## Abstract

A new infrared confocal refractive index measurement (IR-CRIM) method with high precision is proposed for a lens. Based on the property that the maximum point of a confocal axial intensity response curve accurately corresponds to the converging focus of the measuring beam, IR-CRIM can precisely identify the front and back vertices of the test lens, and obtain the optical thickness *d* of the test lens, and resulting calculate the lens refractive index *n* using the ray-tracing algorithm. The broadband refractive index dispersion of the test lens can be acquired by the Cauchy dispersion law and the *n* obtained at several wavelengths. Preliminary experimental results and theoretical analyses indicate that IR-CRIM achieves an accuracy of 6 × 10^{−4} in the wavelength range of 500–1700 nm. It provides a novel approach for the high-precision and direct measurement of the refractive index of a lens in visible and near infrared wavelength range.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Spherical lenses, which are widely used in navigation, wideband observation, weapons systems, and laser manufacturing fields, are fundamental optical components in visible and infrared (IR) systems, and its refractive index is one of the most important parameters that characterize the optical properties of a lens [1], the accurate measurement of lens refractive index, especially in the infrared band, is a difficult measurement problem in optical measurement field. The existing measurement methods for the refractive index mainly include two kinds which are applicable for raw material and manufactured components, respectively.

In the raw material refractive index measurement, the most common measurement methods, prism-coupling [2], minimum deflection angle [3], and Abbe [4] methods, high measurement accuracy, but they can only measure the sampling of the raw material before the lens is manufactured, and then use the measured refractive index of the raw material as that of the manufactured lens. However, the refractive index may change dramatically with the variation of the stress and temperature during the manufacturing and assembling process of a lens [5], so the measured lens refractive index obtained indirectly by the above methods may deviate far from the refractive index of raw material.

In the lens refractive index measurement, a liquid immersion method was proposed [6–8], which immerses a test lens into a liquid mixture, then changes the liquid index until it approximates the refractive index of the test lens, at this time it measures the liquid index to obtain the lens index indirectly, and its measurement accuracy is at 10^{−3} level. An interferometry method was proposed [8–10], which uses an interference fringe to compare the refractive index of the test lens and matching liquid, and then obtains the lens refractive index indirectly with its measurement accuracy of 10^{−3} level. The above two methods are indirect method for the refractive index measurement, and their measurement accuracy depend on the liquid refractive index measuring equipment.

To improve the measurement accuracy and simply the measurement process, a calculation method was proposed [11,12] to measure lens refractive index directly, using the thin lens formula. This method uses interferometry to focus on the test lens with an accuracy of better than 0.02mm, then it calculates the lens refractive index using the thin lens formula, and its measurement accuracy can reach ± 4 × 10^{−4}. It improved the measurement accuracy of lens refractive index dramatically, but it is difficult to achieve the accurate focusing and measurement in infrared band due to the limitation of its measurement principle. Improved tomography focusing methods based on interference principle was proposed [13–19], which can be used for the focusing and measurement of the tested optical components in infrared band. But the focusing accuracy only reached micro meter level, and the measurement accuracy of the refractive index was only about 10^{−3} in infrared band.

Therefore, in order to realize the high precision tomography focusing of lens in visible and infrared band and further improve the measurement accuracy of the lens refractive index, a new infrared confocal refractive index measurement (IR-CRIM) method is proposed in this paper. It uses the confocal method to tomography focus on the test lens, and uses Cauchy dispersion law to obtain the refractive index dispersion in visual and infrared wavelength range. Compared with the existing methods, the proposed IR-CRIM is a simple and direct measurement method, can achieve the high precision measurement of the refractive index in broad wavelength range.

## 2. Measurement principle

The principle of IR-CRIM is presented in Fig. 1. A divergent beam emitted from an infrared spot light source transmits through a beam splitter (BS), and is focused onto the test lens by a broadband IR-convergence system (IR-CS). The light reflected by the test lens is collected by the IR-CS, after being reflected by the BS, and filtered by the pinhole located at the back focus of the IR-CS, and then is received by the detector to obtain the confocal intensity curve *I*_{C}(*z*). The test lens moves along the axial direction, and when its surface (either front or back) approaches the convergence point of the test beam, a “bell-shaped” confocal curve *I*_{C}(*z*) is produced.

When the front vertex P_{A} or back vertex P_{B} of the test lens approaches the convergence point of the test beam, the confocal intensity curve generated by IR-CRIM is given by

Here λ is the wavelength of the test beam, *p*_{1}(*ρ*) and *p*_{2}(*ρ*) are the front and back pupil functions of the IR-CS, *D* and *f* are the effective aperture and focal length of the IR-CS, respectively, *ρ* is the normalized radius of the pupil, *u* is the normalized axial optical coordinate, and *z* is the axial coordinate of the test lens. Assuming that *p*_{1}(*ρ*) = *p*_{2}(*ρ*) = 1, *I*_{C}(*z*) is simplified to

*I*_{CA}(*z*) and *I*_{CB}(*z*) are the axial response curves near the vertices P_{A} and P_{B} of the test lens, respectively, and the maximum points P_{CA} and P_{CB} of *I*_{CA}(*z*) and *I*_{CB}(*z*) correspond exactly to the convergence points of the test beam. Thus, IR-CRIM can precisely identify the vertices P_{A} and P_{B} by the confocal tomography focusing, and obtain the optical thickness *d* of the test lens by measuring the distance between P_{CA} and P_{CB}.

According to Snell’s law, the lens refractive index *n* can be calculated by a ray-tracing algorithm. The ray-tracing principle is shown in Fig. 2.

First, an arbitrary ray in the beam aperture is selected for calculation. Ray-tracing is performed at the first surface of the test lens, and the calculation formula can be expressed as

where *t* is the lens geometrical thickness, *r* is the radius of curvature, *α* is the angle between the selected ray and the optical axis, which can be expressed as a function of *ρ*, *D,* and *f* as

Considering all the rays in the pupil, the calculation needs to integrate *t*(*r*, *d*, *n*, *α*) with respect to *ρ* in pupil plane from 0 to 1.

where *K*(*ρ*) is the normalized intensity distribution function of the test beam in the aperture.

If the intensity of the test beam is uniformly distributed, *K*(*ρ*) = 1, and the integral calculation in Eq. (6) can be simplified according to the mean value theorems for definite integrals, so a single ray can represent the beam throughout the aperture. This representative ray can be selected using the gravity method, and its angle *α* is

From Eqs. (4)-(7), *n* can be calculated.

The above calculation can obtain the refractive index *n* at a particular wavelength. To obtain the refractive index at other wavelengths in a broad range, the Cauchy dispersion equation is used to transform the measured *n* to that corresponding to the required wavelength. The Cauchy dispersion equations are:

The refractive indices *n*_{1},*n*_{2}…*n*_{i} corresponding to several (at least 3) wavelengths are measured, so that Eq. (8) can be solved for the dispersion coefficients *a*, *b* and *c*. The Cauchy dispersion equation of the test lens is thus constructed as

Finally, the refractive index of the test lens for any wavelength in the broad wavelength range can be calculated by Eq. (9).

## 3. Simulation and optimization

#### 3.1 Broadband IR-convergence system design

To measure the refractive indices at different wavelengths, the wavelength of the light source needs to be changed. Therefore, to ensure the accuracy in the refractive index calculation, the confocal tomographic focusing system must have achromatic functionality, and in particular, the aperture angle *α* of IR-CS must be invariant as wavelength varied. Because of limitations of the material, the transmission convergence system cannot achieve achromatic focusing over a broad wavelength range. Therefore a broadband IR-convergence system, based on a double hollow parabolic reflector structure, is proposed. Its schematic is presented in Fig. 3.

The IR-CS system consists of two coaxial parabolic reflectors with holes at center. The divergent beam emitted from the infrared point source transmits through the central hole of reflector 1, and reaches the surface of parabolic reflector 2. The divergent beam is collimated into a parallel beam by reflector 2, and reflected onto the surface of reflector 1, then it is reflected again and converged into a convergent beam by reflector 1. At last, the convergent beam transmits through the central hole of reflector 2, and forms the test beam.

As all components used in the IR-CS are reflective components, there is no chromatic aberration in the full wavelength range. So, the convergence point does not move as the wavelength changes, and the test beam convergence angle remains constant. In addition, because both reflectors are parabolic, the IR-CS does not introduce any wave front aberration, thus providing an ideal convergent beam for the IR-CRIM.

#### 3.2 Optimization of the inner diameter of the hollow beam

Infrared light has a long wavelength and undergoes significant diffraction. When the test beam transmits into the test lens, the lens may introduce significant wave aberration, to broaden the confocal axial response curve, and resulting reduce the chromatographic focusing resolution.

To illustrate the effect of long-wave diffraction aberration, the confocal axial response near the back vertex P_{B} of the test lens is given by

Here *J _{0}* is the zero-order Bessel function,

*Φ*(

*ρ*) is the long-wave diffraction aberration introduced by the test lens,

*v*is the normalized lateral optical coordinate, and

*x*and y are the lateral coordinates of the optical pupil.

To suppress the aberration, a pupil filtering method is used. The test beam is modulated into an annular beam shown in Fig. 4 by the filter, and IR-CRIM uses the hollow optical pencil to focus the back vertex P_{B} of the test lens.

Here, *ε* is the normalized inner radius of the annular beam. The parabolic reflectors used in the IR-CS are hollow, such that the beam can form a hollow optical pencil directly when converged by the IR-CS.

In the case of an annular beam, and assuming that *p*_{1}(*ρ*) = *p*_{2}(*ρ*) = 1, the confocal tomography focusing response is

To analyze the relation between the value of *ε* and the confocal axial tomography focusing capability, response curves for different *ε* are calculated using Eq. (12). The result is shown in Fig. 5.

It can be seen from Fig. 5 that as *ε* increases, the full width at half maximum (FWHM) of the curve decreases and the theoretical resolution is improved, but the response intensity and the signal-to-noise ratio (SNR) decrease. To optimize the focusing accuracy, the sensitivity and signal-to-noise ratio of the system should be considered together in the optimization procedure.

Assuming that SNR = 1/150, the focusing resolution of the system is simulated, and the result is shown in Fig. 6.

As can be seen from Fig. 6, IR-CRIM obtains the best axial focusing resolution when *ε* = 0.71. The FWHM of the axial response curve is 86 μm, the focusing resolution is 538 nm, and the resolution is improved by 16% compared to the solid beam.

## 4. Experiments and analyses

To verify the effectiveness of the proposed IR-CRIM, experimental apparatus was designed as shown in Fig. 7.

Light sources of wavelengths 532 nm, 1064 nm and 1550 nm were used. The emitted beams were coupled to the optical path by two long-wave filters (L-P1 and L-P2), and then focused by an off-axis parabolic mirror to produce a point source. The test lens moved in the axial direction with a precision slider with its axial coordinate *z* monitored by a distance measurement interferometer (DMI). The confocal curve was obtained from axial position *z* and the light intensity signal *I* measured by the detector.

The IR-CRIM system used is shown in Fig. 8.

Test beams of wavelengths 532 nm and 1064 nm are produce by laser sources, and a 1550 nm beam is produced by filtering the broadband light emitted from a black-body light source. For the IR-CS, its effective aperture was 90 mm, its inner diameter was 63.9 mm, its focal length was 600 mm, and its working distance was 175 mm. The diameter of the pinhole was 100 μm, and A P-3257 infrared photo detector (Hamamatsu Corporation), with a detection band of 0.5–12 μm, was used. The DMI used was an XL-80 laser interferometer (Renishaw Corporation) with a relative measurement precision of 1 ppm. The rail was a high-accuracy air-bearing slider, with a straightness of 0.3 μm and an effective moving range of 500 mm.

#### 4.1 Tomography focusing characteristics experiments

To test the tomography focusing characteristics of IR-CRIM, a 20-mm-thick infrared optical flat was used as the sample, and the IR-CRIM focused the front and back surfaces of the flat at a wavelength of 1064 nm.

The test and theoretical curves are shown in Fig. 9.

Figure 9 shows that a bell-shaped confocal response curve is formed on the front and back surfaces of the plate, and the measured response curves coincide well with the theoretical curve. The FWHM of the confocal response curves obtained by the IR-CRIM apparatus at the front and back surfaces are 0.0894 mm and 0.0942 mm, respectively, which are consistent with the FWHM of the theoretical simulation curve, i.e., 0.086 mm.

Thus, the chromatographic focusing curve *I*_{CB} was only slightly broadened (the increase in FWHM was ~5.4%). Long-wave diffraction was effectively suppressed, confirming that the IR-CRIM procedure can chromatographically focus an IR lens precisely.

#### 4.2 Refractive index measurement experiment

To test the lens refractive index measurements accuracy of IR-CRIM in a broad wavelength range, a concave spherical lens made of CaF_{2} was used as the test sample. The lens had a radius of curvature of −499.6632 mm and a geometric thickness of 20.3136 mm. The refractive index calibration value was *n* = 1.42847 at a wavelength of 1064 nm.

The experiment was performed under a pressure of 110350 ± 60 Pa, temperature of 20.5 ± 0.5 °C, and relative humidity of 21 ± 5%.

The lens was focused using the 532-nm, 1064-nm and 1550-nm sources, and the corresponding optical thicknesses measured were *t*_{532} = 20.4897 mm, *t*_{1064} = 20.5712 mm and *t*_{1550} = 20.7277 mm, respectively. The refractive indices calculated using Eq. (8) were *n*_{532} = 1.4352, *n*_{1064} = 1.42841 and *n*_{1550} = 1.4262, deviating from the nominal values by Δ*n*_{532} = 0.0001, Δ*n*_{1064} = 0.00006 and Δ*n*_{1550} = 0.0001. The Cauchy coefficients calculated using Eq. (8) were *a* = 1.4240, *b* = 5635.8 and *c* = −6.95 × 10^{8}. The index dispersion was then calculated using Eq. (9), and the result is shown in Fig. 10.

In the visible to near infrared range of 500–1700 nm, Fig. 10 shows good agreement between the lens refractive index dispersion curve and the nominal curve of CaF_{2}, and their maximum deviation is Δ*n*_{max} = 6 × 10^{−4}.

The measurement error of *n*_{532}, *n*_{1064} and *n*_{1550}, measured by IR-CRIM directly, is much smaller than Δ*n*_{max} = 6 × 10^{−4}, so the Δ*n*_{max} is mainly caused by wavelength transform calculation using Eq. (9). The refractive index *n* at more wavelengths are measured directly when more test wavelengths are used for the measurement, and the fitted experiment curve of refractive index dispersion will be much closer to the standard curve of refractive index dispersion, so the wavelength transformation calculation error decreases. Therefore, IR-CRIM can achieve accuracy better than 6 × 10^{−4} in the range of 500–1700 nm.

## 5. Conclusions

In this paper, we propose a new infrared confocal refractive index measurement with high accuracy for lens refractive index. The proposed IR-CRIM method uses the vertex of a confocal axial intensity curve to focus the front and back vertices of the test lens, to further obtain its optical thickness *d*. Then it calculates the lens refractive index *n* by the ray-tracing algorithm, and the broadband refractive index dispersion of the test lens by using Cauchy dispersion law and several *n* measured at different wavelengths. Compared to the existing measurement methods, IR-CRIM has the following advantages.

- 1) It can test the lens refractive index directly, without the need for sampling or sample preparation, and the measurement process is nondestructive and convenient.
- 2) The measurement accuracy of IR-CRIM is higher than that of the existing other measurement methods. The measurement accuracy of lens refractive index measurement method reported is about 10
^{−3}level [6–12], and the measurement accuracy of IR-CRIM is improved 2 times more than them, moreover the measurement accuracy can be further improved by measuring the refractive index*n*at more wavelengths. - 3) It can measure the refractive index in a broad wavelength range, and in theory, the refractive index corresponding to any wavelength can be obtained using the Cauchy converse formula.

Preliminary experimental results and theoretical analyses indicate that the accuracy of lens refractive index measurement can reach 6 × 10^{−4} within the 500–1700 nm wavelength range. Thus, IR-CRIM provides a novel high-precision and direct measurement approach for the lens refractive index in visible and near infrared wavelength.

## Funding

National Nature Science Foundation of China (NSFC) (No. 51405020, 61327010).

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