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Abstract
A demodulation method for interferometric fiber sensors (IFSs) is proposed in this article. The phase variation induced by the measurands can be estimated by calculating the Fourier phase at the intrinsic spatial frequencies of the fiber sensor. Theoretical analysis of the demodulation method is discussed in detail. Numerical simulations are put forward to demonstrate the consistency of the demodulation results under different wavelength sampling interval and noise level, showing a better stability compared with the conventional peak wavelength tracking technique. The proposed method is also experimentally demonstrated by an inline multimode interferometer based on a single-mode fiber (SMF) offset-splicing structure. Experimental results indicate that the phase response of different cladding modes can be analyzed simultaneously. Simultaneous measurement of strain and temperature is realized in our confirmatory experiment by analyzing the phase sensitivities of two selected cladding modes.
© 2017 Optical Society of America
1. Introduction
Interferometric fiber sensors (IFSs) are widely investigated due to the remarkable superiorities of high sensitivity and precision. The interferences are formed by two or more beams that have certain phase difference [1–3]. Since the optical wavelength is very short, usually several hundreds of nanometers or a few micrometers, very small optical path difference (OPD) variation can be detected by IFSs, offering relatively high sensitivity and resolution in measurement of parameters such as pressure [4–6], refractive index (RI) [7–11], displacement [12,13], strain [14–18], and curvature [19–21], etc.
Wavelength tracking demodulation technique is mostly used for interferometric fiber sensors due to the simple operation and linear response. The mechanism is based on the resonant wavelength shift due to the OPD variation caused by physical parameters. The wavelength shift usually shows a linear response to the measurands. By calibrating the sensitivity, the measurand can be interrogated from the wavelength shift. However, a significant limitation of wavelength demodulation is the accuracy. Since the resonant wavelength is determined by searching the maximum (for resonant peak) or minimum (for resonant dip) intensity value in a certain wavelength window, the results may suffer from unpredictable errors that depend on wavelength sampling rate or spectral noise [22,23].
In order to improve the accuracy, Fast Fourier Transform (FFT) is applied to the optical spectrum to demodulate the sensor signal in the frequency spectrum domain [22–25]. Traditional FFT method to demodulate the fiber sensor is to monitor the peak variations (including both peak intensity variation and peak frequency variation) in the FFT spectrum. It usually requires a relatively large amount of variation of the measurands. But in some sensing applications, parameters that have very small amount of changes should be detected, such as micro pressure or strain sensing. In these applications, the OPD variation introduced by the measurands is small enough to be neglected compared with the initial OPD value. Therefore, the peak location variation in the FFT spectrum is not able to be recognized.
In this article, we propose a demodulation method to calculate the phase variation of the sensor in the Fourier domain. For fiber interferometer, most of the energy is contained in the intrinsic spatial frequencies that are corresponding to the interference components. For this reason, we believe that the information of phase variation between each interference component can be obtained by the Fourier phase at these spatial frequencies. The mechanism is theoretically analyzed. Numerical simulations prove that the demodulation result is insensitive to the spectral noise and sampling interval. Experiment is conducted by testing multimode interferometers to demonstrate the proposed scheme. For multimode interference, phase response of each cladding mode can be obtained by calculating the Fourier phase at each related spatial frequency, so it can be applied to multi-parameters measurement or quasi-distributed sensing system.
2. Theoretical analysis
To theoretically explain the principle of the demodulation method, we firstly use a model of two-beam interference for analysis. The condition of multimode interference is generalized theoretically and demonstrated in the subsequent experiment. For a two-beam interferometer, the spectrum function can be expressed by Eq. (1). The interference function can be regarded as a trigonometric function of the wavelength within a certain range. In the equation, d represents the optical path difference (OPD). A and B are two constants determined by the optical intensity and coupling ratio between two optical paths.
When the OPD of the interferometer is modulated by the environmental parameters, the whole spectrum will show a wavelength shift, as written by Eqs. (2) and (3). △d is the OPD variation, while △λ is the wavelength shift value.
From Eq. (3), the shift value is dependent on the wavelength. Within a certain wavelength window that is not too wide, the shift can be regarded as a constant. The free spectrum range (FSR) variation is approximately presented by Eq. (4). Usually the OPD variation is too small compared with the initial OPD value, thus the FSR variation is small enough to be neglected. According to this, we can conclude that the spectrum experiences a constant wavelength shift without shape distortion, as demonstrated by Eqs. (5) and (6), and △φ is the additional phase variation item introduced by the wavelength shift. It can be seen from the equation that the wavelength is discretely sampled, just like the spectrum data acquired for analysis in the experiment or practical applications. N and △λ0 stand for the number and interval of wavelength sampling points.
From the discussion above, the interference pattern can be regarded as a trigonometric function of the wavelength, as shown in Eq. (1). An intrinsic spatial frequency is related to the interference pattern, as revealed by Eq. (7). When the wavelength sampling interval and wavelength range are fixed, the frequency interval of the FFT result is ascertained to be 1/N△λ0. For a certain fiber sensor structure, the energy is mainly distributed in the few intrinsic spatial frequencies. Therefore, the phase information of the IFS is also promising to be obtained at these frequencies. To seek for a confirmation, we can compute the fast Fourier transform of the spectrum represented by Eq. (5) at the spatial frequency fk. The algorithm is shown as Eq. (8).
According to Eqs. (6) and (7), the spectrum function T '(λ) can be transformed as Eq. (9). In the expression, θC is relevant to the starting wavelength of the spectrum and frequency sampling interval. In applications, these parameters are fixed to be constant at the beginning, so θC is also a constant, as described by Eq. (10).Using the transformed spectrum function of Eq. (9) to calculate the FFT, the real part and the imaginary part of the FFT results at the intrinsic spatial frequency are computed as Eqs. (11)-(13).
The cosinusoidal and sinusoidal value of the Fourier phase at the spatial frequency can be computed from the real and imaginary parts of the FFT result, as expressed in Eq. (14) and (15). Am[F '(fk)] represents the amplitude of the FFT result. When the same computing process is applied to the undisturbed spectrum (initial reference spectrum) T(λj), the FFT phase result can be easily acquired based on the above discussion and are demonstrated in Eq. (16). Comparing the results between the disturbed and undisturbed spectrum of Eqs. (14) and (16), the Fourier phase variation can be concluded as Eq. (17). Obviously, the Fourier phase variation at the intrinsic spatial frequency is just the same as the measurand induced phase variation of the IFS. Therefore, the IFS signal can be demodulated from the Fourier phase variation.3. Numerical simulations
According to the above theoretical analysis, the Fourier phase that contains the measurand information is computed from all the sampling data. In other words, there is no redundant information in the sampled spectrum when demodulating the IFS signal.
By contrast, the conventional method to search the resonant peak wavelength uses the information of only one sampling point. Hence, the accuracy of the wavelength tracking method may be easily affected by sparse wavelength sampling or spectral noise. The noise may come from the thermal and shot noise of the photodetector (PD) or intensity fluctuation of the light source. The mechanism is exhibited by Fig. 1. As shown in Fig. 1(a), the discrete wavelength sampling may cause the fact that the real peak wavelength of the analog spectrum is located between two adjacent sampling points, so the measured peak wavelength comes from one of them. Obviously, the measurement error appears. As for Fig. 1(b), it clearly reveals that the spectral noise may cause a deviation of the peak wavelength. What's more, the measurement error caused by either wavelength sampling or spectral noise is totally random and unpredictable. The essence of the measurement error is the severe information redundancy.
Fig. 1 Mechanism of measurement error of wavelength demodulation caused by (a) sampling interval and (b) spectral noise.
Based on the above discussions, there are reasons to believe that the proposed Fourier phase demodulation method has a better stability. In order to confirm this, we put forward numerical simulations to investigate the performance of these two demodulation methods under different spectral noise level and sampling intervals.
3.1 Performance with spectral noise
As already discussed by other researchers, the acquired spectrum of the sensor structure may suffer from noise limitations [22]. The noise may come from instability of the light source or the shot and thermal noise from the detector in the optical spectrum analyzer (OSA). In the simulations, random white noise with different amplitude coefficient is added on a simulated interference function, as expressed in Eq. (18). In the equation, d is the initial OPD of the IFS, △d is the OPD variation step in the simulations and m stands for step number. As for noise item, rand(λj) is a unit white noise function, and n determines the noise level.
In the simulations, the initial OPD and the OPD variation step are set to be 200μm and 20nm respectively. The wavelength sampling interval is set to be a constant of 0.02nm. The interference coefficient B that determines the extinction ratio is set as 0.45, while the noise coefficient n is set to be 0, 0.005 (SNR = 39.1dB) and 0.01 (SNR = 33.1dB) for three conditions. Firstly, we selected the condition of 0.01 noise coefficient with OPD variation value ranging from 0 to −200nm (m = 0,1,2......10) to verify the intrinsic spatial frequency stability, as demonstrated by Eq. (4). The Fourier spectra under different OPD values (m values) are simulated and plotted in Fig. 2, in which we can observe an unaltered spatial frequency (f = 0.0833). Most of the energy and information are contained in this intrinsic spatial frequency, while other frequencies represent the noise information, as indicated by the noise floor. In the following simulations the Fourier phase variations are all calculated at this spatial frequency.
In each condition the OPD decreases with a step of 20nm. Two demodulation methods (wavelength tracking and Fourier phase) are applied to the simulated spectrum simultaneously. The results are demonstrated in the following Fig. 3. From the spectra shown in Figs. 3(a), 3(c) and 3(e), increasing level of spectra noise can be observed obviously. The corresponding demodulation results using the two methods in three conditions are respectively revealed in Figs. 3(b), 3(d) and 3(f).
Fig. 3 Simulated spectra with OPD variation ranging from 0 to 200nm under noise coefficient of (a) 0; (c) 0.005 and (e) 0.01 and demodulated results using wavelength searching and Fourier phase under noise level of (b) 0; (d) 0.005 and (f) 0.01.
The demodulated signals acquired by the wavelength tracking suffer from obvious fading of accuracy with increasing noise. The evidence can be provided by the decreasing of the R2 value (goodness of fit) from 0.999 to 0.983. By contrast, the Fourier phase results exhibit very good stability. The R2 value maintains at 0.999, indicating a high accuracy of the demodulation results under different levels of spectral noise. According to Eqs. (8) and (18), the errors of the real and imaginary parts of the Fourier number are indicated by Eqs. (19) and (20). Since the expectation of the white noise function and the average number of the cosinusoidal item within the range of j = [1:N] are both 0, the errors can be expected to be approximately equal to 0. Therefore, the noise effect is almost eliminated.
3.2 Performance with wavelength sampling
In line with the noise performance simulations, we still set three conditions with wavelength sampling interval of 0.05nm, 0.2nm and 0.4nm respectively. Under each condition, the OPD variation value changes from 0 to −200nm with 20nm step. The simulated results are demonstrated in Fig. 4, which is similar to the Fig. 3. With the decreasing of wavelength sampling rate (i.e. the increasing of sampling interval), the wavelength demodulation results are suffering from great fading of accuracy. The R2 values are 0.998, 0.982 and 0.946 for the three conditions respectively. The corresponding sensitivities are 7.7, 7.6 and 8.4pm/nm. Meanwhile, the Fourier phase results maintain a stable performance with R2 values keeping at 0.999, and stable values of the fitted sensitivity of −0.232, −0.232 and −0.231deg/nm.
Fig. 4 Simulated spectra with OPD variation ranging from 0 to 200nm under wavelength sampling interval of (a) 0.05nm; (c) 0.2nm and (e) 0.4nm and demodulated results using wavelength searching and Fourier phase under wavelength sampling interval of (b) 0.05nm; (d) 0.2nm and (f) 0.4nm.
3.3 Measurement error analysis
In order to make the comparison between these two methods more clearly, we define a parameter of measurement error (ME) to evaluate the demodulation accuracy under different conditions. The measurement error are defined as the following Eq. (21). In the equation, x represents the parameter which is selected as the symptom to demodulate the OPD variation. More specifically, x stands for wavelength in the wavelength tracking method, and phase for the proposed Fourier demodulation method. xM and xT are respectively the measured value and theoretical value of the parameter. The result errors for noise analysis and wavelength sampling analysis are calculated and plotted in Figs. 5(a) and 5(b). From the figures we can observe that the Fourier phase results maintain a very good stability and high accuracy under different noise level and sampling rate, while the wavelength demodulation results show much bigger error values, and the error is increasing with the aggravation of disturbing factors (noise and wavelength sampling).
Fig. 5 Results error of the two demodulation methods under different (a) noise level and (b) sampling interval.
4. Experimental results and discussion
In the previous theoretical analysis, the proposed demodulation scheme is explained by a model of two-beam interference. The principle can be extended to multi-beam interference, for which the transmission function can be regarded as the summation of several components of two-beam interference, as revealed by Eq. (22). The FFT spectrum of the multi-beam interference function will show a series of intrinsic spatial frequencies, as expressed in Eq. (23). According to the above analysis, the phase information of each interference component △φn is contained in the corresponding spatial frequency. By calculating the Fourier phase variation at these intrinsic spatial frequencies, the response of different interference components can be analyzed simultaneously.
An experiment is carried out by utilizing a multimode interferometer to demonstrate the performance of the proposed method. The sensor is formed by a single mode fiber (SMF) core offset splicing structure. This structure is very simple and commonly used in fiber sensing applications [26,27]. The schematic diagram of the core offset splicing structure can be seen in Fig. 6. In our experiment, the offset value is set to be 6.5μm and the length of the sensor head is 4cm. Cladding modes will be excited at the first splicing point due to the mode field mismatch. At the second splicing point the cladding modes will be re-coupled to the fiber core and interfere with the core mode. In our experiment, we test the sensor response under temperature and strain variation. The experimental setup is demonstrated in Fig. 6. For temperature measurement, the fiber under test (FUT) is placed in a temperature groove. A temperature control (TEC) module (TLTP-TEC2410S, Talent) is utilized to provide different temperature in the groove. For strain sensing, the FUT is held on a two-dimensional (2D) electrical translation stage, as shown in Fig. 6. A movable stage can apply strain to the sensor structure and the moving distance is controlled by a computer.
The sensor spectra under different temperature (20°C to 65°C) are obtained by an optical spectrum analyzer (OSA, Yokogawa 6370c) and plotted in Fig. 7(a). For the sake of accurate Fourier analysis, the spectra are firstly deduced from logarithmic scale to linear scale. The spectrum exhibits a red shift with the increasing of the temperature. The frequency spectra are acquired by taking the FFT of the optical spectra, and the results are plotted in Fig. 7(b). Several peaks can be observed, indicating that several cladding modes are excited by the core offset structure. It can be seen that the locations of these peaks are stable with different temperature, which agrees with our numerical simulation in Fig. 2.
In order to realize the simultaneous measurement of temperature and strain, we select two dominant cladding modes to analyze the response. The corresponding spatial frequencies are 0.035nm−1 and 0.075 nm−1 respectively, as shown in Fig. 7(b). The Fourier phase variations of these two cladding modes are calculated and plotted in Fig. 8. Linear fit shows that the two selected cladding modes have different temperature sensitivities of −1.22deg/°C and −1.47 deg/°C respectively. This difference comes from the diverse variation amount of effective refractive index (ERI) difference between each cladding mode and core mode.
For comparison between the proposed Fourier phase demodulation method and conventional wavelength demodulation method, we select two resonant dips in the optical spectrum to monitor the wavelength shift with the temperature variation. The two selected dips are indicated in Fig. 7. The wavelength demodulation results are shown in the following Fig. 9.
The relationship between the resonant wavelength and temperature can be observed in Fig. 9(a). The sensitivities of the two dips are 45.7pm/°C and 38.6pm/°C, respectively. The partial enlarged scale of the two selected resonant dips are demonstrated in Figs. 9(b) and 9(c). We can see obvious noise in the detailed spectral curves. In our experiment, the spectral noise is mainly due to the low optical power of the super continuum source. The linear fit of the wavelength shift indicates that the result suffers from relatively poor linearity and unpredictable errors. The R2 values are respectively 0.827 and 0.961. However, the Fourier phase sensitivities show great linearity with R2 values of the two modes are both 0.999. Therefore, it is manifested by the comparison that the proposed Fourier phase demodulation results have better performance than the wavelength demodulation, which is in agree with the simulations.
As for strain sensing, the initial distance between the two stages are 20cm. Each time the movable stage is driven by the electric motor to move 20μm, i.e. applying strain of 100με. The optical spectra response under different strain are plotted in Fig. 10(a). The applied strain causes the blue shift of the multimode interference spectrum. We still select the same two cladding modes that we used for temperature measurement to compute the Fourier phase variation. The results are plotted in Fig. 10(b). The sensitivities of the two cladding modes are −0.004deg/με and 0.019deg/με.
Since the two selected cladding modes have different responses to temperature and strain, these two parameters can be measured simultaneously from the phase variation of the two modes by using a matrix. The expressions are shown in Eq. (24) and (25).
We conducted another supplementary experiment to investigate the Fourier phase stability of the SMF offset structure. The spectrum of the sensor is recorded with a time interval of 10 minutes under a stable environment. The computed Fourier phase of the two selected cladding modes are plotted in Fig. 11. The mean square error (MSE) of the two modes are calculated to be 0.03° and 0.06°, respectively.
5. Conclusion
To summarize, we proposed a phase demodulation method for interferometric fiber sensor by analyzing the Fourier phase variations at the intrinsic spatial frequencies. Theoretical analysis indicates that the Fourier phase variations at the spatial frequencies are equal to the phase changes that come from the OPD variation of the IFS. Numerical simulations have proved that the proposed method can maintain great stability and high accuracy under the influence of wavelength sampling and spectral noise. The scheme is also experimentally demonstrated by an inline multimode interference structure. Simultaneous measurement of temperature and strain is realized by monitoring the phase variations of two selected cladding modes. This method also has potential in the applications such as quasi-distributed sensing system using frequency-division multiplexed IFS.
Funding
This work is supported by a grant (No. 61290315, 61290311, 61275083) from Natural Science Foundation of China and a grant from the Fundamental Research Funds for the Central Universities (No. 2017KFYXJJ032)
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