Home | Survey | Topics | Index | References | Dictionary | Contributing | Gallery | Community
| Effective optical particle shape: An equivalent sphere for a spheroid | Prev topic | Next topic |
Chen (1995) used the eikonal approximation (see, for example, Sharma and Somerford 1999) to show that light scattering at angles smaller than ~30° by a spheroid can be well approximated by that of an equivalent sphere under a condition that the minimum radius of curvature am be large as compared to the wavelength, λ, of light:
| amk ≥ 4 | (1) |
where k = 2π / λ is the wavenumber, am = aµ for an oblate spheroid and a / µ2 for a prolate spheroid, where a is the semimajor axis and µ = a / b.
The element [1,1] of the scattering matrix of the spheroid, M11, can be approximated as follows:
| M11(θ, a, b, m) = (α2 / β4 ) M11, Mie( θ, De, me ) | (2) |
where m is the refractive index of the spheroid, and M11, Mie is the [1,1] element of the scattering matrix the equivalent sphere. Here the approximation holds to within 10%.
The attenuation cross section, Cc of the spheroid can be approximated as follows.
| Cc(a, b, m) = (α / β02 ) Cc, Mie(De0, me0 ) | (3) |
where Cc, Mie is the attenuation cross section of the equivalent sphere in Mie theory, and the equivalent sphere diameters, De, De0, and refractive indices, me, me0, are defined as follows:
| De = 2 (β / µ ) a | (4a) |
| De0 = 2 (β0 / µ ) a | (4b) |
and
| me = 1 + [µ / (αβ)] (m - 1) | (5a) |
| me0 = 1 + [µ / (α0β0)] (m - 1) | (5b) |
where parameters α and β, as well as α0 and β0, are defined by equations of the following type:
| α = (cos2u + µ2 sin2u)½ | (6) |
| β = (α2 cos2v + sin2v)½ | (7) |
where the pair of angles (u, v) define a direction in space as follows. For α and β, (u, v) = (Θ, Φ), and can be obtained as follows:
| cosΦ = [cos(θ/2) sinθ0 cosφ0 - sin(θ/2) cosθ0] / sinΘ | (8) |
| cosΘ = cos(θ/2) cosθ0 + sin(θ/2) sinθ0 cosφ0 | (9) |
where (θ, φ) is the scattering direction and (θ0, φ0) define the orientation of the semimajor axis of the spheroid relative to that of the axis of the incident light beam. For α0 and β0, (u, v) = (θ0, φ0).
| CITATION: Jonasz M. 2006. Effective optical particle shape: An equivalent sphere for a spheroid (www.tpdsci.com/Tpc/EfOptPtShpSphSpd.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 19-Sep-2006 Modified: 02-Oct-2006 Peer-reviewed: PENDING |
| Copyright 2005-2008 MJC Optical Technology. All rights reserved. | Terms of use | Menu |