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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 7 — Apr. 4, 2005
  • pp: 2377–2384
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Fiber Bragg-grating strain sensor interrogation using laser radio-frequency modulation

G. Gagliardi, M. Salza, P. Ferraro, and P. De Natale  »View Author Affiliations


Optics Express, Vol. 13, Issue 7, pp. 2377-2384 (2005)
http://dx.doi.org/10.1364/OPEX.13.002377


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Abstract

We demonstrate the possibility of using radio-frequency modulation spectroscopic techniques for interrogation of fiber Bragg-grating (FBG) structures. Sidebands at 2 GHz are superimposed onto the output spectrum of a 1560-nm DFB diode laser. The power reflected by an FBG is demodulated at multiples of the sideband frequency. The sideband-to-carrier beat signal is shown to be extremely sensitive to Bragg wavelength shifts due to mechanical stress. Using this method, both static and dynamic strain measurements can be performed, with a noise-equivalent sensitivity of the order of 150 nε/√Hz, in the quasi-static domain (2 Hz), and 1.6 nε/√Hz at higher frequencies (1 kHz). The measured frequency response is presently limited at 20 kHz only by the test device bandwidth. A long-term reproducibility in strain measurements within 100 nε is estimated from laser frequency drift referred to molecular absorption lines.

© 2005 Optical Society of America

1. Introduction

Over the last decade, fiber Bragg-gratings (FBGs) have been proposed for a variety of sensing applications [1

1. Y. J. Rao and S. Huang, “Applications of Fiber Optic Sensors,” in Fiber Optic Sensors, F.T.S. Yu and S. Yin eds. (Marcel Dekker, Inc., New York, Basel, 2002). [CrossRef]

]. Indeed, thermal and mechanical conditions of an FBG directly affect its reflectivity spectrum. Analogously to conventional electrical probes, FBGs can serve as strain-gauge sensors to perform measurements with improved accuracy and sensitivity, with the advantage of immunity to electromagnetic interference and potential low cost. In addition, the ability to be made into a compact, lightweight, rugged device, small enough to be directly embedded into materials, make them very attractive to create smart monitoring systems capable of operating in different environments. Emerging applications include measurements of stress changes in buildings and airplane bodies, and depth measurements in streams, rivers, and reservoirs for flood control. A number of detection schemes have been developed, mostly using broadband radiation sources combined to different kinds of optical spectrum analyzers, with typical sensitivities in the 1–10 microstrain range [2

2. A. Kersey, M.A. Davis, H.J. Patrick, M. LeBlanc, K.P. Koo, C.G. Askins, M.A. Putnam, and E.J. Friebele, “Fiber Grating Sensors,” J. Lightwave Technol. 15, 1442–1463 (1997). [CrossRef]

]. Even better performances are sometimes required for monitoring special structures, such as high-quality telescopes, long interferometers, and precision tools. A further field of application, in which FBGs may find a significant and immediate impact, is given by geophysics. In particular, the possibility to perform both static and dynamic measurements with ultra-high sensitivity can be of great relevance for the study of the evolution of seismic and volcanic areas [3

3. B.A. Chouet, “Long-period volcano seismicity: its sources and use in eruption forecasting,” Nature 380, 309–316 (1996). [CrossRef]

5

5. P. Ferraro and G. de Natale, “On the possible use of optical fiber Bragg gratings as strain sensors for geodynamical monitoring,” Opt. Laser Eng. 37, 115–130 (2002). [CrossRef]

]. Broadband radiation sources, such as super-luminescent emitting diodes (SLED), fiber-amplified spontaneous emitters (ASE) or super-fluorescent fiber sources, prevent the use of active wavelength tuning for FBG interrogation, unless external modulator or spectral filters are adopted, which reduce the amount of optical power reflected by the grating sensor, thus limiting the achievable sensitivity and increasing the interrogation time. Lower detection limits for strain and temperature measurements were obtained by means of sophisticated techniques relying on scanning Michelson interferometers that provided accurate wavelength measurements of the FBG reflected light [6

6. Y.J. Rao, “In-fibre Bragg grating sensors,” Meas. Sci. Technol. 8, 355–375 (1997). [CrossRef]

]. However, use of phase-shift detection by interferometric techniques may be problematic due to possible drifts in the bias phase itself. Quite recently, alternative approaches with frequency-locked laser systems have been devised, demonstrating sensitivities in the order of few nanostrain [7

7. A. Arie, B. Lissak, and M. Tur, “Static Fiber-Bragg Grating Strain Sensing Using Frequency-Locked Lasers,” J. Lightwave Technol. 17, 1849–1855 (1999). [CrossRef]

9

9. B. Lissak, A. Arie, and M. Tur, “Highly sensitive dynamic strain measurements by locking lasers to fiber Bragg gratings,” Opt. Lett. 23, 1930–1932 (1998). [CrossRef]

]. Nonetheless, in these cases, the bandwidth of the locking feedback loop may severely limit the dynamic response of the system. Furthermore, complicated set-ups are necessary, which are not well-suited for field sensing applications, especially when multiple-point strain monitoring have to be carried out.

In this paper, we report on the implementation of a novel optical method for strain measurements with high sensitivity and reliability. The sensor apparatus includes a DFB diode-laser, emitting around 1560 nm, and a fiber Bragg-grating element. The interrogation technique is based on radio-frequency modulation of laser wavelength (LRFM) with phase-sensitive detection of the FBG reflected radiation. Interaction of carrier and sideband frequencies with the reflective grating provides a beat signal that tracks any wavelength shifts of the Bragg peak. This approach proved to be very suitable for dynamic strain monitoring. Results on sensitivity and reproducibility tests as well as calibration by different methods are described.

2. Experimental arrangement and measurement principle

The optical set-up is depicted in Fig. 1. A DFB diode laser, emitting around 1560.6 nm, with a linewidth of 5 MHz and a maximum power of 20 mW, is used for interrogation of 50-% and 80-% reflectivity FBG elements (AOS GmbH) centered at 1560.78 nm with 0.07-nm and 0.1-nm bandwidth, respectively. A Peltier-based PID controller and a very low noise power supply (rms ripple less than 1 µA in a 150-kHz bandwidth) provide the diode temperature stabilization and injection current. The laser, equipped with a single-mode fiber pig-tail, is directly connected to the FBG via a 3-dB coupler that also collects the radiation reflected by the grating. An internal Faraday isolator ensures optimal isolation of the laser from optical feedback of the FBGs, while FC/APC fiber connectors avoid etalon effects eventually caused by spurious reflections. For testing purposes, a 50-% FBG has been glued onto a piezoelectric actuator (PZT) with a wide electrical bandwidth (DC-40 kHz). Hence, static and dynamic strain can be applied to the fiber grating in a controlled manner, the PZT strain-to-voltage ratio being well known. The interrogation principle relies on heterodyne detection of the FBG reflected radiation when the source is tuned into resonance with it. The laser frequency is modulated at f=2.2 GHz, applying a synthesized radio-frequency (RF) signal (R&S mod. SML03) to the laser current. This generates a pair of sidebands at a distance f from the carrier frequency with a power ratio of 10 %. The choice of the modulation frequency was based on a compromise between a high sensitivity and a reasonable demand on detection electronics. The radiation coming from the grating is detected by an InGaAs PIN photodiode, with a bandwidth of 5 GHz (Thorlabs mod. SIR5-FC), and amplified by a 2400-MHz amplifier. Demodulation at frequency f is performed by a double-balanced mixer (DBM), while proper phase-sensitive detection is accomplished adjusting the local-oscillator (LO) phase. All signals can be continuously acquired through a DAQ interface by a LabView routine running on a laptop computer.

Fig. 1. Sketch of the experimental setup. F-P stands for Fabry-Pèrot, BS for beam-splitter, OI for optical isolator, BT for bias-tee, DBM for double-balanced mixer and PD for photodiode.

The DBM produces a DC output proportional to the superposition of the beat signals of the carrier with the two RF sidebands, whose initial phases differ by 180°. This superposition is extremely sensitive to phase and power differences between the sidebands [11

11. J.A. Silver, “Frequency-modulation spectroscopy for trace species detection: theory and comparison among experimental methods,” Appl. Opt. 31, 707–717 (1992) and references therein. [CrossRef] [PubMed]

]. When the laser wavelength is quasi-resonant with the FBG center and periodically scanned by a triangular wave, the IF port of the DBM produces a dispersive-like signal that vanishes symmetrically with respect to the Bragg center wavelength, as illustrated in Fig. 2. As a consequence, a strain would result in a detuning of the FBG curve center with respect to laser frequency and thus a shift of the zero-crossing point towards longer or shorter wavelengths, according to whether an elongation or a compression takes place. With this in mind, any mechanical or thermal stress acting on the sensing element affects the beat signals and causes a variation of the resulting DBM signal amplitude. The voltage change is detected in a small region around the FBG center, where the frequency dependence is linear (Fig. 2). Nearly-perfect zeroing of the latter occurs only when the FBG is unperturbed. At a first glance, one may note that, in this approach, the DBM signal itself already contains the information on the strain suffered by the FBG. In addition, an important advantage derives from RF modulation and detection, which strongly reduces the amplitude “flicker” noise mainly due to the laser diode. In fact, in the case of laser-based detection, the incident radiation power is relatively high and the detector noise contribution can be considered negligible. From that, a higher detectivity results. At the same time, a cancellation of possible power fluctuations occurs in the phase-sensitive processing, since any such noise would affect both sidebands. Actually, this is completely true only assuming that the residual amplitude modulation (RAM) of the laser can be neglected. Besides its sensitivity, this method also enables to measure strain variations in the whole acoustic range, as a sideband unbalance in the Bragg reflected beam can be detected by the electronics with a very fast response. In principle, two different approaches can be employed, for different strain ranges: a voltage measurement in a narrow window around the FBG center, or a relative frequency measurement by tracking the zero of the dispersive-like curve over a wider laser frequency scan. In the first case, particularly devoted to ultra small strain detection (below the µε level), the ultimate sensitivity and accuracy limits are mainly due to the laser frequency and amplitude stability. Accurate frequency calibrations of the FBG spectra, crucial for a quantitative strain estimate, can be readily carried out thanks to a 1-GHz-FSR Fabry-Pérot interferometer, using the RF sidebands as a precise frequency marker (absolute uncertainty of less than 1 Hz) around the linear slope.

Fig. 2. Example of the signal obtained by high-frequency mixer demodulation upon FBG reflection (maximum at 1560.778 nm). The lower trace corresponds to the transmission of a 1-GHz free-spectral-range Fabry-Pérot interferometer, where RF sidebands at 2.2 GHz are visible.

3. Results and discussion

The strain sensor was characterized by a number of test measurements aimed at verifying the performance in terms of sensitivity, reproducibility and frequency response. The intrinsic wavelength-vs-strain responsivity of the 50-% reflectivity grating was first estimated using an optical spectrum analyzer (OSA). For this purpose, a series of 10-µε steps were applied to the FBG via the PZT and the data measured by the OSA yielded a factor of 0.8±0.1 pm/µε. A similar measurement was also performed using our system. The strain for a given voltage was deduced from the PZT calibration curve, while the corresponding shift of the FBG peak was observed as a change in the DBM output, when the laser was near to the resonance. Spectral calibration of the signals was carried out by means of the Fabry-Pérot interferometer. In this test, the FBG was thermally insulated and the laboratory temperature actively controlled to keep the effect of possible thermal exchanges with the environment as low as possible, within typical measurement times. Figure 3 shows a fairly linear response as a function of strain, with a slope of 0.94±0.02 pm/µε (115±3 MHz/µε). The latter value is in good agreement with that obtained by the OSA and can be assumed as the sensor’s gauge factor. This factor is lower than the one expected from the FBG manufacturer’s specification, probably owing to a limited transfer efficiency of the force exerted by the piezo on the fiber.

Fig. 3. Response of the LRFM system to static deformations directly applied to the 50-% FBG by the PZT. The dashed line corresponds to a weighted linear fit. Vertical bars are standard errors of mean FBG shifts deriving from 10 measured values.

We demonstrated detection of very small static and dynamic strain. The frequency response of the sensor was observed from the noise spectral density of the DBM signal by a fast Fourier transform (FFT) analyzer. As an example, the FFT spectra corresponding to periodic deformations of 3 µε at 2 Hz and 1 kHz are shown in Fig. 4 and 5, respectively,. In the quasi-static case, the signal-to-noise ratio was estimated to be about 39 dB that corresponds to a minimum detectable strain of 150 nε/√Hz. Similarly, at the highest frequency, from a 48-dB signal we extrapolated a lower limit of 1.6 nε/√Hz. The mixer output spectrum was continuously measured in the whole acoustic range, obtaining a nearly-flat frequency response with a roll-off around 10 kHz due to the PZT low-pass filtering. Indeed, the signal-to-noise ratio at 20 kHz was measured to be about 30 dB. The noise baseline visible in both traces of Fig. 4 presently limits the sensitivity and it is likely due to ambient acoustic noise as well as to possible thermally-induced refractive-index changes in the FBG [10

10. S. Knudsen, A.B. Tveten, and A. Dandridge, “Measurements of Fundamental Thermal Induced Phase Fluctuations in the Fiber of a Sagnac Interferometer,” IEEE Photonics Technol. Lett. 7, 90–92 (1995). [CrossRef]

]. In addition, thermal noise may be directly caused by spontaneous heating of the piezo when driven by an ac voltage. On the other hand, the contribution coming from laser amplitude noise is expected to be drastically reduced in the high-frequency demodulation process.

Fig. 4. FFT amplitude spectrum of the sensor output measured by a spectrum analyzer (resolution bandwidth of 50 mHz). The high peak corresponds to excitation of the PZT by a sine wave at 2 Hz with a strain-equivalent amplitude of 3 µε. The red line represents the background when the PZT is off.
Fig. 5. Spectrum given by a 3-µε deformation applied to the FBG at higher acoustic frequencies (1 kHz). In this case, the resolution bandwidth is 50 Hz.

A more accurate calibration of our measurement method was carried out by comparison of the 80-% FBG to an electrical strain gauge, with 1-µε sensitivity. The sensors were fixed on an aluminum plate, very close to each other. In Fig. 6, the time response of the resistive probe and the laser sensor are both shown after excitation of a sound wave at 1.1 kHz in the plate, via a common loud speaker. It is worth noting the difference in their signal-to-noise ratios, corresponding to a sensitivity factor of about 200. Subsequently, variable loads were applied bending the plate with small calibrated weights, for values ranging from 1 µε to 50 µε. The data obtained using the FBG sensor are plotted as a function of the electrical gauge values, in Fig. 7. From a weighted linear fit, we retrieved a slope of 183±19 nε/mV. A valuable alternative to measure the calibration factor is in principle using directly the RF sidebands as a frequency marker, without need for any external reference method.

Fig. 6. LRFM-system time response to excitation of an acoustic wave at 1100 Hz with 20-µε peak-to-peak amplitude (80-% FBG), compared to the output of an electrical strain gauge with 1-µε sensitivity (upper trace).
Fig. 7. Static strain values measured by the laser sensor are plotted vs. the electrical probe values. A weighted linear fit is also represented. The error bars represent the maximum uncertainty due to the electrical gauge reading.

The reproducibility and the ultimate sensitivity limit of this method are both influenced by the laser inherent frequency stability. A source drift would indeed result in a false strain since the FBG-signal heterodyne detection acts as an efficient frequency-to-amplitude conversion. This aspect is particularly relevant for the static regime and thus was thoroughly investigated. For this purpose, a portion of the beam propagating into the fiber was collimated and directed to an external 1-m-long gas cell, containing 13C-enriched (99 %) acetylene at a pressure of 2 Torr. In the spectral range between 6410 and 6420 cm-1, 13C2H2 presents high-J ro-vibrational transitions belonging to the ν 1+ν 3+ν 4ν 4 combination band [12

12. G. di Lonardo, L. Fusina, E. Venuti, J.W. C. Johns, M.I. El Idrissi, J. Liévin, and M. Herman, “The Vibrational Energy Pattern in Acetylene. V. 13C2H2,” J. Chem Phys. 111, 1008–1016 (1999). [CrossRef]

]. The laser was tuned into resonance with a weak absorption linen and its relative wavelength position was continuously monitored using the Fabry-Pérot interferometer for spectral calibration. Over a time interval of about 4 hours, a line position distribution with a root-mean-square of 4 MHz and a maximum deviation of approximately 15 MHz was observed. Using the sensor calibration factor previously measured, the latter value corresponds to a maximum strain-equivalent fluctuation of 100 nε and gives a first indication of the long-term reproducibility level achievable by our system.

The dynamic range is another crucial feature of a strain sensor. In the case of the 80-% FBG, the maximum measurable strain was found to be 70 µε. The upper limit mainly derives from the sensor’s linearity interval, which can be further extended using a wider-bandwidth grating. As an alternative, measurements of larger strain values can be efficiently performed by a simple zero-tracking of the dispersive-like FBG signal recorded along with a laser frequency scan.

4. Conclusions

A novel interrogation technique for FBG sensors, based on a radio-frequency modulated 1560-nm diode laser, was developed. Phase-sensitive detection of the FBG-reflected radiation was carried out in order to generate a voltage signal suitable to transduction of the FBG center wavelength shifts caused by mechanical deformations. We demonstrated that this method can be of potentially high sensitivity in strain measurements. The set-up proved to be capable of static and dynamic strain detection, with a wide frequency response. In particular, the experimental results demonstrated an equivalent-noise sensitivity of 150 nε/√Hz, in the quasi-static domain, and of 1.6 nε/√Hz at higher acoustic frequencies. These limits can be mainly attributed to the ambient acoustic noise as well as to residual laser frequency fluctuations. The evaluation of the sensor’s frequency response was limited at about 20 kHz essentially due to the test device bandwidth. A comparison with conventional sensing devices provided a strain calibration of our system, pointing out a sensitivity gain of a factor 200. Furthermore, a series of test experiments to estimate the reproducibility performance of our method yielded a long-term stability well within 100 nε. The sensor’s minimum dynamic range is presently limited to approximately 60 dB, but can be readily increased by means of a different FBG. These results are, in several cases, competitive with those obtained by other FBG sensors, especially for dynamic strain measurements, usually carried out with sophisticated interferometric techniques [2

2. A. Kersey, M.A. Davis, H.J. Patrick, M. LeBlanc, K.P. Koo, C.G. Askins, M.A. Putnam, and E.J. Friebele, “Fiber Grating Sensors,” J. Lightwave Technol. 15, 1442–1463 (1997). [CrossRef]

]. The system makes use of standard optical telecommunication components and, once calibrated, its operation can be easily automated. Also, a voltage-controlled oscillator (VCO) could be adopted to replace the RF generator. As a further simplification, a two-tone frequency-modulation spectroscopic technique, based on two different sideband frequencies [13

13. G.R. Janik, C.B. Carlisle, and T.F. Gallagher, “Two-tone frequency-modulation spectroscopy,” J. Opt. Soc. Am. B 3, 1070–1074 (1998). [CrossRef]

], would make possible use of lower-frequency response electronics and detectors. In the future, we expect to obtain significant improvements by locking the laser source to molecular spectral lines for high-precision stabilization of long- and short-term frequency fluctuations.

Acknowledgments

The authors wish to thank Domenico Alfieri and Antonio Di Maio for technical support. Also, the authors acknowledge helpful discussions with Torsten Thiel (AOS GmbH, Dresden) and Livio Gianfrani. This work was funded by the Italian Ministry for Education, University and Research in the framework of “Progetto PON-Simona”.

References and links

1.

Y. J. Rao and S. Huang, “Applications of Fiber Optic Sensors,” in Fiber Optic Sensors, F.T.S. Yu and S. Yin eds. (Marcel Dekker, Inc., New York, Basel, 2002). [CrossRef]

2.

A. Kersey, M.A. Davis, H.J. Patrick, M. LeBlanc, K.P. Koo, C.G. Askins, M.A. Putnam, and E.J. Friebele, “Fiber Grating Sensors,” J. Lightwave Technol. 15, 1442–1463 (1997). [CrossRef]

3.

B.A. Chouet, “Long-period volcano seismicity: its sources and use in eruption forecasting,” Nature 380, 309–316 (1996). [CrossRef]

4.

M. Nakano, H. Kumagai, N. Kumazoua, K. Yamaoaka, and B.A. Chouet, “The excitation and characterization frequencies of the long period volcanic event: an approach based on an autoregressive model of a linear dynamic system,” J. Geophys. Res. 103, 10031–10046 (1998). [CrossRef]

5.

P. Ferraro and G. de Natale, “On the possible use of optical fiber Bragg gratings as strain sensors for geodynamical monitoring,” Opt. Laser Eng. 37, 115–130 (2002). [CrossRef]

6.

Y.J. Rao, “In-fibre Bragg grating sensors,” Meas. Sci. Technol. 8, 355–375 (1997). [CrossRef]

7.

A. Arie, B. Lissak, and M. Tur, “Static Fiber-Bragg Grating Strain Sensing Using Frequency-Locked Lasers,” J. Lightwave Technol. 17, 1849–1855 (1999). [CrossRef]

8.

L.A. Ferreira, E.V. Diatzikis, J.L. Santos, and F. Farahi, “Demodulation of fiber Bragg grating sensors based on dynamic tuning of a multimode laser diode,” Appl. Opt. 38, 4751–4759 (1999). [CrossRef]

9.

B. Lissak, A. Arie, and M. Tur, “Highly sensitive dynamic strain measurements by locking lasers to fiber Bragg gratings,” Opt. Lett. 23, 1930–1932 (1998). [CrossRef]

10.

S. Knudsen, A.B. Tveten, and A. Dandridge, “Measurements of Fundamental Thermal Induced Phase Fluctuations in the Fiber of a Sagnac Interferometer,” IEEE Photonics Technol. Lett. 7, 90–92 (1995). [CrossRef]

11.

J.A. Silver, “Frequency-modulation spectroscopy for trace species detection: theory and comparison among experimental methods,” Appl. Opt. 31, 707–717 (1992) and references therein. [CrossRef] [PubMed]

12.

G. di Lonardo, L. Fusina, E. Venuti, J.W. C. Johns, M.I. El Idrissi, J. Liévin, and M. Herman, “The Vibrational Energy Pattern in Acetylene. V. 13C2H2,” J. Chem Phys. 111, 1008–1016 (1999). [CrossRef]

13.

G.R. Janik, C.B. Carlisle, and T.F. Gallagher, “Two-tone frequency-modulation spectroscopy,” J. Opt. Soc. Am. B 3, 1070–1074 (1998). [CrossRef]

OCIS Codes
(060.2370) Fiber optics and optical communications : Fiber optics sensors
(060.2630) Fiber optics and optical communications : Frequency modulation
(230.1950) Optical devices : Diffraction gratings
(280.3420) Remote sensing and sensors : Laser sensors

ToC Category:
Research Papers

History
Original Manuscript: February 22, 2005
Revised Manuscript: March 16, 2005
Published: April 4, 2005

Citation
G. Gagliardi, M. Salza, P. Ferraro, and P. De Natale, "Fiber Bragg-grating strain sensor interrogation using laser radio-frequency modulation," Opt. Express 13, 2377-2384 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-7-2377


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References

  1. Y. J. Rao and S. Huang, �??Applications of Fiber Optic Sensors,�?? in Fiber Optic Sensors, F.T.S. Yu and S. Yin eds. (Marcel Dekker, Inc., New York, Basel, 2002). [CrossRef]
  2. A. Kersey, M.A. Davis, H.J. Patrick, M. LeBlanc, K.P. Koo, C.G. Askins, M.A. Putnam, and E.J. Friebele, �??Fiber Grating Sensors,�?? J. Lightwave Technol. 15, 1442-1463 (1997). [CrossRef]
  3. B.A. Chouet, �??Long-period volcano seismicity: its sources and use in eruption forecasting,�?? Nature 380, 309-316 (1996). [CrossRef]
  4. M. Nakano, H. Kumagai, N. Kumazoua, K. Yamaoaka, B.A. Chouet, �??The excitation and characterization frequencies of the long period volcanic event: an approach based on an autoregressive model of a linear dynamic system,�?? J. Geophys. Res. 103, 10031-10046 (1998). [CrossRef]
  5. P. Ferraro, G. de Natale, �??On the possible use of optical fiber Bragg gratings as strain sensors for geodynamical monitoring,�?? Opt. Laser Eng. 37, 115-130 (2002). [CrossRef]
  6. Y.J. Rao, �??In-fibre Bragg grating sensors,�?? Meas. Sci. Technol. 8, 355-375 (1997). [CrossRef]
  7. A. Arie, B. Lissak, and M. Tur, �??Static Fiber-Bragg Grating Strain Sensing Using Frequency-Locked Lasers,�?? J. Lightwave Technol. 17, 1849-1855 (1999). [CrossRef]
  8. L.A. Ferreira, E.V. Diatzikis, J.L. Santos, and F. Farahi, �??Demodulation of fiber Bragg grating sensors based on dynamic tuning of a multimode laser diode,�?? Appl. Opt. 38, 4751-4759 (1999). [CrossRef]
  9. B. Lissak, A. Arie, and M. Tur, �??Highly sensitive dynamic strain measurements by locking lasers to fiber Bragg gratings,�?? Opt. Lett. 23, 1930-1932 (1998). [CrossRef]
  10. S. Knudsen, A.B. Tveten, and A. Dandridge, �??Measurements of Fundamental Thermal Induced Phase Fluctuations in the Fiber of a Sagnac Interferometer,�?? IEEE Photonics Technol. Lett. 7, 90-92 (1995). [CrossRef]
  11. J.A. Silver, �??Frequency-modulation spectroscopy for trace species detection: theory and comparison among experimental methods,�?? Appl. Opt. 31, 707-717 (1992) and references therein. [CrossRef] [PubMed]
  12. G. di Lonardo, L. Fusina, E. Venuti, J.W. C. Johns, M.I. El Idrissi, J. Liévin, M. Herman, �??The Vibrational Energy Pattern in Acetylene. V. 13C2H2,�?? J. Chem Phys. 111, 1008-1016 (1999). [CrossRef]
  13. G.R. Janik, C.B. Carlisle, and T.F. Gallagher, �??Two-tone frequency-modulation spectroscopy,�?? J. Opt. Soc. Am. B 3, 1070-1074 (1998). [CrossRef]

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