Measuring the radius of curvature of spherical surfaces with actively tunable Fizeau and&#x00A0;Twyman-Green interferometers

Minjae Kim; Arjent Imeri; Syed Azer Reza

doi:10.1364/AO.519083

Applied Optics
Vol. 63,
Issue 15,
pp. 4077-4087
(2024)
•https://doi.org/10.1364/AO.519083

Measuring the radius of curvature of spherical surfaces with actively tunable Fizeau and Twyman-Green interferometers

Minjae Kim, Arjent Imeri, and Syed Azer Reza

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Minjae Kim, Arjent Imeri, and Syed Azer Reza, "Measuring the radius of curvature of spherical surfaces with actively tunable Fizeau and Twyman-Green interferometers," Appl. Opt. 63, 4077-4087 (2024)

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Abstract

Accurate and repeatable measurement of the radius of curvature (RoC) of spherical sample surfaces is of great importance in optics. This importance lies in the ubiquitous use of spherical optical elements such as curved mirrors and lenses. Due to a high measurement sensitivity, interferometric techniques are often deployed for accurate characterization of the sample surface RoC. One method by which a typical commercial Fizeau or Twyman-Green (TG) interferometer measures surface RoC is via scanning between two principal retroreflective optical configurations—namely, the confocal and catseye configurations. Switching between these two configurations is typically achieved by moving an optical head along the axis of the propagating laser beam and the RoC is estimated by measuring the magnitude of mechanical motion to switch between the two principal configurations. In this paper, we propose a motion-free catseye/confocal-imaging-based sample RoC measurement system. The necessity of bulk motion to switch between the two configurations is circumvented via the use of an actively controlled varifocal lens. We demonstrate the usefulness of the proposed innovation in RoC measurements with either the TG or the Fizeau interferometer. Furthermore, we convert a commercial motion-based Zygo RoC measurement system into a motion-free one by introducing a tunable lens inside the apparatus and using it to accurately characterize the RoC of different test samples. We also compute the wavefront aberrations for all spherical sample surfaces from the recorded measurement data.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Sample	Sample Surface Type	Interferometer Type	$f_{B}$ (mm)	$D_{S e p}$ (mm)
1	Concave lens	Fizeau	$250 \pm 1 %$	$160 \pm 1$
2	Concave lens	Fizeau	$250 \pm 1 %$	$160 \pm 1$
3	Convex lens	Fizeau	$250 \pm 1 %$	$120 \pm 1$
4	Convex lens	Fizeau	$250 \pm 1 %$	$120 \pm 1$
5	Convex lens	Fizeau	$250 \pm 1 %$	$120 \pm 1$
6	Concave mirror	TG	NA	NA
7	Concave lens	TG	NA	NA
8	Spherical ball	TG	NA	NA

Sample	$I_{c a t s}$ (mA)	$I_{c o n f}$ (mA)	$D_{c a t s}$ (mm)	$D_{c o n f}$ (mm)	$R_{M e a s} \pm ϵ_{R}$ (mm)	$R_{R e f}$ (mm)	$ϵ_{A v} %$
1	−136.02	102.62	136.67	78.11	$58.56 \pm 0.44$	$57.74 \pm 0.01$	1.42%
2	−128.86	−32.26	136.22	106.64	$29.58 \pm 0.36$	$29.19 \pm 0.01$	1.33%
3	87.74	−17.31	109.68	165.88	$56.20 \pm 0.78$	$55.04 \pm 0.01$	2.10%
4	73.80	−73.87	111.71	201.59	$89.88 \pm 0.92$	$90.24 \pm 0.01$	0.40%
5	73.16	−113.56	112.90	237.46	$124.56 \pm 0.93$	$125.11 \pm 0.01$	0.44%
6	108.12	171.11	559.01	343.40	$215.61 \pm 0.42$	$218.00 \pm 0.01$	1.10%
7	83.79	90.51	738.00	678.04	$59.96 \pm 0.88$	$57.94 \pm 0.01$	3.85%
8	227.22	218.04	255.58	266.74	$11.16 \pm 0.12$	$11.00 \pm 0.01$	1.45%

Sample	$A (µ m)$	$B (µ m)$	$C (µ m)$	$S_{I} (µ m)$	$S_{I I} (µ m)$	$S_{I I I} (µ m)$	$Δ W_{R M S} (µ m)$
1	2.80	−0.25	0.16	22.40	−0.51	0.63	1.09
2	1.40	0.38	0.19	11.20	0.76	0.76	0.65
3	−0.25	0.08	0.05	−2.00	0.17	0.21	0.05
4	3.10	0.55	0.05	24.80	1.10	0.19	1.10
5	2.80	0.05	0.38	22.40	0.11	1.52	1.29

Sample	Sample Surface Type	Interferometer Type	$f_{B}$ (mm)	$D_{S e p}$ (mm)
1	Concave lens	Fizeau	$250 \pm 1 %$	$160 \pm 1$
2	Concave lens	Fizeau	$250 \pm 1 %$	$160 \pm 1$
3	Convex lens	Fizeau	$250 \pm 1 %$	$120 \pm 1$
4	Convex lens	Fizeau	$250 \pm 1 %$	$120 \pm 1$
5	Convex lens	Fizeau	$250 \pm 1 %$	$120 \pm 1$
6	Concave mirror	TG	NA	NA
7	Concave lens	TG	NA	NA
8	Spherical ball	TG	NA	NA

Sample	$I_{c a t s}$ (mA)	$I_{c o n f}$ (mA)	$D_{c a t s}$ (mm)	$D_{c o n f}$ (mm)	$R_{M e a s} \pm ϵ_{R}$ (mm)	$R_{R e f}$ (mm)	$ϵ_{A v} %$
1	−136.02	102.62	136.67	78.11	$58.56 \pm 0.44$	$57.74 \pm 0.01$	1.42%
2	−128.86	−32.26	136.22	106.64	$29.58 \pm 0.36$	$29.19 \pm 0.01$	1.33%
3	87.74	−17.31	109.68	165.88	$56.20 \pm 0.78$	$55.04 \pm 0.01$	2.10%
4	73.80	−73.87	111.71	201.59	$89.88 \pm 0.92$	$90.24 \pm 0.01$	0.40%
5	73.16	−113.56	112.90	237.46	$124.56 \pm 0.93$	$125.11 \pm 0.01$	0.44%
6	108.12	171.11	559.01	343.40	$215.61 \pm 0.42$	$218.00 \pm 0.01$	1.10%
7	83.79	90.51	738.00	678.04	$59.96 \pm 0.88$	$57.94 \pm 0.01$	3.85%
8	227.22	218.04	255.58	266.74	$11.16 \pm 0.12$	$11.00 \pm 0.01$	1.45%

Abstract

Data availability

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Applied Optics