Abstract

The modeling, design, simulation, fabrication, calibration, and testing of a three-element, 15.3  cm fiber Bragg grating strain sensor array with the coherent optical frequency domain reflectometry (C-OFDR) interrogation technique are demonstrated. The fiber Bragg grating array (FBGA) is initially simulated using in-house software that incorporates transfer matrices. Compared to the previous techniques used, the transfer matrix method allows a systemwide approach to modeling the FBGA–C-OFDR system. Once designed and simulated, the FBGA system design is then imprinted into the core of a boron–germanium codoped photosensitive fiber using the phase mask technique. A fiber optic Fabry–Perot interferometric (FPI) strain gauge calibrator is then used to determine the strain gauge factor of a single fiber Bragg grating (FBG), and the results are used on the FBGA. The FPI strain gauge calibrator offers nondestructive testing of the FBG. To test the system, the FBGA is then attached to a 75   cm cantilever beam and interrogated using an incremental tunable laser. Electric strain gauges (ESGs) are then used to independently verify the strain measurements with the FBGA at various displacements of the cantilever beam. The results show that the peak strain error is 18% with respect to ESG results. In addition, good agreement is shown between the simulation and the experimental results.

© 2007 Optical Society of America

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References

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  1. M. Froggatt, "Distributed measurement of the complex modulation of a photo-induced Bragg grating in an optical fiber," Appl. Opt. 35, 5162-5164 (1996).
    [CrossRef] [PubMed]
  2. B. A. Childers, T. L. Brown, J. P. Moore, and K. H. Wood, "Recent developments in the application of optical frequency domain reflectometry to distributed Bragg grating sensing," Proc. SPIE 4578, 19-31 (2002).
    [CrossRef]
  3. R. G. Duncan, B. A. Childers, D. K. Gifford, D. E. Petti, A. W. Hickson, and T. L. Brown, "Distributed sensing technique for test article damage detection and monitoring," Proc. SPIE 5050, 367-375 (2002).
    [CrossRef]
  4. A. M. Abdi and A. R. Kost, "Theoretical suppression of cavity interferences in a fiber Bragg grating array interrogated with coherent optical frequency domain reflectometry," Smart Mater. Struct. 15, 1296-1304 (2006).
    [CrossRef]
  5. M. Froggatt and J. Moore, "Distributed measurement of static strain in an optical fiber with multiple Bragg gratings at nominally equal wavelengths," Appl. Opt. 37, 1741-1746 (1998).
    [CrossRef]
  6. A. C. Brooks, M. E. Froggatt, S. G. Allison, T. C. Moore, Sr., D. A. Hare, C. F. Batten, and D. C. Jegley, "Use of 3000 Bragg grating strain sensors distributed on four eight-meter optical fibers during static load tests of a composite structure," Proc. SPIE 4332, 133-142 (2001).
    [CrossRef]
  7. R. Kashyap, Fiber Bragg Gratings (Academic, 1996).
  8. R. M. Measures, Structural Monitoring with Fiber Optic Technology (Academic, 2001).
  9. A. T. Alavie, R Maaskant, R. Stubbe, A. Othonos, M. Ohn, B. Sahlgren, and R. M. Measures, "Characterization of fiber Bragg grating sensors and their relation to manufacturing technique," Proc. SPIE 2444, 528-535 (1995).
    [CrossRef]
  10. A. M. Abdi and A. Kost, "Infrastructure optics," Proc. SPIE 5384, 218-228 (2004).
    [CrossRef]
  11. E. Udd, Fiber Optic Smart Structures (Wiley, 1995).
  12. A. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge U. Press, 1998).
  13. M. del Vries, M. Nasta, V. Bhatia, T. Tran, J. Greene, and R. O. Claus, "Performance of embedded fatigue loaded reinforced concrete specimen," Smart Mater. Struct. 4, A107-A113 (1995).
    [CrossRef]
  14. R. Passy, N. Gision, J. P. von der Weid, and H. Gilgen, "Experimental and theoretical investigations of coherent OFDR with semiconductor laser sources," J. Lightwave Technol. 12, 1622-1630 (1994).
    [CrossRef]
  15. G. A. Sidney, R. L. Fox, M. E. Froggatt, and B. A. Childers, "Novel piezoelectric actuators for tuning an optical fiber Bragg grating," Opt. Eng. 41, 2448-2455 (2002).
    [CrossRef]

2006 (1)

A. M. Abdi and A. R. Kost, "Theoretical suppression of cavity interferences in a fiber Bragg grating array interrogated with coherent optical frequency domain reflectometry," Smart Mater. Struct. 15, 1296-1304 (2006).
[CrossRef]

2004 (1)

A. M. Abdi and A. Kost, "Infrastructure optics," Proc. SPIE 5384, 218-228 (2004).
[CrossRef]

2002 (3)

G. A. Sidney, R. L. Fox, M. E. Froggatt, and B. A. Childers, "Novel piezoelectric actuators for tuning an optical fiber Bragg grating," Opt. Eng. 41, 2448-2455 (2002).
[CrossRef]

B. A. Childers, T. L. Brown, J. P. Moore, and K. H. Wood, "Recent developments in the application of optical frequency domain reflectometry to distributed Bragg grating sensing," Proc. SPIE 4578, 19-31 (2002).
[CrossRef]

R. G. Duncan, B. A. Childers, D. K. Gifford, D. E. Petti, A. W. Hickson, and T. L. Brown, "Distributed sensing technique for test article damage detection and monitoring," Proc. SPIE 5050, 367-375 (2002).
[CrossRef]

2001 (1)

A. C. Brooks, M. E. Froggatt, S. G. Allison, T. C. Moore, Sr., D. A. Hare, C. F. Batten, and D. C. Jegley, "Use of 3000 Bragg grating strain sensors distributed on four eight-meter optical fibers during static load tests of a composite structure," Proc. SPIE 4332, 133-142 (2001).
[CrossRef]

1998 (1)

1996 (1)

1995 (2)

M. del Vries, M. Nasta, V. Bhatia, T. Tran, J. Greene, and R. O. Claus, "Performance of embedded fatigue loaded reinforced concrete specimen," Smart Mater. Struct. 4, A107-A113 (1995).
[CrossRef]

A. T. Alavie, R Maaskant, R. Stubbe, A. Othonos, M. Ohn, B. Sahlgren, and R. M. Measures, "Characterization of fiber Bragg grating sensors and their relation to manufacturing technique," Proc. SPIE 2444, 528-535 (1995).
[CrossRef]

1994 (1)

R. Passy, N. Gision, J. P. von der Weid, and H. Gilgen, "Experimental and theoretical investigations of coherent OFDR with semiconductor laser sources," J. Lightwave Technol. 12, 1622-1630 (1994).
[CrossRef]

Abdi, A. M.

A. M. Abdi and A. R. Kost, "Theoretical suppression of cavity interferences in a fiber Bragg grating array interrogated with coherent optical frequency domain reflectometry," Smart Mater. Struct. 15, 1296-1304 (2006).
[CrossRef]

A. M. Abdi and A. Kost, "Infrastructure optics," Proc. SPIE 5384, 218-228 (2004).
[CrossRef]

Alavie, A. T.

A. T. Alavie, R Maaskant, R. Stubbe, A. Othonos, M. Ohn, B. Sahlgren, and R. M. Measures, "Characterization of fiber Bragg grating sensors and their relation to manufacturing technique," Proc. SPIE 2444, 528-535 (1995).
[CrossRef]

Allison, S. G.

A. C. Brooks, M. E. Froggatt, S. G. Allison, T. C. Moore, Sr., D. A. Hare, C. F. Batten, and D. C. Jegley, "Use of 3000 Bragg grating strain sensors distributed on four eight-meter optical fibers during static load tests of a composite structure," Proc. SPIE 4332, 133-142 (2001).
[CrossRef]

Batten, C. F.

A. C. Brooks, M. E. Froggatt, S. G. Allison, T. C. Moore, Sr., D. A. Hare, C. F. Batten, and D. C. Jegley, "Use of 3000 Bragg grating strain sensors distributed on four eight-meter optical fibers during static load tests of a composite structure," Proc. SPIE 4332, 133-142 (2001).
[CrossRef]

Bhatia, V.

M. del Vries, M. Nasta, V. Bhatia, T. Tran, J. Greene, and R. O. Claus, "Performance of embedded fatigue loaded reinforced concrete specimen," Smart Mater. Struct. 4, A107-A113 (1995).
[CrossRef]

Brooks, A. C.

A. C. Brooks, M. E. Froggatt, S. G. Allison, T. C. Moore, Sr., D. A. Hare, C. F. Batten, and D. C. Jegley, "Use of 3000 Bragg grating strain sensors distributed on four eight-meter optical fibers during static load tests of a composite structure," Proc. SPIE 4332, 133-142 (2001).
[CrossRef]

Brown, T. L.

R. G. Duncan, B. A. Childers, D. K. Gifford, D. E. Petti, A. W. Hickson, and T. L. Brown, "Distributed sensing technique for test article damage detection and monitoring," Proc. SPIE 5050, 367-375 (2002).
[CrossRef]

B. A. Childers, T. L. Brown, J. P. Moore, and K. H. Wood, "Recent developments in the application of optical frequency domain reflectometry to distributed Bragg grating sensing," Proc. SPIE 4578, 19-31 (2002).
[CrossRef]

Childers, B. A.

B. A. Childers, T. L. Brown, J. P. Moore, and K. H. Wood, "Recent developments in the application of optical frequency domain reflectometry to distributed Bragg grating sensing," Proc. SPIE 4578, 19-31 (2002).
[CrossRef]

G. A. Sidney, R. L. Fox, M. E. Froggatt, and B. A. Childers, "Novel piezoelectric actuators for tuning an optical fiber Bragg grating," Opt. Eng. 41, 2448-2455 (2002).
[CrossRef]

R. G. Duncan, B. A. Childers, D. K. Gifford, D. E. Petti, A. W. Hickson, and T. L. Brown, "Distributed sensing technique for test article damage detection and monitoring," Proc. SPIE 5050, 367-375 (2002).
[CrossRef]

Claus, R. O.

M. del Vries, M. Nasta, V. Bhatia, T. Tran, J. Greene, and R. O. Claus, "Performance of embedded fatigue loaded reinforced concrete specimen," Smart Mater. Struct. 4, A107-A113 (1995).
[CrossRef]

del Vries, M.

M. del Vries, M. Nasta, V. Bhatia, T. Tran, J. Greene, and R. O. Claus, "Performance of embedded fatigue loaded reinforced concrete specimen," Smart Mater. Struct. 4, A107-A113 (1995).
[CrossRef]

Duncan, R. G.

R. G. Duncan, B. A. Childers, D. K. Gifford, D. E. Petti, A. W. Hickson, and T. L. Brown, "Distributed sensing technique for test article damage detection and monitoring," Proc. SPIE 5050, 367-375 (2002).
[CrossRef]

Fox, R. L.

G. A. Sidney, R. L. Fox, M. E. Froggatt, and B. A. Childers, "Novel piezoelectric actuators for tuning an optical fiber Bragg grating," Opt. Eng. 41, 2448-2455 (2002).
[CrossRef]

Froggatt, M.

Froggatt, M. E.

G. A. Sidney, R. L. Fox, M. E. Froggatt, and B. A. Childers, "Novel piezoelectric actuators for tuning an optical fiber Bragg grating," Opt. Eng. 41, 2448-2455 (2002).
[CrossRef]

A. C. Brooks, M. E. Froggatt, S. G. Allison, T. C. Moore, Sr., D. A. Hare, C. F. Batten, and D. C. Jegley, "Use of 3000 Bragg grating strain sensors distributed on four eight-meter optical fibers during static load tests of a composite structure," Proc. SPIE 4332, 133-142 (2001).
[CrossRef]

Ghatak, A.

A. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge U. Press, 1998).

Gifford, D. K.

R. G. Duncan, B. A. Childers, D. K. Gifford, D. E. Petti, A. W. Hickson, and T. L. Brown, "Distributed sensing technique for test article damage detection and monitoring," Proc. SPIE 5050, 367-375 (2002).
[CrossRef]

Gilgen, H.

R. Passy, N. Gision, J. P. von der Weid, and H. Gilgen, "Experimental and theoretical investigations of coherent OFDR with semiconductor laser sources," J. Lightwave Technol. 12, 1622-1630 (1994).
[CrossRef]

Gision, N.

R. Passy, N. Gision, J. P. von der Weid, and H. Gilgen, "Experimental and theoretical investigations of coherent OFDR with semiconductor laser sources," J. Lightwave Technol. 12, 1622-1630 (1994).
[CrossRef]

Greene, J.

M. del Vries, M. Nasta, V. Bhatia, T. Tran, J. Greene, and R. O. Claus, "Performance of embedded fatigue loaded reinforced concrete specimen," Smart Mater. Struct. 4, A107-A113 (1995).
[CrossRef]

Hare, D. A.

A. C. Brooks, M. E. Froggatt, S. G. Allison, T. C. Moore, Sr., D. A. Hare, C. F. Batten, and D. C. Jegley, "Use of 3000 Bragg grating strain sensors distributed on four eight-meter optical fibers during static load tests of a composite structure," Proc. SPIE 4332, 133-142 (2001).
[CrossRef]

Hickson, A. W.

R. G. Duncan, B. A. Childers, D. K. Gifford, D. E. Petti, A. W. Hickson, and T. L. Brown, "Distributed sensing technique for test article damage detection and monitoring," Proc. SPIE 5050, 367-375 (2002).
[CrossRef]

Jegley, D. C.

A. C. Brooks, M. E. Froggatt, S. G. Allison, T. C. Moore, Sr., D. A. Hare, C. F. Batten, and D. C. Jegley, "Use of 3000 Bragg grating strain sensors distributed on four eight-meter optical fibers during static load tests of a composite structure," Proc. SPIE 4332, 133-142 (2001).
[CrossRef]

Kashyap, R.

R. Kashyap, Fiber Bragg Gratings (Academic, 1996).

Kost, A.

A. M. Abdi and A. Kost, "Infrastructure optics," Proc. SPIE 5384, 218-228 (2004).
[CrossRef]

Kost, A. R.

A. M. Abdi and A. R. Kost, "Theoretical suppression of cavity interferences in a fiber Bragg grating array interrogated with coherent optical frequency domain reflectometry," Smart Mater. Struct. 15, 1296-1304 (2006).
[CrossRef]

Maaskant, R

A. T. Alavie, R Maaskant, R. Stubbe, A. Othonos, M. Ohn, B. Sahlgren, and R. M. Measures, "Characterization of fiber Bragg grating sensors and their relation to manufacturing technique," Proc. SPIE 2444, 528-535 (1995).
[CrossRef]

Measures, R. M.

A. T. Alavie, R Maaskant, R. Stubbe, A. Othonos, M. Ohn, B. Sahlgren, and R. M. Measures, "Characterization of fiber Bragg grating sensors and their relation to manufacturing technique," Proc. SPIE 2444, 528-535 (1995).
[CrossRef]

R. M. Measures, Structural Monitoring with Fiber Optic Technology (Academic, 2001).

Moore, J.

Moore, J. P.

B. A. Childers, T. L. Brown, J. P. Moore, and K. H. Wood, "Recent developments in the application of optical frequency domain reflectometry to distributed Bragg grating sensing," Proc. SPIE 4578, 19-31 (2002).
[CrossRef]

Moore, T. C.

A. C. Brooks, M. E. Froggatt, S. G. Allison, T. C. Moore, Sr., D. A. Hare, C. F. Batten, and D. C. Jegley, "Use of 3000 Bragg grating strain sensors distributed on four eight-meter optical fibers during static load tests of a composite structure," Proc. SPIE 4332, 133-142 (2001).
[CrossRef]

Nasta, M.

M. del Vries, M. Nasta, V. Bhatia, T. Tran, J. Greene, and R. O. Claus, "Performance of embedded fatigue loaded reinforced concrete specimen," Smart Mater. Struct. 4, A107-A113 (1995).
[CrossRef]

Ohn, M.

A. T. Alavie, R Maaskant, R. Stubbe, A. Othonos, M. Ohn, B. Sahlgren, and R. M. Measures, "Characterization of fiber Bragg grating sensors and their relation to manufacturing technique," Proc. SPIE 2444, 528-535 (1995).
[CrossRef]

Othonos, A.

A. T. Alavie, R Maaskant, R. Stubbe, A. Othonos, M. Ohn, B. Sahlgren, and R. M. Measures, "Characterization of fiber Bragg grating sensors and their relation to manufacturing technique," Proc. SPIE 2444, 528-535 (1995).
[CrossRef]

Passy, R.

R. Passy, N. Gision, J. P. von der Weid, and H. Gilgen, "Experimental and theoretical investigations of coherent OFDR with semiconductor laser sources," J. Lightwave Technol. 12, 1622-1630 (1994).
[CrossRef]

Petti, D. E.

R. G. Duncan, B. A. Childers, D. K. Gifford, D. E. Petti, A. W. Hickson, and T. L. Brown, "Distributed sensing technique for test article damage detection and monitoring," Proc. SPIE 5050, 367-375 (2002).
[CrossRef]

Sahlgren, B.

A. T. Alavie, R Maaskant, R. Stubbe, A. Othonos, M. Ohn, B. Sahlgren, and R. M. Measures, "Characterization of fiber Bragg grating sensors and their relation to manufacturing technique," Proc. SPIE 2444, 528-535 (1995).
[CrossRef]

Sidney, G. A.

G. A. Sidney, R. L. Fox, M. E. Froggatt, and B. A. Childers, "Novel piezoelectric actuators for tuning an optical fiber Bragg grating," Opt. Eng. 41, 2448-2455 (2002).
[CrossRef]

Stubbe, R.

A. T. Alavie, R Maaskant, R. Stubbe, A. Othonos, M. Ohn, B. Sahlgren, and R. M. Measures, "Characterization of fiber Bragg grating sensors and their relation to manufacturing technique," Proc. SPIE 2444, 528-535 (1995).
[CrossRef]

Thyagarajan, K.

A. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge U. Press, 1998).

Tran, T.

M. del Vries, M. Nasta, V. Bhatia, T. Tran, J. Greene, and R. O. Claus, "Performance of embedded fatigue loaded reinforced concrete specimen," Smart Mater. Struct. 4, A107-A113 (1995).
[CrossRef]

Udd, E.

E. Udd, Fiber Optic Smart Structures (Wiley, 1995).

von der Weid, J. P.

R. Passy, N. Gision, J. P. von der Weid, and H. Gilgen, "Experimental and theoretical investigations of coherent OFDR with semiconductor laser sources," J. Lightwave Technol. 12, 1622-1630 (1994).
[CrossRef]

Wood, K. H.

B. A. Childers, T. L. Brown, J. P. Moore, and K. H. Wood, "Recent developments in the application of optical frequency domain reflectometry to distributed Bragg grating sensing," Proc. SPIE 4578, 19-31 (2002).
[CrossRef]

Appl. Opt. (2)

J. Lightwave Technol. (1)

R. Passy, N. Gision, J. P. von der Weid, and H. Gilgen, "Experimental and theoretical investigations of coherent OFDR with semiconductor laser sources," J. Lightwave Technol. 12, 1622-1630 (1994).
[CrossRef]

Opt. Eng. (1)

G. A. Sidney, R. L. Fox, M. E. Froggatt, and B. A. Childers, "Novel piezoelectric actuators for tuning an optical fiber Bragg grating," Opt. Eng. 41, 2448-2455 (2002).
[CrossRef]

Proc. SPIE (5)

B. A. Childers, T. L. Brown, J. P. Moore, and K. H. Wood, "Recent developments in the application of optical frequency domain reflectometry to distributed Bragg grating sensing," Proc. SPIE 4578, 19-31 (2002).
[CrossRef]

R. G. Duncan, B. A. Childers, D. K. Gifford, D. E. Petti, A. W. Hickson, and T. L. Brown, "Distributed sensing technique for test article damage detection and monitoring," Proc. SPIE 5050, 367-375 (2002).
[CrossRef]

A. T. Alavie, R Maaskant, R. Stubbe, A. Othonos, M. Ohn, B. Sahlgren, and R. M. Measures, "Characterization of fiber Bragg grating sensors and their relation to manufacturing technique," Proc. SPIE 2444, 528-535 (1995).
[CrossRef]

A. M. Abdi and A. Kost, "Infrastructure optics," Proc. SPIE 5384, 218-228 (2004).
[CrossRef]

A. C. Brooks, M. E. Froggatt, S. G. Allison, T. C. Moore, Sr., D. A. Hare, C. F. Batten, and D. C. Jegley, "Use of 3000 Bragg grating strain sensors distributed on four eight-meter optical fibers during static load tests of a composite structure," Proc. SPIE 4332, 133-142 (2001).
[CrossRef]

Smart Mater. Struct. (2)

M. del Vries, M. Nasta, V. Bhatia, T. Tran, J. Greene, and R. O. Claus, "Performance of embedded fatigue loaded reinforced concrete specimen," Smart Mater. Struct. 4, A107-A113 (1995).
[CrossRef]

A. M. Abdi and A. R. Kost, "Theoretical suppression of cavity interferences in a fiber Bragg grating array interrogated with coherent optical frequency domain reflectometry," Smart Mater. Struct. 15, 1296-1304 (2006).
[CrossRef]

Other (4)

E. Udd, Fiber Optic Smart Structures (Wiley, 1995).

A. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge U. Press, 1998).

R. Kashyap, Fiber Bragg Gratings (Academic, 1996).

R. M. Measures, Structural Monitoring with Fiber Optic Technology (Academic, 2001).

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Figures (13)

Fig. 1
Fig. 1

Model of a reflective-type FBG illuminated by a broadband source or scanned by a tunable laser; most of the incident beam is transmitted, but a narrow band with a center wavelength (Bragg wavelength) at λ B is reflected in the direction of the source. The center wavelength is directly related to the period of the FBG and the effective refractive index.

Fig. 2
Fig. 2

Block diagram of a FBGA with C-OFDR interrogation. The tunable laser wavelength scans an optical fiber line composed of an N-element FBGA and a reference reflector, resulting in a modulated interferometric signal. The modulated interferometric signal is detected by a photo-optic detector, and by using a Fourier transform, a bandpass filter, and an inverse Fourier transform, the positional and wavelength information of a desired FBG are determined.

Fig. 3
Fig. 3

Model of an N-element FBGA with a backend reference reflector.

Fig. 4
Fig. 4

Simulation software used to model the FBGA system. (a) The FBGA system is designed on the layout GUI, and (b) the analysis is done on the processor GUI.

Fig. 5
Fig. 5

Design of the three-element, 15.3   cm FBGA, and the resulting interferometric signal in (a) the Fourier domain and (b) the wavelength domain. The operating bandwidth is 1 .6   nm , and the tunable laser wavelength increment is 1 pm.

Fig. 6
Fig. 6

Fiber-optic FPI displacement and strain sensor. R 1 is the reflectance from the front end of an optical fiber, and R 2 is the reflectance from a position that is adjustable. The optical field reflects from both end faces and interferes. One end of a test sensor is attached to surface R 2 , while the other end is fixed. The displacement of surface R 2 strains the test sensor.

Fig. 7
Fig. 7

Physical layout of the FPI strain gauge calibrator. The system is composed of two sensor holders (SH#1 and SH#2) positioned on a groove. SH#2 is kept fixed by friction using an adjustable side screw (tensioner). The test sensor is kept in tension and fixed by SH#1 and SH#2. The FPI displacement sensor is composed of a cleaved front facet of a SMF28 optical fiber and the machine-finished surface of SH#1. The piezoceramic housing (piezo housing) fixes the outer edges of the piezoceramic actuator.

Fig. 8
Fig. 8

Hardware setup used to measure the strain gauge factor of a FBG. The setup includes couplers (C), a high-voltage source (HV), an erbium-doped fiber amplifier (EDFA) broadband source, a fiber-optic photodetector (PD), a computer-based oscilloscope (OSC), and an optical spectrum analyzer (OSA).

Fig. 9
Fig. 9

FBGA and ESG attachment to a cantilever beam host. The tip of the cantilever beam is varied, and the corresponding strain information from the FBGs and ESGs are compared.

Fig. 10
Fig. 10

Fractional change of the center wavelength of the FBG as a function of induced strain using (a) the FPI strain gauge calibrator and (b) the cantilever beam method. Note that the fits are forced to cross the origin.

Fig. 11
Fig. 11

FBGA reflectance spectrum in the (a) wavelength and (b) Fourier domains. Note the decrease in the magnitude of the interference peaks as a function of position. (c) FBGA demodulated spectrum. Note the decrease in the peak magnitude and the distortion of the FBG spectra, particularly for FBG3.

Fig. 12
Fig. 12

(a) ESG and FBGA results as functions of cantilever beam tip displacement and axial position. (b) Percent strain error of each FBG in the array with respect to the ESG results as functions of tip displacement.

Fig. 13
Fig. 13

Simulated FBGA reflectance in (a) the Fourier domain with a 2.5   pm uniform noise added to the wavelength increment. Note the decrease in the magnitude of the interference peaks as a function of position. (b) FBGA demodulated spectrum. Note the decrease in the peak magnitude and the distortion of the FBG spectra, particularly for FBG3.

Equations (27)

Equations on this page are rendered with MathJax. Learn more.

E i ( z ) z = j κ E r ( z ) e j 2 δ z ,
E r ( z ) z = j κ E i ( z ) e j 2 δ z ,
δ = 1 2 ( 2 β K ) ,
= 2 π n ( 1 λ 1 λ B ) ,
λ B = 2 n Λ .
[ E i ( 0 ) E r ( 0 ) ] = [ T ] [ E i ( l ) 0 ] ,
T = [ cosh ( α l ) j δ sinh ( α l ) α j κ sinh ( α l ) α j κ sinh ( α l ) α cosh ( α l ) j δ sinh ( α l ) α ] ,
R ( λ ) = | E r ( 0 ) E i ( 0 ) | 2 = | T 21 ( λ ) T 11 ( λ ) | 2 .
T = [ R r e f ] [ P ] FBG 1 [ P ] FBG 2 [ P ] FBG 3 × FBG N 1 [ P ] FBG N ,
P = exp [ 4 π n λ l p 0 0 4 π n λ l p ] ,
R ref = 1 t [ r 0 0 r ] ,
r = n 1 n + 1 ,
t = 2 n n + 1 ,
f i = 2 n L i λ 2 .
f i j = 2 n | L i L j | λ 2 .
FBG i ( λ ) = | F 1 { F ( R ( λ ) ) REC ( f f i W ) } | ,
REC ( x ) = { 1 | x | 1 2 0 otherwise } ,
W 1 Δ λ .
Δ λ = λ 2 2 n l .
Δ λ B λ B = SGF Δ ε + TGF Δ T ,
SGF = 1 n 2 2 [ ρ 12 ν ( ρ 11 + ρ 12 ) ] .
ϕ = 4 π ( l Δ l ) λ = k z 0 ,
T ( z 0 ) = ( 1 + z 0 2 λ 2 4 π 2 w 0 4 ) 1 .
P r P i R 1 + ( 1 R 1 ) 2 R 2 T ( z 0 ) + 2 R 1 R 2 T ( z 0 ) × ( 1 R 1 ) cos ( ϕ ) .
ν = 2 R 1 R 2 T ( z 0 ) ( 1 R 1 ) R 1 + ( 1 R 1 ) 2 R 2 T ( z 0 ) .
P r P i 2 R 1 ( 1 + cos ϕ ) .
ε = Δ l L g = N λ 2 L g ,

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