Abstract

In order to measure exactly a large thickness of glass plate with a spectrally resolved interferometer using a spectral analyzer with a low resolution of 0.5nm and a supercontinuum light source with a large bandwidth of about 300nm, a new measurement method with a spectrally resolved interferometer is proposed where a variable signal position is generated by moving a reference surface with a piezoelectric stage. It is made clear how to decide the signal position by analyzing amplitude distribution of Fourier transform of the interference signal. Through four-step measurement by using four different optical configurations with four different signal positions the thickness of glass plate can be obtained from a slope of a spectral phase distribution which does not contain the refractive index of glass plate. A small measurement error of 50nm is achieved in measuring 1mm thickness of a glass plate.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Interferometer have been widely used for thickness measurement with rapid progress of precision manufacturing industry. Measurements of thin-film thickness with spectrally resolved interferometers (SRIs) have been reported in many papers [17]. In SRIs many wavelengths contained in the light source are separated with a diffraction grating to get an interference signal distributed along wavelength. Since the interference signal is detected with a line sensor in a spectral analyzer, the maximum measurable thickness is limited by the resolution of the spectral analyzer. When a spectral analyzer with a high resolution of 0.06 nm was used, the thickness of a few millimeters was measured with measurement error of about a few hundred nanometers with the light source whose central wavelength and bandwidth were about 1300 nm and 40 nm, respectively [1,2]. Recently a portable and inexpensive spectral analyzer is widely employed to construct a SRI easily, but the resolution of the spectral analyzer is not so high. Hence a technique for extending the maximum measurable thickness in a SRI with a low resolution spectral analyzer must be proposed by analyzing the amplitude distribution of Fourier transform of the interference signal. Moreover, in many papers [14,6,7] optical path differences contained in an interference signal are extracted from peak positions in the amplitude distribution of Fourier transform of the interference signal. However the peak position does not exactly correspond to an optical path difference which contains a path in a dispersion medium. On the other hand a slop of spectral phase distribution of an interference signal along wavenumber provides a more exact optical path difference than the peak position. And a large bandwidth of a light source is better to get a small measurement error in calculating the slop of spectral phase distribution.

In this paper, in order to measure exactly a large thickness of glass plate with a SRI using a spectral analyzer with a low resolution of 0.5 nm and a supercontinuum light source with a large bandwidth of about 300 nm, a new measurement method with the SRI is proposed where the measurement range is extended by moving a reference surface with a piezoelectric transducer (PZT) stage. First, Fourier transform of an interference signal produced by a rear surface of glass plate is analyzed in details to make it clear how to extend the measurement range. This Fourier transformed signal has a large spread width in the amplitude distribution caused by dispersion effect of the glass plate. The central position of the spread width is called signal position. It is made clear that the amplitude distribution is not symmetric about the signal position. The signal position is adjusted by the position of the reference surface so that the interference signal can be detected with the low resolution spectral analyzer. The measurable maximum thickness is derived by considering the resolution of the spectral analyzer and the dispersion effect of the glass plate. Next, the measurement method is presented where four different optical configurations are used together with four different signal positions. Through this four-step measurement the thickness of the glass plate can be obtained from a slope of a spectral phase distribution which does not contain the refractive index of glass plate. Finally, in experiments the positions of the glass plate and an additional reference surface are calculated roughly from the signal positions, and a dispersion effect of the beam splitter are made clear from a spread width. And then it is confirmed that the detected values of the signal positions agree with the theoretical values. A small measurement error of 50 nm is achieved in measuring 1 mm thickness of a glass plate.

2. Interference signal with a variable signal position

Schematic configuration of a SRI is shown in Fig. 1. An interference signal generated by a rear surface of an object (OB) is analyzed to make it clear how a variable signal position is working well in a SRI with a low resolution spectral analyzer. The variable signal position is provided by changing a position Z of a reference surface (RS) with a piezoelectric transducer (PZT) stage. A beam from supercontinuum light source is divided by a beam splitter (BS) for the object and reference arms, and the two beams from the two arms are combined again by the BS to generate an interference signal. Spectral intensity I(σ) of the supercontinuum light source is shown in Fig. 2, where σ is wavenumber and the spectral range is about from 500 to 800 nm with a central wavelength of 650 nm. The OB is a glass plate of BK7 with thickness T and refractive index n(σ). The interference signal is detected by a spectral analyzer whose resolution Δλ is 0.5 nm. When the position of front surface of the OB is ZF and the position of the RS is Z, the interference signal S(σ) is expressed as

$$S(\sigma )= I(\sigma )+ I(\sigma )\cos \{ 4\pi [{Z_F} + n(\sigma )T - Z]\sigma \}, $$
where the phase π arisen in the reflection by the rear surface of the OB is ignored for the sake of simplicity. An interference signal S(λ) detected with a constant interval of Δλ=0.5 nm is converted to S(σ) with a constant interval of ΔσA by an interpolation formula. Interval Δσ corresponding to two wavelengths of λ and λλ is given by
$$\Delta \sigma \textrm{ = }({1/\lambda } )- [{1/({\lambda + \Delta \lambda } )} ]\cong \Delta \lambda /{\lambda ^2}. $$
By using the weighted average wavelength λA of the spectral intensity as λ in Eq. (2), the constant interval of ΔσAλ/λA2 is decided for the interpolation. A least square line of a0+a1σ is defined for the distribution of n(σ)σ, and it is considered that n(σ)σT = a1+(n(σ)-a1)σT in Eq. (1). Then Fourier transform of S(σ) or Fourier transformed signal which is a function of distance d is given by
$$\begin{aligned} F(d) &= {F_I}(d) + {F_I}({d - \textrm{2}({{Z_F} + {a_1}T - Z} )} )\otimes \Im \{ (1/2)\exp [{j4\pi \{ n(\sigma )\sigma - {a_1}\sigma \} T} ]\} \\ &+ {F_I}({d + \textrm{2}({{Z_F} + {a_1}T - Z} )} )\otimes \Im \{ (1/2)\exp [{ - j4\pi\{ n(\sigma )\sigma - {a_1}\sigma \} T} ]\}, \end{aligned}$$
where $\Im$ and ⊗ mean Fourier transform and convolution, respectively, and FI(d)= $\Im${I(σ)}. From Eq. (3) a variable signal position is given by
$${d_S}\textrm{ = 2}({{Z_F} + {a_1}T - Z} ). $$
The variable signal position is equal to the difference between the optical path of ZF+a1T in the object arm and that of Z in the reference arm. Equation (3) is rewritten as
$$F(d) = {F_I}(d) + {F_I}({d - {d_S}} )\otimes {F_T}(d )+ {F_I}({d + {d_S}} )\otimes {F_T}_{}^ \ast ({ - d} )= {F_I}(d) + {F_S}(d )+ {F_S}_{}^ \ast ({ - d} ), $$
where ${F_T}(d) = \Im \{ (1/2)\exp [{j4\pi \{ n(\sigma )- {a_1}\} T\sigma } ]\}$ and * mean complex conjugate. Figure 3 shows the amplitude distribution of F(d) obtained by simulations, where T=997.4 µm, ZF-Z=-1587 µm, and ΔσA­=0.5×10−3/0.62=1.3×10−3 µm-1. Although the maximum detectable distance dmax in F(d) is 1/(2ΔσA) = 385 µm, the maximum distance for FS(d) is limited to dL=dmax/2 = 193 µm to distinguish the components of FS(d) from noise components at larger distances. The amplitude distribution of F(d) in region of d≥0 is used for the measurement. In the simulations the Sellmeier equation of BK7 glass is used as the refractive index n(σ), and a1=1.540 is obtained from calculating a least square line in the region from σS=1.2 µm-1 to σE=2.1 µm-1. In Fig. 3, dS=-102 µm from Eq. (4), and the spread width WI of FI(d) and WS of FS*(-d) are 12 µm and 68 µm, respectively. Simulation results at different values of thickness T µm make it clear that the WS=12 + 0.056 T µm holds with error less than 0.4 µm in the region of T from 500 µm to 2200 µm. From Eq. (5) and the simulation results shown in Fig. 3 it is concluded that the spread width WS is caused by both WI and dispersion effect of n(σ), and also that the position of dS is the center of width WS.

 figure: Fig. 1.

Fig. 1. Schematic configuration of SRI for measuring thickness of a glass plate.

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 figure: Fig. 2.

Fig. 2. Spectral intensity I(σ) of supercontinuum light source used in experiments.

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 figure: Fig. 3.

Fig. 3. Amplitude distributions of F(d) when dS is a minus value

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An interference signal S(σ) detected in experiments is shown in Fig. 4, and its amplitude distribution F(d) is shown in Fig. 5, where the conditions of T, ZF-Z, and ΔσA are the same as those in Fig. 3. The spread width WI and WS are 12 µm and about 67 µm, respectively, and dS=-102 µm. The values of WI, WS, and dS are almost equal to those in Fig. 3, but many signal components exist in the width of WN=62 µm. It is seemed that the signal components in the region of WN were caused by mechanical vibrations of the interferometer and electronics noise of the spectral analyzer. Components of FS*(-d) exist in the region of distance larger than WI/2+WN=68 µm. Since the dS locates at the center of width WS, the position Z of RS is determined so that |dS|=|2(ZF+a1T-Z)| is larger than WI/2+WN+WS/2=68+(12 + 0.056 T)/2 µm. The relation of WI/2+WN+WS=dL decides the maximum measurable thickness Tm. The solution of 68+(12 + 0.056 T) = 193 µm leads to Tm=2018 µm.

 figure: Fig. 4.

Fig. 4. Detected interference signal S(σ)

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 figure: Fig. 5.

Fig. 5. Amplitude distribution of F(d) at dS<0

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3. Principle of thickness measurement

Figure 6 shows the configuration of a SRI with a low resolution spectral analyzer. An OB and a fixed reference surface 2 (RS2) are contained in the object arm, and the reference arm contains a reference surface 1 (RS1) fixed on a PZT stage. The RS1 is moved by the PZT stage to a specified position to produce an interference signal which can be detected with the low resolution spectral analyzer. Four-step measurement is carried out to make thickness measurement as shown in Fig. 7. In step 1, the RS1 is moved to a position of Z1 and the optical path difference (OPD) is equal to ZF-nB(σ)lɛ-Z1, where ZF is the position of the front surface of OB, lɛ is the difference between the two paths of the object and reference beams propagating in the BS, and nB(σ) is the refractive index of BS. This lɛ is caused by the different lengths of the two sides in the BS. The interference signal except the first term of I(σ) in Eq. (1) is expressed as

$${S_1}(\sigma )= I(\sigma )\cos \{ 4\pi [{Z_F} - {n_B}(\sigma ){l_\varepsilon } - {Z_1}]\sigma \}. $$
Signal processing is carried out in the same way as described in Sec.2. And all of the notations defined in Sec. 2 have suffix of i that means step i hereafter. A rectangular window having the width slightly larger than the spread width WS1 of F1S*(-d) is used to select F1S*(-d) from F1(d) at dS1<0. Inverse Fourier transform is performed on this windowed distribution to get the following distribution:
$${S_{1F}}(\sigma )= I(\sigma )\exp\{ - j4\pi [{Z_F} - {n_B}(\sigma ){l_\varepsilon } - {Z_1}]\sigma \}. $$
The unwrapped phase of S1F(σ) is extracted as
$${\varphi _1}(\sigma )={-} 4\pi [{Z_F} - {n_B}(\sigma ){l_\varepsilon } - {Z_1}]\sigma. $$
In step 2, the RS1 is moved by the PZT stage to a new position of Z2 for reducing the OPD between the two beams reflected from the rear surface of OB and the RS1, and the interference signal is expressed as
$${S_2}(\sigma )= I(\sigma )\cos \{ 4\pi [{Z_F} + n(\sigma )T - {n_B}(\sigma ){l_\varepsilon } - {Z_2}]\sigma - \pi \}, $$
where π arises due to the beam reflected by the rear surface of OB. By the same signal processing as in step1, the unwrapped phase is given by
$${\varphi _2}(\sigma )={-} 4\pi [{Z_F} + n(\sigma )T - {n_B}(\sigma ){l_\varepsilon } - {Z_2}]\sigma + \pi, $$
By combining φ1(σ) and φ2(σ) to eliminate nB(σ)lɛ, a measurement value of D1 is obtained as
$${D_1}(\sigma )= {\varphi _2}(\sigma )- {\varphi _1}(\sigma )= 4\pi ({{Z_2} - {Z_1} - n(\sigma )T} )\sigma + \pi. $$
In step 3, the RS1 is moved to a position of Z3 for reducing the OPD between the two beams reflected from the RS1 and the RS2. The unwrapped phase of interference signal S3(σ) is given by
$${\varphi _3}(\sigma )={-} 4\pi [{Z_{R2}} + ({n(\sigma )- 1} )T - {n_B}(\sigma ){l_\varepsilon } - {Z_3}]\sigma. $$
In step 4, the OB is removed from the object arm and the RS1 is moved to a position of Z4 for compensating the OPD change arisen by the removal of OB. The unwrapped phase of interference signal S4(σ) is given by
$${\varphi _4}(\sigma )={-} 4\pi [{Z_{R2}} - {n_B}(\sigma ){l_\varepsilon } - {Z_4}]\sigma. $$
By eliminating ZR2-nB(σ)lɛ with Eqs. (12) and (13), a measurement value of D2(σ) is expressed as
$${D_2}(\sigma )= {\varphi _4}(\sigma )- {\varphi _3}(\sigma )= 4\pi [{Z_4} - {Z_3} + ({n(\sigma )- 1} )T]\sigma. $$
A measurement value of D3(σ) is obtained from D1(σ) and D2(σ) as follows:
$${D_3}(\sigma )={-} {D_1}(\sigma )- {D_2}(\sigma )\textrm{ = }4\pi [{{Z_1} - {Z_2} + {Z_3} - {Z_4} + T} ]\sigma - \pi. $$
Finally, by denoting the slope of a least square line in the distribution of D3(σ) as slope[D3(σ)], the thickness of T can be obtained as
$$T = slope{\kern 1pt} {\kern 1pt} {\kern 1pt} [{D_3}(\sigma )]/4\pi - ({{Z_1} - {Z_2} + {Z_3} - {Z_4}} ). $$

 figure: Fig. 6.

Fig. 6. Schematic configuration of spectral resolved interferometer using variable difference frequency.

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 figure: Fig. 7.

Fig. 7. Configurations of four-step measurement.

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4. Experimental result

The SRI was constructed as shown in Fig. 6 for single point thickness measurement of a glass plate. The RS1 and the RS2 were one reflecting surface of a glass plate with wedged angle. The RS1 was fixed on a PZT stage with 0.5 nm resolution and 10 nm repeatability of positioning. Both the BS and the OB were BK7 glass and the thickness of OB was about 1 mm. The interference signal was detected with a spectral analyzer with 0.5 nm resolution. Since the interfering optical fields were fed into the spectral analyzer by an optical fiber with a 250 µm core diameter, the spatial resolution of the SRI was 250 µm. The signal processing carried out in step 1 to step 4 was the method described in Secs. 2 and 3 whose contents correspond to step 2. The reference position Zi in step i was determined so that variable signal position dSi is larger than WI/2+WNi+WSi/2.

In step 1, the PZT stage was moved and stopped at position Z1=0.000 µm as an origin for other reference positions. The interference signal S1(σ) detected in the region from σS = 1.2 µm-1 to σE = 2.1 µm-1 is shown in Fig. 8(a). Figure 8(b) shows the amplitude distribution of F1(d) in the region of d>0. WS1 and WN1 were about 13 µm and 19 µm, respectively. Since this WS1 was caused by nB(σ)lɛ as shown by Eq. (6), lɛ was calculated to be about 20 µm from the relation of WS1=13 = 12 + 0.056lɛ µm with error of about 1 µm. Since the amplitude of F1(d) was larger than that of F2(d) shown in Fig. 5, WN1 was smaller than WN2=62 µm. The signal position was -dS1=62 µm which was larger than WI/2+WN1+WS1/2 = 6 + 19+(13/2) = 32 µm. From dS1=2(ZF-a1lɛ-Z1) = 2(ZF-1.54×20-0)=-62 µm, ZF=0 was obtained. A rectangle window existing from d=54 µm to d=69 µm was used to select the component of FS1*(-d), and the unwrapped phase φ1(σ) was obtained after getting inverse Fourier transform of the windowed F1(d). The σ region in the φ1(σ) was from 1.4 µm-1 to 1.95 µm-1 in a large intensity region of I(σ) to reduce noise effects in the φ1(σ).

 figure: Fig. 8.

Fig. 8. (a) Detected interference signal S1(σ) and (b) amplitude distribution of its Fourier transform F1(d).

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In step 2, the PZT stage was moved and stopped at a position Z2=1587.0 µm, which compensated the increase of the optical path by a1T. Detected signal S2(σ) and the amplitude distribution of Fourier transform F2(d) of S2(σ) are the same as Figs. 4 and 5, respectively. Windowed F2(d) was made by a rectangle window from 66 µm to 157 µm to select the component of FS2*(-d) from F2(d), and inverse Fourier transform was performed on the windowed F2(d) to get the unwrapped phase φ2(σ). The dS2 was -102 µm, which agreed with the theoretical value of dS2=2(ZF-a1lɛ+a1T-Z2) = 2(-31 + 1.54×1017.4-1587)=-102 µm, where a measured value of T=1017.4 µm was used.

In step 3 and step 4, the amplitude distributions of F3(d) and F4(d) were detected as shown in Figs. 9(a) and 9(b), respectively. In step 3, the RS2 was used in the object arm and the PZT stage was moved to position Z3=11935.0 µm. In Fig. 9(a) the values of WI/2+WN3, WS3, and dS3 were about 60 µm, 65 µm, and -98 µm, respectively. From dS3=2(ZR2-a1lɛ+(a1-1)T-Z3) = 2(ZR2-31 + 549-11935)=-98 µm, ZR2=11368 µm was obtained. In step 4, the PZT stage was moved back to position Z4=11366.0 µm because the OB was removed from the object arm. The values of WI/2+WN4 and WS4 were about 25 µm and 13 µm, respectively. The dS4 was -56 µm, which almost agreed with dS4=2(ZR2-a1lɛ-Z4) = 2(11368-31-11366)=-58 µm. These results described above are shown at Table 1 as values in measurement of case 1, and the amplitude distributions of Fi(d) (i=1 to 4) have been provided in Figs. 8(b), 5, 9(a), and 9(b), respectively. Distribution of D3(σ) obtained from φ1(σ) to φ4(σ) is shown in Fig. 10. The distribution of D3(σ) was a straight line with small fluctuations generated by noise components in φ1(σ) to φ4(σ). It is estimated that the main source of these noise components were produced by the mechanical vibrations of the interferometer and electronics noise of the spectral analyzer. In order to reduce noise effects, the least square method was used to calculate the slope of D3(σ) in the region of from 1.4 µm-1 to 1.95 µm-1 corresponding to a large intensity region of I(σ). The values of slope[D3(σ)]/4π are given by Table 2, where the measurement was repeated three times as case 1 to case 3. It is estimated that the differences in the values of slope[D3(σ)]/4π were caused by a position change of ZR2 due to removing the OB between step 3 and step 4 and the 10 nm positioning repeatability of the PZT stage. In addition, the value of Z1-Z2+Z3-Z4 was regarded to be a constant value of -1018.000 µm in Eq. (16). The three measured values of T were a little different by less than 55 nm as shown in Table 2.

 figure: Fig. 9.

Fig. 9. Amplitude distribution of Fourier transform of the interference signal detected in (a) step 3 and (b) step 4.

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 figure: Fig. 10.

Fig. 10. D3(σ) calculated from φ1(σ) to φ4(σ) in case 1.

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Tables Icon

Table 1. Values in measurement of case 1.

Tables Icon

Table 2. Measured values in case 1 to 3.

5. Conclusion

The interference signal having the signal position was analyzed for measuring a large thickness of glass plate with the SRI using the spectral analyzer with a low resolution of 0.5 nm and the supercontinuum light source with a large bandwidth of about 300 nm. It was confirmed that the amplitude distribution of the signal component produced by the glass plate is not symmetric about the signal position which is the center of the spread width of the signal component. A measurable maximum thickness was derived by considering the resolution of the spectral analyzer and the signal component produced by the glass plate. In the four-step measurement the four spectral phase distributions of the interference signals were calculated through selecting the required signal components and doing inverse Fourier transform, and the thickness of glass plate could be obtained from the slope of the spectral phase distribution which does not contain the refractive index of glass plate. Also the positions of the reference surface 2 and the front surface of the glass plate and a dispersion effect of beam splitter were obtained from the signal components. And it was confirmed that the detected values of the signal positions agreed with the theoretical ones. A small measurement error of 50 nm was achieved in the measurement of 1 mm thickness of the glass plate.

Funding

National Natural Science Foundation of China (11674111, 11750110426, 61575070, 62005086).

Disclosures

The authors declare no conflicts of interest.

References

1. J. Na, H. Y. Choi, E. S. Choi, C. Lee, and B. H. Lee, “Self-referenced spectral interferometry for simultaneous measurements of thickness and refractive index,” Appl. Opt. 48(13), 2461–2467 (2009). [CrossRef]  

2. S. J. Park, K. S. Park, Y. H. Kim, and B. H. Lee, “Simultaneous Measurements of Refractive Index and Thickness by Spectral-Domain Low Coherence Interferometry Having Dual Sample Probes,” IEEE Photonics Technol. Lett. 23(15), 1076–1078 (2011). [CrossRef]  

3. 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]  

4. S. Maeng, J. Park, O. Byungsun, 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]  

5. 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, 170–174 (2013). [CrossRef]  

6. J. Park, J. Bae, J. A. Kim, and J. Jin, “Physical thickness and group refractive index measurement of individual layers for double-stacked microstructures using spectral-domain interferometry,” Opt. Commun. 431, 181–186 (2019). [CrossRef]  

7. J. Park, H. Mori, and J. Jin, “Simultaneous measurement method of the physical thickness and group refractive index free from a non-measurable range,” Opt. Express 27(17), 24682–24692 (2019). [CrossRef]  

References

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  1. J. Na, H. Y. Choi, E. S. Choi, C. Lee, and B. H. Lee, “Self-referenced spectral interferometry for simultaneous measurements of thickness and refractive index,” Appl. Opt. 48(13), 2461–2467 (2009).
    [Crossref]
  2. S. J. Park, K. S. Park, Y. H. Kim, and B. H. Lee, “Simultaneous Measurements of Refractive Index and Thickness by Spectral-Domain Low Coherence Interferometry Having Dual Sample Probes,” IEEE Photonics Technol. Lett. 23(15), 1076–1078 (2011).
    [Crossref]
  3. 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]
  4. S. Maeng, J. Park, O. Byungsun, 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]
  5. 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, 170–174 (2013).
    [Crossref]
  6. J. Park, J. Bae, J. A. Kim, and J. Jin, “Physical thickness and group refractive index measurement of individual layers for double-stacked microstructures using spectral-domain interferometry,” Opt. Commun. 431, 181–186 (2019).
    [Crossref]
  7. J. Park, H. Mori, and J. Jin, “Simultaneous measurement method of the physical thickness and group refractive index free from a non-measurable range,” Opt. Express 27(17), 24682–24692 (2019).
    [Crossref]

2019 (2)

J. Park, J. Bae, J. A. Kim, and J. Jin, “Physical thickness and group refractive index measurement of individual layers for double-stacked microstructures using spectral-domain interferometry,” Opt. Commun. 431, 181–186 (2019).
[Crossref]

J. Park, H. Mori, and J. Jin, “Simultaneous measurement method of the physical thickness and group refractive index free from a non-measurable range,” Opt. Express 27(17), 24682–24692 (2019).
[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, 170–174 (2013).
[Crossref]

2012 (1)

2011 (1)

S. J. Park, K. S. Park, Y. H. Kim, and B. H. Lee, “Simultaneous Measurements of Refractive Index and Thickness by Spectral-Domain Low Coherence Interferometry Having Dual Sample Probes,” IEEE Photonics Technol. Lett. 23(15), 1076–1078 (2011).
[Crossref]

2010 (1)

2009 (1)

Bae, J.

J. Park, J. Bae, J. A. Kim, and J. Jin, “Physical thickness and group refractive index measurement of individual layers for double-stacked microstructures using spectral-domain interferometry,” Opt. Commun. 431, 181–186 (2019).
[Crossref]

Byungsun, O.

Choi, E. S.

Choi, H. Y.

Eom, T. B.

Jin, J.

J. Park, J. Bae, J. A. Kim, and J. Jin, “Physical thickness and group refractive index measurement of individual layers for double-stacked microstructures using spectral-domain interferometry,” Opt. Commun. 431, 181–186 (2019).
[Crossref]

J. Park, H. Mori, and J. Jin, “Simultaneous measurement method of the physical thickness and group refractive index free from a non-measurable range,” Opt. Express 27(17), 24682–24692 (2019).
[Crossref]

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, 170–174 (2013).
[Crossref]

S. Maeng, J. Park, O. Byungsun, 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]

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]

Kang, C. S.

Kim, J. A.

J. Park, J. Bae, J. A. Kim, and J. Jin, “Physical thickness and group refractive index measurement of individual layers for double-stacked microstructures using spectral-domain interferometry,” Opt. Commun. 431, 181–186 (2019).
[Crossref]

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, 170–174 (2013).
[Crossref]

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]

Kim, J. W.

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, 170–174 (2013).
[Crossref]

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]

Kim, Y. H.

S. J. Park, K. S. Park, Y. H. Kim, and B. H. Lee, “Simultaneous Measurements of Refractive Index and Thickness by Spectral-Domain Low Coherence Interferometry Having Dual Sample Probes,” IEEE Photonics Technol. Lett. 23(15), 1076–1078 (2011).
[Crossref]

Lee, B. H.

S. J. Park, K. S. Park, Y. H. Kim, and B. H. Lee, “Simultaneous Measurements of Refractive Index and Thickness by Spectral-Domain Low Coherence Interferometry Having Dual Sample Probes,” IEEE Photonics Technol. Lett. 23(15), 1076–1078 (2011).
[Crossref]

J. Na, H. Y. Choi, E. S. Choi, C. Lee, and B. H. Lee, “Self-referenced spectral interferometry for simultaneous measurements of thickness and refractive index,” Appl. Opt. 48(13), 2461–2467 (2009).
[Crossref]

Lee, C.

Maeng, S.

Mori, H.

Na, J.

Park, J.

J. Park, J. Bae, J. A. Kim, and J. Jin, “Physical thickness and group refractive index measurement of individual layers for double-stacked microstructures using spectral-domain interferometry,” Opt. Commun. 431, 181–186 (2019).
[Crossref]

J. Park, H. Mori, and J. Jin, “Simultaneous measurement method of the physical thickness and group refractive index free from a non-measurable range,” Opt. Express 27(17), 24682–24692 (2019).
[Crossref]

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, 170–174 (2013).
[Crossref]

S. Maeng, J. Park, O. Byungsun, 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]

Park, K. S.

S. J. Park, K. S. Park, Y. H. Kim, and B. H. Lee, “Simultaneous Measurements of Refractive Index and Thickness by Spectral-Domain Low Coherence Interferometry Having Dual Sample Probes,” IEEE Photonics Technol. Lett. 23(15), 1076–1078 (2011).
[Crossref]

Park, S. J.

S. J. Park, K. S. Park, Y. H. Kim, and B. H. Lee, “Simultaneous Measurements of Refractive Index and Thickness by Spectral-Domain Low Coherence Interferometry Having Dual Sample Probes,” IEEE Photonics Technol. Lett. 23(15), 1076–1078 (2011).
[Crossref]

Appl. Opt. (1)

IEEE Photonics Technol. Lett. (1)

S. J. Park, K. S. Park, Y. H. Kim, and B. H. Lee, “Simultaneous Measurements of Refractive Index and Thickness by Spectral-Domain Low Coherence Interferometry Having Dual Sample Probes,” IEEE Photonics Technol. Lett. 23(15), 1076–1078 (2011).
[Crossref]

Opt. Commun. (2)

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, 170–174 (2013).
[Crossref]

J. Park, J. Bae, J. A. Kim, and J. Jin, “Physical thickness and group refractive index measurement of individual layers for double-stacked microstructures using spectral-domain interferometry,” Opt. Commun. 431, 181–186 (2019).
[Crossref]

Opt. Express (3)

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Figures (10)

Fig. 1.
Fig. 1. Schematic configuration of SRI for measuring thickness of a glass plate.
Fig. 2.
Fig. 2. Spectral intensity I(σ) of supercontinuum light source used in experiments.
Fig. 3.
Fig. 3. Amplitude distributions of F(d) when dS is a minus value
Fig. 4.
Fig. 4. Detected interference signal S(σ)
Fig. 5.
Fig. 5. Amplitude distribution of F(d) at dS<0
Fig. 6.
Fig. 6. Schematic configuration of spectral resolved interferometer using variable difference frequency.
Fig. 7.
Fig. 7. Configurations of four-step measurement.
Fig. 8.
Fig. 8. (a) Detected interference signal S1(σ) and (b) amplitude distribution of its Fourier transform F1(d).
Fig. 9.
Fig. 9. Amplitude distribution of Fourier transform of the interference signal detected in (a) step 3 and (b) step 4.
Fig. 10.
Fig. 10. D3(σ) calculated from φ1(σ) to φ4(σ) in case 1.

Tables (2)

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Table 1. Values in measurement of case 1.

Tables Icon

Table 2. Measured values in case 1 to 3.

Equations (16)

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S ( σ ) = I ( σ ) + I ( σ ) cos { 4 π [ Z F + n ( σ ) T Z ] σ } ,
Δ σ  =  ( 1 / λ ) [ 1 / ( λ + Δ λ ) ] Δ λ / λ 2 .
F ( d ) = F I ( d ) + F I ( d 2 ( Z F + a 1 T Z ) ) { ( 1 / 2 ) exp [ j 4 π { n ( σ ) σ a 1 σ } T ] } + F I ( d + 2 ( Z F + a 1 T Z ) ) { ( 1 / 2 ) exp [ j 4 π { n ( σ ) σ a 1 σ } T ] } ,
d S  = 2 ( Z F + a 1 T Z ) .
F ( d ) = F I ( d ) + F I ( d d S ) F T ( d ) + F I ( d + d S ) F T ( d ) = F I ( d ) + F S ( d ) + F S ( d ) ,
S 1 ( σ ) = I ( σ ) cos { 4 π [ Z F n B ( σ ) l ε Z 1 ] σ } .
S 1 F ( σ ) = I ( σ ) exp { j 4 π [ Z F n B ( σ ) l ε Z 1 ] σ } .
φ 1 ( σ ) = 4 π [ Z F n B ( σ ) l ε Z 1 ] σ .
S 2 ( σ ) = I ( σ ) cos { 4 π [ Z F + n ( σ ) T n B ( σ ) l ε Z 2 ] σ π } ,
φ 2 ( σ ) = 4 π [ Z F + n ( σ ) T n B ( σ ) l ε Z 2 ] σ + π ,
D 1 ( σ ) = φ 2 ( σ ) φ 1 ( σ ) = 4 π ( Z 2 Z 1 n ( σ ) T ) σ + π .
φ 3 ( σ ) = 4 π [ Z R 2 + ( n ( σ ) 1 ) T n B ( σ ) l ε Z 3 ] σ .
φ 4 ( σ ) = 4 π [ Z R 2 n B ( σ ) l ε Z 4 ] σ .
D 2 ( σ ) = φ 4 ( σ ) φ 3 ( σ ) = 4 π [ Z 4 Z 3 + ( n ( σ ) 1 ) T ] σ .
D 3 ( σ ) = D 1 ( σ ) D 2 ( σ )  =  4 π [ Z 1 Z 2 + Z 3 Z 4 + T ] σ π .
T = s l o p e [ D 3 ( σ ) ] / 4 π ( Z 1 Z 2 + Z 3 Z 4 ) .