Abstract

The basic Mie formulas are explained and several useful scattering functions are defined, such as the angular Mie scattering coefficients iθ and the Mie phase function pM. Finally, the scattering pattern for the forward, sideward, and backward area is surveyed, based on data for the refractive index n=1.33.

© 1962 Optical Society of America

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References

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  1. G. Mie, Ann. Physik. (4) 25, 377 (1908).
    [Crossref]
  2. Or rather, the distance of observation is large compared to the radius of the particle (far-field solution).
  3. The am, bm, πm, and τm differ from those tabulated, such as A. Lowan [Tables of Scattering Functions for Spherical Particles’ (U. S. Natl. Bur. Standards, 1949) Appl. Math. Ser. NR. 4], R. O. Gumprecht and C. M. Sliepcevich [Light Scattering Functions for Spherical Particles (University of Michigan Press, Ann Arbor Michigan, 1951)], and R. Penndorf and B. Goldberg [“New tables of scattering functions for spherical particles,” Geophys. Research Paper No. 45, parts 1–5 (1956)], and others. If the author’s own tabulated values are denoted with amT, bmT, πmT, and τmT, the above defined am, bm, πm, and τm can be computed from the tabulated values as follows:am=(-1)miamT,bm=(-1)m+1ibmT,πm=(-1)m+1πmT/m(m+1),τm=(-1)mτmT/m(m+1).
  4. H. E. Van de Hulst, Light Scattering by Small Spheres (John Wiley & Sons, Inc.New York, 1957).
  5. In radar application, iθ is called the bistatic scattering coefficient.
  6. From now on, only i1 and i2 will be written, but the variables still are {α,n,θ}, as expressed by Eq. (4).
  7. S. Chandrasekhar, Radiative Transfer (Oxford University Press, Oxford, 1950), p. 5.
  8. Ch.-M. Chu, G. C. Clark, and S. W. Churchill, “Tables of Angular distribution coefficients for light-scattering by spheres”, Eng. Research Inst., University of Michigan, Ann Arbor (1957).
  9. H. Walter, Optik 14, 130 (1957).
  10. H. Walter, Optik 16, 401 (1959).
  11. R. O. Gumprecht and C. M. Sliepcevich, J. Phys. Chem. 57, 90 (1953).
    [Crossref]

1959 (1)

H. Walter, Optik 16, 401 (1959).

1957 (1)

H. Walter, Optik 14, 130 (1957).

1953 (1)

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

1908 (1)

G. Mie, Ann. Physik. (4) 25, 377 (1908).
[Crossref]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford University Press, Oxford, 1950), p. 5.

Chu, Ch.-M.

Ch.-M. Chu, G. C. Clark, and S. W. Churchill, “Tables of Angular distribution coefficients for light-scattering by spheres”, Eng. Research Inst., University of Michigan, Ann Arbor (1957).

Churchill, S. W.

Ch.-M. Chu, G. C. Clark, and S. W. Churchill, “Tables of Angular distribution coefficients for light-scattering by spheres”, Eng. Research Inst., University of Michigan, Ann Arbor (1957).

Clark, G. C.

Ch.-M. Chu, G. C. Clark, and S. W. Churchill, “Tables of Angular distribution coefficients for light-scattering by spheres”, Eng. Research Inst., University of Michigan, Ann Arbor (1957).

Gumprecht, R. O.

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

Lowan, A.

The am, bm, πm, and τm differ from those tabulated, such as A. Lowan [Tables of Scattering Functions for Spherical Particles’ (U. S. Natl. Bur. Standards, 1949) Appl. Math. Ser. NR. 4], R. O. Gumprecht and C. M. Sliepcevich [Light Scattering Functions for Spherical Particles (University of Michigan Press, Ann Arbor Michigan, 1951)], and R. Penndorf and B. Goldberg [“New tables of scattering functions for spherical particles,” Geophys. Research Paper No. 45, parts 1–5 (1956)], and others. If the author’s own tabulated values are denoted with amT, bmT, πmT, and τmT, the above defined am, bm, πm, and τm can be computed from the tabulated values as follows:am=(-1)miamT,bm=(-1)m+1ibmT,πm=(-1)m+1πmT/m(m+1),τm=(-1)mτmT/m(m+1).

Mie, G.

G. Mie, Ann. Physik. (4) 25, 377 (1908).
[Crossref]

Sliepcevich, C. M.

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

Van de Hulst, H. E.

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

Walter, H.

H. Walter, Optik 16, 401 (1959).

H. Walter, Optik 14, 130 (1957).

Ann. Physik. (4) (1)

G. Mie, Ann. Physik. (4) 25, 377 (1908).
[Crossref]

J. Phys. Chem. (1)

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

Optik (2)

H. Walter, Optik 14, 130 (1957).

H. Walter, Optik 16, 401 (1959).

Other (7)

Or rather, the distance of observation is large compared to the radius of the particle (far-field solution).

The am, bm, πm, and τm differ from those tabulated, such as A. Lowan [Tables of Scattering Functions for Spherical Particles’ (U. S. Natl. Bur. Standards, 1949) Appl. Math. Ser. NR. 4], R. O. Gumprecht and C. M. Sliepcevich [Light Scattering Functions for Spherical Particles (University of Michigan Press, Ann Arbor Michigan, 1951)], and R. Penndorf and B. Goldberg [“New tables of scattering functions for spherical particles,” Geophys. Research Paper No. 45, parts 1–5 (1956)], and others. If the author’s own tabulated values are denoted with amT, bmT, πmT, and τmT, the above defined am, bm, πm, and τm can be computed from the tabulated values as follows:am=(-1)miamT,bm=(-1)m+1ibmT,πm=(-1)m+1πmT/m(m+1),τm=(-1)mτmT/m(m+1).

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

In radar application, iθ is called the bistatic scattering coefficient.

From now on, only i1 and i2 will be written, but the variables still are {α,n,θ}, as expressed by Eq. (4).

S. Chandrasekhar, Radiative Transfer (Oxford University Press, Oxford, 1950), p. 5.

Ch.-M. Chu, G. C. Clark, and S. W. Churchill, “Tables of Angular distribution coefficients for light-scattering by spheres”, Eng. Research Inst., University of Michigan, Ann Arbor (1957).

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Figures (4)

Fig. 1
Fig. 1

Definition of scattering parameters.

Fig. 2
Fig. 2

Intensity functions i1+i2 for n=1.33 as function of size parameter α for scattering angles θ=0, 90, and 180°. Solid curve for data computed in small steps (Δα=0.1); broken curve represents interpolation between available data, it does not represent true course of the intensity functions. Left inset represents Rayleigh region.

Fig. 3
Fig. 3

Intensity functions i1+i2 for n=1.33 as function of size parameter a for selected scattering angles θ between 0 and 40°. Curves are somewhat smoothed because of logarithmic scale.

Fig. 4
Fig. 4

General trend of i1+i2, iθ and pM for n=1.33 as function of size parameter α. Data are smoothed because of large range, but this allows comparison of the different trends of the three most important scattering functions.

Tables (1)

Tables Icon

Table I Occurrence of “break” for n=1.33 (values of α given to the nearest integer).

Equations (21)

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J = J 0 f { λ , r , n , θ } .
J { θ , n } = ( J 0 / 2 k 2 d 2 ) ( i 1 { α , n , θ } + i 2 { α , n , θ } ) .
i 1 = S 1 2 = | 1 ( 2 m + 1 ) ( a m π m + b m τ m ) | 2 i 2 = S 2 2 = | 1 ( 2 m + 1 ) ( a m τ m + b m π m ) | 2 .
i θ { α , n , θ } = ( 2 π α 2 ) - 1 ( i 1 { α , n , θ } + i 2 { α , n , θ } )
d ω = 2 π sin θ d θ .
0 4 π i θ d ω = 2 π 0 π i θ sin θ d θ = ( 1 / α 2 ) 0 π ( i 1 + i 2 ) sin θ d θ = K ,
0 π ( i 1 + i 2 ) sin θ d θ = 2 1 ( 2 m + 1 ) ( a m 2 + b m 2 ) ,
σ θ = π r 2 i θ = ( 2 k 2 ) - 2 ( i 1 + i 2 )
σ = 0 4 π σ θ d ω = 2 π 0 π σ θ sin θ d θ = ( λ 2 / 4 π ) 0 π ( i 1 + i 2 ) sin θ d θ .
κ θ = π r 2 i θ N / ρ a = ( π r 2 N / ρ a ) ( i 1 + i 2 ) .
β θ = π r 2 N i θ = ( N / 2 k 2 ) ( i 1 + i 2 ) .
β = 2 π 0 π β θ sin θ d θ = ( π r 2 N / α 2 ) 0 π ( i 1 + i 2 ) sin θ d θ = π r 2 N K .
p R { cos θ } = 3 4 ( 1 + cos 2 θ ) ,
p R { cos θ } = 1.5.
p { x } = m = 0 ω ˜ m P m { x } ,
0 4 π p { x } d ω 4 π = 1 ,
p M = 4 π i θ / K = 2 ( i 1 + i 2 ) / α 2 K .
p M { cos θ } = 4 π β θ / β = 4 π κ θ / κ = 4 π σ θ / σ .
f = p M / 4 π = i θ / K ,
f { cos θ } = i 1 { θ } + i 2 { θ } i 1 { π / 2 } + i 2 { π / 2 } ,
am=(-1)miamT,bm=(-1)m+1ibmT,πm=(-1)m+1πmT/m(m+1),τm=(-1)mτmT/m(m+1).