- Email: [email protected]
Characterization of dynamic strain measurement using reflection spectrum from a fiber Bragg grating
Optics Communications 270 (2007) 25–30 www.elsevier.com/locate/optcom
Characterization of dynamic strain measurement using reflection spectrum from a fiber Bragg grating Hang-Yin Ling
a,*
, Kin-Tak Lau a, Wei Jin b, Kok-Cheung Chan
c
a
Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hong Kong, China Department of Electrical Engineering, The Hong Kong Polytechnic University, Hong Kong, China JC Optronics Limited, Unit 210B, 2/F, Photonics Center, Hong Kong Science Park, Shatin, Hong Kong, China b
c
Received 27 January 2005; received in revised form 24 August 2006; accepted 24 August 2006
Abstract A simulation method for evaluating dynamic strain distribution along a uniform Fiber Bragg Grating (FBG) using its reflection spectrum is presented in this paper. This method is valid for simulating both uniform and non-uniform strain distributions along the FBG under various dynamic conditions. The reflection spectrum from the FBG, subjected to different strain profiles, is simulated based on the Transfer matrix (T-matrix) approach. Three strain profiles, such as a uniform strain, a linear strain gradient and a quadratic strain field, are selected to demonstrate the viability of the proposed method. The importance of the dynamic simulation of the reflection spectrum on the filter design in a FBG dynamic strain sensing system is highlighted. 2006 Elsevier B.V. All rights reserved. Keywords: Fibre-Bragg grating; Non-uniform strain; Reflection spectrum
1. Introduction The use of Fiber Bragg Gratings (FBGs) for strain and temperature sensing applications has been on the rise in the past decade [1–4] due to their small physical size, insensitivity to electromagnetic interference, lightweight, capability of sensing at high temperature and environmentally unfavourable conditions, and also their multiplexing ability. By considering the constant strain and temperature effects along the grating, the FBGs are broadly exploited to measure the strain and temperature on the basis of the Bragg wavelength shift captured from the reflection spectrum [5,6]. However, direct measurement of the Bragg wavelength shift is no longer valid when the grating is subjected to non-uniform strain. The reason is that non-uniform strain could cause distortions of the spectrum, hence, a single peak of the spectrum cannot be maintained and the *
Corresponding author. Tel.: +852 27667663; fax: +852 23654703. E-mail addresses: [email protected], [email protected]. edu.hk (H.-Y. Ling). 0030-4018/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.08.032
Bragg wavelength becomes undistinguishable. In most cases, such distortions are highly correlated to the non-uniform strain profile along the grating. Consequently, the changes of the shape of the spectrum have found to be very useful for damage assessment of structures [7–9]. A Transfer matrix (T-matrix) method, first introduced by Yamada and Sakuda [10], was invented for the applications of the non-uniform strain sensing by Hung et al. [11]. They applied the T-matrix formulation by approximating the applied strain as a piecewise continuous function, calculating the average period in each grating segment due to the applied strain, and substituting this local period back into the coupling coefficient in the formulation. This method is broadly adopted to calculate the non-uniform strain distribution along the grating for constructing the reflection spectrum with the corresponding strain field. However, only the strain distribution under the static condition has been considered so far [12,13]. It is evidenced that dynamic strain monitoring in various kinds of engineering structures, especially for load-bearing structures such as propeller blades, aircraft’s fuselage and
26
H.-Y. Ling et al. / Optics Communications 270 (2007) 25–30
wing structures, is extremely important for the maintenance of the structural integrity under their service conditions [14]. In this paper, the strain distribution along a grating is simulated as illustrated in the reflection spectrum from the FBG which is subjected to various dynamic strain fields. Firstly, the principles of the FBG dynamic strain sensing and the spectrum simulation using the T-matrix method are reviewed. Then, the proposed method for dynamic strain simulation is verified by three dynamic strain fields (i.e. a uniform strain, a linear strain gradient and a quadratic strain). Finally, the use of the spectrum in dynamic strain simulation for the filter design in a FBG dynamic strain sensing system is discussed. 2. Principle of the dynamic strain sensing using FBGs The FBG is defined as a small periodical perturbation to the effective index of refraction neff of the optical fiber core described by [1] 2p dneff ðzÞ ¼ dneff 1 þ m cos z þ /ðzÞ ; ð1Þ K0 where m is the fringe visibility, K0 is the nominal period, /(z) is the grating chirp and dneff is the ‘‘dc’’ index change spatially averaged over a grating period. By Coupled-Mode theory, the first-order differential equations describing mode propagation through the grating in z-direction are dRðzÞ ¼ i^ rRðzÞ þ ijSðzÞ; dz dS ðzÞ ¼ i^ rSðzÞ ijRðzÞ; dz
ð2aÞ ð2bÞ
where R(z) and S(z) are the amplitudes of the forward- and ^ is the general backward-propagating modes, respectively. r ‘‘dc’’ self-coupling coefficient as a function of the propagating wavelength k, defined as 1 1 2p 1 ^ ¼ 2pneff r ð3Þ þ dneff /0 ðzÞ; k kD k 2 where / 0 (z) = d//dz and / 0 (z) = 0 for the uniform grating, kD = 2neffK0 is the designed wavelength which would be changed when the grating is subjected to various strain condition. In this study, the Bragg wavelength of the grating under strain-free state is 1540.2 nm. j ¼ pk mdneff is the ‘‘ac’’ coupling coefficient, in which dneff ¼ 1:131 104 and m 1. The length of the uniform grating is assumed to be L (L = 10 mm), so the limits of the grating is defined as L/2 6 z 6 L/2. While the boundary conditions of the uniform Bragg grating are R(L/2) = 1 and S(L/2) = 0 [1]. The reflectivity of the Bragg grating, calculated as a function of the wavelength, can be expressed as S ðL=2Þ 2 : rðkÞ ¼ ð4Þ RðL=2Þ
To obtain the reflectivity of the uniform Bragg grating, which is subjected to either uniform or non-uniform strain, the T-matrix formulation is used to model the grating with non-constant properties. In this approach, the grating is assumed to be divided into M smaller sections and with uniform coupling properties. It is important to note that the number of sections M cannot be arbitrarily large since several grating periods are required for complete coupling. Hence, the M is constrained as [15] M
2neff L ; kD
ð5Þ
where M was set to 200 in the present study [15]. By defining Ri and Si as the field amplitudes after traversing the ith grating section, the propagation through this uniform section can be described by, Ri1 Ri ¼ Fi ; ð6Þ Si S i1 where " Fi¼
#
coshðcB DzÞ i cr^B sinh ðcB DzÞ
i cjB sinh ðcB DzÞ
i cjB sinh ðcB DzÞ
cosh ðcB DzÞ þ i cr^B sinh ðcB DzÞ
;
in which Dz is the length of each section and cB ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^2 . As a result, the T-matrix F for entire grating j2 r can be written as, RðL=2Þ RðL=2Þ ¼F ; ð7Þ S ðL=2Þ S ðL=2Þ where F = FM Æ FM1 F1. The reflectivity of the entire grating is then calculated using Eq. (4). For a dynamic strain sensing application, it is assumed that the strain function along the grating (in z-direction) with respect to time is ezz ðz; tÞ ¼ e0 ðzÞ þ em ðzÞf ðtÞ;
ð8Þ
where e0(z) is the DC component of strain, em(z) and f(t) are the amplitude and the profile of the fluctuating strain modulation respectively. The function f(t) has an average value of zero and satisfies the criterion of jf(t 6 1)j. In the case of a sinusoidal strain modulation, f(t) = sin(-t + /), in which and / are the angular oscillation frequency and the phase difference respectively. Note that the sensing principle of the uniform dynamic strain can be found in our previous work [16]. When the grating is subjected to the non-uniform dynamic strain, its reflective spectrum will not only be shifted but also distorted due to the non-uniform changes in both the local index of refraction and the grating period. Under this circumstance, the relationship between the wavelength and the strain along the grating (in z-direction) can be written as [17] kB ðz; tÞ ¼ 2neff ðz; tÞKðz; tÞ ¼ 2neff0 K0 ð1 þ aezz ðz; tÞÞ;
ð9Þ
where a ¼ 1 12 n2eff0 ½p12 mðp11 p12 Þ is the grating gauge factor [18], in which p12 and p11 are the components of the fibre-optic strain tensor and m is the Poisson’s ratio. K0 and
H.-Y. Ling et al. / Optics Communications 270 (2007) 25–30
neff0 are the initial grating period and the initial average effective mode index of refraction at the strain-free state respectively. By substituting Eq. (9) into Eq. (3), in which kD is replaced by kB(z, t), the simulated spectrum can be obtained under different non-uniform dynamic strain conditions. 3. Results and discussion Both positional and shape changes in the reflection spectrum, caused by the strain variation along the grating, should be known first in order to design a suitable filter to capture the spectrum. In particular, the filter must be sensitive to the distortion of the spectrum for measuring the non-uniform strain. Without knowing the spectral changes in advance, the filter cannot be designed properly. Therefore, the spectrum simulation provides a fast and reliable mean for filter design in dynamic strain measurement. To demonstrate the versatility of the simulated spectrum from the grating subjected to different strain profiles, three classical strain profiles [15] (A, B and C) at 100 Hz are discussed as follows. Note that e0(z) is assumed to be zero in all cases. 3.1. Uniform strain [em(z) = C0, dem(z)/dz = 0] Fig. 1(a) shows the reflectivity of the grating under a uniform strain of C0 = 2571 · 106 le at 100 Hz. Note that C0 is a constant value. It is observed that the Bragg wavelength shifts to left and back to centre, and then to
3ms 2ms 2.5ms
4ms
10ms
1ms
5ms
3.5ms
4.5ms 0.5ms
0ms
9.5ms 5.5ms
9ms
8ms
6ms
7ms 8.5ms
27
right and again back to centre periodically within 0.01 s. A sinusoidal change in the Bragg wavelength with respect to time, as illustrated in Fig. 1(b), is due to the grating being suppressed by the periodical change in the constant strain with respect to time. No change of the reflectivity is noticed since the local index of refraction and the grating period of the grating change uniformly under the uniform strain condition. 3.2. Linear strain gradient [em(z) = m0z + C0, dem(z)/ dz = m0] Three cases, the cases of m0 = 40 · 103 with C0 = 0, m0 = 40 · 103 with C0 = 500 · 106 and m0 = 400 · 103 with C0 = 0, are considered to compare the effect of the various strain values under different strain gradients on the reflection spectrum. Note that m0 is the slope of the linear strain gradient. The cases of the same small strain gradient are depicted in Figs. 2 and 3. It is noted that no obvious shift of the Bragg wavelength is observed in Fig. 2(a). As indicated in Fig. 2(b), the reflectivity decreases and increases with time within 0.005 s. The lower the reflectivity is, the greater the distortion of the reflection spectrum is. In the case of C0 = 0, the grating is evenly subjected to positive strain at one end and negative strain at another end resulted in the variation of the local index of refraction as sketched in Fig. 4. Such a variation of the local index of refraction causes the distortion and the reduction of the reflectivity in the reflection spectrum. As the average strain along the grating is equivalent to zero instantaneously, no change in the Bragg wavelength is recorded. The periodic change in the reflectivity of the reflection spectrum with
7.5ms
6.5ms
1.5ms
1
1
0.9
0.8 Reflectivity (R)
0.8
Reflectivity (R)
0.7 0.6 0.5
0.6 0.4 0.2
0.4 0.3
0 1.544
0.2
1.542
0.1 0 1.536
-6
x 10 1.537
1.538
1.539
1.54
1.541
Wavelength (m)
1.542
1.543
1 1.54
0.5
1.538
1.544 -6
x 10
Fig. 1(a). Reflectivity of the grating against wavelength under a uniform constant strain (e0(z) = 2571 · 106 le) at 100 Hz, ( reflection spectrum at strain-free state).
Wavelength (m)
1.536
0
Time (0.01s)
Fig. 1(b). Reflection spectrum of the grating changing with time under a uniform constant strain (e0(z) = 2571 · 106 le) at 100 Hz.
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H.-Y. Ling et al. / Optics Communications 270 (2007) 25–30 10ms
1
0 ms, 5ms
0.9
5ms
0.5ms, 4.5ms
1.5ms, 3.5 ms
9.5ms
4.5ms
0ms
1ms, 4ms
0.8
2ms, 3ms
4.ms
5.5ms 9ms
0.5ms
6ms
1ms
0.6
2.5ms
0.5 0.4
1
3ms
0.3
0.9
2ms
0.2
0.8
0.1
0.7
0 1.539
1.5395
1.54
1.5405
1.541
1.5415 -6
Wavelength (m)
x 10
Fig. 2(a). Reflectivity of the grating against wavelength under a linear small strain gradient (e0(z) = 40 · 103z) at 100 Hz, ( reflection spectrum at strain-free state).
Reflectivity (R)
Reflectivity (R)
0.7
3.5ms
8.5ms
1.5ms
6.5ms 8ms 7ms
2.5ms
7.5ms
0.6 0.5 0.4 0.3 0.2 0.1 0 1.538 1.5385
1.539
1.5395
1.54
1.5405
1.541
1.5415 1.542 -6
Wavelength (m)
x 10
1 Fig. 3(a). Reflectivity of the grating against wavelength under a linear small strain gradient (e0(z) = 40 · 103z + 500 · 106) at 100 Hz, ( reflection spectrum at strain-free state).
Reflectivity (R)
0.8 0.6 0.4
1
0 1.542
0.8
1
1.541 -6
x 10
Wavelength (m)
0.5
1.54 1.539
0
Reflectivity (R)
0.2
0.6 0.4 0.2
Time (0.01s)
Fig. 2(b). Reflection spectrum of the grating changing with time under a linear small strain gradient (e0(z) = 40 · 103z) at 100 Hz.
0 1.542 1.541
respect to time is due to the strain function in Eq. (8). On the other hand, for C0 = 500 · 106 in Fig. 3, the distortion and the change of the reflectivity in the reflection spectrum in different time intervals are similar to that of the C0 = 0. However, the shift of the Bragg wavelength is observed in this case because the average strain along the grating is non-zero. Fig. 5(a) plots the reflection spectrum of the grating under the large strain gradient of m0 = 400 · 103, where the distortion of the reflection spectrum is more apparent, compared to the one shown in Fig. 2(a). Such an observation implies that the distortion of the reflection spectrum becomes more obvious when the
-6
x 10
1 1.54
0.5
1.539 Wavelength (m)
1.538
0
Time (0.01s)
Fig. 3(b). Reflection spectrum of the grating changing with time under a linear small strain gradient (e0(z) = 40 · 103z + 500 · 106) at 100 Hz.
grating is in response to higher strain gradient. Similarly, the cyclic change in the reflectivity of the reflection spectrum from the grating, suppressed under the large strain gradient, is demonstrated in Fig. 5(b).
H.-Y. Ling et al. / Optics Communications 270 (2007) 25–30
29
1 0.8 Reflectivity (R)
δ neff
0.6 0.4 0.2
z 0
Fig. 4. Variation of the local index of reflection along the grating subjected to a linear small strain gradient (e0(z) = 40 · 103z + 500 · 106).
1.542
1 1.54
-6
x 10
0.9
0
Wavelength (m)
0ms, 5ms
0.8
Reflectivity (R)
0.5 1.538
1
Time (0.01s)
Fig. 5(b). Reflection spectrum of the grating changing with time under a linear large strain gradient (e0(z) = 400 · 103z) at 100 Hz.
0.7 0.6 0.5 0.4
0.5ms, 4.5ms
0.3 0.2
1
2.5ms 1ms, 4ms
2ms, 3ms
0.8
0 1.537
1.538
1.539
1.54
1.541
Wavelength (m)
1.542
1.543 -6
x 10
Fig. 5(a). Reflectivity of the grating against wavelength under a linear reflection large strain gradient (e0(z) = 400 · 103z) at 100 Hz, ( spectrum at strain-free state).
Reflectivity (R)
0.1
0.6 0.4 0.2 0
3.3. Quadratic strain field [e0(z) = a0 z2, de0(z)/dz = 2a0z] In this case, a quadratic strain field along the grating is assumed to be e0(z) = z2 by setting a0 to 1. Fig. 6 gives the reflection spectrum of the grating in different time intervals. Multiple peaks can be seen beside the main peak. These multiple peaks appear to the left and right hand sides of the main peak alternatively. The level of the reflectivity of these multiple peaks increases and decreases sinusoidally as shown in Fig. 6. These phenomena are caused by the variation of the strain gradient along the grating. The results of the above simulations with different strain profiles along the grating demonstrated the relationship between the shape, reflectivity, shift of the Bragg wavelength of the reflection spectrum and the strain distribution along the grating. It can be concluded that the severity of distortion of the reflection spectrum is highly dependent on the strain gradient. Also, the shift of the Bragg wavelength linearly correlates to the average strain along the
1.541 -6
x 10
1 1.54
Wavelength (m)
0.5 1.539
0
Time (0.01s)
Fig. 6. Reflection spectrum of the grating changing with time under a quadratic strain field (e0(z) = 800z2) at 100 Hz.
grating. The strain values alternating with time resulted in the cyclic changes in the Bragg wavelength and the deformation of the reflection spectrum. 4. Conclusion A simulation method for evaluating both uniform and non-uniform dynamic strain distributions along the uniform FBG by means of its reflection spectrum and the
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H.-Y. Ling et al. / Optics Communications 270 (2007) 25–30
T-matrix formulation was introduced. The proposed method was validated by applying different strain fields: uniform strain, linear strain gradient and quadratic strain field, along the grating. It was revealed that the filter design in the FBG dynamic strain measurement system mainly relied on the spectrum simulation in which the FBG was subjected to either uniform or non-uniform dynamic strain field. Moreover, minimum detectable strain amplitude and maximum attainable frequency measured by the FBG can be evaluated under versatile filter configurations. It is important to note that this FBG sensing system can be employed for real-time monitoring of structures during their in-service time. This methodology is also applicable for a multiplexed FBG sensor system to extract the preliminary dynamic strain information of the structures. Acknowledgement This research project was funded by The Hong Kong Polytechnic Research Grant A-PF34. References [1] K.O. Hill, G. Meltz, Fiber Bragg grating technology fundamentals and overview, J. Lightwave Technol. 15 (1997) 1263. [2] Y. Yu, H. Tam, W. Chung, M.S. Demokan, Fiber Bragg grating sensor for simultaneous measurement of displacement and temperature, Opt. Lett. 25 (2000) 1141. [3] X. Shu, Y. Liu, D. Zhao, B. Gwandu, F. Floreani, L. Zhang, I. Bennion, Dependence of temperature and strain coefficients on fiber grating type and its application to simultaneous temperature and strain measurement, Opt. Lett. 27 (2002) 701. [4] S. Magne, S. Rougeault, M. Vilela, P. Ferdinand, State-of-strain evaluation with fiber Bragg grating rosettes: application to discrimination between strain and temperature effects in fiber sensors, Appl. Opt. 36 (1997) 9437.
[5] Y.B. Lin, J.C. Chern, K.C. Chang, Y.W. Chan, L.A. Wang, The utilization of fiber Bragg grating sensors to monitor high performance concrete at elevated temperature, Smart Mater. Struct. 13 (2004) 784. [6] K.T. Lau, C.C. Chan, L.M. Zhou, W. Jin, Strain monitoring in composite-strenthened concrete structures using optical fibre sensors, Comput. Part B 32 (2001) 33. [7] M. LeBlanc, S.Y. Huang, M. Ohn, R.M. Measures, Distributed strain measurement based on a fiber Bragg grating and its reflection spectrum analysis, Opt. Lett. 21 (1996) 1405. [8] P. Giaccari, G.R. Dunkel, L. Humbert, J. Botsis, H.G. Limberger, R.P. Salathe, On a direct determination of non-uniform internal strain fields using fibre Bragg gratings, Smart Mater. Struct. 14 (2005) 127. [9] M. Studer, K. Peters, J. Botsis, Method for determination of crack bridging parameters using long optical fiber Bragg grating sensors, Comput. Part B 34 (2003) 347. [10] M. Tamada, K. Sakuda, Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach, Appl. Opt. 26 (1987) 3474. [11] S. Huang, M. LeBlanc, M.M. Ohn, R.M. Measures, Bragg intregrating structural sensing, Appl. Opt. 34 (1995) 5003. [12] S. Huang, M.M. Ohn, M. LeBlanc, R.M. Measures, Continuous arbitrary strain profile measurements with fiber Bragg gratings, Smart Mater. Struct. 7 (1998) 248. [13] F. Casagrande, P. Crespi, A.M. Grassi, A. Lulli, R.P. Kenny, M.P. Whelan, From the reflected spectrum to the properties of a fiber Bragg grating: a genetic algorithm approach with application to distributed strain sensing, Appl. Opt. 41 (2002) 5238. [14] H.Y. Ling, K.T. Lau, L. Cheng, Determination of the dynamic stress profile and delamination detection of composite structures using embedded multiplexed fibre-optic sensors, Comput. Struct. 66 (2004) 317. [15] M. Prabhugoud, K. Peters, Modified transfer matrix formulation for Bragg grating strain sensors, J. Lightwave Technol. 22 (2004) 2302. [16] H.Y. Ling, K.T. Lau, L. Cheng, W. Jin, Viability of using an embedded FBG sensor in a composite structure for dynamic strain measurement, Measurement 39 (2006) 328. [17] K. Peters, M. Studer, J. Botsis, A. Iocco, H. Limberger, R. Salathe, Embedded optical fiber Bragg grating sensor in a nonuniform strain field: measurements and simulations, Exp. Mech. 41 (2001) 19. [18] Y.J. Rao, In-fibre Bragg grating sensors, Meas. Sci. Technol. 8 (1997) 355.
Characterization of dynamic strain measurement using reflection spectrum from a fiber Bragg grating Hang-Yin Ling
a,*
, Kin-Tak Lau a, Wei Jin b, Kok-Cheung Chan
c
a
Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hong Kong, China Department of Electrical Engineering, The Hong Kong Polytechnic University, Hong Kong, China JC Optronics Limited, Unit 210B, 2/F, Photonics Center, Hong Kong Science Park, Shatin, Hong Kong, China b
c
Received 27 January 2005; received in revised form 24 August 2006; accepted 24 August 2006
Abstract A simulation method for evaluating dynamic strain distribution along a uniform Fiber Bragg Grating (FBG) using its reflection spectrum is presented in this paper. This method is valid for simulating both uniform and non-uniform strain distributions along the FBG under various dynamic conditions. The reflection spectrum from the FBG, subjected to different strain profiles, is simulated based on the Transfer matrix (T-matrix) approach. Three strain profiles, such as a uniform strain, a linear strain gradient and a quadratic strain field, are selected to demonstrate the viability of the proposed method. The importance of the dynamic simulation of the reflection spectrum on the filter design in a FBG dynamic strain sensing system is highlighted. 2006 Elsevier B.V. All rights reserved. Keywords: Fibre-Bragg grating; Non-uniform strain; Reflection spectrum
1. Introduction The use of Fiber Bragg Gratings (FBGs) for strain and temperature sensing applications has been on the rise in the past decade [1–4] due to their small physical size, insensitivity to electromagnetic interference, lightweight, capability of sensing at high temperature and environmentally unfavourable conditions, and also their multiplexing ability. By considering the constant strain and temperature effects along the grating, the FBGs are broadly exploited to measure the strain and temperature on the basis of the Bragg wavelength shift captured from the reflection spectrum [5,6]. However, direct measurement of the Bragg wavelength shift is no longer valid when the grating is subjected to non-uniform strain. The reason is that non-uniform strain could cause distortions of the spectrum, hence, a single peak of the spectrum cannot be maintained and the *
Corresponding author. Tel.: +852 27667663; fax: +852 23654703. E-mail addresses: [email protected], [email protected]. edu.hk (H.-Y. Ling). 0030-4018/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.08.032
Bragg wavelength becomes undistinguishable. In most cases, such distortions are highly correlated to the non-uniform strain profile along the grating. Consequently, the changes of the shape of the spectrum have found to be very useful for damage assessment of structures [7–9]. A Transfer matrix (T-matrix) method, first introduced by Yamada and Sakuda [10], was invented for the applications of the non-uniform strain sensing by Hung et al. [11]. They applied the T-matrix formulation by approximating the applied strain as a piecewise continuous function, calculating the average period in each grating segment due to the applied strain, and substituting this local period back into the coupling coefficient in the formulation. This method is broadly adopted to calculate the non-uniform strain distribution along the grating for constructing the reflection spectrum with the corresponding strain field. However, only the strain distribution under the static condition has been considered so far [12,13]. It is evidenced that dynamic strain monitoring in various kinds of engineering structures, especially for load-bearing structures such as propeller blades, aircraft’s fuselage and
26
H.-Y. Ling et al. / Optics Communications 270 (2007) 25–30
wing structures, is extremely important for the maintenance of the structural integrity under their service conditions [14]. In this paper, the strain distribution along a grating is simulated as illustrated in the reflection spectrum from the FBG which is subjected to various dynamic strain fields. Firstly, the principles of the FBG dynamic strain sensing and the spectrum simulation using the T-matrix method are reviewed. Then, the proposed method for dynamic strain simulation is verified by three dynamic strain fields (i.e. a uniform strain, a linear strain gradient and a quadratic strain). Finally, the use of the spectrum in dynamic strain simulation for the filter design in a FBG dynamic strain sensing system is discussed. 2. Principle of the dynamic strain sensing using FBGs The FBG is defined as a small periodical perturbation to the effective index of refraction neff of the optical fiber core described by [1] 2p dneff ðzÞ ¼ dneff 1 þ m cos z þ /ðzÞ ; ð1Þ K0 where m is the fringe visibility, K0 is the nominal period, /(z) is the grating chirp and dneff is the ‘‘dc’’ index change spatially averaged over a grating period. By Coupled-Mode theory, the first-order differential equations describing mode propagation through the grating in z-direction are dRðzÞ ¼ i^ rRðzÞ þ ijSðzÞ; dz dS ðzÞ ¼ i^ rSðzÞ ijRðzÞ; dz
ð2aÞ ð2bÞ
where R(z) and S(z) are the amplitudes of the forward- and ^ is the general backward-propagating modes, respectively. r ‘‘dc’’ self-coupling coefficient as a function of the propagating wavelength k, defined as 1 1 2p 1 ^ ¼ 2pneff r ð3Þ þ dneff /0 ðzÞ; k kD k 2 where / 0 (z) = d//dz and / 0 (z) = 0 for the uniform grating, kD = 2neffK0 is the designed wavelength which would be changed when the grating is subjected to various strain condition. In this study, the Bragg wavelength of the grating under strain-free state is 1540.2 nm. j ¼ pk mdneff is the ‘‘ac’’ coupling coefficient, in which dneff ¼ 1:131 104 and m 1. The length of the uniform grating is assumed to be L (L = 10 mm), so the limits of the grating is defined as L/2 6 z 6 L/2. While the boundary conditions of the uniform Bragg grating are R(L/2) = 1 and S(L/2) = 0 [1]. The reflectivity of the Bragg grating, calculated as a function of the wavelength, can be expressed as S ðL=2Þ 2 : rðkÞ ¼ ð4Þ RðL=2Þ
To obtain the reflectivity of the uniform Bragg grating, which is subjected to either uniform or non-uniform strain, the T-matrix formulation is used to model the grating with non-constant properties. In this approach, the grating is assumed to be divided into M smaller sections and with uniform coupling properties. It is important to note that the number of sections M cannot be arbitrarily large since several grating periods are required for complete coupling. Hence, the M is constrained as [15] M
2neff L ; kD
ð5Þ
where M was set to 200 in the present study [15]. By defining Ri and Si as the field amplitudes after traversing the ith grating section, the propagation through this uniform section can be described by, Ri1 Ri ¼ Fi ; ð6Þ Si S i1 where " Fi¼
#
coshðcB DzÞ i cr^B sinh ðcB DzÞ
i cjB sinh ðcB DzÞ
i cjB sinh ðcB DzÞ
cosh ðcB DzÞ þ i cr^B sinh ðcB DzÞ
;
in which Dz is the length of each section and cB ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^2 . As a result, the T-matrix F for entire grating j2 r can be written as, RðL=2Þ RðL=2Þ ¼F ; ð7Þ S ðL=2Þ S ðL=2Þ where F = FM Æ FM1 F1. The reflectivity of the entire grating is then calculated using Eq. (4). For a dynamic strain sensing application, it is assumed that the strain function along the grating (in z-direction) with respect to time is ezz ðz; tÞ ¼ e0 ðzÞ þ em ðzÞf ðtÞ;
ð8Þ
where e0(z) is the DC component of strain, em(z) and f(t) are the amplitude and the profile of the fluctuating strain modulation respectively. The function f(t) has an average value of zero and satisfies the criterion of jf(t 6 1)j. In the case of a sinusoidal strain modulation, f(t) = sin(-t + /), in which and / are the angular oscillation frequency and the phase difference respectively. Note that the sensing principle of the uniform dynamic strain can be found in our previous work [16]. When the grating is subjected to the non-uniform dynamic strain, its reflective spectrum will not only be shifted but also distorted due to the non-uniform changes in both the local index of refraction and the grating period. Under this circumstance, the relationship between the wavelength and the strain along the grating (in z-direction) can be written as [17] kB ðz; tÞ ¼ 2neff ðz; tÞKðz; tÞ ¼ 2neff0 K0 ð1 þ aezz ðz; tÞÞ;
ð9Þ
where a ¼ 1 12 n2eff0 ½p12 mðp11 p12 Þ is the grating gauge factor [18], in which p12 and p11 are the components of the fibre-optic strain tensor and m is the Poisson’s ratio. K0 and
H.-Y. Ling et al. / Optics Communications 270 (2007) 25–30
neff0 are the initial grating period and the initial average effective mode index of refraction at the strain-free state respectively. By substituting Eq. (9) into Eq. (3), in which kD is replaced by kB(z, t), the simulated spectrum can be obtained under different non-uniform dynamic strain conditions. 3. Results and discussion Both positional and shape changes in the reflection spectrum, caused by the strain variation along the grating, should be known first in order to design a suitable filter to capture the spectrum. In particular, the filter must be sensitive to the distortion of the spectrum for measuring the non-uniform strain. Without knowing the spectral changes in advance, the filter cannot be designed properly. Therefore, the spectrum simulation provides a fast and reliable mean for filter design in dynamic strain measurement. To demonstrate the versatility of the simulated spectrum from the grating subjected to different strain profiles, three classical strain profiles [15] (A, B and C) at 100 Hz are discussed as follows. Note that e0(z) is assumed to be zero in all cases. 3.1. Uniform strain [em(z) = C0, dem(z)/dz = 0] Fig. 1(a) shows the reflectivity of the grating under a uniform strain of C0 = 2571 · 106 le at 100 Hz. Note that C0 is a constant value. It is observed that the Bragg wavelength shifts to left and back to centre, and then to
3ms 2ms 2.5ms
4ms
10ms
1ms
5ms
3.5ms
4.5ms 0.5ms
0ms
9.5ms 5.5ms
9ms
8ms
6ms
7ms 8.5ms
27
right and again back to centre periodically within 0.01 s. A sinusoidal change in the Bragg wavelength with respect to time, as illustrated in Fig. 1(b), is due to the grating being suppressed by the periodical change in the constant strain with respect to time. No change of the reflectivity is noticed since the local index of refraction and the grating period of the grating change uniformly under the uniform strain condition. 3.2. Linear strain gradient [em(z) = m0z + C0, dem(z)/ dz = m0] Three cases, the cases of m0 = 40 · 103 with C0 = 0, m0 = 40 · 103 with C0 = 500 · 106 and m0 = 400 · 103 with C0 = 0, are considered to compare the effect of the various strain values under different strain gradients on the reflection spectrum. Note that m0 is the slope of the linear strain gradient. The cases of the same small strain gradient are depicted in Figs. 2 and 3. It is noted that no obvious shift of the Bragg wavelength is observed in Fig. 2(a). As indicated in Fig. 2(b), the reflectivity decreases and increases with time within 0.005 s. The lower the reflectivity is, the greater the distortion of the reflection spectrum is. In the case of C0 = 0, the grating is evenly subjected to positive strain at one end and negative strain at another end resulted in the variation of the local index of refraction as sketched in Fig. 4. Such a variation of the local index of refraction causes the distortion and the reduction of the reflectivity in the reflection spectrum. As the average strain along the grating is equivalent to zero instantaneously, no change in the Bragg wavelength is recorded. The periodic change in the reflectivity of the reflection spectrum with
7.5ms
6.5ms
1.5ms
1
1
0.9
0.8 Reflectivity (R)
0.8
Reflectivity (R)
0.7 0.6 0.5
0.6 0.4 0.2
0.4 0.3
0 1.544
0.2
1.542
0.1 0 1.536
-6
x 10 1.537
1.538
1.539
1.54
1.541
Wavelength (m)
1.542
1.543
1 1.54
0.5
1.538
1.544 -6
x 10
Fig. 1(a). Reflectivity of the grating against wavelength under a uniform constant strain (e0(z) = 2571 · 106 le) at 100 Hz, ( reflection spectrum at strain-free state).
Wavelength (m)
1.536
0
Time (0.01s)
Fig. 1(b). Reflection spectrum of the grating changing with time under a uniform constant strain (e0(z) = 2571 · 106 le) at 100 Hz.
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H.-Y. Ling et al. / Optics Communications 270 (2007) 25–30 10ms
1
0 ms, 5ms
0.9
5ms
0.5ms, 4.5ms
1.5ms, 3.5 ms
9.5ms
4.5ms
0ms
1ms, 4ms
0.8
2ms, 3ms
4.ms
5.5ms 9ms
0.5ms
6ms
1ms
0.6
2.5ms
0.5 0.4
1
3ms
0.3
0.9
2ms
0.2
0.8
0.1
0.7
0 1.539
1.5395
1.54
1.5405
1.541
1.5415 -6
Wavelength (m)
x 10
Fig. 2(a). Reflectivity of the grating against wavelength under a linear small strain gradient (e0(z) = 40 · 103z) at 100 Hz, ( reflection spectrum at strain-free state).
Reflectivity (R)
Reflectivity (R)
0.7
3.5ms
8.5ms
1.5ms
6.5ms 8ms 7ms
2.5ms
7.5ms
0.6 0.5 0.4 0.3 0.2 0.1 0 1.538 1.5385
1.539
1.5395
1.54
1.5405
1.541
1.5415 1.542 -6
Wavelength (m)
x 10
1 Fig. 3(a). Reflectivity of the grating against wavelength under a linear small strain gradient (e0(z) = 40 · 103z + 500 · 106) at 100 Hz, ( reflection spectrum at strain-free state).
Reflectivity (R)
0.8 0.6 0.4
1
0 1.542
0.8
1
1.541 -6
x 10
Wavelength (m)
0.5
1.54 1.539
0
Reflectivity (R)
0.2
0.6 0.4 0.2
Time (0.01s)
Fig. 2(b). Reflection spectrum of the grating changing with time under a linear small strain gradient (e0(z) = 40 · 103z) at 100 Hz.
0 1.542 1.541
respect to time is due to the strain function in Eq. (8). On the other hand, for C0 = 500 · 106 in Fig. 3, the distortion and the change of the reflectivity in the reflection spectrum in different time intervals are similar to that of the C0 = 0. However, the shift of the Bragg wavelength is observed in this case because the average strain along the grating is non-zero. Fig. 5(a) plots the reflection spectrum of the grating under the large strain gradient of m0 = 400 · 103, where the distortion of the reflection spectrum is more apparent, compared to the one shown in Fig. 2(a). Such an observation implies that the distortion of the reflection spectrum becomes more obvious when the
-6
x 10
1 1.54
0.5
1.539 Wavelength (m)
1.538
0
Time (0.01s)
Fig. 3(b). Reflection spectrum of the grating changing with time under a linear small strain gradient (e0(z) = 40 · 103z + 500 · 106) at 100 Hz.
grating is in response to higher strain gradient. Similarly, the cyclic change in the reflectivity of the reflection spectrum from the grating, suppressed under the large strain gradient, is demonstrated in Fig. 5(b).
H.-Y. Ling et al. / Optics Communications 270 (2007) 25–30
29
1 0.8 Reflectivity (R)
δ neff
0.6 0.4 0.2
z 0
Fig. 4. Variation of the local index of reflection along the grating subjected to a linear small strain gradient (e0(z) = 40 · 103z + 500 · 106).
1.542
1 1.54
-6
x 10
0.9
0
Wavelength (m)
0ms, 5ms
0.8
Reflectivity (R)
0.5 1.538
1
Time (0.01s)
Fig. 5(b). Reflection spectrum of the grating changing with time under a linear large strain gradient (e0(z) = 400 · 103z) at 100 Hz.
0.7 0.6 0.5 0.4
0.5ms, 4.5ms
0.3 0.2
1
2.5ms 1ms, 4ms
2ms, 3ms
0.8
0 1.537
1.538
1.539
1.54
1.541
Wavelength (m)
1.542
1.543 -6
x 10
Fig. 5(a). Reflectivity of the grating against wavelength under a linear reflection large strain gradient (e0(z) = 400 · 103z) at 100 Hz, ( spectrum at strain-free state).
Reflectivity (R)
0.1
0.6 0.4 0.2 0
3.3. Quadratic strain field [e0(z) = a0 z2, de0(z)/dz = 2a0z] In this case, a quadratic strain field along the grating is assumed to be e0(z) = z2 by setting a0 to 1. Fig. 6 gives the reflection spectrum of the grating in different time intervals. Multiple peaks can be seen beside the main peak. These multiple peaks appear to the left and right hand sides of the main peak alternatively. The level of the reflectivity of these multiple peaks increases and decreases sinusoidally as shown in Fig. 6. These phenomena are caused by the variation of the strain gradient along the grating. The results of the above simulations with different strain profiles along the grating demonstrated the relationship between the shape, reflectivity, shift of the Bragg wavelength of the reflection spectrum and the strain distribution along the grating. It can be concluded that the severity of distortion of the reflection spectrum is highly dependent on the strain gradient. Also, the shift of the Bragg wavelength linearly correlates to the average strain along the
1.541 -6
x 10
1 1.54
Wavelength (m)
0.5 1.539
0
Time (0.01s)
Fig. 6. Reflection spectrum of the grating changing with time under a quadratic strain field (e0(z) = 800z2) at 100 Hz.
grating. The strain values alternating with time resulted in the cyclic changes in the Bragg wavelength and the deformation of the reflection spectrum. 4. Conclusion A simulation method for evaluating both uniform and non-uniform dynamic strain distributions along the uniform FBG by means of its reflection spectrum and the
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H.-Y. Ling et al. / Optics Communications 270 (2007) 25–30
T-matrix formulation was introduced. The proposed method was validated by applying different strain fields: uniform strain, linear strain gradient and quadratic strain field, along the grating. It was revealed that the filter design in the FBG dynamic strain measurement system mainly relied on the spectrum simulation in which the FBG was subjected to either uniform or non-uniform dynamic strain field. Moreover, minimum detectable strain amplitude and maximum attainable frequency measured by the FBG can be evaluated under versatile filter configurations. It is important to note that this FBG sensing system can be employed for real-time monitoring of structures during their in-service time. This methodology is also applicable for a multiplexed FBG sensor system to extract the preliminary dynamic strain information of the structures. Acknowledgement This research project was funded by The Hong Kong Polytechnic Research Grant A-PF34. References [1] K.O. Hill, G. Meltz, Fiber Bragg grating technology fundamentals and overview, J. Lightwave Technol. 15 (1997) 1263. [2] Y. Yu, H. Tam, W. Chung, M.S. Demokan, Fiber Bragg grating sensor for simultaneous measurement of displacement and temperature, Opt. Lett. 25 (2000) 1141. [3] X. Shu, Y. Liu, D. Zhao, B. Gwandu, F. Floreani, L. Zhang, I. Bennion, Dependence of temperature and strain coefficients on fiber grating type and its application to simultaneous temperature and strain measurement, Opt. Lett. 27 (2002) 701. [4] S. Magne, S. Rougeault, M. Vilela, P. Ferdinand, State-of-strain evaluation with fiber Bragg grating rosettes: application to discrimination between strain and temperature effects in fiber sensors, Appl. Opt. 36 (1997) 9437.
[5] Y.B. Lin, J.C. Chern, K.C. Chang, Y.W. Chan, L.A. Wang, The utilization of fiber Bragg grating sensors to monitor high performance concrete at elevated temperature, Smart Mater. Struct. 13 (2004) 784. [6] K.T. Lau, C.C. Chan, L.M. Zhou, W. Jin, Strain monitoring in composite-strenthened concrete structures using optical fibre sensors, Comput. Part B 32 (2001) 33. [7] M. LeBlanc, S.Y. Huang, M. Ohn, R.M. Measures, Distributed strain measurement based on a fiber Bragg grating and its reflection spectrum analysis, Opt. Lett. 21 (1996) 1405. [8] P. Giaccari, G.R. Dunkel, L. Humbert, J. Botsis, H.G. Limberger, R.P. Salathe, On a direct determination of non-uniform internal strain fields using fibre Bragg gratings, Smart Mater. Struct. 14 (2005) 127. [9] M. Studer, K. Peters, J. Botsis, Method for determination of crack bridging parameters using long optical fiber Bragg grating sensors, Comput. Part B 34 (2003) 347. [10] M. Tamada, K. Sakuda, Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach, Appl. Opt. 26 (1987) 3474. [11] S. Huang, M. LeBlanc, M.M. Ohn, R.M. Measures, Bragg intregrating structural sensing, Appl. Opt. 34 (1995) 5003. [12] S. Huang, M.M. Ohn, M. LeBlanc, R.M. Measures, Continuous arbitrary strain profile measurements with fiber Bragg gratings, Smart Mater. Struct. 7 (1998) 248. [13] F. Casagrande, P. Crespi, A.M. Grassi, A. Lulli, R.P. Kenny, M.P. Whelan, From the reflected spectrum to the properties of a fiber Bragg grating: a genetic algorithm approach with application to distributed strain sensing, Appl. Opt. 41 (2002) 5238. [14] H.Y. Ling, K.T. Lau, L. Cheng, Determination of the dynamic stress profile and delamination detection of composite structures using embedded multiplexed fibre-optic sensors, Comput. Struct. 66 (2004) 317. [15] M. Prabhugoud, K. Peters, Modified transfer matrix formulation for Bragg grating strain sensors, J. Lightwave Technol. 22 (2004) 2302. [16] H.Y. Ling, K.T. Lau, L. Cheng, W. Jin, Viability of using an embedded FBG sensor in a composite structure for dynamic strain measurement, Measurement 39 (2006) 328. [17] K. Peters, M. Studer, J. Botsis, A. Iocco, H. Limberger, R. Salathe, Embedded optical fiber Bragg grating sensor in a nonuniform strain field: measurements and simulations, Exp. Mech. 41 (2001) 19. [18] Y.J. Rao, In-fibre Bragg grating sensors, Meas. Sci. Technol. 8 (1997) 355.