Abstract

For a plane wave incident on either a Luneburg lens or a modified Luneburg lens, the magnitude and phase of the transmitted electric field are calculated as a function of the scattering angle in the context of ray theory. It is found that the ray trajectory and the scattered intensity are not uniformly convergent in the vicinity of edge ray incidence on a Luneburg lens, which corresponds to the semiclassical phenomenon of orbiting. In addition, it is found that rays transmitted through a large-focal-length modified Luneburg lens participate in a far-zone rainbow, the details of which are exactly analytically soluble in ray theory. Using these results, the Airy theory of the modified Luneburg lens is derived and compared with the Airy theory of the rainbows of a homogeneous sphere.

© 2008 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1980), footnote p. 147.
  2. 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).
    [CrossRef]
  3. 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]
  4. J. M. Gordon, “Spherical gradient-index lenses as perfect imaging and maximum power transfer devices,” Appl. Opt. 39, 3825-3832 (2000).
    [CrossRef]
  5. A. S. Gutman, “Modified Luneburg lens,” J. Appl. Phys. 25, 855-859 (1954).
    [CrossRef]
  6. J. Sochacki, “Exact analytical solution of the generalized Luneburg lens problem,” J. Opt. Soc. Am. 73, 789-795 (1983).
    [CrossRef]
  7. J. Sochacki, “Generalized Luneburg lens problem solution: a comment,” J. Opt. Soc. Am. 73, 1839 (1983).
    [CrossRef]
  8. 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).
    [CrossRef] [PubMed]
  9. J. A. Lock, “Scattering of a plane wave by a Luneburg lens. II. Wave theory,” J. Opt. Soc. Am. A 25, 2980-2990 (2008).
    [CrossRef]
  10. 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).
    [CrossRef]
  11. K. W. Ford and J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (N.Y.) 7, 259-286 (1959).
    [CrossRef]
  12. 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]
  13. M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1980), p. 123.
  14. B. R. Johnson, “Light scattering by a multilayer sphere,” Appl. Opt. 35, 3286-3296 (1996).
    [CrossRef] [PubMed]
  15. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), p. 205.
  16. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), p. 167.
  17. 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] [PubMed]
  18. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), p. 207.
  19. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), p. 175.
  20. F. Michel, G. Reidemeister, and S. Ohkubo, “Luneburg lens approach to nuclear rainbow scattering,” Phys. Rev. Lett. 89, 152701-1-152701-4 (2002).
    [CrossRef]
  21. J. A. Lock and T. A. McCollum, “Further thoughts on Newton's zero-order rainbow,” Am. J. Phys. 62, 1082-1089 (1994).
    [CrossRef]
  22. C. L. Brockman and N. G. Alexopoulos, “Geometrical optics of inhomogeneous particles: glory ray and the rainbow revisited,” Appl. Opt. 16, 166-174 (1977).
    [CrossRef] [PubMed]
  23. J. A. Adam and P. Laven, “Rainbows from inhomogeneous transparent spheres: a ray-theoretic approach,” Appl. Opt. 46, 922-929 (2007).
    [CrossRef] [PubMed]
  24. 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]
  25. 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).
  26. 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).
    [PubMed]
  27. C. L. Adler, J. A. Lock, and R. F. Fleet, “Rainbows in the grass II: arbitrary diagonal incidence,” Appl. Opt. (to be published).
    [PubMed]
  28. V. Khare and H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976-980 (1974).
    [CrossRef]
  29. 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 .
  30. W. T. Grandy, Scattering of Waves from Large Spheres (Cambridge U. Press, 2000).
    [CrossRef]
  31. J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Rep. 356, 229-365 (2002).
    [CrossRef]
  32. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 446, Fig. 10.6.
  33. G. P. Konnen and J. H. de Boer, “Polarized rainbow,” Appl. Opt. 18, 1961-1965 (1979).
    [CrossRef] [PubMed]
  34. 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.

2008

2007

2002

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]

2000

1999

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]

1998

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).
[CrossRef]

1996

1994

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

1992

1991

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] [PubMed]

1983

1979

1977

1974

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

1972

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]

1969

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]

1959

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

1954

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

Abramowitz, M.

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.

Adam, J. A.

Adler, C. L.

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

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).
[PubMed]

Alexopoulos, N. G.

Berry, M. V.

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]

Bohren, C. F.

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

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

Born, M.

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.

Brockman, C. L.

de Boer, J. H.

Fiedler-Ferrari, N.

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] [PubMed]

Fleet, R. F.

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

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).
[PubMed]

Flores, J. R.

Ford, K. W.

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

Gordon, J. M.

Grandy, W. T.

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

Greenwood, A. D.

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]

Gutman, A. S.

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

Hashidate, T.

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).
[CrossRef]

Huffman, D. R.

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

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

Jin, J.-M.

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]

Johnson, B. R.

Khare, V.

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

Konnen, G. P.

Kozaki, S.

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).
[CrossRef]

Laven, P.

Lock, J. A.

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

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).
[CrossRef]

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

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

McCollum, T. A.

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

Michel, F.

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

Motojima, K.

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).
[CrossRef]

Mount, K. E.

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]

Nussenzveig, H. M.

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] [PubMed]

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]

Ohki, M.

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).
[CrossRef]

Ohkubo, S.

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

Reidemeister, G.

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

Sakurai, H.

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).
[CrossRef]

Sochacka, M.

Sochacki, J.

Staronski, R.

Stegun, I. A.

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.

van de Hulst, H. C.

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.

Wheeler, J. A.

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

Wiscombe, W. J.

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] [PubMed]

Wolf, E.

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.

Am. J. Phys.

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

Ann. Phys. (N.Y.)

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

Appl. Opt.

IEEE Antennas Propag. Mag.

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]

Int. J. Electron.

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).
[CrossRef]

J. Appl. Phys.

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

J. Math. Phys.

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. Opt. Soc. Am.

J. Opt. Soc. Am. A

Phys. Rep.

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

Phys. Rev. A

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] [PubMed]

Phys. Rev. Lett.

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]

Rep. Prog. Phys.

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]

Other

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).
[CrossRef]

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.

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Figures (4)

Fig. 1
Fig. 1

Refractive index of a modified Luneburg lens as a function of r a for f = 0.9 , 1.0, and 1.1.

Fig. 2
Fig. 2

Ray trajectories through a modified Luneburg lens for (a) f = 1.0 , (b) f = 0.75 , and (c) f = 1.25 .

Fig. 3
Fig. 3

Ray intensity as a function of the scattering angle θ for f = 0.90 (dot-dashed curve), f = 0.99 (dot-dot-dashed curve), f = 1.0 (solid curve), f = 1.01 (dotted curve), and f = 1.10 (dashed curve). For f = 1.01 the rainbow angle is θ R = 78.61 ° , and for f = 1.10 the rainbow angle is θ R = 55.74 ° .

Fig. 4
Fig. 4

Scattering angle θ as a function of the ray angle of incidence β for f = 0.90 (dot-dashed curve), f = 0.99 (dot-dot-dashed curve), f = 1.0 (solid curve), f = 1.01 (dotted curve), and f = 1.10 (dashed curve).

Equations (80)

Equations on this page are rendered with MathJax. Learn more.

N ( r ) = [ 2 ( r a ) 2 ] 1 2 .
N ( r ) = [ 1 + f 2 ( r a ) 2 ] 1 2 f .
r N ( r ) sin ( ψ ) = b ,
d r d ξ = r [ ( r N b ) 2 1 ] 1 2 .
b = a sin ( β ) .
ξ = ξ α ,
r 2 cos 2 ( ξ ) ( a 2 A ) + r 2 sin 2 ( ξ ) ( a 2 B ) = 1 ,
β = 2 α ,
A = 1 + cos ( β ) ,
B = 1 cos ( β ) .
r ( ξ ) = a sin ( β ) [ 1 cos ( β ) cos ( 2 ξ + β ) ] 1 2
ξ 0 = ( π β ) 2 ,
r 0 = r ( ξ 0 ) = a [ 1 cos ( β ) ] 1 2 .
θ = β for 0 θ π 2 .
I scat ( θ ) = ( I 0 a 2 R 2 ) sin ( β ) cos ( β ) T ( θ ) [ sin ( θ ) d θ d β ] ,
I scat ( θ ) = ( I 0 a 2 R 2 ) cos ( θ ) T ( θ ) for 0 θ π 2 ,
= 0 for π 2 θ π ,
N d s = a 0 π β d ξ sin ( β ) [ 1 2 cos ( β ) cos ( 2 ξ + β ) + cos 2 ( β ) ] [ 1 cos ( β ) cos ( 2 ξ + β ) ] 2 = a [ π 2 + cos ( β ) ] ,
L = a [ π 2 cos ( θ ) ] .
L = a ( 1 + π 2 ) ,
E scat ( θ ) = ( E 0 a R ) ( cos θ ) 1 2 exp { i [ k R + x L ( π 2 cos ( θ ) ) π ] } + i ( E 0 a R ) x L [ J 1 ( x L θ ) ( x L θ ) ] exp ( i k R ) for 0 θ π 2 ,
= 0 for π 2 < θ π ,
x L = 2 π a λ
sin ( 2 α ) = sin ( 2 β ) D ,
cos ( 2 α ) = [ f 2 + cos ( 2 β ) ] D ,
A = ( 1 + f 2 + D ) 2 ,
B = ( 1 + f 2 D ) 2 ,
D = [ f 4 + 2 f 2 cos ( 2 β ) + 1 ] 1 2 .
r ( ξ ) = 2 1 2 f a sin ( β ) [ 1 + f 2 D cos ( 2 ξ 2 α ) ] 1 2
ξ 0 = π 2 + α ,
r 0 = r ( ξ 0 ) = a B 1 2 .
ξ exit = β + 2 α ,
θ = β ξ exit = 2 α .
d θ d β = 2 [ 1 + f 2 cos ( 2 β ) ] D 2 .
cos ( 2 β ) = cos ( θ ) [ 1 f 4 sin 2 ( θ ) ] 1 2 f 2 sin 2 ( θ ) .
D = f 2 cos ( θ ) + [ 1 f 4 sin 2 ( θ ) ] 1 2 .
I scat ( θ ) = ( I 0 a 2 4 R 2 ) { f 2 cos ( θ ) + [ 1 f 4 sin 2 ( θ ) ] 1 2 } 2 [ 1 f 4 sin 2 ( θ ) ] 1 2
I scat ( π 2 ) = I 0 a 2 ε 1 2 ( 2 R 2 ) ,
I scat ( π ) = I 0 a 2 ε 2 R 2 .
N d s = 2 a ξ exit π β d ξ sin ( β ) { ( 1 + f 2 ) [ 1 + f 2 D cos ( 2 ξ 2 α ) ] 2 f 2 sin 2 ( β ) [ 1 + f 2 D cos ( 2 ξ 2 α ) ] 2 } = [ a ( 1 + f 2 ) f ] [ π 2 arctan ( M P ) 1 2 ] + a cos ( β ) ,
M = D + f 2 1 ,
P = D f 2 + 1 .
L = [ 1 + π ( 1 + f 2 ) 4 f ] ,
E scat ( θ ) = ( E 0 a 2 R ) exp ( i Φ ) { f 2 cos ( θ ) + [ 1 f 4 sin 2 ( θ ) ] 1 2 } [ 1 f 4 sin 2 ( θ ) ] 1 4 + i ( E 0 a R ) x L [ J 1 ( x L θ ) ( x L θ ) ] exp ( i k R ) ,
Φ = k R + x L { [ ( 1 + f 2 ) f ] [ π 2 arctan ( M P ) 1 2 ] cos ( β ) } π ,
cos ( 2 β R ) = 1 f 2 ,
sin ( θ R ) = 1 f 2 .
cos ( 2 β ) = f 2 sin 2 ( θ ) ± cos ( θ ) [ 1 f 4 sin 2 ( θ ) ] 1 2 ,
I scat ( θ ) = ( I 0 a 2 4 R 2 ) { f 2 cos ( θ ) + [ 1 f 4 sin 2 ( θ ) ] 1 2 } 2 [ 1 f 4 sin 2 ( θ ) ] 1 2
for 0 β β R ,
= ( I 0 a 2 4 R 2 ) { f 2 cos ( θ ) [ 1 f 4 sin 2 ( θ ) ] 1 2 } 2 [ 1 f 4 sin 2 ( θ ) ] 1 2
for β R β π 2 .
θ R = π 2 2 ε 1 2 .
Φ ± = x L { [ ( 1 + f 2 ) f ] [ π 2 arctan ( M ± P ± ) 1 2 ] cos ( β ) } + ζ ± ,
D ± = f 2 cos ( θ ) ± [ 1 f 4 sin 2 ( θ ) ] 1 2 ,
M ± = D ± + f 2 1 ,
P ± = D ± f 2 + 1 ,
ζ ± = π for 0 β < β R ,
= π 2 for β R < β π 2 ,
z a = K cos ( 2 α ) cos ( β ) sin ( 2 α ) sin ( β ) ,
x a = K sin ( 2 α ) cos ( β ) + cos ( 2 α ) sin ( β ) ,
K = D 2 { 2 [ 1 + f 2 cos ( 2 β ) ] }
x = ± [ ( 4 3 ) ( z z c ) ] 3 2 [ ( f 2 1 ) ( f 2 + 1 ) 1 2 ] .
E scat ( θ ) = E + exp ( i Φ + ) + E exp ( i Φ )
θ = θ R Δ
d Φ ± d β = x L sin ( β ) [ 1 ( f 4 1 ) D 2 ] ,
Φ ( β ) = Φ R 2 1 2 x L ( β β R ) 2 [ f ( f 2 1 ) 1 2 ] 2 3 2 x L ( f 2 + 2 ) ( β β R ) 3 [ 3 f ( f 2 1 ) ( f 2 + 1 ) 1 2 ] + O [ ( β β R ) 4 ] ,
β = β R + ε for the larger - impact - parameter supernmerary ray ,
= β R ε for the smaller - impact - parameter supernumerary ray ,
ε = ( f 4 1 ) 1 4 ( Δ 2 ) 1 2 ± Δ 2 + O ( Δ 3 2 ) .
Φ ± ( θ ) = Φ R x L Δ sin ( β R ) ± 2 x L Δ 3 2 3 h 1 2 + O ( Δ 2 ) ,
h = 4 f 2 [ ( f 2 1 ) 3 2 ( f 2 + 1 ) 1 2 ] .
E ± = ( E 0 a 2 R ) [ ( f 4 1 ) 3 8 ( 2 Δ ) 1 4 ] [ 1 ± ( 2 Δ ) 1 2 ( f 4 1 ) 1 4 ] .
u x L 2 3 Δ h 1 3 ,
u 1 4 π 1 2 Ai ( u ) sin [ ( 2 u 3 2 3 ) + π 4 ] ,
u 1 4 π 1 2 Ai ( u ) cos [ ( 2 u 3 2 3 ) + π 4 ] ,
E scat ( θ ) = ( E 0 k R ) [ 2 3 4 π 1 2 ( f 2 + 1 ) 5 12 ( 2 f ) 1 6 ] exp [ i ( Φ R x L Δ sin ( β R ) 3 π 4 ) ] { ( f 2 1 ) 1 2 x L 7 6 Ai ( u ) + i 2 5 6 [ f ( f 2 + 1 ) ] 1 3 x L 5 6 Ai ( u ) } .
E scat ( θ ) C x H 7 6 Ai ( u ) + i { [ 2 p 2 ( 9 p 2 ) S 2 ] [ 6 p 4 3 ( p 2 1 ) 1 3 S 4 3 ] } x H 5 6 Ai ( u ) ,
S = sin ( β R ) = [ ( p 2 N 2 ) ( p 2 1 ) ] 1 2 ,
C = cos ( β R ) = [ ( N 2 1 ) ( p 2 1 ) ] 1 2 ,

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