Determination of size and complex index of refraction of single particles with elastic light scattering

Mir Seliman Waez; Steven J. Eckels; Christopher M. Sorensen

doi:10.1364/AO.413675

Applied Optics
Vol. 60,
Issue 3,
pp. 600-605
(2021)
•https://doi.org/10.1364/AO.413675

Determination of size and complex index of refraction of single particles with elastic light scattering

Mir Seliman Waez, Steven J. Eckels, and Christopher M. Sorensen

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Mir Seliman Waez, Steven J. Eckels, and Christopher M. Sorensen, "Determination of size and complex index of refraction of single particles with elastic light scattering," Appl. Opt. 60, 600-605 (2021)

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- Physical Optics
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About this Article
History
- Original Manuscript: October 30, 2020
- Revised Manuscript: December 14, 2020
- Manuscript Accepted: December 15, 2020
- Published: January 14, 2021

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Abstract

We show that for spherical particles greater than ca. 5 µm, the differential scattering cross section is only weakly dependent on the real and imaginary parts of the refractive index (${m} = {n} + {i}\kappa$) when integrated over angle ranges near ${37} \pm {5}^\circ$ and ${115} \pm {5}^\circ$, respectively. With this knowledge, we set up an arrangement that collects scattered light in the ranges ${37} \pm {5}^\circ$, ${115} \pm {5}^\circ$, and ${80} \pm {5}^\circ$. The weak functionality on refractive index for the first two angle ranges simplifies the inversion of scattering to the particle properties of diameter and the real and imaginary refractive indices. Our setup also uses a diamond-shaped incident beam profile that allows us to determine when a particle went through the exact center of the beam. Application of our setup to droplets of an absorbing liquid successfully determined the diameter and complex refractive index to accuracies ranging from a few to ten percent. Comparisons to simulated data derived from the Mie equations yielded similar results.

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${2 R}_{a c t u a l}$ ( $\pm 10 µ m$ )	${2 R}_{m e a s}$ (µm)	$κ {k R}_{a c t u a l}$ ( $\pm 9 %$ )	$κ {k R}_{m e a s}$	$n_{a c t u a l}$	$n_{m e a s}$	Discrepancy (%)
${2 R}_{a c t u a l}$ ( $\pm 10 µ m$ )	${2 R}_{m e a s}$ (µm)	$κ {k R}_{a c t u a l}$ ( $\pm 9 %$ )	$κ {k R}_{m e a s}$	$n_{a c t u a l}$	$n_{m e a s}$	$2 R$	$κ k R$	$n$
300	286	0.89	0.72	1.33	1.34	−5	−19	$+ 0.75$
300	287	0.89	0.71	1.33	1.33	−4	−20	0
300	276	0.89	0.75	1.33	1.34	−8	−16	$+ 0.75$
300	289	0.89	0.73	1.33	1.33	−4	−18	0
300	286	0.89	0.73	1.33	1.32	−5	−18	−0.75
300	292	0.77	0.61	1.33	1.31	−3	−20	−1.5
300	297	0.77	0.62	1.33	1.31	−1	−19	−1.5
300	274	0.77	0.68	1.33	1.32	−9	−12	−0.75
300	273	0.77	0.65	1.33	1.32	−9	−15	−0.75
300	292	0.77	0.61	1.33	1.31	−3	−20	−1.5

${2 R}_{a c t u a l}$ ( $\pm 10 µ m$ )	${2 R}_{m e a s}$ (µm)	$κ {k R}_{a c t u a l}$ ( $\pm 9 %$ )	$κ {k R}_{m e a s}$	$n_{a c t u a l}$	$n_{m e a s}$	Discrepancy (%)
${2 R}_{a c t u a l}$ ( $\pm 10 µ m$ )	${2 R}_{m e a s}$ (µm)	$κ {k R}_{a c t u a l}$ ( $\pm 9 %$ )	$κ {k R}_{m e a s}$	$n_{a c t u a l}$	$n_{m e a s}$	$2 R$	$κ k R$	$n$
220	223	0.80	0.70	1.49	1.41	$+ 1$	−12	−5
220	228	0.80	0.74	1.49	1.40	$+ 4$	−7	−6
220	215	0.80	0.74	1.49	1.40	−2	−7	−6
220	217	0.80	0.74	1.49	1.43	−1	−7	−4
220	219	0.80	0.71	1.49	1.41	−0.5	−11	−6
220	208	1.10	1.0	1.49	1.44	−5	−9	−3
220	218	1.10	1.19	1.49	1.43	−1	$+ 8$	−4
220	214	1.10	1.20	1.49	1.45	−3	$+ 9$	−3
220	205	1.10	1.24	1.49	1.49	−7	$+ 13$	0
220	210	1.10	1.18	1.49	1.46	−4	$+ 7$	−2

${2 R}_{a c t u a l}$ ( $\pm 10 µ m$ )	${2 R}_{m e a s}$ (µm)	$κ {k R}_{a c t u a l}$ ( $\pm 9 %$ )	$κ {k R}_{m e a s}$	$n_{a c t u a l}$	$n_{m e a s}$	Discrepancy (%)
${2 R}_{a c t u a l}$ ( $\pm 10 µ m$ )	${2 R}_{m e a s}$ (µm)	$κ {k R}_{a c t u a l}$ ( $\pm 9 %$ )	$κ {k R}_{m e a s}$	$n_{a c t u a l}$	$n_{m e a s}$	$2 R$	$κ k R$	$n$
5	5.8	0.21	0.19	1.44	1.39	16	−11	−3.6
10	9.8	0.14	0.115	1.53	1.54	−2	−18	0.6
10	9.9	0.06	0.046	1.56	1.59	−1	−23	1.9
20	19	0.12	0.10	1.56	1.57	−4	−15	0.6
50	52	0.40	0.33	1.70	1.61	4	−18	−5.5
50	52.5	0.40	0.33	1.50	1.43	5	−18	−5.0
100	102	1.00	0.92	1.60	1.56	2	−8	−2.6
100	96	0.10	0.092	1.60	1.59	−4	−8	−0.6
100	101	0.30	0.28	1.60	1.56	1	−6	−2.6
110	107	1.56	1.65	1.53	1.50	−3	6	−2.0

${2 R}_{a c t u a l}$ ( $\pm 10 µ m$ )	${2 R}_{m e a s}$ (µm)	$κ {k R}_{a c t u a l}$ ( $\pm 9 %$ )	$κ {k R}_{m e a s}$	$n_{a c t u a l}$	$n_{m e a s}$	Discrepancy (%)
${2 R}_{a c t u a l}$ ( $\pm 10 µ m$ )	${2 R}_{m e a s}$ (µm)	$κ {k R}_{a c t u a l}$ ( $\pm 9 %$ )	$κ {k R}_{m e a s}$	$n_{a c t u a l}$	$n_{m e a s}$	$2 R$	$κ k R$	$n$
300	286	0.89	0.72	1.33	1.34	−5	−19	$+ 0.75$
300	287	0.89	0.71	1.33	1.33	−4	−20	0
300	276	0.89	0.75	1.33	1.34	−8	−16	$+ 0.75$
300	289	0.89	0.73	1.33	1.33	−4	−18	0
300	286	0.89	0.73	1.33	1.32	−5	−18	−0.75
300	292	0.77	0.61	1.33	1.31	−3	−20	−1.5
300	297	0.77	0.62	1.33	1.31	−1	−19	−1.5
300	274	0.77	0.68	1.33	1.32	−9	−12	−0.75
300	273	0.77	0.65	1.33	1.32	−9	−15	−0.75
300	292	0.77	0.61	1.33	1.31	−3	−20	−1.5

${2 R}_{a c t u a l}$ ( $\pm 10 µ m$ )	${2 R}_{m e a s}$ (µm)	$κ {k R}_{a c t u a l}$ ( $\pm 9 %$ )	$κ {k R}_{m e a s}$	$n_{a c t u a l}$	$n_{m e a s}$	Discrepancy (%)
${2 R}_{a c t u a l}$ ( $\pm 10 µ m$ )	${2 R}_{m e a s}$ (µm)	$κ {k R}_{a c t u a l}$ ( $\pm 9 %$ )	$κ {k R}_{m e a s}$	$n_{a c t u a l}$	$n_{m e a s}$	$2 R$	$κ k R$	$n$
220	223	0.80	0.70	1.49	1.41	$+ 1$	−12	−5
220	228	0.80	0.74	1.49	1.40	$+ 4$	−7	−6
220	215	0.80	0.74	1.49	1.40	−2	−7	−6
220	217	0.80	0.74	1.49	1.43	−1	−7	−4
220	219	0.80	0.71	1.49	1.41	−0.5	−11	−6
220	208	1.10	1.0	1.49	1.44	−5	−9	−3
220	218	1.10	1.19	1.49	1.43	−1	$+ 8$	−4
220	214	1.10	1.20	1.49	1.45	−3	$+ 9$	−3
220	205	1.10	1.24	1.49	1.49	−7	$+ 13$	0
220	210	1.10	1.18	1.49	1.46	−4	$+ 7$	−2

Abstract

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Figures (4)

Tables (3)

Equations (5)

Applied Optics