F. Michel, G. Reidemeister, and S. Ohkubo, “Luneburg lens approach to nuclear rainbow scattering,” Phys. Rev. Lett. 89, 152701-1-152701-4 (2002).

[CrossRef]

J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Rep. 356, 229-365 (2002).

[CrossRef]

A. D. Greenwood and J.-M. Jin, “A field picture of wave propagation in inhomogeneous dielectric lenses,” IEEE Antennas Propag. Mag. 41, 9-17 (1999).

[CrossRef]

H. Sakurai, T. Hashidate, M. Ohki, K. Motojima, and S. Kozaki, “Electromagnetic scattering by the Luneburg lens reflector,” Int. J. Electron. 84, 635-645 (1998).

J. A. Lock and T. A. McCollum, “Further thoughts on Newton's zero-order rainbow,” Am. J. Phys. 62, 1082-1089 (1994).

[CrossRef]

N. Fiedler-Ferrari, H. M. Nussenzveig, and W. J. Wiscombe, “Theory of near-critical-angle scattering from a curved interface,” Phys. Rev. A 43, 1005-1038 (1991).

[CrossRef]

V. Khare and H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976-980 (1974).

[CrossRef]

M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315-397 (1972), Secs. 6.2, 6.3.

[CrossRef]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125-176 (1969).

[CrossRef]

K. W. Ford and J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (N.Y.) 7, 259-286 (1959).

[CrossRef]

A. S. Gutman, “Modified Luneburg lens,” J. Appl. Phys. 25, 855-859 (1954).

[CrossRef]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 446, Fig. 10.6.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964), pp. 448, 449 and Eqs. (10.4.60), (10.4.62).

C. L. Adler, J. A. Lock, and R. F. Fleet, “Rainbows in the grass II: arbitrary diagonal incidence,” Appl. Opt. (to be published).

J. A. Lock, C. L. Adler, and R. F. Fleet, “Rainbows in the grass I: external-reflection rainbows from pendant droplets,” Appl. Opt. (to be published).

M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315-397 (1972), Secs. 6.2, 6.3.

[CrossRef]

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), p. 175.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), p. 167.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1980), footnote p. 147.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1980), p. 123.

N. Fiedler-Ferrari, H. M. Nussenzveig, and W. J. Wiscombe, “Theory of near-critical-angle scattering from a curved interface,” Phys. Rev. A 43, 1005-1038 (1991).

[CrossRef]

C. L. Adler, J. A. Lock, and R. F. Fleet, “Rainbows in the grass II: arbitrary diagonal incidence,” Appl. Opt. (to be published).

J. A. Lock, C. L. Adler, and R. F. Fleet, “Rainbows in the grass I: external-reflection rainbows from pendant droplets,” Appl. Opt. (to be published).

K. W. Ford and J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (N.Y.) 7, 259-286 (1959).

[CrossRef]

W. T. Grandy, Scattering of Waves from Large Spheres (Cambridge U. Press, 2000).

A. D. Greenwood and J.-M. Jin, “A field picture of wave propagation in inhomogeneous dielectric lenses,” IEEE Antennas Propag. Mag. 41, 9-17 (1999).

[CrossRef]

A. S. Gutman, “Modified Luneburg lens,” J. Appl. Phys. 25, 855-859 (1954).

[CrossRef]

H. Sakurai, T. Hashidate, M. Ohki, K. Motojima, and S. Kozaki, “Electromagnetic scattering by the Luneburg lens reflector,” Int. J. Electron. 84, 635-645 (1998).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), p. 175.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), p. 167.

A. D. Greenwood and J.-M. Jin, “A field picture of wave propagation in inhomogeneous dielectric lenses,” IEEE Antennas Propag. Mag. 41, 9-17 (1999).

[CrossRef]

V. Khare and H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976-980 (1974).

[CrossRef]

H. Sakurai, T. Hashidate, M. Ohki, K. Motojima, and S. Kozaki, “Electromagnetic scattering by the Luneburg lens reflector,” Int. J. Electron. 84, 635-645 (1998).

J. A. Lock, “Scattering of a plane wave by a Luneburg lens. II. Wave theory,” J. Opt. Soc. Am. A 25, 2980-2990 (2008).

J. A. Lock, “Scattering of a plane wave by a Luneburg lens. III. Finely stratified multilayer sphere model,” J. Opt. Soc. Am. A 25, 2991-3000 (2008).

J. A. Lock and T. A. McCollum, “Further thoughts on Newton's zero-order rainbow,” Am. J. Phys. 62, 1082-1089 (1994).

[CrossRef]

C. L. Adler, J. A. Lock, and R. F. Fleet, “Rainbows in the grass II: arbitrary diagonal incidence,” Appl. Opt. (to be published).

J. A. Lock, C. L. Adler, and R. F. Fleet, “Rainbows in the grass I: external-reflection rainbows from pendant droplets,” Appl. Opt. (to be published).

J. A. Lock and T. A. McCollum, “Further thoughts on Newton's zero-order rainbow,” Am. J. Phys. 62, 1082-1089 (1994).

[CrossRef]

F. Michel, G. Reidemeister, and S. Ohkubo, “Luneburg lens approach to nuclear rainbow scattering,” Phys. Rev. Lett. 89, 152701-1-152701-4 (2002).

[CrossRef]

H. Sakurai, T. Hashidate, M. Ohki, K. Motojima, and S. Kozaki, “Electromagnetic scattering by the Luneburg lens reflector,” Int. J. Electron. 84, 635-645 (1998).

M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315-397 (1972), Secs. 6.2, 6.3.

[CrossRef]

N. Fiedler-Ferrari, H. M. Nussenzveig, and W. J. Wiscombe, “Theory of near-critical-angle scattering from a curved interface,” Phys. Rev. A 43, 1005-1038 (1991).

[CrossRef]

V. Khare and H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976-980 (1974).

[CrossRef]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125-176 (1969).

[CrossRef]

H. Sakurai, T. Hashidate, M. Ohki, K. Motojima, and S. Kozaki, “Electromagnetic scattering by the Luneburg lens reflector,” Int. J. Electron. 84, 635-645 (1998).

F. Michel, G. Reidemeister, and S. Ohkubo, “Luneburg lens approach to nuclear rainbow scattering,” Phys. Rev. Lett. 89, 152701-1-152701-4 (2002).

[CrossRef]

F. Michel, G. Reidemeister, and S. Ohkubo, “Luneburg lens approach to nuclear rainbow scattering,” Phys. Rev. Lett. 89, 152701-1-152701-4 (2002).

[CrossRef]

H. Sakurai, T. Hashidate, M. Ohki, K. Motojima, and S. Kozaki, “Electromagnetic scattering by the Luneburg lens reflector,” Int. J. Electron. 84, 635-645 (1998).

J. R. Flores, J. Sochacki, M. Sochacka, and R. Staronski, “Quasi-analytical ray tracing through the generalized Luneburg lens,” Appl. Opt. 31, 5167-5170 (1992).

J. Sochacki, “Exact analytical solution of the generalized Luneburg lens problem,” J. Opt. Soc. Am. 73, 789-795 (1983).

J. Sochacki, “Generalized Luneburg lens problem solution: a comment,” J. Opt. Soc. Am. 73, 1839 (1983).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 446, Fig. 10.6.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964), pp. 448, 449 and Eqs. (10.4.60), (10.4.62).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), p. 207.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), p. 205.

K. W. Ford and J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (N.Y.) 7, 259-286 (1959).

[CrossRef]

N. Fiedler-Ferrari, H. M. Nussenzveig, and W. J. Wiscombe, “Theory of near-critical-angle scattering from a curved interface,” Phys. Rev. A 43, 1005-1038 (1991).

[CrossRef]

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1980), p. 123.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1980), footnote p. 147.

J. A. Lock and T. A. McCollum, “Further thoughts on Newton's zero-order rainbow,” Am. J. Phys. 62, 1082-1089 (1994).

[CrossRef]

K. W. Ford and J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (N.Y.) 7, 259-286 (1959).

[CrossRef]

B. R. Johnson, “Light scattering by a multilayer sphere,” Appl. Opt. 35, 3286-3296 (1996).

J. M. Gordon, “Spherical gradient-index lenses as perfect imaging and maximum power transfer devices,” Appl. Opt. 39, 3825-3832 (2000).

J. R. Flores, J. Sochacki, M. Sochacka, and R. Staronski, “Quasi-analytical ray tracing through the generalized Luneburg lens,” Appl. Opt. 31, 5167-5170 (1992).

C. L. Brockman and N. G. Alexopoulos, “Geometrical optics of inhomogeneous particles: glory ray and the rainbow revisited,” Appl. Opt. 16, 166-174 (1977).

J. A. Adam and P. Laven, “Rainbows from inhomogeneous transparent spheres: a ray-theoretic approach,” Appl. Opt. 46, 922-929 (2007).

[CrossRef]

J. A. Lock, C. L. Adler, and R. F. Fleet, “Rainbows in the grass I: external-reflection rainbows from pendant droplets,” Appl. Opt. (to be published).

C. L. Adler, J. A. Lock, and R. F. Fleet, “Rainbows in the grass II: arbitrary diagonal incidence,” Appl. Opt. (to be published).

G. P. Konnen and J. H. de Boer, “Polarized rainbow,” Appl. Opt. 18, 1961-1965 (1979).

A. D. Greenwood and J.-M. Jin, “A field picture of wave propagation in inhomogeneous dielectric lenses,” IEEE Antennas Propag. Mag. 41, 9-17 (1999).

[CrossRef]

H. Sakurai, T. Hashidate, M. Ohki, K. Motojima, and S. Kozaki, “Electromagnetic scattering by the Luneburg lens reflector,” Int. J. Electron. 84, 635-645 (1998).

A. S. Gutman, “Modified Luneburg lens,” J. Appl. Phys. 25, 855-859 (1954).

[CrossRef]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125-176 (1969).

[CrossRef]

J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Rep. 356, 229-365 (2002).

[CrossRef]

N. Fiedler-Ferrari, H. M. Nussenzveig, and W. J. Wiscombe, “Theory of near-critical-angle scattering from a curved interface,” Phys. Rev. A 43, 1005-1038 (1991).

[CrossRef]

F. Michel, G. Reidemeister, and S. Ohkubo, “Luneburg lens approach to nuclear rainbow scattering,” Phys. Rev. Lett. 89, 152701-1-152701-4 (2002).

[CrossRef]

V. Khare and H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976-980 (1974).

[CrossRef]

M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315-397 (1972), Secs. 6.2, 6.3.

[CrossRef]

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1980), p. 123.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), p. 205.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), p. 167.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), p. 207.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), p. 175.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1980), footnote p. 147.

There are some discrepancies in the literature concerning Eq. . The form given here when applied to p=2 and N=1.333 agrees with Eq. and Table 1 of but does not agree for p=2 and N=1.333 with Eqs. (4.30), (4.31), and (4.34) of the earlier , which were reproduced as Eqs. (5.106) and (5.108) of . The appropriate quantities to expand in Taylor series about the rainbow angle are Eq. (4.29) of and Eq. (5.62) of .

W. T. Grandy, Scattering of Waves from Large Spheres (Cambridge U. Press, 2000).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964), pp. 448, 449 and Eqs. (10.4.60), (10.4.62).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 446, Fig. 10.6.

Most of the results of this paper and of were presented in preliminary form at the Fall Meeting of the Ohio Section of the American Physical Society, Cleveland Ohio, October 14-15, 2005.