Robust motion-free and error-correcting method of estimating the focal length of a lens

Syed Azer Reza; Arslan Anjum

doi:10.1364/AO.56.000342

Applied Optics
Vol. 56,
Issue 2,
pp. 342-353
(2017)
•https://doi.org/10.1364/AO.56.000342

Robust motion-free and error-correcting method of estimating the focal length of a lens

Syed Azer Reza and Arslan Anjum

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Syed Azer Reza and Arslan Anjum, "Robust motion-free and error-correcting method of estimating the focal length of a lens," Appl. Opt. 56, 342-353 (2017)

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Related Topics
Table of Contents Category
- Instrumentation, Measurement, and Metrology
Optics & Photonics Topics
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The topics in this list come from the Optics and Photonics Topics applied to this article.

About this Article
History
- Original Manuscript: October 10, 2016
- Revised Manuscript: November 28, 2016
- Manuscript Accepted: November 28, 2016
- Published: January 10, 2017

Abstract

This paper presents a motion-free technique to characterize the focal length of any spherical convex or concave lens. The measurement test-bench uses a Gaussian laser beam, an electronically controlled variable focus lens (ECVFL), a digital micro-mirror device (DMD), and a standard photo-detector (PD). The method requires measuring beam spot sizes for different focal length settings of the ECVFL and using the measurement data to obtain a focal length estimate through an iterative least-squares-based curve-fitting algorithm. The method is also shown to overcome potential measurement errors that arise due to inaccurate placement of optical components on the test-bench as well as unknown principal plane locations of asymmetric lens samples such as plano-convex lenses. Contrary to the commercially deployed and other proposed methods of focal length characterization, this method does not involve any bulk mechanical motion of optical elements. This approach eliminates measurement errors due to gradual mechanical wear and tear and improves measurement repeatability by minimizing mechanical hysteresis. The compact and fully automated method delivers fast, repeatable, and reliable measurements, which we believe makes it ideal for deployment in industrial lens production units and characterizing lenses used in sensitive imaging systems and various other optical experiments and systems. Measured focal lengths are within the 1% manufacturer-provided tolerance values showing excellent agreement between theory and experiments. We also demonstrate measurement robustness by rectifying discrepancies between known and actual separation distances on the measurement test bench.

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Lens 1 $f_{Stat} = 100 mm \pm 1 %$		Lens 2 $f_{Stat} = 75 mm \pm 1 %$		Lens 3 $f_{Stat} = - 100 mm \pm 1 %$		Lens 4 $f_{Stat} = - 150 mm \pm 1 %$
$f$ (cm)	$w_{out} (f)$ (mm)	$f$ (cm)	$w_{out} (f)$ (mm)	$f$ (cm)	$w_{out} (f)$ (mm)	$f$ (cm)	$w_{out} (f)$ (mm)
13.86	0.319	10.6	0.747	6.93	0.244	14.80	0.119
11.83	0.449	9.47	0.802	6.58	0.343	13.71	0.123
9.47	0.577	8.59	0.864	6.22	0.446	12.79	0.131
7.97	0.726	7.97	0.923	5.87	0.551	12.00	0.137
7.24	0.842	7.55	0.983	5.59	0.642	11.33	0.144
6.58	0.957	7.24	1.035	5.39	0.738	10.73	0.152
5.87	1.093	6.93	1.113			10.20	0.160
5.39	1.202	6.58	1.155			9.71	0.165

Lens Sample	Manufacturer Values of $f_{Stat}$ (mm) ± % Tolerance	Estimated Best-Fit Values of $f_{Stat}$ (mm)	Difference between Estimated and Manufacturer Values of $f_{Stat}$
1	$100 \pm 1 %$	99.1	0.9%
2	$75 \pm 1 %$	75.6	0.8%
3	$- 100 \pm 1 %$	−99.0	1.0%
4	$150 \pm 1 %$	151.1	0.73%

Estimation Property	Sample Lens 1	Sample Lens 2	Sample Lens 3
Goodness of Fit ( $R^{2}$ )	0.99887	0.98200	0.99808
Average Residual $m^{2}$	$1.34 \times 10^{- 08}$	$3.13 \times 10^{- 08}$	$5.98 \times 10^{- 09}$
$f_{Stat}$ Estimation Error	0.38 mm	0.0306 mm	$- 0.34 mm$
Percentage Estimation Error	0.39%	0.04%	0.35%

Estimation Parameters	Best-Known Value	Estimated Value	Estimated Offset $Δ D_{i}$
Distance $D_{1}$	55.5 cm	55.4805 cm	0.0195 cm
Distance $D_{2}$	4.9 cm	4.8996 cm	0.0004 cm
Distance $D_{3}$	8.6 cm	8.6000 cm	0 cm

	Initial Guess and Converged Values
> Initial Guess	${(D_{2})}_{Design}$ ${(D_{3})}_{Design}$	${(D_{2})}_{Des} + 5 mm$ ${(D_{3})}_{Des} - 5 mm$	${(D_{2})}_{Des} - 5 mm$ ${(D_{3})}_{Des} + 5 mm$	${(D_{2})}_{Des} + 10 mm$ ${(D_{3})}_{Des} - 10 mm$	${(D_{2})}_{Des} - 10 mm$ ${(D_{3})}_{Des} + 10 mm$
$D_{2}$	4.9 cm	4.8996 cm	4.8996 cm	4.8996 cm	4.8996 cm
$D_{3}$	8.6 cm	8.6000 cm	8.6000 cm	8.6000 cm	8.6000 cm

	Initial Guess and Converged Values
> Initial Guess	${(D_{1})}_{Design}$	${(D_{1})}_{Des} + 5 mm$	${(D_{1})}_{Des} - 5 mm$	${(D_{1})}_{Des} + 10 mm$	${(D_{1})}_{Des} - 10 mm$
$D_{1}$	55.5 cm	55.4805 cm	55.4805 cm	55.4805 cm	55.4805 cm

Abstract

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Applied Optics