1 Introduction

To characterize the optical and transmission properties of various optically transparent plates and films, their thicknesses and refractive indices are fundamental parameters [1, 2]. For this purpose, the measurement of the thickness and refractive index of transparent materials is very important in various applications [3, 4]. Optical inspection systems are an attractive solution for the inspection process in automated mass production lines due to their high throughput, good reliability, and relatively low cost [5,6,7,8]. Because optical inspection systems have long been crucial tools in the nondestructive characterization of materials and devices, they are widely applied for the measurement of these parameters [1, 9]. As a result, various interesting optical methods of addressing this task have been developed in prior art in recent years. Existing measurement methods include the use of dual-confocal microscopy [1], wavelength-scanning interferometry [2, 3], Fabry–Perot interferometry [10], double-slit interferometry [11], Mach–Zehnder interferometry [12, 13], low-coherence interferometry [14,15,16,17,18], optical diffraction tomography [19, 20], optical coherence tomography [21,22,23], ellipsometry [9, 24, 25], diffractive element [26], triangulation methods [27,28,29], electronically controlled tunable lens [4], and so forth. In general, the optical methods for simultaneous measurement of the thickness and refractive index of materials are convincing, because they are time saving and robust in the measurement process.

The most common optical methods for simultaneous measurement of the thickness and refractive index of materials can be broadly classified as interferometry, ellipsometry, and triangulation methods. To the best of the authors’ knowledge, interferometry and ellipsometry are used to measure thin films and electronic glass (thickness of < 1 mm), and they require complex instrumentation with high resolution and high cost. Therefore, they are not suitable for in-line real-time measurements [2]. In contrast, triangulation methods are particularly attractive for in-line real-time measurements and most popular due to their simpler structures and lower cost. Triangulation methods are widely adopted by laser displacement sensors currently [30, 31]. Triangulation means distance measurement by angle calculation. When using triangulation methods, a light source projects a laser spot onto a measured target. The reflected light falls incident onto a CCD sensor at a certain angle depending on the distance. From the position of the light spot on the CCD sensor and the distance from the sender to the CCD sensor, the distance to the measured target is calculated [32]. However, these techniques based on triangulation methods in prior art are very few and they are difficult to measure a greater thickness (> 1 mm) of transparent plate.

Accordingly, the present study develops a novel optical system based on triangulation methods for simultaneous measurement of the thickness (> 1 mm) and refractive index of a transparent plate. The proposed optical system can measure a transparent plate with thickness of > 1 mm with a simpler structure and lower cost. The proposed optical system is characterized numerically, then verified experimentally using a laboratory-built prototype, and finally compared with a commercial instrument.

2 Proposed optical system with two optical paths

2.1 Structure layout

Figure 1 illustrates the structure of the proposed optical system for simultaneously measuring the thickness and refractive index of a transparent plate with two optical paths. As shown, the light beam emitted from a laser light source (Unice, LHRP-0201, 633 nm) passes through a calcite beam displacer (Thorlabs, BD27), where it is split into two separate polar light beams. It is noted that there are many methods for having two parallel beams, for example, using an unpolarized LED light source, a beam splitter (BS), and a reflective mirror to separate one beam into two parallel beams. The two light beams are then passed through two beam splitters (BS1 and BS2, Thorlabs, CM1-BS013), a triplet achromatic lens (Thorlabs, TRS127-020-A), and a transparent sample and are incident on a first reflective mirror surface. The two light beams reflected from the first reflective mirror surface pass back through the sample, the triplet achromatic lens, and two BSs and are then incident on a polar beam splitter (PBS, Edmund, 4SF2-550), where it is split into two separate optical paths. In this study, the triplet achromatic lens is used as an objective lens. In one optical path (designated as Optical Path I), the light beam emerging from the PBS is passed through a first achromatic lens (Thorlabs, AC254-200A) and is then incident on a first CCD sensor (Basler scA1390-17 fm, referred to hereafter as CCDI). Meanwhile, in the second optical path (designated as Optical Path II), the light beam reflected from the second reflective mirror surface passes through a second achromatic lens (Thorlabs, AC254-200A) and is then incident on a second CCD sensor (Basler scA1390-17 fm, referred to hereafter as CCDII).

Fig. 1
figure 1

Structure of the proposed optical system

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In realizing the simultaneous measurement of the thickness and refractive index of the transparent sample, the two optical paths are combined using a self-written measurement processing algorithm to achieve a rapid measuring capability and a high measuring accuracy. We could understand that there are some birefringence and polarization problems in many samples. Here these problems could be neglected for our testing samples.

2.2 Measuring principle

Figure 2a, b presents the schematic illustrations of the optical paths within the proposed optical system without and with the transparent sample, respectively. As shown in a, when the transparent sample is not put into the proposed optical system (without sample), the light beams in Optical Path I and Optical Path II are finally incident on the center points of CCDI and CCDII, respectively. However, as shown in b, when the transparent sample is put into the proposed optical system (with sample), the light beams in Optical Path I and Optical Path II are finally incident on the off-center points of CCDI and CCDII with shift distances \({\Delta _{\text{I}}}\) and \({\Delta _{{\text{II}}}}\), respectively. From basic geometric principles, the following equations for Optical Paths I and II can be obtained, respectively:

Fig. 2
figure 2

Schematic illustration of optical paths in the proposed optical system: a without sample and b with sample

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$$2{z_{\text{I}}}=\frac{{{\Delta _{\text{I}}}{f_1}}}{{{f_{{\text{2I}}}}}},$$
(1I)
$$2{z_{{\text{II}}}}=\frac{{{\Delta _{{\text{II}}}}{f_1}}}{{{f_{{\text{2II}}}}}},$$
(1II)

where \(2{z_{\text{I}}}\) and \(2{z_{{\text{II}}}}\) are the shift distance of light beam in the triplet achromatic lens, f1 is the focal length of the triplet achromatic lens, and f2 is the effective focal length of the achromatic lens.

Figure 3 presents the detailed illustration of optical path I in Fig. 2b. From the triangles ABC and ABD, the following equations can be obtained, respectively:

Fig. 3
figure 3

Detailed illustration of Optical Path I in Fig. 2b

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$$d\tan {\theta _{\text{I}}}^{\prime}={y_{\text{I}}},$$
(2)
$$d\tan {\theta _{\text{I}}}={y_{\text{I}}}+{z_{\text{I}}},$$
(3)

where d is the thickness of the sample, \({\theta _{\text{I}}}\) is the incidence angle of Optical Path I, and \(\theta _{{\text{I}}}^{\prime }\) is the refractive angle of Optical Path I. Subtracting Eq. (2) from Eq. (3), d can be obtained as:

$$d=\frac{{{z_{\text{I}}}}}{{\tan {\theta _{\text{I}}} - \tan \theta _{{\text{I}}}^{\prime }}}.$$
(4)

In general, the refractive angle \(\theta _{{\text{I}}}^{\prime }\) between two optical media satisfies Snell’s law, i,e.,

$${n_{\text{0}}}\sin {\theta _{\text{I}}}=n\sin \theta _{{\text{I}}}^{\prime }{\text{,}}$$
(5)

where \({n_{\text{0}}}\) is the refractive index of air and n is the refractive index of the sample. Equation (5) can be rewritten as

$$\tan \theta _{{\text{I}}}^{\prime }=\frac{{{n_0}\sin {\theta _{\text{I}}}}}{{\sqrt {{n^2} - {n_0}^{2}{{\sin }^2}{\theta _{\text{I}}}} }}.$$
(6)

Substituting Eq. (6) into Eq. (4), the following equations for Optical Paths I and II can be obtained, respectively:

$$d=\frac{{{z_{\text{I}}}}}{{\tan {\theta _{\text{I}}} - \frac{{{n_0}\sin {\theta _{\text{I}}}}}{{\sqrt {{n^2} - {n_0}^{2}{{\sin }^2}{\theta _{\text{I}}}} }}}},$$
(7I)
$$d=\frac{{{z_{{\text{II}}}}}}{{\tan {\theta _{{\text{II}}}} - \frac{{{n_0}\sin {\theta _{{\text{II}}}}}}{{\sqrt {{n^2} - {n_0}^{2}{{\sin }^2}{\theta _{{\text{II}}}}} }}}}.$$
(7II)

The measuring principle of the proposed optical system is described by the following. When the transparent sample is measured, the light beams in Optical Path I and Optical Path II are altered, which produce centroid’s changes in the positions of the light spots on CCDI and CCDII with shift distances ΔI and ΔII, respectively. Here, the off-center shift distances ΔI and ΔII are obtained by finding the centroids of the images captured by the CCD sensors. From Eq. (1), zI and zII can be obtained. Then substituting zI and zII into Eqs. (7I) and (7II), the thickness d and the refractive index n of the sample can be obtained simultaneously. When the sample is measured, the thickness d and the refractive index n will simultaneously alter the laser beams incident on CCDI and CCDII, which in turn induce changes in the positions of the light spots on CCDI and CCDII.

To accurately and efficiently analyze the relation between position information of light spots and the individual thickness d and refractive index n, the laser beam progression in the proposed optical system is tracked using a skew-ray tracing method and a self-written measurement processing algorithm. Light beam tracing equations must be established via a homogeneous transformation matrix (HTM) and skew-ray tracing method proposed by Lin to model and perform ray tracing for the proposed optical system in this study [33,34,35,36]. The HTM corresponding to the coordinate frame of each optical boundary relative to a reference coordinate system is defined sequentially. A mathematical model is proposed to develop a systematic forward and reverse mathematical derivation. Then, the self-written measurement processing algorithm, which combines the mathematical model and Newton–Raphson method, is used to calculate the individual thickness d and refractive index n of the sample. HTM and the skew-ray tracing method are powerful tools in this field and we only introduce their basic background to avoid repeat in this study. For more comprehensive coverage, the reader is referred to [33,34,35,36].

3 Numerical simulation of the proposed optical system

A series of ray tracing simulations were performed to verify the measuring performance of the proposed optical system and to determine suitable values of the major design parameters. In this section, the ray trace function of ZEMAX software was used to simulate the actual ray propagation in the proposed optical system. The Matlab software was then used to process the images of the laser spot position map and calculate the location of the image centroid. Figure 4 illustrates the detailed optical model constructed using commercial ZEMAX software. Table 1 summarizes the selected values of each design parameter.

Fig. 4
figure 4

ZEMAX optical model of the proposed optical system

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Table 1 Design parameters of the proposed optical system
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Figure 5a, b illustrates the simulation results obtained for the variation of the image centroid with the thickness d (at n = 1.515) and refractive index n (at d = 1 mm) of the sample in the two optical paths, respectively. These figures indicate that the laser spots on CCDI and CCDII show a changing trend when the sample experiences a change in thickness d or refractive index n. Figures 6a, b presents the numerical results obtained for the thickness d (at n = 1.515) and the refractive index n (at d = 1 mm), respectively, given different original values, in which the errors of the point to point for the thickness d and refractive index n are plotted, respectively, too. In Fig. 6, the original values of the thickness and refractive index for the samples are the setting values in the ZEMAX model. The calculated values of the thickness and refractive index for the samples are the calculated values using the self-written measurement processing algorithm. Here, the threshold values in Newton–Raphson method are set as 0.007 and 0.005% for the thickness d and the refractive index n, respectively. It is observed that the maximal calculated errors of the thickness d and refractive index n are 0.0065 and 0.0045%, respectively. These calculated errors are very small; in other words, the numerical results are consistent with the theoretical analysis presented in Sect. 2.2.

Fig. 5
figure 5

Simulation results for variation of image centroid position with a thickness (n = 1.515) and refractive index (d = 1 mm)

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Fig. 6
figure 6

Numerical results: a variation of calculated thickness with original thickness (n = 1.515) and b variation of calculated refractive index with original refractive index (d = 1 mm)

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4 Experimental characterization of prototype model

As shown in Fig. 7, the validity of the proposed optical system was verified by means of a laboratory-built prototype with three micro slide samples. The experiments are conducted in a temperature-controlled laboratory and a commercial instrument (Keyence LK-G80 laser displacement sensor) was used to compare and verify the measured accuracy of the proposed optical system. It is noted that Keyence LK-G80 laser displacement sensor cannot simultaneously measure the thickness and refractive index of the sample.

Fig. 7
figure 7

Laboratory-built prototype of the proposed optical system

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In experimental trials, the thickness d and refractive index n are measured in turn for selected ten points on three samples (Sample A, Sample B, and Sample C) with different thicknesses and refractive indexes. The thickness of Sample B is greater than that of Sample C and the thickness of Sample C is greater than that of Sample A. Figures 8 and 9 present the captured images of the laser spots on CCDI and CCDII before and after inserting the three samples, respectively. Figures 10, 11, and 12 present the measured results obtained for the thickness d and refractive index n of Sample A, Sample B, and Sample C, respectively. In these figures, each measured value is the average value of ten repeated experiments on each point to reduce the random errors and the error bar presents the standard deviation on each point. The average standard deviations for the measured thickness d and refractive index n of Sample A, Sample B, and Sample C are listed in Table 2, respectively. It is observed that compared with the measured results of the commercial instrument, the proposed optical system achieves a high measuring accuracy (less than 3 µm for thickness d and 0.003 for refractive index n) with Sample A, in which the measured errors of the thickness d and refractive index n are less than 0.35 and 0.2%, respectively. For Sample C, the proposed optical system achieves a higher measuring accuracy (less than 2.2 µm for thickness d and 0.003 for refractive index n), in which the measured errors of the thickness d and refractive index n are less than 0.032 and 0.025%, respectively. However, for Sample B, the proposed optical system achieves the highest measuring accuracy (less than 0.4 µm for thickness d and 0.0001 for refractive index n), in which the measured errors of the thickness d and refractive index n are less than 0.019 and 0.007%, respectively. These experimental results are summarized in Table 2 to compare them all together.

Fig. 8
figure 8

Captured images of laser spots on CCDI: a before inserting sample, b after inserting Sample A, c after inserting Sample B, and d after inserting Sample C

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Fig. 9
figure 9

Captured images of laser spots on CCDII: a before inserting sample, b after inserting Sample A, c after inserting Sample B, and d after inserting Sample C

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Fig. 10
figure 10

Experimental results of Sample A: a measured thickness and b measured refractive index

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Fig. 11
figure 11

Experimental results of Sample B: a measured thickness and b measured refractive index

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Fig. 12
figure 12

Experimental results of Sample C: a measured thickness and b measured refractive index

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Table 2 Experimental results and comparison of samples
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From Eqs. (7I) and (7II), zI and zII increase in proportion to the thickness d of the sample, which means the sensitivity of the proposed optical system is better with a greater thickness of sample. In other words, the signal-to-noise ratio (SNR) of the proposed optical system is greater with a greater thickness of sample. This is why the proposed optical system achieves a higher measuring accuracy with a greater thickness of sample. The resulting thicknesses and refractive indices using the proposed optical system are in good agreement with those determined by the commercial instrument. Consequently, it confirms that the proposed optical system is suitable and has potential for simultaneous measurement of the thickness and refractive index of a transparent plate (thickness of > 1 mm) with a simpler structure and lower cost. The proposed optical system is aimed at the specific measurement applications of the car and building windows.

However, the response time of the proposed optical system is less than 47 s now. When compared to commercial products, the proposed system using a laboratory-built prototype spends much time in our measurement process, because we use LabVIEW software and PC to perform the image acquisition and calculate the centroid, and use MATLAB software to calculate the individual thickness d and refractive index n of the sample using a self-written measurement processing algorithm. Therefore, in the future, we will try to organize a new research group, and the proposed system will be combined with a sturdy computer engineering to enable the proposed system for in-line real-time measurement.

5 Conclusions

This study has presented a novel optical system based on triangulation methods for simultaneous measurement of the thickness (> 1 mm) and refractive index of a transparent plate. The performance of the proposed optical system has been evaluated using a laboratory-built prototype. The experimental results have shown that the optical system has a high measuring accuracy. When compared with the commercial instrument, the measured errors of the thickness d and refractive index n for the proposed optical system are less than 0.019 and 0.007%, respectively, with the transparent plant (Sample B).