Abstract

We developed an optical interferometric probe for measuring the geometrical thickness and refractive index of silicon wafers based on a Fizeau-type spectral-domain interferometer, as realized by adopting the optical fiber components of a circulator and a sheet-type beam splitter. The proposed method enables us to achieve a much simpler optical composition and higher immunity to air fluctuations owing to the use of fiber components and a common-path configuration as compared to a bulk-type optical configuration. A femtosecond pulse laser having a spectral bandwidth of 80 nm at a center wavelength of 1.55 µm and an optical spectrum analyzer having a wavelength uncertainty of 0.02 nm were used to acquire multiple interference signals in the frequency domain without a mechanical phase-shifting process. Among the many peaks in the Fourier-transformed signals of the measured interferograms, only three interference signals representing three different optical path differences were selected to extract both the geometrical thickness and group refractive index of a silicon wafer simultaneously. A single point on a double-sided polished silicon wafer was measured 90 times repetitively every two seconds. The geometrical thickness and group refractive index were found to be 476.89 µm and 3.6084, respectively. The measured thickness is in good agreement with that of a contact type method within the expanded uncertainty of contact-type instruments. Through an uncertainty evaluation of the proposed method, the expanded uncertainty of the geometrical thickness was estimated to be 0.12 µm (k = 2).

© 2014 Optical Society of America

1. Introduction

Thickness is one of the essential measurable quantities for quality control in a number of precision industries. To determine the thicknesses of mechanical parts, contact-type methods using various types of styluses have been widely used owing to their easy operation and handling [1]. However, contact-type methods can seriously damage surfaces during the measurements, and their slow measurement speed may be another weak point. Therefore, contact-type methods are generally not feasible for on-line inspections of semiconductors.

Optical interferometry is a good candidate as a noncontact-type method given its high precision [25]. To extract the geometrical thickness from an optical thickness value determined by analyzing interference signals, the refractive index of the silicon wafer should be known. When the refractive index is unknown or given with low accuracy, the measurement precision becomes practically limited. Moreover, optical interferometers using monochromatic light require a substantial amount of time to determine the optical thickness due to the required phase-shifting technique. To overcome these problems, many researches based on spectral domain analysis have been done for decades [6, 7]. Recently, the Korea Research Institute of Standards and Science (KRISS) proposed and realized a Michelson type interferometer which uses a femtosecond-pulse laser [8]. The proposed method can measure the geometrical thickness and group refractive index simultaneously. In addition, a spectral domain analysis can offer high-speed measurements [9]. With the optical properties of the optical comb of a femtosecond-pulse laser, the measurement uncertainty of the geometrical thickness can be less than 0.1 µm (k = 2) [10]. The lateral profiles of the geometrical thickness and group refractive index were also measured in the range of 100 mm along the center line of a silicon wafer [11]. Even if these methods have numerous advantages [811], they are not easy to apply in industry due to the complexity of the optical system and the optical alignments.

In this paper, we propose and realize a novel optical probe based on Fizeau-type interferometry to measure the geometrical thickness and refractive index of a silicon wafer simultaneously. To work with various applications, the optical probe was modified to realize a simple layout with a few optical components. Bulk-type optical components were replaced with optical fiber components, which can improve the signal-to-noise ratio, as the light propagates through optical fibers and fiber components with low loss. It also offers easy alignment, handling and maintenance. Given that the Fizeau-type interferometer layout has a common path configuration, the effects caused by air flow fluctuations are minimized. We can also adopt multi-mode fiber in the detection part to ensure robustness against external vibration. Because the proposed system is compact in size, a multiple-array system including several probes can be realized in a limited space for high productivity. For feasibility tests, 90 repeated measurements of the geometrical thickness and refractive index of a silicon wafer were made at a single point. To verify the geometrical thickness values, they were compared with those obtained by a contact-type metrological standard instrument.

2. Basic principle and experimental layout

Figure 1 shows the general layout of the Michelson’s type optical interferometer. The light emitted by the light source in the figure is divided by a beam splitter. The divided lights are reflected from a reference mirror and a measurement mirror, after which they are combined at the beam splitter. The interference signal is observed at a detector. This process can be expressed by Eq. (1).

I(z)=12I0[1+cos(2πcdf)]=12I0[1+cosφ(z)]
Here, I(z) is the interference intensity, I0 is the background intensity, c is the speed of light in a vacuum, f is the optical frequency, d is the optical path difference, which is a product of the refractive index of the medium N and the geometrical path difference 2z. Lastly, φ(z) is a phase value representing 4π∙Nzf/c.

 

Fig. 1 Optical layout of the Michelson’s type interferometer.

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A spectral domain interferometer is a well-known means of determining an optical path difference in a short time because it does not require a phase-shifting technique. According to Eq. (1), when a light source has a broad spectral bandwidth, a period of the interference spectrum in the spectral domain becomes d/c. The optical path difference can be calculated as d = c·P with P determined by the Fourier transform. The optical path difference can also be obtained from the phase slope of (f)/df by Eq. (2) even if the spectral distribution of the light source is not symmetric [9].

d=c2π|dφ(f)df|

Figures 2(a) and 2(b) show the optical layout of the proposed optical probe based on Fizeau-type interferometry. The optical probe part is movable and consists of a collimating lens and a sheet-type beam splitter, as shown as Fig. 2(d). Because the purpose of our experimental setup is a feasibility check and performance evaluation, it was constructed using commercial optical components having a diameter of 25.4 mm. However, it can be miniaturized easily using small-sized optical components. The light source was a femtosecond pulse laser (M-Comb, MenloSystems) having a spectral bandwidth of about 80 nm at a center wavelength of 1550 nm. The light emitted from the light source traveled into the interferometer part through a circulator (C). The light was collimated by a collimating lens (CL), and then reflected from a sheet-type beam splitter (BP108, Thorlabs), the front and back surfaces of a silicon wafer, and a mirror (M). Among the several interference patterns generated between reflected lights, only three interference patterns were selected to determine the geometrical thickness (T) and group refractive index (N) of the silicon wafer using Eqs. (3a) and (3b). These selected interference patterns contain information about the optical path differences of OPD1, OPD2, and OPD3, which represent L0 (the path between BS and M without a silicon wafer), L1 + N·T + L2 or L0-T + N·T (the path between BS and M through the silicon wafer), and N·T (the path between the front surface and back surface of the silicon wafer), respectively. The reflected lights were combined, after which they traveled to the spectrum analyzer through the circulator with low loss of optical power. To analyze the interference patterns, we adopted an optical spectrum analyzer (AQ6370C, Yokogawa) with a wavelength uncertainty of 0.02 nm. Figure 2(c) shows an image of the optical interferometer part in the experimental setup.

 

Fig. 2 Optical layouts and images of the Fizeau-type interferometer used to measure the geometrical thickness and group refractive index of a silicon wafer (C: circulator, CL: collimating lens, BS: sheet-type beam splitter, M: mirror); (a) optical layout without a silicon wafer, (b) optical layout with a silicon wafer, (c) Top view image of the experimental setup, and (d) Image of the optical probe part.

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T=OPD1OPD2+OPD3
N=OPD3/T

3. Experimental results and discussion

In the experiments, a double-sided polished silicon wafer having the nominal thickness of 475 µm ± 25 µm was used as a target. It was measured 90 times repeatedly with a time interval of 2 s. The optical path differences of L0 were measured 90 times as well before inserting the silicon wafer into the beam path, as shown in Fig. 2(a). To calculate the geometrical thickness using Eq. (3a), L0 was determined in advance by averaging the measured values.

Figures 3(a) and 3(b) show the interference spectrum related to L0 and its Fourier transformed data. Before Fourier transform, the horizontal axes of Figs. 3(a) and 3(c) were changed from wavelength to frequency. The P1 in Fig. 3(b) represents the period of the interference spectrum in a spectral domain. The optical path difference of L0 can be calculated by Eq. (2). To get phase slope of (f)/df, the desired peak located in positive time domain is selected with a sampling window, and then inverse-Fourier transformed. The phase φ(f) can be extracted by taking the imaginary part of the logarithmic function of the inverse-Fourier transformed result. Finally the phase slope can be calculated by the linear fit of φ(f) [10].

 

Fig. 3 Interference spectra and its Fourier transform: (a) interference spectrum obtained without a silicon wafer, (b) Fourier transformed data of the interference spectrum (a), (c) interference spectrum obtained with a silicon wafer, and (d) Fourier transformed data of the interference spectrum (c).

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Figure 3(c) shows interference spectrum generated by the superposition of several optical path differences of L1, L1 + N·T, N·T, and L0-T + N·T. To extract the each optical path difference, the interference spectrum of Fig. 3(c) was Fourier transformed in the same manner as the previous example. Figure 3(d) shows several peaks in the Fourier domain; these peaks can be clarified by blocking beam paths in front of the silicon wafer or the mirror in Fig. 2(b). For example, when blocking the beam path between the silicon wafer and the mirror, the optical path differences of L1, N·T and L1 + N·T can be observed. In Fig. 3(d), the peaks of P2, P3, P4 and P5 contain information about the optical path differences of L1, L0-T + N·T, L1 + N·T, and N·T. In Fig. 3(d), there are also observed some harmonics and small peaks. The small peaks and harmonics in Figs. 3(b) and 3(d) may be caused by the spectral distribution of the light source, its harmonics, and other optical paths such as L2 or N·T + L2. This is likely one of the drawbacks of this method. Therefore, each peak should be identified carefully so as not to select a wrong peak. In our experiments, the interference spectra were acquired with 8196 sampling points within the wavelength range of 1500 nm to 1600 nm. Therefore, the wavelength sampling interval of the spectra became 0.012 nm. To improve the precision in the Fourier domain, a zero padding technique was utilized with a total number of 223.

The mean geometrical thickness of the silicon wafer was 476.89 μm with a standard deviation of 86 nm. To double-check the value, it was also measured 10 times repeatedly by a contact-type standard instrument being traceable to the length standard. This instrument was developed by KRISS and has served as a national standard instrument for measuring the geometrical thicknesses of silicon wafers in Korea. The mean value measured by the contact-type instrument was 476.99 μm. Measured values obtained by both methods were in good agreement within the combined measurement uncertainty of the contact-type standard instrument, 0.1 μm (k = 2). In addition, the group refractive index of the silicon wafer was estimated to be 3.6084.

Table 1 shows the measurement uncertainty components and values [12]. The major uncertainty component was related to the discrete Fourier transform (DFT) algorithm which is used to obtain optical path differences from the interference spectra. The other dominant factor was repeatability of the measurement, which was evaluated from the 90 repeated measurements of each optical path difference. When calculating the geometrical thickness using Eq. (3), the three terms of OPD1, OPD2, and OPD3 were found to be dependent on each other. Therefore, it was necessary to consider the coupling effect on these parameters [10, 13]. Finally, the combined measurement uncertainty was estimated to be 0.12 μm (k = 2) when measuring a silicon wafer with a geometrical thickness of 476.89 μm. When the silicon wafer is tilted during the measurement, the optical power delivered to the detection part can be reduced, introducing addition measurement error. If a multi-mode fiber is used between a circulator port and the collimating lens, a certain level of optical power can be obtained even under vibration conditions. According to a previous study by the authors [11], the measurement error caused by a tilting angle of the silicon wafer of 2° was estimated to be 21 nm. Although this uncertainty source is also considered, the expanded uncertainty will increase slightly. However, the interferometric probe still can be used effectively for thickness measurement of silicon wafers.

Tables Icon

Table 1. Uncertainty Evaluation of T = 476.89 μm

4. Summary

An optical probe based on Fizeau-type interferometry was proposed and realized with a simple layout. It can measure the geometrical thickness and group refractive index of a silicon wafer simultaneously. To realize the simple layout, the original bulk-type optical components were replaced with optical fiber components. Moreover, the proposed optical probe can reduce the optical power loss, resulting in easy alignments and handling for industrial use. Because the Fizeau-type interferometer layout uses a common path to a sheet-type beam splitter, the effect of air flow fluctuations can be minimized. Owing to the compact size of the proposed probe, a multiple array system including several probes can easily be realized.

For a feasibility test, 90 repeated measurements of the geometrical thickness were made. The mean value of the geometrical thickness and the group refractive index of the silicon wafer were 476.89 μm and 3.6084, respectively. The expanded uncertainty of the geometrical thickness was 0.12 μm (k = 2). To double-check the geometrical thickness value, it was also measured by a contact-type standard instrument developed by KRISS. Both obtained values of the geometrical thickness were in good agreement within the expanded uncertainty of the contact-type instrument. Although the proposed method has a simple layout, it is capable of performing similarly to the tools developed in our previous research works. Moreover, the proposed optical probe is applicable to the display and semiconductor industries.

Acknowledgments

This work was supported by National Research Council of Science & Technology through the Convergence Practical Application Program (Grant 13-10-KRISS).

References and links

1. M. Kimura, Y. Saito, H. Daio, and K. Yakushiji, “A new method for the precise measurement of wafer roll off of silicon polished wafer,” Jpn. J. Appl. Phys. 38(1A), 38–39 (1999). [CrossRef]  

2. 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]  

3. 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]  

4. H.-J. Lee and K.-N. Joo, “Optical interferometric approach for measuring the geometrical dimension and refractive index profiles of a double-sided polished undoped Si wafer,” Meas. Sci. Technol. 25(7), 075202 (2014). [CrossRef]  

5. G. D. Gillen and S. Guha, “Use of Michelson and Fabry-Perot interferometry for independent determination of the refractive index and physical thickness of wafers,” Appl. Opt. 44(3), 344–347 (2005). [CrossRef]   [PubMed]  

6. M. Suematsu and M. Takeda, “Wavelength-shift interferometry for distance measurements using the Fourier transform technique for fringe analysis,” Appl. Opt. 30(28), 4046–4055 (1991). [CrossRef]   [PubMed]  

7. V. N. Kumar and D. N. Rao, “Using interference in the frequency domain for precise determination of thickness and refractive indices of normal dispersive materials,” J. Opt. Soc. Am. B 12(9), 1559–1563 (1995). [CrossRef]  

8. 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]  

9. J. Jin, J. W. Kim, C.-S. Kang, J.-A. Kim, and S. Lee, “Precision depth measurement of through silicon vias (TSVs) on 3D semiconductor packaging process,” Opt. Express 20(5), 5011–5016 (2012). [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]  

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. Guide to the expression of uncertainty in measurement, (International organization for standardization, 1993).

13. G. Nam, C.-S. Kang, H.-Y. So, and J. Choi, “An uncertainty evaluation for multiple measurements by GUM, III: using a correlation coefficient,” Accredit. Qual. Assur. 14(1), 43–47 (2009). [CrossRef]  

References

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  1. M. Kimura, Y. Saito, H. Daio, and K. Yakushiji, “A new method for the precise measurement of wafer roll off of silicon polished wafer,” Jpn. J. Appl. Phys. 38(1A), 38–39 (1999).
    [Crossref]
  2. 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]
  3. 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]
  4. H.-J. Lee and K.-N. Joo, “Optical interferometric approach for measuring the geometrical dimension and refractive index profiles of a double-sided polished undoped Si wafer,” Meas. Sci. Technol. 25(7), 075202 (2014).
    [Crossref]
  5. G. D. Gillen and S. Guha, “Use of Michelson and Fabry-Perot interferometry for independent determination of the refractive index and physical thickness of wafers,” Appl. Opt. 44(3), 344–347 (2005).
    [Crossref] [PubMed]
  6. M. Suematsu and M. Takeda, “Wavelength-shift interferometry for distance measurements using the Fourier transform technique for fringe analysis,” Appl. Opt. 30(28), 4046–4055 (1991).
    [Crossref] [PubMed]
  7. V. N. Kumar and D. N. Rao, “Using interference in the frequency domain for precise determination of thickness and refractive indices of normal dispersive materials,” J. Opt. Soc. Am. B 12(9), 1559–1563 (1995).
    [Crossref]
  8. 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]
  9. J. Jin, J. W. Kim, C.-S. Kang, J.-A. Kim, and S. Lee, “Precision depth measurement of through silicon vias (TSVs) on 3D semiconductor packaging process,” Opt. Express 20(5), 5011–5016 (2012).
    [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]
  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. Guide to the expression of uncertainty in measurement, (International organization for standardization, 1993).
  13. G. Nam, C.-S. Kang, H.-Y. So, and J. Choi, “An uncertainty evaluation for multiple measurements by GUM, III: using a correlation coefficient,” Accredit. Qual. Assur. 14(1), 43–47 (2009).
    [Crossref]

2014 (1)

H.-J. Lee and K.-N. Joo, “Optical interferometric approach for measuring the geometrical dimension and refractive index profiles of a double-sided polished undoped Si wafer,” Meas. Sci. Technol. 25(7), 075202 (2014).
[Crossref]

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 (3)

2010 (1)

2009 (1)

G. Nam, C.-S. Kang, H.-Y. So, and J. Choi, “An uncertainty evaluation for multiple measurements by GUM, III: using a correlation coefficient,” Accredit. Qual. Assur. 14(1), 43–47 (2009).
[Crossref]

2005 (1)

2003 (1)

1999 (1)

M. Kimura, Y. Saito, H. Daio, and K. Yakushiji, “A new method for the precise measurement of wafer roll off of silicon polished wafer,” Jpn. J. Appl. Phys. 38(1A), 38–39 (1999).
[Crossref]

1995 (1)

1991 (1)

Chen, L.

Choi, J.

G. Nam, C.-S. Kang, H.-Y. So, and J. Choi, “An uncertainty evaluation for multiple measurements by GUM, III: using a correlation coefficient,” Accredit. Qual. Assur. 14(1), 43–47 (2009).
[Crossref]

Coppola, G.

Daio, H.

M. Kimura, Y. Saito, H. Daio, and K. Yakushiji, “A new method for the precise measurement of wafer roll off of silicon polished wafer,” Jpn. J. Appl. Phys. 38(1A), 38–39 (1999).
[Crossref]

De Nicola, S.

Eom, T. B.

Ferraro, P.

Gillen, G. D.

Griesmann, U.

Guha, S.

Iodice, M.

Jin, J.

Joo, K.-N.

H.-J. Lee and K.-N. Joo, “Optical interferometric approach for measuring the geometrical dimension and refractive index profiles of a double-sided polished undoped Si wafer,” Meas. Sci. Technol. 25(7), 075202 (2014).
[Crossref]

Kang, C.-S.

Kim, J. W.

Kim, J.-A.

Kimura, M.

M. Kimura, Y. Saito, H. Daio, and K. Yakushiji, “A new method for the precise measurement of wafer roll off of silicon polished wafer,” Jpn. J. Appl. Phys. 38(1A), 38–39 (1999).
[Crossref]

Kumar, V. N.

Lee, H.-J.

H.-J. Lee and K.-N. Joo, “Optical interferometric approach for measuring the geometrical dimension and refractive index profiles of a double-sided polished undoped Si wafer,” Meas. Sci. Technol. 25(7), 075202 (2014).
[Crossref]

Lee, S.

Maeng, S.

Nam, G.

G. Nam, C.-S. Kang, H.-Y. So, and J. Choi, “An uncertainty evaluation for multiple measurements by GUM, III: using a correlation coefficient,” Accredit. Qual. Assur. 14(1), 43–47 (2009).
[Crossref]

O, B.

Park, J.

Rao, D. N.

Saito, Y.

M. Kimura, Y. Saito, H. Daio, and K. Yakushiji, “A new method for the precise measurement of wafer roll off of silicon polished wafer,” Jpn. J. Appl. Phys. 38(1A), 38–39 (1999).
[Crossref]

So, H.-Y.

G. Nam, C.-S. Kang, H.-Y. So, and J. Choi, “An uncertainty evaluation for multiple measurements by GUM, III: using a correlation coefficient,” Accredit. Qual. Assur. 14(1), 43–47 (2009).
[Crossref]

Suematsu, M.

Takeda, M.

Wang, Q.

Yakushiji, K.

M. Kimura, Y. Saito, H. Daio, and K. Yakushiji, “A new method for the precise measurement of wafer roll off of silicon polished wafer,” Jpn. J. Appl. Phys. 38(1A), 38–39 (1999).
[Crossref]

Accredit. Qual. Assur. (1)

G. Nam, C.-S. Kang, H.-Y. So, and J. Choi, “An uncertainty evaluation for multiple measurements by GUM, III: using a correlation coefficient,” Accredit. Qual. Assur. 14(1), 43–47 (2009).
[Crossref]

Appl. Opt. (3)

J. Opt. Soc. Am. B (1)

Jpn. J. Appl. Phys. (1)

M. Kimura, Y. Saito, H. Daio, and K. Yakushiji, “A new method for the precise measurement of wafer roll off of silicon polished wafer,” Jpn. J. Appl. Phys. 38(1A), 38–39 (1999).
[Crossref]

Meas. Sci. Technol. (1)

H.-J. Lee and K.-N. Joo, “Optical interferometric approach for measuring the geometrical dimension and refractive index profiles of a double-sided polished undoped Si wafer,” Meas. Sci. Technol. 25(7), 075202 (2014).
[Crossref]

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 (4)

Other (1)

Guide to the expression of uncertainty in measurement, (International organization for standardization, 1993).

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Figures (3)

Fig. 1
Fig. 1 Optical layout of the Michelson’s type interferometer.
Fig. 2
Fig. 2 Optical layouts and images of the Fizeau-type interferometer used to measure the geometrical thickness and group refractive index of a silicon wafer (C: circulator, CL: collimating lens, BS: sheet-type beam splitter, M: mirror); (a) optical layout without a silicon wafer, (b) optical layout with a silicon wafer, (c) Top view image of the experimental setup, and (d) Image of the optical probe part.
Fig. 3
Fig. 3 Interference spectra and its Fourier transform: (a) interference spectrum obtained without a silicon wafer, (b) Fourier transformed data of the interference spectrum (a), (c) interference spectrum obtained with a silicon wafer, and (d) Fourier transformed data of the interference spectrum (c).

Tables (1)

Tables Icon

Table 1 Uncertainty Evaluation of T = 476.89 μm

Equations (4)

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I ( z ) = 1 2 I 0 [ 1 + cos ( 2 π c d f ) ] = 1 2 I 0 [ 1 + cos φ ( z ) ]
d = c 2 π | d φ ( f ) d f |
T = O P D 1 O P D 2 + O P D 3
N = O P D 3 / T

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