2622 OPTICS LETTERS / Vol. 34, No. 17 / September 1, 2009

Analytical solution of four-mode coupling in shearstrain loaded fiber Bragg grating sensors

Mathias S. Müller,* Hala J. El-Khozondar, Thorbjörn C. Buck, and Alexander W. KochInstitute for Measurement Systems and Sensor Technology, Technische Universität München, Theresienstrasse 90/N5,

80333 Munich, Germany*Corresponding author: m.s.mueller@tum.de

Received May 28, 2009; revised July 16, 2009; accepted July 21, 2009;posted August 7, 2009 (Doc. ID 112061); published August 24, 2009

The polarization-dependant reflection spectra of fiber Bragg grating (FBG) sensors in polarization-maintaining fibers are influenced by shear strain. This influence can be evaluated from a tensorial coupled-mode theory approach. Yet, this approach requires the numerical integration of the four coupled-mode equa-tions. We present an easy to handle, completely analytical treatment of the polarization-dependent reflectionspectra of FBGs. We derive the required equations and compare the results to the numerical integration ofthe four tensorial coupled mode equations. © 2009 Optical Society of America

OCIS codes: 060.0060, 060.2370, 060.2300, 060.2420.

Fiber Bragg gratings (FBG) are currently being in-vestigated for their employment as multiparameterstrain sensors [1–5]. FBGs written into polarization-maintaining fibers (PMFs) exhibit two narrow reflec-tion peaks at the Bragg wavelengths �B,p and �B,s cor-responding to the two modes of polarization, p and s.These modes possess two different effective refractiveindices ne,p/s yielding the two differing Bragg wave-lengths from the Bragg condition �B,i=2ne,i�, where� is the grating period. Both ne,i and � are functionsof the strain applied to the fiber at the position of theFBG, represented by the strain tensor e. It has beensuggested to add a second FBG at the same positionbut at a strongly different wavelength such that fourindividual parameters can be extracted from theFBG’s position. Then the principal strains, i.e., thediagonal entries of the strain tensor and the tem-perature, can be reconstructed [6]. A requirement isthat the influence of the off-diagonal strain tensor en-tries, namely the shear strains, on the spectral re-sponse of the FBG are negligible.

However, these shear strains, explicitly the exycomponent, do influence the spectral response, as werecently pointed out [7,8]. This influence may be cal-culated by a tensorial coupled-mode analysis, consid-ering the four full guided modes of the polarization-maintaining fiber. The results show that a polari-zation cross coupling between the modes of orthogo-nal polarization is induced by the shear strain. Thiscross coupling is overlapped by the characteristicforward–backward coupling of the grating. As a re-sult, a complicated spectral response of the reflectedamplitudes of the two polarization directions can beobserved that is strongly distorted with respect to theoriginal, unloaded response. This has implications forthe extraction of the measurand, the Bragg wave-lengths �B,i, and in turn the reconstruction of the di-agonal strain tensor entries and the temperature, aswe discuss in [9].

The tensorial coupled mode analysis [7] or a trans-fer matrix method [8] is able to predict the spectralresponse of the shear strain loaded FBG. However,

this prediction can only be conducted by either nu-

0146-9592/09/172622-3/$15.00 ©

merically integrating the coupled-mode equations ornumerically performing the transfer matrix ap-proach. Yet, to gain further insight into the problemand possibly derive a solution for it, a manageableanalytical solution is required that either approxi-mates the problem well enough or ideally solves it ex-actly. In this work, we demonstrate how such a solu-tion can be constructed and how it approximates ormatches the numerical solution from the tensorialcoupled mode analysis. We therefore consider anFBG that is subjected to a strain tensor with shearstrains. This situation may arise in an applicationwhere the FBG is embedded in a host material. Atreatment of shear strains within embedded FBGshas also been given by van Steenkiste and Springer[10], yet this is not applicable to the case in question,since it does not consider the projection of the funda-mental modes of the polarization-maintaining mea-surement fiber onto the perturbed fundamentalmodes of the shear strain loaded FBG, as will beshown in the following treatment. It is this projectionthat leads to the observed polarization cross cou-pling, and therefore to the distortion of the Bragggrating spectra.

The mechanical loading of the host material willresult in a strain tensor within the FBG that may, ingeneral, take an arbitrary form. From Pockels law ofphotoelasticity [11] the change in the impermeabilitytensor B upon a load represented by a strain e withinthe fiber may be computed. In a weakly guiding opti-cal fiber only the transversal components of the fieldspossess a noteworthy amplitude [12], and thus thelongitudinal field components are neglected.

From Maxwell’s equations a vectorial wave equa-tion may be derived for the transversal field compo-nents Dt= �Dx ,Dy�T, which for homogeneous media inthe direction of propagation takes the form [13]

�Bxx − ne,i−2 Bxy

Bxy Byy − ne,i−2� · Dt = M� i · Dt = 0. �1�

The effective refractive indices of the p- and the

s-polarized mode ne,p and ne,s may be found from the

2009 Optical Society of America

September 1, 2009 / Vol. 34, No. 17 / OPTICS LETTERS 2623

requirement that Eq. (1) possesses a solution andhence the determinant of M� i vanishes. Equation (1)only has nontrivial solutions if M� i exhibits off diago-nal entries, viz. Bxy is nonzero and hence the shearstrain component exy is nonzero. Yet, for the casewhere Bxy is zero, the analytical solution of the prob-lem is known [14,15]. The effective refractive indicesare thus given by [13]

ne,p/s−2 =

�Bxx + Byy� ± ��Bxx − Byy�2 + 4Bxy2

2. �2�

We derive the displacement fields of the fundamentalmodes of the loaded FBG by solving the wave equa-tion (1). This yields for the displacement fields of thep- and s-modes of the loaded fiber

Dp = �−M1,12

M1,11,1T

, Ds = �−M2,12

M2,11,1T

, �3�

where all variables in the coordinate system of theloaded FBG are indicated by the � symbol. To obtainthe polarization, we compute the normalized electricfields Ep ,Es of the modes by

Ej = B · Dj/B · Dj. �4�

The PMF guiding the light to the FBG is assumedto have its fundamental modes polarized parallel tothe coordinate axes x and y (Fig. 1). Thus, the polar-ization of these fundamental modes can be approxi-mated by Ep= �1,0,0�T and Es= �0,1,0�T. The ampli-tudes of the four modes in the PMF are labeled A+= �Ap+,As+�T for the forward-propagating modes andA−= �Ap−,As−�T for the backward-propagating ones.The amplitudes of the modes inside the FBG are la-beled Aj accordingly.

We now know the polarization of the modes of theloaded FBG. To find the amplitudes of the fundamen-tal modes of the loaded FBG upon an arbitrary illu-mination, a projection of the PMF modes onto theFBG modes is required. This projection is found fromthe continuity requirement of the transversal electricfields, yielding

Fig. 1. FBG subject to a mechanical load �e�. The funda-mental modes of the FBG change their polarization due to

¯

the mechanical perturbation B.

A+ = �Ep,x Es,x

Ep,y Es,y� · A+ = Q� · A+. �5�

In the fundamental system of the loaded FBG, thecoupled-mode equations for polarization cross cou-pling are decoupled. We can thus treat the problemfor each mode separately, by considering theforward–backward coupling induced by the gratingstructure alone. For this problem, the analytical so-lution is known, see for example [14]. It is describedby the complex reflection coefficient �i=Ai−/Ai+ formode i, which depends on fixed parameters such asthe length of the grating and on variable parameterssuch as the grating period � and the effective refrac-tive index ne,i of the corresponding mode. If we as-sume that the fixed parameters do not change in anactual measurement situation, the parameters of theanalytical solution in the fundamental system of theloaded fiber may be described by �i=��� ,� ,ne,i�. Forthe two modes of polarization the two complex reflec-tion coefficients may be summarized in the matrix

�� = diag����,�,ne,p�,���,�,ne,s��T. �6�

From these definitions, the amplitudes of the re-flected modes in the fundamental system of theloaded FBG are given by

A− = �� · A+. �7�

Equations (5) and (7) then yield

A− = Q� · A− = Q� �� · A+ = Q� ��Q� −1 · A+ �8�

Equation (8) can be interpreted as follows: Firstthe modes of the fiber guiding the light to the FBGare projected onto the fundamental modes of theloaded FBG. In the system of fundamental modes, nopolarization coupling takes place, and the conven-tional reflection coefficients are employed to computethe amplitudes of the reflected fundamental modes.The reflected fundamental modes are then projectedback onto the fundamental modes of the fiber. Thelast step includes the polarization mode coupling andgenerates the distorted spectra.

To compare the theoretical results with the nu-merically integrated four-mode coupled-mode theorywe compute a wide range of reflection spectra withdifferent parameters of the FBG, all of which show aclose agreement with the derived theory. One par-ticular example is given in Fig. 2. The figure shows asimulation of a 3-mm-long FBG with a refractive in-dex modulation amplitude �n=1�10−4 in a PMFwith a beat length LB of 3.5 mm.

The shear strain is set to exy=0.5�10−3 and exy=1�10−3, which could arise in an application where theFBG is embedded in a laminated composite, such asin [16], with the difference in the transversal far-fieldstrains ed

� being 6400 �m/m [10] and the fibers’ prin-cipal axes being oriented at 45° to the coordinate sys-tem of the far-field strains and thus transformed fol-

lowing the derivation given in [7]. The illuminating

2624 OPTICS LETTERS / Vol. 34, No. 17 / September 1, 2009

amplitudes are chosen to be A+= �1,1�T. The coupled-mode theory solution closely matches the presentedanalytical solution, although a small deviation in thepeak amplitude may be observed.

The results for a different FBG with a beat lengthof 4.2 mm and a refractive index modulation ampli-tude of �n=0.5�10−4 are shown in Fig. 3. Addition-ally we choose the light source polarization only inthe x direction using A+= �1,0�T. The polarizationmode coupling can easily be observed in this case asthe reflected light shows an intensity in both polar-ization directions.

Figure 4 illustrates the results of a grating 7 mmin length, with incident illumination set to A+= �0.3,1�T. The beat length is set to 10.6 mm. Againboth results are in close agreement.

Summing up, we demonstrated the analyticaltreatment of shear strain loaded fiber Bragg gratingsensors. We derived the required transformation rulefrom a vectorial wave equation and demonstratedthat the results closely match those of the tensorialfour-wave coupled-mode approach we previously pre-sented [7]. This analytical treatment may allow the

Fig. 2. Results for a grating with LB=3.5 mm, exy=1�10−3, L=3 mm, and �n=1�10−4.

Fig. 3. Results for a grating with LB=4.2 mm, exy=1−3 −4

�10 , L=3 mm, and �n=0.5�10 .

derivation of algorithms capable of processing thespectral information reflected from shear strainloaded FBGs in such a way that the actual value ofthe shear strain may be extracted. This would allowfor a true reconstruction of the strain tensor at theFBG’s sensor position.

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Fig. 4. Results for a grating with LB=10.6 mm, exy=0.5�10−3, L=7 mm, and �n=0.3�10−4.

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