1. Introduction

There are many non-destructive methods for measuring the refractive index of a lens. Some techniques are based on the liquid immersion method: In [1], a test lens is immersed in a mixed liquid whose refractive index is varied until its index is identical to that of the test lens, the lens refractive index is indirectly obtained by measuring the refractive index of the mixture liquid. This widely practiced method has many limitations and restrictions: for example, the compounded liquids must be of a miscible nature, the search for such constituents and then the preparation of the desired mixture are time-consuming procedures, most of the miscible organic compounds are poisonous in nature, before reaching the desired results one has to make many trials. The non-miscible liquid-immersion techniques use a Murty shearing interferometer [2], a Ronchi grating [3], an acousto-optic grating [4], a Fabry–Perot etalon [5], moire deflectometry [6] or Fourier transform spectrometry [7] to measure the lens refractive index. These methods do not need to iteratively adjust the liquid index, however, they all need to measure the liquid index and their accuracies are limited by the liquid index measurement accuracy. Since the test lens must be immersed in liquid solution, it is difficult to align the lens and to implement the method.

Most of the non-immersive techniques are based on the lens maker's formula. For reaching the final results, one has to measure various parameters like the strength parameters (i.e., $r_1$ and $r_2$), the thickness ($d$) and the focal length ($f$) of the lens. Since the lens formula is a 2-order Eq. about the refractive index, most methods use thin-lens approximation to simplify the calculations. Parameters $r_1$, $r_2$and $f$ can be measured by normal method, Michelson interferometry [8] or digital holographic interferometry [9]. However, omitting $d$ in the thin-lens formula inevitably brings error into the results, in addition, these methods are all sensitive to the measurement error of $r_1$, $r_2$and $f$, thus their accuracies are limited. Laser differential confocal technique [10] is used to precisely identify the position of the test lens, the lens refractive index is obtained by ray tracing facet iterative calculation without thin-lens approximation. Since at least six parameters must be measured, this method is complex and time-consuming.

In general, all the non-immersive methods need to measure lens geometrical parameters. Although they are free from difficulties due to thermal and volatility changes in the immersion liquids, their accuracies are easy to be affected by the focal length measurement accuracy and the figure error of the test lens. It is particularly important to note that all these non-immersive methods are supposed to measure a spherical lens. If the test lens has at least one aspherical surface, a theoretical error will be introduced. In order to accurately measure the refractive index of spherical lens and aspherical lens, a novel non-immersive lens refractive index measurement method is presented in this paper. It is based on point-diffraction interferometry and need not to measure the lens focal length. Only three parameters are necessary even the test lens is a thick lens. Experiments indicate its accuracy is better than 2.2 × 10⁻⁴. This method is also suitable to measure the refractive index of an aspherical lens. There is no theoretical error if the aspherical lens has a spherical or planar surface. Since it is not sensitive to the measurement error of $r_2$, even a thin lens that has two aspherical surfaces can be measured with high accuracy.

2. Measurement principle

Figure 1(a) shows a lens in the air, $d$ is the thickness of the lens, $n$ is the refractive index of the lens, $n^{\prime }$ is the refractive index of the air. $r_1$and$r_2$ are the radii of curvature of the front and back surface, $O_1$and$O_2$ are the front and back vertex point, $P_1$ and ${P^{\prime }}_1$are the object point and the image point respectively. Obviously, if $P_1$is coincident with the vertex point $O_1$ of the front surface, then ${P^{\prime }}_1$ is solely imaged by the refraction of the back surface. Supposing $l$ and $l^{\prime }$represent the object distance and the image distance respectively, $l$ and $r_2$ are negative according to the sign convention. Since $l=-d$, $n^{\prime }=1$, we have

 

Fig. 1 Imaging by the refraction of the back surface of a lens.

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Thus

Supposing the distance between the object point $P_1$ and the image point ${P^{\prime }}_1$ is $\Delta R$(According to the sign convention, if ${P^{\prime }}_1$ falls in the right side of $P_1$, $\Delta R$ is positive), then $l^{\prime }=\Delta R-d$. The one arrow notation of $\Delta R$and $l^{\prime }$in Fig. 1 indicates the direction they are being measured from. Equation (2) is rewritten as

Equation (3) reveals that if an object point-source lies at the lens front vertex, the lens refractive index can be calculated by measuring the lens thickness ($d$), the radius of curvature of the back surface ($r_2$) and the distance between the object point and its image point ($\Delta R$). Since there is no thin-lens approximation in the derivation process, both thick lens and thin lens can be tested in this way (The only exception is for the lens with $r_2\text{=}-d$, Eq. (3) cannot be applied since $d+r_2=0$).

The refractive index measurement for a planar-convex or planar-concave lens could be further simplified. Supposing the object point-source is put in contact with the vertex of the curve side of the lens as Figs. 1(b) and 1(c) show ($r_1$ in Fig. 1(c) is negative according to the sign convention), since $r_2\text{=}\infty$, Eq. (3) is simplified as

The planar-convex or planar-concave lens act as plane-parallel-plate if the object point $P_1$ lies at the vertex of the curve side of the lens, thus Eq. (4) looks as the same with the Eq. in [11]. Since Eqs. (3) and (4) are independent of the parameters of the front surface of the lens, even an aspherical lens could be tested if one of its surfaces is spherical or planar.

During the measurement, it is easy to measure $d$and $r_2$, thus the crucial point is how to measure the distance $\Delta R$.

2.1 Measuring the distance between the object point and its image point

The distance between the object point and its image point is measured by point-diffraction longitudinal interferometry, its principle has already been proposed in [11] by us. For the reader's convenience, we give a concise depiction here. As shown in Fig. 2(a), the center of the exit pupil is at the origin of the $(x,y,z)$ coordinate system, a point source $P_0$ fixed on $z$-axis is located at $(0,0,r)$ and acts as the reference. Another object point-source $P_1$ is fixed at the coordination of $(0,0,R)$. Both $P_0$ and $P_1$ produce spherical wavefronts in the exit pupil. The difference of their spherical wavefronts at the exit pupil ($xoy$plane) is $\Delta w\left(x,y\right)$,

 

Fig. 2 Principle to measure the distance between the object point and its image point.

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Equation (5) is a 2-order approximation about $\left(x^2+y^2\right)/R^2$ and $\left(x^2+y^2\right)/r^2$, thus it provides an accurate estimation for$\Delta w\left(x,y\right)$, especially when the distance $R$ and $r$ is not much larger than $x$ and $y$.

Supposing ${P^{\prime }}_1$is the image point of $P_1$ with respect to a test lens, the longitudinal distance between the object point $P_1$and its image point ${P^{\prime }}_1$ is $\Delta R$,thus the coordination of ${P^{\prime }}_1$ is $\left(0,0,R-\Delta R\right)$, as shown in Fig. 2(b). The wavefront difference between the image point ${P^{\prime }}_1$ and the fixed point source $P_0$ at the exit pupil is $\Delta w^{\prime }\left(x,y\right)$,

Coefficients $k^{\prime }$and $k$ can be measured by interferometry. Supposing the difference between $k^{\prime }$and $k$ is$\Delta k$,

From Eq. (7),

2.2 Calibrating the distance$R$

If ${P^{\prime }}_1$is the image point of the object point$P_1$ with a glass plate, as shown by Fig. 3, the distance $\Delta R$ is connected with the thickness $d$ and refractive index $n$ of the glass plate. $\Delta R$ is easily found by use of Snell law for small angle of incidence to be [12]

 

Fig. 3 The distance between the object point and its image point with respect to a glass plate.

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From Eqs. (8) and (9), if a glass plate is inserted as the test lens, we have

Equation (10) reveals that the distance $R$ from the object point $P_1$ to the exit pupil can be calibrated by inserting a glass plate with a known refractive index ($n$) and thickness ($d$) in front of $P_1$ and measuring the coefficients $k$,$k^{\prime }$ through two interferometric measurements.

3. Experiments

With two single mode fibers to simulate the ideal point-sources, a lens refractive index measurement system as Fig. 4 is proposed to verify the above principle. The output beam from a laser source passes through a half-wave plate, and is split into two orthogonally polarized beams by a polarization prism beam splitter (BS). One beam is reflected from a retroreflector mounted on a piezoelectric phase shifter and the other beam is reflected from another retroreflector. Both beams pass twice through quarter-wave plates to rotate their polarizations, before the two beams come out of the prism BS finally. After passing through polarizers, the two beams are coupled into fiber 1 and fiber 2 separately by microscope objectives. The tip of fiber 1 (object point $P_1$) is put in contact with the front vertex point $O_1$ of the lens under test, thus its image point ${P^{\prime }}_1$ is solely refracted by the back surface of the lens. The tip of fiber 2 (object point $P_0$) is fixed and acts as the reference. The spherical wavefront from the image point ${P^{\prime }}_1$ and the object point $P_0$ interfere with each other through a pellicle BS. A CCD camera without lens is used to image the fringe patterns. The relative intensity between the two beams can be adjusted by the angular orientation of the half-wave plate and the polarizers.

 

Fig. 4 Experimental system for the lens refractive index measurement.

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It is obvious that if the image point ${P^{\prime }}_1$ is conjugated to the object point $P_0$ with the pellicle BS, the fringe patterns are null thus it is difficult to locate the center of the exit pupil from the fringe patterns. But if ${P^{\prime }}_1$ has a longitudinal defocus with $P_0$, the spherical wavefronts of ${P^{\prime }}_1$ and $P_0$ will produce the longitudinal interference. A series of concentric rings will be observed on the CCD camera, the center of the concentric rings can be looked as the center of the exit pupil.

In our experiments, the light source is a single-longitudinal-mode solid laser with a wavelength of 532 nm. It has enough coherence length (≥20 m) to facilitate the adjustment. The single mode fiber has a core diameter of 3.5 μm, the aperture angle of the spherical wave diffracted from the fiber tip is about 7°. To assure point contact with the vertex of a concave surface, the tip of fiber 1 should be polished to a convex sphere with a small radius of curvature. The CCD camera has a pixel resolution of 1024 × 1024 and each pixel has a dimension of 4.65 μm × 4.65 μm. Because the CCD sensor area is constrained within a square of 5 mm × 5 mm, while the spherical wavefront diffracted from the point sources spreads out with the enlarging distance, to ensure the sampled wavefront has enough representativeness, the CCD camera could not be placed far from the image point unless a CCD camera with large sensor size is used.

The tip of fiber 1 (object point$P_1$) is placed about 120 mm to the CCD camera. A quartz glass plate ($n=1.46079$@ 532nm,$d=4.040mm$) is used to calibrate the actual distance $R$. It is obvious that the thickness of the pellicle BS will introduce error for the preset distance$R$. However, the Φ25 mm pellicle BS is made of extremely thin ($d\text{=}2$μm) nitrocellulose membrane with a refraction index of 1.5. The membrane is stretched over a black anodized aluminum frame, resulting in a uniform surface (less than λ/2 of variation at 635 nm across diameter). From Eq. (9), the longitudinal distance error introduced by the pellicle BS is only 0.67 μm thus can be neglected with respect to the preset distance$R$.

A biconvex lens made with BK7 glass is firstly measured in our experiments, its $r_2$ is measured by a spherometer to be −76.3495 mm, $d$is measured by a micrometer to be 7.992mm. The refractive index of the lens under test is derived through the following interferometric measurements.

First, as Fig. 5(a) shows, the tip of fiber 2 (point source $P_0$) is placed to a defocus position with respect to the tip of fiber 1 (object point$P_1$), where a series of concentric rings could be observed on the CCD camera. Then fiber 2 is finely adjusted until the center of the concentric rings coincides with the center of the CCD camera. The well-known Hariharan five-step algorithm [13] is applied to process the fringe patterns. The coefficient $k$is derived by polynomial fitting the unwrapped phases to Eq. (5) (a constant term should be included in the polynomials to reveal the initial phase difference of the two point sources).

 

Fig. 5 Processes for the lens refractive index measurement.

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Second, as Fig. 5(b) shows, the quartz glass plate is inserted between the object point$P_1$ and the pellicle BS. The orientation of the glass plate is finely adjusted until the concentric rings appear again and the center of the rings coincides with the center of the CCD camera. It is important not to do any adjustment for the fibers after inserting the glass plate. The coefficient $k^{\prime }$ can be derived by the polynomial fitting of Eq. (6) after applying the second phase shifting interferometric measurement. The distance $R$is calculated by Eq. (10). With 10 repeated measurements, the average $\overline{R}$ is calculated as 118.2130 mm, the standard deviation is 0.012 mm. The following tests are all applied under the calibrated distance$R$, which means the fibers remain fixed throughout the measurements of samples.

Third, as Fig. 5(c) shows, the glass plate is taken out. The lens under test is put in contact with the tip of fiber 1. The position and orientation of the lens under test is finely adjusted until its front vertex point $O_1$coincides with the tip of fiber 1 (object point $P_1$) and the center of the concentric rings coincides with the center of the CCD camera. ${P^{\prime }}_1$is the image point of the object point $P_1$ with respect to the lens under test. Then the third phase shifting interferometric measurement is applied to derive the coefficient $k^{\prime }$. The longitudinal distance $\Delta R$ between $P_1$and ${P^{\prime }}_1$ is calculated by Eq. (8) with the distance $R$pre-calibrated in the second step. Figures 5(d)-5(f) show the corresponding interferograms captured in the step 1~3 respectively.

With the measured $\Delta R$ from the above processes. The refractive index of the test lens is calculated by Eq. (3), the measurement results are shown in Table 1.

Table 1. Lens refractive index measurement results

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It should be noted that, once the distance $R$and coefficient $k$ are calculated, they can be stored and used for any following tests. In other words, with a pre-calibrated distance $R$, the parameter$\Delta R$can be derived through a single interference measurement, thus the above procedure is greatly simplified in applications.

3.1 Refractive index measurement for thick lens

Three parameters ($d$, $r_2$, $\Delta R$) must be measured to derived the refractive index of a thick lens. Ball lens is a kind of special thick lens, since $r_2=-d/2$, Eq. (3) is rewritten as

In our experiment, the ball lens under test is made of BK7 glass, its thickness $d$ (i.e., diameter of the ball lens) is measured by a micrometer to be 10.005 mm. As Fig. 6(a) shows, the image point ${P^{\prime }}_1$ is now a virtual point which falls in the left side of the object point $P_1$, thus $\Delta R$ is negative in Eq. (11), the measurement results are shown in Table 1.

 

Fig. 6 Refractive index measurement for ball lens and planar-cylindrical lens.

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3.2 Refractive index measurement for aspherical lens

Most aspherical lens have a single aspherical surface in consideration of the manufacturing cost, the other surface is usually spherical or planar. All the aspherical lens with a spherical or planar surface can be measured by this method. Taking the spherical or planar side as the back surface, the refractive index of the aspherical lens can be calculated from Eq. (3) or Eq. (4). Cylindrical lens is a typical aspherical lens which have curvature in only one direction, it is not easy to measure the refractive index of a cylindrical lens by normal non-immersive methods. A planar-convex cylindrical lens made of BK7 glass is tested in our experiments, the tip of fiber 1 is put in contact with the vertex point of the cylindrical side of the lens, as shown by Fig. 6(b). From Eq. (4), only two parameters ($d$, $\Delta R$) are needed to calculated the refractive index of a planar-cylindrical lens. The measurement results are also listed in Table 1.

4. Error analysis and discussions

Partially differentiating Eq. (3) with respect to $d$, $r_2$ and $\Delta R$ respectively, we have

$\partial n/\partial d$,$\partial n/\partial r_2$and $\partial n/{\partial }_{\Delta R}$ are the sensitivity coefficients for the measurement error of $d$, $r_2$ and $\Delta R$ respectively. From Eq. (1), we have

Table 2 shows the sensitivity coefficients at different $d/r_2$. It is obvious that the accuracy of the measured lens refractive index is determined by the relative error of $d$,$r_2$ and $\Delta R$. Although $\Delta R$ is an independently measured parameter, it dependents on the other two parameters ($d$, $r_2$) as Eq. (15) shows, thus the refractive index measured by this method is actually determined by $d$ and $r_2$ of the test lens. One can expect a higher accuracy for the lens with a larger $d$ or $r_2$, since the measured $d$ or $r_2$ usually have a smaller relative error, the data in Table 1 also verify this point.

Table 2. Sensitivity coefficients at different $d/r_2$

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From Eq. (8),

Equation (19) reveals that the relative error of the measured $\Delta R$ is determined by the relative error of $R$and $\Delta k$. Enlarging the distance $R$is the most immediate way to lower its relative errors in order to improve the accuracy of $\Delta R$. However, due to the diffracted wavefront of the point sources spreading bigger and bigger along the axis, enlarging the distance $R$ is not always feasible unless a CCD camera with large sensor size is used. Since$\Delta k\text{=}{k}^{\prime }-k$, and $k$,$k^{\prime }$ are all derived through interferometry, the relative error of $\Delta k$ is intrinsically the relative error of the wavefront difference measured by the interferometry. Improving the precision of the interferometry can also improve the accuracy of $\Delta R$. Since interferometric system is easily affected by vibration, temperature drift, and air turbulence, environmental condition controlling is necessary in order to acquire a better accuracy.

For a thin lens, $d/r_2\approx 0$, the corresponding sensitivity coefficient $\partial n/\partial r_2\approx 0$, which means the measured lens refractive index is insensitive to the error of $r_2$. For example, substituting the measured data of$n$,$d$and $r_2$ of the biconvex lens listed in Table 1 into Eq. (17), we have $d/r_2\approx -0.1047$, the sensitivity coefficient $\partial n/\partial r_2\approx 7.95\times {10}^{-4}$. For comparison, if the refractive index of the biconvex lens is calculated by the thin-lens formula, then $n=1+r/\left(2f\right)$, the sensitivity coefficient $\partial n/\partial r=1/\left(2f\right)$$=\left(n-1\right)/r$$\approx 6.80\times {10}^{-3}$. It means: For this biconvex lens, if the measurement error of $r_2$ is as big as 1 mm (corresponding to the relative error of 1.3%), the refractive index error by the thin-lens formula is in the order of 10⁻³, while the error of this method is still in the order of 10⁻⁴. In practice, it is easy to measure $r_2$ with a precision better than 0.1% even with a normal method. Since $d$ is the only parameter which can be directly measured by a micrometer or other precision tool, one can usually expect its precision far exceeds the precision of $r_2$. Thus it is quite normal that the accuracy of this method is better than all the methods based on the thin-lens formula [8~9] in which at least two strength parameters ($r_1$, $r_2$) must be measured. Its insensitivity to the error of $r_2$ is also very useful for measuring a thin lens that has two aspherical surfaces. If taking the radius of curvature of the best-fit spherical surface as the radius of the aspherical surface, the measurement error for the refractive index of the aspherical lens is still acceptable.

For a thick lens, the sensitivity coefficients vary at different ratio of $d$ to $r_2$. If $d/r_2=-2$,the lens is a ball lens, the sensitivity coefficients listed in Table 2 are derived from Eq. (11). For a lens with $d/r_2=-1$, Eq. (3) cannot be applied since $d+r_2=0$. One can avoid this situation by using the other side of the lens as the back surface.

5. Conclusion

A novel non-immersive method for measuring the lens refractive index with the fiber point-diffraction longitudinal interferometry is proposed in this paper. The lens imaging process is simplified to the single refraction of the back surface if the object point is put in contact with the vertex of the front surface. The lens refractive index is derived through measuring the thickness ($d$), the radius of curvature of the back surface ($r_2$), and the distance between the object point and its image point ($\Delta R$), which is measured interfeorometrically based on the modeling of the longitudinal interferometry of two point sources.

Experimental results for thin lens and thick lens reveal that the measurement accuracy of this method is better than 2.2 × 10⁻⁴ under normal laboratory environment. The accuracy is mainly determined by the relative error of $d$,$r_2$and $\Delta R$. Detailed error analysis reveals that the accuracy of the measured refractive index can be further improved by enlarging the distance$R$ or by using a CCD camera with a larger sensor size.

Comparing with the other non-immersive methods, this method need not to measure the lens focal length. The lens refractive index measurement is simplified to measure just three parameters. Since there is no thin-lens approximation in the derivation of the lens refractive index, this method is suitable for both thin lens and thick lens. Because the lens front surface is excluded in the imaging process, even an aspherical lens could be tested by this method, it totally avoids the theoretical error when measuring an aspherical lens with a spherical or planar surface. Even for a thin lens with two aspherical surfaces, the measurement error is still acceptable since this method is insensitive to the measurement error of $r_2$. It is important to note that this method can be extended to measure the refractive index of an irregular sample with a single polished plane, providing its thickness is always measured from the contact point to the plane. This characteristic greatly simplifies the sample preparations in the refractive index measurement.