Abstract

The glory ray and the rainbow are analyzed by considering the scattering of light from inhomogeneous particles. It is shown that melting ice crystals may be strong contributors to the glory ray. Geometrical optics is used to investigate and catalog a wide variety of particle inhomogeneities which support rainbow and glory rays.

© 1977 Optical Society of America

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References

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  1. H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  2. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  3. C. L. Brockman, “High Frequency Electromagnetic Wave Backscattering from Radially Inhomogeneous Dielectric Spheres,” Ph.D. thesis, UCLA, 1974.
  4. H. M. Nussenzweig, J. Math. Phys. 10, 125 (1969).
    [CrossRef]
  5. H. Inada, M. A. Plonus, IEEE Trans. Antennas Propag. AP-18, 89 (1970).
    [CrossRef]
  6. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965).

1970

H. Inada, M. A. Plonus, IEEE Trans. Antennas Propag. AP-18, 89 (1970).
[CrossRef]

1969

H. M. Nussenzweig, J. Math. Phys. 10, 125 (1969).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965).

Brockman, C. L.

C. L. Brockman, “High Frequency Electromagnetic Wave Backscattering from Radially Inhomogeneous Dielectric Spheres,” Ph.D. thesis, UCLA, 1974.

Inada, H.

H. Inada, M. A. Plonus, IEEE Trans. Antennas Propag. AP-18, 89 (1970).
[CrossRef]

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Nussenzweig, H. M.

H. M. Nussenzweig, J. Math. Phys. 10, 125 (1969).
[CrossRef]

Plonus, M. A.

H. Inada, M. A. Plonus, IEEE Trans. Antennas Propag. AP-18, 89 (1970).
[CrossRef]

Van de Hulst, H. C.

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

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965).

IEEE Trans. Antennas Propag.

H. Inada, M. A. Plonus, IEEE Trans. Antennas Propag. AP-18, 89 (1970).
[CrossRef]

J. Math. Phys.

H. M. Nussenzweig, J. Math. Phys. 10, 125 (1969).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965).

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

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

C. L. Brockman, “High Frequency Electromagnetic Wave Backscattering from Radially Inhomogeneous Dielectric Spheres,” Ph.D. thesis, UCLA, 1974.

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Figures (14)

Fig. 1
Fig. 1

Ray behavior inside a dielectric sphere.

Fig. 2
Fig. 2

Front axial ray.

Fig. 3
Fig. 3

Geometry of rays in an inhomogeneous spherical particle.

Fig. 4
Fig. 4

Possible geometry of a p = 2, ray for m < −1.

Fig. 5
Fig. 5

Possible geometry of a p = 2, ray for m > 0.

Fig. 6
Fig. 6

Ray trajectory.

Fig. 7
Fig. 7

Ray trajectory.

Fig. 8
Fig. 8

Examples of glory ray behavior.

Fig. 9
Fig. 9

Example of internally refracted ray behavior.

Fig. 10
Fig. 10

δ vs β for case 1—g < 1.

Fig. 11
Fig. 11

δ vs α for case 2—1 < g <g1.

Fig. 12
Fig. 12

δ vs α for case 3—g1 < g < g2.

Fig. 13
Fig. 13

δ vs α for case 4—g2 < g < g3.

Fig. 14
Fig. 14

δ vs α for cases 5—g3 < g.

Tables (2)

Tables Icon

Table I Glory Ray Summary for g = 2

Tables Icon

Table II Limiting Values of g

Equations (33)

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r 1 a = 1 - g 1 + g
r 1 b = g - 1 g + 1
E s / E i t 1 a ( r 2 a ) p t 2 a
E s / E i t 1 b ( r 2 b ) p t 2 b
r 1 a = 1 - M 1 + M r 2 a = M - 1 M + 1 t 1 a = 2 M + 1 t 2 a = 2 M M + 1 ,
r 1 b = g 2 - M g 2 + M r 2 b = M - g 2 M + g 2 t 1 b = 2 g 2 M + g 2 t 2 b = 2 M M + g 2 ,
M = g cos β cos α ,
n ( r ) r × s = constant = n ( r ) r sin ϕ ¯ = c ¯ ,
sin α = g sin β .
θ ¯ = 2 m + 1 ( π 2 - β ) ,             m > - 1.
θ p = ( p + 1 ) θ ¯ ,
θ p = p + 1 m + 1 ( π 2 - β ) .
θ B = 2 α + ( p + 1 ) θ ¯ - π .
θ B = ( 2 N - 1 ) π ,
2 α + 2 ( p + 1 m + 1 ) ( π 2 - β ) = 2 N π .
θ B = 2 α + ( p + 1 m + 1 ) ( π - 2 β ) - π .
δ = 2 α + ( p + 1 m + 1 ) ( π - 2 β ) - 2 N π ,
ξ = { sin β sin [ ( m + 1 ) θ + β ] } 1 / ( m + 1 ) .
M = p + 1 m + 1 ,
cos α = [ g 2 - 1 ( p + 1 m + 1 ) 2 - 1 ] 1 / 2 .
cos α = [ g 2 - 1 ( p + 1 ) 2 - 1 ] 1 / 2 .
- 1 + p + 1 g m r < ,
m r = - + ( p + 1 ) cos α r g cos α r .
- 1 < m p < - 1 + p + 1 2 N .
- 1 < m r - 1 + p + 1 g ,
- 1 + ( p + 1 ) cos α r ( g 2 - sin 2 α r ) 1 / 2 < m p < - 1 + p + 1 2 N ,
- 1 + ( p + 1 ) cos α r ( g 2 - sin 2 α r ) 1 / 2 m p - 1 + ( p + 1 ) ( 1 - 2 π sin - 1 1 g ) 2 N - 1 ,
g 1 = 1 sin ( π 4 N ) .
- 1 + ( p + 1 ) cos α r ( g 2 - sin 2 α r ) 1 / 2 m r p - 1 + p + 1 2 N ,
g 2 = 2 N - 1 1 - 2 π sin - 1 1 g 2 .
g 2 ( 2 N - 1 ) + 2 π .
- 1 + ( p + 1 ) cos α r ( g 2 - sin 2 α r ) 1 / 2 m r p - 1 + p + 1 2 N - 1 ( 1 - 2 π sin - 1 1 g ) .
g 3 = 2 N .

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