Improvement in the measurement of focal length using spot patterns and spherical aberration

Yobani Mejía

doi:10.1364/AO.52.005577

Applied Optics
Vol. 52,
Issue 23,
pp. 5577-5584
(2013)
•https://doi.org/10.1364/AO.52.005577

Improvement in the measurement of focal length using spot patterns and spherical aberration

Yobani Mejía

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Yobani Mejía, "Improvement in the measurement of focal length using spot patterns and spherical aberration," Appl. Opt. 52, 5577-5584 (2013)

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Abstract

Using an optical setup that includes a square array of $3 \times 3$ holes, we used nine meridional rays to measure the effective focal length of a lens. We observed the selected meridional rays as a spot pattern on a diffuse screen. First, we generated a regular square spot pattern (reference pattern) without a lens to test, and then we generated two spot patterns in two different axial positions when the lens being tested refracts the rays. By selecting two sets of four rays of each spot pattern, we were able to measure the difference of the longitudinal (primary) spherical aberration in two positions. With this difference we were able to improve the calculation of the effective focal length. To determine the method’s precision, we first simulated the relative error in the effective focal length considering the error in the measurement of the ray heights. Then we determined the experimental relative error by means of the standard deviation of the focal lengths obtained for each spot (in the image of reference and for the images at the two different locations) for both sets of four spots. The experimental results agree very well with the simulation. The error analysis allows us to establish under what conditions it is possible to obtain relative errors of less than 1% in the effective focal length.

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Lens Number	$f_{587}$ (mm)	Diameter (mm)	Company
1	+200.00	30.0	Edmund Optics
2	−100.38	49.2	Melles–Griot
3	+100.00	25.4	ThorLabs
4	+75.00	25.4	ThorLabs
5	+40.00	25.4	ThorLabs

$Δ z$ (mm)	$f \pm δ f (p)$ (mm)	$ε_{f} (p)$ (%)	$f \pm δ f (l)$ (mm)	$ε_{f} (l)$ (%)
10.0	201.90±8.52	4.22	200.50±2.70	1.35
40.0	201.85±1.42	0.70	199.59±1.00	0.50
70.0	200.93±0.82	0.41	200.05±0.67	0.34
100.0	200.16±0.63	0.32	200.11±0.54	0.27

	Spot Number	$y_{0}$ (pixels)	$y_{1}$ (pixels)	$y_{2}$ (pixels)	$f$ (mm)	$f \pm δ f$ (mm)	$ε_{f}$ (%)
$p$	1	106.97	78.73	73.64	210.22	201.90±8.51	4.22
	2	106.36	78.84	73.39	195.32
	3	106.14	78.42	73.37	210.41
	4	105.98	78.44	72.91	191.65
$l$	1	150.80	110.98	103.43	199.71	200.50±2.70	1.35
	2	150.33	110.86	103.43	202.18
	3	150.76	111.13	103.73	203.62
	4	149.87	110.59	102.96	196.50

	Spot Number	$y_{0}$ (pixels)	$y_{1}$ (pixels)	$y_{2}$ (pixels)	$f$ (mm)	$f \pm δ f$ (mm)	$ε_{f}$ (%)
$p$	1	106.97	78.73	25.45	200.77	200.16±0.63	0.32
	2	106.36	78.84	25.49	199.38
	3	106.14	78.42	25.56	200.80
	4	105.98	78.44	25.36	199.68
$l$	1	150.80	110.98	35.75	200.45	200.11±0.54	0.27
	2	150.33	110.86	35.46	199.37
	3	150.76	111.13	35.69	199.84
	4	149.87	110.59	35.84	200.77

	Spot Number	$y_{0}$ (pixels)	$y_{1}$ (pixels)	$y_{2}$ (pixels)	$f$ (mm)	$f \pm δ f$ (mm)	$ε_{f}$ (%)
$p$	1	90.13	130.18	183.07	−100.70	−99.39±1.06	1.07
	2	90.69	131.09	185.68	−98.18
	3	89.90	129.61	183.52	−98.53
	4	90.22	130.21	183.45	−100.15
$l$	1	126.98	183.08	260.04	−97.50	−98.52±0.80	0.81
	2	126.74	183.09	258.19	−99.73
	3	127.77	185.60	262.40	−98.31
	4	127.89	185.08	261.77	−98.55

	Spot Number	$y_{0}$ (pixels)	$y_{1}$ (pixels)	$y_{2}$ (pixels)	$f$ (mm)	$f \pm δ f$ (mm)	$ε_{f}$ (%)
$p$	1	39.64	21.70	−23.02	97.50	97.72±0.17	0.18
	2	39.88	21.56	−23.22	97.95
	3	39.42	21.67	−22.76	97.61
	4	39.34	21.52	−22.73	97.81
$l$	1	55.86	30.47	−32.84	97.06	97.01±0.38	0.39
	2	55.75	30.57	−33.05	96.39
	3	56.01	30.27	−33.12	97.18
	4	56.32	30.51	−33.09	97.42

	Spot Number	$y_{0}$ (pixels)	$y_{1}$ (pixels)	$y_{2}$ (pixels)	$f$ (mm)	$f \pm δ f$ (mm)	$ε_{f}$ (%)
$p$	1	48.54	18.68	−74.09	73.27	73.16±0.13	0.17
	2	48.94	18.97	−74.64	73.19
	3	48.34	18.52	−73.90	73.22
	4	48.40	19.03	−73.86	72.94
$l$	1	68.47	25.87	−106.56	72.38	72.38±0.17	0.24
	2	68.44	26.01	−106.59	72.26
	3	68.99	25.82	−107.11	72.66
	4	68.84	26.26	−107.17	72.23

	Spot Number	$y_{0}$ (pixels)	$y_{1}$ (pixels)	$y_{2}$ (pixels)	$f$ (mm)	$f \pm δ f$ (mm)	$ε_{f}$ (%)
$p$	1	48.54	−26.08	−70.54	38.21	38.09±0.08	0.20
	2	48.94	−25.28	−70.34	38.02
	3	48.34	−25.39	−69.88	38.03
	4	48.40	−25.67	−70.12	38.11
$l$	1	68.47	−41.15	−106.65	36.58	36.75±0.11	0.31
	2	68.44	−41.82	−106.77	36.88
	3	68.99	−40.37	−105.95	36.82
	4	68.84	−40.72	−106.35	36.71

	Theoretical		Experimental
Lens Number	$f_{470}$ (mm)	$Δ f_{p l}$ (mm)	$f \pm δ f$ (mm)	$ε_{f}$ (%)	$Δ f_{p l}$ (mm)
1	200.08	—	200.14±0.83	0.41	—
2	−99.07	0.92	−100.26±1.70	1.69	0.87
3	98.35	0.69	98.42±0.45	0.46	0.70
4	73.72	0.92	73.93±0.25	0.34	0.77
5	39.34	1.75	39.44±0.16	0.40	1.34

Abstract

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Tables (9)

Equations (6)

Applied Optics