1. Introduction

An optical resonator is a device consisting of two light reflective surfaces. In the case that the resonator length is an integer multiple of the half-wavelength of light, a large electric field intensity is confined within the optical resonator. Optical resonators are used for laser oscillators and spectrometers with a higher wavelength resolution than that of typical spectrometers [1,2]. A Fabry-Perot etalon (FP etalon) is one of the well-used optical resonators with two parallel partially reflective surfaces [3–6]. Based on the sensitive spectral responses of the FP etalon, several research works have been reported in which FP etalons were applied for precision measurement of various physical quantities, e.g., temperature [7,8], pressure [9,10], distance [11,12] and angle [13–15]. In the measurement systems using FP etalons, lasers are typically employed as the light source for multi-beam interference, which requires the light source to have a long enough coherence length that could cover the large number of round trips made by the higher order internally reflected beams within the two surfaces of the FP etalon. The light source is also required to have a good directionality for maintaining the condition of constructive interference between the beams. In addition, it is necessary to use a focusing lens to focus the transmitted beams from the etalon so that the beams can be coupled into a single-mode optical fiber located at the focal point of the focusing lens for the measurement by OSA. A good directionality of a beam is an important condition for the fiber coupling. In the case of a monochromatic continuous wave (CW) laser, light intensity-based measurements can be conducted by detecting the intensity change of the multi-beam interference light outputted from the cavity. Meanwhile, optical frequency-based measurements can be carried out by employing multiple CW laser sources with different wavelengths [16], or a wavelength-scanning laser source [17–19], as well as a white light source [20–22].

Recently, sensors employing mode-locked femtosecond lasers have been developed to improve the performance of conventional optical sensors [23–25]. The mode-locked femtosecond laser is an ultrashort pulsed laser that contains a frequency spectrum with equally spaced optical modes [26]. It combines the characteristics of a CW laser source, i.e., directionality, coherence and high intensity with that of a white light source having a broad optical spectrum. Such unique characteristics of mode-locked femtosecond lasers have been utilized in various absolute measurement systems [27–31] to overcome the limitations of systems using conventional CW laser and white light sources [32–36]. The wide and stable spectrum of a mode-locked femtosecond laser has also been utilized to construct spectral response-based measurement systems using FP etalons, such as a FP etalon angle sensor [37].

For a measurement system using a FP etalon, the uncertainty in the cavity length value is one of the most critical issues since the spectral response of the etalon can change sensitively as the cavity length value changes. It is thus necessary to determine the cavity length precisely for a high-resolution sensor using a FP etalon. The peak-to-peak method (the period tracking method) is the most well-used method for the absolute cavity length measurement by taking use of the spectral peaks in the fringe spectrum of the FP spectral response where the cavity length d can be obtained from the wavelengths λ₁ and λ₂ (or frequencies f₁ and f₂) of two neighboring peaks with a peak-to-peak spacing of the free spectral range (FSR) based on the relationship of d = λ₁ λ₂ ∕ 2n (λ₂ − λ₁) or d = c ∕ 2n (f₁ − f₂) at normal incidence. Assume the center wavelength of the input light is 1500 nm, the difference between λ₂ and λ₁ i.e., the wavelength FSR, is approximately 1 nm for d = 1 mm. In this case, if there is a measurement error δλ οf 0.02 nm in FSR, which is a practical limit of wavelength measurement by using a commercially available optical spectrum analyzer (OSA) [38], a relative measurement error of δλ∕FSR = 2% or an absolute uncertainty of 20 µm can be caused in the measurement of d. This value is consistent with that experimentally demonstrated in [39]. To improve the accuracy of the peak-to-peak method, Yi, et al., introduced a Fourier transform technique on the data analysis [40], and Guan et al., added a virtually variable reference FP interferometer in the measurement system [41]. However, the measurement accuracy of these methods was still limited to a level of sub-micrometer while the associated data processing and/or system configuration were getting more complicated. Recently, Sweeney et al. proposed a hybrid method for high-precision measurement of Fabry-Perot cavity displacements, i.e., the variations of absolute cavity length by combining the conventional peak-to-peak method (the period tracking method) with a phase tracking method. The former provided a coarse measurement of the initial cavity length, based on which the latter measured the cavity displacements with a nanometric precision [42]. An averaging operation was employed in the peak-to-peak method by using multiple peak data to reduce the measurement error of the initial absolute cavity length. However, due to the low accuracy of the conventional peak-to-peak method, the measurement accuracy of absolute cavity length was still limited on the order of sub-micrometer. In addition, none of these provided uncertainty analysis on the cavity length measurement results based on the “Guides to the Expression of Uncertainty in Measurement (GUM)” following the law of propagation of uncertainty [43], which is an internationally authorized way for reliable identification of the accuracy of a measurement method. It should be acknowledged that Huang et al. analyzed the measurement resolution of the peak-to-peak method following the general law of error propagation [44], which has a similar form as the law of propagation of uncertainty in GUM. However, only the influence of the OSA resolution was taken into account. Although it was reasonable for estimating the measurement resolution for the cavity length, it has not been followed the guidance provided by GUM and therefore could not be treated as a comprehensive uncertainty analysis for reliable identification of the accuracy of cavity length measurement.

In this paper, the wide spectral range associated with a good spatial coherence of a mode-locked femtosecond laser is utilized in the establishment of an improved peak-to-peak method for a further enhancement of the measurement accuracy of the cavity length measurement. In this new method, a femtosecond laser beam is projected into the cavity at normal incidence for obtaining a fringe spectrum of the etalon spectral response in the frequency domain with a set of equally spaced spectral peaks by an OSA. Instead of using two neighboring spectral peaks with a peak-to-peak spacing of FSR, a pair of peaks across an integer multiple k of FSR, i.e., with a peak-to-peak spacing of kFSR, is employed for evaluating the etalon cavity length d with a reduced measurement error based on the fact that the measurement error is inversely proportional to the peak-to-peak spacing kFSR, i.e., the larger k, the smaller measurement error. The proposal of this scheme is the most novelty of this paper, which has never been reported before. Assuming that the obtainable total number of spectral peaks is N within the operating bandwidth of the FP etalon determined by the finite spectral range of the femtosecond laser, there are maximumly N – k pairs of peaks that can be employed to evaluate a mean value of d for further reduction of the measurement error. However, although an increase of k will result in a reduction of the measurement error on one hand based on the nature of the peak-to-peak method, the effect of the averaging operation will also be reduced on the other hand due to the decrease in the averaging number of N – k. To solve this trade-off problem, an optimized peak-to-peak number k is then selected together with an optimized averaging number of N – k under the constrain of a fixed N to achieve a minimum measurement error of d, which is the second novelty point of this paper. An optical setup is constructed to carry out experiments for verifying the feasibility of the proposed method. Finally, the measurement uncertainty analysis is conducted based on GUM, where a combined uncertainty is obtained to estimate the closeness of the true value of cavity length with respect to the measured cavity length at a certain level of confidence or probability.

2. Principle of the improved peak-to-peak method

Figure 1 shows a schematic of the beam propagation in a FP etalon when a femtosecond laser is projected into the etalon. The incident beam is reflected repeatedly at the parallel reflective surfaces of the FP etalon and produces partially penetrating beams at each reflection. The generated beams by the FP etalon are divided into two groups according to the propagating direction, which are the transmitted beams and the reflected beams. When the transmitted beams are focused by a focusing lens, a multi-beam interference light can be observed at the focal plane of the lens. The interference condition is determined by the phase difference between neighboring transmitted beams. As expressed in Fig. 1, the phase difference (δ ) is related to the optical path difference, which is expressed by the following equation [4]:

 

Fig. 1. Beam propagations at a Fabry-Perot etalon and its fringe spectrum.

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Denote the obtainable total number of spectral peaks by N within the operating bandwidth, which is determined by the finite spectral range of the femtosecond laser. Figure 2 illustrates the fringe spectrum in a more detail where the peaks are numbered from 1, 2, … to N. The spacing between two neighboring peaks is equal to the free spectral range (FSR). Considering the m^th and the (m + k)^th peaks, Eq. (2) can be rewritten as:

 

Fig. 2. Obtainable fringe spectrum and the determination of $k$.

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By subtracting Eq. (3) from Eq. (4), the cavity length d can be evaluated as follows:

It should be noted that the error of Δf_{k_m} is not considered in Eqs. (5) and (6), which can be introduced from the identification errors of the peak frequencies f_m and f_m+k. On the other hand, as can be seen in Eq. (5), the accuracy of the cavity length measurement is basically determined by that of Δf_{k_m}, or the accuracies of the peak frequencies f_m and f_m+k, which are identified from the fringe spectrum. In practice, Δf_{k_m} has an identification error ε_{k_m} that can be caused by a number of factors, such as the resolution and accuracy of the OSA, the fitting accuracy of the spectral peak, etc. Without losing the generality, Δf_{k_m} can be expressed without losing the generality:

In this case, Eq. (5) can then be rewritten as:

Meanwhile, any two of the spectral peaks in the fringe spectrum can be selected for measurement of the cavity length as shown in Eq. (5). For the conventional peak-to-peak method, the cavity length is measured by using two neighboring peaks with a peak-to-peak spacing of Δf_{1_m} or FSR where k is equal to 1 [39]. On the other hand, as can be seen in Eq. (10), the measurement error δd_{k_m} in the cavity length d is inversely proportional to k. This means that the conventional peak-to-peak method has the maximum measurement error δd_{k_m} where k takes the minimum number of 1. Therefore, two peaks with a much larger spacing, i.e., k ≫ 1 are selected in the proposed method, which is referred to the improved peak-to-peak method. Although k can be as large as N − 1, where m = 1, corresponding to a minimum measurement error δd_{N-1_1}, there is only one sample of d can be obtained since there is only one pair of the spectral peaks at the two ends of the operating bandwidth can be utilized. The error can be further reduced by averaging the cavity length samples obtained under the same condition of m = 1 and k = N - 1 from multiple measurements of fringe spectrum by OSA. This method is simple in calculation and can improve the measurement accuracy compared to the conventional peak-to-peak method using two neighboring peaks. It can be an effective method if the OSA for the fringe spectrum measurement has a high speed and a high accuracy across the entire operating spectral bandwidth determined by the femtosecond laser source. The femtosecond laser source is also required to have a uniform spectrum across the spectral bandwidth or to have high intensities at the two ends of the operating bandwidth. However, it typically takes more than 1 minute for the OSA to acquire one set of a wide-range fringe spectrum measurement in high precision mode. Therefore, acquiring multiple sets of fringe spectrum measurement data (typically several tens of sets) for such an averaging operation would require a long time, which is too long from the point of view of practical applications, such as displacement and strain measurements. Furthermore, the averaging operation can only reduce random errors such as electronic noises or mechanical vibrations. The influence of thermal drift, a slowly time-varying systematic error, on the measurements would increase significantly over such a long measurement time, which cannot be reduced by the averaging operation. In addition, a femtosecond laser source typically does not have a uniform spectrum, often with low intensities in the lower and/or the upper limit of the bandwidth.

In the proposed method, k is taken to be larger than 1 and smaller than N so that an averaging operation using N − k samples of d can be carried out to further reduce the measurement error of cavity length. As illustrated in Fig. 2, the N − k samples of d can be denoted by d_{k_m}(m=1,2, … N − k). d_{k_m} can then be expressed by

The mean cavity length ${\bar{d}_k}$ can then be obtained by the following equation.

As can be seen in Eq. (12), for a fixed N, a decrease of k will increase the averaging number of N − k, which reduce the measurement error in the cavity length. However, as shown in Eq. (10), a decrease of k will increase the measurement error in the cavity length due to the decrease of kFSR in Δf_{k_m}. To solve this trade-off problem, Eq. (11) is substituted into Eq. (12) to evaluate the error $\delta {{\bar{d}}_k}$ in $\mathrm{\bar{d}}$_k as follows:

As stated before. ε_{k_m} is a random error with a very small amplitude. Therefore, $\mathop \sum \limits_{m = 1}^{N - k} {\varepsilon _{k_m}}$ in Eq. (14), i.e., the sum of ε_{k_m} (m=1, 2, N-k), would oscillate randomly within a sufficiently small amplitude range when k is changed. In other words, the $\mathrm{\bar{d}}$ change of $\mathop \sum \limits_{m = 1}^{N - k} {\varepsilon _{k_m}}$ associated with the change of k is negligible. Consequently, based on Eq. (14), the error δ_k is expected to be the smallest when:

In other words, N∕2 is the optimized number of k for the improved peak-to-peak method where a minimum measurement error of the cavity length can be achieved. It should be noted that the optimized number of k may vary within a small range around N∕2 if the oscillation amplitude of ε_{k_m} is large in a practical measurement system. The improved peak-to-peak method only needs to acquire one set of the fringe spectrum measurement by the OSA, which is advantageous to measurement time. A large number of spectral peaks of N can be detected over the operating bandwidth by utilizing the wide spectral range of a mode-locked femtosecond laser, in which N/2 pairs of spectral peaks with a peak-to-peak spacing of N/2 multiples of FSR can be used for the averaging operation. As a result, the proposed method has the advantages of both high accuracy and high speed. It should be noted that since the averaging operation is made with the spectral peaks across the operating bandwidth, not only the random sampling errors of the OSA but also the residual calibration errors of the OSA can be reduced. This is another advantage of the proposed method.

3. Experiments

3.1 Construction of the optical setup

Figure 3 shows a schematic of the experimental setup for feasibility tests of the improved peak-to-peak method using a mode-locked femtosecond laser. The femtosecond laser (C-fiber HP, Menlo systems) with a bandwidth between 182.8 and 206.7 THz, and a repetition frequency of 100 MHz, is projected into the FP etalon by a collimator lens. In the experiment, an air-gapped FP etalon (Light Machinery, reflectivity 70%, FSR 200 GHz, operating bandwidth between 187.4 and 199.8 THz) with a designed cavity length of 750 ± 2 µm was used as the specimen [45]. The FP etalon was mounted on a rotation stage with a tilt angle control to align an incident angle precisely, which was equivalent to the reflection angle at the cavity for the air-typed FP etalon. From the repeated reflections at the FP etalon, both transmitted beams and reflected beams were generated. The reflected beams were guided to a beam profiler (BP209IR1/M, Thorlabs) through a beam splitter and a focusing lens (focal length 31 mm) for alignment of angular position of FP etalon cavity with respect to the optical axis of the femtosecond laser. The transmitted beams were collected by a single-mode optical fiber located at the focal point of the focusing lens for the cavity length measurement based on the improved peak-to-peak method. The fringe spectrum of the transmitted beams was detected by the OSA (AQ6370C, Yokogawa. Co) connected with the optical fiber.

 

Fig. 3. A schematic of the experimental setup for feasibility tests of the improved peak-to-peak method.

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The incident angle to the FP etalon was precisely aligned to be approximately zero using the reflected beams and the beam profiler. An autocollimation method was employed for the measurement of the incident angle. Figure 4 shows a schematic of the alignment method. First, a reference beam was made by placing a retroreflector at an optical axis like shown in Fig. 4 (a) and its beam spot position was measured by the beam profiler placed at the focal point of the focusing lens. Then, the beam spot of the beam reflected at the surface of the FP etalon was measured with the identical configuration. The incident angle is a composition of the x-axis rotation denoted as the rotation angle, and the y-axis angle denoted as the tilt angle. In this case, the rotation angle θ_x and the tilt angle θ_y can be expressed by using the beam spot positions measured by the beam profiler, as the following equations:

 

Fig. 4. A schematic of the autocollimation method (a) detections of a reference beam and the reflected beam, (b) positions of the detected beam spots at the image plane.

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Figure 5 shows the measured Y and Z -directional positions of a reference beam and the reflected beam. The beam spot data for both the reference beam and the reflected beam were measured for 120 s by the beam profiler. Note that before the measurement of the reflected beam spot, the position of the FP etalon was precisely adjusted to make both Δs_H and Δs_V to be the smallest. From the averaged beam spot positions evaluated from the results in Fig. 5, θ_x and θ_y were evaluated to be 0.0122 degrees and 0.0140 degrees, respectively, from which the reflection angle θ was identified to be 0.0186 degrees.

 

Fig. 5. Results of beam spots detected for 120 s (a) horizontal $k$ position of beam spots (b) vertical position of beam spots.

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3.2 Detection of peak frequencies

The measurement of the fringe spectrum for the FP etalon was made at a wavelength bandwidth between 1500 nm and 1600 nm, a resolution of 0.02 nm, and a sampling interval of 0.004 nm. It took more than 1 minute for such a measurement. The obtained values were converted into the frequency domain (from 187 to 200 THz). Figure 6(a) shows an obtained transmitted spectrum. From the obtained fringe spectrum, 64 peak frequencies were obtained in the overall range. Figure 6(b) shows the bandwidth between 192.1 and 192.5 THz, which is a part of the spectrum in (a), for showing the shape of the spectrum. To evaluate the mean cavity length from this spectrum, all the available peak frequencies should be detected by employing a precise peak evaluation method. The effect of the asymmetricity of the spectrum on the peak evaluation should be also considered [46]. Therefore, in this paper, the peak frequencies were detected based on the Savitky-Golay differentiation filter. This filtering method evaluates the derivative line using intensities of adjacent sampling points weighted with provided convolution coefficients. Based on this, the spectrum was differentiated to a cubic polynomial slope using intensities of 6 adjacent sampling frequencies by the following equation [47]:

 

Fig. 6. Measurement results (a) obtained entire fringe spectrum and (b) the fringe spectrum between 192.1 and 192.5 THz, (c) a first derivative spectrum of (a), (d) the derivative spectrum between 192.1 and 192.5 THz, (e) evaluated peak frequencies and (f) a schematic of the detection of peak frequencies.

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3.3 Evaluation of the mean cavity length

The evaluation of the mean cavity length based on the proposed mechanism was conducted. Based on N = 62, the most precise mean cavity length was expected to be obtained at k = 31 according to Eq. (15). A refractive index value of the FP etalon medium was set to be 1.000273 [48]. Figure 7 shows evaluated cavity length samples d_{k_m} at k = 31 based on Eq. (8) with the comparisons of samples at k = 1 and k = 10. As shown in the figure, the standard deviation of the cavity length samples was 45.580 nm for k = 31, which was significantly reduced from 952.973 nm for k = 1, and 176.521 nm for k=10. This result is consistent with Eq. (9) where the deviation of the cavity length samples is inversely proportional to k. In addition, all the cavity length samples in the figure showed random distributions, which are consistent with that of the identification error ε_{k_m} in Eq. (9). The randomness of the cavity length samples thus demonstrated the validity of the assumption for the randomness of ε_{k_m}, which is employed to derive the optimized k in Eq. (15).

 

Fig. 7. Evaluated cavity length samples at (a) k = 31, (b) k = 10, and (c) k = 31.

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Next, 100 times repeated measurements were conducted to observe errors in the mean cavity length. The fringe spectra were repetitively detected by the OSA at a same configuration and the mean cavity length values were evaluated from each differentiated fringe spectrum. The measurements were conducted in a laboratory environment, in which room temperature was controlled at 17 °C ± 0.5 K. Figure 8(a) shows the obtained mean cavity lengths at k = 31. The mean cavity lengths at k = 1 and k = 10 are shown in Fig. 8(b) and (c) respectively for a comparison. Figure 8(d) shows the cavity lengths at k = 61, representing the lowest δd_{k_m} obtainable without mean calculation. The standard deviation of the mean cavity length values at k =31 was evaluated to be 3.841 nm, which was smaller than 9.277 nm at k = 1, 5.307 nm at k = 10, and 9.316 nm at k = 61. The standard deviations of the mean cavity length for entire k values are compared in Fig. 8(e). The reason for the difference in mean cavity lengths at each k value was that the values for a fixed trial in Fig. 8 represents ${\bar{d}_k}$ expressed by Eq. (13), where $\delta {\bar{d}_k}$ is the mean measurement error of the real cavity length d defined by Eq. (14). In the proposed, the minimum $\delta {\bar{d}_k}$ can be reached by taking k = N / 2 as presented by Eq. (15). In other words, the mean measurement error gets minimum value at k = 31 and gets minimum at k = 1 or k = N - 1 = 61. From this perspective, since the mean cavity length of each trial number is a difference between the real cavity length d and the error $\delta {\bar{d}_k}$, the mean cavity length ${\bar{d}_k}$ takes its maximum value when k = 31 and takes its minimum value when k = 1 or k = 61, provided $\delta {\bar{d}_k}$ is a positive value. In addition, the standard deviation of ${\bar{d}_k}$ for the group of the trials reaches to is minimum and maximum at k = 31 and at k = 1 or k = 61, respectively, because the mean measurement error $\delta {\bar{d}_k}$ at k = 31 is the lowest. This is consistent with the results in Fig. 8. Thus, an improvement on the precision of the mean cavity length by the optimization of k value was verified. By choosing the mean cavity length at k = 31, the cavity length of the FP etalon was evaluated to be 751.354 µm. This value was consistent with the designed cavity length of 750 ± 2 µm by the manufacturer [45], indicating the feasibility of the measurement result by the proposed method. Meanwhile, the obtained errors from k = 20 to k = 40 varied from 3.841 nm to 4.018 nm and these changes were relatively stable compared to the decrease in the errors from k = 1 and 61.

 

Fig. 8. Results of repeated measurements for (a) k = 31, (b) k = 1, (c) = 10, (d) k = 61, and (e) standard deviations of mean cavity length for entire k values.

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4. Uncertainty analysis

Generally, it is very important to verify the measurement accuracy of the FP cavity length. A conventional way for the verification is to compare directly with the result made by a reference method or instrument with a higher accuracy. However, this was difficult in this kind of high-precision measurement of cavity length where such a reference is not available. In the previous section, the measurement result of cavity length by the proposed method was verified by comparison with the cavity length value provided by the etalon manufacturer. However, the tolerance range of the manufacturer value was ± 2 µm, which was too large to accurately verify the measured cavity length value by the proposed method. For this reason, in this section, the authors carry out the uncertainty analysis based on “Guides to the Expression of Uncertainty in Measurement (GUM)” [43], which an internationally authorized way to evaluate the reliability of a measurement result without directly comparing with the true value. A combined uncertainty value obtained by the uncertainty analysis based on GUM provides an estimation on the closeness of the true value of cavity length with respect to the measured cavity length at a certain level of probability or confidence. In other words, the combined uncertainty value is a quantitative measure for the accuracy of the measurement result.

The mean cavity length values at k = 31 and the standard deviation obtained in the repeated measurements were used in the uncertainty analysis. Denote standard uncertainty of the mean cavity length by u_mean. The standard uncertainty u_mean is contributed by standard uncertainties of cavity length samples d_{k_m} used in the averaging operation, which can be expressed as follows:

The standard uncertainty u_{d_m} then can be evaluated by the following equation:

4.1 Uncertainty of refractive index

First of all, the uncertainty of the refractive index of the FP etalon medium ${u_n}$ was considered. The refractive index can be influenced by changes in temperature, pressure, and humidity of the FP etalon cavity. However, it was difficult to measure atmospheric conditions inside the FP etalon because the cavity was in closed space by the structure of the FP etalon. Therefore, an uncertainty value of refractive index of air at room temperature was then referenced from literature. Atmospheric pressure at laboratory environment was considered to be 1 atm and relative humidity was controlled below 50%. Temperature changes during the measurement were controlled to be 17 °C ± 0.5 K. Thus, u_n was selected to be 5 × 10⁻⁷ [48].

4.2 Uncertainty of reflection angle at the cavity

The uncertainty of the reflection angle at the FP etalon cavity u_θ was then evaluated. Since the reflection angle was a composition of the tilt angle and the rotation angle, u_θ is contributed by uncertainty of the tilt angle u_{θ_x} and uncertainty of the rotation angle u_{θ_y}. Therefore, uncertainties u_{θ_x} and u_{θ_y} were firstly evaluated. As described in the earlier section, the angular position of the FP etalon cavity was aligned by the autocollimation method, which considered the reference beam made by the retroreflector as the beam parallel to an optical axis. However, due to a beam deviation of the retroreflector, an angular error between the reference beam and the optical axis can be occurred. Denote the angular error at the retroreflector by θ_retro, equations for the rotation and the tilt angles in Eqs. (16) and (17) can be modified as follows:

The maximum θ_retro provided by the manufacturer was 3 arcseconds [49]. Based on Eqs. (26) and (27), the standard uncertainty u_{θ_m} is contributed by three parameters, which are uncertainty of θ_retro, uncertainties of a difference in the beam spot positions Δs_H and Δs_V, and uncertainty of the focal length F. Since Δs_H and Δs_V were evaluated by subtracting coordinate data of reference beam spots with a reflected beam spots, uncertainties of reference beam spot positions and reflected beam spot positions were separately evaluated. The uncertainty of F, which is a distance error between the measurement plane of the beam profiler and the focusing lens, was evaluated based on uncertainty of a measurement scale used for an alignment of the focal distance.

The standard uncertainty of the rotation angle u_{θ_x} can be expressed as follows:

Then, u_θ was evaluated using the obtained uncertainties u_{θ_x} and u_{θ_y} based on Eq. (18), which can be evaluated by following equations:

Table 1. Summary of uncertainty sources for the rotation angle and the tilt angle

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Table 2. Summary of uncertainty sources for the reflection angle at the cavity

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4.3 Uncertainty of spectral peak detection

The uncertainty of the peak-to-peak spacing u_{Δf_m} was evaluated. In this paper, spectral peaks were determined by evaluations of first derivatives becoming zero. Therefore, sampling accuracy of the OSA and intensity fluctuations of the mode-locked femtosecond laser could influence errors in Δf_{k_m}. Meanwhile, it was complicated to consider all factors that cause the intensity fluctuations on each sampling frequency. For instance, chromatic aberration at the focusing lens and thermal expansions of the components used in the experimental setup could cause errors due to difference in coupling ratio for each frequency.

In correspondence to the above problem, a Monte Carlo estimation was conducted to evaluate u_{Δf_m} [50]. Contributions for the estimation were investigated by errors in the spectral peaks observed in the 100 times repeated measurements of the earlier section. From the experiment result, standard deviations for N number of spectral peaks in the fringe spectrum were evaluated. Figure 9 summarizes the obtained standard deviations for each spectral peak. According to the results, the uncertainties obtained from the repeated measurement varied from 0.52 × 10⁻⁴ to 1.69 × 10⁻⁴ THz.

 

Fig. 9. Standard deviations of cavity lengths for 100 times repeated measurements.

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The Monte Carlo estimation for the evaluation of u_{Δf_m} was conducted based on the obtained standard deviations in Fig. 9. In the estimation, a trial number of 10⁶ and a significant figure of 10⁻⁶ were employed. Random samples for total N spectral peaks were generated individually in the range of each standard deviation value. Obtained f_m+k and f_m were then subtracted to evaluate random samples of Δf_{k_m} for N − k pairs of peaks. Figure 10(a) shows a distribution of one of the results of the Monte Carlo estimation (f_m+k = 197.016051 THz, f_m = 190.833692 THz) and other results were shown as the estimated uncertainties in Fig. 10(b). As shown in the figure, the distribution of estimated values showed a Gaussian distribution, in which standard deviations of the distribution were equivalent to standard uncertainties. Based on this, the standard uncertainties u_{Δf_m} for the N – k pairs of peaks obtained by the Monte Carlo estimation were from 5.112 × 10⁻⁵ to 1.972 × 10⁻⁴ THz

 

Fig. 10. Results of Monte Carlo estimation (a) estimated one of the distributions of $\Delta f$ (${f_{m + k}}$ = 197.016051 THz and ${f_m}$ = 190.833692 THz) and (b) entire uncertainties.

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4.4 Combined uncertainty of the mean cavity length

Table 3 and Fig. 11 summarize the standard uncertainties of cavity length samples d_{k_m} evaluated by the uncertainty sources and the sensitivity coefficients obtained in the previous section. Thus, based on Eq. (20), the standard uncertainty of the mean cavity length u_mean was then evaluated. The standard uncertainty was evaluated to be 3.144 nm. On the other hand, based on GUM, an expanded uncertainty with a 95% confidence is equal to twice of the standard uncertainty [43]. Therefore, the expanded uncertainty of u_mean (95% confidence) was evaluated to be 6.288 nm. The obtained standard deviation of the mean cavity length was within the estimated expanded uncertainty. Meanwhile,

 

Fig. 11. Standard uncertainties of d_{k_m}.

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Table 3. Summary of uncertainty sources for the cavity length ${{\boldsymbol d}_{{\boldsymbol k}\_{\boldsymbol m}}}$

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5. Conclusions

An improved peak-to-peak measurement method for the cavity length d of a Fabry-Perot etalon using a mode-locked femtosecond laser has been proposed. In the proposed method, the fringe spectrum of the transmitted beams from the etalon, which consists of N spectral peaks equally spaced with the free spectral range (FSR) of the etalon, is employed. At first, a sample of d is calculated by using a pair of spectral peaks with a peak-to-peak spacing of an integer multiple k (k ≫ 1 of FSR, i.e., kFSR) with a reduced measurement error, which is an advantage of the proposed method compared with the conventional peak-to-peak method with k = 1. This is also the most novelty of this paper. A mean value of d is then calculated based on an averaging operation on the N − k samples of d obtained from the N − k pairs of spectral peaks with the same peak-to-peak spacing kFSR for further reduction of the measurement error. An analysis has been made to demonstrate that N∕2 is the optimized number of k for obtaining the minimum measurement error of d through balancing the contributions of the peak-to-peak spacing and the averaging operation. This is the second novelty point of this paper.

Based on the proposed method, measurement experiments on an air-gapped FP etalon has been carried out using a constructed optical setup. The detected spectral peaks have been treated by employing the Savitky-Golay differentiation filtering, based on which the peak frequencies have been evaluated. 62 spectral peaks (N = 62) were identified in the fringe spectrum, which was limited by the spectral range of the femtosecond laser used in the experiment. Repeated measurements have been carried out to quantify the precision of the mean cavity length calculated at the optimized k of 31. The mean cavity length was evaluated to be 751.354 µm. The standard deviation of the mean cavity length was evaluated to be 3.841 nm, which was confirmed to be the smallest value compared to those with other k values. Based on GUM and the Monte Carlo estimation, the combined uncertainty of the cavity length measurement was evaluated to be 6.288 nm with a 95% confidence, which means the true value of the cavity length is estimated to be located within the range of 751.354 µm ± 6.288 nm with a 95% confidence. The nanometric measurement uncertainty has demonstrated the effectiveness of the proposed method for high-precision measurement of etalon cavity length. It should be noted that the proposed method can also be applied to the cavity length measurement of FP interferometric sensors, such as fiber-optic extrinsic FP interferometric sensors.