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Abstract
Measurements of phase refractive indexes of thick objects by using fitting method have been reported in many papers, but the fitting method produces errors in the fitted coefficients of a fitting function. In this paper it is made clear that the thickness of object, for which the phase refractive index can be measured exactly, is limited by the errors. Phase refractive indexes of three kinds of glass plates of 1 mm thickness are measured directly from spectral phases detected with a spectrally resolved interferometer. Instead of using the fitting method, the 2π phase ambiguity contained in a detected spectral phase is determined by using an assumption that an actual refractive index measured in experiment is almost the same as well-known data of the refractive index. The actual refractive indexes measured with an error less than 8 × 10−5 are slightly different from the well-known data about the slope and the constant value in the distributions.
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1. Introduction
Phase refractive index is an important property in optical materials. It has been measured with different techniques using intensity detection of light coming out from a material. As a technique using phase detection, interferometers are widely used to measure phase refractive index and group refractive index [1–10]. In refractive index measurement a value of thickness of an object is required. A measurement error in phase refractive index is caused by the measurement error in thickness. It is derived that measurement error ratio in phase refractive index is equal to that in thickness, as shown in Appendix 1. If the error ratio in refractive index is 10−4, the measurement error in thickness is 0.1 µm at 1 mm thickness. Hence it is desirable that the thickness is measured exactly with an interferometer although a micrometer is used for thickness measurement [5,6]. In order to measure the thickness with a spectrally resolved interferometer, the refractive index of an object must be eliminated in a detected distribution by detecting a few distributions with different optical configurations. When the detected distributions are Fourier transforms of the interference signals, an exact thickness cannot be measured due to dispersive effect of an object [7,8]. It is better that the detected distributions are spectral phases calculated from the interference signals. Spectral phase is calculated exactly from an interference signal through Fourier transform and band-pass filtering [9–12]. When a cosine waveform contained in an interference signal is extracted by detecting the maximum and minimum values of the interference signal, the spectral phase calculated by arccosine function is influenced by detection error in the maximum and minimum values [1,2]. Phase refractive index is obtained from a detected spectral phase together with the measured thickness. However, because the spectral phase contains 2π phase ambiguity, determination of the phase ambiguity is required to measure exactly a phase refractive index.
In order to determine the phase ambiguity, a fitting method is employed where the detected spectral phase is fitted with a fitting function. In Refs. [1,2], the fitting function is a polynomial function and the 2π phase ambiguity is determined at one wavenumber from the fitted coefficients of the polynomial function. At this one wavenumber the interference signal has a maximum or minimum value. On the other hand, in Ref. [10] the 2π phase ambiguity is determined from the fitted value of a fitting function at the wavenumber of zero. Because there is some difference between the fitted distribution of a fitting function and the distribution of the detected spectral phase, the fitting method involves with an error in the phase ambiguity. In Ref. [3] the refractive index at one wavelength is measured with another instrument to determine the phase ambiguity instead of employing the fitting method. In Refs. [4,5], the derivative of the detected spectral phase is calculated to eliminate the phase ambiguity and get a group refractive index. The group refractive index is fitted with a fitting function which is the derivative of a Sellmeier equation. The coefficients in the Sellmeier equation or phase refractive index are decided by this fitting, but the fitting method produces errors in the fitted coefficients. Then, the phase refractive index expressed by the fitted coefficients is compensated by using a known value of the refractive index at one wavelength. This method has a disadvantage that an actual refractive index cannot be obtained directly from the detected spectral phase. The fitting method is not employed in Ref. [6] where the phase ambiguity is obtained by using an assumption that the refractive indexes at different two wavenumbers are the same value. At the two wavenumbers, adjacent two peaks of the interference signal exist. It seems that the above assumption causes an error in the determination of phase ambiguity.
In this paper, characteristics of the fitting method are examined by simulations because the fitting method produces errors in the fitted coefficients [1,2,4,5,10]. It is made clear that the thickness of an object, for which the refractive index can be measured exactly, is limited by an error in the fitted value of phase ambiguity. The maximum thickness for exact measurement is presented at three kinds of glass plates in the simulations. Also the characteristics are investigated at different fitting functions. Spectral phases are detected in four optical configurations with the spectral resolved interferometer [12] to measure an exact thickness for 1 mm-thickness glass plate. Since the fitting method cannot be employed for 1 mm-thickness glass plate, the phase ambiguity is determined by using an assumption that the refractive index obtained from experiment is almost the same as the well-known data of refractive index. Phase refractive indexes of three kinds of glass plates are measured with an error less than 8 × 10−5. These actual phase refractive indexes measured in the experiment are different from the well-known data by less than 2 × 10−4.
2. Principle of measurements
Figure 1 shows a configuration of spectral resolved interferometer (SRI) for thickness and refractive index measurements of an object (OB) with a large thickness T. A beam splitter (BS) divides a beam from a supercontinuum light source for two arms of interferometer. One beam is reflected in the object arm, and another beam is reflected by the reference surface 1 (RS1) in the reference arm. These two reflected beams are interfered and an interference signal is detected with a low-resolution spectral analyzer (SPA). The position of RS1 is adjusted with a piezoelectric transducer (PZT) stage for three different reflected beams in the object arm, so that one reflected beam in the object arm produces the interference signal. The measurement consists of four measurement steps. In step 1, the position of RS1 is Z1 to detect an interference signal S1(σ) generated by the two beams from the front surface of OB and the RS1 with the low-resolution SPA. A spectral phase of the interference signal wrapped within 2π is calculated through Fourier transform and frequency filtering. The spectral phase of the interference signal unwrapped from σS is given by
In step 3, the position of RS1 is Z3 to detect the interference signal S3(σ) produced by RS2. The spectral phase of interference signal is given by
In step 4, OB is removed from the object arm, and the position of RS1 is Z4 to detect the interference signal S4(σ) generated by RS2. The spectral phase of interference signal is given by
By combining the spectral phases in four measurement steps, an equation is expressed as
Thickness is obtained from
where the notation of slope{φT(σ)} means the slope of a least square line calculated for the distribution of φT(σ). Z1, Z2, Z3, and Z4 are known values provided by a controller of the PZT stage.In order to make refractive index measurement, the following phase distribution is made from φ1(σ) and φ2(σ):
By adding 2πpA to φ21(σ) to get the condition of 0≤φM(σS) < 2π, a spectral phase for refractive index measurement is obtained as
where pA = -INT{φ21(σS)/2π}, and INT{x} means the integer part of x. Thus, refractive index is given byIn Eq. (9), T is measured with Eq. (6). The phase ambiguity of 2πp must be determined to get a distribution of n(σ) with Eq. (9). When thickness T is a small value, the integer value of p can be determined with the fitting method as reported in Ref. [10]. In next section characteristics of the fitting method are examined to make clear a maximum value of T where p can be determined by the fitting method.
3. Characteristics of fitting method
3.1 Error in determination of phase ambiguity
In order to examine the characteristics of fitting method, three kinds of situations are examined in simulations. It is well known that the refractive index of glass plates is expressed as Sellmeier equation or Cauchy’s equation. In these simulations, Sellmeier equation of BK7 glass is used as refractive index nBK7(σ) to make a detected phase distribution φM(σ). The φM(σ) is fitted with a fitting function made by using Cauchy’s equation.
First, distribution of nBK7(σ)σ in the range from σS = 1.4 µm-1 to σE = 1.95 µm-1 is fitted with a fitting function f1(σ) = (b0 + b2σ2)σ. The fitted values of b0 and b2 are denoted by b0f and b2f, respectively. The distributions of f1f(σ) = (b0f + b2fσ2)σ and nBK7(σ)σ are shown schematically in Fig. 2(a) with green dashed curve and dark solid curve, respectively, where b0f = 1.504419 and b2f = 0.004264. Figure 2(b) shows the distribution of D1f(σ)=nBK7(σ)σ-f1f(σ) in the fitting range from σS = 1.4 µm-1 to σE = 1.95 µm-1. Root mean square value of D1f(σ) is denoted by RMSD{D1f(σ)}, and RMSD{D1f(σ)}=4.52 × 10−5. The distribution of f1f(σ) is produced under the constrain of f1(0) = 0 in the fitting.
Next, the distribution of nBK7(σ)σ is fitted with a fitting function f2(σ) = (b0 + b2σ2)σ-p, in order to examine if a fitted value of p is equal to zero or not. The fitted values are b0f = 1.507774, b2f = 0.003872, and pf = 0.003727. The distribution of f2f(σ) = (b0f + b2fσ2)σ-pf is shown by the green dashed curve in Fig. 3(a). The distribution of D2f(σ)=nBK7(σ)σ-f2f(σ) is shown in Fig. 3(b), and RMSD{D2f(σ)}=4.63 × 10−6. The fitted values in f2f(σ) are different from those in f1f(σ) because the term of p in f2(σ) works in the fitting so that RMSD{D2f(σ)} is smaller than RMSD{D1f(σ)}. It is made clear that the fitting function f2(σ) produces an error of ef = 0.003727 at σ=0 µm-1 against nBK7(σ)σ=0 at σ=0 µm-1.
Finally, a spectral phase φM(σ)/2π=2TnBK7(σ)σ-pC is fitted with a fitting function f2(σ) = 2 T(b0 + b2σ2)σ-p. When T = 1 mm and σS = 1.4 µm-1, φM(σS)/2π=0.54 and pC = INT{2TnBK7(σS)σS}=4235. In fitting φM(σ)/2π with f2(σ), the fitted values in f2f(σ) are b0f = 1.507774, b2f = 0.003872, and pf = 4242.45. Figure 4 shows the distributions of φM(σ)/2π and f2f(σ) with dark solid curve and green dashed curve, respectively. The difference between -pC = -4235 and f2f(0)=-pf = -4242.45 is equal to 7.45. This difference is caused by the fitting characteristic shown in Fig. 3 as explained below. When nBK7(σ)σ of Fig. 3 changes to 2TnBK7(σ)σ, the fitting function changes to f2(σ) = 2 T(b0 + b2σ2)σ-p. In these changes, f2f(0)=-pf = -2Tef is produced. Since the distribution of φM(σ)/2π=2TnBK7(σ)σ-pC shown in Fig. 4(a) is obtained by shifting the distribution of 2TnBK7(σ)σ by -pC along the vertical axis, f2f(σ) is also shifted by -pC and f2f(0)=-2Tef -pC is produced as shown in Fig. 4(a). Figure 4(b) shows D2f(σ) = [φM(σ)/2π]-f2f(σ) whose distribution is the same as that of Fig. 3(b) except the values along the vertical axis. RMSD{D2f(σ)}=9.3 × 10−3 in Fig. 4(b) is 2 T times larger than RMSD{D2f(σ)} =4.63 × 10−6 in Fig. 3(b). Since the phase ambiguity is calculated by rounding off the value of pf, the condition of |pf-pC|=|2Tef|=0.00745T < 0.5 is required to get the value of pC. This condition decides a maximum thickness Tmax for which the refractive index can be measured exactly. The values of ef and RMSD{D2f(σ)}at quartz glass and soda lime glass were also examined with the same simulation as in Fig. 3. In these simulations, refractive index nQZ(σ) of quartz glass was Sellmeier equation, and refractive index nSL(σ) of soda lime was expressed as nSL(σ)=a0-a1σ-2 + a2σ2 [13] whose coefficients of a0, a1, and a2 were obtained by fitting the experimental data reported in Ref. [14] with the fitting function of fSL(σ)=a0-a1σ-2 + a2σ2. The values of ef, Tmax, and RMSD{D2f(σ)} at BK7 glass, quartz glass, and soda lime glass are shown in Table 1.
3.2 Values of ef in different fitting functions
Values of ef were examined for two different fitting functions of f2(σ) = 2 T(b0 + b2σ2)σ-p and f3 (σ) = 2 T(b0 + b2σ2 + b4σ4)σ-p with the same simulation as in Fig. 3. Table 2 shows the values of ef, Tmax, and RMSD{D3f(σ)} at f3(σ). In Tables 1 and 2, the values of ef at BK7 glass and quartz glass are almost the same, and they are larger than the absolute value of ef at soda lime glass. Since nSL(σ) is given by a polynomial function while nBK7(σ) and nQZ(σ) are given by Sellmeier equation, the distribution of nSL(σ)σ is well fitted by the fitting functions of f2(σ) and f3(σ). Hence the values of ef at soda lime glass are smaller. It is made clear in Tables 1 and 2 that the distribution of φM(σ)/2π is well fitted with f3(σ) than f2(σ) because the values of RMSD{D3f(σ)} are smaller than the values of RMSD{D2f(σ)}. But large values of ef occur at f3(σ). These results mean that the three terms of f3(σ) work well to make the value of RMSD smaller in the fitting range from σS to σE while the fitted value of pf becomes more different from the value of pC. Therefore, f2(σ) is better than f3(σ).
Table 1. Values of ef, Tmax, and RMSD for different glass plates at f2(σ) = 2 T(b0 + b2σ2)σ-p.
Table 2. Values of ef, Tmax, and RMSD for different glass plates at f3(σ) = 2 T(b0 + b2σ2 + b4σ4)σ-p.
4. Measurement for glass plate with large thickness
4.1 Measurement of thickness
Experiments were carried out with the SRI shown in Fig. 1. RS1 and RS2 were wedged glass plates. The RS1 was fixed on a PZT stage with positioning repeatability of 10 nm. The length of lε in BS of BK7 glass was about 20 µm. OB was a BK7 glass plate with 1 mm thickness. The light source has a spectral range from 500 nm to 800 nm. Interference signals were detected with a SPA of 0.2 nm wavelength resolution. Spline interpolation was performed to convert the interference signals detected in wavelength domain to the one in wavenumber domain with a constant interval of 0.001 µm-1. Thickness measurement consists of four measurement steps. Figure 5 shows the detected interference signal S1(σ) in step 1, where the value of Z1 was 0 µm and the distance of Z1-ZF was about 15 µm. Since the optical path difference (OPD) contained in S1(σ) was a small value of about 2(ZF-nB(σ)lε-Z1)≈-90 µm, the intensity distribution shows clearly the phase change of S1(σ). Figure 6(a) shows the amplitude distribution of the Fourier transform of S1(σ), and frequency component of F1(f) was selected by a rectangle window from -97.6 µm to -82.3 µm. Detected spectral phase φ1(σ) was obtained from the inverse Fourier transform of F1(f), as shown in Fig. 6(b). In step 2, the RS1 was moved to the position of Z2 = 1580 µm to make a small OPD between Z2 and ZR = ZF + n(σ)T, and the spectral phase φ2(σ) was obtained with the same signal processing as described above. In step 3, the RS1 was moved to the position of Z3 = 11047 µm for the detection of φ3(σ). In step 4, the OB was removed from the object arm, and the position of RS1 was changed to Z4 = 10487 µm to get the spectral phase φ4(σ). Figure 7 shows the distribution of φT(σ) given by Eq. (6) with Z1-Z2 + Z3-Z4=-1020 µm, and the slope of least square line in φT(σ) provided T = 1017.460 µm. The above measurements were repeated three times as case 1 to case 3, and the measured values of T are shown in Table 3. The difference of T among the three cases is less than 38 nm, which means that the measurement error in T was less than 38 nm. It is estimated that the main sources of the measurement error in T were caused by removing the OB between step 3 and step 4 and the 10 nm positioning repeatability of the PZT stage.
Fig. 6. Distributions of (a) amplitude of Fourier transform of S1(σ) and (b) the spectral phase φ1(σ).
Table 3. Measured values of T in three cases.
4.2 Measurement of refractive index
Figure 8 shows the distribution of φ21(σ)/2π calculated with Eq. (7) and Z2-Z1 = 1580 µm from the spectral phases of φ1(σ) and φ2(σ) detected in case 1 at BK7 glass plate. Since φ21(σS)/2π was equal to 4423.86 at σS = 1.4 µm-1, the value of pA in Eq. (8) was -4423 to satisfy the condition of 0≤φM(σS) < 2π. The distribution of φM(σ)/2π was calculated with Eq. (8) from the distribution of Fig. 8, as shown in Fig. 9 where pA = -4423 and φM(σS)/2π=0.86. In order to determine the phase ambiguity 2πp containing in the detected distribution of φM(σ) at a glass plate with large thickness, it is assumed that the refractive index n(σ) measured in experiments with Eq. (9) is almost equal to well-known data or equation neq(σ) of the refractive index. Therefore the following difference function is defined:
where np(σ, p)=φM(σ)/4πTσ+(p/2Tσ), and φM(σ) and T are the known values. A minimum value of D(p) is searched by changing the value of p, and the refractive index n(σ)=np(σ, p) is determined at the minimum value. Figure 10 shows the distribution of φM(σ)/4πTσ obtained from the distribution of Fig. 9 with T = 1017.460 µm in case 1. Figure 11 shows the distribution of D(p) with neq(σ)=nBK7(σ) for the detected distribution of Fig. 10. A minimum value of D(p) existed at p = 4309, as shown in Fig. 11. In case 2 and 3, a minimum value of D(p) also existed at p = 4309. It became clear that p is equal to 4309 when the value of T is about 1017.4 µm. Refractive indexes n(σ) measured in case 1 to case 3 are shown in Fig. 12 with red dashed curves. The blue solid curve in Fig. 12 is nBK7(σ). In order to make clear the difference between the measured refractive index n(σ) and the theoretical refractive index nBK7(σ), RMSD{n(σ)-nBK7(σ)} was calculated. The values of RMSD{n(σ)-nBK7(σ)} in case 1 to case 3 were 5.5 × 10−5, 5.1 × 10−5, and 7.9 × 10−5, respectively. Figure 13 shows difference Di1(σ) between the two measured distributions of refractive index in case i and case 1. In Fig. 13, the blue distribution is D21(σ) from case 2 and case 1, and the red distribution is D31(σ) from case 3 and case 1. When the detected spectral phase is expressed by φM(σ)/2π=a0 + a1σ+a2σ2 +… and the differences in T and a1 between case i and case 1 are denoted by ΔT and Δa1, respectively, a constant value of Di1(σ) is approximately expressed as CD = -a1ΔT/2T2 + Δa1/2 T by ignoring terms of small values in a constant value of Di1(σ), as shown in Appendix 2. Value of a1 was obtained by fitting φM(σ)/2π with f(σ)=a0 + a1σ. At D21(σ), the constant value of CD = -1.7 × 10−5 was calculated from a1 = 3090.0 in case 1, Δa = -0.1 and ΔT = 23 nm in case 2. This constant value is nearly equal to the average value of -2 × 10−5 at D21(σ) shown in Fig. 13. This analysis makes it clear that Di1(σ) is nearly equal to a constant value and the small fluctuations from the constant value are caused by the differences in the terms of a0 and am of m > 1 between case i and case 1. Thus, Fig. 13 makes clear that the three measured refractive indexes have different constant term and the refractive index n(σ) of BK7 glass plate can be measured with an error less than about 6 × 10−5. Since the refractive index n(σ) is directly measured from the detected spectral phase φM(σ), n(σ) is regarded as an actual refractive index. From Fig. 12 it is confirmed that the slope of n(σ) is slightly smaller than the slope of nBK7(σ), and n(σ) is a little larger than nBK7(σ) in the region of σ less than 1.55 µm-1.Fig. 12. Refractive index n(σ) of BK7 glass plate (red dashed curve) measured in (a) case 1, (b) case 2, and (c) case 3. Blue solid curve is refractive index nBK7(σ). RMSD{n(σ)-nBK7(σ)} in case 1 to case 3 are equal to 5.5 × 10−5, 5.1 × 10−5, and 7.9 × 10−5, respectively.
Fig. 13. Distributions of D21(σ) (blue curve) and D31(σ) (green curve) calculated from the measured results shown in Fig. 12.
4.3 Measurement of other objects
The object was a fused silica glass plate. The measured values of T in three cases are shown in Table 4. Measurement error in T was less than 45 nm. A minimum value of D(p) existed at p = 4332 in three cases, where neq(σ)=nFS(σ). Figure 14 shows refractive indexes measured in case 1 to case 3 with red dashed curves, and the blue solid curves are nFS(σ). The distributions of refractive indexes measured in three cases are almost parallel to the distribution of nFS(σ). The values of RMSD{n(σ)-nFS(σ)} in case 1 to case 3 were 4.9 × 10−5, 1.2 × 10−4, and 5.5 × 10−5, respectively. Figure 15 shows difference D21(σ) and D31(σ) with blue and red curves, respectively. These distributions of D21(σ) and D31(σ) make it clear that the three measured refractive indexes have different constant term and the refractive index n(σ) of fused silica glass plate can be measured with an error less than about 8 × 10−5. From Fig. 14 it is confirmed that the actual refractive index n(σ) of the fused silica glass plate is almost the same as that of nFS(σ) except a small deviation less than 1.2 × 10−4 from nFS(σ).
Fig. 14. Refractive index n(σ) of fused silica glass plate (red dashed curve) measured in (a) case 1, (b) case 2, and (c) case 3. Blue solid curve is refractive index nFS(σ). RMSD{n(σ)-nFS(σ)} in case 1 to case 3 are 4.9 × 10−5, 1.2 × 10−4, and 5.5 × 10−5, respectively.
Fig. 15. Distributions of D21(σ) (blue curve) and D31(σ) (green curve) calculated from the measured results shown in Fig. 14.
Table 4. Measured values of T at fused silica glass plate in three cases.
The object was a soda lime glass plate. The values of T measured in three cases are shown in Table 5. Measurement error in T was less than 45 nm. A minimum value of D(p) existed at p = 2881 in three cases, where neq(σ)=nSL(σ). Figure 16 shows refractive indexes measured in case 1 to case 3 with red dashed curves, and the blue solid curves are nSL(σ). The values of RMSD{n(σ)-nSL(σ)} in case 1 to case 3 were 2.1 × 10−4, 2.1 × 10−4, and 2.2 × 10−4, respectively. Figure 17 shows D21(σ) and D31(σ) with blue and red curves, respectively. These distributions of D21(σ) and D31(σ) make it clear that the three measured refractive indexes have different constant terms and the refractive index n(σ) of soda lime glass plate can be measured with an error less than about 4 × 10−5. From Fig. 16, it is confirmed that the slope of n(σ) is slightly smaller than the slope of nSL(σ), and n(σ) is equal to nSL(σ) around σ=1.6 µm-1.
Fig. 16. Refractive index n(σ) of soda lime glass plate (red dashed curve) measured in (a) case 1, (b) case 2, and (c) case 3. Blue solid curve is refractive index nSL(σ). RMSD{n(σ)-nSL(σ)} in case 1 to case 3 are 2.1 × 10−4, 2.1 × 10−4, and 2.2 × 10−4, respectively.
Fig. 17. Distributions of D21(σ) (blue curve) and D31(σ) (green curve) calculated from the measured results shown in Fig. 16.
Table 5. Measured values of T at soda lime glass plate in three cases.
5. Conclusion
The characteristics of the fitting method were examined with simulations. It was made clear that the thickness of object, for which the phase refractive index can be measured exactly, is limited to Tmax because the fitting method produces errors in the fitted coefficients of a fitting function. It was shown that the polynomial fitting function with three terms provides a larger value of Tmax than that with four terms. Values of the thickness Tmax at the glass plates of BK7, fused silica, and soda lime were less than about 120 µm. In order to measure refractive indexes of the three kinds of glass plates of about 1 mm thickness without using the fitting method, the 2π phase ambiguity contained in the detected spectral phase was determined by using an assumption that an actual refractive index measured in experiment is almost the same as the well-known data of refractive index. The thicknesses of the glass plates could be measured with an error less than 45 nm from the four spectral phases detected with different optical configurations of the interferometer. The refractive indexes of the three glass plates could be measured with an error less than 8 × 10−5 from the spectral phase through the determination of the phase ambiguity. It was made clear that the actual refractive indexes measured in the experiments were slightly different from the well-known data of refractive indexes about the slope and the constant value in the distributions.
Appendix 1
The refractive index is given by
where C = 2Tn(σ)σ is a measured value without any error and C can be regarded as a constant value. When error Δn in n(σ) is caused by error ΔT in a measured value of T, the following equation holds under the condition of ΔT/T<<1:Hence,
From C = 2Tn(σ)σ, the following equation is obtained:
Equation (14) means that the error ratio in T are the same as that in Δn. The minus sign means that the measured refractive index decreases when ΔT increases.
Appendix 2
From φM(σ)/2π=a0 + a1σ+a2σ2 + ·×× and n(σ) = [φM(σ)/2π]/2Tσ+p/2Tσ, a constant value in n(σ) is given by
When T and a1 change to T + ΔT and a1 + Δa1 at case 2 or case 3, a constant value in Di1(σ) is expressed as
By ignoring a small value of Δa1ΔT/2T2, CD = -a1ΔT/2T2 + Δa1/2 T is obtained.
Funding
National Natural Science Foundation of China (62005086).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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