EFPI sensor utilizing optical spectrum analyzer with tunable laser: detection of baseline oscillations faster than spectrum acquisition rate

Transcription

1 EFPI sensor utilizing optical spectrum analyzer with tunable laser: detection of baseline oscillations faster than spectrum acquisition rate Nikolai Ushakov *, Leonid Liokumovich Department of Radiophysics, St. Petersburg State Polytechnical University ABSTRACT A novel approach for extrinsic Fabry-Perot interferometer baseline measurement has been developed. The principles of frequency-scanning interferometry are utilized for registration of the interferometer spectral function, from which the baseline is demodulated. The proposed approach enables one to capture the absolute baseline variations at frequencies much higher than the spectral acquisition rate. Despite the conventional approaches, associating a single baseline indication to the registered spectrum, in the proposed method a modified frequency detection procedure is applied to the spectrum. This provides an ability to capture the baseline variations which took place during the spectrum acquisition. The limitations on the parameters of the possibly registered baseline variations are formulated. The experimental verification of the proposed approach for different perturbations has been performed. Keywords: fiber-optic sensors, frequency detection, signal processing, extrinsic Fabry-Perot interferometer, spectral measurements, phase measurements, EFPI, frequency-scanning interferometry. 1. INTRODUCTION Fiber-optic interferometric sensors have been a subect of an extensive study of academia and industry during the last two decades 1. Their immunity to electromagnetic radiation, low cost, small dimensions, ability to operate in harsh environments and high performance make them attractive for a great diversity of applications for measurement of temperature 2, strain 3, pressure 4, humidity 5, electric field 6 and micro-displacements 7-9 in such areas as oil and gas exploitation 4, structure health monitoring 3, nuclear energetics 10 and fundamental science 8. One of the most popular schemes of fiber-optic interferometric sensors is extrinsic Fabry-Perot interferometer (EFPI), which, along with the fiber Bragg gratings (FBG) is most commonly used for strain, temperature, pressure and vibration sensing. Sensors based on both structures are characterized by tiny dimensions and multiplexing abilities, however, the resolution is higher for EFPI sensors, and the acquisition rate is higher for FBG sensors. Among the EFPI baseline measurement techniques one can distinguish two basic classes: tracking only the baseline variations and capturing the absolute baseline value, which includes conventional white-light 11 techniques utilizing tunable read-out interferometer and approaches based on the registration and subsequent analysis of the interferometer spectral function. The variation-tracking approaches are characterized by relatively high operation frequencies and resolutions 12. One of the most accurate spectral registering methods is frequency scanning interferometry, demonstrating picometer-level resolutions, high absolute accuracies and large dynamic measurement range 7. However, the measurement rate doesn t exceed several Hz, while in some applications higher speeds along with absolute value and high resolution are essential. In the current paper a novel signal processing approach is proposed, enabling one to overcome this disadvantage and track much faster baseline fluctuations than the spectrum acquisition rate. In frequency scanning techniques the optical spectral function of the interferometer is registered, which, for a low-finesse case, is given by expression below * Corresponding author: n.ushakoff@spbstu.ru S FP (L, λ) = S 0 (L, λ) + S(L, λ), (1) S(L, λ) = S m cos(4πnl/λ + γ(l, λ)), (2) γ(l, λ) = arctan(2lλ/πnw 2 ) + φ, (3)

2 where S m = 2(R 1 R 2 * ) 1/2, R 1 is reflecting coefficient of the first mirror and R 2 * is reflecting coefficient of the second mirror with taking into account the light losses due to beam divergence; n is the refractive index of the media inside the interferometer; λ is the free-space light wavelength; the first term in γ(l, λ) is a Gouy phase shift 13, an additional phase term φ is induced by the mirrors; w is an effective radius of a Gaussian beam at the output of the first fiber, close to the fiber mode field radius. One of the most attractive approach for estimation of the baseline L from the registered spectrum is to approximate its variable component S (λ) with analytical expression (2) by means of least-squares fitting. Such fitting returns the global minimum of the residual norm, given by expression R ( L) = S ( λ) S( L, λ) = [ S' S ( L) ] 2 ' i i, (4) where discretization of the wavelength scale was taken into account: λ i = λ 0 + i, is the step between the spectral points, i = M/2, M/2, M is the number of points in digitized spectrum; S i =S (λ i ), S i (L)=S(λ i, L). As shown in 7, when only the slopes of the S (λ) function are processed, the resolution and robustness of the approximation approach are dramatically enhanced, resulting in attained baseline resolution about 30 pm for the baseline value L 200 µm. i 2. SIGNAL PROCESSING As was already mentioned in the introduction, the sample rate of conventional spectral function-registering measurements is equal to spectrum acquisition rate, which doesn t exceed several Hz. In the current paper we propose a method of registering much more rapid fluctuations of the interferometer baseline. Throughout the paper the following notation will be used: the length of the temporal interval of the spectrum acquisition T M ; the width of the wavelength scanning range Λ = M. Let us consider that the wavelength-scanning interferometry methods are used for interrogating the sensor, therefore, at each particular temporal moment t i [ T M /2; T M /2] (or i-th spectrum point) the wavelength λ i can be written as λ i = λ 0 + k λ t i. It should be noted that the zero time moment t = 0 corresponds to i = 0 (the middle point of the spectrum) and the step between t i moments is equal to 1/f D, where f D is the sample rate of the acquired photodetector signal. On this basis one can take into account the variation of the interferometer baseline during the spectrum acquisition time T M. Denoting the corresponding EFPI spectrum as S (λ i, L(t i )), and expressing it according to the (2), the following formula can be obtained S ( λ L( t )) ( t ) 4πnL i 4πnL 4πn δl i = SM cos + γ( L( ti ), λi ) = SM cos + + γ( L( ti ), λ ), (5) λi λi λi 0 i, i i where L 0 is the mean value of the interferometer baseline during spectrum acquisition, δl(t i ) is the baseline variation with respect to the mean L 0, further denoted as δl i for simplicity. For further convenience we transition from the wavelength λ i to the optical frequency ν = ν 0 + k ν t in the EFPI spectrum expressions, as was done in, for instance 14. It should be noted that the uniform grids of the wavelengths λ i and the optical frequencies ν do not correspond to each other, since the uniform wavelength stepping with produces a nonuniform frequency grid (ν i = c/nλ i ), and vice versa. Hereinafter, the index i will define the uniform wavelength grid, and the index will define the uniform frequency grid. On this basis the expression for the interferometer spectral function transforms to the following form 4πnL 4πn δl n cos 0 M. (6) c c c 0 ( L ) = S ν + ν + γ( L + δl, c ν ) S cos π ( L + δl ) ν + γ S 0 M 4 0 The structure of the expression (6) is quite similar to the one of the frequency-modulated signal with respect to the ν, except for the phase term γ, providing an undesired phase modulation. However, considering weak dependency γ (L) and assuming δl<<l 0, the influence of this additional phase modulation is quite weak and the influence of the ( t )

3 baseline variations δl on the γ can be neglected. For the case of the constant interferometer baseline during the spectrum acquisition (6) transforms to n cos. (7) c ( L ) S π L ν + γ S 0 = M 4 0 This signal is quasi-harmonic since the argument variations are related not only to the equivalent frequency 2nL 0 /c, but also to the non-uniform term γ, which behavior must be precisely calculated or can be verified experimentally. Even in presence of the δl perturbation the signal S remains quasi-harmonic, its argument can be obtained by means of the Hilbert transform. Comparing the expressions (6) and (7), the signal processing for obtaining the δl from the measured spectrum S can be divided into the two following steps: Find the average baseline value L 0 by means of approximating the measured spectrum S i by the analytical expression (2) (the expression (8) can be applied to S as well, depending on in which units the abscissa axis is represented). A detailed description and analysis of the approximation method used in the current study is presented in 7. In the context of the frequency detection task this first step is essential for finding the carrier frequency, necessary for the following demodulation of the baseline variations. By means of the Hilbert transform calculate the analytical signal for the measured spectral function S. After that obtain the argument of the analytical signal, to which apply the standard unwrapping procedure, based on the Itoh-criterion 15. This will produce the continuous argument ψ. Then the desired difference of the arguments is calculated as Ψ = ψ 4πn/cL 0 ν γ, (8) from which the baseline variation δl can be found according to the expression δl Ψ c =. (9) 4πnν The use of the first step, obtaining the L 0 with very high accuracy enables one to find the non-perturbation part of the ψ argument with much greater precision than detrending of the ψ and other simple methods deleting the regular components of ψ. For proper performance of the proposed approach the limits on the spectrum and the amplitude of the baseline variation δl must be formulated. The applicability criterion is that the spectral components of the S temporal representation do not decrease below zero and satisfy the Nyquist limit. For simplicity let us consider the limitations for the case of harmonic oscillation of the interferometer baseline δl = L m cos(2πf L t ). On this basis and taking into account that the frequency scanning range k ν T M is much smaller than the central frequency ν 0, the total phase of S can be expressed as 4πnL k 4πnν 0 ( t ) = t + L cos( 2πf t ) = 2πf t + ψ cos( πf t ) 0 ν ψ m L Sc m 2 L, (10) c c where f Sc is an equivalent carrier frequency of the S, given by formula 2nL k c 0 ν f S c =. (11) Therefore, estimating the S spectrum width according to the Carson s bandwidth rule as a sum of perturbation frequency f L and frequency deviation f F = 4πnν 0 f L L m /c, the restriction on the baseline variations can be formulated as f L + f F < f Sc, f L + f F < f D f Sc. Typically, the second inequality is fulfilled by default, so only the first one is relevant, expressed in terms of the current system as follows: 4πnν f L 2 +. (12) c 0 Lm 2nL0k 2 0 m 2 4 ν nl k L = f + π < λ L n 2 λ0 c λ 0

4 On this basis, for the practical values L 0 ~ 1000 µm, device parameters k λ = 2.4 µm/s, λ 0 = 1.55 µm (see section 3 of the current paper) the carrier frequency f Sc estimated by expression (11) is about 2 khz, therefore, for the signal frequency f L = 100 Hz the limitation on the baseline oscillations amplitude providing the absence of the spectrum aliasing is L m 2.3 µm. The relations of the amplitude L m and frequency f L limits for different baseline values L 0 are illustrated in figure 1. Figure 1. Limits of the amplitude and frequency of the harmonic baseline variation for different baseline values. It should be noted that in practical optical spectrum analyzers the uniform wavelength grid is generally used, therefore, the corresponding optical frequency scale ν i in (6) and (7) will be related to temporal moments as ν i = c/(λ 0 + k λ t i ), resulting in incorrect calculation of the analytical signals phases and therefore, improper performance of the signal processing. In order to overcome this problem, two possible solutions can be proposed: - Utilization of non-uniform fast Fourier transform (NUFFT) algorithms 16 for analytical signal calculation; - Interpolation of the initially registered spectrum S (λ i ) with the uniform wavelength scale to spectrum S (c/ν ) with the uniform frequency scale before analytic signal calculation. An inverse interpolation will be needed for the calculated δl signal in order to obtain the signal δl i (and L i = L 0 + δl i ), uniformly sampled with respect to time. After calculating the baseline variation with uniform temporal sampling δl i it can be filtered by a low-pass filter with cut-off frequency f Sc. The reason for doing so is that on the one hand, for proper performance, the frequency of registered δl i is limited by f Sc and on the other hand, the sample rate of δl i is again equal f D, which usually is much higher than the working frequency band. 3. EXPERIMENTAL DEMONSTRATION The proposed approach was implemented and tested experimentally. Spectra measurements were performed using the optical sensor interrogator National Instruments PXIe 4844, installed on PXI chassis PXIe 1065, controlled by PXIe 8106 controller. Spectrometer parameters are the following: scanning range nm (spectral interval width Λ = 80 nm), spectral step = 4 pm (number of spectral points M = 20001), scanning speed k λ = 2400 nm/s, spectrum acquisition time T M 0.035s, output power P 0.05mW. Examined interferometer was formed by the fiber end packaged in FC/PC connector and an external mirror, adusted to the PZT actuator. The controlling voltage for the PZT was generated by the PXIe 5421 signal generator, installed on the same PXI chassis. The experimental setup is schematically illustrated in figure 2.

5 Figure 2. Experimental setup. The efficiency of the utilized PZT actuator was approximately 100nm/V in the frequency range Hz. The mean baseline value was set to about 720 µm, resulting in the S i carrier frequency f Sc 1350 Hz. The parameters of the PZT supplying voltage were the following: - The peak-to-peak amplitude was varied from 0.1 to 6 V, resulting in peak-to-peak value of the EFPI baseline variations from 20 nm to 1.2 µm; - the frequency was varied from 50 to 1000 Hz; - the oscillation shape was either harmonic or triangular. When processing the experimental signals, the influence of the γ(l, λ i ) phase term must properly be taken into account. The regular component of γ i (the first term in (3)) can be calculated analytically according to L 0, obtained at the first step of the processing, while the second term φ i is more complex to be calculated analytically, however, it can be examined experimentally with the use of the above mentioned signal processing. For that we have registered the EFPI spectrum in the absence of the baseline fluctuations. After that the φ i phase shift was found according to the expression (8), where only the first term in γ was involved. The wavelength dependency of the obtained phase shift, averaged with respect to 100 spectra is illustrated in figure 3. Figure 3. Experimentally obtained phase shift. During the processing of experimental signals this phase shift was taken into account as φ i in (3). As can be concluded with the use of (9), if not compensated, it would provoke a ghost signal with frequency ~ 30 Hz and amplitude ~ 13 nm. Below we would like to demonstrate the results of applying the proposed signal processing approach to experimentally registered EFPI spectral functions. Three signals were used to modulate the EFPI baseline: a harmonic, 500 Hz, ~90 nm amplitude; b harmonic, 1000 Hz, ~15 nm amplitude; c triangular, 100 Hz, ~30 nm amplitude. In the figures below the following functions are illustrated: Originally registered EFPI spectra (figure 4). It can be seen from the figure 4 (a) that the baseline oscillation with the largest amplitude produces some distortions of the EFPI spectrum, which takes the form of angular-modulated signal. The rest two spectra (figures 4 (b,c)) look pretty like the harmonic signals. Fourier transforms of the EFPI spectra (figure 5). For harmonic perturbations with large frequency the modulation-induced spectral components can clearly be observed (again, for the signal with greater amplitude these components are much stronger).

6 Temporal dependencies of demodulated EFPI baseline variations are illustrated in figure 6. After applying the proposed signal processing approach, the baseline oscillations are adequately registered. Fourier transforms of the demodulated EFPI baseline variations are shown in figure 7 (the Blackman window was applied to the signals before calculating FFTs). Figure 4. EFPI spectral functions for different baseline oscillations. Figure 5. FFTs of the EFPI spectral functions for different baseline oscillations. Figure 6. Demodulated baseline oscillations.

7 Figure 7. Spectra of the demodulated baseline oscillations (rms values). 4. CONCLUSION In the current paper a novel approach enabling one to overcome the conventional limitations of the frequencyscanning absolute interferometry, where from a single acquired interferometer spectrum a single baseline value is registered. Instead, using the proposed signal processing approach, one is able to track the fast deviations of the absolute baseline value that took place during the spectrum acquisition. The upper frequency limit of the registered oscillations is defined by the baseline value and is provoked by the no-aliasing conditions. The lower limit is defined by the duration of spectrum acquisition and is related to that at least one period of the registered oscillation must be presented at the temporal interval of the spectrum acquisition (the problem of the perturbation linearly varying during the spectrum acquisition was studied and solved in 17 ). REFERENCES [1] Udd, E., [Fiber Optic Sensors: an Introduction for. Engineers and Scientists, 2-nd edition], John Wiley & Sons, Inc., Hoboken, New Jersey, (2011). [2] Zhang, G., Yang, M. and Wang, M., Large temperature sensitivity of fiber-optic extrinsic Fabry-Perot interferometer based on polymer-filled glass capillary, Opt. Fib. Technol. 19, (2013). [3] Schilder, C., Kohlhoff, H., Hofmann, D., Basedau, F., Habel, W.R., Baeßler, M., Niederleithinger, E., Georgi, S. and Herten, M., Static and dynamic pile testing of reinforced concrete piles with structure integrated fibre optic strain sensors, Proc. SPIE 8794, (2013). [4] Pechstedt, R.D., Fibre optic pressure and temperature sensor for applications in harsh environments, Proc. SPIE 8794, (2013). [5] Willsch, R., Ecke, W., Schwotzer, G. and H. Bartelt, Nanostructure-based Optical Fibre Sensor Systems and Examples of their Application, Proc. SPIE 6585, 65850B (2007). [6] Liokumovich, L.B., Medvedev, A.V. and Petrov, V.M., Fiber-optic polarization interferometer with an additional phase modulation for electric field measurements, Opt. Mem. Neur. Netw. 22(1), (2013). [7] Ushakov, N., Liokumovich, L. and Medvedev, A., EFPI signal processing method providing picometer-level resolution in cavity length measurement, Proc. SPIE 8789, 87890Y (2013). [8] Petrov, V., Petrov, M., Bryksin, V., Petter, J. and Tschudi, T., Optical detection of the Casimir force between macroscopic obects, Opt. Lett. 31(21), (2006). [9] Zhou, X., Yu, Q., Wide-Range Displacement Sensor Based on Fiber-Optic Fabry Perot Interferometer for Subnanometer Measurement, IEEE Sens. Jour. 11(7), (2011). [10] Cheymol, G., Villard, J.F., Gusarov, A. and Brichard, B., Fibre Optic Extensometer for High Radiation and High Temperature Nuclear Applications, IEEE Trans. Nucl. Sci. 60 (5), (2013). [11] Yuan, Y., Wu, B., Yang, J. and Yuan, L., Tunable optical-path correlator for distributed strain or temperaturesensing application, Opt. Lett. 35(20), (2010).

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