Abstract

Analytic expressions for scattering by a single-layered sphere when either the core is tiny or the shell thickness is small are presented within the framework of a perturbative approach. In this approach, the contribution of the core or the shell (each acting as a perturbation) to the scattering is separated out, thus facilitating a deeper insight into the nature of its effect. This is illustrated by the application of the analytic expressions to the well-known practical case of absorption of visible light by water droplets contaminated with graphitic carbon (soot) in the atmosphere.

© 1986 Optical Society of America

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References

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  1. R. W. Fenn, H. Oser, “Scattering properties of concentric soot–water spheres for visible and infrared light,” Appl. Opt. 4, 1504–1509 (1965).
    [CrossRef]
  2. R. E. Danielson, D. R. Moore, H. C. van de Hulst, “The transfer of visible radiation through clouds,”J. Atmos. Sci. 26, 1078–1087 (1969).
    [CrossRef]
  3. H. Rosen, D. A. Hansen, R. L. Dod, T. Novakov, “Soot in urban atmospheres: determination by an optical absorption technique,” Science 208, 741–744 (1980).
    [CrossRef] [PubMed]
  4. T. P. Ackerman, O. B. Toon, “Absorption of visible radiation in atmosphere containing mixtures of absorbing and nonabsorbing particles,” Appl. Opt. 20, 3661–3668 (1981).
    [CrossRef] [PubMed]
  5. P. Chylek, V. Ramaswamy, R. J. Cheng, “Effect of graphitic carbon on the albedo of clouds,”J. Atmos. Sci. 41, 3076–3084 (1984).
    [CrossRef]
  6. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  7. R. Bhandari, “Scattering coefficients for a multilayered sphere: analytic expressions and algorithms,” Appl. Opt. 24, 1960–1967 (1985).
    [CrossRef] [PubMed]
  8. See, for example, D. M. Roessler, D. S. Y. Wang, M. Kerker, “Optical absorption by randomly oriented carbon spheroids,” Appl. Opt. 22, 3648–3651 (1983), and Ref. 5.
    [CrossRef] [PubMed]
  9. We point out here that a peak similar to the one corresponding to m2= 2.0 − i 0.66 in Fig. 8 should be present in Fig. 4 of Ref. 5, even though the refractive indices used there for carbon (1.94 − i 0.66) and water (1.33 − i 10−9) are slightly different. Our calculations bear this out.
  10. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York1957); P. Chylek, J. T. Kiehl, M. K. W. Ko, “Narrow resonance structure in the Mie scattering characteristics,” Appl. Opt. 17, 3019–3021 (1978).
    [CrossRef] [PubMed]
  11. H. S. Bennett, G. J. Rosasco, “Resonances in the efficiency factors for absorption: Mie scattering theory,” Appl. Opt. 17, 491–493 (1978).
    [CrossRef] [PubMed]
  12. M. G. Andreasen, “Back-scattering cross section of a thin, dielectric spherical shell,”IRE Trans. Antennas Propag. AP-5, 267–270 (1957).
    [CrossRef]
  13. M. G. Andreasen, “Radiation from a radial dipole through a thin dielectric spherical shell,”IRE Trans. Antennas Propag. AP-5, 337–342 (1957).
    [CrossRef]

1985 (1)

1984 (1)

P. Chylek, V. Ramaswamy, R. J. Cheng, “Effect of graphitic carbon on the albedo of clouds,”J. Atmos. Sci. 41, 3076–3084 (1984).
[CrossRef]

1983 (1)

1981 (1)

1980 (1)

H. Rosen, D. A. Hansen, R. L. Dod, T. Novakov, “Soot in urban atmospheres: determination by an optical absorption technique,” Science 208, 741–744 (1980).
[CrossRef] [PubMed]

1978 (1)

1969 (1)

R. E. Danielson, D. R. Moore, H. C. van de Hulst, “The transfer of visible radiation through clouds,”J. Atmos. Sci. 26, 1078–1087 (1969).
[CrossRef]

1965 (1)

1957 (2)

M. G. Andreasen, “Back-scattering cross section of a thin, dielectric spherical shell,”IRE Trans. Antennas Propag. AP-5, 267–270 (1957).
[CrossRef]

M. G. Andreasen, “Radiation from a radial dipole through a thin dielectric spherical shell,”IRE Trans. Antennas Propag. AP-5, 337–342 (1957).
[CrossRef]

Ackerman, T. P.

Andreasen, M. G.

M. G. Andreasen, “Back-scattering cross section of a thin, dielectric spherical shell,”IRE Trans. Antennas Propag. AP-5, 267–270 (1957).
[CrossRef]

M. G. Andreasen, “Radiation from a radial dipole through a thin dielectric spherical shell,”IRE Trans. Antennas Propag. AP-5, 337–342 (1957).
[CrossRef]

Bennett, H. S.

Bhandari, R.

Cheng, R. J.

P. Chylek, V. Ramaswamy, R. J. Cheng, “Effect of graphitic carbon on the albedo of clouds,”J. Atmos. Sci. 41, 3076–3084 (1984).
[CrossRef]

Chylek, P.

P. Chylek, V. Ramaswamy, R. J. Cheng, “Effect of graphitic carbon on the albedo of clouds,”J. Atmos. Sci. 41, 3076–3084 (1984).
[CrossRef]

Danielson, R. E.

R. E. Danielson, D. R. Moore, H. C. van de Hulst, “The transfer of visible radiation through clouds,”J. Atmos. Sci. 26, 1078–1087 (1969).
[CrossRef]

Dod, R. L.

H. Rosen, D. A. Hansen, R. L. Dod, T. Novakov, “Soot in urban atmospheres: determination by an optical absorption technique,” Science 208, 741–744 (1980).
[CrossRef] [PubMed]

Fenn, R. W.

Hansen, D. A.

H. Rosen, D. A. Hansen, R. L. Dod, T. Novakov, “Soot in urban atmospheres: determination by an optical absorption technique,” Science 208, 741–744 (1980).
[CrossRef] [PubMed]

Kerker, M.

Moore, D. R.

R. E. Danielson, D. R. Moore, H. C. van de Hulst, “The transfer of visible radiation through clouds,”J. Atmos. Sci. 26, 1078–1087 (1969).
[CrossRef]

Novakov, T.

H. Rosen, D. A. Hansen, R. L. Dod, T. Novakov, “Soot in urban atmospheres: determination by an optical absorption technique,” Science 208, 741–744 (1980).
[CrossRef] [PubMed]

Oser, H.

Ramaswamy, V.

P. Chylek, V. Ramaswamy, R. J. Cheng, “Effect of graphitic carbon on the albedo of clouds,”J. Atmos. Sci. 41, 3076–3084 (1984).
[CrossRef]

Roessler, D. M.

Rosasco, G. J.

Rosen, H.

H. Rosen, D. A. Hansen, R. L. Dod, T. Novakov, “Soot in urban atmospheres: determination by an optical absorption technique,” Science 208, 741–744 (1980).
[CrossRef] [PubMed]

Toon, O. B.

van de Hulst, H. C.

R. E. Danielson, D. R. Moore, H. C. van de Hulst, “The transfer of visible radiation through clouds,”J. Atmos. Sci. 26, 1078–1087 (1969).
[CrossRef]

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York1957); P. Chylek, J. T. Kiehl, M. K. W. Ko, “Narrow resonance structure in the Mie scattering characteristics,” Appl. Opt. 17, 3019–3021 (1978).
[CrossRef] [PubMed]

Wang, D. S. Y.

Appl. Opt. (5)

IRE Trans. Antennas Propag. (2)

M. G. Andreasen, “Back-scattering cross section of a thin, dielectric spherical shell,”IRE Trans. Antennas Propag. AP-5, 267–270 (1957).
[CrossRef]

M. G. Andreasen, “Radiation from a radial dipole through a thin dielectric spherical shell,”IRE Trans. Antennas Propag. AP-5, 337–342 (1957).
[CrossRef]

J. Atmos. Sci. (2)

R. E. Danielson, D. R. Moore, H. C. van de Hulst, “The transfer of visible radiation through clouds,”J. Atmos. Sci. 26, 1078–1087 (1969).
[CrossRef]

P. Chylek, V. Ramaswamy, R. J. Cheng, “Effect of graphitic carbon on the albedo of clouds,”J. Atmos. Sci. 41, 3076–3084 (1984).
[CrossRef]

Science (1)

H. Rosen, D. A. Hansen, R. L. Dod, T. Novakov, “Soot in urban atmospheres: determination by an optical absorption technique,” Science 208, 741–744 (1980).
[CrossRef] [PubMed]

Other (3)

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

We point out here that a peak similar to the one corresponding to m2= 2.0 − i 0.66 in Fig. 8 should be present in Fig. 4 of Ref. 5, even though the refractive indices used there for carbon (1.94 − i 0.66) and water (1.33 − i 10−9) are slightly different. Our calculations bear this out.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York1957); P. Chylek, J. T. Kiehl, M. K. W. Ko, “Narrow resonance structure in the Mie scattering characteristics,” Appl. Opt. 17, 3019–3021 (1978).
[CrossRef] [PubMed]

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Figures (15)

Fig. 1
Fig. 1

A single-layered sphere.

Fig. 2
Fig. 2

Absorption cross section of water droplet (m2 = 1.33 − i0.0) per unit volume of the core (m1 = 2.0 − imI, mI > 0) denoted by σabs/V as a function of its volume fraction F. The radius of the water droplet r2 = 5 μm and the wavelength of light λ = 0.5 μm. Each curve is labeled by the value of mI. mI = 0.66 corresponds to graphitic carbon (soot).

Fig. 3
Fig. 3

σabs/V as a function of the radius r2 of the water droplet. The soot core (m1 = 2.0 −1 i0.66) has a fixed radius r1 = 0.0046 μm.

Fig. 4
Fig. 4

Same as Fig. 3, but for larger water droplets.

Fig. 5
Fig. 5

A plot of 1/|D1(h)|2 versus the size parameter x2 = 2πr2/λ depicting the sharp resonant behavior when the refractive index m2 = 5.0. 1/|D1(h)|2 is the modulus square of the internal field coefficient associated with the scattering amplitude a1(h) for a homogenous spherical particle.

Fig. 6
Fig. 6

The lower end of the curve indicates the approach to the Rayleigh-scattering limit for the composite particle (soot core + water shell) in air.

Fig. 7
Fig. 7

The lower end of this curve corresponds essentially to Rayleigh absorption by the carbon particle (radius r1 = 0.0046 μm) in air.

Fig. 8
Fig. 8

Absorption cross section of water droplet (m1 = 1.33 −i0.0) per unit volume of the shell (m2 = 2.0 −imI, mI > 0) denoted by σabs/V as a function of its volume fraction F. The radius of the water droplet r2 = 5 Am and the wavelength of light λ = 0.5 μm. Each curve is labeled by the value of mI. mI = 0.66 corresponds to graphitic carbon (soot).

Fig. 9
Fig. 9

Ripple structure in σabs/V for a soot-coated water droplet that is determined by the resonances in the internal field of a homogeneous water droplet of the same size. The volume fraction F of soot is 10−7. The step size is 10−2μm.

Fig. 10
Fig. 10

Same as Fig. 9, except that the step size is 4 × 10−4 m.

Fig. 11
Fig. 11

Same as Fig. 10, except that a further reduction in step size to 2 × 10−4 m reveals three more peaks labeled 1, 2, and 3. Peak 2 lies just beyond 5 μm.

Fig. 12
Fig. 12

Peak 2 of Fig. 11 shown in greater detail. It is due to a resonance in the electric mode, n = 76.

Fig. 13
Fig. 13

Same as Fig. 12, except that F = 10−4 (flat curve), F = 10−5 (the broad-peak curve), F = 10−6 (the narrow-peak curve).

Fig. 14
Fig. 14

Same as Fig. 11, except that F = 10−4.

Fig. 15
Fig. 15

Geometry of a spherical shell of refractive index m2. The refractive indices of the core and the external medium are m1 and m3, respectively.

Equations (82)

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a n = N n D n ,
b n = M n C n ,
N n = | ψ n ( x 2 ) m 2 ψ n ( m 2 x 2 ) m 2 χ n ( m 2 x 2 ) 0 ψ n ( x 2 ) ψ n ( m 2 x 2 ) χ n ( m 2 x 2 ) 0 0 m 2 ψ n ( m 2 x 1 ) m 2 χ n ( m 2 x 1 ) m 1 ψ n ( m 1 x 1 ) 0 ψ n ( m 2 x 1 ) χ n ( m 2 x 1 ) ψ n ( m 1 x 1 ) | ,
D n = | ζ n ( x 2 ) m 2 ψ n ( m 2 x 2 ) m 2 χ n ( m 2 x 2 ) 0 ζ n ( x 2 ) ψ n ( m 2 x 2 ) χ n ( m 2 x 2 ) 0 0 m 2 ψ n ( m 2 x 1 ) m 2 χ n ( m 2 x 1 ) m 1 ψ n ( m 1 x 1 ) 0 ψ n ( m 2 x 1 ) χ n ( m 2 x 1 ) ψ n ( m 1 x 1 ) | ,
M n = | ψ n ( x 2 ) ψ n ( m 2 x 2 ) χ n ( m 2 x 2 ) 0 ψ n ( x 2 ) m 2 ψ n ( m 2 x 2 ) m 2 χ n ( m 2 x 2 ) 0 0 ψ n ( m 2 x 1 ) χ n ( m 2 x 1 ) ψ n ( m 1 x 1 ) 0 m 2 ψ n ( m 2 x 1 ) m 2 χ n ( m 2 x 1 ) m 1 ψ n ( m 1 x 1 ) | ,
C n = | ζ n ( x 2 ) ψ n ( m 2 x 2 ) χ n ( m 2 x 2 ) 0 ζ n ( x 2 ) m 2 ψ n ( m 2 x 2 ) m 2 χ n ( m 2 x 2 ) 0 0 ψ n ( m 2 x 1 ) χ n ( m 2 x 1 ) ψ n ( m 1 x 1 ) 0 m 2 ψ n ( m 2 x 1 ) m 2 χ n ( m 2 x 1 ) m 1 ψ n ( m 1 x 1 ) | ,
x i = 2 π r i / λ ,
ψ n ( z ) = z j n ( z ) , χ n ( z ) = - z n n ( z ) , ζ n ( z ) = z h n 2 ( z ) ,
σ ext = λ 2 2 π n = 1 ( 2 n + 1 ) Re ( a n + b n ) , σ sca = λ 2 2 π n = 1 ( 2 n + 1 ) ( a n 2 + b n 2 ) , σ abs = σ ext - σ sca = λ 2 2 π n = 1 ( 2 n + 1 ) × [ Re ( a n + b n ) - ( a n 2 + b n 2 ) ] .
a n = a n ( h ) [ 1 - ( β n / α n ) N ^ n ( h ) / N n ( h ) 1 - ( β n / α n ) D ^ n ( h ) / D n ( h ) ] ,
a n ( h ) = N n ( h ) / D n ( h )
N n ( h ) = ψ n ( x 2 ) ψ n ( m 2 x 2 ) - m 2 ψ n ( x 2 ) ψ n ( m 2 x 2 ) , D n ( h ) = ζ n ( x 2 ) ψ n ( m 2 x 2 ) - m 2 ζ n ( x 2 ) ψ n ( m 2 x 2 ) , N ^ n ( h ) = ψ n ( x 2 ) χ n ( m 2 x 2 ) - m 2 ψ n ( x 2 ) χ n ( m 2 x 2 ) , D ^ n ( h ) = ζ n ( x 2 ) χ n ( m 2 x 2 ) - m 2 ζ n ( x 2 ) χ n ( m 2 x 2 ) , α n = m 2 ψ n ( m 1 x 1 ) χ n ( m x 1 ) - m 1 ψ n ( m 1 x 1 ) χ n ( m 2 x 1 ) , β n = m 2 ψ n ( m 1 x 1 ) ψ n ( m 2 x 1 ) - m 1 ψ n ( m 1 x 1 ) ψ n ( m 2 x 1 ) .
ψ n ( z ) = z n + 1 / G 1 n , ψ n ( z ) = ( n + 1 ) z n / G 1 n , χ n ( z ) = G 2 n / z n , χ n ( z ) = - n G 2 n / z n + 1 ,
G 1 n = 1 × 3 × 5 ( 2 n + 1 ) , G 2 n = 1 × 3 × 5 ( 2 n - 1 ) .
β n / α n = - ( 2 n + 1 ) ( n + 1 ) ( m 2 x 1 ) 2 n + 1 ( m 1 2 - m 2 2 ) ( G 1 n ) 2 [ n m 1 2 + m 2 2 ( n + 1 ) ] .
a n = a n ( h ) { 1 - ( β n / α n ) [ N ^ n ( h ) / N n ( h ) - D ^ n ( h ) / D n ( h ) ] } .
a n = a n ( h ) + f n ,
f n = m 2 ( - i β n / α n ) ( - 1 ) / [ D n ( h ) ] 2 ,
b n = b n ( h ) + g n ,
g n = i m 2 ( γ n / δ n ) / [ C n ( h ) ] 2 ,
γ n / δ n = - ( m 2 x 1 ) 2 n + 3 ( m 1 2 / m 2 2 - 1 ) G 1 n 2 ( 2 n + 3 ) .
b n ( h ) = M n ( h ) / C n ( h )
C n ( h ) = m 2 ζ n ( x 2 ) ψ n ( m 2 x 2 ) - ζ n ( x 2 ) ψ n ( m 2 x 2 ) .
f n = δ n 1 f 1 ,             n = 1 , 2 ,
g n = 0 ,             n = 1 , 2 , ,
σ ext = σ ext ( h ) + 3 λ 2 2 π Re ( f 1 ) ,
σ sca = σ sca ( h ) + 3 λ 2 2 π 2 Re { [ a 1 ( h ) ] * f 1 } ,
σ abs = σ abs ( h ) + 3 λ 2 2 π Re ( f 1 { 1 - 2 [ a 1 ( h ) ] * } ) .
σ abs = ( 3 λ 2 ) 2 π m 2 Im [ - ( 2 / 3 ) ( m 2 x 1 ) 3 ( m 1 2 - m 2 2 ) m 1 2 + 2 m 2 2 ] 1 D 1 ( h ) 2 ,
σ abs / V = - 6 π m 2 4 λ Im [ ( m 1 2 - m 2 2 ) m 1 2 + 2 m 2 2 ] / D 1 ( h ) 2 ,
D 1 ( h ) = ζ 1 ( x 2 ) ψ 1 ( m 2 x 2 ) - m 2 ζ 1 ( x 2 ) ψ 1 ( m 2 x 2 ) .
D 1 ( h ) = exp ( - i x 2 ) [ sin ( m 2 x 2 ) - i m 2 cos ( m 2 x 2 ) ] .
1 / D 1 ( h ) 2 = 1 / [ 1 + ( m 2 2 - 1 ) cos 2 ( m 2 x 2 ) ] ,
p = [ 1 / ( 2 m 2 ) ] ln [ ( m 2 + 1 ) / ( m 2 - 1 ) ] .
σ abs / V = - 54 π m 2 2 λ ( 2 + m 2 2 ) 2 Im { m 1 2 - m 2 2 m 1 2 + 2 m 2 2 } ,
ψ n ( m 2 x 1 ) = ψ n ( m 2 x 2 - m 2 ) = ψ n ( m 2 x 2 ) - ( m 2 ) ψ n ( m 2 x 2 ) + ( - m 2 ) 2 ψ n ( m 2 x 2 ) / 2 +
ψ n ( m 2 x 1 ) = ψ n ( m 2 x 2 - m 2 ) = ψ n ( m 2 x 2 ) - ( m 2 ) ψ n ( m 2 x 2 ) + ( - m 2 ) 2 ψ n ( m 2 x 2 ) / 2 + ,
= x 2 - x 1 = 2 π ( r 2 - r 1 ) / λ .
G n ( z ) + [ 1 - n ( n + 1 ) / z 2 ] G n ( z ) = 0 ,
ψ n ( m 2 x 1 ) = ψ n ( m 2 x 2 ) - m 2 ψ n ( m 2 x 2 ) + m 2 2 2 t n ( m 2 x 2 ) ψ n ( m 2 x 2 ) / 2 + ,
ψ n ( m 2 x 1 ) = ψ n ( m 2 x 2 ) - m 2 t n ( m 2 x 2 ) ψ n ( m 2 x 2 ) + m 2 2 2 [ t n ( m 2 x 2 ) ψ n ( m 2 x 2 ) ] / 2 + ,
t n ( m 2 x 2 ) = n ( n + 1 ) / ( m 2 x 2 ) 2 - 1.
N n = | ψ n ( x 2 ) m 2 ψ n ( m 2 x 2 ) m 2 χ n ( m 2 x 2 ) 0 ψ n ( x 2 ) ψ n ( m 2 x 2 ) χ n ( m 2 x 2 ) 0 0 [ m 2 ψ n ( m 2 x 2 - m 2 2 ψ n ( m 2 x 2 ) ] [ m 2 χ n ( m 2 x 2 - m 2 2 χ n ( m 2 x 2 ) ] [ m 1 ψ n ( m 1 x 2 ) - m 1 2 ψ n ( m 1 x 2 ) ] 0 [ ψ n ( m 2 x 2 ) - m 2 t n ( m 2 x 2 ) ψ n ( m 2 x 2 ) ] [ χ n ( m 2 x 2 ) - m 2 t n ( m 2 x 2 ) χ n ( m 2 x 2 ) ] [ ψ n ( m 1 x 2 ) - m 1 t n ( m 1 x 2 ) ψ n ( m 1 x 2 ) ] |
N n = m 2 N n ( h ) - ( m 2 2 - m 1 2 ) R n ,
N n ( h ) = ψ n ( m 1 x 2 ) ψ n ( x 2 ) - m 1 ψ n ( m 1 x 2 ) ψ n ( x 2 ) ,
R n = [ n ( n + 1 ) / ( m 2 2 m 1 x 2 2 ) ] ψ n ( x 2 ) ψ n ( m 1 x 2 ) + ψ n ( x 2 ) ψ n ( m 1 x 2 ) .
D n = m 2 D n ( h ) - ( m 2 2 - m 1 2 ) S n ,
D n ( h ) = ψ n ( m 1 x 2 ) ζ n ( x 2 ) - m 1 ψ n ( m 1 x 2 ) ζ n ( x 2 ) ,
S n = [ n ( n + 1 ) / ( m 2 2 m 1 x 2 2 ) ] ζ n ( x 2 ) ψ n ( m 1 x 2 ) + ζ n ( x 2 ) ψ n ( m 1 x 2 ) .
a n = N n / D n = a n ( h ) + a ^ n ,
a n ( h ) = N n ( h ) / D n ( h )
a ^ n = - i ( m 2 2 - m 1 2 ) { [ ψ n ( m 1 x 2 ) ] 2 + [ n ( n + 1 ) / ( m 2 2 x 2 2 ) ] [ ψ n ( m 1 x 2 ) ] 2 } / [ D n ( h ) ] 2 .
b n = b n ( h ) + b ^ n ,
b n ( h ) = M n ( h ) / C n ( h ) ,
M n ( h ) = m 1 ψ n ( m 1 x 2 ) ψ n ( x 2 ) - ψ n ( m 1 x 2 ) ψ n ( x 2 ) ,
C n ( h ) = m 1 ψ n ( m 1 x 2 ) ζ n ( x 2 ) - ψ n ( m 1 x 2 ) ζ n ( x 2 ) ,
b ^ n = - i ( m 2 2 - m 1 2 ) { [ ψ n ( m 1 x 2 ) ] 2 } / [ C n ( h ) ] 2 .
σ ext = σ ext ( h ) + σ ^ ext ,
σ ^ ext ( h ) = λ 2 2 π n ( 2 n + 1 ) Re [ a n ( h ) + b n ( h ) ] ,
σ ext = λ 2 2 π n ( 2 n + 1 ) Re ( a ^ n + b ^ n ) ,
σ sca = σ sca ( h ) + σ ^ sca ,
σ sca ( h ) = λ 2 2 π n ( 2 n + 1 ) [ a n ( h ) 2 + b n ( h ) 2 ] ,
σ ^ sca = λ 2 2 π n ( 2 n + 1 ) 2 Re [ a ^ n a n ( h ) * + b ^ n b n ( h ) * ] ,
σ abs = σ ext - σ sca = [ σ ext ( h ) - σ sca ( h ) ] + λ 2 2 π n ( 2 n + 1 ) × Re { a ^ n [ 1 - 2 a n ( h ) * ] + b ^ n [ 1 - 2 b n ( h ) * ] } .
σ abs = - λ 2 A / ( 2 π ) ,
A = n ( 2 n + 1 ) Im [ ( m 2 2 - m 1 2 ) ( { [ ψ n ( m 1 x 2 ) ] 2 + [ n ( n + 1 ) / ( m 2 2 x 2 2 ) ] [ ψ n ( m 1 x 2 ) ] 2 } / D n ( h ) 2 + [ ψ n ( m 1 x 2 ) ] 2 / C n ( h ) 2 ) ] .
σ abs / V = π A / ( λ x 2 2 ) ,
n ^ × [ E 3 ( r 2 ) - E 2 ( r 2 ) ] = 0 ,
n ^ × [ E 2 ( r 1 ) - E 1 ( r 1 ) ] = 0 ,
n ^ × [ H 3 ( r 2 ) - H 2 ( r 2 ) ] = 0 ,
n ^ × [ H 2 ( r 1 ) - H 1 ( r 1 ) ] = 0 ,
H 2 ( r 1 ) = H 2 ( r 2 ) - Δ r · H 2 ( r 2 ) .
H 1 ( r 1 ) = H 1 ( r 2 ) - Δ r · H 1 ( r 2 ) ,
n ^ × [ H 3 ( r 2 ) - H 1 ( r 2 ) ] = Δ r n ^ × [ ( n ^ · ) ( H 2 - H 1 ) ] r = r 2 .
( A · B ) = ( B · ) A + ( A · ) B + B × ( × A ) + A × ( × B )
× H = i ( ω / c ) E
n ^ × [ H 3 ( r 2 ) - H 1 ( r 2 ] = Δ r ( i ω / c ) { n ^ × [ 2 E 2 ( r 2 ) - 1 E 1 ( r 2 ) ] } × n ^ + Δ r n ^ × [ ( H 2 - H 1 ) · n ^ ] r = r 2 - ( Δ r / r 2 ) n ^ × [ H 3 ( r 2 ) - H 1 ( r 2 ) ] .
n ^ × [ H 3 ( r 2 ) - H 1 ( r 2 ) ] = Δ r ( i ω / c ) { n ^ × [ 2 E 3 ( r 2 ) - 1 E 1 ( r 2 ) ] } + Δ r μ 3 ( 1 / μ 2 - 1 / μ 1 ) n ^ × ( H 3 · n ^ ) r = r 2 .
n ^ × [ E 3 ( r 2 ) - E 1 ( r 2 ) ] = Δ r ( - i ω / c ) × { n ^ × [ μ 2 H 3 ( r 2 ) - μ 1 H 1 ( r 2 ) ] } × n ^ + Δ r 3 ( 1 / 2 - 1 / 1 ) n ^ × ( E 3 · n ^ ) r = r 2 .
n ^ × [ H 3 ( r 2 ) - H 1 ( r 2 ) ] = Δ r ( i ω / c ) × { n ^ × [ 2 E 3 ( r 2 ) - 1 E 1 ( r 2 ) ] } × n ^ ,
n ^ × [ E 3 ( r 2 ) - E 1 ( r 2 ) ] = Δ r 3 ( 1 / 2 - 1 / 1 ) × n ^ × ( E 3 · n ^ ) r = r 2 .
n ^ × [ H 3 ( r 2 ) - H 1 ( r 2 ) ] = ( Δ r ) ( i ω / c ) ( 2 - 1 ) ( n ^ × E 3 ) × n ^ .

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