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Finite element model for embedded fiber Bragg grating sensor
View the table of contents for this issue, or go to the journal homepage for more
2006 Smart Mater. Struct. 15 550
(http://iopscience.iop.org/0964-1726/15/2/038)
Home Search Collections Journals About Contact us My IOPscience
Received 29 March 2005, in final form 18 January 2006Published 23 February 2006Online at stacks.iop.org/SMS/15/550
AbstractThis paper presents an integrated formulation for the calculation of thespectral response of a fiber Bragg grating sensor embedded in a host materialsystem, as a function of the loading applied to the host structure. Inparticular, the calculation of the transverse strain sensitivity of a fiber Bragggrating sensor through the calculation of the change in effective index(or indices) of refraction of the fiber cross-section due to the applied load ispresented in detail. For the calculation of the fiber propagation constants, atwo-step finite element formulation is used incorporating the optical,geometric and material properties of the cross-section. Once the propagationconstants and principal optical axes are known along the fiber, a modifiedtransfer matrix method is applied to calculate the spectral response of theFBG. It is shown that the FE formulation yields close agreement withprevious methods for benchmark diametrical compression cases. However,the current method provides the potential to evaluate the effects of high straingradients across the optical fiber core present in some loading applications.
1. Introduction
Optical fiber Bragg grating (FBG) sensors have been widelyembedded in composite material systems for the measurementof curing stresses, interlaminar stresses, delamination, crackgrowth, and other phenomena [1]. Their unique ability to beembedded within fiber-reinforced composites with a minimumperturbation to the surrounding host material makes themattractive for the above applications. One particular field ofcurrent interest is long-term health monitoring of compositestructures, for example aircraft or FRP reinforced concretestructures. Therefore, a clear understanding of the spectralresponse of the FBG as a function of the behavior of thestructure is required.
A variety of models have been developed for thestress transfer in isotropic materials embedded with opticalfibers [2–4]. The goal of such models is to calculate thestrain in the optical fiber, and hence the sensor response,due to loading applied to the host material system. Kollarand Van Steenkiste developed a model of the strain transferbetween a laminated composite and an optical fiber embedded
1 Author to whom any correspondence should be addressed.
between two laminae [5]. The optical fiber is treated as anelliptical inclusion in the composite laminate cross-sectionand the average fiber strain due to the surrounding strain inthe laminate is calculated analytically. Later, Prabhugoudand Peters modeled a unidirectional composite by applying acombination of the finite element method and optimal shear-lag theory [6]. While applying varying levels of detail tothe host structure, each of these models considers the opticalfiber as an isotropic or orthotropic, homogeneous fiber withconstant strain across the cross-section of the fiber. Althoughthis assumption produces excellent results for the mechanicalresponse of embedded fibers [7], it is not evident that the sameis true for the optical response of the FBG due to the opticalnon-homogeneities of the optical fiber.
Once the strain field applied to the optical fiber is known,the shift in Bragg wavelength of the FBG, �λB, is calculatedas [8]
�λB = �(2�neff) = 2��neff + 2neff�� (1)
where� is the period of the grating and�neff is the change ineffective refractive index of the fundamental core mode due tothe applied strain. For example, in the case of an unconstrained
Figure 1. Typical optical fiber cross-sections: (a) circular core,step-index (no birefringence), (b) elliptical core, step-index(geometrical birefringence), (c) bow-tie fiber with pre-stressedregions (geometrical and residual stress birefringence).
fiber subjected to an axial load,
�λB = λBεz
[1 − n2
eff
2{p11 − ν(p11 + p12)}
](2)
where εz is the axial strain in the fiber, p11 and p12 arephotoelastic constants for silica, and ν is the Poisson’s ratioof the fiber [9].
Although the term �� in (1) can be directly relatedto the resulting axial strain in the FBG, the term �neff ismore difficult to calculate. This change is due both togeometrical changes in the cross-section and the strain-opticeffect. In addition, induced birefringence in the fiber separatesthe single Bragg reflection peak into two peaks [10]. Thisbirefringence is highly dependent on the index of refractionand material distribution throughout the fiber cross-section.Examples of typical optical fiber cross-sections are shown infigure 1. The circular and elliptical step-index cross-sectionsshown in figures 1(a) and (b) are mechanically isotropicwith an elastic modulus Ecore = Eclad and Poisson’s ratioνcore = νclad. Optically, they are non-homogeneous withtwo index of refraction regions: ncore in the core and nclad inthe cladding. Whereas the circular fiber propagates light atone propagation constant, β1, per mode, the elliptical fiberpropagates identical modes along two orthogonal axes withtwo different propagation constants, β1 and β2. The thirdoptical fiber, referred to as the polarization maintaining (PM)fiber, shown in figure 1(c), includes regions of a separatematerial (and is therefore both mechanically and opticallynon-homogeneous) called the stress applying part (SAP). Thepurpose of these regions is to provide stress to the fiber coreas the fiber is drawn, inducing residual birefringence due tothe thermal expansion mismatch of the silica and SAP. Severalother configurations for the PM fiber exist other than the bow-tie form shown in figure 1(c).
To calculate �neff due to an applied strain field, Kimet al considered the fibers shown in figure 1 to be opticallyand mechanically homogeneous with isotropic or transverselyisotropic material properties [11]. Based on the model of aninclusion in a composite laminate, the remote strains wereanalytically linked to the principal strains in the fiber core and�β in the direction of propagation calculated. As in latermodels, the assumption is made that most of the energy ofthe fundamental mode propagating in the fiber is containedin the core, therefore the principal strains at the center of thefiber are sufficient to estimate �neff. In addition, none of theabove models account for the birefringence due to the change
in the geometry of the fiber, although this effect is considerablysmaller than the strain-optic effect. Sirkis related the change inBragg wavelengths, �λB,1 and �λB,2, to the principal strainsat the center of the core, ε1, ε2, and ε3, as [12]
�λB,1
λB,1= ε1 − n2
o
2(p11ε2 + p12ε3 + p12ε1)
�λB,2
λB,2= ε1 − n2
o
2(p12ε2 + p11ε3 + p12ε1)
(3)
where ε2 and ε3 are in the plane of the fiber cross-section,ε1 is in the axial direction and no is the effective refractiveindex of the fundamental mode before strain is applied.This formulation will be referred to as the center strainapproximation in this paper. Wagreich et al demonstratedthe linear dependence of �λB,1 and �λB,2 on the appliedload for a circular core fiber (figure 1(a)) under diametricalcompression [10]. Lawrence et al [13] modeled the mechanicalnon-homogeneities in a PM fiber with elliptical SAP using afinite element analysis to calculate the strains at the center ofthe fiber due to applied transverse loading. Their main goal wasto calculate the calibration matrix for the transverse sensitivityof an FBG sensor in a PM fiber. Later Bosia et al [14]also modeled the mechanical non-homogeneities in a bowtietype PM fiber using finite element analysis to calculate theprincipal strains at the center of the core and hence the shift inBragg wavelength due to applied transverse loading using (3).Both the experimental and numerical studies of [13] and [14]demonstrated that for a PM fiber the shift in Bragg wavelengthis nonlinear with transverse load for certain loading angles.Gafsi and El-Sherif expanded the center strain formulationto include variations of refractive indices along the axis ofthe fiber by introducing (3) into the coupled mode equationsdescribing the spectral response of the FBG [15]. However, inall the above methods, there is still a discrepancy between theexperimentally measured and theoretical sensitivity to appliedstrain.
The goal of this paper is to derive a finite element (FE)formulation to predict the optical response of an embeddedFBG sensor as a function of the loading applied to the hoststructure. The formulation incorporates both the mechanicaland optical non-homogeneities of the optical fiber. Firstly, theFE formulation calculates the change in index of refractiondistribution throughout the cross-section of the fiber due tothe resulting mechanical stresses. From the updated indexof refraction distribution, the propagation constants of thefundamental modes, as well as the propagation axes, areobtained. Previous work by Huang has approached a similarproblem for planar waveguides analytically [16]. As can beobserved from [16], analytical solutions are only obtainablefor a few loading conditions. In the current formulation, thepropagation constants are then introduced into a discretizedversion of the coupled mode equations to determine thespectral response of the FBG.
The FE model can be applied to FBGs embedded ina variety of host material systems for which extensive FEmodeling has already been performed. Examples include fiberreinforced composites and concrete structures. The currentmodel also allows one to accurately calculate the sensitivity ofthe FBG to transverse strains and is applicable to various fiber
551
M Prabhugoud and K Peters
types, including bowtie and panda PM fibers. The formulationalso includes the effect of rotating polarization axes due tosignificant strain amplitudes, especially when the fiber isembedded near a stress concentration or failure location.
2. Finite element formulation
The propagation of a given guided mode through an opticalfiber can be characterized through the mode distribution in thecross-section of the optical fiber and the propagation constant,β , for a given frequency. The mode propagation constantis related to the effective index of refraction, neff, for theparticular mode through
β = 2π
λneff (4)
where λ is the propagating wavelength [17]. Exact solutionsfor the propagation characteristics of optical fibers, obtainedby solving wave equations, are limited to relatively simplegeometries (e.g., circular or elliptical cross-sections) with anaxisymmetric index of refraction distribution in the core. Thus,to calculate the propagation characteristics of an optical fiberwith an arbitrary cross-sectional shape or arbitrary variationof refractive index in the core, cladding, and SAP, one needsto adopt a numerical method such as the finite elementmethod [18]. Current finite element methods for optical fiberwaveguides can be classified into vector methods and scalarmethods.
Vector finite element methods are applicable to all valuesof refractive index difference between the core and thecladding. The main disadvantages of these methods are thelarge computational effort required and the appearance ofspurious modes in the solution. The spurious modes can beeliminated using a penalty approach [18]. Different variationsof the vector formulation are based on the components ofthe electric field, �E , or magnetic field, �H , considered. Forexample, in the formulation of Yeh et al, the axial componentsof �E and �H field are considered [19]. All other componentsare then expressed in terms of these axial components usingMaxwell’s equations. A minimizing functional is obtainedby applying the vector wave equation along with a continuitycondition at the core-cladding interface. In the work ofKoshiba, the minimizing functional is obtained from thecomplete �E or �H field satisfying the vector wave equation [18].Different approaches have also been proposed to addressthe open boundary problem, for example applying the FEformulation to the core and appropriate boundary conditionto the core–cladding interface. This reduces the numberof elements required in the cladding to obtain an accuratesolution [20, 21].
Scalar finite element methods, on the other hand, are onlyapplicable to weakly guiding fibers, i.e. for which the variationof the refractive index is negligible over a distance of onewavelength [17]. However, such an assumption is reasonablefor most fibers into which FBGs are written, within their elasticstrain limit. The advantages of a scalar method are that nospurious solutions appear (since only linearly polarized modesare captured) and only one component of the �E or �H fieldis considered, reducing the size of the required system ofequations to solve for the propagation constant. For this reason,in this paper we derive a sensor element based on a scalarformulation without imposing the assumption of axisymmetry.
Figure 2. Schematic of the procedure for calculation of FBG spectralresponse for a sensor embedded in a host material system.
2.1. Overview
In the current analysis, the prediction of the FBG spectralresponse is performed through the following steps (seefigure 2):
(i) The surrounding host composite material and opticalfiber sensor are meshed using a commercial FE package(e.g., ANSYS for the current work). The chosen sensormesh is shown in figure 3, where the fiber is dividedinto segments of length �z in the axial direction andeach cross-section is meshed using 2D plane stresstriangular elements. An example element is also shownin figure 3. For the purpose of later calculations, this‘3D propagation’ element is characterized by its stiffnessproperties, indices of refraction, and length, �z. Axis 1is along the propagation direction and coincides with theglobal propagation axis, z. Axes p and q are the localoptical axes of propagation.
(ii) Using the thermo-mechanical FE model, the nodaldisplacements are obtained due to the external appliedloads. From the nodal displacements, strain componentsin each element are also calculated.
(iii) For each 2D triangular element (as shown in figure 3), theindex of refraction change due to the applied strain field iscalculated in local optical axes using a linear strain-opticlaw (see section 2.2). The updated indices of refractionare then transformed from the local axes to the globalstructural axes.
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Finite element model for embedded FBG sensor
Figure 3. Discretization of optical fiber into FBG sensor elements.Also shown are definition of local polarization axes p and q , globalfiber axes, X and Y , and local principal strain axes, 1–2–3. z is thedirection of propagation along the optical fiber.
(iv) For each sensor segment, the propagation constant forthe optical fiber about the global structural axis iscalculated using the optical FE formulation including theupdated index of refraction distribution and the nodaldisplacements.
(v) The propagation constant/effective index of refraction attwo other angles with respect to the global structuralaxis is calculated. From these values the maximumand mininum propagation constants are calculated forthe cross-section as well as the global optical axescorresponding to these extrema (see appendix C).
(vi) The FBG spectral response is calculated from the localaxial strain, effective indices of refraction, and curvatureof each segment using the modified T -matrix method (seesection 2.4).
The details of some of these calculations are given below.
2.2. Calculation of indices of refraction for an element
Each element is assumed to be optically isotropic with an indexof refraction in the unstressed state of no
e . The displacementfield vector for light propagating through the element in the1 direction is given as [11]
{D} = Ap{sp} sin
[ωt − 2πn p
e
λx1
]
+ Aq{sq} sin
[ωt − 2πnq
e
λx1
](5)
where sp and sq are orthogonal unit vectors in the 2–3 planein the direction of the principal optical axes, n p
e and nqe are the
element indices of refraction about these axes, ω is the angularfrequency of the wave, and Ap and Aq are the amplitudes ofthe displacement vector components. We can write the waveequation for this displacement field vector as [11]
{s} × ({s} × [B]{D})+ 1
(ne)2{D} = 0 (6)
where ne = n pe or nq
e , {s} is the unit vector in the propagationdirection, and [B] is the material dielectric impermeability
tensor,
[B] =[ B1 B6 B5
B6 B2 B4
B5 B4 B3
]. (7)
Writing {D} in terms of its components {D} = (0, D2, D3),{s} = (1, 0, 0), and evaluating (6) yields the matrix equation,
[B2 − 1/n2
e B4
B4 B3 − 1/n2e
]{D2
D3
}= 0. (8)
The non-trivial solutions to (8), n pe and nq
e , are
1
(n p,qe )2
=(B2 + B3)±
√(B2 − B3)2 + 4B2
4
2. (9)
These solutions correspond to the indices of refraction aboutthe principal optical axes p and q in the 2–3 plane shownin figure 3. These axes will be determined later. For anoptically isotropic material, B1 = B2 = B3 = 1/(no
e)2,
B4 = B5 = B6 = 0. Therefore, n pe = nq
e = noe .
Once strain is applied to the element, the dielectricimpermeability tensor change is defined by the linear strain-optic equation,
�Bi =6∑
j=1
pi jε j (10)
where [p] is the strain-optic tensor and the compact notationis used for the strain components (ε1 = ε11, ε2 = ε22, ε3 =ε33, ε4 = γ23, ε5 = γ31, ε6 = γ12) [11]. Writing Bi =Bo
i + �Bi , expanding the solution of (9) and applying theisotropic properties to Bo
i , we find
1
(n p,qe )2
= 1
noe
+ �B2 +�B3
2
± 12
√(�B2 −�B3)2 + 4�B2
4 . (11)
For an optically isotopic material, the strain-optic tensor, [p],reduces to
(12)Substituting (12) into (10) into (11) yields n p
e and nqe for an
element in the stressed state.1
(n p,qe )2
= 1
(noe)
2+ p12ε1 + (p11 + p12)
2(ε2 + ε3)
± (p11 − p12)
2
√(ε2 − ε3)2 + ε2
4 . (13)
Kim et al considered the same formulation for the opticalfiber reduced to a single optically homogeneous element andderived a linearized form of (13) to calculate the sensitivityof the FBG to transverse strain [11]. To model a polarizationmaintaining fiber (such as figure 1(c)) they considered the fiberto be initially optically orthotropic. This approach producesidentical results to the center strain approximation of (3).
The angle of orientation of the element principal opticalaxes is identical to the principal strain directions [22].Although the index of refraction is not a true tensor quantity,it can be represented by an ellipse in the 2–3 plane with the
553
M Prabhugoud and K Peters
major and minor axes of length n pe and nq
e in the principal straindirections [22]. Therefore, to calculate ne about the global axesX and Y , we find
nXe = n p
e nqe√
(n pe cosψe)2 + (nq
e sinψe)2
nYe = n p
e nqe√
(n pe sinψe)2 + (nq
e cosψe)2
(14)
where ψe is the angle required to rotate the p axis to the Xaxis.
2.3. Calculation of propagation constants for a sensorsegment
Once the index of refraction for each element is known aboutboth the X and Y axes, the propagation constants βmax andβmin and the orientation of the principal optical axes for thecomplete cross-section must be calculated. The propagationcharacteristics for linearly polarized (LP) modes propagatingthrough a waveguide of arbitrary cross-section and arbitraryvariation of refractive index are determined by solving thescalar wave equation over the cross-section of the fiber, heredefined as the region� (refer to appendix A for derivation),∂
∂x
[pzx
∂
∂x(px (x, y))
]+ ∂
∂y
[pzy
∂
∂y(py (x, y))
]
+ (qk20 − β2) (x, y) = 0 (15)
where the field (x, y) and the coefficients px , pzx , py, pzy ,and q are defined in table 1 for the fundamental L P01
x andL P01
y modes. The functional for (15) is given by
F =∫ ∫
�
δ { ∂∂x
[pzx (x, y)
∂
∂x(px (x, y) (x, y))
]
+ ∂
∂y
[pzy(x, y)
∂
∂y(py(x, y) (x, y))
]
+ (q(x, y)k20 − β2) (x, y)
}dx dy. (16)
Taking the first variation of (16) and reducing, one obtains
δF =∫ ∫
�
[pzx
∂
∂x(px )
∂
∂x(δ )+ pzy
∂
∂y(py )
∂
∂y(δ )
+ (β2 − qk20)(δ )
]dx dy
−∫
�
{δ [
pzx∂
∂x(px )+ pzy
∂
∂y(py )
]}d� = 0 (17)
where � is the boundary of the region �. Since we areonly concerned with propagated modes, i.e. modes that arefully contained in the optical fiber, we apply the boundarycondition = 0 on �. Discretizing the region into triangularelements as shown in figure 3 and noting that the coefficientspx , pzx , py, pzy , and q are constant for each element, (17)reduces to∑
e
∫ ∫
�e
[pe
zx pex
∂
∂x
∂
∂x(δ )+ pe
zy pey
∂
∂y
∂
∂y(δ )
+ (β2 − qek20)(δ )
]dx dy = 0 (18)
where the superscript e refers to the values for a given element.We expand in each element,
e = [ N1 N2 N3 ]
[ 1
2
3
]= {N}T{ } (19)
Figure 4. Definition of triangular element with nodal coordinates.
Table 1. Parameters of scalar wave equation for fundamental LPmodes.
Mode px pzx py pzy q
LP01x Ex n2
x 1/n2z 1 1 n2
x
LP01y Ey 1 1 n2
y 1/n2z n2
y
where N1, N2, and N3 are element shape functions and 1, 2,and 3 are nodal values of . Substituting (19) into (18), weobtain the global matrix equation,
[K ]{ } − β2[M]{ } = 0 (20)
with
[K ] =∑
e
∫ ∫
�e
[qek2
0{N}{N}T − pezx pe
x {Nx }{Nx }T
− pezy pe
y{Ny}{Ny}T]
dx dy
[M] =∑
e
∫ ∫
�e
[{N}{N}T
]dx dy.
(21)
The required derivatives of the shape function for the triangularelement defined in figure 4 are given in appendix B. Aftercalculating the matrices [K ] and [M], (20) is solved for theeigenvalues, β , at a fixed wavelength. Knowing that theeffective refractive indices for the LP01 modes lie between therefractive index of the core and cladding, it is computationallyefficient to step the effective refractive index from the claddingvalue to the core value and solve for values of β satisfying
Det{[K ] − β2[M]} = 0. (22)
The value of β for the fundamental LP01 mode is thus thelowest value of β obtained. This procedure is repeated overthe range of wavelengths required.
The propagation constants computed correspond to thetwo orthogonal axes X and Y chosen as the optical fiber globalaxes in figure 3. However, these axes do not necessarilycorrespond to the optical propagation axes for the fiber cross-section. The propagation constant β , or similarly the effectiveindex of refraction neff via (4), varies as an ellipse with theorientation of the axes X and Y , as shown in figure 5. Theorthogonal optical propagation axes, X ′ and Y ′, correspondto the maximum and minimum values of β or neff on theglobal index ellipse, as also indicated in figure 5. θ is defined
554
Finite element model for embedded FBG sensor
Figure 5. Global index ellipse showing variation of neff for a givenmode with rotation of global fiber axes X–Y . X ′–Y ′ corresponds toglobal propagation axes of the optical fiber.
as the angle of rotation from the X–Y system to the X ′–Y ′system. In order to calculate nmax
eff and nmineff , one could rotate
the axes X and Y , perform the FE calculations for everyorientation and search for nmax
eff or nmineff . However, as this must
be performed for every segment of the FBG, a more efficientmethod is to calculate neff about three different axes (not co-linear) and solve for the three unknowns nmax
eff , nmineff , and θ . This
calculation is detailed in appendix C.
2.4. Calculation of the sensor response
Once the propagation constants and optical axes are known foreach fiber segment, the spectral response of the FBG must becalculated. The T -matrix approximation, first introduced byYamada and Sakuda, is widely used to calculate the spectralresponse of Bragg gratings with non-constant properties [23].The primary advantage of the T -matrix method is that itis computationally efficient as compared to direct numericalintegration of the coupled mode equations for the FBG. Asecond method would be to expand the scalar wave equationof (15) to include variations in the z-direction and formulatean FE solution for the full 3D problem for β(z). However,the T -matrix method has been shown both numerically andexperimentally to converge rapidly to the solution of thecoupled mode equations [24–26] and an FE solution to the3D problem would require discretization in the z-directionanyway. Thus, no further accuracy would be gained using 3Delements at the cost of considerably more computational effort.
The T -matrix approximation divides the FBG into Msmaller sections each with uniform coupling properties.Defining Ri and Si to be the field amplitudes of the forward andbackward propagating modes after traversing the i th section,the propagation through this uniform section is described bythe 2 × 2 matrix [Fi ] defined as
{Ri
Si
}= [Fi ]
{Ri−1
Si−1
}. (23)
The components of [Fi ] are given by [27],
[Fi ] =[ cosh(γB�z) − i σ
γBsinh(γB�z) −i κ
γBsinh(γB�z)
i κγB
sinh(γB�z) cosh(γB�z)+ i σγB
sinh(γB�z)
]
(24)where γB = √
κ2 − σ 2 and σ and κ are the couplingcoefficients of the grating. For the entire grating, the transfermatrix, [F], can be written
{R(−L/2)S(−L/2)
}= [F]
{R(L/2)S(L/2)
}(25)
where L is the length of the grating and [F] =[FM ][FM−1] · · · [F1]. To apply the T -matrix approach usingthe results obtained in the previous section, the matrix [Fi ]from (24) is evaluated for each segment of the grating. Itis assumed that the rate of rotation of the global opticalaxes (dθ/dz) (see figure 5) is not too large, otherwise thepolarization state would need to be considered in the transfermatrix. In order to incorporate the propagation constant of eachsegment from the finite element solution into (24), the couplingcoefficient σ is evaluated as
σ = β + 2π
λδneff − π
�(z)(26)
where δneff is the mean mode effective index of refractionvariation and β = βmax or βmin depending on the modeconsidered. For ease of calculation, it is generally assumedthat the variation of β is small over the bandwidth of theFBG; therefore, the value of β corresponding to the Braggwavelength is used. �(z) is a modified period function usedto match the coupling properties of the grating and is derivedin [24],
�(z) ≡ �0[1 + ε1(z)+ zε′1(z)]. (27)
The solution of �(z) in [24] includes terms multiplied by theeffective strain-optic coefficient pe. However, these terms takeinto account the Poisson contraction of the fiber due to an axialload and the change in index of refraction of the material withstrain. Since both of these effects are already included in theFE formulation, they are dropped from the function �(z) inthis paper. For this formulation, the value of � is calculatedfor each grating segment using the average strain ε1 and straingradient�ε1/�z between the two faces of the element shownin figure 2.
The reflectivity r(λ) of the Bragg grating can thenbe calculated as a function of wavelength by applying theboundary conditions S(L/2) = 0 and R(L/2) = 1, andcalculating the field amplitudes at z = −L/2 using (25). r(λ)is then defined as the ratio [27]
r(λ) =∣∣∣∣∣
S(−L/2)
R(−L/2)
∣∣∣∣∣2
. (28)
2.5. Additional comments
Some additional comments are required at this point to discusseffects neglected in the current FE/T -matrix formulation.Firstly, the effect of bending on the FBG, although onlypronounced once the curvature is significant, can easily beincluded in the FE formulation. Such a high curvature
555
M Prabhugoud and K Peters
Figure 6. Cross-section of circular core, step-index fiber with appliedtransverse load. Core size is exaggerated to show dimensions.
in the FBG would be expected for some damage detectionapplications. The fiber bending acts to reduce thereflectivity of the grating, particularly in regions of maximumreflectivity [28]. Prabhugoud and Peters modeled the loss ofreflectivity through the bending power loss term, �P(λ, z),as a function of the localized curvature, ρ [6]. The form of�P(λ, z) was determined experimentally in [6]. Since thisterm acts to reduce the power of all modes propagating throughthe grating in either direction, the �P can be evaluated foreach sensor segment and the optical transfer matrix reducedaccording to
Secondly, by reducing the Maxwell’s equations to the 2Dwave propagation problem of (15), the effect of the shearstrains γ12 and γ13 has been eliminated. However, it hasbeen demonstrated by several authors that unless these shearcomponents are extremely large, as in the case of a tiltedgrating, this effect is negligible (see [6]). In addition, thepresence of a coating on the fiber would reduce this effect evenfurther. Should it be desired to include this effect, however, onecan modify the coupling coefficient κ of (24) for each segmentaccording to
κ = π
λν(γ )δneff (30)
where ν(γ ) is the fringe visibility of the grating [27]. For smallshear strains, ν(γ ) � cos γ .
Finally, although neff is assumed to change throughout theoptical fiber due to the applied stress, the amplitude of theindex modulation δneff is assumed to remain constant along thegrating due to the fact that �δneff � �neff. This assumptionis consistent with previous solutions [15].
3. Numerical examples
In order to check the validity of the FE approach presentedin section 2, several benchmark cases for which the centerstrain approximation should work well were modeled. Threedifferent fiber cross-sections were considered: a circular, step-index fiber for which birefringence is due to the applied
Table 2. Parameters of circular core, step-index fiber.
loading; an elliptical step-index fiber for which birefringenceis due to the geometry of the fiber and the applied loading; anda PM fiber for which the birefringence is due to the geometryof the fiber, the residual stresses in the fiber, and the appliedloading. For each example, the loading is through diametricalcompression. We have assumed a constant period of the gratingof � = 530 nm.
3.1. Circular core, step-index fiber
The geometry of the circular core step-index fiber cross-section, along with the applied transverse load, is shown infigure 6. Table 2 lists the material and geometric parametersof the fiber for this simulation. As observed from table 2,the fiber is mechanically isotropic. This particular examplewas chosen since analytical solutions are known for boththe propagation constant at zero applied load [17] and thestrain field throughout the fiber cross-section due to diametricalcompression [29].
During the solution process, the calculated index ofrefraction for each segment (for diametrical compression, allsegments are the same), nmax
eff and nmineff , are plotted as a function
of the normalized frequency
V = 2π
λrcore
√n2
core − n2clad (31)
for a fixed value of the applied load P . The obtainedsolutions are plotted in figure 7. One observes that multiplemode solutions appear for large values of V . To check theconvergence of the FE solution to the exact solution, twodifferent mesh sizes were considered as shown in figure 7.The FE solution was found to converge to the analytical,exact solution for the fine mesh size which was used for allfurther simulations. One exception is the LP11 mode whichis not found by the fine mesh simulation. This is probablydue to the axisymmetric boundary conditions, = 0, thatwere applied on the outer edge of the fiber, since the LP11
mode is not axisymmetric. However, for FBG sensors thefiber diameter is chosen to be small such that only the firstLP01 mode propagates and contributes to the Bragg reflection;therefore, the higher modes will not even be considered. Forthe following examples, we therefore consider the case V = 2.
Similar plots to the one shown in figure 7 are thengenerated for the range of applied loads required. From thevariation of effective indices of refraction with diametricalload, we calculate the new Bragg wavelengths as
λmaxB = 2nmax
eff �
λminB = 2nmin
eff �.(32)
556
Finite element model for embedded FBG sensor
Figure 7. Variation of effective index of refraction with normalizedfrequency for circular cross-section, step-index optical fiber. Circlesrepresent exact solution. Triangles represent results of coarse meshsimulations and squares represent results of fine mesh (270 coreelements, 452 cladding elements) simulations.
Figure 8. Variation of Bragg wavelength with applied diametricalload for FBG in circular core, step-index optical fiber. Squaresrepresent result of center strain approximation (CSA). FE result isplotted as a solid line.
Figure 8 plots the obtained linear dependence of each Braggwavelength with applied transverse load for the circular, step-index fiber. The center strain approximation of (3) is alsoplotted in figure 8 for comparison. As observed, the FEsolution matches well with the solution obtained from thecenter strain approximation and correctly evaluates the inducedbirefringence due to the applied loading.
3.2. Elliptical core, step index fiber
As a second benchmark case, we consider an elliptical corefiber that has initial birefringence due to the geometricalproperties of the fiber. For this example, loading was onlyapplied along the principal axes of the fiber since off-axisloading will be considered in the next example. The geometry
Figure 9. Cross-section of a elliptical core, step-index fiber withapplied transverse load. Core size is exaggerated to showdimensions.
0 2 4 6 8 101530
1532
1534
1536
1538
1540λ
P (N/mm)
FECSA
B
Slow axis
Fast axis
(nm
)
Figure 10. Variation of Bragg wavelength with applied diametricalload for elliptical core, step-index optical fiber. Squares represent theresult of the center strain approximation (CSA). The FE result isplotted as a solid line.
of the fiber cross-section modeled with the applied transverseload directed along the major axis of the elliptical core is shownin figure 9. The fiber parameters used for this simulationare listed in table 3. Figure 10 plots the linear variationof Bragg wavelength with applied load obtained from theFE solution as well as the results using the center strainapproximation. Again the FE solution matches well with thesolution obtained from the center strain approximation. Thedual Bragg wavelengths in figure 10 at zero load are due to thegeometrical birefringence of the optical fiber.
3.3. PM fiber
For the final numerical simulation, we consider an FBGwritten into a PM fiber incorporating SAP regions as shown infigure 11. The core and cladding regions are circular, whereasthe SAP region is elliptical. This example demonstrates the
557
M Prabhugoud and K Peters
Figure 11. Cross-section of PM fiber with applied transverse load.Core size is exaggerated to show dimensions.
Table 3. Parameters of elliptical core, step-index fiber.
ability of the FE model to capture birefringence due to thefiber geometry, residual stresses due to the thermal expansioncoefficient mismatch between the SAP and the silica, and theapplied loading. The PM fiber was meshed with 272 coreelements, 640 inner cladding elements, 376 SAP elements, and490 outer cladding elements. The material and geometricalparameters of the PM fiber used for the simulations are given intable 4. Since exact optical properties of the SAP are difficult toobtain, n, p11, and p12 for the SAP were assumed to be sameas for the cladding. Simulations were performed for severalvalues of these parameters; however, no noticeable differenceswere obtained for the LP01 modes, presumably due to the lowenergy density of these modes in the SAP. Therefore, onlythe mechanical properties of the SAP are of importance whencalculating the Bragg wavelengths. To simulate the role of theSAP during solidification of the optical fiber, the geometryof figure 11 was modeled and afterwards a thermal loadingof �T = −800 ◦C was applied. The effect of the index ofrefraction change due to temperature was not included, sincethe final temperature was assumed to be room temperature, atwhich the index values of table 4 were originally measured.For completeness, the change in index of refraction due toa combined thermal loading (the thermo-optic effect) andapplied strain is given in appendix D.
Figure 12 demonstrates that the azimuthal variation ofeffective refractive index, obtained from the FE simulation, isan ellipse. Also plotted for comparison is an ellipse oriented atangle θ to the global X axis with nmax
eff and nmineff as major and
minor axis lengths. Figures 13(a) and 14(a) plot the variationof Bragg wavelengths with applied transverse load for twodifferent loading angles. One notes the nonlinear variationof Bragg wavelength with applied transverse load as also
Figure 12. Azimuthal variation of global effective refractive indexfor P = 10 N mm−1 and γ = 54◦ (squares). Ellipse oriented at θwith major and minor axes nmax
observed in [14]. For comparison, the results using the centerstrain approximation of (3) are also plotted. For this exampleof a PM fiber in diametrical compression there is no significantdifference in the calculations using the current FE formulationand the center strain approximation. Even for the case wheresignificant strain gradients exist across the fiber cross-section,the difference between the center strain approximation and FEsolution is small.
To demonstrate the importance of calculating the globaloptical axes for each segment, the global slow axis is alsoplotted versus applied load in figures 13(b) and 14(b). For thefirst case, γ = 36◦, the slow axis is originally at θ = 0◦ dueto the residual stresses. As the applied loading increases, theapplied strain overcomes the residual stresses and θ asymptotesto the loading angle γ . Similarly, for the second case, γ = 72◦,the slow axis starts at θ = 0◦ and increases towards theapplied loading angle. However, once γ = 45◦ the residualstresses are effectively balanced by the applied loading and forhigher loads the fast and the slow axes are reversed (as seen atP = 2 N mm−1 in figure 14(b)).
558
Finite element model for embedded FBG sensor
0 1 2 31535.4
1535.45
1535.5
1535.55
1535.6
1535.65
1535.7
1535.75
P (N/mm)
FECSA
Slow axis
Fast axis
λ B(n
m)
0 2 4 6 8 10
0
6
12
18
24
30
36
θ
P (N/mm)
(deg
rees
)
(a)
(b)
Figure 13. (a) Variation of Bragg wavelength with applieddiametrical load at γ = 36◦ for PM fiber. Squares represent the resultof the center strain approximation (CSA), while FE results areplotted as a solid line. (b) Variation of the orientation of global indexellipse with the applied load. The variation of θ is plotted over agreater range of P to demonstrate asymptotic behavior.
To demonstrate the complete procedure outlined insection 2.1, we also simulated the spectral response of anFBG, in the PM fiber considered above, to varying diametricalload along the length of the grating. The following uniformgrating parameters were assumed L = 8 mm, δneff = 0.5 ×10−04,� = 530 nm, and �z = 0.5 mm. Figure 15(a) plotsthe linear variation of applied diametrical load along the lengthof the grating. Two different loading angles, γ = 36◦ andγ = 72◦, were considered. Figures 15(b) and (c) plot thespectra obtained before and after loading is applied. One canobserve the split peak at zero load due to the residual strain.We have assumed 50% power coupling into each of the LP01
x
and LP01y modes. As seen in figures 15(b) and (c), at γ = 36◦
the peak splitting due to the birefringence is increased by thespectral bandwidth increase due to the load distribution alongthe grating. However, for γ = 72◦ the induced birefringence is
0 1 2 31535.4
1535.44
1535.48
1535.52
1535.56
1535.6
1535.64
P (N/mm)
FECSA
Slow axis
Fast axisλ B(n
m)
0 2 4 6 8 10-48
-36
-24
-12
0
12
24
36
48
P (N/mm)
θ(d
egre
es)
(a)
(b)
Figure 14. (a) Variation of Bragg wavelength with applieddiametrical load at γ = 72◦ for PM fiber. Squares represent the resultof the center strain approximation (CSA), while FE results areplotted as a solid line. (b) Variation of the orientation of global indexellipse with the applied load. The variation of θ is plotted over agreater range of P to demonstrate asymptotic behavior.
in the opposite sense to that of the load distribution; therefore,the two peaks appear to have coalesced.
4. Conclusions
The FE formulation for the response of an FBG due to appliedloading demonstrated and verified in this paper would allow therapid calculation of the FBG response for complicated loadingconditions. The above numerical studies on diametricalcompression benchmark cases have demonstrated the accuracyof the FE formulation. Future studies will demonstrate thecalculation of the spectral response of an FBG embedded in acomplex material system such as a fiber reinforced composite.Furthermore, the accuracy of the center strain approximationwill be evaluated for various loading conditions in order toderive guidelines for FBG sensor data analysis.
559
M Prabhugoud and K Peters
-4 -2 0 2 4
0
0.5
1
1.5
2
2.5
3
P (
N/m
m)
z (mm)1535 1535.25 1535.5 1535.75 15360
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Wavelength (nm)
r
1535 1535.25 1535.5 1535.75 15360
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Wavelength (nm)
r
(a) (b)
(c)
Figure 15. (a) Linear variation of applied diametrical load along the length of the FBG. FBG spectra obtained before (dashed line) and after(solid line) application of transverse load at (b) γ = 36◦ and (c) γ = 72◦.
Acknowledgments
The authors would like to acknowledge the National ScienceFoundation for their financial support of this work throughgrant No CMS 0219690.
Appendix A. Derivation of scalar wave equation
Maxwell’s equations for source free, time harmonic fieldsare [17]
∇ × �E = −jωµ0 �H (A.1)
∇ × �H = −jωε0[ε] �E (A.2)
∇ · �D = ε0∇ · ([εr ] �E) = 0 (A.3)
∇ · �H = 0 (A.4)
where �E(x, y, z) = �E(x, y)e−jβz and �H (x, y, z) =�H (x, y)e−jβz are the electric field and magnetic fields
respectively, ε0 and µ0 are the free space permittivity andpermeability constants, and [εr ] is the material permittivitytensor given by
[εr ] =
n2x 0 0
0 n2y 0
0 0 n2z
. (A.5)
Taking the curl of (A.1) and substituting (A.2) we obtain
∂2 Ey
∂x∂y− ∂2 Ex
∂y2− ∂2 Ex
∂z2+ ∂2 Ez
∂x∂z= k2
0n2x Ex (A.6)
∂2 Ex
∂x∂y− ∂2 Ey
∂x2− ∂2 Ey
∂z2+ ∂2 Ez
∂y∂z= k2
0n2y Ey (A.7)
∂2 Ex
∂x∂z− ∂2 Ez
∂x2− ∂2 Ez
∂y2+ ∂2 Ey
∂y∂z= k2
0n2z Ez (A.8)
560
Finite element model for embedded FBG sensor
where k0 = 2π/λ. Noting that for the LP01x mode, Ey = 0 and
using (A.3) yields
Ez = − j
βn2z
∂
∂x(n2
x Ex ). (A.9)
Substituting (A.9) into (A.6) we obtain
∂
∂x
[ 1
n2z
∂
∂x(n2
x Ex )]
+ ∂2 Ex
∂y2+ k2
0n2x Ex − β2Ex = 0. (A.10)
Similarly for the LP01y mode, Ex = 0 and
∂
∂y
[ 1
n2z
∂
∂y(n2
y Ey)]
+ ∂2 Ey
∂x2+ k2
0n2y Ey − β2 Ey = 0. (A.11)
Thus, the scalar wave equation allows us to solve for the fieldsEx and Ey independently. Although the scalar wave equationderived above is similar to the one derived by Koshiba [18],here we include the gradient of the refractive index in theformulation.
Appendix B. Shape function derivatives for thetriangular element
The shape functions for the triangular, constant index ofrefraction element shown in figure 4 are [18]
∫ ∫
�e{N}{N}Tdx dy = Ae
12
[ 2 1 11 2 11 1 2
]
∫ ∫
�e
{Nx }{Nx }Tdx dy = 1
4Ae
[ b21 b1b2 b1b3
b1b2 b22 b2b3
b1b3 b2b3 b23
]
∫ ∫
�e{Ny}{Ny}Tdx dy = 1
4Ae
[ c21 c1c2 c1c3
c1c2 c22 c2c3
c1c3 c2c3 c23
]
(B.1)
where
Ae = 12
∣∣∣∣∣1 1 1x1 x2 x3
y1 y2 y3
∣∣∣∣∣b1 = y2 − y3
b2 = y3 − y1
b3 = y1 − y2
c1 = x3 − x2
c2 = x1 − x3
c3 = x2 − x1
.
Appendix C. Calculation of global index ellipse
As verified by the results of the numerical example consideredin section 3, the variation of effective index of refraction foreach segment is an ellipse. The magnitude and orientation ofthe index ellipse is not known a priori, but must be determinedfrom the three indices calculated about the X , Y , and M (atan angle γ to the X axis) axes, as shown in figure 5. Thesethree known indices are referred to here as n0
eff, n90eff, and nγeff.
The indices to be obtained are labelled nmaxeff and nmin
eff and arethe maximum (major axis) and minimum (minor axis) effectiverefractive indices, shown in figure 5. Thus, the followingmethod is used to solve for the three unknowns (θ, nmax
eff , andnmin
eff ): we define X ′ and Y ′ to be the global propagation axesfor the cross-section for which nmax
eff and nmineff are calculated.
The index ellipse defined in the optical propagation coordinatesystem is given by
(X ′
nmaxeff
)2
+(
Y ′
nmineff
)2
= 1. (C.1)
Points A, B and C lie on the ellipse as shown in figure 5.Defining m1 = tan(θ),m2 = tan(90 − θ), and m3 = tan(|γ −θ |), the lengths OA, OB and OC are given by(
n0eff
)2 = (x ′1)
2 + (y ′1)
2 = (1 + m21)
×(
nmaxeff nmin
eff
)2
(nmin
eff
)2 +(
m1nmaxeff
)2(C.2)
(n90
eff
)2 = (x ′2)
2 + (y ′2)
2 = (1 + m22)
×(
nmaxeff nmin
eff
)2
(nmin
eff
)2 +(
m2nmaxeff
)2(C.3)
(nγeff
)2 = (x ′3)
2 + (y ′3)
2 = (1 + m23)
×(
nmaxeff nmin
eff
)2
(nmin
eff
)2 +(
m3nmaxeff
)2. (C.4)
Equations (C.2)–(C.4) are a coupled algebraic system ofequations for the three unknowns. Rewriting (C.2) for nmax
eff ,substituting into (C.3), and solving for nmin
eff , one obtains
(nmin
eff
)2 =(
n0eff
)2(n90
eff
)2[m22 − m2
1](
n0eff
)2[1 + m22] −
(n90
eff
)2[1 + m21]. (C.5)
Equation (C.5) gives the relationship between θ and nmineff .
Substituting (C.5) into (C.2) one obtains a similar relationshipbetween θ and nmax
eff ,
(nmax
eff
)2 =(
n0eff
)2(n90
eff
)2[m22 − m2
1](
n90eff
)2m2
2[1 + m21] −
(n0
eff
)2m2
1[1 + m22]. (C.6)
Thus, substituting (C.5) and (C.6) into (C.4) one obtains anequation for θ , which can be solved iteratively.
Appendix D. Refractive indices for an elementsubjected to thermal load
For FBG sensor problems including thermal loading, (9) canbe expanded to include a linear thermo-optic effect [11],
�Bi = Wi�T + pi j (ε j − α j�T ) (D.1)
where {α} are the coefficients of thermal expansion of thesensor in the local optical coordinates. For an isotropic sensor(α1 = α2 = α3 = α and α4 = α5 = α6 = 0). The coefficientsWi , defined as
Wi =(∂Bi
∂T
)
σ=const.
(D.2)
561
M Prabhugoud and K Peters
are measured during iso-stress conditions. For an opticallyisotropic material, the non-zero coefficients are thus evaluatedas
W1 = W2 = W3 = ∂
∂T
(1
(noe)
2
)= − 2
(noe)
3
(∂no
e
∂T
). (D.3)
A typical value for the thermo-optic coefficient (∂noe)/(∂T )
for silica is given by Kim et al as 1.2 × 10−5 ◦C−1 [11].Applying (D.1), the local principal indices of refraction(formerly (13)) would thus be modified to
1
(n p,qeff )
2= 1
(noe)
2+ p12ε1 + (p11 + p12)
2(ε2 + ε3)
± (p11 − p12)
2
√ε2
4 + (ε2 − ε3)2
− 2
(noe)
3
(∂no
e
∂T
)�T + (p11 + p12)α�T . (D.4)
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