Abstract

We present exact and approximate analytic expressions for the time-averaged electromagnetic energy within dielectric spheres on the basis of rigorous Mie theory. Such information is of importance for the study of photochemical reactions within atmospheric water spheres. Numerical results show that on the average the energy inside a cloud droplet is enlarged by a factor exceeding 2 compared with that of a sphere of the same radius of the surrounding medium. In regions of resonance peaks the electromagnetic energy may be increased by more than 2 orders of magnitude.

© 1987 Optical Society of America

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References

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  1. T. E. Graedel, C. J. Weschler, “Chemistry within aqueous atmospheric aerosols and raindrops,” Rev. Geophys. Space Phys. 19, 505–539 (1981).
    [Crossref]
  2. W. L. Chameides, “The photochemistry of a remote marine stratiform cloud,”J. Geophys. Res. 89, 4739–4755 (1984).
    [Crossref]
  3. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), pp. 119–130.
  4. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2d ed. (Cambridge U. Press, Cambridge, 1952), p. 134.
  5. J. V. Dave, “Subroutines for computing the parameters of the electromagnetic radiation scattered by a sphere,” (IBM Scientific Center, Palo Alto, California, 1968).
  6. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 477–482.
  7. W. M. Irvine, J. B. Pollack, “Infrared optical properties of water and ice spheres,” Icarus 8, 324–360 (1968).
    [Crossref]
  8. G. J. Rosasco, H. S. Bennett, “Internal field resonance structure Implications for optial absorption and scattering by microscopic particles,”J. Opt. Soc. Am. 68, 1242–1250 (1978).
    [Crossref]
  9. P. R. Conwell, P. W. Barber, C. K. Rushforth, “Resonant spectra of dielectric spheres,” J. Opt. Soc. Am. A 1, 62–67 (1984).
    [Crossref]
  10. P. Chylek, “Partial-wave resonances and the ripple structure in the Mie normalized extinction cross section,”J. Opt. Soc. Am. 66, 285–287 (1976).
    [Crossref]
  11. J. R. Probert-Jones, “Resonance component of backscattering by large dielectric spheres,” J. Opt. Soc. Am. A 1, 822–830 (1984).
    [Crossref]
  12. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions, (Elsevier, New York, 1969), pp. 24–27.
  13. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 88.

1984 (3)

W. L. Chameides, “The photochemistry of a remote marine stratiform cloud,”J. Geophys. Res. 89, 4739–4755 (1984).
[Crossref]

P. R. Conwell, P. W. Barber, C. K. Rushforth, “Resonant spectra of dielectric spheres,” J. Opt. Soc. Am. A 1, 62–67 (1984).
[Crossref]

J. R. Probert-Jones, “Resonance component of backscattering by large dielectric spheres,” J. Opt. Soc. Am. A 1, 822–830 (1984).
[Crossref]

1981 (1)

T. E. Graedel, C. J. Weschler, “Chemistry within aqueous atmospheric aerosols and raindrops,” Rev. Geophys. Space Phys. 19, 505–539 (1981).
[Crossref]

1978 (1)

G. J. Rosasco, H. S. Bennett, “Internal field resonance structure Implications for optial absorption and scattering by microscopic particles,”J. Opt. Soc. Am. 68, 1242–1250 (1978).
[Crossref]

1976 (1)

P. Chylek, “Partial-wave resonances and the ripple structure in the Mie normalized extinction cross section,”J. Opt. Soc. Am. 66, 285–287 (1976).
[Crossref]

1968 (1)

W. M. Irvine, J. B. Pollack, “Infrared optical properties of water and ice spheres,” Icarus 8, 324–360 (1968).
[Crossref]

Barber, P. W.

P. R. Conwell, P. W. Barber, C. K. Rushforth, “Resonant spectra of dielectric spheres,” J. Opt. Soc. Am. A 1, 62–67 (1984).
[Crossref]

Bennett, H. S.

G. J. Rosasco, H. S. Bennett, “Internal field resonance structure Implications for optial absorption and scattering by microscopic particles,”J. Opt. Soc. Am. 68, 1242–1250 (1978).
[Crossref]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 477–482.

Chameides, W. L.

W. L. Chameides, “The photochemistry of a remote marine stratiform cloud,”J. Geophys. Res. 89, 4739–4755 (1984).
[Crossref]

Chylek, P.

P. Chylek, “Partial-wave resonances and the ripple structure in the Mie normalized extinction cross section,”J. Opt. Soc. Am. 66, 285–287 (1976).
[Crossref]

Conwell, P. R.

P. R. Conwell, P. W. Barber, C. K. Rushforth, “Resonant spectra of dielectric spheres,” J. Opt. Soc. Am. A 1, 62–67 (1984).
[Crossref]

Dave, J. V.

J. V. Dave, “Subroutines for computing the parameters of the electromagnetic radiation scattered by a sphere,” (IBM Scientific Center, Palo Alto, California, 1968).

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions, (Elsevier, New York, 1969), pp. 24–27.

Graedel, T. E.

T. E. Graedel, C. J. Weschler, “Chemistry within aqueous atmospheric aerosols and raindrops,” Rev. Geophys. Space Phys. 19, 505–539 (1981).
[Crossref]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 477–482.

Irvine, W. M.

W. M. Irvine, J. B. Pollack, “Infrared optical properties of water and ice spheres,” Icarus 8, 324–360 (1968).
[Crossref]

Pollack, J. B.

W. M. Irvine, J. B. Pollack, “Infrared optical properties of water and ice spheres,” Icarus 8, 324–360 (1968).
[Crossref]

Probert-Jones, J. R.

J. R. Probert-Jones, “Resonance component of backscattering by large dielectric spheres,” J. Opt. Soc. Am. A 1, 822–830 (1984).
[Crossref]

Rosasco, G. J.

G. J. Rosasco, H. S. Bennett, “Internal field resonance structure Implications for optial absorption and scattering by microscopic particles,”J. Opt. Soc. Am. 68, 1242–1250 (1978).
[Crossref]

Rushforth, C. K.

P. R. Conwell, P. W. Barber, C. K. Rushforth, “Resonant spectra of dielectric spheres,” J. Opt. Soc. Am. A 1, 62–67 (1984).
[Crossref]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 88.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), pp. 119–130.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2d ed. (Cambridge U. Press, Cambridge, 1952), p. 134.

Weschler, C. J.

T. E. Graedel, C. J. Weschler, “Chemistry within aqueous atmospheric aerosols and raindrops,” Rev. Geophys. Space Phys. 19, 505–539 (1981).
[Crossref]

Icarus (1)

W. M. Irvine, J. B. Pollack, “Infrared optical properties of water and ice spheres,” Icarus 8, 324–360 (1968).
[Crossref]

J. Geophys. Res. (1)

W. L. Chameides, “The photochemistry of a remote marine stratiform cloud,”J. Geophys. Res. 89, 4739–4755 (1984).
[Crossref]

J. Opt. Soc. Am. (2)

P. Chylek, “Partial-wave resonances and the ripple structure in the Mie normalized extinction cross section,”J. Opt. Soc. Am. 66, 285–287 (1976).
[Crossref]

G. J. Rosasco, H. S. Bennett, “Internal field resonance structure Implications for optial absorption and scattering by microscopic particles,”J. Opt. Soc. Am. 68, 1242–1250 (1978).
[Crossref]

J. Opt. Soc. Am. A (2)

P. R. Conwell, P. W. Barber, C. K. Rushforth, “Resonant spectra of dielectric spheres,” J. Opt. Soc. Am. A 1, 62–67 (1984).
[Crossref]

J. R. Probert-Jones, “Resonance component of backscattering by large dielectric spheres,” J. Opt. Soc. Am. A 1, 822–830 (1984).
[Crossref]

Rev. Geophys. Space Phys. (1)

T. E. Graedel, C. J. Weschler, “Chemistry within aqueous atmospheric aerosols and raindrops,” Rev. Geophys. Space Phys. 19, 505–539 (1981).
[Crossref]

Other (6)

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions, (Elsevier, New York, 1969), pp. 24–27.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 88.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), pp. 119–130.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2d ed. (Cambridge U. Press, Cambridge, 1952), p. 134.

J. V. Dave, “Subroutines for computing the parameters of the electromagnetic radiation scattered by a sphere,” (IBM Scientific Center, Palo Alto, California, 1968).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 477–482.

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Figures (4)

Fig. 1
Fig. 1

Extinction, absorption cross sections, Qext and Qabs, and normalized electromagnetic energy Wn(a) versus size parameter x for 1 ≤ x ≤ 50.

Fig. 2
Fig. 2

Same as Fig. 1 except for 47 ≤ x ≤ 48. First resonance peaks of the a56, b57, and a57 coefficients are identified.

Fig. 3
Fig. 3

Percentage error δW(a) of the electromagnetic energy arising from comparison of Eq. (11) and expression (16) with the general expression (8) versus mi. Five curves for various size parameters, x = 10, 20, 30, 40, 50, are shown.

Fig. 4
Fig. 4

Percentage error δW(a) of the approximate formula (16) versus size parameter x for several values of mi: 10−6, 10−4, 10−3, and 10−2.

Equations (19)

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E θ = E 0 exp ( i ω t ) cos ϕ n = 1 m B n × { d n P n 1 ( cos θ ) sin θ j n ( ρ ) + i c n ρ d P n 1 ( cos θ ) d θ d [ ρ j n ( ρ ) ] d ρ } , E ϕ = - E 0 exp ( i ω t ) sin ϕ n = 1 m B n × { d n d P n 1 ( cos θ ) d θ j n ( ρ ) + c n ρ P n 1 ( cos θ ) sin θ d [ ρ j n ( ρ ) ] d ρ } , E r = E 0 exp ( i ω t ) cos ϕ n = 1 m B n ( n + 1 ) i c n ρ P n 1 ( cos θ ) j n ( ρ ) ,
H θ = H 0 exp ( i ω t ) sin ϕ n = 1 m 2 B n × { c n P n 1 ( cos θ ) sin θ j n ( ρ ) + i d n ρ d P n 1 ( cos θ ) d θ d [ ρ j n ( ρ ) ] d ρ } , H ϕ = H 0 exp ( i ω t ) cos ϕ n = 1 m 2 B n × { c n d P n 1 ( cos θ ) d θ j n ( ρ ) + i d n ρ P n 1 ( cos θ ) sin θ d [ ρ j n ( ρ ) ] d ρ } , H r = H 0 exp ( i ω t ) sin ϕ n = 1 m 2 B n n ( n + 1 ) i d n ρ P n 1 ( cos θ ) j n ( ρ ) ,
c n = i ζ n ( x ) ψ n ( y ) - m ψ n ( y ) ζ n ( x ) , d n = i m ψ n ( y ) ζ n ( x ) - ψ n ( y ) ζ n ( x ) ,
W ( a ) = 0 π 0 2 π 0 a 1 4 ( E 0 E 0 * + E ϕ E ϕ * + E r E r * ) + μ 1 4 ( H θ H θ * + H ϕ H ϕ * + H r H r * ) r 2 sin θ d θ d ϕ d r ,
W ( a ) = 0 a n = 1 [ ( 2 n + 1 ) W n , 0 j n ( ρ ) j n ( ρ * ) + ( n + 1 ) W n , 1 j n - 1 ( ρ ) j n - 1 ( ρ * ) + n W n , 1 j n + 1 ( ρ ) j n + 1 ( ρ * ) ] r 2 d r ,
W n , 0 = π 2 E 0 2 2 ( m 2 + m * 2 2 m m * d n d n * + m 2 m * 2 c n c n * ) , W n , 1 = π 2 E 0 2 2 ( m 2 + m * 2 2 m m * c n c n * + m 2 m * 2 d n d n * )
0 x t J ν ( α t ) J ν ( α * t ) d t = x α 2 - α * 2 × [ J ν ( α x ) d J ν ( α * x ) d x - J ν ( α * x ) d J ν ( α x ) d x ]
W ( a ) = 3 4 W 0 n = 1 2 n + 1 y 2 - y * 2 { α n [ A n * ( y ) y - A n ( y ) y * ] + β n [ A n * ( y ) y * - A n ( y ) y ] } ,
A n ( y ) = ψ n ( y ) ψ n ( y ) , W 0 = 2 3 π a 3 E 0 2 2 , α n = m 2 + m * 2 2 [ m d n ψ n ( y ) ] [ m d n ψ n ( y ) ] * + m m * [ m c n ψ n ( y ) ] [ m c n ψ n ( y ) ] * , β n = m 2 + m * 2 2 [ m c n ψ n ( y ) ] [ m c n ψ n ( y ) ] * + m m * [ m d n ψ n ( y ) ] [ m d n ψ n ( y ) ] * .
m c n ψ n ( y ) = ψ n ( x ) - a n ζ n ( x ) , m d n ψ n ( y ) = m ψ n ( x ) - m b n ζ n ( x ) .
lim m i 0 W ( a ) = 3 4 W 0 n = 1 2 n + 1 y 2 γ n [ 1 + A n 2 ( y ) - n ( n + 1 ) y 2 ] ,
γ n = m 2 [ m c n ψ n ( y ) ] [ m c n ψ n ( y ) ] * + m 2 [ m d n ψ n ( y ) ] [ m d n ψ n ( y ) ] * .
W wa ( a ) 3 8 W 0 i m i x 3 n = 1 ( 2 n + 1 ) ( c n c n * + d n d n * ) × [ ψ n * ( ψ ) ψ n ( y ) - ψ n ( y ) ψ n * ( y ) ] m m * .
c n ψ n ( y ) = ψ n ( x ) - a n ζ n ( x ) , m d n ψ n ( y ) = ψ n ( x ) - b n ζ n ( x )
W wa ( a ) 3 8 W 0 2 m r m i x 3 n = 1 ( 2 n + 1 ) + [ Re ( a n ) + Re ( b n ) - a n a n * - b n b n * ] .
W wa ( a ) 3 8 W 0 m r m i x Q abs .
Re ( a n ) m r ± Im ( a n ) m i Re ( a n ) m r , Re ( b n ) m r ± Im ( b n ) m i Re ( b n ) m r ;
Im ( a n ) m r ± Re ( a n ) m i Im ( a n ) m r , Im ( b n ) m r ± Re ( b n ) m i Im ( b n ) m r .
Q abs 8 x 3 m i .

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