Abstract

The well-known resonance structure in the scattering amplitudes for dielectric spheres (Mie scattering) is known to arise from poles in the complex size parameter plane, with the sharpest resonances being due to poles that are close to the real axis. It is also well known that these resonances are damped by the presence of an absorption component in the refractive index. In this paper we show that for small absorption components, the poles move away from the real axis linearly. This result may then be used to study changes in the height, width, and area under the resonances. Perhaps the most interesting result concerns the area, which remains constant over a large range of k.

© 1988 Optical Society of America

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References

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  2. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).
  3. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  4. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
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  8. W. M. Irvine, “Light scattering by spherical particles: radiation pressure, asymmetry factor, and extinction cross section,”J. Opt. Soc. Am. 55, 16–21 (1965).
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  10. S. T. Shipley, J. A. Weinman, “A numerical study of scattering by large dielectric spheres,”J. Opt. Soc. Am. 68, 130–134 (1978).
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  11. 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]
  12. 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]
  13. A. Ashkin, J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354 (1977).
    [CrossRef]
  14. A. Ashkin, J. M. Dziedzic, “Observation of optical resonances of dielectric spheres by light scattering,” Appl. Opt. 20, 1803–1814 (1981).
    [CrossRef] [PubMed]
  15. P. Chylek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
    [CrossRef]
  16. R. Thurn, W. Kiefer, “Structural resonances observed in the Raman spectra of optically levitated liquid drops,” Appl. Opt. 24, 1515–1519 (1985).
    [CrossRef] [PubMed]
  17. R. Fuchs, K. L. Kliewer, “Optical modes of vibration in an ionic crystal sphere,”J. Opt. Soc. Am. 58, 319–330 (1968).
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  18. G. J. Rosasco, H. S. Bennett, “Internal field resonance structure: implications for optical absorption and scattering by microscopic particles,”J. Opt. Soc. Am. 68, 1242–1250 (1978).
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  19. P. R. Conwell, P. W. Barber, C. K. Rushforth, “Resonant spectra of dielectric spheres,” J. Opt. Soc. Am. A 1, 62–67 (1984).
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  20. J. R. Probert-Jones, “Resonance component of backscatter by large dielectric spheres,” J. Opt. Soc. Am. A 1, 822–830 (1984).
    [CrossRef]
  21. P. Chylek, V. Ramaswamy, A. Ashkin, J. M. Dziedzic, “Simultaneous determination of refractive index and size of spherical dielectric particles from light scattering data,” Appl. Opt. 22, 2302–2307 (1983).
    [CrossRef] [PubMed]
  22. S. C. Hill, C. K. Rushforth, R. E. Benner, P. R. Conwell, “Sizing dielectric spheres and cylinders by aligning measured and computed resonance locations: algorithm for multiple orders,” Appl. Opt. 24, 2380–2390 (1985).
    [CrossRef] [PubMed]
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  24. W. K. H. Panopky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1962).
  25. R. Loudon, The Quantum Theory of Light (Oxford U. Press, Oxford, 1973).
  26. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
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    [CrossRef] [PubMed]
  28. A. Bott, W. Zdunkowski, “Electromagnetic energy within dielectric spheres,” J. Opt. Soc. Am. A 4, 1361–1365 (1987).
    [CrossRef]
  29. G. Viera, M. A. Box, “Information content analysis of aerosol remote-sensing experiments using singular function theory. I. Extinction measurements,” Appl. Opt. 26, 1312–1327 (1987).
    [CrossRef] [PubMed]
  30. N. G. Alexopoulos, N. K. Uzunoglu, “Electromagnetic scattering from active objects: invisible scatterers,” Appl. Opt. 17, 235–239 (1978).
    [CrossRef] [PubMed]
  31. M. Kerker, “Electromagnetic scattering from active objects,” Appl. Opt. 17, 3337–3339 (1978).
    [CrossRef] [PubMed]
  32. M. Kerker, “Resonances in electromagnetic scattering by objects with negative absorption,” Appl. Opt. 18, 1180–1189 (1979).
    [CrossRef] [PubMed]
  33. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).

1987 (3)

1986 (1)

1985 (2)

1984 (2)

1983 (1)

1981 (1)

1980 (1)

H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

1979 (1)

1978 (7)

1977 (1)

A. Ashkin, J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354 (1977).
[CrossRef]

1976 (1)

1968 (1)

1966 (1)

1965 (1)

Alexopoulos, N. G.

Ashkin, A.

Attard, P.

Barber, P. W.

Benner, R. E.

Bennett, H. S.

Bohren, C. F.

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

Bott, A.

Box, M. A.

Bryant, G.

Bryant, H. C.

Chylek, P.

Conwell, P. R.

Cox, A. J.

Deirmendjian, D.

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

Dziedzic, J. M.

Fuchs, R.

Hill, S. C.

Huffman, D. R.

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

Irvine, W. M.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

Kerker, M.

Kiefer, W.

Kiehl, J. T.

P. Chylek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
[CrossRef]

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]

Kliewer, K. L.

Ko, M. K. W.

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]

P. Chylek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
[CrossRef]

Loudon, R.

R. Loudon, The Quantum Theory of Light (Oxford U. Press, Oxford, 1973).

McKellar, B. H. J.

Nussenzveig, H. M.

H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

Panopky, W. K. H.

W. K. H. Panopky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1962).

Phillips, M.

W. K. H. Panopky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1962).

Probert-Jones, J. R.

Ramaswamy, V.

Rosasco, G. J.

Rushforth, C. K.

Shipley, S. T.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Thurn, R.

Uzunoglu, N. K.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Viera, G.

Weinman, J. A.

Wiscombe, W. J.

H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

Zdunkowski, W.

Appl. Opt. (10)

N. G. Alexopoulos, N. K. Uzunoglu, “Electromagnetic scattering from active objects: invisible scatterers,” Appl. Opt. 17, 235–239 (1978).
[CrossRef] [PubMed]

H. S. Bennett, G. J. Rosasco, “Resonances in the efficiency factors for absorption: Mie scattering theory,” Appl. Opt. 17, 491–493 (1978).
[CrossRef] [PubMed]

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]

M. Kerker, “Electromagnetic scattering from active objects,” Appl. Opt. 17, 3337–3339 (1978).
[CrossRef] [PubMed]

M. Kerker, “Resonances in electromagnetic scattering by objects with negative absorption,” Appl. Opt. 18, 1180–1189 (1979).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, “Observation of optical resonances of dielectric spheres by light scattering,” Appl. Opt. 20, 1803–1814 (1981).
[CrossRef] [PubMed]

R. Thurn, W. Kiefer, “Structural resonances observed in the Raman spectra of optically levitated liquid drops,” Appl. Opt. 24, 1515–1519 (1985).
[CrossRef] [PubMed]

S. C. Hill, C. K. Rushforth, R. E. Benner, P. R. Conwell, “Sizing dielectric spheres and cylinders by aligning measured and computed resonance locations: algorithm for multiple orders,” Appl. Opt. 24, 2380–2390 (1985).
[CrossRef] [PubMed]

G. Viera, M. A. Box, “Information content analysis of aerosol remote-sensing experiments using singular function theory. I. Extinction measurements,” Appl. Opt. 26, 1312–1327 (1987).
[CrossRef] [PubMed]

P. Chylek, V. Ramaswamy, A. Ashkin, J. M. Dziedzic, “Simultaneous determination of refractive index and size of spherical dielectric particles from light scattering data,” Appl. Opt. 22, 2302–2307 (1983).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. A (5)

Phys. Rev. A (1)

P. Chylek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
[CrossRef]

Phys. Rev. Lett. (2)

A. Ashkin, J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354 (1977).
[CrossRef]

H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

Other (9)

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

W. K. H. Panopky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1962).

R. Loudon, The Quantum Theory of Light (Oxford U. Press, Oxford, 1973).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

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

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

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

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Tables (4)

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Table 1 Basic Pole and Resonance Parameters

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Table 2 Terms in the Resonance Taylor Expansiona

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Table 3 Pole Imaginary Component versus k

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Table 4 Extinction Resonance Height versus k

Equations (43)

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a n = p n p n + i q n
b n = r n r n + i s n ,
p n = ψ n ( x ) ψ n ( m x ) m ψ n ( m x ) ψ n ( x ) , q n = χ n ( x ) ψ n ( m x ) m ψ n ( m x ) χ n ( x ) , r n = m ψ n ( x ) ψ n ( m x ) ψ n ( m x ) ψ n ( x ) , s n = m χ n ( x ) ψ n ( m x ) ψ n ( m x ) χ n ( x ) .
p n p n ( x = x R ) = γ
q n ( x x R ) q n ( x = x R ) = β ( x x R ) ,
a n = α x z ,
1 = α x R ( z r + i z i ) = α / ( i z i ) ; α = i z i .
a n = i z i x ( x R + i z i )
a n = γ γ + i β ( x x R ) .
z i = γ / β .
p n γ + i k p n i + k 2 p n r ,
q n β ( x x R ) + i k q n i + k 2 q n r
p n i = p m | k = 0 ,
p n r = 1 2 2 p m 2 | k = 0
a n = α x z = α ( x x R ) ( z r z r ) i z i .
a n = γ + i k p n i + k 2 p n r γ + i k p n i + k 2 p n r + i β ( x x R ) k q n i + i k 2 q n r .
α = α + k p n i / β i k 2 p n r / β ,
z r = z r k p n i / β k 2 q n r / β ,
z i = z i k q n i / β + k 2 p n r / β .
z i = z i k q n i / β ,
Re a n = ( x z r ) Re α z i Im α ( x z r ) 2 + z i 2
| a n | 2 = α 2 ( x z r ) 2 + z i 2 .
H e = Im α / z i
z i z i k q n i / β
H s = | α | 2 / z i 2
H e 2 ,
W s = 2 z i .
W e = W s [ 1 + ( Re α / Im α ) 2 ] 1 / 2 .
A e = π Im α
π z i
A s = π | α | 2 / z i
z i 2 z i k q n i / β .
A e H e = A s H s = π z i .
H e = ( 1 k q n i / β z i ) 1 .
k ~ z i / 15 .
γ = ψ n ( x ) ψ n ( m x ) m ψ n ( x ) ψ n ( m x )
β = ( 1 m 2 ) [ χ n ( x ) ψ n ( m x ) + n ( n + 1 ) m χ n ( x ) ψ n ( m x ) / m 2 x 2 ] .
ψ n ( z ) = [ n ( n + 1 ) / z 2 1 ] ψ n ( z ) .
β γ = ( 1 m 2 ) { ψ n 2 ( m x ) [ ψ n ( x ) χ n ( x ) ψ n ( x ) χ n ( x ) ] + n ( n + 1 ) ψ n 2 ( m x ) [ ψ n ( x ) χ n ( x ) ψ n ( x ) χ n ( x ) ] / x 2 } = ( m 2 1 ) [ ψ n 2 ( m x ) + n ( n + 1 ) ψ n 2 ( m x ) / x 2 ] .
ψ n ( x ) χ n ( x ) ψ n ( x ) χ n ( x ) = 1.
q n i = χ n ( x ) ψ n ( m x ) + m x χ n ( x ) ψ n ( m x ) + [ 1 n ( n + 1 ) / m 2 x 2 ] x χ n ( x ) ψ n ( x ) .
q n i γ = m x [ ψ n ( m x ) + ψ n ( m x ) / 2 m x ] 2 + m x ψ n 2 ( m x ) [ 1 ( n + 1 2 ) 2 / m 2 x 2 ] .
m x > n .

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