Monotonic responses from monochromatic optical particle counters

J. C. Barnard; L. C. Harrison

doi:10.1364/AO.27.000584

Applied Optics
Vol. 27,
Issue 3,
pp. 584-592
(1988)
•https://doi.org/10.1364/AO.27.000584

Monotonic responses from monochromatic optical particle counters

J. C. Barnard and L. C. Harrison

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J. C. Barnard and L. C. Harrison, "Monotonic responses from monochromatic optical particle counters," Appl. Opt. 27, 584-592 (1988)

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Abstract

Mie theory is used to investigate the degree to which a response from an optical particle counter (OPC) can be made monotonic when sizing particles of a given refractive index. This investigation is restricted to OPC designs that use monochromatic illumination and have coaxial light-collection optics. By systematically exploring the angular limits of the collection optics, it is found that, for certain limits, a very nearly monotonic response can be achieved for a nonresonant cavity OPC for a range of refractive indices. In contrast, for a resonant cavity instrument, this investigation did not reveal any angular limits where the response is close to being monotonic, unless the refractive index of the particles being sized has a large imaginary component.

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0.1	5.6	12.3	20.6	30.8	43.3	58.6	77.4	100.6	128.9
0.6	6.2	13.1	21.5	31.9	44.7	60.3	79.5	103.1	132.1
1.1	6.8	13.9	22.5	33.1	46.1	62.1	81.7	105.8	135.4
1.6	7.5	14.6	23.5	34.3	47.6	63.9	83.9	108.5	138.7
2.2	8.1	15.4	24.4	35.5	49.0	65.7	86.1	111.2	142.0
2.7	8.8	16.3	25.4	36.7	50.6	67.5	88.4	114.0	145.5
3.3	9.5	17.1	26.5	38.0	52.1	69.4	90.7	116.9	149.0
3.8	10.2	18.0	27.5	39.3	53.7	71.4	93.1	119.8	152.6
4.4	10.9	18.8	28.6	40.6	55.3	73.4	95.5	122.8	156.2
5.0	11.6	19.7	29.7	41.9	56.9	75.4	98.0	125.8	160.0

The Monotonicity Parameter, s, the Average Relative Error, E_r, the Maximum Relative Error, E_m, and the Size Parameter at which E_m occurs, x_m, for the Improved Truncation Angles, θ_f, θ_r, of a Nonresonant Cavity Instrument.
Refractive Index (m)	θ_f(°)	θ_r(°)	s	E_r(%)	E_m(%)	x_m
1.33	19	96	−0.20	0.9	20.5	10.8
1.50	18	99	−0.40	0.8	26.6	5.0
1.50 – 1.0i	20	100	0.40	0.0	0.0	---
1.95 – .66i	20	100	0.28	0.0	0.0	---

The Monotonicity Parameter, s, the Average Relative Error, E_r, the Maximum Relative Error, E_m, and the Size Parameter at which E_m occurs, x_m, for the Compromise Truncation Angles θ_f=18°, θ_r=99° of a Nonresonant Cavity Instrument. (Also shown – for the sake of comparison – are s, E_r, E_m, and x_m for a forward-scattering OPC with θ_f=4°, θ_r=22°, and a wide-angle OPC with θ_f=45°, θ_r=135°)
m = 1.33
Instrument	θ_f(°)	θ_f(°)	s	E_r(%)	E_m(%)		x_m
Compromise	18	99	−0.25	0.9	26.8		10.8
Forward-Scattering	4	22	−3.45	7.8	117.2		5.2
Wide-Angle	45	135	−16.0	10.4	29.7		15.6
m = 1.50
Compromise	18	99	−0.40	0.8	26.6		5.0
Forward-Scattering	4	22	−5.53	8.5	111.6		3.6
Wide-Angle	45	135	−5.28	3.9	27.6		8.8
M = 1.50 – 1.0i
Compromise	18	99	0.42	0.0	0.0		---
Forward-Scattering	4	22	−0.08	0.4	13.1		31.2
Wide-Angle	45	135	0.37	0.0	0.0		---
m = 1.95 – 0.66i
Compromise	18	99	0.30	0.0	0.0	---
Forward-Scattering	4	22	−0.13	0.7	16.5	29.8
Wide-Angle	45	135	−0.16	0.2	39.2	2.0

The Monotonicity Parameter, s, the Average Relative Error, E_r, the Maximum Relative Error, E_m, and the Size Parameter at which E_m occurs, x_m, for the Improved Truncation Angles θ_f, θ_r, of a Resonant Cavity Instrument.
Refractive Index (m)	θ_f(°)	θ_r(°)	s	E_r(%)	E_m(%)	x_m
1.33	5	60	−2.70	3.0	55.4	7.3
1.50	5	158	−3.33	4.6	49.2	5.6
1.50 – 1.0i	5	166	1.00	0.0	0.0	---
1.95 – .66i	5	167	0.97	0.0	0.0	----

The Monotonicity Parameter, s, the Average Relative Error, E_r, the Maximum Relative Error, E_m, and the Size Parameter at which E_m occurs, x_m, for the Compromise Truncation Angles θ_f=7°, θ_r=71° of a Nonresonant Cavity Instrument. (Also shown for the sake of comparison are s, E_r, E_m, and x_m for a forward-scattering OPC with θ_f=4°, θ_r=22°, and a wide-angle OPC with θ_f=35°, θ_r=120°)
m = 1.33
Instrument	θ_f(°)	θ_r(°)	s	E_r(%)	E_m(%)	x_m
Compromise	7	71	−3.09	2.7	53.5	7.3
Forward-Scattering	4	22	−8.09	10.6	124.6	5.0
Wide-Angle	35	120	−25.07	21.3	68.3	11.4
m = 1.50
Compromise	7	71	−3.54	5.9	55.3	5.6
Forward-Scattering	4	22	−12.76	16.9	113.1	3.6
Wide-Angle	35	120	−13.10	11.0	71.2	5.0
m = 1.50 – 1.0i
Compromise	7	71	0.43	0.0	0.0	---
Forward-Scattering	4	22	−0.25	0.2	7.7	30.0
Wide-Angle	35	120	−0.09	0.2	47.8	1.8
m = 1.95 – .66i
Compromise	7	71	0.35	0.0	0.0	-----
Forward-Scattering	4	22	−0.28	0.6	15.4	30.0
Wide-Angle	35	120	−0.60	1.0	122.8	1.6

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Applied Optics