Abstract

We propose and realize a modified spectral-domain interferometer to measure the physical thickness profile and group refractive index distribution of a large glass substrate simultaneously. The optical layout was modified based on a Mach-Zehnder type interferometer, which was specially adopted to be insensitive to mechanical vibration. According to the measurement results of repeated experiments at a length of 820 mm along the horizontal axis, the standard deviations of the physical thickness and group refractive index were calculated to be 0.173 μm and 3.4 × 10−4, respectively. To verify the insensitivity to vibration, the physical thickness values were monitored at a stationary point while the glass panel was swung at an amplitude exceeding 20 mm. The uncertainty components were evaluated, and the combined measurement uncertainty became 161 nm (k = 1) for a glass panel with a nominal thickness of 0.7 mm.

© 2015 Optical Society of America

1. Introduction

The size of the glass substrates used in flat panel displays has increased, whereas panel thicknesses are declining. For example, eighth-generation glass panels can be greater than 2 m with a physical thickness of less than 0.7 mm [1]. If the physical thickness is not uniformly distributed over the entire area of the large glass panel, the fabricated pixels can be damaged during the assembling process, or the image quality can be degraded due to the effects of unwanted diffraction. Therefore, precise measurements of the thickness profile are necessary to improve the throughput.

Optical interferometry is a well-established non-contact method which is used to measure optical thicknesses with good precision [2–5 ]. However, to make full use of the merits of optical interferometers, environmental parameters must be well stabilized during the measurements and the refractive index of the medium should be known in advance or monitored in real time. However, the display manufacturing process cannot easily meet the operation conditions of optical interferometry. First, there are numerous mechanical vibration sources caused by the high-speed transport and delivery of large glass panels. In addition, large and thin glass panels can have their own vibration modes owing to their flexibility. Second, the refractive indices of glass panels can differ slightly with each production lot according to the material composition ratio. Therefore, to measure the physical thickness profile of large glass panels, alternative methods such as chromatic confocal microscopy [6,7 ] or an optical triangular method [8] are utilized, as these methods are relatively robust against vibration and do not require the refractive index values. However, such alternative methods cannot easily meet the performance levels required for recently developed large and thin glass panels in terms of the measurement resolution, repeatability, and accuracy.

In this paper, a vibration-insensitive interferometric method is proposed and realized to measure the physical thickness profile and the group refractive index distribution of a large glass panel. The proposed method was modified from previous works by the authors [9–12 ] with a transmission-type Mach-Zehnder interferometer configuration to ensure insensitivity to vibration, because it is impossible to use the previous methods under severe vibration. The advantages of the proposed method include the following: (1) it has a transmission type layout which allows the interference signal to be obtained with large swings as strong as several tens of mm; (2) the variation of the physical thickness becomes less than 0.l μm despite the fact that the swing amplitude exceeds ± 10 mm; and (3) the method can determine the physical thickness regardless of the material composition of the sample. To verify the proposed method, the physical thickness profiles of a large glass panel 1000 mm (x) × 880 mm (y) × 0.7 mm (z) in size were measured ten times consecutively at a length of 820 mm along the horizontal axis. At the same time, the group refractive index distributions were obtained. To assess the vibration-insensitive characteristics, the variation in the physical thickness was measured at a fixed position when the glass panel was swung with an amplitude of ± 20 mm at a bottom point. The proposed method determines the physical thickness with good precision when operating under a harsh vibration condition, making it feasibile for use during in-line inspections.

2. Basic principles

The spectral-domain interferometer with a broadband source gives the optical path difference as determined by analyzing the interference spectrum. The intensity distribution of the interference spectrum, I(f, L) can be expressed by Eq. (1),

I(f,L)=I0(f){1+cos(2πfLc)}
where f is the optical frequency, I 0(f) is the spectral intensity distribution of the light source, L is the optical path difference (OPD), and c is the speed of light in a vacuum. Here, I(f, L) appears to be a sinusoidal form in the frequency domain at a certain value of L, and the period equals c/L in units of Hz. The value of L can be determined according to the following steps. First, the raw interference spectrum is Fourier-transformed, after which data near a certain peak containing information about L is selected. After the inverse Fourier-transform of the selected data, the slope of the phase term pertaining to the wavenumber (2πf/c) in Eq. (1) can be calculated, resulting in the value of L [10]. Therefore, the spectral-domain interferometer allows high-speed measurements in conjunction with good precision because the OPDs can be determined by a single acquisition of the interference spectrum.

Despite the advantages of the spectral-domain interferometer, the refractive index of the medium should be known to extract the path difference from an OPD. To overcome this problem, several studies regarding simultaneous measurements of the physical thickness and group refractive index have been proposed and realized by KRISS [9–12 ]. In this paper, a modified method based on the optical layout presented in earlier work by the authors is proposed for physical thickness measurements of large glass panels, as shown in Fig. 1 . The collimated light coming from an IR broadband source travels to a beamsplitter(BS1) and is then divided into two beams. One light beam travels along a path with a length of L1 in a counterclockwise direction, and the other light travels along a path with a length of L2 in a clockwise direction, as shown in Fig. 1. Beam paths with lengths of L1 and L2 represent the reference path and the measurement path, respectively. At the spectrometer shown in Fig. 1, interference spectra are observed in the wavelength range of 1512 nm to 1600 nm.

 

Fig. 1 Optical layout of interferometer setup for the simultaneous measurement of the physical thickness and group refractive index

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When a sample is inserted into the measurement path, the light is transmitted and reflected. Light as denoted by W1 in Fig. 1 exists in the form of a beam which is transmitted directly through the sample, while light as denoted by W2 is a beam transmitted through the sample after double reflections on both surfaces of the sample. The light travelling along the reference path is defined as W0. Therefore, three different OPDs can be expressed as shown by Eqs. (2a), (2b), and (2c) ,

OPD1=L2L1
OPD2(x)=2N(x)T(x)
OPD3(x)={L2T(x)+N(x)T(x)}L1
where OPD1 is an OPD without the sample, OPD2 is an OPD between W1 and W2, and OPD3 is an OPD between W0 and W1. Using all three OPDs, both the physical thickness T(x) and the group refractive index N(x) can be calculated using Eqs. (3a) and (3b) .
T(x)=OPD2(x)2{OPD3(x)OPD1}
N(x)=OPD2(x)2T(x)
In this work, the significant difference from our previous work is the vibration-insensitive characteristic. Figure 2(a) describes the optical rays transmitted through a large glass panel when the sample swings at an angle amount of θ at pivot point O. In this simulation, we assume that the glass is a homogeneous rigid body having a uniform thickness distribution along the y-axis. Therefore, according to Snell’s law, the angle of refraction θʹ is represented as Eq. (4).
θ=sin1(sinθN)
When the sample swings, the incident beam propagates through the sample along the path with a length of Tʹ not T, as expressed in Eq. (5).

 

Fig. 2 (a) Schematic diagram of the change in the optical path caused by variations in the sample swing, and (b) thickness variation caused by the swing amplitude, u s

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T=T(1cosθ)

The transmitted beams of W1ʹ and W2ʹ in Fig. 2(a) have different paths from W1 and W2. To calculate the difference in the physical thickness caused by the swing of the sample, the changes of both OPD2 and OPD3 must be calculated in terms of T, N, and θ. The change of OPD2 resulting from the interference between W1ʹ and W2ʹ, ΔOPD2, can be determined by Eq. (6). With a constant value of L1, the change in OPD3, ΔOPD3, is also obtained from the light of W1ʹ in a similar manner. Equation (7) is used to determine ΔOPD3. With ΔOPD2 and ΔOPD3, the difference in the physical thickness, ΔT, can also be expressed in terms of T, N, and θ, as shown in Eq. (8).

ΔOPD2=2NT2Tsinθsinθ2NT
ΔOPD3=N(TT)+T(11cosθ)+Tsin(θθ)tanθ
ΔT=ΔOPD22ΔOPD3

Figure 2(b) shows the calculation result of ΔT in terms of the swing amplitude u s at the bottom point of a large glass panel (h = 880 mm) according to Eq. (8). According to Fig. 2(b), even if u s is increased to 20 mm, ΔT becomes less than 200 nm. However, in our previous work with a reflection-type configuration, the reflected light can stray owing to the large reflection angle when the sample swings considerably. Accordingly, the effect on the sample swing was significant in terms of the optical power in our previous work.

3. Experimental setup and results

Figures 3(a) and (b) show images of the thickness measurement system with a stroke of 1000 mm on the x-axis. The sample was a large glass panel 1000 mm (x) × 880 mm (y) × 0.7 mm (z) in size. A superluminescent diode (SLD) emitting at a center wavelength of 1550 nm with a bandwidth of 90 nm and an output power of 2.5 mW was adopted as a light source. The detector was a grating-based spectrometer with a sampling rate of 970 Hz. The interference spectra were obtained during synchronization with the encoder signal of the moving stage. The optical axis of a measurement probe was located 25 mm above the bottom edge of the glass panel, as depicted in Fig. 3(c). The sample was fixed using a pair of rubber pads to prevent slippage and damage during the movements. The entire measurement area was divided into two regions, A (without sample, 40 mm) and B (with sample, 820 mm), as shown in Fig. 3(c). To measure OPD1, the acquisition of the interference spectra was started 40 mm ahead of the sample. In addition, the moving velocity of the sample reached 50 mm/s in a section moving at a constant velocity with acceleration of 50 mm/s2.

 

Fig. 3 (a) Photo of the system used to measure the thickness of a large glass panel, (b) photo of the inside of the interferometer setup, and (c) experimental method to acquire the thickness profile data of a large glass panel

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The thickness profiles were measured ten times consecutively along the x-axis to evaluate the measurement repeatability. The sampling interval of the interference spectra was 1 mm, and total number of samples was 860. Figure 4(a) shows the measurement results of the physical thickness profile of region B of the glass panel shown in Fig. 3(c), at a length of 820 mm. The physical thickness profiles obtained from ten individual measurements were in good agreement, with standard deviation of 173 nm on average at every measured point. The inlet of Fig. 4(a) shows an enlarged plot of thickness profiles as an example. Figure 4(b) shows the averaged group refractive index distribution with the mean value of 1.515 and the peak-to-valley of 5.47 × 10−3.

 

Fig. 4 Results of ten consecutive measurements of a large glass panel (a) physical thickness profile, (b) group refractive index distribution

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To investigate the vibration-insensitive characteristics of the proposed system, the physical thickness was measured 10000 times repeatedly at a fixed location of the glass panel in two cases. In the first case, the glass panel was stationary, while in the second the glass panel was swung at point O shown in Fig. 5(c) . For a stationary condition, the mean value of the physical thickness was 712.262 μm with a standard deviation of 21 nm. Even if the large glass panel appears to remain still, the physical thickness values contains the variance caused by air fluctuations, stage control noise, or the vibration modes of the panel, as shown in Fig. 5 (a). On the other hand, Fig. 5(b) shows the variation in the physical thickness while swinging the glass at an initial amplitude of 20 mm. The thickness variation reached a maximum of approximately 260 nm (P.V. value) at the beginning and decreased naturally as the sample became stationary.

 

Fig. 5 Repeatable thickness measurement results with (a) no initial swing and (b) a large amount of swing, and (c) optical layout of the sample swing condition

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Table 1 shows the uncertainty components of the physical thickness measurements by the proposed method [13]. As shown in Table 1, the uncertainty of each OPD originated from the uncertainties of the discrete Fourier Transform (DFT) algorithm, the measurement repeatability, the refractive index of air, and the spectrometer. Among such factors affecting the uncertainty, the main factor was the DFT algorithm, which was utilized to determine the OPDs from the interference spectra. The uncertainty related to the DFT algorithm was evaluated by a numerical simulation. With the ideally created interference spectra, the errors of the analysis algorithm were calculated according to the OPDs. The uncertainty of the refractive index of air was estimated to exceed 10−6 under the environmental conditions of our laboratory. This value, shown in Table 1, is negligible. The uncertainty related to the spectrometer was determined in terms of the accuracy and repeatability of the wavelength and in terms of the thermal drift. Because the physical thickness is calculated arithmetically using OPD1, OPD2, and OPD3, as in Eq. (3a), the coupling effect among the three OPDs should be considered to calculate the uncertainty of the physical thickness of the glass panel at a stationary point [10]. The other important factor was the physical thickness variation of the glass panel under swing, as shown in the experimental result in Fig. 5(b), which gives a measurement uncertainty of 152 nm. As a result, in Table 1, the combined uncertainty of the physical thickness was calculated to be 161 nm (k = 1) for a glass panel with a physical thickness of 712.262 μm by using the uncertainty in swing conditions and the uncertainty at a stationary point.

Tables Icon

Table 1. Uncertainty evaluation of T = 712.262 μm

4. Summary

A modified spectral-domain interferometer capable of measuring the physical thickness profile and group refractive index distribution of a large glass panel was proposed and realized. The proposed method was operated based on the Mach-Zehnder interferometer layout to improve the vibration-insensitive characteristic under harsh environmental conditions. The advantages of the proposed method are as follows: (1) The transmission-type layout allows for the interference signal to be obtained with large swing as strong as several tens of mm, (2) the physical thickness variation is less than 0.l μm despite the fact that the swing amplitude exceeds ± 10 mm, and (3) the proposed method can determine the physical thickness regardless of the material composition of the sample.

To verify the proposed method, physical thickness profiles of a large glass panel 1000 mm (x) × 880 mm (y) × 0.7 mm (z) in size were measured ten times consecutively at a length of 820 mm along the horizontal axis. At the same time, the group refractive index distributions were obtained. The standard deviation in the physical thickness profile of the glass panel was 173 nm on average in the thickness results obtained at all measured points. The refractive index distribution varied within approximately the deviation of 5.5 × 10−3 (P.V. value). To assess the vibration-insensitive characteristic, the physical thickness variation was measured at a fixed position when the glass panel was swung with an amplitude of ± 20 mm at a bottom point. The physical thickness variation was less than 260 nm, much lower than the swing amplitude. According to the uncertainty evaluation, the overall combined uncertainty was estimated to be 161 nm (k = 1) under the swing condition. Among the many uncertainty components, the major uncertainty factors were the DFT algorithm, measurement repeatability, and the uncertainty of the thickness variation under the swing condition. It is expected that the proposed method can offer physical thickness measurements with good precision when operating under harsh vibration conditions. This capability is applicable to in-line inspections.

Acknowledgments

This work was supported by the National Research Council of Science & Technology through the Convergence Commercialization Project (CCP-13-14-KRISS).

References and links

1. H. Minami, F. Matsumoto, and S. Suzuki, “Prospects of LCD Panel Fabrication and Inspection Equipment Amid Growing Demand for Increased Size,” Hitachi Rev. 56(3), 63–69 (2007).

2. J.-A. Kim, C.-S. Kang, T. B. Eom, J. Jin, H. S. Suh, and J. W. Kim, “Quadrature laser interferometer for in-line thickness measurement of glass panels using a current modulation technique,” Appl. Opt. 53(20), 4604–4610 (2014). [CrossRef]   [PubMed]  

3. J.-A. Kim, J. W. Kim, T. B. Eom, J. Jin, and C.-S. Kang, “Vibration-insensitive measurement of thickness variation of glass panels using double-slit interferometry,” Opt. Express 22(6), 6486–6494 (2014). [CrossRef]   [PubMed]  

4. G. Coppola, P. Ferraro, M. Iodice, and S. De Nicola, “Method for measuring the refractive index and the thickness of transparent plates with a lateral-shear, wavelength-scanning interferometer,” Appl. Opt. 42(19), 3882–3887 (2003). [CrossRef]   [PubMed]  

5. J. Park, L. Chen, Q. Wang, and U. Griesmann, “Modified Roberts-Langenbeck test for measuring thickness and refractive index variation of silicon wafers,” Opt. Express 20(18), 20078–20089 (2012). [CrossRef]   [PubMed]  

6. A. Miks, J. Novak, and P. Novak, “Analysis of method for measuring thickness of plane-parallel plates and lenses using chromatic confocal sensor,” Appl. Opt. 49(17), 3259–3264 (2010). [CrossRef]   [PubMed]  

7. D. W. Zhou, T. Gambaryan-Roisman, and P. Stephan, “Measurement of water falling film thickness to flat plate using confocal chromatic sensoring technique,” Exp. Therm. Fluid Sci. 33(2), 273–283 (2009). [CrossRef]  

8. J. P. Peterson and R. B. Peterson, “Laser triangulation for liquid film thickness measurements through multiple interfaces,” Appl. Opt. 45(20), 4916–4926 (2006). [CrossRef]   [PubMed]  

9. J. Jin, J. W. Kim, C.-S. Kang, J.-A. Kim, and T. B. Eom, “Thickness and refractive index measurement of a silicon wafer based on an optical comb,” Opt. Express 18(17), 18339–18346 (2010). [CrossRef]   [PubMed]  

10. S. Maeng, J. Park, B. O, and J. Jin, “Uncertainty improvement of geometrical thickness and refractive index measurement of a silicon wafer using a femtosecond pulse laser,” Opt. Express 20(11), 12184–12190 (2012). [CrossRef]   [PubMed]  

11. J. Park, J. Jin, J. W. Kim, and J.-A. Kim, “Measurement of thickness profile and refractive index variation of a silicon wafer using the optical comb of a femtosecond pulse laser,” Opt. Commun. 305(15), 170–174 (2013). [CrossRef]  

12. J. Jin, S. Maeng, J. Park, J.-A. Kim, and J. W. Kim, “Fizeau-type interferometric probe to measure geometrical thickness of silicon wafers,” Opt. Express 22(19), 23427–23432 (2014). [CrossRef]   [PubMed]  

13. Guide to the expression of uncertainty in measurement, (International Organization for Standardization, 1993).

References

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  1. H. Minami, F. Matsumoto, and S. Suzuki, “Prospects of LCD Panel Fabrication and Inspection Equipment Amid Growing Demand for Increased Size,” Hitachi Rev. 56(3), 63–69 (2007).
  2. J.-A. Kim, C.-S. Kang, T. B. Eom, J. Jin, H. S. Suh, and J. W. Kim, “Quadrature laser interferometer for in-line thickness measurement of glass panels using a current modulation technique,” Appl. Opt. 53(20), 4604–4610 (2014).
    [Crossref] [PubMed]
  3. J.-A. Kim, J. W. Kim, T. B. Eom, J. Jin, and C.-S. Kang, “Vibration-insensitive measurement of thickness variation of glass panels using double-slit interferometry,” Opt. Express 22(6), 6486–6494 (2014).
    [Crossref] [PubMed]
  4. G. Coppola, P. Ferraro, M. Iodice, and S. De Nicola, “Method for measuring the refractive index and the thickness of transparent plates with a lateral-shear, wavelength-scanning interferometer,” Appl. Opt. 42(19), 3882–3887 (2003).
    [Crossref] [PubMed]
  5. J. Park, L. Chen, Q. Wang, and U. Griesmann, “Modified Roberts-Langenbeck test for measuring thickness and refractive index variation of silicon wafers,” Opt. Express 20(18), 20078–20089 (2012).
    [Crossref] [PubMed]
  6. A. Miks, J. Novak, and P. Novak, “Analysis of method for measuring thickness of plane-parallel plates and lenses using chromatic confocal sensor,” Appl. Opt. 49(17), 3259–3264 (2010).
    [Crossref] [PubMed]
  7. D. W. Zhou, T. Gambaryan-Roisman, and P. Stephan, “Measurement of water falling film thickness to flat plate using confocal chromatic sensoring technique,” Exp. Therm. Fluid Sci. 33(2), 273–283 (2009).
    [Crossref]
  8. J. P. Peterson and R. B. Peterson, “Laser triangulation for liquid film thickness measurements through multiple interfaces,” Appl. Opt. 45(20), 4916–4926 (2006).
    [Crossref] [PubMed]
  9. J. Jin, J. W. Kim, C.-S. Kang, J.-A. Kim, and T. B. Eom, “Thickness and refractive index measurement of a silicon wafer based on an optical comb,” Opt. Express 18(17), 18339–18346 (2010).
    [Crossref] [PubMed]
  10. S. Maeng, J. Park, B. O, and J. Jin, “Uncertainty improvement of geometrical thickness and refractive index measurement of a silicon wafer using a femtosecond pulse laser,” Opt. Express 20(11), 12184–12190 (2012).
    [Crossref] [PubMed]
  11. J. Park, J. Jin, J. W. Kim, and J.-A. Kim, “Measurement of thickness profile and refractive index variation of a silicon wafer using the optical comb of a femtosecond pulse laser,” Opt. Commun. 305(15), 170–174 (2013).
    [Crossref]
  12. J. Jin, S. Maeng, J. Park, J.-A. Kim, and J. W. Kim, “Fizeau-type interferometric probe to measure geometrical thickness of silicon wafers,” Opt. Express 22(19), 23427–23432 (2014).
    [Crossref] [PubMed]
  13. Guide to the expression of uncertainty in measurement, (International Organization for Standardization, 1993).

2014 (3)

2013 (1)

J. Park, J. Jin, J. W. Kim, and J.-A. Kim, “Measurement of thickness profile and refractive index variation of a silicon wafer using the optical comb of a femtosecond pulse laser,” Opt. Commun. 305(15), 170–174 (2013).
[Crossref]

2012 (2)

2010 (2)

2009 (1)

D. W. Zhou, T. Gambaryan-Roisman, and P. Stephan, “Measurement of water falling film thickness to flat plate using confocal chromatic sensoring technique,” Exp. Therm. Fluid Sci. 33(2), 273–283 (2009).
[Crossref]

2007 (1)

H. Minami, F. Matsumoto, and S. Suzuki, “Prospects of LCD Panel Fabrication and Inspection Equipment Amid Growing Demand for Increased Size,” Hitachi Rev. 56(3), 63–69 (2007).

2006 (1)

2003 (1)

Chen, L.

Coppola, G.

De Nicola, S.

Eom, T. B.

Ferraro, P.

Gambaryan-Roisman, T.

D. W. Zhou, T. Gambaryan-Roisman, and P. Stephan, “Measurement of water falling film thickness to flat plate using confocal chromatic sensoring technique,” Exp. Therm. Fluid Sci. 33(2), 273–283 (2009).
[Crossref]

Griesmann, U.

Iodice, M.

Jin, J.

Kang, C.-S.

Kim, J. W.

Kim, J.-A.

Maeng, S.

Matsumoto, F.

H. Minami, F. Matsumoto, and S. Suzuki, “Prospects of LCD Panel Fabrication and Inspection Equipment Amid Growing Demand for Increased Size,” Hitachi Rev. 56(3), 63–69 (2007).

Miks, A.

Minami, H.

H. Minami, F. Matsumoto, and S. Suzuki, “Prospects of LCD Panel Fabrication and Inspection Equipment Amid Growing Demand for Increased Size,” Hitachi Rev. 56(3), 63–69 (2007).

Novak, J.

Novak, P.

O, B.

Park, J.

Peterson, J. P.

Peterson, R. B.

Stephan, P.

D. W. Zhou, T. Gambaryan-Roisman, and P. Stephan, “Measurement of water falling film thickness to flat plate using confocal chromatic sensoring technique,” Exp. Therm. Fluid Sci. 33(2), 273–283 (2009).
[Crossref]

Suh, H. S.

Suzuki, S.

H. Minami, F. Matsumoto, and S. Suzuki, “Prospects of LCD Panel Fabrication and Inspection Equipment Amid Growing Demand for Increased Size,” Hitachi Rev. 56(3), 63–69 (2007).

Wang, Q.

Zhou, D. W.

D. W. Zhou, T. Gambaryan-Roisman, and P. Stephan, “Measurement of water falling film thickness to flat plate using confocal chromatic sensoring technique,” Exp. Therm. Fluid Sci. 33(2), 273–283 (2009).
[Crossref]

Appl. Opt. (4)

Exp. Therm. Fluid Sci. (1)

D. W. Zhou, T. Gambaryan-Roisman, and P. Stephan, “Measurement of water falling film thickness to flat plate using confocal chromatic sensoring technique,” Exp. Therm. Fluid Sci. 33(2), 273–283 (2009).
[Crossref]

Hitachi Rev. (1)

H. Minami, F. Matsumoto, and S. Suzuki, “Prospects of LCD Panel Fabrication and Inspection Equipment Amid Growing Demand for Increased Size,” Hitachi Rev. 56(3), 63–69 (2007).

Opt. Commun. (1)

J. Park, J. Jin, J. W. Kim, and J.-A. Kim, “Measurement of thickness profile and refractive index variation of a silicon wafer using the optical comb of a femtosecond pulse laser,” Opt. Commun. 305(15), 170–174 (2013).
[Crossref]

Opt. Express (5)

Other (1)

Guide to the expression of uncertainty in measurement, (International Organization for Standardization, 1993).

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Figures (5)

Fig. 1
Fig. 1 Optical layout of interferometer setup for the simultaneous measurement of the physical thickness and group refractive index
Fig. 2
Fig. 2 (a) Schematic diagram of the change in the optical path caused by variations in the sample swing, and (b) thickness variation caused by the swing amplitude, u s
Fig. 3
Fig. 3 (a) Photo of the system used to measure the thickness of a large glass panel, (b) photo of the inside of the interferometer setup, and (c) experimental method to acquire the thickness profile data of a large glass panel
Fig. 4
Fig. 4 Results of ten consecutive measurements of a large glass panel (a) physical thickness profile, (b) group refractive index distribution
Fig. 5
Fig. 5 Repeatable thickness measurement results with (a) no initial swing and (b) a large amount of swing, and (c) optical layout of the sample swing condition

Tables (1)

Tables Icon

Table 1 Uncertainty evaluation of T = 712.262 μm

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

I ( f , L ) = I 0 ( f ) { 1 + cos ( 2 π f L c ) }
OPD 1 = L 2 L 1
OPD 2 ( x ) = 2 N ( x ) T ( x )
OPD 3 ( x ) = { L 2 T ( x ) + N ( x ) T ( x ) } L 1
T ( x ) = OPD 2 ( x ) 2 { OPD 3 ( x ) OPD 1 }
N ( x ) = OPD 2 ( x ) 2 T ( x )
θ = sin 1 ( sin θ N )
T = T ( 1 cos θ )
Δ OPD 2 = 2 N T 2 T sin θ sin θ 2 N T
Δ OPD 3 = N ( T T ) + T ( 1 1 cos θ ) + T sin ( θ θ ) tan θ
Δ T = Δ OPD 2 2 Δ OPD 3

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