For example, the error of KV at a point caustic is manifest in Fig. 4, and, for small l, the corresponding error in Kl is also clear in Fig. 8. However, when l becomes large, the fact that εlP goes to zero [see relation (5.6)] does not mean that the error in Kl approaches zero. It simply means that Kl approaches KW, and these are then both in error by the same unknown term that is O(k-1). In this vein, it follows that there will be an error component in Kl that, for sufficiently large l, is like that of KW near the angle caustics—see the discussion at relation (3.24)—and this is what is sought here.
  26. For large γ, Rl(γ, 0)≈γ exp(iπ/4)∫0∞u exp(-i2πγu3/3)du-i∫0∞u exp(i2πγu3/3)du, and this can be evaluated by use of ∫0∞tn exp(iat3)dt=exp[iπ(n+1)sgn(a)/6]×Γ[(n+1)/3]/[3|a|(n+1)/3]. One can find the width of the error in this limit by considering Rl(γ, tγ-1/3), and the result is consistent with relation (3.24).
  27. For free space, W(x, y; x′, p′)=(x′-x)1-p′2-yp′, and, together with Eqs. (3.7) and (3.8), this leads to the angular spectrum solution for the propagator. See, for example, L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Sec. 3.2.5. Note that the evanescent part of the spectrum is absent in the asymptotic solution; the homogeneous component is exact, however, and this is all that is required in an asymptotic solution.
  28. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1980), p. 342. In geometrical optics, with the rays written as y(x), the Lagrangian and the Hamiltonian take the forms L(y, y˙, x)=n(x, y)1+y˙2 and H(y, p, x)=-n2(x, y)-p2, respectively.
  29. Without guidance, computer algebra is of little help in deriving Eq. (6.7). The problem with selecting the desired roots from among those of a quartic can be avoided by introduction of S≔(x′-x)2/(ay-p2). With this, y′=y′(x, y, p; x′) can be written as p=[4(y′-y)-aS]/(4S), where only the positive root is valid, and S is found in terms of y and y′ by solution of a quadratic: (y′-y-aS/4)2=ayS-(x′-x)2. It follows that S=(4/a)[y+y′±Z(y′)]. The substitution of the expression for p into L is simplified if the expression in Eq. (6.6) is first written in terms of S. It is also interesting that the angle characteristic is so simple for this example. It follows that, on solving p′=p′(x, y, p; x′) for y as a function of p and p′ and substituting the result into L+yp-y′p′, one obtains T(x, p; x′, p′)=a(x′-x)24(p′-p)-(p′3-p3)3a. Note, however, that T is defined only when (x′-x)×(p′-p)>0.

1998

1997

1969

1928

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. A.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 133.

Buchdahl, H. A.

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

Courant, R.

R. Courant, D. Hilbert, Methods of Mathematical Physics, (Interscience, New York, 1962), Vol. 2, pp. 32–39.

Delos, J. B.

J. B. Delos, “Semiclassical calculation of quantum mechanical wavefunctions,” Adv. Chem. Phys., I. Prigogine, S. A. Rice, eds. (Wiley Interscience, 1986), Vol. 67, pp. 161–213.

Fedoriuk, M. V.

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

Forbes, G. W.

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1980), p. 342. In geometrical optics, with the rays written as y(x), the Lagrangian and the Hamiltonian take the forms L(y, y˙, x)=n(x, y)1+y˙2 and H(y, p, x)=-n2(x, y)-p2, respectively.

Gutzwiller, M. C.

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

Hilbert, D.

R. Courant, D. Hilbert, Methods of Mathematical Physics, (Interscience, New York, 1962), Vol. 2, 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. Calif. Press, Berkeley, Calif., 1964), pp. 111–115.

Mandel, L.

For free space, W(x, y; x′, p′)=(x′-x)1-p′2-yp′, and, together with Eqs. (3.7) and (3.8), this leads to the angular spectrum solution for the propagator. See, for example, L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Sec. 3.2.5. Note that the evanescent part of the spectrum is absent in the asymptotic solution; the homogeneous component is exact, however, and this is all that is required in an asymptotic solution.

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, UK1995), pp. 169–187.

Wolf, E.

For free space, W(x, y; x′, p′)=(x′-x)1-p′2-yp′, and, together with Eqs. (3.7) and (3.8), this leads to the angular spectrum solution for the propagator. See, for example, L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Sec. 3.2.5. Note that the evanescent part of the spectrum is absent in the asymptotic solution; the homogeneous component is exact, however, and this is all that is required in an asymptotic solution.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 133.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Proc. Natl. Acad. Sci. USA

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

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

It may be surprising that ∂2V/∂y∂y′ depends on b, which can be seen from Eq. (3.11) to be ∂3W/∂p′3 evaluated at the caustic. One can alternatively derive Eq. (3.15) by including the cross term A(y-y0)(p′-p0′) in Eq. (3.11) and performing a Legendre transformation to derive the expansion for V.

The Fourier transform can be done asymptotically, and the associated saddle points stay within our allowed region for p when |y′-y0′|<αk-1/2; this means that we can use the fact that the Fourier transform of a cubic phase is given in terms of an Airy function by ∫ exp[i(xt+at3/3)]dt=2π|a|-1/3Ai(xa-1/3), where a and x are real and a-1/3 is the real root when a<0.

Since we are to consider the Fourier transform of this difference only in an asymptotically small neighborhood of y1′, it can be shown that it is only necessary to fit the component of the error that is so evident in Fig. 4.

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

J. B. Delos, “Semiclassical calculation of quantum mechanical wavefunctions,” Adv. Chem. Phys., I. Prigogine, S. A. Rice, eds. (Wiley Interscience, 1986), Vol. 67, pp. 161–213.

If the Taylor expansion for g(x) about the origin is written as g(x)=Σjgjxj, then ∫g(x)exp(ikx2/2)dx=2πi/k[g0+ig2/k+O(k-2)].

Kl may work in such a case because, in place of just the two options employed by KM, it makes up the field by use of a continuum of asymptotic representations.

The curvature of a three-dimensional parametric curve r(t) is given by c=|r˙×r¨|/|r˙|3, and, for the plane curve in the scaled phase space considered here, this becomes l(y˙p¨-y¨p˙)[y˙2+l2p˙2]-3/2, where the dot denotes (d/dp).

To be consistent with the caustic phase shifts of Section 3, the phase-space curve shown in Fig. 3 must be regarded as if it originated as a vertical line segment. [As it rotates away from the vertical in a clockwise sense, one half of the regular phase shift associated with rotating fully through the vertical is acquired, and this is the exp(-iπ/4) factor that is present in the Rayleigh–Sommerfeld solution and therefore in both Eqs. (2.3) and (3.2).] Under propagation, the curve rotates further: first through the horizontal, and then the upper branch is again passed through the vertical. [In terms of Eq. (3.16), this means that the upper branch of Fig. 3 has M=1, while M=0 for the lower branch, and M˜=1 for the whole parabola used in deriving Eq. (3.17).] In terms of Eq. (2.3), this means that ∂p/∂p′<0 and, in fact, that ∂p/∂p′ has phase exp(-iπ/2).

It is important not to confuse the differences between various asymptotic estimates and their errors—i.e., their differences from an exact solution. Nevertheless, we estimate the errors here by considering no more than the differences between asymptotic estimates. For example, the error of KV at a point caustic is manifest in Fig. 4, and, for small l, the corresponding error in Kl is also clear in Fig. 8. However, when l becomes large, the fact that εlP goes to zero [see relation (5.6)] does not mean that the error in Kl approaches zero. It simply means that Kl approaches KW, and these are then both in error by the same unknown term that is O(k-1). In this vein, it follows that there will be an error component in Kl that, for sufficiently large l, is like that of KW near the angle caustics—see the discussion at relation (3.24)—and this is what is sought here.

For large γ, Rl(γ, 0)≈γ exp(iπ/4)∫0∞u exp(-i2πγu3/3)du-i∫0∞u exp(i2πγu3/3)du, and this can be evaluated by use of ∫0∞tn exp(iat3)dt=exp[iπ(n+1)sgn(a)/6]×Γ[(n+1)/3]/[3|a|(n+1)/3]. One can find the width of the error in this limit by considering Rl(γ, tγ-1/3), and the result is consistent with relation (3.24).

For free space, W(x, y; x′, p′)=(x′-x)1-p′2-yp′, and, together with Eqs. (3.7) and (3.8), this leads to the angular spectrum solution for the propagator. See, for example, L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Sec. 3.2.5. Note that the evanescent part of the spectrum is absent in the asymptotic solution; the homogeneous component is exact, however, and this is all that is required in an asymptotic solution.

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1980), p. 342. In geometrical optics, with the rays written as y(x), the Lagrangian and the Hamiltonian take the forms L(y, y˙, x)=n(x, y)1+y˙2 and H(y, p, x)=-n2(x, y)-p2, respectively.

Without guidance, computer algebra is of little help in deriving Eq. (6.7). The problem with selecting the desired roots from among those of a quartic can be avoided by introduction of S≔(x′-x)2/(ay-p2). With this, y′=y′(x, y, p; x′) can be written as p=[4(y′-y)-aS]/(4S), where only the positive root is valid, and S is found in terms of y and y′ by solution of a quadratic: (y′-y-aS/4)2=ayS-(x′-x)2. It follows that S=(4/a)[y+y′±Z(y′)]. The substitution of the expression for p into L is simplified if the expression in Eq. (6.6) is first written in terms of S. It is also interesting that the angle characteristic is so simple for this example. It follows that, on solving p′=p′(x, y, p; x′) for y as a function of p and p′ and substituting the result into L+yp-y′p′, one obtains T(x, p; x′, p′)=a(x′-x)24(p′-p)-(p′3-p3)3a. Note, however, that T is defined only when (x′-x)×(p′-p)>0.

A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, Cambridge, UK1995), pp. 169–187.

See Sec. 5 of Ref. 1.

A. W. Conway, J. L. Synge, eds., The Mathematical Papers of Sir William Rowan Hamilton (Cambridge U. Press, Cambridge, UK, 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, 1980), p. 133.

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. Calif. Press, Berkeley, Calif., 1964), pp. 111–115.

R. Courant, D. Hilbert, Methods of Mathematical Physics, (Interscience, New York, 1962), Vol. 2, pp. 32–39.

A differential ray bundle has a momentum caustic when the rays in the bundle become parallel. Momentum caustics carry a sign: They are positive when the rays go from being divergent to convergent and negative otherwise.

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Figures (12)

Fig. 1
Fig. 1

Thick curve representing a ray that is specified with respect to a Cartesian coordinate system as shown. The ray momentum, p, is just the product of the local refractive index and the y component of the unit ray tangent. L is the ray’s optical length between the two reference planes, and all the variables denoted by script letters are regarded as functions of x, y, p, and x.

Fig. 2
Fig. 2

Final transverse position and momentum of the members of a ray family define a parametric curve in phase space. Point caustics occur at values of y where the curve is vertical and are flagged here by black circles. White circles mark the momentum caustics that are associated with values of p at which the tangent is horizontal. For a propagator, the ray family contains all the rays through (x, y). That is, x, y, and x are fixed, and p is adopted here as the parameter.

Fig. 3
Fig. 3

Parabolic approximation of a phase-space curve in the neighborhood of a caustic. The parabolas at the point and angle caustics in this segment of a phase-space curve can be written as y=y0+(b/2)(p-p0)2, and p=p1+(c/2)(y-y1)2.

Fig. 4
Fig. 4

Comparison of the Airy function and an asymptotic approximation of the form A(u)π-1/2H(u)u-1/4 sin[2u3/2/3+π/4]. The error E(u)=A(u)-Ai(-u) is shown as the solid curve, and the algebraic approximation given in relation (3.22) is also shown for comparison.

Fig. 5
Fig. 5

Schematic of Maslov’s method. One first divides the phase-space curve into segments containing only one type of caustic. Smooth switching functions are used in reconstituting the total wave field. The shaded regions at the caustics all have equal area and serve as guides in configuring the switching functions.

Fig. 6
Fig. 6

Approximation of each segment of the phase-space curve between adjacent point and angle caustics by an ellipse that is aligned to the coordinate axes. The area of the shaded sections is 42/(3k)0.3λ, and their widths are given in terms of the semiaxes.

Fig. 7
Fig. 7

Three-dimensional plot of κ(l, p) as defined in condition (5.1). This example results from a phase-space curve with two point caustics and one angle caustic. The white curve on the surface locates the maximum of κ for each fixed value of p, and it is given by Eq. (5.2). According to Eq. (5.3), C(p) is found by evaluation of κ along this white curve; this function is also sketched on the right-hand wall of the boundary box.

Fig. 8
Fig. 8

Plots of Kl(B, u) for various values of B. This function evidently differs significantly from Ai(-u) only near the origin. For small B, there is a global relative difference of just 5B3/48 [see Eq. (5.5)].

Fig. 9
Fig. 9

Plot showing that rays from (x, y) are each parabolic and that the caustic is itself a parabola. The locus of this point caustic is given by y=(x-x)2/(4y).

Fig. 10
Fig. 10

Phase-space curves for the rays from (x, y)=(0, 1), where μ=1. The curves are given for two different distances of propagation.

Fig. 11
Fig. 11

Comparison of the exact propagator K(0, 1; 1, y) and KM(0, 1; 1, y) for the case in which μ=1 and k=100. (All but the last argument of these functions are suppressed in the figure.) Recall that the point caustic sits at (y, p)=(1/4, 5-1/2/2) and that the angle caustic sits at (y, p)=(1/2, 0); see Fig. 10. In this case, 6/(YP)107, and Kl is found to be almost indistinguishable from K on this scale.

Fig. 12
Fig. 12

Comparison of the exact propagator K(0, 1; 2, y) and Kl(0, 1; 2, y) for the case in which k=10. Again, all but the last argument of these functions are suppressed. In this case, the point caustic sits at (y, p)=(1, 1/2) and the angle caustic sits at (y, p)(1.101, 0.647); see Fig. 10. Since 6/(YP)1000, it is not surprising to see gross errors in KM for this case. Note, in |KM|, the characteristic failure associated with a point caustic that now spreads beyond the neighboring angle caustic. A similar phenomenon occurs in the Fourier domain, and these problems flag an essential limitation of KM. In contrast, the error in Kl never exceeds roughly 20%, and it is evident that the error in |Kl| is no more than roughly 5%.

Equations (87)

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[2+n2(x, y)k2]U(x, y)=0,
U(x, y)=-U(x, y)K(x, y; x, y)dy.
Kl(x, y; x, y)=k2πexp(-iπ/4)ϕ1lyp+i pp1/2×exp-k2l(y-y)2exp{ik[L+p(y-y)]}dp,
ϕ(x, y, p; x)n2(x, y)-p2n2(x, y)-p21/2.
K˜l(x, y; x, p)=k2πexp(-iπ/4)ϕyp+il pp1/2×exp-kl2(p-p)2exp[ik(L-yp)]dpk2π Kl(x, y; x, y)exp(-ikyp)dy.
p=-Vy(x, y; x, y),
p=Vy(x, y; x, y).
KV(x, y; x, y)=jk2πϕV(j)2V(j)yy(x, y; x, y)1/2×exp-i π212+M(j)×exp[ikV(j)(x, y; x, y)].
ϕV(j)(x, y; x, y)=-V(j)x(x, y; x, y)V(j)x(x, y; x, y),
W(x, y; x, p)=V(x, y; x, Y)-Yp,
Vy[x, y; x, Y(x, y; x, p)]-p=0.
p=-Wy(x, y; x, p),
y=-Wp(x, y; x, p).
KW(x, y; x, y)=k2π K˜W(x, y; x, p)exp(ikyp)dp,
K˜W(x, y; x, p)=jk2πϕW(j)2W(j)yp(x, y; x, p)1/2×exp-i π2M˜(j)×exp[ikW(j)(x, y; x, p)],
ϕW(j)=-W(j)x(x, y; x, p)W(j)x(x, y; x, p).
y=y0+(b/2)(p-p0)2;
W(p)=w-y0(p-p0)-(b/6)(p-p0)3,
p=p0+σ[(2/b)(y-y0)]1/2,
V(j)(y)=(w+y0p0)+p0(y-y0)+(-1)j(2/3)×[(2/b)(y-y0)]1/2(y-y0).
2Vyy=-pyy=-ppy pyy=2Wyppyy
2V(j)yy=F/[2b(y-y0)]1/2.
KV(y)=-i exp(ikw)Fπ 2k2|b|1/3H[(y-y0)b]×[(2k2/b)1/3(y-y0)]-1/4 exp(ikp0y)×sin23[(2k2/b)1/3(y-y0)]3/2+π4,
KW(y)=-i exp(ikw)F2k2|b|1/3×exp(ikp0y)Ai[-(2k2/b)1/3(y-y0)],
E(u)H(u)sin23u3/2+π4πu-Ai(-u).
p=p1+(c/2)(y-y1)2,
K˜W(p)=-i exp(ikv)Gπ 2k2|c|1/3H[(p-p1)c]×[(2k2/c)1/3(p-p1)]-1/4 exp(-iky1p)×sin23[(2k2/c)1/3(p-p1)]3/2+π4,
K˜V(p)=-i exp(ikv)G2k2|c|1/3×exp(-iky1p)Ai[-(2k2/c)1/3(p-p1)].
E(u)Eˆ(u)H(u)0.21u-0.32 exp(-2.1u)-H(-u)0.38 exp(1.1u),
Eˆ(u)exp(iqu)du=0.21Γ(0.68)(2.1-iq)0.68-0.38(1.1+iq),
KW(y)-KV(y)KV(y)W[(kc/2)1/3(y-y1)],
CV[2m](x, y; x, y)=k2πϕV[2m]2V[2m]yy(x, y; x, y)1/2×exp-i π212+M[2m]×exp{ikV[2m](x, y; x, y)}.
C˜W[2m+1](x, y; x, p)=k2πϕW[2m+1]2W[2m+1]yp(x, y; x, p)1/2×exp-i π2M˜[2m+1]×exp{ikW[2m+1](x, y; x, p)},
KM(x, y; x, y)=mCV[2m](x, y; x, y)e[2m](y)+k2π C˜W[2m+1](x, y; x, p)×e˜[2m+1](p)exp(ikyp)dp,
e[2m](y)=1,yinsegment2m,1-e˜[2m-1]V[2m]y(x, y; x, y),yT(2m-1, 2m),1-e˜[2m+1]V[2m]y(x, y; x, y),yT(2m, 2m+1),0otherwise.
p=p2+a(y-y2),
V(y)=v+p2(y-y2)+(a/2)(y-y2)2,
W(p)=(v-y2p2)-y2(p-p2)-(p-p2)2/(2a).
KV(y)=kF|a|2π1/2 exp(-iπ/4)exp[ikV(y)].
KM(y)=e[0](y)KV(y)+k2π K˜W(p)×e˜[1](p)exp(ikyp)dp.
M(δ) KM(y2+δ)-KV(y2+δ)KV(y2+δ)
=s(δ)-ika2π  exp[-ika(t-δ)2/2]s(t)dt,
M(δ)=s(δ)-ika2π d exp[-ika(t-δ)2/2]dt+-dd exp[-ika(t-δ)2/2]s(t)dt.
M(uD/ka)=σ(u)-Di2π 1 exp[-iD2(v-u)2/2]dv-Di2π -11 exp[-iD2(v-u)2/2]σ(v)dv.
k>[2/(2-1)]3/2/(YP)6/(YP).
k>200/(YP).
kκ(l, p)l3yp2+l pp2-3×yp2pp2-pp2yp22
lmax(p)yppp.
C(p)κ[lmax(p), p]=18yppp-3×yp2pp2-pp2yp22.
Kl(B, u)12π1-iBt exp[-B(t2-u)2/4]×exp[i(ut-t3/3)]dt,
Kl(B, u)=Ai(-u)[1->B3]+O(B4).
lP5b2/(24kl3).
Kl(y)KV(y)=Rl(γ, s)1+iγv×exp[-iπγ(v-s)2(2v+s)/3]×exp[-π(v-s)2]dv,
Rl(γ, Δ/γ)=1-5γ2/[48π(1+iΔ)3]+O(γ4).
lA(y-y1)5c2l324k[1+c2l2(y-y1)2]-3/2,
l=Y/P,
k>6/(YP).
px=12[n2(x, y)-p2]-1/2 n2y(x, y),
yx=[n2(x, y)-p2]-1/2p,
Lx=[n2(x, y)-p2]-1/2n2(x, y),
x[n2(y)-p2]=n2yyx-2p px=0.
p(x, y, p; x)=p+μ(x-x)2μy-p2,
y(x, y, p; x)=y+p(x-x)μy-p2+μ(x-x)24(μy-p2),
L(x, y, p; x)=μy(x-x)μy-p2+μp(x-x)22(μy-p2)+μ2(x-x)312(μy-p2)3/2.
V(j)(y)=μ3[(y+y)-(-1)jZ(y)]1/2×[2(y+y)+(-1)jZ(y)],
2V(j)yy=μ(x-x)24Z(y)[y+y-(-1)jZ(y)]-3/2.
Q(g)g2[(μg/2-p)2-μy]+(x-x)2,
g+(3p+p2+8µy)/(2µ).
W(j)(p)=-yp+(μy-p2)gj+¾ μpgj2- μ2gj3.
2W(j)yp=p(j)p=1-μgj2½ μgj-pQ(gj).
g(y)g(y)+k2μy=-f(x)f(x).
f(x)=A+ exp(ikμux)+A- exp(-ikμux)u0,B+ exp(-k-μux)+B- exp(k-μux)u<0.
g(y)g(y)+k2μ(y-u)=0,
g(y)=αAi[(k2μ)1/3(u-y)]+βBi[(k2μ)1/3(u-y)].
U(x, y)=-0w(u)Ai[(k2μ)1/3(u-y)]×exp[-k-μu(x-x)]du+0w(u)Ai[(k2μ)1/3(u-y)]×exp[ikμu(x-x)]du,
U(x, y)=-w(u)Ai[(k2μ)1/3(u-y)]du.
(k2μ)2/3-Ai[(k2μ)1/3(u-y)]Ai[(k2μ)1/3(u-y)]dy=δ(u-u).
w(u)=(k2μ)2/3-U(x, y)Ai[(k2μ)1/3(u-y)]du.
U(x, y)=(k2μ)2/3-U(x, y)-0Ai[(k2μ)1/3(u-y)]×Ai[(k2μ)1/3(u-y)]exp[-k-μu(x-x)]du+0Ai[(k2μ)1/3(u-y)]Ai[(k2μ)1/3(u-y)]×exp[ikμu(x-x)]dudy.
K(x, y; x, y)=(k2μ)2/30{Ai[(k2μ)1/3(-u-y)]×Ai[(k2μ)1/3(-u-y)]×exp[-kμu(x-x)]+Ai[(k2μ)1/3(u-y)]×Ai[(k2μ)1/3(u-y)]exp[ikμu(x-x)]}du.
 exp[i(xt+at3/3)]dt=2π|a|-1/3Ai(xa-1/3),
g(x)exp(ikx2/2)dx=2πi/k[g0+ig2/k+O(k-2)].
Rl(γ, 0)γ exp(iπ/4)0u exp(-i2πγu3/3)du-i0u exp(i2πγu3/3)du,
0tn exp(iat3)dt=exp[iπ(n+1)sgn(a)/6]×Γ[(n+1)/3]/[3|a|(n+1)/3].
p=[4(y-y)-aS]/(4S),
(y-y-aS/4)2=ayS-(x-x)2.
T(x, p; x, p)=a(x-x)24(p-p)-(p3-p3)3a.

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