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 0.5 nm and a supercontinuum light source with a large bandwidth of about 300 nm, 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 50 nm is achieved in measuring 1 mm thickness of a glass plate.
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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 [1–7]. 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 [1–4,6,7] 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
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.
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
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).
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.
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.
National Natural Science Foundation of China (11674111, 11750110426, 61575070, 62005086).
The authors declare no conflicts of interest.
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