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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 16 — Aug. 11, 2003
  • pp: 1918–1924
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Temperature insensitive measurements of static displacements using a fiber Bragg grating

Yinian Zhu, Ping Shum, Chao Lu, M. B. Lacquet, P. L. Swart, A. A. Chtcherbakov, and S. J. Spammer  »View Author Affiliations


Optics Express, Vol. 11, Issue 16, pp. 1918-1924 (2003)
http://dx.doi.org/10.1364/OE.11.001918


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Abstract

We report a chirped fiber Bragg grating transducer for the measurement of acceleration, in which a cantilever beam and fiber Bragg grating are used. The cantilever induces strain on the grating resulting in a Bragg grating wavelength modification that is subsequently detected. The output signal is insensitive to temperature variations and for a temperature change from -20 °C to 40 °C, the output signal fluctuated less than 5 % without any temperature compensation schemes. Because the accelerometer does not utilize the complex demodulation techniques it is potentially inexpensive. For the experimental system a linear output range of 8 g could be detected.

© 2003 Optical Society of America

1. Introduction

For more than two decades, fiber Bragg gratings (FBGs) have been a subject of research and development [1

1. K. O. Hill, F. Fujii, D. C. Johnson, and B. S. Kawasaki, “Photosensitivity on optical fiber waveguides: application to reflection filter fabrication,” Appl. Physics Lett. 32, 647–649 (1978). [CrossRef]

]. Because the optical line-width of the reflected Bragg wavelength can be in the order of 0.1 nm, the main area of interest has been in optical fiber communication systems such as amplifiers and dispersion compensation devices [2

2. V. Mizrahi, “Components and devices for optical communications based on UV-written-fiber phase gratings,” Optical Fiber Communications Conference, San Jose (1993).

,3

3. J. A. R. Williams, I. Bennion, K. Sugden, and N. J. Doran, “Fiber dispersion compensation using a chirped in-fiber Bragg grating,” Electron. Lett. 30, 985–987 (1994). [CrossRef]

]. However, many optical fiber sensor applications are also based on this technology. Typically these sensors are embedded or epoxied onto a transducer to provide measurements of a variable. Any change in the fiber grating’s properties, such as strain, pressure, temperature or vibration, will change the reflected Bragg wavelength [4

4. A. D. 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 Tech. 15, 1442–1462 (1997). [CrossRef]

8

8. R. T. Jones, T. A. Berkoff, D. G. Dellemore, D. A. Early, J. S. Sirks, M. A. Putnam, E. J. Friebele, and A. D. Kersey, “Cantilever plate deformation monitoring using wavelength division multiplexed fiber Bragg grating sensor,” SPIE , 2718, 258–268 (1996). [CrossRef]

]. With these sensors, the signal information is recovered and accomplished using optical instrumentation to detect strain induced wavelength changes of a FBG device.

Accelerometers are used throughout engineering to perform measurements such as vibration, angles of inclination, even recording, platform stabilization, vehicle suspension control, earthquake monitoring, and pacemaker control. While accelerometer designs may encompass many disciplines and schemes, the typical accelerometer uses capacitance to sense acceleration. Although FBG sensors seem to be very promising for many strain-mapping applications, it may also desirable, in some applications, to monitor other parameters such as acceleration, temperature, or acoustic signals, and FBG elements can be incorporated into various transducer arrangements to detect such measurands. An example of this device is the use of FBG sensors for pressure sensing [9

9. M. G. Xu, H. Geiger, and J. P. Dakin, “Fiber grating pressure sensor with enhanced sensitivity using a glass-bubble housing,” Electron. Lett. 32, 128–129 (1996). [CrossRef]

].

Because acceleration will result in a wavelength shift of the grating, previously reported sensors make use of wavelength demodulation schemes such as scanning Fabry-Perot filters, spectrum analyzers or interferometric techniques [10

10. T. A. Beroff and A. D. Kersey, “Experimental demonstration of a fiber Bragg grating accelerometer,” IEEE Photon. Tech. Lett. 8, 1677–1679 (1996). [CrossRef]

,11

11. M. D. Todd, G. A. Johnson, B. A. Althouse, and S. T. Vohra, “Flexural beam-based fiber Bragg grating accelerometers,” IEEE Photon. Tech. Lett. 10, 1605–1607 (1998). [CrossRef]

]. In additional to the acceleration-induced wavelength shift, the Bragg shift is also temperature dependent. Additional temperature compensation techniques are therefore necessary adding to system cost and complexity. This paper reports on the measurement of acceleration utilizing FBG that is not based on a wavelength detection scheme. This novel accelerometer configuration ensures automatic temperature insensitivity without any additional components and circuitry.

2. Sensing principle of chirped FBG

FBGs are formed by generating a permanent periodic modulation of the fiber’s core refractive index using an ultraviolet laser with a wavelength around 248 nm. This period index modulation with period Λ can be written into Ge-doped single-mode fiber using various methods such as interferometry [12

12. G. Meltz, W. W. Morey, and W. H. Glenn, “Formation of Bragg gratings in optical fiber by a transverse holographic method,” Opt. Lett. 14, 823–825 (1989). [CrossRef] [PubMed]

] or phase marks [13

13. K. O. Hill, B. Malo, F. Bilodeau, D. C. Johnson, and J. Albert, “Bragg gratings fabricated in mono-mode photosensitive optical fiber by UV exposure through a phase mask,” Appl. Phys. Lett. 62, 1035–1037 (1993). [CrossRef]

]. According to Bragg’s law, the grating will only reflect a specific wavelength, Bragg wavelength, which is given by [14

14. R. Kashyap, Fiber Bragg Grating (Academic Press, 1999), Chap. 4.

]

λB=2nΛ
(1)

where n is the effective core index of refraction. Any change in the refractive index or the index modulation pitch will result in a shift of the Bragg wavelength. Sensing occurs through the monitoring of the variation in the grating’s peak wavelength. The peak wavelength varies as the grating spacing changes. This change may be due to a change in the local ambient temperature or due to a strain field being applied to the fiber. The sensitivity of the grating’s tuning variability and potential uses of the devices have been studied and reported. For example, The FBGs were firstly used as an external tuning mirror for an argon ion laser by Hill et al. in 1978 [1

1. K. O. Hill, F. Fujii, D. C. Johnson, and B. S. Kawasaki, “Photosensitivity on optical fiber waveguides: application to reflection filter fabrication,” Appl. Physics Lett. 32, 647–649 (1978). [CrossRef]

]. Subsequently, the FBGs were used for external tuning of a semiconductor laser and then for strain and temperature sensing [15

15. J. Dunphy, G. Meltz, F. Lamm, and W. Morey, “Fiber-optic strain sensor multi-function, distributed optical fiber sensor for composite cure and response monitoring,” SPIE , 1370, 116–118 (1991).

].

The strain and temperature dependent variations on the Bragg wavelength as a result of the photo-elastic and thermo-optic effect, respectively, is given by [4

4. A. D. 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 Tech. 15, 1442–1462 (1997). [CrossRef]

]

ΔλB=2nΛ({1(n22)[P12ν(P11+P12)]}Δε+[a+(dndT)n]ΔT)
(2)

where P ij is the Pockel’s coefficients of the stress-opitc tensor, ν is Poisson’s ratio, a is the coefficient of thermal expansion of the material, ΔT is the temperature change, and Δε is the strain variation. The first term in Eq. (2) incorporating the refractive index, Pockel’s coefficients and Poisson’s ratio, is known as the photo-elastic coefficient. Assuming bulk values for the strain-optic coefficients and low levels of germanium doping in the core, the photo-elastic coefficient is approximately equal to 0.22 [3

3. J. A. R. Williams, I. Bennion, K. Sugden, and N. J. Doran, “Fiber dispersion compensation using a chirped in-fiber Bragg grating,” Electron. Lett. 30, 985–987 (1994). [CrossRef]

]. This results in a theoretical strain dependent wavelength shift of approximately 1.2 pm/µε at 1550 nm.

A FBG can be considered as a narrow band-stop optical filter. The characteristics of this filter can be greatly changed through the external perturbations interacting with the grating, which cause a change in the grating spacing and therefore the filter’s peak wavelength and bandwidth [16

16. A. Othonos and K. Kalli, Fiber Bragg Gratings: Fundamentals and Applications in Telecommunications and Sensing (Artech House, 1999), Chap. 3.

]. In this case as constructed, the Bragg grating has a uniform spacing. To reiterate, the spacing varies as an external perturbation interacts with the grating. Alternatively, it is possible to transform a traditional fixed-spacing Bragg grating into a grating with variable spacing by attaching it to a mechanical element that undergoes a nonlinear change of shape. For example, Hill et al. [17

17. K. O. Hill, F. Bilodeau, B. Molo, T. Kitagawa, S. Theriault, D. C. Johnson, and J. Albert, “Chirped in-fiber Bragg gratings for compensation of optical-fiber dispersion,” Opt. Lett. 19, 1314–1316 (1994). [CrossRef] [PubMed]

] demonstrated grating chirped by a simple strain of a fiber that was rigidly attached to a cantilever beam of a non-uniform cross section. For a configuration with the beam length of l, beam thickness d, and beam width b that will vary linearly with distance x from the beam’s waist, applied force F at the deflector point (x=1) creates a strain distribution in the beam. According to the pure bending theory, the axial component of strain is given by [18

18. K. H. Huebner, The finite Element Method for Engineer (John Wiley & Sons, 1975), Chap. 2.

]

ε(x)=12F(lx)ν(x)Emb(x)d3(x)
(3)

where E is Young’s modulus of the material, and ν(x) is the distance from the mid-plane.

The proposed accelerometer is based on a cantilever structure consisting of a mass of m at the end of a rectangular shaped beam. To measure acceleration the grating is attached to the surface of the beam. If the structure is subjected to an acceleration of a, the beam will bend resulting in a strain gradient along of the beam. The resulting strain at a given position x on the beam is given by (0<x<l) [19

19. P. L. Fuhr, S. J. Spammer, and Y. Zhu, “A novel signal demodulation technique for chirped Bragg grating strain sensors,” Smart Mater. Struct. 9, 85–94 (2000). [CrossRef]

]

ε(x)=6F(lx)mbd3Ea
(4)

From Eq. (3) it is clear that the strain will vary linearly along the length of the cantilever beam for an application. If this linear strain gradient is applied to a uniform period grating it will result in a linear chirped grating with an FWHM line-width proportional to the strain gradient. In the case of sensing using a chirped Bragg grating, the width of the packet of propagating light, which is the width of reflection for chirped grating, also varies as the chirp varies. In the case where the traditional grating is attached to a beam that is undergoing a linear bend, the grating’s chirp will vary in a linear manner, which in turn implies that the packet width varies linearly with linear grating chirp. Monitoring the width of the packet will enable the strain level to be determined. In addition, while the peak wavelength varies, the variation in the packet width is essentially insensitive to temperature change. Therefore, monitoring the ambient temperature variation is hidden within the peak wavelength variation whereas the strain level may be ascertained by monitoring the packet width.

The thermo-optic effect is the primary contributor to the temperature sensitivity of bare fiber Bragg gratings resulting in [4

4. A. D. 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 Tech. 15, 1442–1462 (1997). [CrossRef]

]

δλBλB=(1pe)ε0.78ε6.7×106°C
(5)

up to 85 °C. A typical value for the thermal response at 1550 nm is 0.01 nm/°C. At higher temperatures, the sensitivity increase and the response become slight non-linear. The line-width difference between a chirped grating and a unchirped grating shown in Fig. 1 [20

20. Y. Zhu, B. M. Lacquet, P. L. Swart, and S. J. Spammer, “Realization of chirped fiber Bragg gratings by using differently tapered transducers and loading procedures,” Meas. Sci. Tech. 12, 922–926 (2001).

].

Fig. 1. Reflection spectra of both the uniform and the chirped Bragg gratings.

Considering Fig. 1 it can be seen that the total amount or reflected power for the chirped grating is more compare to the uniform period grating. By simply detecting this amount of reflected power the acceleration could be measured.

3. Experimental measurements

The fiber Bragg grating used in our experimental setup is 50 mm long with a center wavelength of 1549.7 nm, and an FWHM line-width of 0.13 nm. It was epoxy onto the surface of a stainless steel beam with a length of 50 mm, width of 5 mm and thickness of 1 mm. A super luminescent (SLD) was used to illuminate the fiber via Port 1 of an optical circulator in which light goes out through Port 2 and reflective light is back to Port 2 while the output from Port 3 was measured with an optical spectrum analyzer (ANDO AQ-6316B) as well as a pin-photodiode (PD). The experiment arrangement is shown in Fig. 2. Firstly the grating spectra and output optical power was measured for different static displacements of the cantilever end. For every increase in displacement of the beam tip, the line-width of the grating spectrum widens as a result of an increasing strain gradient on the grating. The resulting detector output signal for displacements varying from +0.9 mm to -0.6 mm is depicted in Fig. 3. It can be seen that the output is linear except in the case of large displacements. This is attributed to the non-linearity of the stress-strain relationship of the beam material above approximately 1200 µε. The fact that the zero deflection position is necessary in order to ensure that directional information of the acceleration could be obtained.

Fig. 2. Experimental setup.
Fig. 3. Fiber Bragg grating output for beam tip displacement.

Because the acceleration output is not dependent on an absolute wavelength detection technique, it should be insensitive to temperature fluctuations. From Eq. (2) it is clearly shown that the wavelength shift is temperature and strain dependent. If the acceleration is subjected to temperature variations the Bragg wavelength will therefore shift as a result of the thermo-optic effect and the strain-induced thermal expansion of the beam. However, this wavelength shift will not affect the output signal because the photodiode detects the total reflected power. Measurements were taken for the zero beam deflection while the temperature was varied between -20 °C to 40 °C. The corresponding wavelength shift with temperature is shown in Fig. 4.

Fig. 4. Dependence of Bragg wavelength shift on temperature.

The acceleration output for a constant beam deflection as function of temperature is depicted in Fig. 5. The output variation over the whole temperature range was less than 5 mV, and this explains that the measurements of signals from the static displacement are temperature insensitive. Finally the acceleration output was compared to the output of a piezo-resistive acceleration (ICSensors 3021, sensitivity: 8.5 mV/g, maximum acceleration: 2 g). The amplified output signals from the piezo-resistive accelerations are -6.8 V. This corresponds to an acceleration of 1.48 g for a give an amplifier gain of 540.

Fig. 5. Dependence of sensor output signal on temperature.

4. Conclusion

The feasibility of a fiber Bragg grating accelerometer has been investigated by measuring the reflected optical power of a strain induced chirped fiber Bragg grating. The grating was attached on the surface of a cantilever beam that has a varying linear strain distribution along the beam length. Because the detection of the bending-induced strain on the grating is not measured with any wavelength monitoring techniques, the sensor is insensitive to wavelength shift dependent temperature variations. For a temperature range from -20 °C to 40 °C, the output signal variation did not exceed 5 mV.

To obtain a linear output signal for a positive and negative acceleration, it is necessary to use a chirped fiber Bragg grating or a pre-strained uniform fiber Bragg grating with a linear strain distribution. Although the linear elastic range of the beam material used is approximately 1200 µε, other material could be used resulting in a higher acceleration range of sensor. Considering Fig. 4, the linear output range is approximately 100 mV for this special experiment. This would imply a maximum linear acceleration range of approximately 8 g with a dynamic range 26 dB. This by no means the optimum system design and these figures could improve significantly if a more thorough accelerometer design is conducted.

This research work was supported by Telkom SA Ltd., ATC (Pty) Ltd., THRIP, NRF, and the Rand Afrikaans University of South Africa.

References and links

1.

K. O. Hill, F. Fujii, D. C. Johnson, and B. S. Kawasaki, “Photosensitivity on optical fiber waveguides: application to reflection filter fabrication,” Appl. Physics Lett. 32, 647–649 (1978). [CrossRef]

2.

V. Mizrahi, “Components and devices for optical communications based on UV-written-fiber phase gratings,” Optical Fiber Communications Conference, San Jose (1993).

3.

J. A. R. Williams, I. Bennion, K. Sugden, and N. J. Doran, “Fiber dispersion compensation using a chirped in-fiber Bragg grating,” Electron. Lett. 30, 985–987 (1994). [CrossRef]

4.

A. D. 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 Tech. 15, 1442–1462 (1997). [CrossRef]

5.

S. M. Melle, T. Alavic, S. Karr, T. coroy, K. Lui, and R. M. Measures, “A Bragg grating-tuned fiber laser strain sensor system,” IEEE Photon. Tech. Lett. 5, 516–518 (1992). [CrossRef]

6.

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

7.

W. W. Morey, G. Meltz, and W. H. Glenn, “Fiber optic Bragg grating sensors,” SPIE , 1169, 98–107 (1989).

8.

R. T. Jones, T. A. Berkoff, D. G. Dellemore, D. A. Early, J. S. Sirks, M. A. Putnam, E. J. Friebele, and A. D. Kersey, “Cantilever plate deformation monitoring using wavelength division multiplexed fiber Bragg grating sensor,” SPIE , 2718, 258–268 (1996). [CrossRef]

9.

M. G. Xu, H. Geiger, and J. P. Dakin, “Fiber grating pressure sensor with enhanced sensitivity using a glass-bubble housing,” Electron. Lett. 32, 128–129 (1996). [CrossRef]

10.

T. A. Beroff and A. D. Kersey, “Experimental demonstration of a fiber Bragg grating accelerometer,” IEEE Photon. Tech. Lett. 8, 1677–1679 (1996). [CrossRef]

11.

M. D. Todd, G. A. Johnson, B. A. Althouse, and S. T. Vohra, “Flexural beam-based fiber Bragg grating accelerometers,” IEEE Photon. Tech. Lett. 10, 1605–1607 (1998). [CrossRef]

12.

G. Meltz, W. W. Morey, and W. H. Glenn, “Formation of Bragg gratings in optical fiber by a transverse holographic method,” Opt. Lett. 14, 823–825 (1989). [CrossRef] [PubMed]

13.

K. O. Hill, B. Malo, F. Bilodeau, D. C. Johnson, and J. Albert, “Bragg gratings fabricated in mono-mode photosensitive optical fiber by UV exposure through a phase mask,” Appl. Phys. Lett. 62, 1035–1037 (1993). [CrossRef]

14.

R. Kashyap, Fiber Bragg Grating (Academic Press, 1999), Chap. 4.

15.

J. Dunphy, G. Meltz, F. Lamm, and W. Morey, “Fiber-optic strain sensor multi-function, distributed optical fiber sensor for composite cure and response monitoring,” SPIE , 1370, 116–118 (1991).

16.

A. Othonos and K. Kalli, Fiber Bragg Gratings: Fundamentals and Applications in Telecommunications and Sensing (Artech House, 1999), Chap. 3.

17.

K. O. Hill, F. Bilodeau, B. Molo, T. Kitagawa, S. Theriault, D. C. Johnson, and J. Albert, “Chirped in-fiber Bragg gratings for compensation of optical-fiber dispersion,” Opt. Lett. 19, 1314–1316 (1994). [CrossRef] [PubMed]

18.

K. H. Huebner, The finite Element Method for Engineer (John Wiley & Sons, 1975), Chap. 2.

19.

P. L. Fuhr, S. J. Spammer, and Y. Zhu, “A novel signal demodulation technique for chirped Bragg grating strain sensors,” Smart Mater. Struct. 9, 85–94 (2000). [CrossRef]

20.

Y. Zhu, B. M. Lacquet, P. L. Swart, and S. J. Spammer, “Realization of chirped fiber Bragg gratings by using differently tapered transducers and loading procedures,” Meas. Sci. Tech. 12, 922–926 (2001).

OCIS Codes
(060.2370) Fiber optics and optical communications : Fiber optics sensors
(230.1480) Optical devices : Bragg reflectors

ToC Category:
Research Papers

History
Original Manuscript: June 4, 2003
Revised Manuscript: July 28, 2003
Published: August 11, 2003

Citation
Yinian Zhu, Ping Shum, Chao Lu, M. Lacquet, P. Swart, A. Chtcherbakov, and S. Spammer, "Temperature insensitive measurements of static displacements using a fiber Bragg grating," Opt. Express 11, 1918-1924 (2003)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-16-1918


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References

  1. K. O. Hill, F. Fujii, D. C. Johnson, and B. S. Kawasaki, �??Photosensitivity on optical fiber waveguides: application to reflection filter fabrication,�?? Appl. Physics Lett. 32, 647-649 (1978). [CrossRef]
  2. V. Mizrahi, �??Components and devices for optical communications based on UV-written-fiber phase gratings,�?? Optical Fiber Communications Conference, San Jose (1993).
  3. J. A. R. Williams, I. Bennion, K. Sugden, and N. J. Doran, �??Fiber dispersion compensation using a chirped in-fiber Bragg grating,�?? Electron. Lett. 30, 985-987 (1994). [CrossRef]
  4. A. D. 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 Tech. 15, 1442-1462 (1997). [CrossRef]
  5. S. M. Melle, T. Alavic, S. Karr, T. coroy, K. Lui, and R. M. Measures, �??A Bragg grating-tuned fiber laser strain sensor system,�?? IEEE Photon. Tech. Lett. 5, 516-518 (1992). [CrossRef]
  6. Y. J. Rao, �??In-fiber Bragg grating sensors,�?? Meas. Sci. Tech. 8, 355-375 (1997). [CrossRef]
  7. W. W. Morey, G. Meltz, and W. H. Glenn, �??Fiber optic Bragg grating sensors,�?? SPIE, 1169, 98-107 (1989).
  8. R. T. Jones, T. A. Berkoff, D. G. Dellemore, D. A. Early, J. S. Sirks, M. A. Putnam, E. J. Friebele, and A. D. Kersey, �??Cantilever plate deformation monitoring using wavelength division multiplexed fiber Bragg grating sensor,�?? SPIE, 2718, 258-268 (1996). [CrossRef]
  9. M. G. Xu, H. Geiger, and J. P. Dakin, �??Fiber grating pressure sensor with enhanced sensitivity using a glass-bubble housing,�?? Electron. Lett. 32, 128-129 (1996). [CrossRef]
  10. T. A. Beroff and A. D. Kersey, �??Experimental demonstration of a fiber Bragg grating accelerometer,�?? IEEE Photon. Tech. Lett. 8, 1677-1679 (1996). [CrossRef]
  11. M. D. Todd, G. A. Johnson, B. A. Althouse, and S. T. Vohra, �??Flexural beam-based fiber Bragg grating accelerometers,�?? IEEE Photon. Tech. Lett. 10, 1605-1607 (1998). [CrossRef]
  12. G. Meltz, W. W. Morey, and W. H. Glenn, �??Formation of Bragg gratings in optical fiber by a transverse holographic method,�?? Opt. Lett. 14, 823-825 (1989). [CrossRef] [PubMed]
  13. K. O. Hill, B. Malo, F. Bilodeau, D. C. Johnson, and J. Albert, �??Bragg gratings fabricated in mono-mode photosensitive optical fiber by UV exposure through a phase mask,�?? Appl. Phys. Lett. 62, 1035-1037 (1993). [CrossRef]
  14. R. Kashyap, Fiber Bragg Grating (Academic Press, 1999), Chap. 4.
  15. J. Dunphy, G. Meltz, F. Lamm, and W. Morey, �??Fiber-optic strain sensor multi-function, distributed optical fiber sensor for composite cure and response monitoring,�?? SPIE, 1370, 116-118 (1991).
  16. A. Othonos and K. Kalli, Fiber Bragg Gratings: Fundamentals and Applications in Telecommunications and Sensing (Artech House, 1999), Chap. 3
  17. K. O. Hill, F. Bilodeau, B. Molo, T. Kitagawa, S. Theriault, D. C. Johnson, and J. Albert, �??Chirped in-fiber Bragg gratings for compensation of optical-fiber dispersion,�?? Opt. Lett. 19, 1314-1316 (1994). [CrossRef] [PubMed]
  18. K. H. Huebner, The finite Element Method for Engineer (John Wiley & Sons, 1975), Chap. 2.
  19. P. L. Fuhr, S. J. Spammer, and Y. Zhu, �??A novel signal demodulation technique for chirped Bragg grating strain sensors,�?? Smart Mater. Struct. 9, 85-94, (2000). [CrossRef]
  20. Y. Zhu, B. M. Lacquet, P. L. Swart, and S. J. Spammer, �??Realization of chirped fiber Bragg gratings by using differently tapered transducers and loading procedures,�?? Meas. Sci. Tech. 12, 922-926 (2001).

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