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Abstract
Precise measurements of the geometrical thickness of a sample and its refractive index are important for materials science, engineering, and medical diagnosis. Among the possible non-contact evaluation methods, optical interferometric techniques possess the potential of providing superior resolution. However, in the optical frequency region, the ambiguity in the absolute phase-shift makes it difficult to measure these parameters of optically thick dispersive materials with sufficient resolution. Here, we demonstrate that dual frequency-comb spectroscopy can be used to precisely determine the absolute sample-induced phase-shift by analyzing the data smoothness. This method enables simultaneous determination of the geometrical thickness and the refractive index of a planar sample with a precision of five and a half digits. The thickness and the refractive index at 193.414 THz (λ = 1550 nm) of a silicon wafer determined by this method are 0.5204737(19) mm and 3.475625(58), respectively, without any prior knowledge of the refractive index.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The geometrical thicknesses and the complex refractive indices of layers (hereafter referred to as d and N = n + ik, respectively) are fundamental parameters for materials science, and thus ultra-precise determination of these parameters in a wide range of materials is important. Laser interferometry is commonly used for the dimensional metrology of optically thin samples (i.e. layers whose thickness is comparable with the wavelength of light, λ). In this method, d and n(ω) of a planar sample can be determined by analyzing the absolute sample-induced phase change ΔΦabs(ω), where ω is the optical angular frequency. However, in conventional interferometry of optically thick samples (i.e. d>λ), we need to consider the ambiguity in the value of ΔΦabs(ω) [1], because in this case, ΔΦabs(ω) is usually not identical to the measured phase change Δϕexp(ω) but can differ by multiples of 2π:
The development of the optical frequency comb (OFC) technology has revolutionized the field of precision metrology [17–20]. Dual-comb spectroscopy (DCS), which uses two OFCs [21–26], enables us to unambiguously determine ΔΦabs(ω) in open-air ranging applications [27–30] by combining time-of-flight and interferometric measurements, and makes absolute distance measurements with a dynamic range from kilometers to nanometers possible. This concept has also been applied to materials science in order to determine d and ng(ω) of samples like wafers [12,16,31,32]. However, because the distortion of the optical wave-packet obstructs the determination of ΔΦabs(ω) due to the dispersion of n, the same precision as in open-air ranging has not been achieved.
Here, we demonstrate the ultra-precise determination of d and n(ω) of an optically thick dispersive material by DCS. We determine ΔΦabs by analyzing the smoothness of the n(ω) and k(ω) spectra derived from the experimental data (i.e. the derived spectra should exhibit no artificially induced oscillating behavior with respect to frequency). We show that d and n(ω) of a silicon wafer can be simultaneously determined with a precision of five and a half digits: d = 520473.7 ± 1.5 nm and n = 3.47563 ± 0.00001 at 193.414 THz with a relative uncertainty of ∼3×10−6. This precision of n(ω) is the best value that has so far been reported for a planar sample, and is comparable with the precision of the interferometric measurement of prism-shaped samples [5]. Moreover, the value of n(ω) is consistent with the literature value within the previously reported uncertainty.
2. DCS measurement
Figure 1 shows our DCS system consisting of two OFCs (S-comb and L-comb). In this work we use DCS as described in Ref. [33]. We used two OFCs based on erbium (Er)-doped-fiber-based mode-locked lasers with slightly different repetition rates (fr and fr-Δfr). The two OFCs are referred to as the signal (S) comb and the local (L) comb. In our experiment, fr was ∼48 MHz and Δfr was set to about 69 Hz. We used two output beams from each laser, which were further amplified by Er-doped fiber amplifiers (EDFAs) and their spectral bandwidths were broadened by highly nonlinear fibers (HNLFs): Two of the four output beams were used to measure the interferograms as explained below. The other two output beams were used to detect the carrier-envelope offset frequencies of the S- and the L-combs, fS,ceo and fL,ceo, by f–2f interferometers. We also detected the beat notes between a 1.54-µm continuous-wave (CW) laser and one of the comb modes for both the S-comb and the L-comb (fS,beat and fL,beat). fS,ceo and fL,ceo were phase-locked to the radio frequency (RF) reference signals generated by function generators. Each carrier-envelope offset frequency was phase-locked via a feedback to the current of the corresponding pump laser diode (LD). fS,beat and fL,beat were also phase-locked to the RF reference signals via feedback loops [to the electro-optic phase modulator (EOM), to the piezo-electronic transducer, and to the Peltier element in each oscillator] with different time constants. We controlled the values of fS,ceo–fL,ceo and fS,beat–fL,beat to obtain multiples of Δfr in order to satisfy the condition for coherent averaging [34]. As the CW laser is stabilized to an ultra-stable cavity, the relative line width between the S-comb and the L-comb is below 1 Hz, which is the resolution limit of the spectrum analyzer.
Fig. 1. Experimental DCS setup. LD: laser diode, EOM: electro-optic modulator, PC: personal computer, S: sample, LPF: low-pass filter, AMP: amplifier, BS: beam splitter. The symbol “×2” indicates the frequency doubling in a periodically poled lithium niobate crystal.
The S-comb beam for the measurement of the interferogram (Fig. 1; red thick line) was first divided into two paths by a beam splitter (BS1). The paths for the transmitted and reflected beams are referred to as the reference (ref.) and sample paths, respectively. The beam that entered the sample path, was reflected by a mirror and passed through the sample (S), and then was reflected by another mirror to combine it again with the beam from the ref. path by BS2. The difference between the sample and ref. path lengths allows us to temporally separate their interference signals, which are generated by combining these beams with the L-comb beam. The combined S-comb beam is combined with the L-comb beam (Fig. 1; blue thick line) by BS3. After that, the bandwidth of the combined beam (192.5–197.5 THz) was spatially filtered by a grating pair and an optical slit to avoid aliasing effects. Note that the obtained spectral bandwidth is significantly narrower than fr2/2Δfr, which is the theoretical maximum of the optical bandwidth that can be mapped to the RF domain in a DCS measurement [25]. Finally, the beam was detected by an InGaAs detector. The signal was electrically filtered by a low-pass filter (LPF) with a cutoff frequency of 21.4 MHz, amplified by an amplifier (AMP), and sampled by a 14-bit digitizer. The LPF served as an anti-aliasing filter.
For the demonstration of our method, we used an undoped silicon wafer with a nominal thickness of 525 ± 25 µm as sample. The complex refractive index of silicon is frequency dependent around 192.5–197.5 THz [35]. We mounted the sample on a mirror mount and fixed this holder on a motorized translation stage. By moving the translation stage, we can easily obtain the interferograms with and without the sample. The angle of the sample surface was carefully aligned to achieve normal incidence of the optical beam within an uncertainty of ≈ 5 mrad. The uncertainty was estimated by observing the spot of the light reflected from the sample surface (on a plane ∼300 mm in front of the sample) and measuring the displacement of this spot with respect to the incident S-comb beam. After the alignment, the spot position was almost identical with the position of the incident beam ∼300 mm in front of the sample. All measurements were performed in ambient atmosphere at room temperature (23 ± 2 °C).
For the data acquisition process we first moved the sample to the measurement position. Then, we measured about 60 interferograms (recording time: 60/Δfr ≈ 0.87 s). The averaged interferogram is referred to as U1,w/(t) and contains the sample-related information of the first dataset. As 0.87 s is shorter than the coherence time of our DCS system, it is possible to average these 60 interferograms in the time domain. Then we removed the sample from the measurement position by moving the translation stage and again measured about 60 interferograms, and the averaged interferogram is referred to as U1,w/o(t). We repeated the whole procedure for about 10 hours to obtain 5500 pairs of Uw/(t) and Uw/o(t). To identify each single dataset, we use the notation Ui,w/(t) and Ui,w/o(t) with i = 1 to 5500.
3. Data analysis methods
3.1. Criterion for the correct value of M
Figure 2 describes the underlying problem in the DCS measurement; the phase ambiguity of the experimental Δϕexp(ω) data [Fig. 2(a); red curves] and the resulting possible candidates for ΔΦabs(ω) [Fig. 2(b)]. Here, we explain how to solve this problem of ambiguity in ΔΦabs(ω) obtained by DCS. As a general approach, we first construct a continuous Δϕ(ω) curve [Fig. 2(a); green curve] by unwrapping the data segments starting from a selected data segment, and define the M(ω) of this starting segment as M. Therefore, we consider the equation ΔΦabs(ω)=Δϕ(ω) + 2πM with M being unknown. Our solution is based on a single simple criterion that states that the N(ω) spectrum [ = n(ω) + ik(ω)] is the smoothest when we derive n(ω) and k(ω) from the experimental data using the correct values of M and d. When the n(ω) and k(ω) spectra are calculated with incorrect values of M and d, these spectra contain an artificial oscillating component. This oscillating behavior with respect to frequency is caused by the incorrect analysis of the effect of the multiple internal reflections in Δϕexp(ω) and in the power transmittance T(ω). Since the complex refractive indices of most naturally occurring materials do not exhibit any oscillating behavior with respect to frequency, the smoothness of n(ω) and k(ω) can be used as an indicator of the correctness of M and d.
Fig. 2. (a) Schematic of Δϕexp(ω) obtained in a DCS experiment (red lines). Because the range of Δϕexp is restricted to 0–2π, Δϕexp(ω) is discontinuous and the recorded data is divided into locally continuous data segments. Due to the ultra-high frequency resolution of DCS, a continuous Δϕ(ω) curve can be obtained by unwrapping the data segments starting from a selected data segment (green line). (b) As long as the value of M(ω) of the starting segment of the unwrapping procedure, which is defined as M, is not known exactly, there are several possible candidates for ΔΦabs (green lines distinguished by the different possible values of M denoted by M1, M2, …, Mn). To determine ΔΦabs(ω), we have to determine M.
Figure 3 shows the flowchart of our analytical procedure. First, we measure the interferograms with and without the sample, Uw/(t) and Uw/o(t), respectively [Fig. 3(a)]. Then, by Fourier transform (FT) of each interferogram, Δϕ(ω) and T(ω) are derived [Fig. 3(b)]. Here, Δϕ(ω)=ϕw/(ω)-ϕw/o(ω) and T(ω)=Aw/(ω)/Aw/o(ω), where ϕw/(ω) [ϕw/o(ω)] and Aw/(ω) [Aw/o(ω)] are the phase and the power amplitude of Uw/(t) [Uw/o(t)], respectively. N(ω) can be calculated from T(ω), Δϕ(ω), M, and d by solving the analytical Eqs. (10) and (11) in Section 3.3 for numerous pairs of (Mj,dj) [Fig. 3(c)]. The pairs (Mj,dj) describe the considered candidates for the correct combination of M and d, where j runs over all elements of a two-dimensional array that consists of all possible combinations of the candidate values of M and d. To evaluate the smoothness of n(ω) and k(ω) derived for each pair (Mj,dj), we use the following two functions En(M,d) and Ek(M,d) (See Section 3.4 for details):
Fig. 3. Flow chart of the analytical procedure. (a) The interferograms with and without the sample, Uw/(t) and Uw/o(t). Uw/(t) contains the echo signals due to the multiple internal reflections. (b) Δϕ(ω) and T(ω) calculated by Fourier transform of each interferogram. Δϕ(ω) and T(ω) show oscillating behaviors with respect to frequency due to multiple internal reflections. The colored curves visualize the ambiguity in the value ΔΦabs(ω). (c) By using T(ω), Δϕ(ω), Mj, and dj, n(ω) and k(ω) are calculated for each pair (Mj,dj) as shown by the curves with different colors. (d) We consider the pair (Mj,dj) that minimize the value of En(M,d)+Ek(M,d) (explained in the text) as the correct values.
Here, the subscript i identifies the discrete frequency values of the FT data in the range from the lowest to the highest frequency in the observable spectral region. The correct M and d values are those that minimize En(M,d)+Ek(M,d) [Fig. 3(d)].
Note that a similar analytical procedure has been proposed to determine d in the terahertz (THz) frequency range [36,37], and this idea has been applied to the characterization of various materials [38–41]. In the THz frequency range, the value of M can be easily determined by extrapolating Δϕ(ω) to ω=0, where ΔΦabs should be always zero. This is possible, because the wavelength of THz waves (on the order of millimeter) is comparable to the optical path length through the sample in the case of typical values of d and n. This results in a phase rotation of only a few times inside the sample. In addition, the relatively broad spectra that can be obtained by THz time-domain spectroscopy (typically spanning more than one octave), allow an easy extrapolation of Δϕ(ω) to ω=0. On the other hand, in the optical frequency range, the measurable wavelength (∼1.55 µm) is in general much smaller than the thickness of the sample (≈500 µm in our case). In this case, the phase of the light rotates many times inside the sample, and thus M becomes very large. This leads to the ambiguity in ΔΦabs. Moreover, because the spectra obtained by DCS are typically relatively narrow, the determination of M is quite difficult compared to the case of THz time-domain spectroscopy. This distinct difference between the determination of M and d at terahertz and at optical frequencies makes the determination of N(ω) in the optical frequency range more difficult.
3.2. Calculation of the transmittance and phase difference spectra
As shown in Fig. 4(a), the interferogram contains two interference signals (that related to the sample path and that related to the ref. path), which are temporally separated due to the difference between the optical path lengths of the sample and ref. paths. To use the sample-path and ref.-path information separately, we extracted 7500 data points (corresponding to 221 ps) around the maximum intensity of each interference-signal peak and replaced the other data points with zeros. When the sample is located at the measurement position, the obtained interferograms containing only the data from the sample path and the ref. path are defined as ${U_{\textrm{i,w/}}^{\textrm{sam}}(t )}$ and ${U_{\textrm{i,w/}}^{\textrm{ref}}(t )}$, respectively. When the sample has been removed by using the translation stage, the obtained interferograms are defined as ${U_{\textrm{i,w/o}}^{\textrm{sam}}(t )}$ and ${U_{\textrm{i,w/o}}^{\textrm{ref}}(t )}$, respectively. By using these definitions, the transmittance Ti(ω) and the phase difference Δϕi,exp(ω) can be written as follows:
Fig. 4. (a) Typical averaged interferogram including the information of the sample, Uw/(t). The features related to the sample can be confirmed in the red curve in (b). (b) Typical averaged interferogram data in the vicinity of the interference signals related to the sample path in the cases with (red) and without (blue) the silicon (Si) wafer. Data are offset for clarity. (c) Averaged power transmittance T(ω) and (d) unwrapped phase difference, Δϕ(ω), derived from Uw/(t) and Uw/o(t).
Here, F means Fourier transform. As shown in Eqs. (4) and (5), we use the ref.-path information to cancel fluctuations in our optical setup, and we also use it as the reference phase.
3.3. Calculation of the complex refractive index
For the calculation of N(ω), we require T(ω), Δϕ(ω), M, and d. Thus, N(ω) depends on ΔΦabs(ω) since ΔΦabs(ω) = 2πM +Δϕ(ω), where Δϕ(ω) is the unwrapped curve of the measured phase difference defined in Eq. (5).
T(ω) and ΔΦabs(ω) are connected with N(ω)=n(ω)+ik(ω) and d as shown in the following equation [32]:
3.4. Characterization of the levels of smoothness of n(ω) and k(ω)
For a correct characterization of the smoothness, it is important that the evaluation function only considers the artificially induced oscillating component in n(ω) and k(ω), that is, the oscillating behavior as a function of frequency that is caused by an incorrect analysis of the effect of multiple internal reflections in the T(ω) and Δϕ(ω) spectra. In undoped silicon, n(ω) linearly increases with ω in the measured frequency range. If we use incorrect values for M and d, n(ω) increases with ω but also contains a superimposed oscillating component, i.e. n(ω) can be expressed as n(ω)=α+βω+f(ω), where α and β are constants and f(ω) is the oscillating component. Therefore, by using the second-order derivative, we can evaluate f(ω). Here, we consider that the second-order derivative is sufficient to evaluate the levels of smoothness of n(ω) and k(ω), because the second-order derivatives of n(ω) and k(ω) oscillate around zero. This means that all components except the oscillating component have been successfully removed from the data.
4. Results
4.1. Analysis of the interferogram data
Figure 4(a) shows a typical averaged interferogram obtained with the sample, Uw/(t) (time used for averaging: ≈ 0.87 s). The peak around 10.3 ns (11.2 ns) corresponds to the interference signal between the L-comb beam and the S-comb beam that has passed through the ref. (sample) path. Figure 4(b) shows an enlarged view of the interference signals related to the sample path for two conditions: with (red data, Uw/) and without the sample (blue data, Uw/o). The interference signal in Uw/(t) has several equally-spaced echo signals, which are a result of the multiple reflections inside the sample [31]. The averaged T(ω) and the Δϕ(ω) spectra derived from the interferogram data [each spectrum was averaged 1833 times] are plotted in Figs. 4(c) and 4(d), respectively. In Fig. 4(d), the unwrapping procedure for Δϕ(ω) starts at 193.414 THz where Δϕexp(ω) is close to 0 rad. Here, we selected 193.414 THz as the starting frequency, but since the choice of the starting point does not affect the analytical results, we are also able to choose other frequencies. The important feature of the T(ω) and Δϕ(ω) spectra is that both show an oscillating behavior with respect to frequency, corresponding to the etalon fringes from multiple internal reflections.
Hereafter, we explain the protocol used to precisely determine N(ω). In this paper, we prepared about 25 million pairs of (Mj,dj) (251 ≤ Mj≤1500; 515 µm ≤ dj≤ 535 µm with a discretization of 0.001 µm). We calculated n(ω) and k(ω) for each pair (Mj,dj) and evaluated E(M,d)=En(M,d)+Ek(M,d). Figure 5(a) shows the two-dimensional colormap of E(M,d) on a logarithmic scale. E(M,d) has a clear global minimum at M = 831 and d = 520.474 µm. Figure 5(b) plots the line profile of E(M,d) at M = 831 as a function of d, and the oscillation of E observed in this figure has a period of λ0/2nair (see Supplement 1), where λ0 is the center wavelength of the interference signal. E(M,d) at M = 831 exhibits a global minimum at d = 520.474 µm, and the values of the local minima gradually increase as the absolute difference between d and 520.474 µm becomes larger. Figure 5(c) plots the line profile of E(M,d) at d = 520.474 µm as a function of M; this curve has a minimum at M = 831 and increases monotonically as the absolute difference between M and 831 becomes larger. Figure 5(d) shows E(M,d) in the vicinity of the global minimum. Note that E(M,d) as a function of M has a well-defined minimum as shown in Fig. 5(c). In addition, E(M,d) for various values of d show a similar tendency around M = 831. This fact suggests that the correct value of M can be identified quickly by using appropriate minimum-seeking algorithms.
Fig. 5. (a) The two-dimensional colormap of E(M,d), which is used to evaluate the smoothness of n(ω) and k(ω), on a logarithmic scale. The region enclosed by the white dashed–dotted lines corresponds to the region shown in (d). (b) Cross section of the data in (a) at M = 831. (c) Cross section of the data in (a) at d = 520.474 µm. (d) The three-dimensional image of E(M,d) in the vicinity of the global minimum.
To evaluate the precision of our evaluation method, we estimated the values of the standard deviation of M and d for various averaging conditions of T(ω) and Δϕ(ω). We divided our 5500 spectra of T(ω) and Δϕ(ω) into subsets consisting of either Nave= 125, 250, 500, 1100, 1375, or 1833 spectra, which corresponds to either 44, 22, 11, 5, 4, or 3 subsets, respectively. Then, we averaged T(ω) and Δϕ(ω) over each subset for a selected Nave to determine the mean and the standard deviation of M and d (for example, Nave = 1375 allows us to determine the standard deviation using four data points for M). Figures 6(a) and 6(b) show the Nave dependences of M and d, respectively. As shown in Fig. 6(a), the averaged M values are always closest to 831, independent of Nave. In addition, the uncertainty in M is smaller than ±1 for Nave = 1100 and the uncertainty in M vanishes for Nave = 1375 because all four subsets result in the same value of M. This means that we can determine M uniquely when Nave ≥ 1375, and thus we can precisely determine ΔΦabs(ω). In Fig. 6(b), the mean value of d is 520.473–520.474 µm and its error bar is less than 5 nm for all considered Nave values. The error bar gradually becomes smaller for larger Nave values up to Nave = 1100, and it is smaller than 1.5 nm for Nave≥1100. Our obtained value is consistent with the nominal value of the sample (525 ± 25 µm), supporting the validity of our method.
Fig. 6. The Nave dependence of (a) M and (b) that of d. The error bars represent the standard deviation. (c) n(ω) and (d) k(ω) of the undoped silicon wafer for Nave = 1833. (e) Nave dependence of the standard deviation of n at 193.414 THz including the uncertainty of M (red dots) and calculated using M = 831 without uncertainty (blue dots). The black lines are proportional to Nave-1/2.
4.2. Complex refractive index of silicon
Since d in our experiment was determined precisely when Nave = 1833, we consider the n(ω) and k(ω) calculated from the three pairs of T(ω) and Δϕ(ω) obtained for Nave = 1833. The averaged n(ω) and k(ω) are plotted in Figs. 6(c) and 6(d), respectively. It is found that both n(ω) and k(ω) are rather smooth despite the oscillating behavior of the underlying data shown in Figs. 4(c) and 4(d). This smoothness indicates that we correctly accounted for the effect of multiple internal reflections. In addition, n(ω) increases with frequency (normal dispersion), while k(ω) is almost zero. Because these behaviors are consistent with the literature [35], it is proven that our evaluation method is valid.
As shown in the inset of Fig. 6(c) and in Fig. 6(d), even though the amplitudes of the oscillating component in the n(ω) and k(ω) spectra are minimized, a small residual oscillating component with an amplitude on the order of ∼10−5 is still observed. The origin of this small component is considered to be the uncertainty of T(ω). In contrast to k(ω), the uncertainty of T(ω) has only a weak influence on the uncertainty of n(ω), since n(ω) does not directly depend on T(ω) as shown in Eq. (10).
The red dots in Fig. 6(e) show the values of the standard deviation of n(ω) at 193.414 THz, σn, obtained for different Nave values using the corresponding standard deviation of M, σM: we calculated n(ω) and k(ω) of each subset based on the Nave-times averaged T(ω) and Δϕ(ω) spectra, and then evaluated σn using the n(ω) values of the subsets for the selected Nave. We find that σn gradually decreases in the range of Nave = 125–1100 and it dramatically decreases between Nave = 1100 and 1375 (more than 100 times smaller) because the value of M is uniquely determined for Nave≥1375. In the case of the blue dots, we first fixed the value of M to 831 and calculated d, n(ω), and k(ω) for all 5500 pairs of T(ω) and Δϕ(ω) without averaging, and then we selected Nave to define the subsets and averaged n(ω) over each subset. Finally, we evaluated σn using the averaged n(ω) values of the subsets. The blue dots in Fig. 6(e) thus correspond to the value of σn calculated using M = 831 and σM = 0. In this case, σn gradually decreases in the entire range of Nave and seems to approach a constant value for Nave≥1375. More details of σn are provided in the Discussion.
From Figs. 6(c) and 6(e), we find that n = 3.47563 ± 0.00001 at 193.414 THz for Nave = 1833. This value is consistent with the refractive index of silicon at 193.414 THz reported previously in [35], n = 3.4763 ± 0.002, which is the calibrated value at 23 °C. This quantitative agreement proves that our method can be used to determine n(ω) precisely.
5. Discussion
Firstly, we discuss the impact of the unique determination of M on the precision of n(ω). Because the leading term of n in Eq. (10) is 2πM, it is considered that σn is mainly determined by σM as long as M has not been determined uniquely; σn∼(2πσMc)/dω. For the presently used experimental conditions, σn is evaluated to be 3×10−3σM, which governs the degree of uncertainty of the red dots in Fig. 6(e) for Nave≤1100. Once M has been uniquely determined, σn dramatically decreases by more than two orders of magnitude. This drastic decrease in σn is the main impact of the unique determination of M.
Secondly, we discuss the Nave dependence of σn using the uniquely determined M [Fig. 6(e); blue dots]. When σM = 0, σn is proportional to the uncertainty of Δ(ω), σΔϕ(ω). In our experiment, σΔϕ(ω) is proportional to $N_{\textrm{ave}}^{ - 0.5}$ for Nave≤1100 and tends to be constant for Nave > 1100 (σΔϕ(ω)∼10 mrad, data not shown). This lower limit of σΔϕ(ω) determines the lower limit of σn, which is 1.2×10−5. Furthermore, the standard deviation of d, σd, does also not decrease further when Nave > 1100 [see Fig. 6(b)]. This indicates that the lower limits of σn and σd in our experiment are not determined by the stability of our DCS system, but other experimental uncertainties due to changes in environmental conditions (such as temperature, pressure, and humidity), which lead to changes in nair and n(ω). For instance, a temperature variation of 0.3 K causes a change in the refractive index of silicon on the order of 5.6×10−5, which may limit σn. A detailed list of the contributions to the experimental uncertainty is provided in Supplement 1, and the results are discussed at the end of this section.
Thirdly, we discuss why we were able to precisely determine d even at Nave≤1100 where σM ≠ 0. As mentioned above, the local minima of E(M,d) appear at an interval of λ0/2nair when plotted as a function of d. As the slope of E(M,d) is quite steep around each local minimum [see Fig. 5(b)], σd becomes much smaller than λ0/2nair (∼0.7 µm) even when we choose one of these local minima instead of the global minimum. In addition, because λ0/2nair is larger than the uncertainty of d for Nave = 125 in our experiment, we can choose the correct minimum, i.e., the global minimum, for all values of Nave as shown in Fig. 6(b).
Fourthly, we discuss the experimental requirements for a unique determination of M by our method. Our analytical protocol considers the smoothness of N(ω) to decide whether the values of d and M are correct. An N(ω) spectrum derived for an certain pair (Mj,dj) can contain an artificial oscillating component due to the actual interference between multiple-reflected optical pulses inside the sample. This means that the period of the artificial oscillating component in the derived N(ω) spectrum is determined by the optical-path difference between two adjacent optical pulses where one of these pulses is the result of multiple internal reflection. This optical-path difference is equal to 2n(ω)d. Because σd is small even at smaller Nave values as mentioned above, the uncertainty in 2n(ω)d is governed by σn when σM ≠ 0. When the value of M changes by +1, the value of n(ω) in our experiment changes on the order of 0.003, which causes a change in the period of the artificial oscillating component in n(ω) by 0.09%. As a result, the period of the oscillating behavior of Δϕ(ω) with respect to frequency that is inversely estimated from the calculated n(ω), is not consistent with the measured oscillation period of Δϕ(ω). Because of this difference, the difference between the phases of the inversely calculated Δϕ(ω) and the measured Δϕ(ω) increases by ∼0.6 mrad per oscillation. Because the number of oscillations in the measured Δϕ(ω) spectrum is ∼60 within the measured frequency region [see Fig. 4(d)], the maximum phase difference becomes ∼36 mrad. Thus, when σΔϕ(ω) is smaller than ∼36 mrad, it is possible to uniquely determine M. By accumulating the data at each frequency, we verified that σΔϕ(ω)≈10 mrad. This suggests that an ultra-precise determination of n(ω) requires a precise determination of Δϕ(ω), where σΔϕ(ω) is small enough to distinguish the small phase change associated with a change in M by ±1. A mathematical description of the criterion for the minimum of the evaluation functions is provided in Supplement 1.
Finally, we discuss the contributions of different factors to the measurement uncertainty, such as the refractive index of air (which can change due to fluctuations of temperature, pressure and humidity). In this study, the uncertainty in the measurement of d is considered to be limited mainly by the change in nair due to changes in temperature [Supplement 1, Table S1(a)]: The change in nair corresponds to the difference between the nair at the time of measuring the interferogram with the sample and the nair at the time of the following measurement without the sample (time difference about one minute). On the other hand, the uncertainty in the measurement of n(ω) is considered to be limited mainly by the uncertainty in the measurement of the temperature of the sample [Supplement 1, Table S1(b)]. This work is a proof of concept, and the factors that limit the uncertainty, were not investigated prior to the experiments. However, the results of Table S1 in Supplement 1 show that the limiting factors can be identified relatively easily. Table S2 in Supplement 1 shows the expected contributions to the measurement uncertainty in the case that the temperature, pressure, and humidity can be measured with a similar degree of precision as that of a typical length measurement. Table S2 in Supplement 1 suggests that both d and n can be measured with a relative uncertainty of 3×10−6 to 4×10−6.
6. Conclusion
In summary, we have demonstrated a method that is able to simultaneously determine d and n(ω) of an undoped silicon wafer with standard deviations of σd = 1.5 nm and σn = 1.2×10−5, respectively. We determined the absolute phase spectrum by DCS and a simple criterion with respect to data smoothness. DCS can be used to measure both amplitude and phase with high frequency resolution and accuracy. The dramatic improvement in the precision of the absolute sample-induced phase change ΔΦabs(ω) that is achieved by our method, allows us to measure d and n(ω) simultaneously with degrees of precision that were previously unachievable.
Funding
Japan Society for the Promotion of Science (JP18H02040, JP23560048, JP26800217); Core Research for Evolutional Science and Technology (JPMJCR19J4).
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.
Supplemental document
See Supplement 1 for supporting content.
References
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