Analytical inversions in remote sensing of particle size distributions. 3: Angular and spectral scattering in the Rayleigh-Gans-Born approximation for particles of various geometrical shapes

Alain L. Fymat

doi:10.1364/AO.18.000126

Applied Optics
Vol. 18,
Issue 1,
pp. 126-130
(1979)
•https://doi.org/10.1364/AO.18.000126

Analytical inversions in remote sensing of particle size distributions. 3: Angular and spectral scattering in the Rayleigh-Gans-Born approximation for particles of various geometrical shapes

Alain L. Fymat

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Alain L. Fymat, "Analytical inversions in remote sensing of particle size distributions. 3: Angular and spectral scattering in the Rayleigh-Gans-Born approximation for particles of various geometrical shapes," Appl. Opt. 18, 126-130 (1979)

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Abstract

Analytical inverse formulas are provided for reconstructing the size distribution of particulates whose scattering patterns can be adequately described by the Rayleigh-Gans-Born (or Shifrin) approximation. The formulas hold for arbitrary polarization states at incidence and scattering of light and for angular or spectral measurements. The particle shapes considered are spheres, spherical shells, thin disks (which may be randomly oriented), and thin rods. Circular cylinders and ellipsoids can also be encompassed by our formulas if these particles are described in terms of equivalent spheres having the same volume.

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1. Sphere	G(u), u = 2ka sin½θ
2.
(a) Spherical shell	E(u)
(b) Stratification of shells	exp(−¼u²) (internal Gaussian distribution of polarizabilities)
3.
(a) Circular cylinders	F(u sinβ)E(½v cosβ), v = 2kl sin½θ
(b) Very long cylinders	F(u)E(¼w), w = 2kl sin∊
(c) Thin disk	F(u sinβ)
(d) Thin rod	E(½v cosβ)
4. Randomly oriented	${\bar{R}}^{2}$ = ∫R²(θ,φ)dω/4π
(a) Thin rods	${\bar{R}}^{2}$ = 2Si(v)/v − E²(½v)
(b) Thin disks	${\bar{R}}^{2}$ = (2/u²) [1 − F(2u)]
5. Ellipsoid	G(t), t = 2k·OC·sin½θ

	Shape	F	H(θ,ρ)	ρ	ν	y	t	h(ρ)
1.	Sphere	$(\frac{k}{4 π^{3}}) sin \frac{θ}{2}$	$r^{5} J_{3 / 2}^{2} (r t)$	r	³/₂	rt	$2 k sin \frac{θ}{2}$	2π/r⁵
2. (a)	Spherical shell	$(\frac{9 k}{4 π^{3}}) sin \frac{θ}{2}$	$r^{5} J_{1 / 2}^{2} (r t)$	r	1	rt	$2 k sin \frac{θ}{2}$	2π/r⁵
3. (c)	Thin disk	${(\frac{k sin β}{π l})}^{2} {sin}^{2} \frac{θ}{2}$	$a^{2} J_{1}^{2} (a t)$	a	1	at	$2 k sin \frac{θ}{2} sin β$	2π/a²
3. (d)	Thin rod	$(\frac{2 k cos β}{π^{3} a^{4}}) sin \frac{θ}{2}$	$l J_{1 / 2}^{2} (l t)$	l	½	lt	$k sin \frac{θ}{2} cos β$	2π/l

1. Sphere	G(u), u = 2ka sin½θ
2.
(a) Spherical shell	E(u)
(b) Stratification of shells	exp(−¼u²) (internal Gaussian distribution of polarizabilities)
3.
(a) Circular cylinders	F(u sinβ)E(½v cosβ), v = 2kl sin½θ
(b) Very long cylinders	F(u)E(¼w), w = 2kl sin∊
(c) Thin disk	F(u sinβ)
(d) Thin rod	E(½v cosβ)
4. Randomly oriented	${\bar{R}}^{2}$ = ∫R²(θ,φ)dω/4π
(a) Thin rods	${\bar{R}}^{2}$ = 2Si(v)/v − E²(½v)
(b) Thin disks	${\bar{R}}^{2}$ = (2/u²) [1 − F(2u)]
5. Ellipsoid	G(t), t = 2k·OC·sin½θ

	Shape	F	H(θ,ρ)	ρ	ν	y	t	h(ρ)
1.	Sphere	$(\frac{k}{4 π^{3}}) sin \frac{θ}{2}$	$r^{5} J_{3 / 2}^{2} (r t)$	r	³/₂	rt	$2 k sin \frac{θ}{2}$	2π/r⁵
2. (a)	Spherical shell	$(\frac{9 k}{4 π^{3}}) sin \frac{θ}{2}$	$r^{5} J_{1 / 2}^{2} (r t)$	r	1	rt	$2 k sin \frac{θ}{2}$	2π/r⁵
3. (c)	Thin disk	${(\frac{k sin β}{π l})}^{2} {sin}^{2} \frac{θ}{2}$	$a^{2} J_{1}^{2} (a t)$	a	1	at	$2 k sin \frac{θ}{2} sin β$	2π/a²
3. (d)	Thin rod	$(\frac{2 k cos β}{π^{3} a^{4}}) sin \frac{θ}{2}$	$l J_{1 / 2}^{2} (l t)$	l	½	lt	$k sin \frac{θ}{2} cos β$	2π/l

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Applied Optics