Abstract

The intensity and the degree of polarization of the radiation scattered by a large sphere were computed using the Mie theory at sufficiently small interval of the scattering angle for obtaining a complete picture of all the characteristics of the field of the scattered radiation. The results are presented for four different sizes of the water sphere (radius = 6.25 μ, 12.5 μ, 25.0 μ, and 50.0 μ) assumed to be illuminated by an unpolarized beam of monochromatic radiation with wavelength 0.4 μ. A detailed comparison is then made between the results obtained using the exact Mie theory and those obtained using an approximate approach based on the application of the other laws of the geometrical and physical optics. The angular positions of the primary and secondary rainbows, as well as those of their supernumerary bows as obtained using the approximate method, agree with those obtained from the Mie theory only if the size parameter of the sphere is of the order of 800. Besides the phenomenon of glory which is not amenable to explanation in terms of the geometrical and physical optics, the Mie computations bring out several distinct maxima and minima whose occurrence cannot be explained in likewise manner.

© 1969 Optical Society of America

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References

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  1. H. C. Van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., New York, 1957).
  2. G. Mie, Ann. Phys. 25, 377 (1908).
    [CrossRef]
  3. R. O. Gumprecht, C. M. Sliepcevich, J. Phys. Chem. 57, 90 (1953).
    [CrossRef]
  4. T. S. Fahlen, H. C. Bryant, J. Opt. Soc. Amer. 58, 304 (1968).
    [CrossRef]
  5. R. Hickling, J. Opt. Soc. Amer. 58, 455 (1968).
    [CrossRef]
  6. F. W. J. Olver, Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, Eds. (Applied Mathematics Series 55, U. S. National Bureau of Standards, Washington, D.C., 1964).
  7. H. C. Bryant, A. J. Cox, J. Opt. Soc. Amer. 56, 1529 (1966).
    [CrossRef]
  8. L. Infeld, Quart. J. Appl. Math. 5, 113 (1947).
  9. A. L. Aden, J. Appl. Phys. 22, 601 (1951).
    [CrossRef]
  10. D. Deirmendjian, R. Clasen, “Light Scattering on Partially Absorbing Homogeneous Spheres of Finite Size” (R–393–PR, The Rand Corporation, Santa Monica, Calif.1962).
  11. G. W. Kattawar, G. N. Plass, Appl. Opt. 6, 1377 (1967).
    [CrossRef] [PubMed]
  12. R. O. Gumprecht, C. M. Sliepcevich, Tables of Functions of First and Second Derivatives of Legendre Polynomials (University of Michigan Press, Ann Arbor, Michigan, 1951).
  13. J. V. Dave, “Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere (Rep. No. 320–3237, IBM Scientific Center, Palo Alto, Calif., 1968).
  14. B. Lyot, Ann. Observ. Paris–Meudon 8, 125 (1929).

1968 (2)

T. S. Fahlen, H. C. Bryant, J. Opt. Soc. Amer. 58, 304 (1968).
[CrossRef]

R. Hickling, J. Opt. Soc. Amer. 58, 455 (1968).
[CrossRef]

1967 (1)

1966 (1)

H. C. Bryant, A. J. Cox, J. Opt. Soc. Amer. 56, 1529 (1966).
[CrossRef]

1953 (1)

R. O. Gumprecht, C. M. Sliepcevich, J. Phys. Chem. 57, 90 (1953).
[CrossRef]

1951 (1)

A. L. Aden, J. Appl. Phys. 22, 601 (1951).
[CrossRef]

1947 (1)

L. Infeld, Quart. J. Appl. Math. 5, 113 (1947).

1929 (1)

B. Lyot, Ann. Observ. Paris–Meudon 8, 125 (1929).

1908 (1)

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

Aden, A. L.

A. L. Aden, J. Appl. Phys. 22, 601 (1951).
[CrossRef]

Bryant, H. C.

T. S. Fahlen, H. C. Bryant, J. Opt. Soc. Amer. 58, 304 (1968).
[CrossRef]

H. C. Bryant, A. J. Cox, J. Opt. Soc. Amer. 56, 1529 (1966).
[CrossRef]

Clasen, R.

D. Deirmendjian, R. Clasen, “Light Scattering on Partially Absorbing Homogeneous Spheres of Finite Size” (R–393–PR, The Rand Corporation, Santa Monica, Calif.1962).

Cox, A. J.

H. C. Bryant, A. J. Cox, J. Opt. Soc. Amer. 56, 1529 (1966).
[CrossRef]

Dave, J. V.

J. V. Dave, “Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere (Rep. No. 320–3237, IBM Scientific Center, Palo Alto, Calif., 1968).

Deirmendjian, D.

D. Deirmendjian, R. Clasen, “Light Scattering on Partially Absorbing Homogeneous Spheres of Finite Size” (R–393–PR, The Rand Corporation, Santa Monica, Calif.1962).

Fahlen, T. S.

T. S. Fahlen, H. C. Bryant, J. Opt. Soc. Amer. 58, 304 (1968).
[CrossRef]

Gumprecht, R. O.

R. O. Gumprecht, C. M. Sliepcevich, J. Phys. Chem. 57, 90 (1953).
[CrossRef]

R. O. Gumprecht, C. M. Sliepcevich, Tables of Functions of First and Second Derivatives of Legendre Polynomials (University of Michigan Press, Ann Arbor, Michigan, 1951).

Hickling, R.

R. Hickling, J. Opt. Soc. Amer. 58, 455 (1968).
[CrossRef]

Infeld, L.

L. Infeld, Quart. J. Appl. Math. 5, 113 (1947).

Kattawar, G. W.

Lyot, B.

B. Lyot, Ann. Observ. Paris–Meudon 8, 125 (1929).

Mie, G.

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

Olver, F. W. J.

F. W. J. Olver, Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, Eds. (Applied Mathematics Series 55, U. S. National Bureau of Standards, Washington, D.C., 1964).

Plass, G. N.

Sliepcevich, C. M.

R. O. Gumprecht, C. M. Sliepcevich, J. Phys. Chem. 57, 90 (1953).
[CrossRef]

R. O. Gumprecht, C. M. Sliepcevich, Tables of Functions of First and Second Derivatives of Legendre Polynomials (University of Michigan Press, Ann Arbor, Michigan, 1951).

Van de Hulst, H. C.

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

Ann. Observ. Paris–Meudon (1)

B. Lyot, Ann. Observ. Paris–Meudon 8, 125 (1929).

Ann. Phys. (1)

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

Appl. Opt. (1)

J. Appl. Phys. (1)

A. L. Aden, J. Appl. Phys. 22, 601 (1951).
[CrossRef]

J. Opt. Soc. Amer. (3)

T. S. Fahlen, H. C. Bryant, J. Opt. Soc. Amer. 58, 304 (1968).
[CrossRef]

R. Hickling, J. Opt. Soc. Amer. 58, 455 (1968).
[CrossRef]

H. C. Bryant, A. J. Cox, J. Opt. Soc. Amer. 56, 1529 (1966).
[CrossRef]

J. Phys. Chem. (1)

R. O. Gumprecht, C. M. Sliepcevich, J. Phys. Chem. 57, 90 (1953).
[CrossRef]

Quart. J. Appl. Math. (1)

L. Infeld, Quart. J. Appl. Math. 5, 113 (1947).

Other (5)

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

F. W. J. Olver, Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, Eds. (Applied Mathematics Series 55, U. S. National Bureau of Standards, Washington, D.C., 1964).

D. Deirmendjian, R. Clasen, “Light Scattering on Partially Absorbing Homogeneous Spheres of Finite Size” (R–393–PR, The Rand Corporation, Santa Monica, Calif.1962).

R. O. Gumprecht, C. M. Sliepcevich, Tables of Functions of First and Second Derivatives of Legendre Polynomials (University of Michigan Press, Ann Arbor, Michigan, 1951).

J. V. Dave, “Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere (Rep. No. 320–3237, IBM Scientific Center, Palo Alto, Calif., 1968).

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Figures (14)

Fig. 1
Fig. 1

Path of a light ray through a sphere according to geometrical optics.

Fig. 2
Fig. 2

Scattered intensity and the degree of polarization as a function of the scattering angle. r = 6.25 μ, λ = 0.4 μ, x ≑ 98.2, m = 1.342 − 0.0i, θ = 0.0° (0.2°) 90.0°.

Fig. 3
Fig. 3

Scattered intensity as a function of the scattering angle. r = 6.25μ, λ = 0.4μ, x ≑ 98.2, m = 1.342 − 0.0i θ = 0.0° (0.2°) 20.0°. Solid curve: Mie theory; broken curve: diffraction theory.

Fig. 4
Fig. 4

Scattered intensity and the degree of polarization as a function of the scattering angle. r = 6.25μ, λ = 0.4 μ, x ≑ 98.2, m = 1.342 − 0.0i, θ = 90.0° (0.2°) 180.0°.

Fig. 5
Fig. 5

Scattered intensity and the degree of polarization as a function of the scattering angle. r = 6.25μ, λ = 0.4μ, x ≑ 98.2, m = 1.342 − 0.0i, θ = 160.0° (0.2°) 180.0°.

Fig. 6
Fig. 6

Scattered intensity as a function of the scattering angle. r = 50.0μ, λ = 0.4μ, x ≑ 785.4, m = 1.342 − 0.0i, θ = 1.00° (0.02°) 3.00°. Solid curve: Mie theory; broken curve diffraction theory.

Fig. 7
Fig. 7

Scattered intensity and the degree of polarization as a function of the scattering angle. r = 12.5μ, λ = 0.4μ, x ≑ 196.3, m = 1.342 − 0.0i, θ = 110.0° (0.1°) 170.0°.

Fig. 8
Fig. 8

Scattered intensity and the degree of polarization as a function of the scattering angle. r = 25.0 μ, λ = 0.4μ, x ≑ 392.7, m = 1.342 − 0.0i, θ = 105.00° (0.05°) 135.00°.

Fig. 9
Fig. 9

Scattered intensity and the degree of polarization as a function of the scattering angle. r = 25.0μ, λ = 0.4μ, x ≑ 392.7, m = 1.342 − 0.0i, θ = 135.00° (0.05°) 165.00°.

Fig. 10
Fig. 10

Scattered intensity and the degree of polarization as a function of the scattering angle. r = 50.0μ, λ = 0.43μ, x ≑ 785.4, m = 1.342 − 0.0i, θ = 100.00° (0.02°) 130.00°.

Fig. 11
Fig. 11

Scattered intensity and the degree of polarization as a function of the scattering angle. r = 50.0μ, λ = 0.4μ, x ≑ 785.4, m = 1.342 − 0.0i, θ = 135.00° (0.02°) 155.00°.

Fig. 12
Fig. 12

Scattered intensity and the degree of polarization as a function of the scattering angle. r = 12.5μ, λ = 0.4μ, x ≑ 193.3, m = 1.342 − 0.0i, θ = 170.0° (0.1°) 180.0°.

Fig. 13
Fig. 13

Scattered intensity and the degree of polarization as a function of the scattering angle. r = 25.0μ, λ = 0.4μ, x ≑ 3927, m = 1.342 − 0.0i, θ = 175.00° (0.05°) 180.00°.

Fig. 14
Fig. 14

Scattered intensity and the degree of polarization as a function of the scattering angle. r = 50.0μ, λ = 0.4μ, x ≑ 785.4, m = 1.342 − 0.0i, θ = 178.00° (0.02°) 180.000.

Tables (3)

Tables Icon

Table I Positions of the Primary Rainbow and its Supernumeraries as Obtained Using the Laws of Geometric and Physical Opticsa

Tables Icon

Table II Positions of the Secondary Rainbow and its Supernumeraries as Obtained Using the Laws of Geometric and Physical Opticsa

Tables Icon

Table III Notations as Used by Deirmendjian and Clasen and by Kattawar and Plass

Equations (21)

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I a = x 4 [ J 1 ( x sin θ ) / x sin θ ] 2 ,
cos τ = ( 1 / m ) cos τ .
θ = 2 τ - 2 p τ .
sin τ m = [ ( m 2 - 1 ) / ( p 2 - 1 ) ] 1 2 .
Δ δ = 2 x [ sin τ a - sin τ b - p m ( sin τ a - sin τ b ) ] .
Δ δ = 2 π ( K + 1 4 ) .
I j ( x , m , θ ) = S j ( x , m , θ ) S j * ( x , m , θ ) ,             ( j = 1 , 2 ) ,
S 1 ( x , m , θ ) = n = 1 ( 2 n + 1 ) n ( n + 1 ) [ a n ( x , m ) π n ( μ ) + b n ( x , m ) τ n ( μ ) ] .
S 2 ( x , m , θ ) = n = 1 ( 2 n + 1 ) n ( n + 1 ) [ a n ( x , m ) τ n ( μ ) + b n ( x , m ) π n ( μ ) ] ,
a n ( x , m ) = ψ n ( m x ) ψ n ( x ) - m ψ n ( m x ) ψ n ( x ) ψ n ( m x ) ξ n ( x ) - m ψ n ( m x ) ξ n ( x ) ,
b n ( x , m ) = m ψ n ( m x ) ψ n ( x ) - ψ n ( m x ) ψ n ( x ) m ψ n ( m x ) ξ n ( x ) - ψ n ( m x ) ξ n ( x )
π n ( μ ) = d P n ( μ ) / d μ ,
τ n ( μ ) = μ π n ( μ ) - ( 1 - μ 2 ) [ d π n ( μ ) / d μ ] .
d [ ln ψ n ( m x ) ] / d ( m x ) = ψ n ( m x ) / ψ n ( m x )
a n ( x , m ) = { [ A n ( m x ) / m ] + ( n / x ) } Re [ ξ n ( x ) ] - Re [ ξ n - 1 ( x ) ] { [ A n ( m x ) / m ] + ( n / x ) } ξ n ( x ) - ξ n - 1 ( x )
b n ( x , m ) = [ m A n ( m x ) + ( n / x ) ] Re [ ξ n ( x ) ] - Re [ ξ n - 1 ( x ) ] [ m A n ( m x ) + ( n / x ) ] ξ n ( x ) - ξ n - 1 ( x ) ,
A n ( m x ) = - n m x + j n - 1 ( m x ) j n ( m x ) = - n m x + 1 ( n / m x ) - A n - 1 ( m x ) ,
π n ( - μ ) = ( - 1 ) n - 1 π n ( μ ) ,
τ n ( - μ ) = ( - 1 ) n τ n ( μ ) .
I ( x , m , θ ) = 1 2 [ I 1 ( x , m , θ ) + I 2 ( x , m , θ ) ] ,
P ( x , m , θ ) = I 1 ( x , m , θ ) - I 2 ( x , m , θ ) I 1 ( x , m , θ ) + I 2 ( x , m , θ )

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