Visibility of large spheres observed with a laser velocimeter: a simple model

W. M. Farmer

doi:10.1364/AO.19.003660

Applied Optics
Vol. 19,
Issue 21,
pp. 3660-3667
(1980)
•https://doi.org/10.1364/AO.19.003660

Visibility of large spheres observed with a laser velocimeter: a simple model

W. M. Farmer

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W. M. Farmer, "Visibility of large spheres observed with a laser velocimeter: a simple model," Appl. Opt. 19, 3660-3667 (1980)

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Abstract

A model is developed to express the signal visibility for large refracting or reflecting spheres observed with a fringe-type laser velocimeter. The model yields a closed form solution for the visibility when scattering is observed off axis in the near forward, 90°, or near backward direction. Results show that particle size range can be adjusted by aperture adjustments at the scattered light receiver. Comparison of model predictions with recently published numerical computations shows good agreement.

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Observation direction	Argument of visibility function	Value of D/δ for first zero in visibility	Approximate limits in particle size range
Observation direction	Argument of visibility function	Value of D/δ for first zero in visibility	Minimum	Maximum
Near forward scatter (β ≲ 30°)	$\frac{π (d / δ) (m / m - 1)}{4 F (1 + \frac{1}{2} {tan}^{2} β)}$	4.88 [(m − 1)/m]F (1 + ½ tan²β)	0.4F [(m − 1)/m] (1 + ½ tan²β)	4F[(m − 1)/m] (1 + ½ tan²β)
90° Side scatter	$\frac{2 π (D / δ)}{4 F}$	3.45F	0.282F	2.82F
Near back scatter	$\frac{π (D / δ)}{4 F (1 + \frac{1}{2} {tan}^{2} β)}$	4.88F(1 + ½ tan²β)	0.4F(1 + ½ tan²β)	4F(1 + ½ tan²β)

Observation angle (deg)	Collection aperture f/No.	Index of refraction	First zero in visibility (D/δ)		% Difference approximate-numerical solution
Observation angle (deg)	Collection aperture f/No.	Index of refraction	Approximate solution	Numerical solution	% Difference approximate-numerical solution
10	2.8^a	1.33	3.47	3.65^a	−4.9^a
10	2.8	1.5	4.63	4.70	−1.5
20	2.8^a	1.33	3.64	4.0^a	−9.0^a
20	2.8	1.5	4.86	5.0	−2.9
30	2.8	1.33	3.98	4.5	−11.5
30	2.8	1.4	4.55	4.8	−5.2
30	2.8	1.5	5.31	5.3	+0.18
90	2.2	1.33	7.74	7.59^a	1.94^a

Observation direction	Argument of visibility function	Value of D/δ for first zero in visibility	Approximate limits in particle size range
Observation direction	Argument of visibility function	Value of D/δ for first zero in visibility	Minimum	Maximum
Near forward scatter (β ≲ 30°)	$\frac{π (d / δ) (m / m - 1)}{4 F (1 + \frac{1}{2} {tan}^{2} β)}$	4.88 [(m − 1)/m]F (1 + ½ tan²β)	0.4F [(m − 1)/m] (1 + ½ tan²β)	4F[(m − 1)/m] (1 + ½ tan²β)
90° Side scatter	$\frac{2 π (D / δ)}{4 F}$	3.45F	0.282F	2.82F
Near back scatter	$\frac{π (D / δ)}{4 F (1 + \frac{1}{2} {tan}^{2} β)}$	4.88F(1 + ½ tan²β)	0.4F(1 + ½ tan²β)	4F(1 + ½ tan²β)

Observation angle (deg)	Collection aperture f/No.	Index of refraction	First zero in visibility (D/δ)		% Difference approximate-numerical solution
Observation angle (deg)	Collection aperture f/No.	Index of refraction	Approximate solution	Numerical solution	% Difference approximate-numerical solution
10	2.8^a	1.33	3.47	3.65^a	−4.9^a
10	2.8	1.5	4.63	4.70	−1.5
20	2.8^a	1.33	3.64	4.0^a	−9.0^a
20	2.8	1.5	4.86	5.0	−2.9
30	2.8	1.33	3.98	4.5	−11.5
30	2.8	1.4	4.55	4.8	−5.2
30	2.8	1.5	5.31	5.3	+0.18
90	2.2	1.33	7.74	7.59^a	1.94^a

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Applied Optics