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.