Abstract

For x/n ≫ 1, the following relations between the Mie-scattering functions an (x,m) and bn(x,m) are satisfied: a1 (x,m) = b2 (x,m) = a3 (x,m) = ⋯ = an−1(x,m) = bn(x,m) and b1(x,m) = a2 (x,m) = b3 (x,m) = ⋯ = bn−1 (x,m) = an(x,m) for arbitrary refractive index m. By use of these relations, the Van de Hulst and Deirmendjian conjectures about the x → ∞ behavior of the scattering functions or their linear and bilinear combinations, as well as several new relations, are rigorously proved.

© 1973 Optical Society of America

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References

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  1. G. Mie, Ann. Phys. (Leipz.) 25, 377 (1908).
    [Crossref]
  2. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  3. Reference 2, p. 107.
  4. W. M. Irvine, J. Opt. Soc. Am. 55, 16 (1965).
    [Crossref]
  5. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier, New York, 1969), p. 41.
  6. B. M. Herman, Q. J. R. Meteorol. Soc. 88, 143 (1962).
    [Crossref]
  7. A. L. Aden, J. Appl. Phys. 22, 601 (1951).
    [Crossref]
  8. Reference 2, p. 276.
  9. Reference 5, p. 27.
  10. B. M. Herman and L. J. Battan, Q. J. R. Meteorol. Soc. 87, 223 (1961).
    [Crossref]

1965 (1)

1962 (1)

B. M. Herman, Q. J. R. Meteorol. Soc. 88, 143 (1962).
[Crossref]

1961 (1)

B. M. Herman and L. J. Battan, Q. J. R. Meteorol. Soc. 87, 223 (1961).
[Crossref]

1951 (1)

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

1908 (1)

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

Aden, A. L.

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

Battan, L. J.

B. M. Herman and L. J. Battan, Q. J. R. Meteorol. Soc. 87, 223 (1961).
[Crossref]

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier, New York, 1969), p. 41.

Herman, B. M.

B. M. Herman, Q. J. R. Meteorol. Soc. 88, 143 (1962).
[Crossref]

B. M. Herman and L. J. Battan, Q. J. R. Meteorol. Soc. 87, 223 (1961).
[Crossref]

Irvine, W. M.

Mie, G.

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

van de Hulst, H. C.

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

Ann. Phys. (Leipz.) (1)

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

J. Appl. Phys. (1)

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

J. Opt. Soc. Am. (1)

Q. J. R. Meteorol. Soc. (2)

B. M. Herman, Q. J. R. Meteorol. Soc. 88, 143 (1962).
[Crossref]

B. M. Herman and L. J. Battan, Q. J. R. Meteorol. Soc. 87, 223 (1961).
[Crossref]

Other (5)

Reference 2, p. 276.

Reference 5, p. 27.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier, New York, 1969), p. 41.

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

Reference 2, p. 107.

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Figures (12)

Fig. 1
Fig. 1

Real part of a1(x), b2(x), and a3(x) for completely reflecting sphere.

Fig. 2
Fig. 2

Imaginary part of a1(x), b2(x), and a3(x) for completely reflecting sphere.

Fig. 3
Fig. 3

Real and imaginary part of a1(x) for m = ∞. In the right-hand side Rea1(x), Reb2(x), and Rea3(x) are shown in larger scale.

Fig. 4
Fig. 4

Re(a1 + b1) for completely reflecting sphere converges to 1.

Fig. 5
Fig. 5

Re(a1 + b1) with real refractive index oscillates between 0 and 2.

Fig. 6
Fig. 6

Im(a1 + b1) with real refractive index oscillates between −1 and 1.

Fig. 7
Fig. 7

Oscillations of |a1|2 + |b1|2 and Re(a1b1*) for real m.

Fig. 8
Fig. 8

Re(a1 + b1) for complex m.

Fig. 9
Fig. 9

Im(a1 + b1) for complex m.

Fig. 10
Fig. 10

|a1|2 + |b1|2 for complex m.

Fig. 11
Fig. 11

Re(a1b1*) and |a1b1|2 for complex m.

Fig. 12
Fig. 12

| a 1 1 2 | and | b 1 1 2 | for complex m.

Tables (1)

Tables Icon

Table I Values of the real and imaginary parts of a1(x), b2(x), and a3(x) for completely reflecting sphere.

Equations (67)

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S 1 ( θ , x , m ) = n = 1 2 n + 1 n ( n + 1 ) [ a n ( x , m ) π n ( θ ) + b n ( x , m ) τ n ( θ ) ] ,
S 2 ( θ , x , m ) = n = 1 2 n + 1 n ( n + 1 ) [ b n ( x , m ) π n ( θ ) + a n ( x , m ) τ n ( θ ) ] ,
a n ( x , m ) = ψ n ( x ) ψ n ( m x ) m ψ n ( m x ) ψ n ( x ) ζ n ( x ) ψ n ( m x ) m ψ n ( m x ) ζ n ( x ) ,
b n ( x , m ) = m ψ n ( x ) ψ n ( m x ) ψ n ( m x ) ψ n ( x ) m ζ n ( x ) ψ n ( m x ) ψ n ( m x ) ζ n ( x ) ,
π n ( θ ) = d P n ( cos θ ) d cos θ ,
τ n ( θ ) = π n ( θ ) cos θ sin 2 θ d π n ( θ ) d cos θ .
Q ext ( x , m ) = 2 x 2 n = 1 ( 2 n + 1 ) Re ( a n + b n ) ,
Q sc ( x , m ) = 2 x 2 n = 1 ( 2 n + 1 ) ( | a n | 2 + | b n | 2 ) .
cos θ = 4 x 2 Q sc [ n = 1 n ( n + 2 ) n + 1 Re ( a n a n + 1 * + b n b n + 1 * ) + n = 1 2 n + 1 n ( n + 1 ) Re ( a n b n * ) ] ,
Q b ( x , m ) = 1 x 2 | n = 1 ( 1 ) n ( 2 n + 1 ) ( a n b n ) | 2 .
a n ( x , m ) = A n ( x , m ) A n ( x , m ) i C n ( x , m ) ,
b n ( x , m ) = B n ( x , m ) B n ( x , m ) i D n ( x , m ) ,
A n = n ( m 2 1 ) x J n + 1 2 ( m x ) J n + 1 2 ( x ) + J n 1 2 ( m x ) J n + 1 2 ( x ) m J n + 1 2 ( m x ) J n 1 2 ( x ) ,
C n = n ( m 2 1 ) x J n + 1 2 ( m x ) N n + 1 2 ( x ) + J n 1 2 ( m x ) N n + 1 2 ( x ) m J n + 1 2 ( m x ) N n 1 2 ( x ) ,
B n = m J n 1 2 ( m x ) J n + 1 2 ( x ) J n + 1 2 ( m x ) J n 1 2 ( x ) ,
D n = m J n 1 2 ( m x ) N n + 1 2 ( x ) J n + 1 2 ( m x ) N n 1 2 ( x ) ,
A n + 1 = ( n + 1 ) ( m 2 1 ) x J n + 3 2 ( m x ) J n + 3 2 ( x ) + J n + 1 2 ( m x ) J n + 3 2 ( x ) m J n + 3 2 ( m x ) J n + 1 2 ( x ) .
J n + 3 2 ( x ) = 2 n + 1 x J n + 1 2 ( x ) J n 1 2 ( x ) ,
A n + 1 = m J n 1 2 ( m x ) J n + 1 2 ( x ) J n + 1 2 ( m x ) J n 1 2 ( x ) + 1 x ( n + 1 ) ( m 2 1 ) J n 1 2 ( m x ) J n 1 2 ( x ) 1 x 2 ( n + 1 ) ( 2 n + 1 ) ( m 2 1 ) m × [ J n + 1 2 ( m x ) J n 1 2 ( x ) + J n 1 2 ( m x ) J n + 1 2 ( x ) ] + 1 x 3 ( n + 1 ) ( 2 n + 1 ) 2 ( m 2 1 ) m J n + 1 2 ( m x ) J n + 1 2 ( x ) .
A n + 1 = m J n 1 2 ( m x ) J n + 1 2 ( x ) J n + 1 2 ( m x ) J n 1 2 ( x ) .
A n + 1 ( x , m ) = B n ( x , m ) .
C n + 1 ( x , m ) = D n ( x , m ) ,
A n + 2 ( x , m ) = A n ( x , m ) ,
C n + 2 ( x , m ) = C n ( x , m ) ,
B n + 2 ( x , m ) = B n ( x , m ) ,
D n + 2 ( x , m ) = D n ( x , m ) .
a n + 1 ( x , m ) = b n ( x , m ) ,
a n + 2 ( x , m ) = a n ( x , m ) ,
b n + 1 ( x , m ) = a n ( x , m ) ,
b n + 2 ( x , m ) = b n ( x , m ) ,
a 1 ( x , m ) = b 2 ( x , m ) = a 3 ( x , m ) = b 4 ( x , m ) = ,
b 1 ( x , m ) = a 2 ( x , m ) = b 3 ( x , m ) = a 4 ( x , m ) = .
Re a 1 ( x ) ~ sin 2 x ,
Im a 1 ( x ) ~ sin x cos x ,
Re b 1 ( x ) ~ cos 2 x ,
Im b 1 ( x ) ~ sin x cos x .
lim x Re [ a n ( x ) + b n ( x ) ] = 1 ,
lim x Im [ a n ( x ) + b n ( x ) ] = 0 ,
lim x ( | a n ( x ) | 2 + | b n ( x ) | 2 ) = 1 ,
lim x | a n ( x ) b n ( x ) | = 1 ,
lim x Re [ a n ( x ) b n * ( x ) ] = 0 .
Re ( a n a n + 1 * ) = Re ( a n b n * ) ,
Re ( b n b n + 1 * ) = Re ( b n a n * ) = Re ( a n b n * ) .
Im a n ( x + π / 4 ) = Re a n ( x ) 1 2 ,
Re a 1 ( x , m ) ~ ( sin m x cos x m cos m x sin x ) 2 sin 2 m x + m 2 cos 2 m x ,
Im a 1 ( x , m ) ~ ( sin m x cos x m cos m x sin x ) ( sin m x sin x + m cos m x cos x ) sin 2 m x + m 2 cos 2 m x ,
Re b 1 ( x , m ) ~ ( m sin m x cos x cos m x sin x ) 2 m 2 sin 2 m x + cos 2 m x ,
Im b 1 ( x , m ) ~ ( m sin m x cos x cos m x sin x ) ( m sin m x sin x + cos m x cos x ) m 2 sin 2 m x + cos 2 m x .
[ lim x Re ( a n + b n ) , lim x Im ( a n + b n ) , lim x ( | a n | 2 + | b n | 2 ) , lim x | a n b n | , lim x Re ( a n b n * ) , when n is an arbitrary but finite integer ]
m = μ i κ .
Re a 1 = Re A 1 ( Re A 1 Im C 1 ) + Im A 1 ( Im A 1 + Re C 1 ) ( Re A 1 Im C 1 ) 2 + ( Im A 1 + Re C 1 ) 2 ,
Im a 1 = Im A 1 ( Re A 1 Im C 1 ) Re A 1 ( Im A 1 + Re C 1 ) ( Re A 1 Im C 1 ) 2 + ( Im A 1 + Re C 1 ) 2 ,
Re b 1 = Re B 1 ( Re B 1 Im D 1 ) + Im B 1 ( Im B 1 + Re D 1 ) ( Re B 1 Im D 1 ) 2 + ( Im B 1 + Re D 1 ) 2 ,
Im b 1 = Im B 1 ( Re B 1 Im D 1 ) Re B 1 ( Im C 1 + Re D 1 ) ( Re B 1 Im D 1 ) 2 + ( Im B 1 + Re D 1 ) 2 .
Re a 1 ( x , m ) ~ cos 2 x + ( μ 2 + κ 2 ) sin 2 x + μ 2 κ sin x cos x ( 1 + μ ) 2 + κ 2 ,
Re b 1 ( x , m ) ~ sin 2 x + ( μ 2 + κ 2 ) cos 2 x + μ + 2 κ sin x cos x ( 1 + μ ) 2 + κ 2 ,
Im a 1 ( x , m ) ~ ( 1 μ 2 κ 2 ) sin x cos x + κ ( cos 2 x sin 2 x ) ( 1 + μ ) 2 + κ 2 ,
Im b 1 ( x , m ) ~ ( μ 2 + κ 2 1 ) sin x cos x k ( cos 2 x sin 2 x ) ( 1 + μ ) 2 + κ 2 .
lim x Re ( a n + b n ) = 1 ,
lim x Im ( a n + b n ) = 0 ,
lim x ( | a n | 2 + | b n | 2 ) = 1 2 ( 1 + | m 1 m + 1 | 2 ) ,
lim x | a n b n | = | m 1 m + 1 | ,
lim x Re ( a n b n * ) = Re m | m + 1 | 2 ,
lim x | a n 1 2 | = lim x | b n 1 2 | = 1 2 | m 1 m + 1 | .
lim x a 1 ( x , m ) = lim x b 2 ( x , m ) = lim x a 3 ( x , m ) = lim x b 4 ( x , m ) = , lim x b 1 ( x , m ) = lim x a 2 ( x , m ) = lim x b 3 ( x , m ) = lim x a 4 ( x , m ) = .
lim x [ a n ( x , m ) + b n ( x , m ) ] = lim x [ a 1 ( x , m ) + b 1 ( x , m ) ] , lim x [ | a n ( x , m ) | 2 + | b n ( x , m ) | 2 ] = lim x [ | a 1 ( x , m ) | 2 + | b 1 ( x , m ) | 2 ] , lim x Re [ a n ( x , m ) a n + 1 * ( x , m ) ] = lim x Re [ b n ( x , m ) b n + 1 * ( x , m ) ] = lim x Re [ a n ( x , m ) b n * ( x , m ) ] = lim x Re [ a 1 ( x , m ) b 1 * ( x , m ) ] ,
lim x | a n ( x , m ) b n ( x , m ) | = lim x | a 1 ( x , m ) b 1 ( x , m ) | .