Simultaneous measurement of refractive index and dispersion using optical coherence tomography for restoration of transparent works of art

Abstract

We describe a novel in situ non-invasive and non-contact technique for simultaneous measurement of refractive index and dispersion of transparent solid materials using optical coherence tomography (OCT). The technique requires multi-angle OCT imaging. It gives typical precision of 0.02 for the refractive index and 0.03 $\mu$m$^{-1}$ for dispersion. The method can be applied to the restoration of modern sculptures made from plastics where it is important to have precise measurement of the refractive index of an object using a non-contact method, so that a resin-based adhesive with a matching refractive index could be found to restore a scratched or broken transparent works of art.

1 Introduction

Refractive index is an important optical parameter that is used for optical design [1], sorting [2] and quantification of materials (e.g. for diagnosis or assessing the purity of materials) [3]. There are different techniques for measuring refractive index of materials which can be grouped into invasive and non-invasive methods. Invasive methods of measuring refractive index require sample extraction and examples include Abbe refractometer [4] and Becke line test [5]. On the other hand, non-invasive methods do not require sample extraction and an example of such methods includes optical coherence tomography (OCT) [6]. OCT images 2D and 3D surface and subsurface microstructure of transparent and semi-transparent materials based on a Michelson interferometer. OCT records the optical path length which is given by the group refractive index multiplied by the actual physical path length. Therefore, the positions of subsurface structures in OCT images are distorted, appearing at a deeper level than in the physical object. The group refractive index of a material can be calculated by dividing the optical thickness of the sample by its true physical thickness [7]. Non-invasive refractive index measurement techniques like OCT are important in areas where sample extraction is prohibited. One such area is in the restoration of broken transparent works of art where there is a need to match the refractive index of the art object with an adhesive resin to achieve a seamless restoration [8,9,10]. In this paper, we will focus on the application of OCT refractive index measurement to the restoration of modern plastic sculptures.

Since the early twentieth century, transparent or translucent plastics were used in works of art, e.g. some of the prominent modern sculptures were made during the Constructivism art movement. Transparent or translucent plastic art is found to be particularly difficult to preserve, as any scratches or slight damage is obvious to the viewer. One of the innovative restoration methods involves finding a stable and preferably reversible (in a safe solvent) resin adhesive with a refractive index matching that of the object [8,9,10]. It was found that the restoration is likely to be seamless if the difference in the phase refractive indices is within 0.02 [8]. Here, we present a non-invasive method of measuring the refractive indices of transparent objects using OCT that can be used to complement the restoration of plastic art objects. OCT is mostly used to measure group refractive index which is often assumed to be equal to the phase refractive index. However, for plastic materials such as PMMA, the difference between phase and group refractive indices in the visible region ($\sim$550 nm) is about 0.04 [11]. Therefore, precise knowledge of the phase index of an object is required in this region, and for OCT measurements, it is necessary to distinguish between the phase and group refractive indices.

Different techniques have been developed to measure refractive indices of materials using OCT. The most common method relies on the comparison between the optical path length and the physical path length [12, 13], which provides high-precision measurements but requires a reference surface placed behind the sample. This is not always possible for in situ contactless measurements. A method that avoids the need for a reference surface is the focus tracking method [14]. However, it assumes that phase and group refractive indices are the same. Tomlins et al. [15] demonstrated a method for simultaneous measurement of phase refractive index and physical thickness of a sample by taking OCT images of the sample at different angles of incidence. However, the technique is only applicable to samples with uniform thickness and parallel surfaces, making it not applicable to most real-world scenarios. Refractive index of samples with known geometry can also be retrieved from OCT images of the sample using the inverse correction method [16], where the distorted OCT image of the sample is numerically corrected to the known geometry. However, the precise actual geometry of a sample may not always be known a priori. Other multi-angle OCT methods used back-projection methods similar to that used in computed tomography (CT) to generate an approximate refractive index map that either ignored the refraction effects that change the direction of the beam [17, 18] or assumed that the phase and group indices are the same [19].

In this paper, a method for non-invasively measuring both the group and phase refractive indices and therefore the dispersion of materials is presented. This only requires normal OCT images taken over multiple angles of incidence. Since a seamless restoration requires matching the resin refractive index to that of the plastic sculpture in the visible part of the spectrum where humans can see, we used our method to determine the refractive index of a mock plastic sample with our in-house developed 550 nm OCT. We also tested this method with a more accessible commercial OCT (Thorlabs Callisto 930 nm OCT) to evaluate the refractive index for 550 nm by extrapolating from the measured phase refractive index and dispersion at 930 nm. While we focus on applications in the restoration of modern plastic sculptures, our method can be applied to a variety of applications involving transparent solid materials.

2 Methods

Our method is based on the propagation of light in a wedge shape or sample with two surfaces that are ‘flat’ within the OCT field of view. By ‘flat’ we mean having a large radius of curvature. The illustration in Fig. 1a shows that when an OCT beam is incident at an angle of $\theta _i$ on a sample with true wedge angle $\beta _T$, the beam of light is refracted with the sample’s phase refractive index $n_p$ according to Snell’s law. The distance the beam of light travels in the sample is then distorted by the sample’s group refractive index $n_g$ which changes the true wedge angle to the apparent wedge angle $\beta _A$ on the OCT image. As shown in appendix A, the apparent wedge angle is related to the true wedge angle and all the other parameters by

$$\begin{aligned} \cot {\beta _A}=\cot {\beta _T}\sqrt{1-\Bigg(\frac{\sin {\theta _i}}{n_{p}}\Bigg)^{2}}(n_{g}\cos {\theta _i )^{-1}}-\Bigg(1-\frac{1}{n_{p}n_{g}}\Bigg)\tan {\theta _i} \end{aligned}$$

(1)

Fig. 1

Measurement of refractive index and dispersion of a plastic mock sample. a An illustration of light propagation within a wedge-shaped sample as imaged by an OCT. The physical object (black) has a true wedge angle $\beta _{T}$ that becomes the apparent wedge angle $\beta _{A}$ on an OCT image (illustrated in red). b Mock plastic sample as was scanned under OCT. The scanning direction across the edge is shown by the red double arrow. c An example of OCT images of the plastic sample shown in (b). The first straight portion from the corner as marked on the top right part was used for the analysis

The incidence angle $\theta _i$ was measured from the slope of the sample top surface in the OCT image, whereas the apparent angle $\beta _A$ was measured as the angle between the sample top and bottom surfaces in the OCT image. All evaluations were made in MATLAB by selecting a region of interest in the OCT image with the top and bottom surfaces visible. The detected sample surfaces were then corrected for curvature distortion by measuring against a thick optically flat piece of glass. Curvature distortion is caused by the effect of the Petzval field curvature of the objective lens which makes a flat surface appear curved in an OCT image. By fitting a first-order polynomial to the corrected surfaces, the slope for both surfaces was measured and $\theta _i$ and $\beta _A$ values were retrieved. The uncertainties in $\theta _i$ and $\beta _A$ were estimated from the error in the slopes. These angles are fed into our model to evaluate the index of refraction and dispersion parameters.

We used a constrained nonlinear minimization solver in MATLAB called fmincon to solve the nonlinear Eq. 1. Since the parameters $n_p$ and $n_g$ are coupled, we set a constraint that allows $n_g$ to exceed $n_p$ by up to 0.05. This value was chosen as the upper bound of the variation between the two parameters in the visible region for transparent plastics based on data found in Ref [11]. OCT images were taken at various $\theta _i$ values between zero and the largest angle of inclination before the bottom surface disappears on the OCT image. The incident angle within this range was chosen randomly by rotating the OCT probe. Solving the constrained problem in MATLAB yielded the solution for $\beta _T$, $n_p$, and $n_g$.

3 Visible range OCT at 550 nm

Since the restoration must appear invisible to the human observer, measurements of the refractive indices are best performed in the visible region [8,9,10]. Since there are no commercial OCT systems operating in the visible range, an in-house developed OCT system operating at 550 nm (Fig. 2) was used. The 550 nm Fourier domain OCT consists of a Michelson interferometer in free-space, a broadband supercontinuum laser source (NKT SuperK Extreme EXU-6 OCT), a spectrograph with a 1800 l/mm grating and a 4096 pixels line camera (e2V AViiVA 4010 EM4 with a camera link frame grabber NI PCIe-1433) with a Pentax 50 mm f/1.2 lens.

Fig. 2

Schematic diagram of the 550 nm Fourier domain OCT system

4 Refractive index measurement using 550 nm OCT

Our method was used to measure the group and phase refractive indices of a plastic sample similar to those used in modern plastic sculptures as shown in Fig. 1b. We first used the visible region in-house developed 550 nm OCT to scan the sample at various angles of incidence. While measurements at three incidence angles uniquely determine the three parameters, between 10–15 OCT images were used to estimate the best-fit parameters and uncertainties. The cotangent of the apparent wedge angle $\beta _A$ was computed for all the measurements and used alongside the angle of incidence data to find the true wedge angle $\beta _T$ and refractive indices $n_p$ and $n_g$ by solving Eq. 1 as described in Sect. 2. The uncertainties in fitted parameters were estimated by simulation where the data points were randomly varied around the fit (the red curve in Fig. 3) while maintaining the average and standard deviation (determined from the 10–15 actual measurements) of their distribution around this fit. For each set of simulated data points, the solver was called and a new set of solutions for $\beta _T$, $n_p$ and $n_g$ was found. The procedure was repeated 20 times, and the standard deviation of all the solutions was recorded as the error for the measurement. Figure 3 shows the measured data and the fit for the plastic sample studied here with fitted values of $71.1 \pm 0.6$ degrees, $1.477 \pm 0.016$, $1.522 \pm 0.016$ for $\beta _T$, $n_p$ and $n_g$, respectively.

To check our results, we measured the refractive index of the material in the visible region using the oil matching method by removing tiny pieces and immersing them in standard refractive index immersion oils. The oil matching method gave the phase refractive index at 589 nm of 1.492± 0.004 which agrees with the result from our method of $1.477 \pm 0.016$ using the 550 nm OCT.

Fig. 3

Measurement of refractive indices of a plastic sample. The fitted line uses the fitted parameters

5 Refractive index measurement using 930 nm commercial OCT

To verify our method of determining the phase and group refractive indices of transparent material, we used known samples of BK7 and S-11 glass prisms and the values are given in Table 1. The measured phase and group refractive indices agreed well with the quoted values for these materials. We also report the fitted true wedge angle $\beta _T$ for the standard samples and compare that with quoted values from the manufacturers. We further estimated dispersion from the measured phase and group refractive indices using the relation

$$\begin{aligned} n_g=n_p\bigg (1+\frac{\lambda D}{n_p}\bigg )^{-1} \end{aligned}$$

(2)

where $D=\frac{\textrm{d}n_p}{\textrm{d}\lambda }$ is the dispersion and $\lambda$ is the wavelength (here taken as the centre wavelength of the OCT). Here, we also found that our estimates of the dispersion agreed with the quoted measurements as shown in Table 1.

Table 1 Refractive index and dispersion measurements of standard samples and a mock sample at 930 nm

Since most of the commercial OCTs operate in the near-infrared (NIR) region, we wanted to see if extrapolated measurements of phase refractive index from NIR to the visible region would be of sufficient accuracy. We, therefore, measured the phase and group refractive indices of the same mock plastic sample using a commercial Thorlabs Callisto 930 nm OCT system and used Eq. 2 to estimate the dispersion. The results are shown in Table 1 where the uncertainty in dispersion is estimated through error propagation. We then extrapolated the phase refractive index value from 930 nm to 550 nm and compared this extrapolated value with the one directly measured with the 550 nm OCT system. Here, we found the phase refractive index of $1.477 \pm 0.015$ at 550 nm, which agrees with that measured directly by the 550 nm OCT system. These results, therefore, demonstrate the applicability of our method in estimating the phase refractive index of transparent samples in the visible region using readily available commercial OCTs operating in the NIR.

The accuracy of our method mainly depends on how well the sample is scanned and the uncertainties in measuring wedge angles from OCT images. As the method relies on the change of the apparent wedge angle with sample inclination, small true wedge angles will change less compared to larger angles. For smaller angles, the relative error will therefore be larger. For very large angles, the visibility of the second surface is reduced in OCT images which will increase the uncertainty in detecting the sample surface and subsequent measurement of either the angle of incidence or the apparent wedge angle. In our study, we scanned true wedge angles between 40 and 80 degrees and the errors in the measured refractive indices were up to 0.02.

Our method also requires that all scans be made on the same sample position while only changing the inclination angle between the sample surface and the OCT beam. In our non-contact scanning, we rotated the probe, as described in Sect. 2 while leaving the sample in one position.

6 Conclusions

In conclusion, a new method for simultaneous refractive index and dispersion measurement based on OCT images taken at multiple angles of incidence has been demonstrated in this paper. Non-contact refractive index measurement of transparent objects is important in the conservation of modern sculptures made of plastics where refractive index matching between an object and the resin adhesive is required for seamless restoration. This method was demonstrated to assist such processes especially since non-contact measurement of the refractive index is required to avoid scratches. Unlike previous OCT-based methods, the new technique simultaneously measures the phase and group refractive indices from which the dispersion of an object can be calculated without the need for reference surfaces. The results obtained indicate that this method is reliable in measuring phase refractive index and dispersion of objects in situ, giving the required accuracy in the refractive index of less than 0.02 which is within the range of allowable differences in refractive index between the object and adhesive to give a seamless restoration.

Due to the lack of commercial OCTs operating in the visible region, we have demonstrated that our method can be used on commercial OCTs operating in the near-infrared region and extrapolate the measured phase refractive index to the visible region. The current technique is limited to samples with segments of surfaces that are ‘flat’ in the OCT field of view and has been studied on wedge angles over 40 degrees. This method is applicable to most modern plastic sculptures as the objects usually have smooth surfaces and are large compared to the OCT field of view and therefore can satisfy the criteria of having segments of ‘flat’ surfaces. Future work will extend the method to objects with surfaces of all shapes including those with irregular surfaces. This could be achieved by imaging around a corner of a transparent object to record its 3D surface around the corner before applying the current method.

Appendix A: Derivation of Eq. 1

We use the schematic shown in Fig. 1a where we can show from the Sine rule applied to triangle POQ that

$$\begin{aligned} \frac{\sin {\beta _{T}}}{PQ}=\frac{\sin {\psi }}{PO} \end{aligned}$$

(A1)

where

$$\begin{aligned} \psi =\frac{\pi }{2}-\beta _{T}+\theta _{r} \end{aligned}$$

(A2)

and therefore

$$\begin{aligned} \sin {\psi } =\cos {\theta _{r}}\cos {\beta _{T}}+\sin {\theta _{r}\sin {\beta _{T}}} \end{aligned}$$

(A3)

Substituting Eqs. A3 into A1 and making PO the subject yield

$$\begin{aligned} PO =PQ(\cos {\theta _{r}}\cot {\beta _{T}}+\sin {\theta _{r}}) \end{aligned}$$

(A4)

Similarly, applying the sine rule to triangle POQ’ we can show that

$$\begin{aligned} \frac{\sin {\beta _{A}}}{PQ'}=\frac{\sin {\phi }}{PO} \end{aligned}$$

(A5)

where

$$\begin{aligned} \phi =\frac{\pi }{2}+\theta _{i}-\beta _{A} \end{aligned}$$

(A6)

and hence

$$\begin{aligned} \sin {\phi } =\cos {\theta _{i}}\cos {\beta _{A}}+\sin {\theta _{i}}\sin {\beta _{A}} \end{aligned}$$

(A7)

Substituting Eqs. A4 and A7 into Eq. A5 and making $\sin {\beta _{A}}$ subject yield

$$\begin{aligned} \sin {\beta _{A}}=\frac{PQ'(\cos {\theta _{i}}\cos {\beta _{A}}+\sin {\theta _{i}}\sin {\beta _{A}})}{PQ(\cos {\theta _{r}}\cot {\beta _{T}}+\sin {\theta _{r}})}. \end{aligned}$$

(A8)

Since $PQ'=n_{g}PQ$, Eq. A8 simplifies to,

$$\begin{aligned} \sin {\beta _{A}}=\frac{n_{g}(\cos {\theta _{i}}\cos {\beta _{A}}+\sin {\theta _{i}}\sin {\beta _{A}})}{\cos {\theta _{r}}\cot {\beta _{T}}+\sin {\theta _{r}}}. \end{aligned}$$

(A9)

Dividing both sides of Eq. A9 by $\sin {\beta _{A}}$ and rearranging the equation yield

$$\begin{aligned} \cot {\beta _{A}}=\frac{\cos {\theta _{r}}\cot {\beta _{T}}+\sin {\theta _{r}}}{n_{g}\cos {\theta _{i}}}-\tan {\theta _{i}} \end{aligned}$$

(A10)

Simplifying further by substituting $\theta _{r}=\sin ^{-1}(\frac{\sin {\theta _{i}}}{n_{p}}),$ we get

$$\begin{aligned} \cot {\beta _{A}}=\frac{\cos {[\sin ^{-1}(\frac{\sin {\theta _{i}}}{n_{p}})]}\cot {\beta _{T}}}{n_{g}\cos {\theta _{i}}} -\left(1-\frac{1}{n_{p}n_{g}}\right)\tan {\theta _{i}} \end{aligned}$$

(A11)

Finally, Eq. A11 simplifies to Eq. 1.