Nondestructive measurement of the refractive index distribution of a glass molded lens by two-wavelength wavefronts

Tomohiro Sugimoto

doi:10.1364/AO.55.008116

Applied Optics
Vol. 55,
Issue 28,
pp. 8116-8125
(2016)
•https://doi.org/10.1364/AO.55.008116

Nondestructive measurement of the refractive index distribution of a glass molded lens by two-wavelength wavefronts

Tomohiro Sugimoto

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Tomohiro Sugimoto, "Nondestructive measurement of the refractive index distribution of a glass molded lens by two-wavelength wavefronts," Appl. Opt. 55, 8116-8125 (2016)

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- Instrumentation, Measurement, and Metrology
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About this Article
History
- Original Manuscript: July 7, 2016
- Revised Manuscript: August 31, 2016
- Manuscript Accepted: September 5, 2016
- Published: September 30, 2016

Abstract

This paper presents a nondestructive and non-exact-index-matching method for measuring the refractive index distribution of a glass molded lens with high refractivity. The method measures two-wavelength wavefronts of a test lens immersed in a liquid with a refractive index dispersion different from that of the test lens and calculates the refractive index distribution by eliminating the refractive index distribution error caused by the shape error of the test lens. The estimated uncertainties of the refractive index distributions of test lenses with $n_{d} \approx 1.77$ and $n_{d} \approx 1.85$ were $1.9 \times 10^{- 5} RMS$ and $2.4 \times 10^{- 5} RMS$ , respectively. I validated the proposed method by evaluating the agreement between the estimated uncertainties and experimental values.

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Property	Variables	Value
Wavelength [nm]	$λ_{1}$	632.81
Wavelength [nm]	$λ_{2}$	457.93
Thickness [mm]	$s_{5} (0,0, λ)$	10
	$s (0,0, λ)$	1.05
	$s_{6} (0,0, λ)$	10
	$s_{7} (0,0, λ)$	15
	$s_{8} (0,0, λ)$	60
	$s_{9} (0,0, λ)$	1.23
Refractive index	$n_{L} (λ_{1})$	1.76696
	$n_{L} (λ_{2})$	1.81743
	$n_{0} (λ_{1})$ , $n (x, y, λ_{1})$	1.84697
	$n_{0} (λ_{2})$ , $n (x, y, λ_{2})$	1.87250
	$n_{W} (λ_{1})$	1.45699
	$n_{W} (λ_{2})$	1.46496
	$n_{A} (λ_{1})$	1.45699
	$n_{A} (λ_{2})$	1.46496
Designed radius of curvature of test lens [mm]	$R_{1}$	−6.145
	$R_{2}$	144.534
Shape error of test lens [mm]	$Δ R_{1}$	0 to 0.2
Entrance pupil diameter [mm]	–	9

Type of Factor	Factor	Uncertainty of Factor (Sample A/B)	Sensitivity Coefficient	Uncertainty of RID of Sample A [RMS]	Uncertainty of RID of Sample B [RMS]
Invariable	Distance between each component’s surface	3 μm	${- Δ N (λ_{1}) n_{j} (λ_{2}) [s_{k} (x, y, λ_{2}) - s_{k} (x_{0}, y_{0}, λ_{2})] / s_{k} (x_{0}, y_{0}, λ_{2}) + Δ N (λ_{2}) n_{j} (λ_{1}) [s_{k} (x, y, λ_{1}) - s_{k} (x_{0}, y_{0}, λ_{1})] / s_{k} (x_{0}, y_{0}, λ_{1})} / Δ ν s_{eff} (x, y)$	$2.1 \times 10^{- 8}$	$5.5 \times 10^{- 7}$
	Refractive index of each component	$1 \times 10^{- 3}$	${- Δ N (λ_{1}) [s_{k} (x, y, λ_{2}) - s_{k} (x_{0}, y_{0}, λ_{2})] + Δ N (λ_{2}) [s_{k} (x, y, λ_{1}) - s_{k} (x_{0}, y_{0}, λ_{1})]} / Δ ν s_{eff} (x, y)$	$1.3 \times 10^{- 7}$	$3.6 \times 10^{- 6}$
	Wavefront measured with SHS	4.6 nm RMS/19 nm RMS	$[Δ N (λ_{1}) - Δ N (λ_{2})] / Δ ν s_{eff} (x, y)$	$1.1 \times 10^{- 6}$	$4.2 \times 10^{- 7}$
	Wavelength in air	0.01 nm	${Δ N (λ_{1}) [d Δ W (x, y, λ_{2}) / d λ] - Δ N (λ_{2}) [d Δ W (x, y, λ_{1}) / d λ]} / Δ ν s_{eff} (x, y)$	$3.4 \times 10^{- 7}$	$6.0 \times 10^{- 7}$
	Refractive index of liquid	$3.4 \times 10^{- 5} / 2.8 \times 10^{- 5}$	${- Δ N (λ_{1}) Σ [s_{l} (x, y, λ_{2}) - s_{l} (x_{0}, y_{0}, λ_{2})] + Δ N (λ_{2}) Σ [s_{l} (x, y, λ_{1}) - s_{l} (x_{0}, y_{0}, λ_{1})]} / Δ ν s_{eff} (x, y)$	$3.6 \times 10^{- 6}$	$3.2 \times 10^{- 6}$
	Thickness of test lens	3 μm	${- Δ N (λ_{1}) n_{0} (λ_{2}) {[d s (x, y, λ_{2}) / d s (0,0, λ_{2})]}^{2} / 2 + Δ N (λ_{2}) n_{0} (λ_{1}) {[d s (x, y, λ_{1}) / d s (0,0, λ_{1})]}^{2} / 2} / Δ ν s_{eff} (x, y)$	$5.0 \times 10^{- 10}$	$2.4 \times 10^{- 8}$
	Shape of each surface of test lens	1 μm	${- Δ N (λ_{1}) [n_{0} (λ_{2}) \sec (d s (x, y, λ_{2}) / d s (0,0, λ_{2})) - n_{L} (λ_{2}) \sec (d s_{l} (x, y, λ_{2}) / d s_{l} (0,0, λ_{2}))] + Δ N (λ_{2}) [n_{0} (λ_{1}) \sec (d s (x, y, λ_{1}) / d s (0,0, λ_{1})) - n_{L} (λ_{1}) \sec (d s_{l} (x, y, λ_{1}) / d s_{l} (0,0, λ_{1}))]} / Δ ν s_{eff} (x, y)$	$2.8 \times 10^{- 9}$	$7.7 \times 10^{- 8}$
	Refractive index of test lens	$5 \times 10^{- 5} / 1 \times 10^{- 5}$	$[Δ N (λ_{1}) s (x, y, λ_{2}) - Δ N (λ_{2}) s (x, y, λ_{1})] / Δ ν s_{eff} (x, y)$	$8.9 \times 10^{- 6}$	$2.0 \times 10^{- 6}$
	Tilt of test lens	1°	$(\sec θ - 1) {- Δ N (λ_{1}) n_{0} (λ_{2}) [s (x, y, λ_{2}) - s (x_{0}, y_{0}, λ_{2})] + Δ N (λ_{2}) n_{0} (λ_{1}) [s (x, y, λ_{1}) - s (x_{0}, y_{0}, λ_{1})]} / Δ ν s_{eff} (x, y)$	$2.3 \times 10^{- 8}$	$9.7 \times 10^{- 7}$
Variable	Refractive index of each component	$1 \times 10^{- 4}$	${Δ N {(λ_{1})}^{2} {[s_{k} (x, y, λ_{2}) - s_{k} (x_{0}, y_{0}, λ_{2})]}^{2} + Δ N {(λ_{2})}^{2} {[s_{k} (x, y, λ_{1}) - s_{k} (x_{0}, y_{0}, λ_{1})]}^{2}}^{1 / 2} / Δ ν s_{eff} (x, y)$	$2.3 \times 10^{- 8}$	$6.2 \times 10^{- 7}$
	Wavefront measured with SHS	5 nm RMS	$[Δ N {(λ_{1})}^{2} + Δ N {(λ_{2})}^{2}]^{1 / 2} / Δ ν s_{eff} (x, y)$	$1.8 \times 10^{- 6}$	$5.8 \times 10^{- 6}$
	Wavelength in air	0.01 nm	${Δ N {(λ_{1})}^{2} {[d Δ W (x, y, λ_{2}) / d λ]}^{2} + Δ N {(λ_{2})}^{2} {[d Δ W (x, y, λ_{1}) / d λ]}^{2}}^{1 / 2} / Δ ν s_{eff} (x, y)$	$8.2 \times 10^{- 7}$	$2.6 \times 10^{- 7}$
	Refractive index of liquid	$3 \times 10^{- 6} / 6 \times 10^{- 6}$	${Δ N {(λ_{1})}^{2} {[Σ (s_{l} (x, y, λ_{2}) - s_{l} (x_{0}, y_{0}, λ_{2}))]}^{2} + Δ N {(λ_{2})}^{2} {[Σ (s_{l} (x, y, λ_{1}) - s_{l} (x_{0}, y_{0}, λ_{1}))]}^{2}}^{1 / 2} / Δ ν s_{eff} (x, y)$	$9.5 \times 10^{- 7}$	$4.9 \times 10^{- 6}$
	Refractive index of test lens	$3 \times 10^{- 5} / 1 \times 10^{- 5}$	$[Δ N {(λ_{1})}^{2} s {(x, y, λ_{2})}^{2} + Δ N {(λ_{2})}^{2} s {(x, y, λ_{1})}^{2}]^{1 / 2} / Δ ν s_{eff} (x, y)$	$8.9 \times 10^{- 6}$	$8.1 \times 10^{- 6}$
The other	Index dispersion variation	0.02	$Δ n (x, y, λ_{1}) [n (x, y, λ_{1}) - 1)] Δ N (λ_{1}) s (x, y, λ_{2}) / Δ ν s_{eff} (x, y)$	$4.8 \times 10^{- 6}$	Negligible
The other	Reproducibility	$1.3 \times 10^{- 5} / 2.0 \times 10^{- 5}$	1	$1.3 \times 10^{- 5}$	$2.0 \times 10^{- 5}$
Combined standard uncertainty				$1.9 \times 10^{- 5}$	$2.4 \times 10^{- 5}$

Property	Variables	Value
Wavelength [nm]	$λ_{1}$	632.81
Wavelength [nm]	$λ_{2}$	457.93
Thickness [mm]	$s_{5} (0,0, λ)$	10
	$s (0,0, λ)$	1.05
	$s_{6} (0,0, λ)$	10
	$s_{7} (0,0, λ)$	15
	$s_{8} (0,0, λ)$	60
	$s_{9} (0,0, λ)$	1.23
Refractive index	$n_{L} (λ_{1})$	1.76696
	$n_{L} (λ_{2})$	1.81743
	$n_{0} (λ_{1})$ , $n (x, y, λ_{1})$	1.84697
	$n_{0} (λ_{2})$ , $n (x, y, λ_{2})$	1.87250
	$n_{W} (λ_{1})$	1.45699
	$n_{W} (λ_{2})$	1.46496
	$n_{A} (λ_{1})$	1.45699
	$n_{A} (λ_{2})$	1.46496
Designed radius of curvature of test lens [mm]	$R_{1}$	−6.145
	$R_{2}$	144.534
Shape error of test lens [mm]	$Δ R_{1}$	0 to 0.2
Entrance pupil diameter [mm]	–	9

Type of Factor	Factor	Uncertainty of Factor (Sample A/B)	Sensitivity Coefficient	Uncertainty of RID of Sample A [RMS]	Uncertainty of RID of Sample B [RMS]
Invariable	Distance between each component’s surface	3 μm	${- Δ N (λ_{1}) n_{j} (λ_{2}) [s_{k} (x, y, λ_{2}) - s_{k} (x_{0}, y_{0}, λ_{2})] / s_{k} (x_{0}, y_{0}, λ_{2}) + Δ N (λ_{2}) n_{j} (λ_{1}) [s_{k} (x, y, λ_{1}) - s_{k} (x_{0}, y_{0}, λ_{1})] / s_{k} (x_{0}, y_{0}, λ_{1})} / Δ ν s_{eff} (x, y)$	$2.1 \times 10^{- 8}$	$5.5 \times 10^{- 7}$
	Refractive index of each component	$1 \times 10^{- 3}$	${- Δ N (λ_{1}) [s_{k} (x, y, λ_{2}) - s_{k} (x_{0}, y_{0}, λ_{2})] + Δ N (λ_{2}) [s_{k} (x, y, λ_{1}) - s_{k} (x_{0}, y_{0}, λ_{1})]} / Δ ν s_{eff} (x, y)$	$1.3 \times 10^{- 7}$	$3.6 \times 10^{- 6}$
	Wavefront measured with SHS	4.6 nm RMS/19 nm RMS	$[Δ N (λ_{1}) - Δ N (λ_{2})] / Δ ν s_{eff} (x, y)$	$1.1 \times 10^{- 6}$	$4.2 \times 10^{- 7}$
	Wavelength in air	0.01 nm	${Δ N (λ_{1}) [d Δ W (x, y, λ_{2}) / d λ] - Δ N (λ_{2}) [d Δ W (x, y, λ_{1}) / d λ]} / Δ ν s_{eff} (x, y)$	$3.4 \times 10^{- 7}$	$6.0 \times 10^{- 7}$
	Refractive index of liquid	$3.4 \times 10^{- 5} / 2.8 \times 10^{- 5}$	${- Δ N (λ_{1}) Σ [s_{l} (x, y, λ_{2}) - s_{l} (x_{0}, y_{0}, λ_{2})] + Δ N (λ_{2}) Σ [s_{l} (x, y, λ_{1}) - s_{l} (x_{0}, y_{0}, λ_{1})]} / Δ ν s_{eff} (x, y)$	$3.6 \times 10^{- 6}$	$3.2 \times 10^{- 6}$
	Thickness of test lens	3 μm	${- Δ N (λ_{1}) n_{0} (λ_{2}) {[d s (x, y, λ_{2}) / d s (0,0, λ_{2})]}^{2} / 2 + Δ N (λ_{2}) n_{0} (λ_{1}) {[d s (x, y, λ_{1}) / d s (0,0, λ_{1})]}^{2} / 2} / Δ ν s_{eff} (x, y)$	$5.0 \times 10^{- 10}$	$2.4 \times 10^{- 8}$
	Shape of each surface of test lens	1 μm	${- Δ N (λ_{1}) [n_{0} (λ_{2}) \sec (d s (x, y, λ_{2}) / d s (0,0, λ_{2})) - n_{L} (λ_{2}) \sec (d s_{l} (x, y, λ_{2}) / d s_{l} (0,0, λ_{2}))] + Δ N (λ_{2}) [n_{0} (λ_{1}) \sec (d s (x, y, λ_{1}) / d s (0,0, λ_{1})) - n_{L} (λ_{1}) \sec (d s_{l} (x, y, λ_{1}) / d s_{l} (0,0, λ_{1}))]} / Δ ν s_{eff} (x, y)$	$2.8 \times 10^{- 9}$	$7.7 \times 10^{- 8}$
	Refractive index of test lens	$5 \times 10^{- 5} / 1 \times 10^{- 5}$	$[Δ N (λ_{1}) s (x, y, λ_{2}) - Δ N (λ_{2}) s (x, y, λ_{1})] / Δ ν s_{eff} (x, y)$	$8.9 \times 10^{- 6}$	$2.0 \times 10^{- 6}$
	Tilt of test lens	1°	$(\sec θ - 1) {- Δ N (λ_{1}) n_{0} (λ_{2}) [s (x, y, λ_{2}) - s (x_{0}, y_{0}, λ_{2})] + Δ N (λ_{2}) n_{0} (λ_{1}) [s (x, y, λ_{1}) - s (x_{0}, y_{0}, λ_{1})]} / Δ ν s_{eff} (x, y)$	$2.3 \times 10^{- 8}$	$9.7 \times 10^{- 7}$
Variable	Refractive index of each component	$1 \times 10^{- 4}$	${Δ N {(λ_{1})}^{2} {[s_{k} (x, y, λ_{2}) - s_{k} (x_{0}, y_{0}, λ_{2})]}^{2} + Δ N {(λ_{2})}^{2} {[s_{k} (x, y, λ_{1}) - s_{k} (x_{0}, y_{0}, λ_{1})]}^{2}}^{1 / 2} / Δ ν s_{eff} (x, y)$	$2.3 \times 10^{- 8}$	$6.2 \times 10^{- 7}$
	Wavefront measured with SHS	5 nm RMS	$[Δ N {(λ_{1})}^{2} + Δ N {(λ_{2})}^{2}]^{1 / 2} / Δ ν s_{eff} (x, y)$	$1.8 \times 10^{- 6}$	$5.8 \times 10^{- 6}$
	Wavelength in air	0.01 nm	${Δ N {(λ_{1})}^{2} {[d Δ W (x, y, λ_{2}) / d λ]}^{2} + Δ N {(λ_{2})}^{2} {[d Δ W (x, y, λ_{1}) / d λ]}^{2}}^{1 / 2} / Δ ν s_{eff} (x, y)$	$8.2 \times 10^{- 7}$	$2.6 \times 10^{- 7}$
	Refractive index of liquid	$3 \times 10^{- 6} / 6 \times 10^{- 6}$	${Δ N {(λ_{1})}^{2} {[Σ (s_{l} (x, y, λ_{2}) - s_{l} (x_{0}, y_{0}, λ_{2}))]}^{2} + Δ N {(λ_{2})}^{2} {[Σ (s_{l} (x, y, λ_{1}) - s_{l} (x_{0}, y_{0}, λ_{1}))]}^{2}}^{1 / 2} / Δ ν s_{eff} (x, y)$	$9.5 \times 10^{- 7}$	$4.9 \times 10^{- 6}$
	Refractive index of test lens	$3 \times 10^{- 5} / 1 \times 10^{- 5}$	$[Δ N {(λ_{1})}^{2} s {(x, y, λ_{2})}^{2} + Δ N {(λ_{2})}^{2} s {(x, y, λ_{1})}^{2}]^{1 / 2} / Δ ν s_{eff} (x, y)$	$8.9 \times 10^{- 6}$	$8.1 \times 10^{- 6}$
The other	Index dispersion variation	0.02	$Δ n (x, y, λ_{1}) [n (x, y, λ_{1}) - 1)] Δ N (λ_{1}) s (x, y, λ_{2}) / Δ ν s_{eff} (x, y)$	$4.8 \times 10^{- 6}$	Negligible
The other	Reproducibility	$1.3 \times 10^{- 5} / 2.0 \times 10^{- 5}$	1	$1.3 \times 10^{- 5}$	$2.0 \times 10^{- 5}$
Combined standard uncertainty				$1.9 \times 10^{- 5}$	$2.4 \times 10^{- 5}$

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