## Abstract

A fiber-optic extrinsic Fabry-Perot interferometer strain sensor (EFPI-S) of l_{s}=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 µ*ε*/Volt (range 70 µε).

© 2001 Optical Society of America

## 1. Introduction

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 [1]. Adaptive structures made of composite materials with integrated piezoelectric actuators such as deformable parabolic antennas for space applications [2] 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 [3]. 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 [3], a SLD based dual wavelength scheme with spectrum analyzer readout for absolute sensing (AEFPI) [4] and wavelength scanning in combination with a reference EFPI [5]. 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 [1] 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 [6]. 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

A schematic of the EFPI sensor system with a new version of the 3-λ demodulation unit [1] and details of the EFPI - sensing element is shown in figure 1.

In contrast to the Epoxy based construction described in [3][4][5] 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 l_{S}=24.5 mm and is defined by the distance between fusion points FA. The EFPI-gap width at zero strain (ε=0) is L_{0}≈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 Θ_{1,2,3} (Θ_{1}=0°) in order to adjust for transmission wavelengths λ_{1}=1333.8, λ_{2}=1343 and λ_{3}=1352.2 nm. Θ_{1}(λ_{1}), Θ_{3}(λ_{3}) are fixed whereas Θ_{2}(λ_{2}) is adjustable. The selected wavelengths provide quadrature interference signals for the given initial EFPI-gap width L_{0}. 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 (l_{s}=24.5 mm, and 50 mm) epoxied to the top electrode of a 1 mm thick PZT-sheet actuator.

The PZT-ceramics plate (70×25×1mm^{3}, type PIC151 of company PI Ceramics) is characterized by the cross-expansion coefficient d_{31}=-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 R_{1,2} (silica-glass - air boundary: R_{1}≈R_{2}=R≈3.3 %) the normalized interferometer output intensity i_{r}=I_{r}/I_{0} (I_{0}=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 $\sigma \approx \delta \lambda \u20442\u2044\sqrt{2\mathrm{ln}2}$ and center wavelength λ_{i} the output signals of the 3-λ unit may be written as:

with i=1,2,3, Φ=Φ_{0}+ΔΦ_{i}(t), Φ_{0}=4π L_{0}/λ_{i}=initial phase with EFPI reflector distance L_{0} typically between 10 and 40 µm ≪ coherence length L_{c}≈λ^{2}/δλ≈130 µm ΔΦ_{i}=4πΔL/λ_{i}=measurand induced phase shift and fringe contrast

${{\mathrm{R}}_{2}}^{\text{eff}}$ 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 [7]. With L_{0}=23 µm the fringe contrast µ≈65 %, with the coherence length playing the dominant role with respect to fringe contrast reduction (${{\mathrm{R}}_{2}}^{\text{eff}}$ (L_{0})≈0.95 R_{2}). λ_{2} can be fine tuned by tilting the mechanically coupled GRIN-lens - photodiode combination for obtaining the required quadrature conditions: ΔΦ_{12}=Φ_{1}(λ_{1})-Φ_{2}(λ_{2})=ΔΦ_{23}=Φ_{2}(λ_{2})-Φ_{3}(λ_{3})=*π*/2. The phase difference of two signals with different wavelengths λ_{i}, λ_{j}=λ_{i}+Δλ is ΔΦ_{ji}=-4πL_{0}/λ_{Syn}. With synthetic wavelength λ_{syn}=λ_{i}λ_{j}/Δλ, Δλ=9.2 nm << λ_{1} ≈λ_{2}=λ the quadrature condition ΔΦ_{ij}=(2N+1) π/2, N=0, 1, 2, … yields the reflector distance L_{0}=(2N+1) λ^{2}/8Δλ^{Q}≈23 µm for N=0 and λ_{syn}=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 u_{2} 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 [1] $f\left(\mathit{\delta}\Delta {\Phi}_{\mathit{ij}}\right)=\mathrm{tan}[\frac{\pi}{4}+m\frac{\pi \Delta \lambda}{2{\lambda}_{2}}]$. f(δΔΦ_{ij})=1 for ΔΦ_{12}=ΔΦ_{23}=π/2 (m=ΔΦ(t)=0) and f(δΔΦ_{ij})>0.9 for ΔΦ(t)<2π. The EFPI strain sensitivity is given by

yielding a gauge factor S_{Φ}=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 [1]. 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 ΔΦ_{ij}=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 m_{0}≠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 Φ_{0} with m=m_{0}. Because the nominal phase difference ΔΦ_{i,j}=π/2 changes by δ(ΔΦ_{ij})/ΔΦ=2πΔλ/λ≈2.7°/2 π (per fringe) the unambiguous range is limited to about δ(ΔΦ_{12})<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 S_{U}=S_{Φ} ΔΦ/ΔU=188.25 (11.7) µε/Volt (with S_{Φ} 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 U_{pp}=1800 Volt at 10 Hz is ΔL_{max}=13.18 µm (Fig. 3) corresponding to the measured phase modulation of ΔΦ_{max}=123.2 rad or 19.6 fringes (ΔΦ_{max}/2π). The corresponding strain ε_{max}=ΔL/l_{S}=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=d_{31} E l_{S}=9.8 µm and keeping in mind the large signal behavior (hysteresis) of piezoceramics. The displacement modulation between L_{0} and L_{1}=L_{0}+ΔL_{max} is accompanied by a fringe contrast modulation µ(L_{1})/µ(L_{0})=0.5. Within the experimental uncertainty this value agrees with the theoretical value µ_{1}/µ_{0}=0.52, as obtained with equ.(2). Fig. 4 shows the resolution limit with the increased scale factor S_{U}=11.7 µ*ε*/Volt. This result is obtained with an extremely low PZT excitation voltage of only U_{pp}=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=l_{S}=24.5 mm). Channel 4 shows the lowpass filtered (f_{cuttoff}=100 Hz) phase demodulation signal (DAC-output voltage), which corresponds to a displacement amplitude ΔL_{min}=110 pm (ε_{min}=ΔL/l_{S}=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 U_{rms}=73 µV corresponding to δε_{rms}=0.85 nm/m and δL_{rms}=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.

## 5. Conclusion

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.

## Acknowledgements

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).

## References

**1. **M. Schmidt and N. Fürstenau, “Fiber-optic extrinsic Fabry-Perot interferometer sensors with three-wavelength digital phase demodulation,” Opt. Lett. **24**, 599–601 (1999) [CrossRef]

**2. **T. Melz, M. Frövel, V. Krajenski, M.A. de la Torre, H. Hanselka, and J. M. Pintado, “Modelling and control of adaptive mechanical structures”, in: Gabbert, Ulrich (Eds.): Fortschritt-Berichte VDI : ser. no. 11, Schwin-gungstechnik **268**, 449–458 (1998)

**3. **K.A. Murphy, M.A. Gunther, A. M. Vengsarkar, and R.O. Claus, “Quadrature phase shifted extrinsic Fabry-Perot optical fiber sensors,” Opt. Lett. **16**, 273–275 (1991) [CrossRef] [PubMed]

**4. **V. Bhatia, K.A. Murphy, R.O. Claus, M. E. Jones, J. L. Grace, T.A. Tran, and J.A. Greene, “Multiple strain state measurements using conventional and absolute optical fiber-based extrinsic Fabry-Perot interferometric strain sensors,” Smart Materials Struct. **4**, 240–245 (1995) [CrossRef]

**5. **T. Li, R.G. May, A. Wang, and R.O. Claus, “Optical scanning extrinsic Fabry-Perot interferometer for absolute microdisplacement measurement,” Appl. Opt. **36**, 8858–8861 (1997) [CrossRef]

**6. **M. Schmidt, N. Fürstenau, W. Bock, and W. Urbanczyk, “Fiber-optic polarimetric strain sensor with three-wavelength digital phase demodulation,” Opt. Lett. **25**1334–1336 (2000) [CrossRef]

**7. **N. Fürstenau, M. Schmidt, H. Horack, W. Goetze, and W. Schmidt, “Extrinsic Fabry-Perot interferometer vibration and acoustic Sensor systems for airport ground traffic monitoring,” IEE Proc. Optoelectron. **144**, 134–144 (1997) [CrossRef]