## Abstract

An optical method to resolve the non-measurable thickness problem caused by the overlap of optical path differences within a specific thickness range when measuring the physical thickness of a sample using a spectral-domain interferometer is proposed and realized. Optical path differences can be discerned by inserting a correction glass piece into the measurement path, thus increasing the measurement optical path length. To verify the proposed method, 0.2-mm-thick N-BK7 glass was used as a sample, with physical thickness and group refractive index measurements conducted according to three different correction glass elements with corresponding nominal thicknesses of 3.0 mm, 3.5 mm, and 4.0 mm. Through uncertainty evaluations according to the correction glass used, the physical thicknesses of the sample were found to be in good agreement within measurement uncertainties of less than 100 nm, results comparable to those of previous works which did not use any correction glass.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Optical interferometers inherently measure the optical thicknesses of optically transparent materials with high precision as a typical non-contact and non-destructive method instead of measuring the physical thickness [1–13]. To decouple the physical thickness and group refractive index from the optical thickness, a measurement method using a reflection-type spectral-domain interferometer was proposed in 2010, and both the physical thickness and group refractive index of a silicon wafer were successfully measured [14]. The simultaneous measurement of the physical thickness and group refractive index was realized by analyzing multiple interference spectra acquired before and after sample insertion into the measurement arm. The measurement uncertainty of the proposed method in the previous work was improved twenty-fold using a femtosecond pulse laser with a spectral bandwidth ten times wider [15], with a measurement system for the thickness profile and group refractive index distribution of a silicon wafer with a diameter of 100 mm also implemented [16]. The fundamental principle for the simultaneous measurement of the physical thickness and group refractive index was also applied to a transmission-type spectral-domain interferometer with vibration-insensitive characteristics. With the transmission-type spectral-domain interferometer, both the physical thickness profile and group refractive index distribution of a 1 m large bare glass substrate were successfully measured even under a harsh environment during a high-speed transport condition [17]. In addition, various measurement methods of the physical thickness and refractive index of a single substrate based on a spectral-domain interferometer have been reported [18–20].

The simultaneous measurement methods of the physical thickness and refractive index proposed in previous works use three different optical path differences (OPDs) [15–17]; one (OPD_{1}) of the three OPDs is obtained without a sample, while the other OPDs (OPD_{2}, OPD_{3}) are obtained with a sample in the measurement path. The OPD_{2} and OPD_{3} correspond to double the optical thickness of a sample and the OPD between a reference beam and a measurement beam transmitted directly through a sample, respectively. Two OPDs but not OPD_{1} vary with the sample thickness, becoming equal at a specific value of the sample thickness. In this case, the two OPDs cannot be distinguished individually, meaning that the sample thickness cannot be measured. Hence, despite its advantages of our previous works, it inherently has the non-measurable thickness problem.

In this study, a method to resolve the non-measurable thickness problem is proposed and realized by inserting a correction glass into the measurement path. It leads that two overlapped OPDs can be clearly separated from each other. Of course, there can be a few methods to resolve the non-measurable thickness problem, such as the use of a delay line in one path of the interferometer. Among them, the use of a correction glass has strengths in terms of simplicity. Also, the proposed method does not degrade the thickness measurement performance and can be easily applied to our previous works. More importantly, the influence of the correction glass on the physical thickness and group refractive index of the sample glass is completely cancelled out in principle, meaning that the accurate information about the physical thickness, group refractive index, and alignment conditions of the correction glass is unnecessary in the proposed method. To check for the effects of the insertion of a correction glass on the measurement uncertainty, uncertainty evaluations of the measurement of the physical thickness of a 0.2-mm-thick glass of the type highly required recently in smart display devices were performed for three correction glasses of different thicknesses. The proposed method is applicable to various types of measurement methods based on spectral-domain interferometers by resolving the fundamental problem which occurs during the simultaneous measurement of the physical thickness and group refractive index.

## 2. Measurement principle

Figure 1 presents a schematic diagram of the reference and measurement paths in a transmission-type optical interferometer. For a clear mathematical expression of OPDs, the measurement path length *l*_{m} is assumed to be longer than the reference path length *l*_{r}. The physical thickness of a sample, T, can be determined by a combination of three different types of OPDs (denoted here as OPD_{1}, OPD_{2}, and OPD_{3}) measured through a Fourier analysis of the interference spectra acquired before and after sample insertion into the measurement path [15–17], where OPD_{1} is the OPD between a reference beam(A) and a measurement beam(B) without a sample, OPD_{2} is the OPD between a measurement beam(B`) transmitted directly through the sample and a measurement beam(C) transmitted through the sample after reflections inside the sample, and OPD_{3} is the OPD between the reference beam(A) and the measurement beam(B`). According to these definitions, the three OPDs are expressed as Eqs. (1) – (3). In addition, both the physical thickness and group refractive index can be calculated using Eqs. (4) – (5) with all three OPDs.

According to Eqs. (2) and (3), both OPD_{2} and OPD_{3} depend on the physical thickness of the sample, as depicted in Fig. 2(a). Specifically, each OPD is linearly proportional to T, and the slopes of OPD_{2} and OPD_{3} with reference to T are 2·N and N – 1, respectively. Theoretically, OPD_{3} when T = 0 results in OPD_{1} according to Eq. (3), which corresponds to the *y*-intercept in the OPD_{3} graph. Due to the different slopes, two OPD lines cross at a non-measurable thickness, T_{c}, which can be expressed as Eq. (6). In other words, if the sample thickness T is identical to T_{c}, it is difficult to determine the physical thickness and the group refractive index of the sample because OPD_{2} and OPD_{3} cannot be identified individually.

*ν*in the full-width at half-maximum (FWHM) about central frequency

*ν*

_{0}. If the spectral profile of the light source is Gaussian, because of the Fourier-transform relation, Δ

*τ*which is the FWHM in the Fourier-transformed amplitude is determined by the simple inverse relation Δ

*τ*= 1/Δ

*ν*. In our case, Δ

*ν*and

*ν*

_{0}are approximately 6.25 THz (∼50 nm in wavelength) and 193 THz (∼1550 nm in wavelength), respectively. As a result, ΔOPD corresponding to the FWHM in OPD scale is calculated by

*c*·Δ

*τ*and found to be approximately 48 µm, where

*c*is the speed of light in vacuum. Considering ΔOPD, there is a certain range of thickness that cannot be measured around T

_{c}, which is defined as the ‘non-measurable thickness range’ in this study. To specify the non-measurable thickness range, the required minimum difference between OPD

_{2}and OPD

_{3}to identify each other clearly is defined as

*ε*, as depicted in Fig. 2(b). According to Fig. 2(b), the non-measurable thickness range can be determined to be the thickness range specified using two points of intersection when |OPD

_{2}- OPD

_{3}| equals to

*ε*. Using Eqs. (2) and (3), |OPD

_{2}- OPD

_{3}| can be expressed as Eq. (7) depending on the range of T. As shown in Fig. 2(a), the difference between two OPDs decreases as T increases in the region of 0 ≤ T ≤ T

_{c}, which means that the slope to T within this region in Fig. 2(b) should be negative. On the other hand, the slope to T in the region of T ≥ T

_{c}is positive because the difference increases as T increases. Using Eq. (7), the thickness values at two intersecting points can be derived, which result in (OPD

_{1}–

*ε*)/(N + 1) and (OPD

_{1}+

*ε*)/(N + 1), respectively. As a result, half-width of non-measurable thickness range around T

_{c}, which is defined as ΔT, is expressed as Eq. (8) using Eq. (6). In this study, the value of ΔOPD was used for

*ε*because two OPDs can be fully distinguished from each other with a separation distance greater than ΔOPD.

_{s}and a group refractive index of N

_{s}is inserted into the measurement path, as shown in Fig. 1. After inserting the correction glass, OPD

_{1}in Eq. (1) increases by ΔOPD

_{1}corresponding to T

_{s}·(N

_{s}– 1), which leads to OPD

_{1,s}, as expressed in Eq. (9). According to Eq. (3), OPD

_{3}also increases by ΔOPD

_{1}.

Graphs of |OPD_{2} - OPD_{3}| before and after the insertion of the correction glass are presented in Fig. 3. Due to the increase of OPD_{3} by the insertion of the correction glass, the previous graph is shifted to the right, which results in a change of the non-measurable thickness from T_{c} to T_{c,s}. As a result, the sample thickness T can be measured from the area outside of the non-measurable thickness range, as depicted in Fig. 3. In addition, according to Eq. (4), the influence of the insertion of the correction glass on thickness determination is cancelled out because there remains only sample glass parameters in OPD_{3} – OPD_{1,s}, not the correction glass parameters.

## 3. Experimental setup and measurement results

Figure 4(a) presents the optical configuration of the proposed thickness measurement system, which was implemented in the form of a Mach-Zehnder interferometer with two beam splitters and two mirrors. As a light source, a superluminescent diode laser with a center wavelength of 1550 nm and a spectral bandwidth of 50 nm in the FWHM was used, and a collimated beam with a 0.76 mm diameter through a fiber collimator is incident on a beam splitter (BS1). The reference path is a counterclockwise path from BS1 to the other beam splitter (BS2) through a mirror (M1), and the measurement path is a clockwise path from BS1 to BS2 through the other mirror (M2). In the middle of the vertical reference path, a reference path bracket is installed, as shown in Fig. 4(a), and a through hole with a diameter of 12 mm exists inside the reference bracket, as shown on the left in Fig. 4(b). The reference path bracket is used to check whether or not OPD_{2} and OPD_{3} overlap by eliminating only OPD_{3} through the reference beam block. On the other hand, in the middle of the vertical measurement path, a measurement path bracket is installed, as shown in Fig. 4(a). Among the four slots from A to D inside the measurement path bracket, as shown on the right in Fig. 4(b), the three slots of A, B, and C contain correction glasses with nominal thicknesses of 3.0 mm, 3.5 mm, and 4.0 mm, respectively, while slot D is a through hole. As depicted in Fig. 4(a), a sample is placed in the middle of the horizontal measurement path. Two beams which propagate along the reference and measurement paths meet at BS2, and the interference spectra generated by the two beams are acquired using a spectrometer. Every interference spectrum is sampled at 512 points from 1508.01 nm to 1596.44 nm. Figure 4(c) shows the experimental setup of the proposed measurement system presented in Fig. 4(a). All optical components were fixed on an 8-mm-thick base plate for structural stability, and the base plate was assembled inside a housing.

For a feasibility test of the proposed method using the thickness measurement system shown in Fig. 4(c), an N-BK7 glass with a nominal thickness of 0.2 mm was used as a test sample. The OPD_{1} which causes the sample thickness to fall into the non-measurable thickness range is calculated and found to be approximately 500 µm according to Eq. (6). Through an initial system alignment step, OPD_{1} was set to a certain value close to the calculated one. After the completion of the system alignment step, OPD_{1} was measured consecutively 100 times, which resulted in an average of 518.391 µm and a standard deviation of 13 nm. Figure 5(a) shows the interference spectrum acquired after sample insertion under such conditions without a correction glass, and the Fourier-transformed amplitude of Fig. 5(a) was obtained as shown in Fig. 5(b). Using the reference path bracket, it was confirmed that OPD_{2} and OPD_{3} were perfectly overlapped in the form of a single peak.

As explained in chapter 2, *ε* was determined to be approximately 50 µm, which is the FWHM of an OPD signal at the Fourier-transformed amplitude. This was utilized to establish the non-measurable thickness range. To measure the sample thickness within the non-measurable thickness range, a 4-mm-thick correction glass in slot C of the measurement path bracket was inserted into the middle of the measurement path, which resulted in an increase in OPD_{1}. OPD_{1,s} was measured 100 times consecutively before and after the sample measurement step, with the average OPD_{1,s} calculated to be 2579.622 µm and 2579.641 µm. Because the change from OPD_{1} to OPD_{1,s} is reflected in the change of OPD_{3} according to Eq. (3), the OPD_{3} signal can be separated from the OPD_{2} signal. Figure 6(a) presents an interference spectrum acquired when inserting a 0.2-mm-thick sample and a 4-mm-thick correction glass; the Fourier-transformed amplitude of Fig. 6(a) is depicted in Fig. 6(b). As shown in Fig. 6(b), the OPD_{3} signal was shifted further to the right, unlike in Fig. 5(b), and both OPDs could finally be measured individually. Through 100 consecutive measurements of OPD_{2} and OPD_{3} after the insertion of the correction glass, the sample thickness was measured and found to be 212.459 µm on average with a standard deviation of 26 nm. To compensate for the drift effect of OPD_{1,s} when determining the thickness, the average of two OPD_{1,s} values measured before and after the sample measurement step was used as OPD_{1,s}. According to Eq. (5), the group refractive index was measured and found to be 1.520 on average with a standard deviation of 1×10^{−4}.

## 4. Discussion

To resolve the non-measurable thickness range problem which occurs when using spectral-domain interferometers, a correction glass with a nominal thickness of T_{s} and a group refractive index of N_{s} is inserted into the measurement path, which results in an increase in OPD_{3} by T_{s}·(N_{s} - 1). The amount of the increase in OPD_{3} should be greater than *ε*, corresponding to the minimum separation interval between OPD_{2} and OPD_{3}. The non-measurable thickness ranges calculated using three correction glasses with different thicknesses are clearly visualized in Fig. 7 in consideration of the initial OPD_{1} of 0.5 mm and the glass sample used. In Fig. 7, the minimum measurable thickness T_{min} can be theoretically estimated on the basis of the FWHM of an OPD signal, and the maximum measurable thickness T_{max} is determined through the wavelength resolution of the spectrometer in use. In each case, it was confirmed that the sample thickness of around 0.2 mm was far enough away from the calculated non-measurable thickness range. To check the consistency of the thickness values measured using different correction glasses under the circumstances in which sample thickness measurements become possible, two other correction glasses with nominal thicknesses of 3.0 mm and 3.5 mm were used to measure the thickness of the same sample. All physical thickness and group refractive index measurement results using three different correction glasses, including the 4-mm-thick correction glass, are summarized in Table 1.

In order to evaluate the agreement in the thickness measurement results using different correction glasses quantitatively, uncertainty evaluations of thickness measurements based on the proposed method were conducted. As with the uncertainty evaluation procedures reported in previous works, the combined measurement uncertainty of the sample thickness T can be estimated through uncertainty calculations of individual OPDs used to measure the physical thickness according to Eq. (4) [15,17]. As major factors affecting the OPD measurement uncertainties, the DFT algorithm and measurement repeatability can be considered. Besides these factors, the refractive index of air and the spectrometer performance metrics of the wavelength accuracy, wavelength repeatability, and thermal drift can also be taken into account as minor factors. Tables 2 to 4 present the uncertainty evaluation results according to correction glasses with thicknesses of 4.0 mm, 3.5 mm, and 3.0 mm, respectively. For each case, the uncertainty calculation procedure is performed in exactly the same way. Also, the uncertainty of each OPD is derived by taking the square root of the total sum of all squared uncertainty values from the factors mentioned earlier. For example, the uncertainty (=55 nm) of OPD_{1} in Table 2 is calculated by using the uncertainty (=52 nm) by DFT algorithm, the uncertainty (=13 nm) by measurement repeatability, and the uncertainty (=3 nm) by the refractive index of air. Finally, the combined uncertainty of T is calculated using the uncertainties of all three OPDs and the correlation coefficients, as reported in the previous work [15]. In these uncertainty tables, the uncertainty values by minor factors were omitted to emphasize only the influences by the major factors because the contribution to the combined measurement uncertainty is negligible and the uncertainty values are identical regardless of the correction glass thickness. In addition, because the effect of the correction glass on the physical thickness measurement of the sample glass is cancelled out according to Eq. (4), the uncertainty sources by the correction glass weren’t considered in uncertainty evaluation of Table 2 to 4. According to the uncertainty evaluation results, the combined measurement uncertainties when using the three different correction glasses were calculated to be 83 nm, 83 nm, and 84 nm, respectively, which shows that the measurement performance does not deteriorate in comparison with the outcomes in previous works, even if a correction glass is inserted. Considering the combined measurement uncertainties, the thickness measurement results were in good agreement within each measurement uncertainty value, as shown in Fig. 8. Also, all the sample thicknesses measured using three different correction glasses are included within the nominal thickness range of 0.2 mm ± 0.025 mm in consideration of the thickness tolerance of ± 0.025 mm provided from manufacturer. Therefore, when using different correction glasses to deal with various samples with different thicknesses, it is believed that good measurement reliability between sample thicknesses measured using different correction glasses is secured. Here, the 0.2-mm-thick glass can be the optimal sample to identify the marginal performance of the proposed method because the uncertainty of the OPD measurement increases dramatically when the OPD value decreases [15].

## 5. Summary

In this paper, an optical method to resolve the non-measurable thickness problem which arises during thickness measurements using spectral-domain interferometers was proposed and realized. Specifically, overlapped OPDs within a certain range of the sample thickness can be separated from each other by inserting a correction glass into the measurement path, which makes the sample thickness measurable. The proposed thickness measurement system was implemented in the form of a transmission-type interferometer, and a reference path bracket for blocking the reference beam was installed in the middle of the reference path to check whether or not OPD_{2} and OPD_{3} overlapped. In the measurement path, a measurement path bracket mounting three correction glasses with nominal thicknesses of 3.0 mm, 3.5 mm, and 4.0 mm was installed to change the non-measurable thickness range. The minimum separation interval between OPD_{2} and OPD_{3} for determining the half-width of the non-measurable thickness range was set as the FWHM of a signal corresponding to a certain OPD in the Fourier-transformed amplitude of the interference spectrum, and the half-width of the non-measurable thickness range was calculated to be approximately 20 µm. By applying three correction glasses with different thicknesses of 3.0 mm, 3.5 mm, and 4.0 mm, the corresponding physical thicknesses of the sample with a nominal thickness of 0.2 mm were measured to be 212.445 µm, 212.426 µm, and 212.459 µm on average through 100 consecutive measurements. The combined thickness measurement uncertainty was evaluated to be less than 100 nm, indicating that the thickness measurement results corresponding to different correction glasses were in good agreement within the measurement uncertainties.

## Funding

Korea Research Institute of Standards and Science (19011044).

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