Abstract

A survey is given of the applications of complex angular momentum theory to Mie scattering, with special emphasis on the recent treatments of the rainbow and the glory. The theory yields uniform asymptotic expansions of the scattering amplitudes for rainbows of arbitrary order, for size parameters ≳ 50, in close agreement with the exact results. The Airy theory fails for parallel polarization in the primary bow and for both polarizations in higher-order rainbows. The theory provides for the first time a complete physical explanation of the glory. It leads to the identification of the dominant contributions to the glory and to asymptotic expressions for them. They include a surface-wave contribution, whose relevance was first conjectured by van de Hulst, and the effect of complex rays in the shadow of the tenth-order rainbow. Good agreement with the exact results is obtained. Physical effects that play an important role include axial focusing, cross polarization, orbiting, the interplay of various damping effects, and geometrical resonances associated with closed or almost closed orbits. All significant features of the glory pattern found in recent numerical studies are reproduced.

© 1979 Optical Society of America

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References

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  1. A. Sommerfeld, Optics (Academic, New York, 1954), p. 247.
  2. H. C. van de Hulst, “A Theory of the Anti-coronae,” J. Opt. Soc. Am. 37, 16–22 (1947).
    [Crossref]
  3. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  4. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen”, Ann. Phys. (Leipzig) 25, 377–445 (1908).
  5. H. C. Bryant and A. J. Cox, “Mie theory and the glory,” J. Opt. Soc. Am. 56, 1529–1532 (1966).
    [Crossref]
  6. S. T. Shipley and J. A. Weinman, “A numerical study of scattering by large dielectric spheres,” J. Opt. Soc. Am. 68, 130–134 (1978).
    [Crossref]
  7. N. A. Logan, “Survey of Some Early Studies of the Scattering of Plane Waves by a Sphere,” Proc. IEEE 53, 773–785 (1965).
    [Crossref]
  8. W. Franz, Theorie der Beugung Elektromagnetischer Wellen (Springer-Verlag, Berlin, 1957).
    [Crossref]
  9. J. B. Keller, “A Geometrical Theory of Diffraction,” in Calculus of Variations and its Applications, Proceedings of Symposia in Applied Mathematics, edited by L. M. Graves (McGraw-Hill, New York, 1958), Vol 8.
    [Crossref]
  10. V. A. Fock, Diffraction of Radio Waves Around the Earth’s Surface (Publishers of the USSRAcademy of Sciences, Moscow, 1946).
  11. B. van der Pol and H. Bremmer, “The Diffraction of Electromagnetic Waves from an Electrical Point Source round a Finitely Conducting Sphere, with Applications to Radio-Telegraphy and the Theory of the Rainbow,” Phil. Mag. 24, 141–176, 825–864 (1937);Phil. Mag. 25, 817–837 (1938).
  12. H. M. Nussenzveig, “High-Frequency Scattering by an Impenetrable Sphere,” Ann. Phys. (N.Y.) 34, 23–95 (1965).
    [Crossref]
  13. K. W. Ford and J. A. Wheeler, “Semiclassical Description of Scattering,” Ann. Phys. (N.Y.) 7, 259–286 (1959).
    [Crossref]
  14. H. M. Nussenzveig, “Applications of Regge Poles to Short-Wavelength Scattering,” in Methods and Problems of Theoretical Physics, edited by J. E. Bowcock (North-Holland, Amsterdam, 1970), p. 203–232.
  15. P. J. Debye, “Das Elektromagnetische Feld um Einen Zylinder und die Theorie des Regenbogens,” Physik. Z. 9, 775–778 (1908).
  16. H. M. Nussenzveig, “High-Frequency Scattering by a Transparent Sphere. I. Direct Reflection and Transmission,” J. Math. Phys. 10, 82–124 (1969).
    [Crossref]
  17. M. V. Berry, “Waves and Thorn’s Theorem,” Adv. Phys. 25, 1–26 (1976).
    [Crossref]
  18. C. B. Boyer, The Rainbow: From Myth to Mathematics (Thomas Yoseloff, New York, 1959).
  19. H. M. Nussenzveig, “The Theory of the Rainbow,” Sci. Am. 236, 116–127 (1977).
    [Crossref]
  20. 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]
  21. V. Khare and H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
    [Crossref]
  22. C. Chester, B. Friedman, and F. Ursell, “An Extension of the Method of Steepest Descents,” Proc. Camb. Phil. Soc. 53, 599–611 (1957).
    [Crossref]
  23. For rainbows formed very close to the backward direction there is an additional enhancement factor O(β1/2) of due to axial focusing (cf. Sec. III).
  24. The contrary statement in Ref. 19 is an uncalled-for editorial insertion.
  25. J. Bricard, “Contribution à l’Étude des Brouillards Naturels,” Ann. Phys. 14, 148–236 (1940).
  26. J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
    [Crossref]
  27. V. Khare, “Short-Wavelength Scattering of Electromagnetic Waves by a Homogeneous Dielectric Sphere,” Ph.D. thesis, University of Rochester (1975) (unpublished).
  28. V. Khare and H. M. Nussenzveig, (unpublished).
  29. V. Khare and H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
    [Crossref]
  30. For the early history of the subject, cf. J. M. Pernter and F. M. Exner, Meteorologische Optik (Braumüller, Vienna, 1910).
  31. T. S. Fahlen and H. C. Bryant, “Optical back scattering from single water droplets,” J. Opt. Soc. Am. 58, 304–310 (1968).
    [Crossref]
  32. J. V. Dave, “Scattering of visible light by large water spheres,” Appl. Opt. 8, 155–164 (1969).
    [Crossref] [PubMed]
  33. M. J. Saunders, “Near-field backscattering measurements from a microscopic water droplet,” J. Opt. Soc. Am. 60, 1359–1365 (1970).
    [Crossref]
  34. V. Khare and H. M. Nussenzveig, “The Theory of the Glory,” in Statistical Mechanics and Statistical Methods in Theory and Application, edited by U. Landman (Plenum, New York, 1977), 723–764.
    [Crossref]
  35. P. Walstra, “Light Scattering by Dielectric Spheres: Data on the Ripple in the Extinction Curve,” Proc. Koninkl. Nederl. Akad. Wetensch. B 67, 491–499 (1964).
  36. A. Ashkin and J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354 (1977).
    [Crossref]
  37. H. M. Nussenzveig (unpublished).
  38. H. M. Nussenzveig and W. J. Wiscombe, (unpublished).

1978 (1)

1977 (3)

H. M. Nussenzveig, “The Theory of the Rainbow,” Sci. Am. 236, 116–127 (1977).
[Crossref]

V. Khare and H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[Crossref]

A. Ashkin and J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354 (1977).
[Crossref]

1976 (2)

M. V. Berry, “Waves and Thorn’s Theorem,” Adv. Phys. 25, 1–26 (1976).
[Crossref]

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[Crossref]

1974 (1)

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

1970 (1)

1969 (3)

J. V. Dave, “Scattering of visible light by large water spheres,” Appl. Opt. 8, 155–164 (1969).
[Crossref] [PubMed]

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. M. Nussenzveig, “High-Frequency Scattering by a Transparent Sphere. I. Direct Reflection and Transmission,” J. Math. Phys. 10, 82–124 (1969).
[Crossref]

1968 (1)

1966 (1)

1965 (2)

H. M. Nussenzveig, “High-Frequency Scattering by an Impenetrable Sphere,” Ann. Phys. (N.Y.) 34, 23–95 (1965).
[Crossref]

N. A. Logan, “Survey of Some Early Studies of the Scattering of Plane Waves by a Sphere,” Proc. IEEE 53, 773–785 (1965).
[Crossref]

1964 (1)

P. Walstra, “Light Scattering by Dielectric Spheres: Data on the Ripple in the Extinction Curve,” Proc. Koninkl. Nederl. Akad. Wetensch. B 67, 491–499 (1964).

1959 (1)

K. W. Ford and J. A. Wheeler, “Semiclassical Description of Scattering,” Ann. Phys. (N.Y.) 7, 259–286 (1959).
[Crossref]

1957 (1)

C. Chester, B. Friedman, and F. Ursell, “An Extension of the Method of Steepest Descents,” Proc. Camb. Phil. Soc. 53, 599–611 (1957).
[Crossref]

1947 (1)

1940 (1)

J. Bricard, “Contribution à l’Étude des Brouillards Naturels,” Ann. Phys. 14, 148–236 (1940).

1937 (1)

B. van der Pol and H. Bremmer, “The Diffraction of Electromagnetic Waves from an Electrical Point Source round a Finitely Conducting Sphere, with Applications to Radio-Telegraphy and the Theory of the Rainbow,” Phil. Mag. 24, 141–176, 825–864 (1937);Phil. Mag. 25, 817–837 (1938).

1908 (2)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen”, Ann. Phys. (Leipzig) 25, 377–445 (1908).

P. J. Debye, “Das Elektromagnetische Feld um Einen Zylinder und die Theorie des Regenbogens,” Physik. Z. 9, 775–778 (1908).

Ashkin, A.

A. Ashkin and J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354 (1977).
[Crossref]

Berry, M. V.

M. V. Berry, “Waves and Thorn’s Theorem,” Adv. Phys. 25, 1–26 (1976).
[Crossref]

Boyer, C. B.

C. B. Boyer, The Rainbow: From Myth to Mathematics (Thomas Yoseloff, New York, 1959).

Bremmer, H.

B. van der Pol and H. Bremmer, “The Diffraction of Electromagnetic Waves from an Electrical Point Source round a Finitely Conducting Sphere, with Applications to Radio-Telegraphy and the Theory of the Rainbow,” Phil. Mag. 24, 141–176, 825–864 (1937);Phil. Mag. 25, 817–837 (1938).

Bricard, J.

J. Bricard, “Contribution à l’Étude des Brouillards Naturels,” Ann. Phys. 14, 148–236 (1940).

Bryant, H. C.

Chester, C.

C. Chester, B. Friedman, and F. Ursell, “An Extension of the Method of Steepest Descents,” Proc. Camb. Phil. Soc. 53, 599–611 (1957).
[Crossref]

Cox, A. J.

Dave, J. V.

Debye, P. J.

P. J. Debye, “Das Elektromagnetische Feld um Einen Zylinder und die Theorie des Regenbogens,” Physik. Z. 9, 775–778 (1908).

Dziedzic, J. M.

A. Ashkin and J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354 (1977).
[Crossref]

Exner, F. M.

For the early history of the subject, cf. J. M. Pernter and F. M. Exner, Meteorologische Optik (Braumüller, Vienna, 1910).

Fahlen, T. S.

Fock, V. A.

V. A. Fock, Diffraction of Radio Waves Around the Earth’s Surface (Publishers of the USSRAcademy of Sciences, Moscow, 1946).

Ford, K. W.

K. W. Ford and J. A. Wheeler, “Semiclassical Description of Scattering,” Ann. Phys. (N.Y.) 7, 259–286 (1959).
[Crossref]

Franz, W.

W. Franz, Theorie der Beugung Elektromagnetischer Wellen (Springer-Verlag, Berlin, 1957).
[Crossref]

Friedman, B.

C. Chester, B. Friedman, and F. Ursell, “An Extension of the Method of Steepest Descents,” Proc. Camb. Phil. Soc. 53, 599–611 (1957).
[Crossref]

Keller, J. B.

J. B. Keller, “A Geometrical Theory of Diffraction,” in Calculus of Variations and its Applications, Proceedings of Symposia in Applied Mathematics, edited by L. M. Graves (McGraw-Hill, New York, 1958), Vol 8.
[Crossref]

Khare, V.

V. Khare and H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[Crossref]

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

V. Khare, “Short-Wavelength Scattering of Electromagnetic Waves by a Homogeneous Dielectric Sphere,” Ph.D. thesis, University of Rochester (1975) (unpublished).

V. Khare and H. M. Nussenzveig, (unpublished).

V. Khare and H. M. Nussenzveig, “The Theory of the Glory,” in Statistical Mechanics and Statistical Methods in Theory and Application, edited by U. Landman (Plenum, New York, 1977), 723–764.
[Crossref]

Logan, N. A.

N. A. Logan, “Survey of Some Early Studies of the Scattering of Plane Waves by a Sphere,” Proc. IEEE 53, 773–785 (1965).
[Crossref]

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen”, Ann. Phys. (Leipzig) 25, 377–445 (1908).

Nussenzveig, H. M.

V. Khare and H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[Crossref]

H. M. Nussenzveig, “The Theory of the Rainbow,” Sci. Am. 236, 116–127 (1977).
[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. M. Nussenzveig, “High-Frequency Scattering by a Transparent Sphere. I. Direct Reflection and Transmission,” J. Math. Phys. 10, 82–124 (1969).
[Crossref]

H. M. Nussenzveig, “High-Frequency Scattering by an Impenetrable Sphere,” Ann. Phys. (N.Y.) 34, 23–95 (1965).
[Crossref]

H. M. Nussenzveig, “Applications of Regge Poles to Short-Wavelength Scattering,” in Methods and Problems of Theoretical Physics, edited by J. E. Bowcock (North-Holland, Amsterdam, 1970), p. 203–232.

V. Khare and H. M. Nussenzveig, (unpublished).

V. Khare and H. M. Nussenzveig, “The Theory of the Glory,” in Statistical Mechanics and Statistical Methods in Theory and Application, edited by U. Landman (Plenum, New York, 1977), 723–764.
[Crossref]

H. M. Nussenzveig (unpublished).

H. M. Nussenzveig and W. J. Wiscombe, (unpublished).

Pernter, J. M.

For the early history of the subject, cf. J. M. Pernter and F. M. Exner, Meteorologische Optik (Braumüller, Vienna, 1910).

Saunders, M. J.

Shipley, S. T.

Sommerfeld, A.

A. Sommerfeld, Optics (Academic, New York, 1954), p. 247.

Ursell, F.

C. Chester, B. Friedman, and F. Ursell, “An Extension of the Method of Steepest Descents,” Proc. Camb. Phil. Soc. 53, 599–611 (1957).
[Crossref]

van de Hulst, H. C.

H. C. van de Hulst, “A Theory of the Anti-coronae,” J. Opt. Soc. Am. 37, 16–22 (1947).
[Crossref]

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

van der Pol, B.

B. van der Pol and H. Bremmer, “The Diffraction of Electromagnetic Waves from an Electrical Point Source round a Finitely Conducting Sphere, with Applications to Radio-Telegraphy and the Theory of the Rainbow,” Phil. Mag. 24, 141–176, 825–864 (1937);Phil. Mag. 25, 817–837 (1938).

Walker, J. D.

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[Crossref]

Walstra, P.

P. Walstra, “Light Scattering by Dielectric Spheres: Data on the Ripple in the Extinction Curve,” Proc. Koninkl. Nederl. Akad. Wetensch. B 67, 491–499 (1964).

Weinman, J. A.

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.

H. M. Nussenzveig and W. J. Wiscombe, (unpublished).

Adv. Phys. (1)

M. V. Berry, “Waves and Thorn’s Theorem,” Adv. Phys. 25, 1–26 (1976).
[Crossref]

Am. J. Phys. (1)

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[Crossref]

Ann. Phys. (1)

J. Bricard, “Contribution à l’Étude des Brouillards Naturels,” Ann. Phys. 14, 148–236 (1940).

Ann. Phys. (Leipzig) (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen”, Ann. Phys. (Leipzig) 25, 377–445 (1908).

Ann. Phys. (N.Y.) (2)

H. M. Nussenzveig, “High-Frequency Scattering by an Impenetrable Sphere,” Ann. Phys. (N.Y.) 34, 23–95 (1965).
[Crossref]

K. W. Ford and J. A. Wheeler, “Semiclassical Description of Scattering,” Ann. Phys. (N.Y.) 7, 259–286 (1959).
[Crossref]

Appl. Opt. (1)

J. Math. Phys. (2)

H. M. Nussenzveig, “High-Frequency Scattering by a Transparent Sphere. I. Direct Reflection and Transmission,” J. Math. Phys. 10, 82–124 (1969).
[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. Opt. Soc. Am. (5)

Phil. Mag. (1)

B. van der Pol and H. Bremmer, “The Diffraction of Electromagnetic Waves from an Electrical Point Source round a Finitely Conducting Sphere, with Applications to Radio-Telegraphy and the Theory of the Rainbow,” Phil. Mag. 24, 141–176, 825–864 (1937);Phil. Mag. 25, 817–837 (1938).

Phys. Rev. Lett. (3)

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

A. Ashkin and J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354 (1977).
[Crossref]

V. Khare and H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[Crossref]

Physik. Z. (1)

P. J. Debye, “Das Elektromagnetische Feld um Einen Zylinder und die Theorie des Regenbogens,” Physik. Z. 9, 775–778 (1908).

Proc. Camb. Phil. Soc. (1)

C. Chester, B. Friedman, and F. Ursell, “An Extension of the Method of Steepest Descents,” Proc. Camb. Phil. Soc. 53, 599–611 (1957).
[Crossref]

Proc. IEEE (1)

N. A. Logan, “Survey of Some Early Studies of the Scattering of Plane Waves by a Sphere,” Proc. IEEE 53, 773–785 (1965).
[Crossref]

Proc. Koninkl. Nederl. Akad. Wetensch. B (1)

P. Walstra, “Light Scattering by Dielectric Spheres: Data on the Ripple in the Extinction Curve,” Proc. Koninkl. Nederl. Akad. Wetensch. B 67, 491–499 (1964).

Sci. Am. (1)

H. M. Nussenzveig, “The Theory of the Rainbow,” Sci. Am. 236, 116–127 (1977).
[Crossref]

Other (15)

C. B. Boyer, The Rainbow: From Myth to Mathematics (Thomas Yoseloff, New York, 1959).

For the early history of the subject, cf. J. M. Pernter and F. M. Exner, Meteorologische Optik (Braumüller, Vienna, 1910).

V. Khare, “Short-Wavelength Scattering of Electromagnetic Waves by a Homogeneous Dielectric Sphere,” Ph.D. thesis, University of Rochester (1975) (unpublished).

V. Khare and H. M. Nussenzveig, (unpublished).

V. Khare and H. M. Nussenzveig, “The Theory of the Glory,” in Statistical Mechanics and Statistical Methods in Theory and Application, edited by U. Landman (Plenum, New York, 1977), 723–764.
[Crossref]

H. M. Nussenzveig (unpublished).

H. M. Nussenzveig and W. J. Wiscombe, (unpublished).

W. Franz, Theorie der Beugung Elektromagnetischer Wellen (Springer-Verlag, Berlin, 1957).
[Crossref]

J. B. Keller, “A Geometrical Theory of Diffraction,” in Calculus of Variations and its Applications, Proceedings of Symposia in Applied Mathematics, edited by L. M. Graves (McGraw-Hill, New York, 1958), Vol 8.
[Crossref]

V. A. Fock, Diffraction of Radio Waves Around the Earth’s Surface (Publishers of the USSRAcademy of Sciences, Moscow, 1946).

A. Sommerfeld, Optics (Academic, New York, 1954), p. 247.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

For rainbows formed very close to the backward direction there is an additional enhancement factor O(β1/2) of due to axial focusing (cf. Sec. III).

The contrary statement in Ref. 19 is an uncalled-for editorial insertion.

H. M. Nussenzveig, “Applications of Regge Poles to Short-Wavelength Scattering,” in Methods and Problems of Theoretical Physics, edited by J. E. Bowcock (North-Holland, Amsterdam, 1970), p. 203–232.

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Figures (8)

FIG. 1
FIG. 1

Tangential incident ray IT gives rise to surface wave along TA, critically refracted to the inside at A, “totally” reflected at B, and critically refracted to the outside at C, traveling as surface wave along CT′ and reemerging tangentially as the scattered ray T′S. * This path with two shortcuts contributes to the p = 2 Debye term.

FIG. 2
FIG. 2

As θ approaches θR from the lighted side, the two real saddle points λ ¯ and λ ¯ move towards confluence; for θ in the dark side, they separate along complex conjugate directions.

FIG. 3
FIG. 3

Comparison between the exact Mie solution for |S2S2,0|2(—), the intensity | S 2 , 2 ( R ) | 2 of the rainbow term according to complex angular momentum theory (---), and the Airy approximation (–‥–), for N = 1.33 and β = 1500.

FIG. 4
FIG. 4

The closed orbit associated with tangential incidence for N ≈ 1.330 07. The numbers are the values of p at the vertices. The directions of the adjoining arrows are those of the angles ζp defined by (3.7) and (3.9). Their lengths give a qualitative indication of the ordering of surface-wave contributions (---) by increasing ζp and of rainbow contributions (—) by increasing R,P/p [cf. (3.8)]. The rainbow angles θR,p are shifted from the corresponding ζp (this is indicated for θR,11). This ordering does not take into account reflection damping, which suppresses high-p contributions.

FIG. 5
FIG. 5

Contributions to |SM(β)|2 and to |SE(β)|2 from Debye terms of various orders p (the values of p are indicated) for N ≈ 1.33007 and β = 150, 500, and 1500: —rainbow terms; ---surface-wave terms. For β = 150, there are contributions (– ‥ –) from values of p that do not appear in Fig. 4.

FIG. 6
FIG. 6

Behavior of |SM(β)|2 for N ≈ 1.33007 near β = 1500: —exact; –‥– approximation by the two leading Debye terms in Fig. 5(p = 11 + p = 24);---- approximation by the sum of all Debye terms shown in Fig. 4, without summing over Δp = 48. The inset shows an amplification (—) of the spike at position A, together with the result obtained by summing over Δp = 48 (----).

PLATE 107
PLATE 107

(H. M. Nussenzveig, p. 1068). Both the primary and secondary bows are visible in this photograph taken in Sweden on 16 July 1978. That the sky is darkest between the two bows (Alexander’s dark band) is easily visible as is a single pronounced supernumerary bow lying to the inside of the primary bow. Photograph courtesy of Alistair B. Fraser.

PLATE 108
PLATE 108

(H. M. Nussenzveig, p. 1068). The glory is seen here around the shadow of an airplane. Three orders can be seen (on the original). Photograph courtesy of Alistair B. Fraser.

Equations (34)

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S j ( β , θ ) = 1 2 l = 1 { [ 1 S l ( j ) ( β ) ] t l ( cos θ ) + [ 1 S l ( i ) ( β ) ] p l ( cos θ ) } , ( i , j = 1 , 2 ; i j ) ,
p ν ( cos θ ) = [ P ν 1 ( cos θ ) P ν + 1 ( cos θ ) ] / sin 2 θ ,
t ν ( cos θ ) = cos θ p ν ( cos θ ) + ( 2 ν + 1 ) P ν ( cos θ ) ,
S l ( j ) ( β ) = ζ l ( 2 ) ( β ) ζ l ( 1 ) ( β ) ( ln ζ l ( 2 ) ( β ) N η j ln ψ l ( α ) ln ζ l ( 1 ) ( β ) N η j ln ψ l ( α ) ) ,
η 1 = 1 , η 2 = N 2 , α = N β
i j ( β , θ ) = | S j ( β , θ ) | 2 ,
b l = [ l + ( 1 / 2 ) ] / k .
λ = l + 1 / 2
l = 0 ϕ ( l + 1 2 , r ) = m = ( ) m 0 ϕ ( λ , r ) exp ( 2 i m π λ ) d λ ,
S l ( j ) ( β ) = ζ l ( 2 ) ( β ) ζ l ( 1 ) ( β ) R 22 ( j ) + ζ l ( 2 ) ( β ) ζ l ( 1 ) ( β ) ζ l ( 1 ) ( α ) ζ l ( 2 ) ( α ) × T 21 ( j ) T 12 ( j ) ( p = 1 P ρ j p 1 + ρ j p 1 ρ j ) ,
ρ j = [ ζ l ( 1 ) ( α ) / ζ l ( 2 ) ( α ) ] R 11 ( j )
S j ( β , θ ) = S j , o ( β , θ ) + p = 1 P S j , p ( β , θ ) + remainder ,
β c β 1 / 3 l β + c β 1 / 3 , c = O ( 1 ) .
= θ θ R
λ ¯ = β sin θ 1 = N β sin θ 2 ; λ ¯ = β sin θ 1 = N β sin θ 2 ,
sin θ 1 R , p = ( p 2 N 2 ) 1 / 2 / ( p 2 1 ) 1 / 2 ( p = 2 , 3 , ) ,
S j , p ( R ) ( β , ) = β 7 / 6 exp [ 2 β A p ( ) ] × { c j , p ( β , ) A i [ ( 2 β ) 2 / 3 ζ p ( ) ] + d j , p ( β , ) β 1 / 3 A i [ ( 2 β ) 2 / 3 ζ p ( ) ] } ,
{ A p ( ) 2 3 [ ζ p ( ) ] 3 / 2 } = 1 2 [ p N ( cos θ 2 ± cos θ 2 ) ( cos θ 1 ± cos θ 1 ) ] ,
c j , p ( β , ) = c j , p ( o ) ( ) + β 1 c j , p ( 1 ) ( ) + , d j , p ( β , ) = d j , p ( o ) ( ) + β 1 d j , p ( 1 ) ( ) + ,
A i ( z ) z 1 / 4 exp ( 2 3 z 3 / 2 ) / 2 π , ( z 1 ) .
P ( N , β , θ ) = 2 β 2 [ i 1 ( N , β , θ ) + i 2 ( N , β , θ ) ] / Q s c a ( N , β ) ,
S 1 ( β , π ) = S M ( β ) + S E ( β ) = S 2 ( β , π ) ,
S j axial ( β , π ) S j , 0 ( g ) ( β , π ) + S j , 2 ( g ) ( β , π ) ,
| ρ j | = | R 11 ( j ) | = 1 b j β 1 / 3 ,
| ρ j | p exp ( p b j β 1 / 3 ) ,
N = [ cos ( 11 π / 48 ) ] 1 1.33007 ,
ζ p π p θ t ( mod 2 π ) , 0 ζ p < θ t = 2 cos 1 ( 1 / N ) .
R , p = π θ R , p | ζ p | ( N 2 1 ) 1 / 2 / p , ( p 1 ) ,
ζ p π p θ t ( mod 2 π ) , θ t ζ p < 0 .
S j , 2 ( res ) ( β , π ) = β 4 / 3 n r n j exp ( i λ n j ζ 2 ) × [ 1 + c n j β 2 / 3 + O ( β 1 ) ] ,
Δ β = 5 π / [ 22 ( N 2 1 ) 1 / 2 ] 0.814
S j S j , 2 ( res ) + S j , 11 ( R ) + S j , o ( g ) + S j , 2 ( g ) .
u = β ( π θ )
S 1 ( β , θ ) 2 S M ( β ) J 1 ( u ) + 2 S E ( β ) J 1 ( u ) / u ,