A fiber-optic extrinsic Fabry-Perot interferometer strain sensor (EFPI-S) of ls=2.5 cm sensor length using three-wavelength digital phase demodulation is demonstrated to exhibit <50 pm displacement resolution (< 2nm/m strain resolution) when measuring the cross expansion of a PZT-ceramic plate. The sensing (single-mode downlead-) and reflecting fibers are fused into a 150/360 µm capillary fiber where the fusion points define the sensor length. Readout is performed using an improved version of the previously described three-wavelength digital phase demodulation method employing an arctan-phase stepping algorithm. In the present experiments the strain sensitivity was varied via the mapping of the arctan - lookup table to the 16-Bit DA-converter range from 188.25 µε/V (6 Volt range 1130 µε) to 11.7 µε/Volt (range 70 µε).
© 2001 Optical Society of America
We report on experiments with an optimized extrinsic Fabry-Perot interferometer (EFPI) strain gauge with extremely high sensitivity for “smart structures” applications using a new version of the three-wavelength digital phase demodulation system . Adaptive structures made of composite materials with integrated piezoelectric actuators such as deformable parabolic antennas for space applications  require sensors for permanent shape monitoring. Conventional resistive strain gauges often cannot provide the required resolution (sub- µε) and stability and exhibit too high noise levels. Fiber-optic sensors have the advantage of intrinsic full electromagnetic immunity so that they may be operated in the high electric field environment of PZT-actuators, even glued to the PZT’s electrodes, which is not possible with resistive strain gauges. Originally the EFPI - concept was described by Murphy et. al . This sensor typically is of l=5-10 mm sensing length with lead- and reflecting fibers glued into a precision capillary fiber using Epoxy. The authors proposed several methods for unambiguous fringe demodulation and stabilization of the operating point, such as producing quadrature phase shifted signals by using two EFPI’s with suitably adjusted gap length differences , a SLD based dual wavelength scheme with spectrum analyzer readout for absolute sensing (AEFPI)  and wavelength scanning in combination with a reference EFPI . Only limited linearity of the sensor characteristics and no detailed information on the achievable resolution and processing speed has been reported so far. In a previous publication  we reported initial results on the linearity of EFPI-displacement vs. strain characteristics using a three-wavelength (3-λ-) digital phase demodulation scheme with microcontroller based real-time phase calculation with up to 80 kHz sampling rate using an arctan-phase stepping algorithm and a resistive strain gauge reference sensor. Application of the 3-λ-technique to polarimetric strain sensor readout has been described in . After describing the improved 3-λ design in section 2, the principle of operation is briefly reviewed in section 3, followed by experimental results obtained with a low resolution – high range and a high resolution – low range setting in section 4.
2. Experimental Setup
In contrast to the Epoxy based construction described in  our EFPI-S strain gauge is made of all - silica technology without any glue using a commercial fusion splicer for fusing the single mode downlead sensing (SMF:9/125 µm) and multimode reflecting fiber section (MMF:100/140 µm) into the capillary fiber (CF1:150/360 µm, fusion points FA). EFPI-S sensing length is lS=24.5 mm and is defined by the distance between fusion points FA. The EFPI-gap width at zero strain (ε=0) is L0≈23 µm. The new version of the 3-λ phase demodulation system is based on a single interference filter as compared to the earlier version with three filters for producing the three different transmission wavelengths. This solution provides significantly improved wavelength stability and reproducibility. Instead of the filters in this case the collinearly aligned collimating (GRIN-)lens - photodiode - units are tilted by angles Θ1,2,3 (Θ1=0°) in order to adjust for transmission wavelengths λ1=1333.8, λ2=1343 and λ3=1352.2 nm. Θ1(λ1), Θ3(λ3) are fixed whereas Θ2(λ2) is adjustable. The selected wavelengths provide quadrature interference signals for the given initial EFPI-gap width L0. The ELED light source (output power from pigtail 15 µW) is connected to the EFPI sensing element via a 2×2 3dB-directional coupler and a single mode (SM) fiber-optic downlead. The reflected interference signal passes the same downlead fiber and directional coupler before being split into three components by means of a 3×3 coupler with only one input port used. The 3-λ phase demodulation is independent of input intensity and fringe contrast after initial adjustment of the three interference signals for equal fringe ampitudes and offsets via the photodiode preamplifier gain.
Fig. 2 shows a photograph of the experimental arrangement with two EFPI’s (ls=24.5 mm, and 50 mm) epoxied to the top electrode of a 1 mm thick PZT-sheet actuator.
The PZT-ceramics plate (70×25×1mm3, type PIC151 of company PI Ceramics) is characterized by the cross-expansion coefficient d31=-2.14 10-10 m/V yielding 385 µε at E=1.8 kV/mm.
3. Principle of Operation
For our case of low mirror reflectivities R1,2 (silica-glass - air boundary: R1≈R2=R≈3.3 %) the normalized interferometer output intensity ir=Ir/I0 (I0=input intensity into the FP) may be approximated by the cosinusoidal characteristic of a two beam interferometer. Including a fringe visibility factor for taking into account the spectral width δλ (fwhm)/λ≈1 % of the interference filters, assuming a Gaussian spectral distribution with standard deviation and center wavelength λi the output signals of the 3-λ unit may be written as:
with i=1,2,3, Φ=Φ0+ΔΦi(t), Φ0=4π L0/λi=initial phase with EFPI reflector distance L0 typically between 10 and 40 µm ≪ coherence length Lc≈λ2/δλ≈130 µm ΔΦi=4πΔL/λi=measurand induced phase shift and fringe contrast
is the effective reflectivity of the reflector opposite of the fiber tip which decreases with increasing ΔL due to beam divergence. An equation for a quantitative estimate is derived in . With L0=23 µm the fringe contrast µ≈65 %, with the coherence length playing the dominant role with respect to fringe contrast reduction ( (L0)≈0.95 R2). λ2 can be fine tuned by tilting the mechanically coupled GRIN-lens - photodiode combination for obtaining the required quadrature conditions: ΔΦ12=Φ1(λ1)-Φ2(λ2)=ΔΦ23=Φ2(λ2)-Φ3(λ3)=π/2. The phase difference of two signals with different wavelengths λi, λj=λi+Δλ is ΔΦji=-4πL0/λSyn. With synthetic wavelength λsyn=λiλj/Δλ, Δλ=9.2 nm << λ1 ≈λ2=λ the quadrature condition ΔΦij=(2N+1) π/2, N=0, 1, 2, … yields the reflector distance L0=(2N+1) λ2/8ΔλQ≈23 µm for N=0 and λsyn=196 µm (λ=1343 nm, Δλ=9.2 nm). The EFPI-gap width is adjusted during sensor manufacture by adjusting the Lissajou ellipses u1 vs u2 and u3 vs u2 on the oscilloscope for a symmmetrical shape (horizontal/vertical main axes).
The equation for phase calculation is derived by combination of the three trigonometric equations (2) for the respective three wavelengths yielding
with fringe order m=0, 1, 2, …. The dephasing factor  . f(δΔΦij)=1 for ΔΦ12=ΔΦ23=π/2 (m=ΔΦ(t)=0) and f(δΔΦij)>0.9 for ΔΦ(t)<2π. The EFPI strain sensitivity is given by
yielding a gauge factor SΦ=4.362 µm/m/rad for λ=1343 nm.
Details of the real time calculation of µ and Φ(t) using the Siemens microcontroller SAB167 are described in . For optimizing processing speed all real time calculations are performed using Integer values. An analog output signal is produced via the 16-Bit digital-to-analog converter (DAC) with ±3 Volt range. The <0.1 mV/digit resolution value is smaller than the electronic noise amplitude of ca. 0.24 mVrms at 1 kHz bandwidth. The strain resolution of the analog output is determined by the mapping of the arctan-lookup table (unambiguous phase range -π/2≤Φ≤π/2) to the ±3 Volt DAC-output range. For the low (high) resolution high (small) strain case the Φ-range is mapped to the integer array 0–393 (0–6284) via multiplication by 250 (4000), yielding 4.00 (0.25) mrad/digit. With 65536 digits/6 Volt we get dU/dΦ=144 mV/2π rad (2.3 V/2π rad). Before starting demodulation after turn – on the fringe order m has to be provided as initialization value. For a “virgin” sensor with ΔΦij=π/2 we have m=0. After fixing the sensor to a structure m depends on the structural loading including initial stress due to the adhesion so that the initial m usually is m0≠0. m has to be monitored continuously in order to extend the unambiguous arctan(Φ) - range (±π/2). This is performed by monitoring the nominator and denominator functions for zero - values and, depending on the sign of the zero crossing, adding or subtracting m π for obtaining the Φ(t) – value relative to Φ0 with m=m0. Because the nominal phase difference ΔΦi,j=π/2 changes by δ(ΔΦij)/ΔΦ=2πΔλ/λ≈2.7°/2 π (per fringe) the unambiguous range is limited to about δ(ΔΦ12)<π corresponding to 66 fringes (20 λ) or a maximum strain of 1700 µm/m.
4. Experimental Results
Examples of dynamic strain measurement with the 24.5 mm EFPI-S are shown in fig. 3 and fig. 4. The scale factors for the low (high) resolution cases are SU=SΦ ΔΦ/ΔU=188.25 (11.7) µε/Volt (with SΦ given in equ. (1) and ΔΦ/ΔU=0.144 (2.3) V/2π). The maximum displacement and strain respectively as measured with a PZT excitation voltage Upp=1800 Volt at 10 Hz is ΔLmax=13.18 µm (Fig. 3) corresponding to the measured phase modulation of ΔΦmax=123.2 rad or 19.6 fringes (ΔΦmax/2π). The corresponding strain εmax=ΔL/lS=527.2 µm/m is somewhat higher as compared to the above estimated theoretical PZT-value (400 µε at 1.8 kV). However the correlation is reasonably good when considering the theoretical PZT elongation with respect to the sensor length Δl=d31 E lS=9.8 µm and keeping in mind the large signal behavior (hysteresis) of piezoceramics. The displacement modulation between L0 and L1=L0+ΔLmax is accompanied by a fringe contrast modulation µ(L1)/µ(L0)=0.5. Within the experimental uncertainty this value agrees with the theoretical value µ1/µ0=0.52, as obtained with equ.(2). Fig. 4 shows the resolution limit with the increased scale factor SU=11.7 µε/Volt. This result is obtained with an extremely low PZT excitation voltage of only Upp=40 mV, corresponding to a theoretical PZT-cross- strain amplitude of 7.6 nm/m and a calculated expansion amplitude of 186 pm (for l=lS=24.5 mm). Channel 4 shows the lowpass filtered (fcuttoff=100 Hz) phase demodulation signal (DAC-output voltage), which corresponds to a displacement amplitude ΔLmin=110 pm (εmin=ΔL/lS=4.5 nm/m), again in resonable agreement with the theoretical estimate of the PZT expansion.
The statistical noise contribution of the strain gauge signal (upper curve) can be seen to be <100 µV. Without PZT-excitation we measured Urms=73 µV corresponding to δεrms=0.85 nm/m and δLrms=21 pm. Assuming white noise as input to the low-pass filter of our demodulation unit the corresponding power spectral density for f=100 Hz bandwidth is obtained as 5.9 µV/√Hz or 69 pm/m/√Hz.
It has been demonstrated that an EFPI - strain gauge in combination with a three-wavelength digital phase demodulation is able to monitor structural deformations of <50 pm. Based on these results initial experiments have been successfully performed with EFPI strain gauges glued to adaptive PZT-controlled CFRP-antenna reflector structures. The results agree with those on the PZT surface and confirm the potential of this sensor for smart structures applications.
We appreciate the support of this work by the European Commission under contract number BR PR-CT98-0751 (BRITE-Euram project “Smart Structures”) and by the consortium members (companies RAMBOLL (DK), FORCE (DK), DRI (DK), Sensortec (FRG), BAM (FRG), OSMOS-Dehacom (FRG), Autostrade (I); http://smart.ramboll.dk).
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