Detection limit analysis of optical fiber sensors based on interferometers with the Vernier-effect

Yulong Li; Yanhui Li; Yi Liu; Yan Li; Shiliang Qu

doi:10.1364/OE.469791

1. Introduction

Optical fiber sensors based on different kinds of fiber-inline interferometers such as Fabry-Perot interferometers (FPIs) [1,2], Mach-Zehnder interferometers (MZIs) [3,4], Michelson interferometers (MIs) [5,6] and Sagnac Interferometers (SIs) [7,8] have been reported. The parameters to be measured can be demodulated by monitoring the wavelength shift of the interference fringes. Their sensitivities are mainly limited by the interference mechanisms and response characteristics of materials used in the interferometers. Vernier-effect provides a generic way for these interferometer-based sensors to further improve the sensitivities greatly by magnifying the spectrum shift. It has been widely employed in optical fiber temperature [9–11], strain [12,13], airflow [14], RI [15], and curvature [13] sensors. The Vernier effect can be achieved by adding a reference interferometer in original interferometer-based sensing system, such as cascading two FPIs [9,12,15], two MZIs [13], or two SIs [10,11]. The optical path differences of the two interferometers must be close, but not the same. The spectrum is the superposition of two kinds of interference fringes. Due to the small fringe interval difference, a large envelope emerges in the spectrum because of Vernier effect. Sensitivity can be amplified by an order of magnitude by monitoring the wavelength shift of the envelope [12]. However, the reported fiber sensors based on the Vernier-effect only focus on the sensitivity amplification, and few papers discuss the influence of the Vernier-effect on detection limit (DL).

DL is the smallest change of the environment parameter that can be measured accurately by the sensor. It is more important than sensitivity for evaluating the actual performance of the sensor. DL of sensors based on spectrum peak shifts is determined by the wavelength resolution (R) and the sensitivity (S), which is $DL = {R / S}$ . The wavelength resolution is generally defined as the uncertainty of the central wavelength with a 95% confidence interval when the environmental parameter is fixed. It depends on unstable fluctuations of the spectrum dip, which is mainly caused by the noise of the spectrometer, the light source, and the sensor structure.

In this paper, the wavelength resolution is found to be related to the wavelength interval of data points, signal-to-noise ratio (SNR) and full width at half maxima (FWHM) of the spectrum dip, environment noises of the sensor structure, and thermal noises of the optical spectrum analyzer (OSA). In measurement, the sensor can effectively distinguish the variation of the parameter only if the central wavelength shift breaks the limitation of the wavelength resolution. Though the Vernier-effect magnifies the sensitivity, the wavelength resolution gets deteriorated, which results in a large value of R. We fabricated a gas pressure fiber sensor based on two parallel FPIs with Vernier-effect and investigate its actual DL to compare with the typical FPI sensor. The actual pressure test results show that the wavelength resolution of the Vernier sensor will degenerate compared with the single-FPI sensor, and the degeneration even exceeds the magnification of the sensitivity. It results in the DL of the Vernier sensor being larger than that of the single-FPI sensor. The DL of the vernier sensor continues to degenerate when the contrast ratio of vernier envelope is reduced or the number of fringes contained in the envelope is decreased. Therefore, the Vernier-effect cannot be used to improve the DL of the interferometer-based fiber sensors.

2. Sample preparation

2.1 Structure of the sensors

A typical single-FPI sensor and a Vernier sensor are fabricated for experiments. The structure of the single-FPI sensor contains a single-mode fiber (SMF), a graded index multimode fiber (GIMMF) 500 µm long, a thick core hollow core fiber (HCF) 330 µm long with an inner diameter of 80 µm, and a thin core HCF 1 cm long with an inner diameter of 5 µm.

The length of the GIMMF and the HCF can be precisely controlled by using the optical fiber cutter and the two-dimensional displacement platform under the observation of the microscope. When the GIMMF was spliced to the SMF, the fusion splicer was set as the default automatic mode. When the thick core HCF was spliced to the GIMMF, the current intensity was set to 2.5 mA and the discharge time was set to 0.3s. At the splicing point between the thick core HCF and thin core HCF, the current intensity was reduced to 1.3 mA to avoid the collapse of the thin core HCF. The end of the HCF is damaged to reduce the reflection light at the end face. The GIMMF is used as a collimator to reduce the transmission loss of the light in the thick core HCF which forms a FP cavity. The gas in the environment can get into the FP cavity through the thin core HCF. The image of the single-FPI sensor is shown in the photo above in Fig. 1(a), and its reflection spectrum is shown in Fig. 1(b), which shows obvious interference fringe. There is a background signal contained in the interference fringes, which is formed by the weak high order modes excited in the GIMMF. However, the amplitude of the background signal is much smaller than that of the interference fringe.

Fig. 1. (a) The image of reference FPI and sensing FPI. (b) Reflection spectrum of the sensing FPI. (c) Connection of the parallel structure of reference FPI and sensing FPI. (d) Reflection spectrum of the parallel structure

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The vernier sensor is composed of two FPIs (a reference FPI and a sensing FPI) in parallel, and the single-FPI sensor can be used as the sensing FPI directly. The structure of the reference interferometer contains a SMF, a GIMMF 500 µm long, a thick core HCF 310 µm long, and a SMF 1 cm long. The image of the reference interferometer is shown in the photo below in Fig. 1(a). Then the single-FPI sensor is parallel connected with the reference interferometer by a 50:50 fiber coupler as show in Fig. 1(c), and the reflection spectrum of the parallel structure is shown in Fig. 1(d). The envelope of the interference fringe is a periodic function of wavelength λ. There are obvious nodes and antinodes in the spectrum, which are formed by the vernier-effect. As the structure parameters of the two FPIs are almost identical, the Vernier envelope in the spectrum shows good quality.

2.2 Calibration experiment

In the spectrum of the Vernier sensor, the second envelope has the most regular shape, so we choose that for gas pressure experiment test. Before the experiment, the sensing interferometer of the Vernier sensor is connected with the pressure pump. Under standard atmospheric pressure, the pressure of the pump is increased from 0MPa to 1MPa at an equal interval of 0.1MPa. The OSA is used to scan the spectrum of the Vernier sensor at different pressures sequentially. Then the single-FPI sensor is tested by using the same method. It can be seen from Fig. 2(a) and 2(b) that the central wavelength of the single-FPI sensor and the Vernier sensor both have good linear relationships with pressure. The gas pressure sensitivities of the single-FPI sensor and the Vernier sensor are 4.25nm/MPa and 67.56nm/MPa, respectively. It can be seen that the Vernier effect increases the sensitivity by about 15.89 times for the single-FPI sensor.

Fig. 2. (a) Spectra measured in the pressure calibration experiment. (b) Sensitivity fitting of the single-FPI sensor and the Vernier sensor.

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3. Sensitivity amplification principle of Vernier-effect

According to the principle of double-beam FP interference, the central wavelength of the interference valleys for the single-FPI sensor can be expressed as [16]

(1)$${\lambda _m} = \frac{{\textrm{4}nL}}{{\textrm{2}m\textrm{ + 1}}},$$

where, m is the order of the interference valley, n is the refractive index (RI) of the air in the FP cavity, and L represents the length of the FP cavity. The value of n increases as the gas press increasing, which results in the shift of the interference valley. The gas pressure sensitivity of can be expressed as

(2)$$S = \frac{{\partial {\lambda _m}}}{{\partial P}} = {\lambda _m}\left( {\frac{1}{L}\frac{{\partial L}}{{\partial P}} + \frac{1}{n}\frac{{\partial n}}{{\partial P}}} \right) \approx \frac{{{\lambda _m}}}{n} \cdot \frac{{\partial n}}{{\partial P}}.$$

According to the principle of the Vernier-effect [16,17,18], the Vernier envelope reaches its valley when the following condition is satisfied:

(3)$${\lambda _{m^{\prime}}} = \frac{{4({{n_1}{L_1} - {n_2}{L_2}} )}}{{2m^{\prime} + 1}},$$

where, n₁ is RI of the air in the sensing FP cavity with a length of L₁, and n₂ is RI of the air in the reference FP cavity with a length of L₂. m is the order of the envelope valley.

The gas pressure sensitivity of can be expressed as

(4)$$S = \frac{{\partial {\lambda _{m^{\prime}}}}}{{\partial P}} = {\lambda _{m^{\prime}}}\frac{1}{n}\frac{{\partial n}}{{\partial P}}\frac{{{n_1}{L_1}}}{{{n_1}{L_1} - {n_2}{L_2}}}.$$

Comparing Eq. (4) to Eq. (2) we can see that, if the values of λ_m and λ_m’ are chosen to be close to each other, the gas pressure sensitivity is magnified by ${{{n_1}{L_1}} / {({{n_1}{L_1} - {n_2}{L_2}} )}}$. The theoretical sensitivity magnification for the fabricated single-FPI sensor and Vernier sensor in this paper can be calculated to be 17 times, which is very close to the actual measurement result. However, although the vernier effect amplifies the sensitivity of the sensor, it also increases the wavelength resolution of the sensor, which is discussed in section 5.

4. Central wavelength demodulation method

The typical pressure demodulation method is to substitute the central wavelength of a certain order valley in the spectrum into the objective function to inversely derive the pressure. There are generally two ways to determine the central wavelength. The first method is to directly select the wavelength of the smallest vertical axis value at the valley. As shown in Fig. 3(a), the wavelength of the smallest value in the valley is 1484.4nm. This method is relatively simple, but the accuracy is not satisfactory. Due to the interval of the data points collected by the spectrometer, this method is only useful when the change of pressure is large enough to cause data points at other wavelengths to float up and down to form the smallest value at a new wavelength.

Fig. 3. (a) Data points of the interference valley. (b) The fitted curve after Lorentz fitting of the interference valley. (c) Data points of the envelope valley. (d) The fitted curve after Lorentz fitting of the envelope valley.

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For example, we set the wavelength interval of the data points collected by OSA to be 0.2nm. Under standard atmospheric pressure, we use the single-FPI sensor to measure the increased pressures of 0.02MPa and 0.022MPa respectively. The spectrum data points at the two pressure values are recorded and shown in Fig. 3(a). The experimental results show that, the central wavelength detected by using the above method doesn’t move when the pressure changes from 0.02 MPa to 0.022 MPa. Although the data points to the left of the center all rise and the data points to the right of the center all fall, which forms a trend of wavelength moving, it does not cause the smallest value shift to a new wavelength point. Therefore, the sensor cannot accurately distinguish these two pressures.

Assume that the vertical axis variation of the data point under two pressures is $\Delta y$, the smallest value of the valley is ${y_{central}}$, and the corresponding central wavelength is ${x_{central}}$. The variation of the gas pressure is not enough to change the central wavelength when

(5)$${y_{central}}\textrm{ + }\Delta {y_{central}} \le {y_{central - 1}}\textrm{ + }\Delta {y_{central - 1}},$$

(6)$${y_{central}}\textrm{ + }\Delta {y_{central}} \le {y_{central + 1}}\textrm{ + }\Delta {y_{central + 1}},$$

where ${y_{central\textrm{ - 1}}}$ and ${y_{central\textrm{ + 1}}}$ are the vertical axis values of the data points to the left and right of the smallest value data point respectively.

The second method is to select data points near the valley for curve fitting, and determine the central wavelength by calculating the symmetry axis of the curve. In nonlinear fitting, Lorentz fitting and Gauss fitting are often used for single peak fitting because of their high fitting accuracy [19,20]. Take Lorentz fitting as an example, and the data points near the valley are processed by:

(7)$$y = {y_0} + \frac{{2A}}{\pi }\frac{w}{{4{{(x - {x_c})}^2} + {w^2}}},$$

where, ${y_0}$ is the offset value, w is the FWHM of the curve, A is the area, and ${x_c}$ is the central value of x in the curve. We use the least square method to determine the central wavelength of the curve. The result is shown in Fig. 3(b). It can be seen that the central wavelength obtained by Lorentz fitting shifts significantly from 1484.427 nm to 1484.433 nm, so the two pressures can be effectively distinguished. Therefore, curve fitting is helpful for the sensor to break the limitation of experimental instruments. It is the same for the Vernier sensor [21], and the comparison of two methods is shown in Figs. 3(c) and 3(d). The data points are chosen by finding the minimum position in each interference valley from the actual spectrum directly. It is worth noting that the data points of the envelope are the points at valleys of inside fringes, so the wavelength interval of the data points is much larger for the Vernier sensor. The curve fitting is necessary for the Vernier sensor to improve its performance. Therefore, the following data processing in this paper adopts the curve fitting method to determine the central wavelength.

5. Evaluation of wavelength resolution in actual measurement

It is evident that the Vernier sensor is much more sensitive than the FP sensor. However, at the same time, the Vernier effect also causes the increase of the wavelength resolution. In actual measurement, each data point near the spectrum valley has a slight deviation along vertical axis due to the influence of noise, resulting in an uncertain error in curve fitting. Noise sources include the spectrometer, the light source, and the sensor structure. The uncertain of the fitting curve causes random shifts of the demodulated central wavelength of the spectrum valley. The central wavelength uncertainty could be regarded as the wavelength resolution. It is analyzed that the size of this random deviation is related to four factors. The first factor is the number of data points that make up the spectrum valley. The second factor is the difference of random error of data points between the Vernier envelope and the interference fringe. Since the fringes inside an envelope are the superposition of two sets of interference fringes, the single-point noise of the vernier envelope increases. The third factor is the FWHM of the fitted curve. The fourth factor is the contrast ratio of the spectrum valley.

5.1 Impact of the number of data points in curve fitting

We obtain the central wavelength of the spectrum valley by applying Lorentzian fitting to the data points of the spectrum. It is necessary to discuss the effect of the number of data points in the fitting process.

Combined with Eq. (7), the error of the fitted curve from the actual data points can be expressed as

(8)$${r_i}(b) = {y_i} - ({b_1} + \frac{{2{b_2}}}{\pi }\frac{{{b_3}}}{{4{{(x - {b_4})}^2} + {b_3}^2}}),$$

where, i is the serial number of the sample points, and b is the undetermined coefficient vector in Eq. (7). According to the principle of nonlinear least squares, the objective function to be optimized is

(9)$$\phi (b) = \frac{1}{2}\sum\limits_{i = 0}^m {{r_i}^2(b)} ,$$

where, m is the number of sample points, and the previous constant 1/2 is just to facilitate the calculation of the subsequent derivation. When the objective function reaches the minimum value, b can be considered as the optimal solution of the fitting function. The following is the first derivative of the objective function:

(10)$$\frac{{\partial \phi }}{{\partial {b_j}}} = \frac{\partial }{{\partial {b_j}}}\frac{1}{2}\sum\limits_{i = 0}^m {{r_i}^2(b)} = \sum\limits_{i = 0}^m {{r_i}(b)} \frac{{\partial {r_i}(b)}}{{\partial {b_j}}},$$

(11)$$\nabla \phi = {J_r}{(b)^T}r(b),$$

where, ${J_r}$ is the Jacobian matrix of r to b

(12)$${J_r} = \left[ \begin{array}{l} \frac{{\partial {r_1}}}{{\partial {b_1}}}\textrm{ }\frac{{\partial {r_1}}}{{\partial {b_2}}}\textrm{ }\frac{{\partial {r_1}}}{{\partial {b_3}}}\textrm{ }\frac{{\partial {r_1}}}{{\partial {b_4}}}\\ \frac{{\partial {r_2}}}{{\partial {b_1}}}\textrm{ }\frac{{\partial {r_2}}}{{\partial {b_2}}}\textrm{ }\frac{{\partial {r_2}}}{{\partial {b_3}}}\textrm{ }\frac{{\partial {r_2}}}{{\partial {b_4}}}\\ \textrm{ } \vdots \textrm{ }\quad \vdots \textrm{ } \quad\vdots \textrm{ } \qquad\vdots \\ \frac{{\partial {r_m}}}{{\partial {b_1}}}\textrm{ }\frac{{\partial {r_m}}}{{\partial {b_2}}}\textrm{ }\frac{{\partial {r_m}}}{{\partial {b_3}}}\textrm{ }\frac{{\partial {r_m}}}{{\partial {b_4}}} \end{array} \right].$$

We set the function to be solved as F(b). The condition for b to be the optimal solution is

(13)$$F(b) = \nabla \phi (b) = {J_r}{(b)^T}r(b) = 0,$$

Because of the nonlinearity of r(b), it can only be solved by iteration. For a one-variable function, suppose b is the root of f(x) obtained by solving, and the distance between b and the real root b₀ of the equation is Δb. Based on the principle of Gauss-Newton method

(14)$$0\textrm{ = }f({b_0}) = f(b + \Delta b) = f(b) + \Delta b{f^{\prime}}(b) + O({(\Delta b)^2}).$$

After ignoring higher order terms, we can deduce that

(15)$$\Delta b \approx{-} \frac{{f(b)}}{{{f^{\prime}}(b)}}.$$

For multivariate functions, the form is the same

(16)$$\Delta b ={-} {J_F}^{ - 1}F(b).$$

In order to solve Eq. (13), we need continue to calculate the derivative of ${J_r}(b)$

(17)$$\begin{aligned} {J_F}{(b)_{ij}} = \frac{{\partial {F_i}}}{{\partial {b_j}}} &= \frac{\partial }{{\partial {b_j}}}{({J_r}{(b)^T}r(b))_i}\\& \textrm{ = }\frac{\partial }{{\partial {b_j}}}\sum\limits_{k = 1}^m {\frac{{\partial {r_k}}}{{\partial {b_j}}}} {r_k}\\& \textrm{ = }\sum\limits_{k = 1}^m {\frac{{\partial {r_k}}}{{\partial {b_i}}}} \frac{{\partial {r_k}}}{{\partial {b_j}}} + \sum\limits_{k = 1}^m {\frac{{{\partial ^2}{r_k}}}{{\partial {b_i}\partial {b_j}}}} {r_k}, \end{aligned}$$

where, i, j are the serial numbers of undetermined parameters b, and k is the serial number of the sample points. The second-order derivative is usually very troublesome to solve, and as the optimization progresses, the value of the error function r(b) is also decreasing, so it is decided to ignore the second term in the above formula, that is

(18)$${J_F}(b) \approx \sum\limits_{k = 1}^m {\frac{{\partial {r_k}}}{{\partial {b_i}}}} \frac{{\partial {r_k}}}{{\partial {b_j}}} = {J_r}{(b)^T}{J_r}(b).$$

Substitute Eq. (13) and Eq. (18) into Eq. (16) to get the final iterative formula

(19)$${b^{(t + 1)}} = {b^{(t)}} + \Delta b,$$

(20)$$\begin{aligned} \Delta b &={-} {({J_r}{(b)^T}{J_r}(b))^{ - 1}}({J_r}{(b)^T}r(b))\\& \textrm{ = } - {J_r}{(b)^{ - 1}}r(b), \end{aligned}$$

where, t is the iteration order. In curve fitting, if there are more data points, the coincidence between the fitted curve and the actual spectrum will be better, and the error r(b) between the data points and the fitted curve will be smaller. It can be concluded from Eq. (20) that the error of the optimal solution of the objective function $\Delta b$ will also be reduced in the iterative process. Therefore, more data points can reduce the uncertain error in curve fitting induced by noises of data points. As a result, the wavelength resolution of the sensor is improved.

To verify this opinion, we designed a set of comparative experiments with different scanning wavelength intervals of OSA to observe the influence of the number of data points on the wavelength resolution for the single-FPI sensor. First, the wavelength interval was set at 0.2 nm, and the OSA was used to scan the spectra of the sensor placed in the pump whose pressure switches between 0.02Mpa and 0.022Mpa in turn for 100 times. We use Lorentz fitting method to determine the central wavelength of interference valley. After 100 repeated measurements, we can get the normal distribution of the demodulated central wavelength values, and calculate the mean µ and standard deviation σ. The upper and lower limits of the confidence interval can be obtained corresponding to the confidence level and σ. In this paper, the 95% confidence interval is related to (µ–1.96σ, µ+1.96σ). This confidence interval is regarded as the wavelength resolution of the sensor. Based on the calculating results, we found that the wavelength resolution was smaller than 0.0057 nm as shown in Fig. 4(a). Then we reduced the wavelength interval to 0.4 nm and performed the same operation. We found that the wavelength resolution degenerated to about 0.0082 nm as shown in Fig. 4(b). Obviously, there is an overlap between two wavelength uncertainty intervals when the number of data points is reduced by half. It is invalid to distinguish two pressures. Then the wavelength interval was set at 0.2 nm again, and the fitting range of the interference valley was reduced from 3.4 nm to 1.2 nm. The demodulated central wavelength values are shown in Fig. 4(c). Comparing Fig. 4(c) with Fig. 4(a), we can see that, the fitting range of the interference valley also influences the wavelength resolution. Smaller fitting range uses fewer data points, which results in the lager wavelength uncertainty. At last, we used Gauss fitting method to determine the central wavelength of interference valley to show the impact of the different fitting methods on the wavelength resolution, and the results are shown in Fig. 4(d). The wavelength interval and the fitting range are consistent with the parameters in Fig. 4(a). It can be seen that, the central wavelength uncertainties demodulated by the two fitting methods are similar.

Fig. 4. (a), (b) The demodulated central wavelength values of the interference valley at two different pressures when the scanning wavelength interval of OSA is 0.2 nm and 0.4 nm, respectively. (c) The demodulated central wavelength values of the interference valley when the fitting range was reduced to 1.2 nm. (d) The demodulated central wavelength values of the interference valley by using Gauss fitting method.

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The number of data points for the single-FPI sensor can be increased by reducing the scanning wavelength interval of OSA. Therefore, the wavelength resolution of the single-FPI sensor can be improved by employing the OSA with a better performance. The data points of the Vernier sensor are the extreme points of fringes contained in the envelope, and the number of the data points cannot be changed after the sensor is made. In general, the number of fringes involved in the envelope is much smaller than the number of data points scanned by OSA in each interference fringe, which results in the degeneration of the wavelength resolution in curve fitting demodulation for the Vernier sensor.

5.2 Comparison of the single-point noise of the Vernier and fringe valleys

The Vernier envelope are formed by superposition of two groups of interference fringes. Therefore, the noises of data points in the envelope valley are also the superposition of noises of data points of two interference fringes. Generally, the errors of the data points for two interference fringes are Gaussian distribution ${\varepsilon _i}(y)$ and ${\varepsilon _i}^{\prime}(y)$

(21)$${\varepsilon _i}(y) = \frac{1}{{\sqrt {2\pi } {\sigma _i}}}\exp \left( { - \frac{{{{(y - {\mu_i})}^2}}}{{2{\sigma_i}^2}}} \right),$$

(22)$${\varepsilon _i}^{\prime}(y) = \frac{1}{{\sqrt {2\pi } {\sigma _i}^{\prime}}}\exp \left( { - \frac{{{{(y - {\mu_i}^{\prime})}^2}}}{{2{\sigma_i}{{^{\prime}}^2}}}} \right).$$

Sum of two Gaussian distribution functions is still a Gaussian distribution function, and the error distribution of data points contained in the vernier ${\varepsilon _I}(y)$ is

(23)$${\varepsilon _I}(y) = \frac{1}{{\sqrt {2\pi } {\sigma _I}}}\exp \left( { - \frac{{{{(y - {\mu_I})}^2}}}{{2{\sigma_I}^2}}} \right),$$

where

(24)$${\sigma _I}^2 = {\sigma _i}^2 + {\sigma _i}{^{'2}}.$$

According to Eq. (24), it can be found that the superposition of two interference fringes will lead to the increase of the error of the Vernier data points. This makes the noise of the data points in the Vernier envelope naturally larger than that in the interference fringe. In experiment, the single-FPI sensor and the Vernier sensor were placed in the pump individually. The pressure of the pump was kept at 0.02Mpa and the spectra of the single-FPI sensor and the Vernier sensor were measured for 30 times. The date points at the minimum positions from the interference valley and Vernier envelope valley were all recorded, and their Y-axis values are shown in Figs. 5(a) and 5(b) respectively. The experiment results confirm the theoretical analysis.

Fig. 5. (a), (b) Y-axis values of the data points at the minimum positions from the interference valley and Vernier envelope valley respectively.

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5.3 Influence of FWHM

The impact of the FWHM on DL for the fiber sensors based on the peak wavelength demodulation has been reported and analyzed theoretically and experimentally [22,23]. The results show that the resonant peak with a narrower linewidth filters the spectral noise more effectively, which leads to lower spectral deviation from the actual center of the peak. Monte Carlo Simulation Results provide an approximate relationship between FWHM, SNR and the standard deviation of the central wavelength for the spectrum valley. When the number of data points is the same, the standard deviation has a linear relationship with FWHM and an exponential relationship with SNR [22], and the numerical results can be approximately expressed as

(25)$$\sigma \approx \frac{{\textrm{FWHM}}}{{4.5(\textrm{SN}{\textrm{R}^{0.25}})}},$$

where, σ is the standard deviation of the central wavelength shift, which determines the wavelength resolution. Ideally, both the FWHM values of the interference valley and the Vernier envelope valley are equal to one-half of their free spectrum ranges (FSRs). Since the FSR of the vernier envelope is much larger than that of the interference fringe, the wavelength resolution of the Vernier sensor degenerates significantly. In the best case, the Vernier envelope is the superposition of two interference fringes with the same intensity. Under this condition, the signal strength of the Vernier envelope is about twice that of the interference fringe, and the amplitude of the lower envelope is approximately equal to the amplitude of the interference fringe. It can be seen from Eq. (24) that noise of the envelope is about $\sqrt 2 $ times as that of the interference fringe. The relationship of the standard deviations between the Vernier sensor ${\sigma _e}$ and the single-FPI sensor ${\sigma _S}$ can be deduced as

(26)$${\sigma _e} \approx {\sigma _S} \cdot \frac{{\textrm{FS}{\textrm{R}_e}}}{{\textrm{FS}{\textrm{R}_S}}} \cdot {\left( {\frac{{\textrm{SN}{\textrm{R}_S}}}{{\textrm{SN}{\textrm{R}_e}}}} \right)^{0.25}} \approx \frac{{1.09 \times \textrm{FS}{\textrm{R}_1}}}{{|{\textrm{FS}{\textrm{R}_1} - \textrm{FS}{\textrm{R}_2}} |}}{\sigma _S}\textrm{ = }\frac{{1.09 \times {n_1}{L_1}}}{{|{{n_1}{L_1} - {n_2}{L_2}} |}}{\sigma _S}.$$

Based on Eq. (2), Eq. (4) and Eq. (26), DL_s of the single-FPI sensor and DL_e of the Vernier sensor can be calculated to be

(27)$$\textrm{D}{\textrm{L}_s} = \frac{{{R_S}}}{{{S_S}}} = \frac{{{\sigma _S}}}{{{S_S}}},$$

(28)$$\textrm{D}{\textrm{L}_e} = \frac{{{R_e}}}{{{S_e}}} = \frac{{{\sigma _e}}}{{{S_e}}} \approx 1.09D{L_S}.$$

It is worth noting that the above results are based on the ideal situation. In practice, the Vernier envelope is the superposition of two interference fringes with a small intensity difference. Antinodes and nodes in the Vernier spectrum get deteriorated. As a result, the amplitude of the lower envelope is smaller than the amplitude of the interference fringe, which induces the increase of the ratio value of the DL between the Vernier sensor and the single-FPI sensor. In order to compare the DL of two sensors in experiment, we used the Vernier sensor to measure pressures of 0.02 MPa and 0.022 MPa respectively. The pressure of the pump switches again between 0.02 MPa and 0.022 MPa in turn for 100 times. The data points of the envelope valley and Lorentz fitting curve of the Vernier sensor at the pressure of 0.022 MPa are shown in Fig. 6(a). The number of the data points used in curve fitting for the Vernier sensor is similar to that for the single-FPI sensor. The demodulated central wavelength values of the envelope valley in the whole measurement process are show in Fig. 6(b). Analyzed key parameters are listed in Table 1 by comparing Fig. 6(b) with Fig. 4(a). Although the magnification of sensitivity leads to an increase in the average wavelength shift, the wavelength resolution also degenerates to a large extent. The wavelength resolution of the single-FPI sensor is about 0.0057 nm, while the wavelength resolution of the Vernier sensor is about 0.24 nm. Considering the sensitivities, we can obtain that the actual DL of the single-FPI sensor and the Vernier sensor are 0.00134 MPa and 0.00358 MPa respectively. The DL of the former is much smaller than that of the Vernier sensor. The ratio value of the DL between the Vernier sensor and the single-FPI sensor reaches 2.67. The single-FPI sensor is able to distinguish the little pressure variation of 0.002 MPa, while the Vernier sensor losses the function.

Fig. 6. (a) Data points and Lorentz fitting curve of the vernier sensor. (b) The demodulated central wavelength values of the envelope valley in the pressure measurement process.

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Table 1. Key parameters of the single-FPI sensor and the Vernier sensor

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5.4 Influence of envelope contrast

In order to show the influence of the Vernier envelope contrast on wavelength resolution and DL, we reduce the contrast of the Vernier envelope to another two different values as shown in Figs. 7(a) and 7(b) by using a fiber attenuator to increase the loss of the reference FPI. The pressure measurement performance of the Vernier sensor with these two contrast values was also tested. The results are shown in Figs. 7(c) to 7(f). Analyzed key parameters are listed in Table 2.

Fig. 7. (a) and (b) Spectra of the Vernier sensor with the envelope contrast values of 0.027 mW and 0.013 mW respectively; (c) and (d) Data points and Lorentz fitting curve of the Vernier sensor with the envelope contrast values of 0.027 mW and 0.013 mW respectively. (e) and (f) The demodulated central wavelength values of the envelope valley with contrast values of 0.027 mW and 0.013 mW respectively in the pressure measurement process.

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Table 2. Key parameters of the Vernier sensor with contrast of 0.027 mw and 0.013 mw

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The wavelength resolution of the Vernier sensor with the envelope contrast of 0.027 mW is about 0.585 nm. When the envelope contrast is reduced to 0.013 mW, the wavelength resolution of the Vernier sensor degenerates to 2.091 nm. For the pressure variation of 0.01 MPa, the Vernier sensor with the envelope contrast of 0.027 mW can just distinguish the pressure change, while the Vernier sensor with the envelope contrast of 0.013 mW losses the function. The ratio value of the DL between the Vernier sensor and the single-FPI sensor is increased to 23.09. Through the comparison of the performance of the Vernier sensor with three different contrast values, it can be found that, the wavelength resolution increases significantly with the decrease of the envelope contrast, resulting in the degeneration of DL, which is very detrimental to the function of the sensor.

5.5 Discussion

Due to the limit of the number of fitting data points, single point noise, FWHM and valley contrast, actual DL of the single-FPI sensor is better than the Vernier sensor. Typically, the number of data points in a fringe is much larger than the number of fringes contained in an envelope. The ratio value of the DL between the Vernier sensor and the single-FPI sensor can be much larger than 23.09. Even when they are the same, the theoretical DL of the Vernier sensor is still 1.09 times as that of the single-FPI sensor. The Vernier caliper, as a comparison, can improve the DL of the length measurement, because the error of the length measurement induced by noises is less than the division value of the caliper. Once the distinguishable error exceeds the division value, the smaller division value will lose its significance. The same is true for the fiber Vernier sensor. Although the sensitivity is greatly improved compared with the single-FPI sensor, the noises are also greatly amplified, resulting in the seriously degeneration of the wavelength resolution in actual measurement. It should be noted that, in this paper, the Vernier sensor is only compared with the single-FPI sensor that works as the sensing FPI to consist the Vernier sensor. We didn’t compare all the Vernier sensors with all the single-FPI sensors. If the structural parameters of the single-FPI sensors and the Vernier sensors are completely irrelevant, the comparison of their DL values makes no sense.

In order to improve the DL of the fiber Vernier sensor, we can use better insulation material to isolate the reference interferometer of the Vernier sensor from the ambient noises such as thermal noise and vibration noise. We can increase the number of fringes contained in an envelope by decreasing the FSR of the interference fringe. For this purpose, the reference interferometer and sensing interferometer with both longer interference length are suggested to be employed to constitute the fiber Vernier sensor. For example, a section of weak-light-confined hollow core Bragg fiber (HCBF) is spliced between two SMFs to form the FPI [24]. A strong bandgap effect caused by inner four bilayers forms a forbidden bandgap in radial direction, which means most of the energy in such a wavelength is transmitted in an axial direction, resulting in the low transmission loss in the specific wavelength. Therefore, a long length FPI with a good interference quality can be achieved. Besides, the envelope contrast of the Vernier sensor should be increased by improving both the consistency and the interference quality for the reference interferometer and sensing interferometer.

6. Conclusion

Theoretical analysis and experimental results demonstrate that the DL of optical fiber sensor based on Vernier effect is not improved with the increase of sensitivity, and even degenerates. DL is determined by the wavelength resolution and the sensitivity, and the wavelength resolution of the Vernier sensor is affected by the number of data points, the superimposed noises of data points, FWHM and contrast of the envelope. In actual measurement, it is suggested that researchers should consider the limitation factors of Vernier sensors and make a reasonable choice between the typical single-FPI sensor and the Vernier sensor. If the proposed Vernier sensor is composed of two FPIs, and the sensing FPI can work alone as an independent sensor, the single-FPI sensor is recommended for its better DL and simple demodulation process. It is more meaningful to research how to improve the actual DL of the Vernier sensor rather than simply introducing the Vernier effect to the FPI sensor.

Funding

National Natural Science Foundation of China (11874010, 11874133); Natural Science Foundation of Shandong Province (ZR2021MF111).

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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Sensor type	Resolution	Pressure change	Average central wavelength shift
Single-FPI sensor	0.0057nm	0.002MPa	0.006nm
Vernier sensor	0.242nm	0.002MPa	0.116nm

Sensor type	Resolution	Pressure change	Average central wavelength shift
Single-FPI sensor	0.0057nm	0.002MPa	0.006nm
Vernier sensor	0.242nm	0.002MPa	0.116nm

Detection limit analysis of optical fiber sensors based on interferometers with the Vernier-effect

Abstract

1. Introduction

2. Sample preparation

2.1 Structure of the sensors

2.2 Calibration experiment

3. Sensitivity amplification principle of Vernier-effect

4. Central wavelength demodulation method

5. Evaluation of wavelength resolution in actual measurement

5.1 Impact of the number of data points in curve fitting

5.2 Comparison of the single-point noise of the Vernier and fringe valleys

5.3 Influence of FWHM

5.4 Influence of envelope contrast

5.5 Discussion

6. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Tables (2)

Equations (28)

Optics Express

Contrast ratio	Resolution	Pressure change shift	Average central wavelength
0.027 mW	0.585 nm	0.01 MPa	0.644nm
0.013 mW	2.091 nm	0.01 MPa	0.565nm