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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  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).
  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).
  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).
  7. N. A. Logan, "Survey of Some Early Studies of the Scattering of Plane Waves by a Sphere," Proc. IEEE 53, 773–785 (1965).
  8. W. Franz, Theorie der Beugung Elektromagnetischer Wellen (Springer-Verlag, Berlin, 1957).
  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.
  10. V. A. Fock, Diffraction of Radio Waves Around the Earth's Surface (Publishers of the USSR Academy 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); 25, 817–837 (1938).
  12. H. M. Nussenzveig, "High-Frequency Scattering by an Impenetrable Sphere," Ann. Phys. (N.Y.) 34, 23–95 (1965).
  13. K. W. Ford and J. A. Wheeler, "Semiclassical Description of Scattering," Ann. Phys. (N.Y.) 7, 259–286 (1959).
  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).
  17. M. V. Berry, "Waves and Thom's Theorem," Adv. Phys. 25, 1–26 (1976).
  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).
  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).
  21. V. Khare and H. M. Nussenzveig, "Theory of the rainbow," Phys. Rev. Lett. 33, 976–980 (1974).
  22. C. Chester, B. Friedman and F. Ursell, "An Extension of the Method of Steepest Descents," Proc. Camb. Phil. Soc. 53, 599–611 (1957).
  23. For rainbows formed very close to the backward direction there is an additional enhancement factor of θ(β1/2) 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'Éude 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).
  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).
  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).
  32. J. V. Dave, "Scattering of visible light by large water spheres," Appl. Opt. 8, 155–164 (1969).
  33. M. J. Saunders, "Near-field backscattering measurements from a microscopic water droplet," J. Opt. Soc. Am. 60, 1359–1365 (1970).
  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.
  35. P. Walstra, "Light Scattering by Dielectric Spheres: Data on the Ripple in the Extinction Curve," Proc. Koninki. 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).
  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).

V. Khare and H. M. Nussenzveig, "Theory of the glory," Phys. Rev. Lett. 38, 1279–1282 (1977).

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

1976 (2)

M. V. Berry, "Waves and Thom's Theorem," Adv. Phys. 25, 1–26 (1976).

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

1974 (1)

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

1970 (1)

1969 (3)

J. V. Dave, "Scattering of visible light by large water spheres," Appl. Opt. 8, 155–164 (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).

H. M. Nussenzveig, "High-Frequency Scattering by a Transparent Sphere. I. Direct Reflection and Transmission," J. Math. Phys. 10, 82–124 (1969).

1968 (1)

1966 (1)

1965 (2)

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

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

1964 (1)

P. Walstra, "Light Scattering by Dielectric Spheres: Data on the Ripple in the Extinction Curve," Proc. Koninki. 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).

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).

1947 (1)

1940 (1)

J. Bricard, "Contribution à l'Éude des Brouillards Naturels," Ann. Phys. 14, 148–236 (1940).

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).

Berry, M. V.

M. V. Berry, "Waves and Thom's Theorem," Adv. Phys. 25, 1–26 (1976).

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); 25, 817–837 (1938).

Bricard, J.

J. Bricard, "Contribution à l'Éude 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).

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).

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 USSR Academy 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).

Franz, W.

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

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).

Hulst, H. C. van de

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

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

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.

Khare, V.

V. Khare and H. M. Nussenzveig, "Theory of the glory," Phys. Rev. Lett. 38, 1279–1282 (1977).

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

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.

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).

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).

H. M. Nussenzveig, "The Theory of the Rainbow," Sci. Am. 236, 116–127 (1977).

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

H. M. Nussenzveig, "High-Frequency Scattering by a Transparent Sphere. I. Direct Reflection and Transmission," J. Math. Phys. 10, 82–124 (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).

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

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.

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.

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).

Pol, B. van der

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); 25, 817–837 (1938).

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).

Walker, J. D.

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

Walstra, P.

P. Walstra, "Light Scattering by Dielectric Spheres: Data on the Ripple in the Extinction Curve," Proc. Koninki. 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).

Wiscombe, W. J.

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

Adv. Phys. (1)

M. V. Berry, "Waves and Thom's Theorem," Adv. Phys. 25, 1–26 (1976).

Am. J. Phys. (1)

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

Ann. Phys. (1)

J. Bricard, "Contribution à l'Éude 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).

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

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).

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).

J. Opt. Soc. Am. (5)

Phys. Rev. Lett. (3)

V. Khare and H. M. Nussenzveig, "Theory of the glory," Phys. Rev. Lett. 38, 1279–1282 (1977).

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

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

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).

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).

Proc. Koninki. Nederl. Akad. Wetensch. (1)

P. Walstra, "Light Scattering by Dielectric Spheres: Data on the Ripple in the Extinction Curve," Proc. Koninki. 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).

Other (16)

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).

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

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

H. M. Nussenzveig (unpublished).

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

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

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

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.

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

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); 25, 817–837 (1938).

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

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

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

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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