- Email: [email protected]
Effects of distributed birefringence on fiber Bragg grating under non-uniform transverse load
ARTICLE IN PRESS Optics & Laser Technology 40 (2008) 1037– 1040
Contents lists available at ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Effects of distributed birefringence on fiber Bragg grating under non-uniform transverse load$ Yiping Wang a,b, Na Chen a, Binfeng Yun a, Zhuyuan Wang a, Changgui Lu a, Yiping Cui a, a b
Advanced Photonics Center, Southeast University, Nanjing 210096, China School of Physics Science and Technology, Nanjing Normal University, Nanjing 210097, China
a r t i c l e in fo
abstract
Article history: Received 28 August 2006 Received in revised form 28 November 2007 Accepted 7 March 2008 Available online 15 May 2008
Many theoretical and experimental studies have been developed to characterize the spectral response of an optical fiber Bragg grating (FBG) in axial strain fields in recent years. However, comparatively few works were devoted to the evolution of the spectrum when a FBG is subjected to non-uniform transverse load. In this paper, the effects of distributed birefringence on FBG under non-uniform transverse load are analyzed and a numerical simulation based on the piecewise-uniform approach is also discussed to simulate the responses of FBG under some typical non-uniform transverse strain fields. Experiment was carried out using different loads applied at different locations of the FBG. Good agreements between experimental results and numerical simulations have been obtained. & 2008 Elsevier Ltd. All rights reserved.
Keywords: Distributed birefringence Fiber bragg grating Non-uniform transverse load
1. Introduction Fiber Bragg gratings (FBGs) have been intensively studied as optical sensors for many years owing to their fiber-based, wavelength-encoded characters [1,2]. Since the wavelength is the parameter modulated by the measurand, any changes in fiber properties, such as strain or temperature will change the Bragg wavelength. Most current FBG sensors are used to measure axial strain. For simultaneously measuring temperature and multidimensional strain components, FBG sensors fabricated in Hi–bi fiber [3] have been proposed in recent years. The goal of multiparameter sensing with FBG written in Hi–bi fiber has been addressed by several researchers. For example Urbanczyk et al. [4] demonstrated a sensor to simultaneously determine temperature and strain sensitivities using a two-mode Bragg grating imprinted in a bow-tie fiber. Guanghui Chen et al. [5] described a sensor to simultaneously measure strain and temperature using FBG written in novel Hi–bi optical fiber. Abe et al. [6] developed a sensor to simultaneously measure three parameters using superimposed Bragg gratings in Hi–bi fiber. In general, for the FBG fabricated in low birefringence fiber, the spectral response characteristics by virtue of axial strain have been well investigated. However, the past studies have shown that
$ Jiangsu Province Natural Science Foundation of China under project BK2004207. The national Science Fund of China for Distinguished Young Scholars under project 60125513. Corresponding author. Tel.: +1186 25 83601769x11; fax: +1186 25 83601769x17. E-mail address: [email protected] (Y. Cui).
0030-3992/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2008.03.006
the characteristics of a FBG under transverse load are quite different from axial load [7–11]. In the case of transverse load, the stress distribution along the two perpendicular directions across the cross-section of the fiber is different, which causes that the propagation constants of the two polarization modes xHE11 and yHE11 to be different. This called ‘‘stress induced birefringence effects’’ will cause the unique Bragg grating condition to break down and even produce two distinct Bragg wavelengths leading to spectrum of FBG broadening and peak splitting eventually. However, in some practical applications, FBG is under nonuniform transverse force, especially for FBG embedded in composite materials. The existence of transverse strain gradients will cause the distributed birefringence on FBG and the spectrum of FBG will become more complex. So far, some work has been done on the non-uniform axial strain fields [12–14], but the response of FBG in non-uniform transverse strain fields needs additional investigation. This paper presents a theoretical model and analyzes the response of FBG in non-uniform transverse strain fields. It also deals with the simulation of the response of FBG reflective spectrums under some typical non-uniform transverse strain fields. The experiment has been carried out and the results are compared to the theoretical predictions of the grating response.
2. Theoretical model A FBG is a periodic perturbation of refractive index along the fiber length and acts to couple a forward propagating fiber core mode to a backward propagating core mode at a particular
ARTICLE IN PRESS 1038
Y. Wang et al. / Optics & Laser Technology 40 (2008) 1037–1040
wavelength given by lB ¼ 2nL
(1)
where lB, L and n are the reflected Bragg wavelength, effective refractive index, and grating period, respectively. Once transverse load is applied to the FBG, the difference between the effective refractive indices of the two orthogonal modes of the fiber will be produced. The reflection spectrum of a FBG will even consist of two Bragg peaks when the transverse load is large enough. For simplicity of expression, we assume the principal optical axis to be aligned with x, y, and z Cartesian axes and x, y denote the two orthogonal axes directions, respectively, z is along the fiber axial direction, as shown in Fig. 1(a). Then the wavelength response of the FBG can be expressed as [15] n2 Dlx ¼ lx z p11 x þ p12 ðz þ yÞ (2) 2 n2 p11 y þ p12 ðz þ xÞ Dly ¼ ly z 2
(3)
where p11 and p12 are the strain-optic coefficients, n is the average refractive index along the two orthogonal axes of the fiber, Dlx and Dly are the wavelength shifts of the two polarized directions. Generally, the length of FBG is much longer than the diameter of the fiber and loading situations assumed to be contained in a single plane is reasonable. Purely diametric compression load on a glass cylinder could be modeled as a line force since both optical fiber and compression platform are hard media. The plane strain elasticity solutions for stress along the central axis in a disk have been solved from the theory of elasticity [16]: sx ¼
f plb
sy ¼
3f plb
(4)
(5)
where f is the force, l is the fiber length under load, b is the radius of the optical fiber. The friction of compression platform that we used here caused the fiber to be under a loading state of plane strain (ends-fixed) and the components of strain normal to this plane are zero (ex ¼ 0). Under this assumption, Eqs. (2) and (3) can be changed by Hook’s law: n2 ð1 þ uÞ Dlx ¼ lx 0 (6) ð1 uÞðp11 sx þ p12 sy Þ uðsx þ sy Þ 2E n2 ð1 þ uÞ ð1 uÞðp11 sy þ p12 sx Þ uðsx þ sy Þ Dly ¼ ly 0 2E
(7)
Fig. 1. (a) View of the FBG subjected to a non-uniform transverse force and (b) cross-section of FBG under transverse compression.
where shear stress is ignored and E and u are, respectively, Young’s modulus and Poisson’s coefficient of the optical fiber, sx and sy are the stress components in the x, y principal directions (see Fig. 1(b)), respectively.
3. Numerical simulation The piecewise-uniform approach [17] is adopted to simulate the spectra of FBG. Assuming the applied transverse load is along the direction of y axis (see Fig. 1(a)), the presence of strain gradients makes the transverse strain non-uniform along the fiber. So the grating is divided into a series of m smaller subgratings and the strain applied on each subgrating can be treated as uniform (see Fig. 1(b)). Define Ri and Si to be the field amplitudes after traversing the section i. Then the propagation through each uniform section can be described by a matrix Ti defined such that " # " # Ri Ri1 ¼ T ðiÞ (8) Si Si1 where ^
ðiÞ T ðiÞ 11 ¼ T 22 ¼ coshðgB DzÞ i
ðiÞ T ðiÞ 12 ¼ T 21 ¼ i
s sinhðgB DzÞ, gB
k sinhðgB DzÞ gB
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^
gB ¼
k2 s2
(9)
Here d is detuning, s and k are, respectively, ‘‘dc’’ and ‘‘ac’’ coupling coefficients. s^ ¼ d þ s, Dz is the length of each subgrating and L is the length of FBG. Because ex ¼ 0 under the loading state of plane, Dz ¼ L/m. Once all the matrices for the individual sections are known, the output amplitudes is: " # " # R0 Rm ¼T (10) S0 Sm where T ¼ T(m) T(m1) y T(i) y T(1), R0 are S0 describe boundary conditions and R0 ¼ 1, S0 ¼ 0. The ‘‘dc’’ and ‘‘ac’’ coupling coefficients s and k are related to the stress-induced refractive index changes. Using Eqs. (6)–(10), we can simulate the spectral response of FBG under some typical non-uniform transverse strain fields. Assuming the period of the grating is 519 nm, n0 ¼ 1.498, the average index change is 104, and the length L of the grating is 6 mm, L is divided to 20 segments. The fiber length under load is 20 mm. Since the FBG is written in the fiber, the FBG length under load is also 6 mm. Young’s modulus E ¼ 74.52 GPa, Poisson’s coefficient u ¼ 0.17, and the strain-optic coefficients p11 and p12 are 0.113 and 0.252, respectively. Define F as the applied transverse load and f0 denote the variable in force axis (0–80 N). Fig. 2(a) shows the response of the FBG for uniform transverse load. The peak will be obviously broadened and then split with the increase of the load. The applied transverse load causes a shift in x-polarization wavelength and it is 10 times higher than y-polarization wavelength calculated from equations described above. We can observe that the FBG wavelength variation for the y-polarization is almost insensitive to the force and almost unchanged. Fig. 2(b) shows a broadening of the spectrum of FBG with the increase of the load when the transverse strain distribution is non-uniform. Once the strain gradient is too severe, besides the split of the original FBG, the bifurcation of spectrum for x-polarization is distinguishable.
ARTICLE IN PRESS Y. Wang et al. / Optics & Laser Technology 40 (2008) 1037–1040
1039
Fig. 2. The reflective spectra of FBG under different condition of transverse force, where wavelength runs from 1.554 to 1.556 um and the force F form 0 to 80 N. The fiber length under load is 20 mm. The reflective spectra for the force distribution: (a) Uniform transverse strain: F ¼ f0; (b) non-uniform transverse strain with strain gradients: zo10 mm, F ¼ f0; 10 mmo zo20 mm, F ¼ 2f0; (c) linear strain: F ¼ f0(1z/L); (d) nonlinear strain: F ¼ f0[1(z/L)2].
Fig. 3. Experimental setup: (a) optical configuration and (b) mechanical configuration.
Seen from Fig. 2(c) to (d), besides the x-polarization peak shifting to longer wavelength, the spectrum for x-polarization is obviously broadened. The spectrum shape and the degree of broadening are dependent on the transverse strain distribution. It should be noted that the piecewise-uniform approach will cause more numerical error to a certain extent in the case of (c) and (d) than in the case of Fig. 2(a) and (b) due to the rougher sampling segments. But the model discussed here employing 20 segments per 6 mm length makes the piecewise-uniform approach accurate enough.
4. Experimental investigations A simple experiment was carried out to investigate the spectral response of FBG under non-uniform transverse force, and the setup is shown in Fig. 3(a). Light from the broadband light source (BBS) was coupled to the FBG which has an unstrained Bragg wavelength of 1555.6 nm through a 3db coupler. The reflected light from the test grating was sent back to optical spectrum
analyzer (OSA, 86142B by Agilent). The compression device is shown in Fig. 3(b), where the bottom compression area is a polished aluminum plate with a width of 18 mm, while the top compression device are two pieces of copper blocks which have the same width of 9 mm. The heights of them are chosen to be different, which is convenient for applying different forces. Denote the different applied forces by f1 and f2. The FBG was written with a length of 10 mm and was placed parallel to the balance reference fiber. In the experiment, the copper blocks were carefully set on the two sides of the center of the FBG symmetrically. The force applied on the higher copper block was increased gradually from 0 to 45 N and the force applied on the lower one was increased from 0 to15 N. The experimental result was displayed and recorded with OSA, presented in Fig. 4. Under such applied non-uniform force, the spectral response of the FBG exhibited obviously birefringence effect which induced broadening and split of the peak at the Bragg reflection wavelength. Besides the birefringence effect, because of the existence of the transverse strain gradient, the bifurcation of spectrum for x-polarization was distinguishable, shown in Fig. 4. Here the gradient load could be considered as an
ARTICLE IN PRESS 1040
Y. Wang et al. / Optics & Laser Technology 40 (2008) 1037–1040
increased from 15 N to 45 N, the experimental result under this strain gradient exhibited obviously split of the Bragg wavelength and distinguishable bifurcation of spectrum for x-polarization. References
Fig. 4. Experimental results of FBG under non-uniform transverse strain.
addition of two uniform transverse load of 15 N and load of 45 N which distributed from the center to the different ends of the FBG. The calculated sensitivity of the reflection Bragg wavelength changes are dlx/dF ¼ 0.0076 nm/N for x axis and dly/dF ¼ 0.00062 nm/N for y axis. The wavelength changes have been measured to be dlx/dF ¼ 0.00069 nm/N and dly/dF ¼ 0.00058 nm/N derived from the features of split center wavelength for x axis and wavelength for y axis, respectively. The split space of slow axis was measured to be 0.208 nm and the theoretically prediction was 0.228 nm. This result correspond very well with the theoretical prediction. 5. Conclusion A study of spectral response of FBG under non-uniform transverse stress has been presented with theoretical model and numerical simulations. The experimental result has been carried out to verify the theoretical predictions. With the transverse force
[1] Kerdey AD, Davis MA, Patrick HJ, LeBlane M, Koo KP, Member, IEEE, et al. Fiber grating sensors. J Lightwave Technol 1997;15:1442. [2] Hill KO, Meltz G. Fiber bragg grating technology fundamentals and overview. J Lightwave Technol 1997;15:1263. ˜ O O, FERREIRA L, ARAU ´ JO F, SANTOS J. Applications of fibre optic grating [3] FRAZA technology to multi-parameter measurement. Fiber Integrated Opt 2005;24(3–4):227. [4] Urbanczyk, Waclaw, Chmielewska, Ewa, Bock, Wojtek J. Measurements of temperature and strain sensitivities of a two-mode Bragg grating imprinted in a bow-tie fibre. Meas Sci Technol 2001;12(7):800. [5] Guanghui Chen, Liying Liu, Hongzhi Jia, Jimin Yu, Lei Xu, Wencheng Wang. Simultaneous strain and temperature measurements with fiber Bragg grating written in novel Hi-Bi optical fiber. IEEE Photonics Technol Lett 2005;16(1):221. [6] Abe Ilda, Kalinowski, Hypolito J, Fraza˜o Orlando, Santos Jose´ L, Nogueira Roge´rio N, et al. Superimposed bragg gratings in high-birefringence fibre optics: simultaneous measurements of three parameters. Meas Sci Technol 2004;15(8):1453. [7] Wagreich RB, Atia WA, Singh H, Sirkis JS. Effects of diametric load on fiber bragg gratings fabricated in low birefringent fiber. Electron Lett 1996;32:1223. [8] Gafsi R, Sherif MAE. Analysis of induced-birefringence effects on fiber bragg gratings. Opt Fiber Technol 2000;6:299. [9] Lawrence CM, Nelson DV. Measurement of transverse strains with fiber gratings. SPIE 1997;3042:218. [10] Abe Ilda, Fraza˜o Orlando, Schiller Marcelo, Nogueira W, Roge´rio N, Kalinowski, et al. Bragg gratings in normal and reduced diameter high birefringence fiber optics. Meas Sci Technol 2006;17(6):1477. [11] Abe I, Kalinowski HJ, Nogueira R, Pinto J, L Frazao O. Production and characterization of bragg gratings written in high birefringence fibre optics. IEE Proc Circuits Devices Syst 2003;150:495. [12] Peters K, Pattis P, Botsis J, Giaccari P. Experimental verification of response of embedded optical fiber Bragg grating sensors in non-homogeneous strain fields. Opt Lasers Eng 2000;33:107. [13] Libo Y, Ansari F. White light interferometric fiber optic distribution strain sensing system. Sensors Actuators (A) 1997;63:177. [14] Huang S, Ohn M, LeBlanc M, Measures R. Continuous arbitrary strain profile measurements with fiber bragg gratings. Smart Mater Struct 1998; 7:248. [15] Sirkis JS. Unified approach to phase–strain–temperature models for smart structure interferometric optical fiber sensors: part 1, development. Opt Eng 1993;32:752. [16] Eisenberg MA. Introduction to the mechanics of solids. Addison-Wesley, Massachusetts. [17] Erdoga T. Fiber grating spectra. J lightwave Technol 1997;15:1277.
Contents lists available at ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Effects of distributed birefringence on fiber Bragg grating under non-uniform transverse load$ Yiping Wang a,b, Na Chen a, Binfeng Yun a, Zhuyuan Wang a, Changgui Lu a, Yiping Cui a, a b
Advanced Photonics Center, Southeast University, Nanjing 210096, China School of Physics Science and Technology, Nanjing Normal University, Nanjing 210097, China
a r t i c l e in fo
abstract
Article history: Received 28 August 2006 Received in revised form 28 November 2007 Accepted 7 March 2008 Available online 15 May 2008
Many theoretical and experimental studies have been developed to characterize the spectral response of an optical fiber Bragg grating (FBG) in axial strain fields in recent years. However, comparatively few works were devoted to the evolution of the spectrum when a FBG is subjected to non-uniform transverse load. In this paper, the effects of distributed birefringence on FBG under non-uniform transverse load are analyzed and a numerical simulation based on the piecewise-uniform approach is also discussed to simulate the responses of FBG under some typical non-uniform transverse strain fields. Experiment was carried out using different loads applied at different locations of the FBG. Good agreements between experimental results and numerical simulations have been obtained. & 2008 Elsevier Ltd. All rights reserved.
Keywords: Distributed birefringence Fiber bragg grating Non-uniform transverse load
1. Introduction Fiber Bragg gratings (FBGs) have been intensively studied as optical sensors for many years owing to their fiber-based, wavelength-encoded characters [1,2]. Since the wavelength is the parameter modulated by the measurand, any changes in fiber properties, such as strain or temperature will change the Bragg wavelength. Most current FBG sensors are used to measure axial strain. For simultaneously measuring temperature and multidimensional strain components, FBG sensors fabricated in Hi–bi fiber [3] have been proposed in recent years. The goal of multiparameter sensing with FBG written in Hi–bi fiber has been addressed by several researchers. For example Urbanczyk et al. [4] demonstrated a sensor to simultaneously determine temperature and strain sensitivities using a two-mode Bragg grating imprinted in a bow-tie fiber. Guanghui Chen et al. [5] described a sensor to simultaneously measure strain and temperature using FBG written in novel Hi–bi optical fiber. Abe et al. [6] developed a sensor to simultaneously measure three parameters using superimposed Bragg gratings in Hi–bi fiber. In general, for the FBG fabricated in low birefringence fiber, the spectral response characteristics by virtue of axial strain have been well investigated. However, the past studies have shown that
$ Jiangsu Province Natural Science Foundation of China under project BK2004207. The national Science Fund of China for Distinguished Young Scholars under project 60125513. Corresponding author. Tel.: +1186 25 83601769x11; fax: +1186 25 83601769x17. E-mail address: [email protected] (Y. Cui).
0030-3992/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2008.03.006
the characteristics of a FBG under transverse load are quite different from axial load [7–11]. In the case of transverse load, the stress distribution along the two perpendicular directions across the cross-section of the fiber is different, which causes that the propagation constants of the two polarization modes xHE11 and yHE11 to be different. This called ‘‘stress induced birefringence effects’’ will cause the unique Bragg grating condition to break down and even produce two distinct Bragg wavelengths leading to spectrum of FBG broadening and peak splitting eventually. However, in some practical applications, FBG is under nonuniform transverse force, especially for FBG embedded in composite materials. The existence of transverse strain gradients will cause the distributed birefringence on FBG and the spectrum of FBG will become more complex. So far, some work has been done on the non-uniform axial strain fields [12–14], but the response of FBG in non-uniform transverse strain fields needs additional investigation. This paper presents a theoretical model and analyzes the response of FBG in non-uniform transverse strain fields. It also deals with the simulation of the response of FBG reflective spectrums under some typical non-uniform transverse strain fields. The experiment has been carried out and the results are compared to the theoretical predictions of the grating response.
2. Theoretical model A FBG is a periodic perturbation of refractive index along the fiber length and acts to couple a forward propagating fiber core mode to a backward propagating core mode at a particular
ARTICLE IN PRESS 1038
Y. Wang et al. / Optics & Laser Technology 40 (2008) 1037–1040
wavelength given by lB ¼ 2nL
(1)
where lB, L and n are the reflected Bragg wavelength, effective refractive index, and grating period, respectively. Once transverse load is applied to the FBG, the difference between the effective refractive indices of the two orthogonal modes of the fiber will be produced. The reflection spectrum of a FBG will even consist of two Bragg peaks when the transverse load is large enough. For simplicity of expression, we assume the principal optical axis to be aligned with x, y, and z Cartesian axes and x, y denote the two orthogonal axes directions, respectively, z is along the fiber axial direction, as shown in Fig. 1(a). Then the wavelength response of the FBG can be expressed as [15] n2 Dlx ¼ lx z p11 x þ p12 ðz þ yÞ (2) 2 n2 p11 y þ p12 ðz þ xÞ Dly ¼ ly z 2
(3)
where p11 and p12 are the strain-optic coefficients, n is the average refractive index along the two orthogonal axes of the fiber, Dlx and Dly are the wavelength shifts of the two polarized directions. Generally, the length of FBG is much longer than the diameter of the fiber and loading situations assumed to be contained in a single plane is reasonable. Purely diametric compression load on a glass cylinder could be modeled as a line force since both optical fiber and compression platform are hard media. The plane strain elasticity solutions for stress along the central axis in a disk have been solved from the theory of elasticity [16]: sx ¼
f plb
sy ¼
3f plb
(4)
(5)
where f is the force, l is the fiber length under load, b is the radius of the optical fiber. The friction of compression platform that we used here caused the fiber to be under a loading state of plane strain (ends-fixed) and the components of strain normal to this plane are zero (ex ¼ 0). Under this assumption, Eqs. (2) and (3) can be changed by Hook’s law: n2 ð1 þ uÞ Dlx ¼ lx 0 (6) ð1 uÞðp11 sx þ p12 sy Þ uðsx þ sy Þ 2E n2 ð1 þ uÞ ð1 uÞðp11 sy þ p12 sx Þ uðsx þ sy Þ Dly ¼ ly 0 2E
(7)
Fig. 1. (a) View of the FBG subjected to a non-uniform transverse force and (b) cross-section of FBG under transverse compression.
where shear stress is ignored and E and u are, respectively, Young’s modulus and Poisson’s coefficient of the optical fiber, sx and sy are the stress components in the x, y principal directions (see Fig. 1(b)), respectively.
3. Numerical simulation The piecewise-uniform approach [17] is adopted to simulate the spectra of FBG. Assuming the applied transverse load is along the direction of y axis (see Fig. 1(a)), the presence of strain gradients makes the transverse strain non-uniform along the fiber. So the grating is divided into a series of m smaller subgratings and the strain applied on each subgrating can be treated as uniform (see Fig. 1(b)). Define Ri and Si to be the field amplitudes after traversing the section i. Then the propagation through each uniform section can be described by a matrix Ti defined such that " # " # Ri Ri1 ¼ T ðiÞ (8) Si Si1 where ^
ðiÞ T ðiÞ 11 ¼ T 22 ¼ coshðgB DzÞ i
ðiÞ T ðiÞ 12 ¼ T 21 ¼ i
s sinhðgB DzÞ, gB
k sinhðgB DzÞ gB
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^
gB ¼
k2 s2
(9)
Here d is detuning, s and k are, respectively, ‘‘dc’’ and ‘‘ac’’ coupling coefficients. s^ ¼ d þ s, Dz is the length of each subgrating and L is the length of FBG. Because ex ¼ 0 under the loading state of plane, Dz ¼ L/m. Once all the matrices for the individual sections are known, the output amplitudes is: " # " # R0 Rm ¼T (10) S0 Sm where T ¼ T(m) T(m1) y T(i) y T(1), R0 are S0 describe boundary conditions and R0 ¼ 1, S0 ¼ 0. The ‘‘dc’’ and ‘‘ac’’ coupling coefficients s and k are related to the stress-induced refractive index changes. Using Eqs. (6)–(10), we can simulate the spectral response of FBG under some typical non-uniform transverse strain fields. Assuming the period of the grating is 519 nm, n0 ¼ 1.498, the average index change is 104, and the length L of the grating is 6 mm, L is divided to 20 segments. The fiber length under load is 20 mm. Since the FBG is written in the fiber, the FBG length under load is also 6 mm. Young’s modulus E ¼ 74.52 GPa, Poisson’s coefficient u ¼ 0.17, and the strain-optic coefficients p11 and p12 are 0.113 and 0.252, respectively. Define F as the applied transverse load and f0 denote the variable in force axis (0–80 N). Fig. 2(a) shows the response of the FBG for uniform transverse load. The peak will be obviously broadened and then split with the increase of the load. The applied transverse load causes a shift in x-polarization wavelength and it is 10 times higher than y-polarization wavelength calculated from equations described above. We can observe that the FBG wavelength variation for the y-polarization is almost insensitive to the force and almost unchanged. Fig. 2(b) shows a broadening of the spectrum of FBG with the increase of the load when the transverse strain distribution is non-uniform. Once the strain gradient is too severe, besides the split of the original FBG, the bifurcation of spectrum for x-polarization is distinguishable.
ARTICLE IN PRESS Y. Wang et al. / Optics & Laser Technology 40 (2008) 1037–1040
1039
Fig. 2. The reflective spectra of FBG under different condition of transverse force, where wavelength runs from 1.554 to 1.556 um and the force F form 0 to 80 N. The fiber length under load is 20 mm. The reflective spectra for the force distribution: (a) Uniform transverse strain: F ¼ f0; (b) non-uniform transverse strain with strain gradients: zo10 mm, F ¼ f0; 10 mmo zo20 mm, F ¼ 2f0; (c) linear strain: F ¼ f0(1z/L); (d) nonlinear strain: F ¼ f0[1(z/L)2].
Fig. 3. Experimental setup: (a) optical configuration and (b) mechanical configuration.
Seen from Fig. 2(c) to (d), besides the x-polarization peak shifting to longer wavelength, the spectrum for x-polarization is obviously broadened. The spectrum shape and the degree of broadening are dependent on the transverse strain distribution. It should be noted that the piecewise-uniform approach will cause more numerical error to a certain extent in the case of (c) and (d) than in the case of Fig. 2(a) and (b) due to the rougher sampling segments. But the model discussed here employing 20 segments per 6 mm length makes the piecewise-uniform approach accurate enough.
4. Experimental investigations A simple experiment was carried out to investigate the spectral response of FBG under non-uniform transverse force, and the setup is shown in Fig. 3(a). Light from the broadband light source (BBS) was coupled to the FBG which has an unstrained Bragg wavelength of 1555.6 nm through a 3db coupler. The reflected light from the test grating was sent back to optical spectrum
analyzer (OSA, 86142B by Agilent). The compression device is shown in Fig. 3(b), where the bottom compression area is a polished aluminum plate with a width of 18 mm, while the top compression device are two pieces of copper blocks which have the same width of 9 mm. The heights of them are chosen to be different, which is convenient for applying different forces. Denote the different applied forces by f1 and f2. The FBG was written with a length of 10 mm and was placed parallel to the balance reference fiber. In the experiment, the copper blocks were carefully set on the two sides of the center of the FBG symmetrically. The force applied on the higher copper block was increased gradually from 0 to 45 N and the force applied on the lower one was increased from 0 to15 N. The experimental result was displayed and recorded with OSA, presented in Fig. 4. Under such applied non-uniform force, the spectral response of the FBG exhibited obviously birefringence effect which induced broadening and split of the peak at the Bragg reflection wavelength. Besides the birefringence effect, because of the existence of the transverse strain gradient, the bifurcation of spectrum for x-polarization was distinguishable, shown in Fig. 4. Here the gradient load could be considered as an
ARTICLE IN PRESS 1040
Y. Wang et al. / Optics & Laser Technology 40 (2008) 1037–1040
increased from 15 N to 45 N, the experimental result under this strain gradient exhibited obviously split of the Bragg wavelength and distinguishable bifurcation of spectrum for x-polarization. References
Fig. 4. Experimental results of FBG under non-uniform transverse strain.
addition of two uniform transverse load of 15 N and load of 45 N which distributed from the center to the different ends of the FBG. The calculated sensitivity of the reflection Bragg wavelength changes are dlx/dF ¼ 0.0076 nm/N for x axis and dly/dF ¼ 0.00062 nm/N for y axis. The wavelength changes have been measured to be dlx/dF ¼ 0.00069 nm/N and dly/dF ¼ 0.00058 nm/N derived from the features of split center wavelength for x axis and wavelength for y axis, respectively. The split space of slow axis was measured to be 0.208 nm and the theoretically prediction was 0.228 nm. This result correspond very well with the theoretical prediction. 5. Conclusion A study of spectral response of FBG under non-uniform transverse stress has been presented with theoretical model and numerical simulations. The experimental result has been carried out to verify the theoretical predictions. With the transverse force
[1] Kerdey AD, Davis MA, Patrick HJ, LeBlane M, Koo KP, Member, IEEE, et al. Fiber grating sensors. J Lightwave Technol 1997;15:1442. [2] Hill KO, Meltz G. Fiber bragg grating technology fundamentals and overview. J Lightwave Technol 1997;15:1263. ˜ O O, FERREIRA L, ARAU ´ JO F, SANTOS J. Applications of fibre optic grating [3] FRAZA technology to multi-parameter measurement. Fiber Integrated Opt 2005;24(3–4):227. [4] Urbanczyk, Waclaw, Chmielewska, Ewa, Bock, Wojtek J. Measurements of temperature and strain sensitivities of a two-mode Bragg grating imprinted in a bow-tie fibre. Meas Sci Technol 2001;12(7):800. [5] Guanghui Chen, Liying Liu, Hongzhi Jia, Jimin Yu, Lei Xu, Wencheng Wang. Simultaneous strain and temperature measurements with fiber Bragg grating written in novel Hi-Bi optical fiber. IEEE Photonics Technol Lett 2005;16(1):221. [6] Abe Ilda, Kalinowski, Hypolito J, Fraza˜o Orlando, Santos Jose´ L, Nogueira Roge´rio N, et al. Superimposed bragg gratings in high-birefringence fibre optics: simultaneous measurements of three parameters. Meas Sci Technol 2004;15(8):1453. [7] Wagreich RB, Atia WA, Singh H, Sirkis JS. Effects of diametric load on fiber bragg gratings fabricated in low birefringent fiber. Electron Lett 1996;32:1223. [8] Gafsi R, Sherif MAE. Analysis of induced-birefringence effects on fiber bragg gratings. Opt Fiber Technol 2000;6:299. [9] Lawrence CM, Nelson DV. Measurement of transverse strains with fiber gratings. SPIE 1997;3042:218. [10] Abe Ilda, Fraza˜o Orlando, Schiller Marcelo, Nogueira W, Roge´rio N, Kalinowski, et al. Bragg gratings in normal and reduced diameter high birefringence fiber optics. Meas Sci Technol 2006;17(6):1477. [11] Abe I, Kalinowski HJ, Nogueira R, Pinto J, L Frazao O. Production and characterization of bragg gratings written in high birefringence fibre optics. IEE Proc Circuits Devices Syst 2003;150:495. [12] Peters K, Pattis P, Botsis J, Giaccari P. Experimental verification of response of embedded optical fiber Bragg grating sensors in non-homogeneous strain fields. Opt Lasers Eng 2000;33:107. [13] Libo Y, Ansari F. White light interferometric fiber optic distribution strain sensing system. Sensors Actuators (A) 1997;63:177. [14] Huang S, Ohn M, LeBlanc M, Measures R. Continuous arbitrary strain profile measurements with fiber bragg gratings. Smart Mater Struct 1998; 7:248. [15] Sirkis JS. Unified approach to phase–strain–temperature models for smart structure interferometric optical fiber sensors: part 1, development. Opt Eng 1993;32:752. [16] Eisenberg MA. Introduction to the mechanics of solids. Addison-Wesley, Massachusetts. [17] Erdoga T. Fiber grating spectra. J lightwave Technol 1997;15:1277.