Inversion of light-scattering measurements for particle size and optical constants: theoretical study

Matthew R. Jones; Bill P. Curry; M. Quinn Brewster; Keng H. Leong

doi:10.1364/AO.33.004025

Applied Optics
Vol. 33,
Issue 18,
pp. 4025-4034
(1994)
•https://doi.org/10.1364/AO.33.004025

Inversion of light-scattering measurements for particle size and optical constants: theoretical study

Matthew R. Jones, Bill P. Curry, M. Quinn Brewster, and Keng H. Leong

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Matthew R. Jones, Bill P. Curry, M. Quinn Brewster, and Keng H. Leong, "Inversion of light-scattering measurements for particle size and optical constants: theoretical study," Appl. Opt. 33, 4025-4034 (1994)

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Abstract

We invert the Fredholm equation representing the light scattered by a single spherical particle or a distribution of spherical particles to obtain the particle size distribution function and refractive index. We obtain the solution by expanding the distribution function as a linear combination of a set of orthonormal basis functions. The set of orthonormal basis functions is composed of Schmidt–Hilbert eigenfunctions and a set of supplemental basis functions, which have been orthogonalized with respect to the Schmidt–Hilbert eigenfunctions by using the Gram–Schmidt orthogonalization procedure. We use the orthogonality properties of the basis functions and of the eigenvectors of the kernel covariance matrix to obtain the solution that minimizes the residual errors subject to a trial function constraint. The inversion process is described, and results from the inversion of several simulated data sets are presented.

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Diameter (μm)	Size Parameter	Optical Properties
		1.1 + i 10⁻³		1.5 + i 10⁻³		2.0 + i 10⁻³
		C_avg (μm²)	C_20° (μm²)	C_avg (μm²)	C_20° (μm²)	C_avg (μm²)	C_20° (μm²)
0.1	0.5	3 × 10⁻⁷	4 × 10⁻⁷	7 × 10⁻⁶	8 × 10⁻⁶	2 × 10⁻⁵	3 × 10⁻⁵
0.5	2.3	3 × 10⁻⁴	3 × 10⁻³	1 × 10⁻²	8 × 10⁻²	4 × 10⁻²	0.1
1.0	4.7	4 × 10⁻³	0.1	0.1	0.4	0.1	0.3
2.0	9.4	1 × 10⁻²	0.2	0.2	0.6	0.2	0.7
3.0	14	3 × 10⁻²	0.6	0.3	3	0.4	1
4.0	19	0.1	0.9	0.5	2	0.6	2
5.0	23	0.1	2	0.6	4	0.9	4
6.0	28	0.2	2	1	5	1	5
7.0	33	0.3	4	1	6	2	5
8.0	38	0.3	4	2	8	2	7
9.0	42	0.6	8	2	10	2	10
10.0	47	06	7	2	11	3	12

Parameter	Value
Parameter	Actual	Retrieved
Diameter (μm)	4.5	4.3
Size parameter	21.2	20.1
Optical properties	1.45 + i 7.5 × 10⁻⁵	1.42 + i 1.0 × 10⁻⁴

Parameter	Value
Parameter	Actual	Retrieved
Diameter (μm)	7.6	7.6
Size parameter	27.9	28.0
Optical properties	1.58 + i 5.0 × 10⁻⁴	1.58 + i 1.0 × 10⁻⁵

Case	Mean Size Parameter		Geometric Standard Deviation		Refractive Index
Case	Retrieved	Actual	Retrieved	Actual	Retrieved	Actual
1	10.1	10.0	1.08	1.11	1.33	1.33
2	19.7	20.0	1.11	1.11	1.46	1.45
3	23.2	20.0	1.09	1.35	1.45	1.45
4	48.9	35.0	1.03	1.11	1.56	1.55
5	47.4	35.0	1.04	1.22	1.56	1.55
6	55.0	55.0	1.03	1.02	1.33	1.33

Diameter (μm)	Size Parameter	Optical Properties
		1.1 + i 10⁻³		1.5 + i 10⁻³		2.0 + i 10⁻³
		C_avg (μm²)	C_20° (μm²)	C_avg (μm²)	C_20° (μm²)	C_avg (μm²)	C_20° (μm²)
0.1	0.5	3 × 10⁻⁷	4 × 10⁻⁷	7 × 10⁻⁶	8 × 10⁻⁶	2 × 10⁻⁵	3 × 10⁻⁵
0.5	2.3	3 × 10⁻⁴	3 × 10⁻³	1 × 10⁻²	8 × 10⁻²	4 × 10⁻²	0.1
1.0	4.7	4 × 10⁻³	0.1	0.1	0.4	0.1	0.3
2.0	9.4	1 × 10⁻²	0.2	0.2	0.6	0.2	0.7
3.0	14	3 × 10⁻²	0.6	0.3	3	0.4	1
4.0	19	0.1	0.9	0.5	2	0.6	2
5.0	23	0.1	2	0.6	4	0.9	4
6.0	28	0.2	2	1	5	1	5
7.0	33	0.3	4	1	6	2	5
8.0	38	0.3	4	2	8	2	7
9.0	42	0.6	8	2	10	2	10
10.0	47	06	7	2	11	3	12

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Equations (30)

Applied Optics