Abstract

It is shown that the cross section for scattering in the long-wavelength limit from a lossy dielectric sphere varies as λ<sup>-2</sup> and not as λ<sup>-1</sup>, as is suggested in standard treatments of the subject. The λ<sup>-1</sup> variation is in conflict with the analytic properties of the scattering amplitudes as a function of <i>k</i> = 2πλ<sup>-1</sup>, and this conflict is resolved when the wavelength dependence of the dielectric constant is taken into account.

© 1982 Optical Society of America

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1964), Secs. 6.13, 10.3, and 14.21.
  2. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Sec. 3.9.2.
  3. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Sec. 9.14.
  4. Ref. 1, Sec. 4.21, gives the classical derivation. Ref. 3, Sec. 9.14, gives a derivation that parallels more closely the quantum-mechanical optical theorem.
  5. See, e.g., R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), Sec. 4.2.2, especially Eq. (4.34), and the statement immediately following it. Note that Newton's Aii and the Si(0) of this Letter (and of Ref. 1) are related by Aii = ik-1Si(00), so in relating Newton's result for Aii to a statement about S(0) the terms real and imaginary and the terms even and odd must be interchanged.
  6. See, e.g., Ref. 3, Sec. 7.10. The fact that Im ∊(k) and Re S(0) are both odd functions of K is no accident. If we construct a material that is a dilute suspension of N of our scatterers per unit volume, the dielectric constant ∊¯ of that material is related to S(0) by ∊¯(k) = 1 + (4πNi/k2)S(0). That Im ∊¯ is odd in k implies that Re S(0) is odd and vice versa.
  7. See, e.g., Ref. 3, Sec. 7.10.
  8. Indeed, by using the dispersion relation for S(0) (e.g., Ref. 5), one can show by similar manipulations that [equation], where Im S¯(0, k′) = Im S(0, k′) - k3 limk′→0 [Im S(0, k′)/k3], exhibiting the λ-2 dependence of Ctot and identifying the coefficient in a different way. This and similar relations will be described in more detail elsewhere.9
  9. B. H. J. McKellar, M. A. Box, and C. Bohren, J. Opt. Soc. Am. (to be published).

Bohren, C.

B. H. J. McKellar, M. A. Box, and C. Bohren, J. Opt. Soc. Am. (to be published).

Box, M. A.

B. H. J. McKellar, M. A. Box, and C. Bohren, J. Opt. Soc. Am. (to be published).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Sec. 9.14.

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Sec. 3.9.2.

McKellar, B. H. J.

B. H. J. McKellar, M. A. Box, and C. Bohren, J. Opt. Soc. Am. (to be published).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1964), Secs. 6.13, 10.3, and 14.21.

Other (9)

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1964), Secs. 6.13, 10.3, and 14.21.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Sec. 3.9.2.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Sec. 9.14.

Ref. 1, Sec. 4.21, gives the classical derivation. Ref. 3, Sec. 9.14, gives a derivation that parallels more closely the quantum-mechanical optical theorem.

See, e.g., R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), Sec. 4.2.2, especially Eq. (4.34), and the statement immediately following it. Note that Newton's Aii and the Si(0) of this Letter (and of Ref. 1) are related by Aii = ik-1Si(00), so in relating Newton's result for Aii to a statement about S(0) the terms real and imaginary and the terms even and odd must be interchanged.

See, e.g., Ref. 3, Sec. 7.10. The fact that Im ∊(k) and Re S(0) are both odd functions of K is no accident. If we construct a material that is a dilute suspension of N of our scatterers per unit volume, the dielectric constant ∊¯ of that material is related to S(0) by ∊¯(k) = 1 + (4πNi/k2)S(0). That Im ∊¯ is odd in k implies that Re S(0) is odd and vice versa.

See, e.g., Ref. 3, Sec. 7.10.

Indeed, by using the dispersion relation for S(0) (e.g., Ref. 5), one can show by similar manipulations that [equation], where Im S¯(0, k′) = Im S(0, k′) - k3 limk′→0 [Im S(0, k′)/k3], exhibiting the λ-2 dependence of Ctot and identifying the coefficient in a different way. This and similar relations will be described in more detail elsewhere.9

B. H. J. McKellar, M. A. Box, and C. Bohren, J. Opt. Soc. Am. (to be published).

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