Parametric analysis of elasto-optic birefringentaxis alignment in eccentrically coatedpolarization-maintaining optical fiber

Grieg A. Olson and Jerry L. Metcalf

When high-birefringent fibers are used in polarization-sensitive devices, one must ensure the properalignment of the optical axes. One nondestructive alignment method involves squeezing the fiber andnoting the change in the resultant birefringent axes. We use the finite-element method to determine thechange in the principal stress direction, and, hence, the extrinsic birefringent axis orientation, as afunction of the fiber coating eccentricity. Fiber eccentricity can shift the extrinsic birefringent axis bymore than 1.5°, whch becomes important in high-performance device applications.

Key words: PM couplers, orientation, finite-element analysis, birefringence, elasto-optic, stress.

Introduction

Environmental conditions that cause an optical fiberto bend, twist, or deform excessively will introducesmall amounts of birefringence thereby affecting theoutput polarization state of the transmitted light.Polarization-maintaining (PM) optical fibers havebeen developed to be environmentally insensitive byintentionally inducing an intrinsic birefringence to alevel significantly greater than would normally becaused by any external perturbation.' The fabricationof highly birefringent PM fibers normally involvescreating a geometric asymmetry, as well as producinga large thermal stress anisotropy in the fiber corematerial.

Applications for PM fibers usually require PMcouplers fabricated from compatible fiber. To makePM couplers, one must take precautions to ensurethat the principal optical axes of the fiber are accu-rately aligned with the transverse device axes to avoidpolarization cross-coupling.2'3 Various alignment tech-niques have been considered. These techniques in-clude fabricating a nonround fiber,4 using a techniqueof looking through the side of the fiber through amicroscope and crossed polarizers,5 2 using a patternrecognition system,6 or aligning the fiber end to end.2

G. A. Olson is with the 3M Fiber Optics Laboratory, A147-2N-02,6801 River Place Boulevard, Austin, Texas 78726. J. L. Metcalf iswith 3M-CAE Applications & Development, 3M Center, 260-6A-13,St. Paul, Minnesota 55144.

Received 25 March 1991.0003-6935/92/091234-05$05.00/0.© 1992 Optical Society of America.

A technique based on the elasto-optic effect has beendemonstrated by Carrara et al.7 and is the method ofinterest here.

This elasto-optic procedure involves applying alateral force to the fiber and noting the change in theresultant birefringent axes. The new birefringentaxes within the loaded section will be at an angle fromthe unperturbed orientation that can be derived froma vectorial analysis of the intrinsic and the extrinsicbirefringence characteristics. The change in the angleof the birefringent axes is determined by measuringthe effect that this perturbation has on polarized lightpropagating within the fiber. The perturbation willcause cross-polarization coupling of polarized lightthat has been launched into one of the fiber's polariza-tion eigenmodes; if the angle of the birefringent axesis not perturbed, the cross-polarization coupling willbe absent. If the axis of the birefringence componentproduced by the externally applied stress (the extrin-sic birefringence) is parallel to the intrinsic birefrin-gent axis, the net birefringent angle will be unper-turbed and the cross-polarization coupling will beminimized.

For the elasto-optic method of orientation to beuseful, one needs the ability to predict the angle of theextrinsic birefringent axes. In a cylindrically symmet-rical fiber, the extrinsic birefringent axis will beparallel to the applied force as long as the fiber issqueezed between two parallel plates. However, if thefiber is not symmetric, the extrinsic birefringencemay not be parallel to the applied force. The mostpronounced asymmetry is that of the protective buffercoating, which may not be well centered upon the

1234 APPLIED OPTICS / Vol. 31, No. 9 / 20 March 1992

fiber. The objective of this study is to evaluate thedirectional sensitivity of the extrinsically producedbirefringent axes, based on varying degrees of eccen-trically coated fibers.

Analysis

Figure 1 illustrates the geometric variables thatdefine the eccentricity of any particular case study ofinterest. Variable e defines the fiber core radial offsetfrom the outer coating diametric center. The variableo defines the core center angular offset. It wassufficient to consider 0 between 0° and 90° during thisstudy because of 1/4 symmetry.

Figure 1 also illustrates the location of the appliedload (F), as well as the birefringent angle (a), whichcorresponds to the direction of the calculated princi-pal extrinsic stress vector (). In general, the modalbirefringence is related to the change in the fiberstress state based on the stress-optics theory, wherebythe change in the principal indices of refraction arelinearly proportional to the change in the principalstresses 8 :

B = (2'rr/X)(n1- n2) = C(2A)( 1 - c2), (1)

where B is the birefringence; X is the light-sourcewavelength; n,, n2 are the principal indices of refrac-tion; C is the material-dependent stress-optic coeffi-cient; al is the maximum principal stress; and oJ2 is theminimum principal stress.

Determination of the fiber stress state was per-formed by using finite-element methods of analysis.The finite-element procedure involves breaking upthe continuum into small areas represented by math-ematical functions known as elements. Connectivityof adjacent elements is established through commonnode points that represent the locations of the un-knowns (degrees of freedom) that are being solved.Appropriate boundary and loading conditions arespecified for the modal along with the material prop-erty definitions. Using matrix methods, we solved thegoverning differential equations of elasticity for the

Glass

Fig. 1. Eccentrically coated optical fiber and the definitions ofvarious geometric variables used in the finite element analysis.

nodal values of displacement and stress. Nonlineari-ties and discontinuities in geometry and materialproperties are easily handled, which distinguishes thefinite-element method from other numerical proce-dures, such as finite difference and boundary-elementmethods.

To facilitate modeling the different fiber configura-tions based on varying eccentricity values of e and 0,we used a parametric model definition. A parametricinput file was created and used to generate thefinite-element models for three values of I: 10, 20,and 30 pm. For each value of i, 0 was varied from 00to 90° in 5° increments. This design matrix resulted in57 different models' being created and analyzed. Thefinite-element model contains 720 elements withapproximately 1550 degrees of freedom. In additionto the variables of eccentricity, other design con-straints were defined as variable input parameters,although maintained as fixed constants throughoutthis study. The parametric modeling variables andtheir values are listed in Table I.

A preliminary study was conducted to verify theparametric modeling format, as well as the accuracyof the finite-element numerical solution. A hypotheti-cal case study was established so that the theoreticalclosed-form solutions could be obtained and directlycompared with the results determined from a finite-element analysis. The plane stress distribution alongthe x and y axes for a homogeneous cylinder of radiusr under a vertical loading F acting along the y axis is9

F 4 r2 x' 11 (r + X)2J

F = 4r 2'rr (r 2+ x)2I (2)

To satisfy the hypothetical case study parameters,we defined the parametric model variables as = 0Pm, = 00, E = = E = 10,000 MPa, and v = V2 =V3 = 0.3. A comparison of the theoretical and theanalytical results is presented in Fig. 2 for a concen-trated vertical force of 15 N. The maximum errorbetween the theoretical and the finite-element analy-sis (FEA) results is less than 2.0%.

During testing, the lateral loads are applied to thefiber by squeezing two parallel plates together asillustrated in Fig. 1. Closed-form solutions for thebirefringence under this loading have been calculated

Table 1. Finite-element Model Parametric Variable Definitions

Variable Value Definition

10-30 pLm Fiber core t-offset dimension (see Fig. 1)0 0-90 Fiber angular offset dimension (see Fig. 1)dfib 80 p.m Optical-fiber diameterdl 149 ,um Primary (inner) coating diameterde2 208 Am Secondary (outer) coating diameterE, 71724 MPa Optical-fiber elastic modulusE2 2.6 MPa Primary coating elastic modulusE3 600.0 MPa Secondary coating elastic modulusv, 0.16 Optical-fiber Poisson's ratioV2 0.30 Primary coating Poisson ratioV3 0.30 Secondary coating Poisson ratioa 10 Pm Vertical displacement load

20 March 1992 / Vol. 31, No. 9 / APPLIED OPTICS 1235

fiber only when the interface element experiences acompressive stress state. Initially the fiber makes apoint contact with the top and the bottom plates. Asforce is applied and the fiber coating deforms, morearea of contact is made, and the total applied force isdistributed over the area remaining in contact. Thesolution for contact simulation requires multipleiterations to obtain a converged solution. Conver-gence is satisfied when the stress state for all inter-face elements does not change between successiveiterations.

Ah FA IDuring the development of the parametric finite-(TY - Theoretical element model, a sensitivity study was conducted to

____. __, __.__,__.6 , . _,_.determine the appropriate number of interface con-0.0 0.2 0.4 0.6 0.8 1.0 tact elements to incorporate into the final analysisx/r model. The objective of this effort was to optimize the

number of contact elements included in the model)retical and FEA stress results for a homogeneous and determine what nodal spacings would reasonably

*ius r. simulate the contact condition between the plates andthe fiber coating material. The final model, as shown

tric fibers,"7," but these solutions deal with in Fig. 3, contains 26 symmetrically placed contactapplied point load at the top and bottom of elements spanning an arc of 150 on the top and theIn reality, elastic deformation will distrib- bottom surfaces. A nonuniform nodal spacing concen-'ces over a given contact area between the trates more of the elements near 90°, because thethe outer coating material. This effect will highest compressive forces occur in this region. For a'ated as a result of the significantly lower typical top-plate displacement of 8 = 10 pm, 10 of the'the outer coating compared with the fused 26 contact elements transmit nodal compressiveThe force distribution was included in the forces. For a displacement of = 15 and = 20 m,

;ing nonlinear interface contact elements. the number of contact elements in compression in-illustrates the locations of the interface creases to 18 and 20, respectively. The maximum,ments defined for the parametric model displacement considered during this study was 8 = 20the deformation of the fiber under load. Pm.

'1'he dashed line indicates the original fiber shapewithout load, and the top plate displacement is &. Theparallel plates are assumed to be perfectly rigid, andthe interface elements simulate the contact responsebetween the plates and the outer coating material.Nodal forces are transmitted from the plates to the

InterfaceContact

- T 1/1 Elements

Fig. 3. Finite-element model of an eccentricially coated opticalfiber showing the deformation induced by compressive loading.The dashed line is the original fiber shape.

Analysis Results

For a concentrically coated optical fiber, the fast axisof the extrinsic birefringence is in the direction of theapplied compressive force, or, in reference to Fig. 1,vertical. The change in orientation of this axis that isdue to an externally applied load for an eccentricallycoated fiber will correspond to the direction of theprincipal stresses that develop. This value is refer-enced as the extrinsic birefringent angle (a) (see Fig.1) and corresponds to the direction of the minimumprincipal stress vector ( 2), or (!/ 2 - a) from themaximum principal stress vector (). For an ideal(symmetric) fiber, a = 0. Fiber asymmetry causes thebirefringent axis to deviate from the vertical, result-ing in t c 0.

As Fig. 4 shows, the maximum change in thebirefringent angle occurs at an azimuthal angle ofe = 45°. This is due primarily to the influence of thefiber shear stress, which is also a maximum at =45°. The effect of the shear stress distribution (T,) isfurther illustrated by considering the influence ofshear in determining the principal stress direction.Relationships defining the magnitudes and the direc-tions of the principal stress values are commonlyderived from a Mohr circle representation of a two-dimensional stress state. The principal stress values

1236 APPLIED OPTICS / Vol. 31, No. 9 / 20 March 1992

5'

Stress(MPa)

Fig. 2. The4cylinder of ra

for symmea perfectlythe fiber. Iute the forplates andbe exaggerstiffness ofsilica fiber.FEA by usFigure 3 icontact eEand shows

6

Txy

(MPa)6x- Gy(MPa)

00 1 5 30 45 60 75 90 Fig.5. FEA component stress results ( = 30 pum,8 = 10 pum).

0Fig. 4. FEA result summary of extrinsic birefringent angle aversus fiber eccentricity (t, 0) = 10 Gum.

are given by

(1, 02) = (a 2 ) ± [(ax 2 a. ) - .2], (3).

and the principal stress direction is given by

tan 2+ = 2_v (4)(U. -a,)

Angle 4) of Eq. (4) defines the principal stress planeat which the associated shear stress is zero andresults in the maximum computed normal stress atthat location. Subscripts x and y correspond to stressvalues aligned with a referenced Cartesian coordinatesystem. With the Cartesian coordinates aligned to thedirection of the load in Fig. 1, the principal stressdirection 4) will correspond directly to the extrinsicbirefringent angle (a).

The coordinate stress components (o, oy, and Tx)that are used to determine the principal stress direc-tion 4), and hence, a, are illustrated in Fig. 5 for aconstant -offset condition of 30 m. The stressdifferential (a - o) remains nearly constant for aneccentrically coated optical fiber, varying by only±7%. Thus, the contribution from the shear stress

value (xy) has a dominant effect on the resultingprincipal stress direction. This may be of interest forour application, or for future efforts, which wouldsuggest that the measurable sensitivity of the extrin-sic birefringent angle a could be increased by impos-ing an alternative loading condition that would ele-vate the fiber shear stresses.

A secondary set of FEA's was conducted to investi-gate the effect on the birefringent angle a by varyingthe vertical displacement load. The objective of thiseffort was to validate an assumption that the accu-racy of the analytically predicted values of a will not

be influenced by the value of the vertical displace-ment loading. This assumption was based on thetheoretical derivation that a depends only on theprincipal stress direction. As expected, the analysisshowed that the principal stress direction does notsignificantly change as a function of loading. Al-though some minor principal stress differences mayexist between the load cases, the slight variations arewithin the range of numerical stability for solutionsbased on finite-element methods. The maximum devi-ation in a is 1.8% based on a load variation of 400%.These results confirm the assumption of load insensi-tivity.

Conclusions and Summary

The objective of this study has been to determine theeffect of fiber eccentricity on the angle ax. Althoughnot of consideration during this study, the influenceof the phase retardation angle may also be character-ized by the same analytical procedures. The totalphase retardation angle is dependent on the principalstress difference (ar, - u2), and these values are calcu-lated during the FEA.

The fiber coating nonconcentricity affects the orien-tation accuracy, causing a small error in the align-ment. For poorly coated fibers, this error could bemore than a degree. The maximum change in theextrinsic birefringent axis will occur at an azimuthalangle of 0 = 450, for any given degree of eccentricity.As the c-offset variable of eccentricity is increased, thedeviation of the birefringent angle becomes moresignificant. The maximum computed change in thebirefringent axis (at 0 = 45°) was 1.508° for a e offsetof 30 ,um, which is a much larger offset than normallyencountered in routine fiber manufacturing. Themaximum deviation is 0.582° at 20-[um offset and0.138° at 10-pm offset. With the usual coating quality(less than 10 jim of eccentricity) the induced orienta-tion error will be of the order of 0.10. The effect ofsuch a misalignment on the performance of couplershas been calculated in detail." An approximate resultthat we obtained by using the typical coupler parame-

20 March 1992 / Vol. 31, No. 9 / APPLIED OPTICS 1237

1.6

1.2

0.8

0.4

0.0

ters shows that high performance PM couplers (polar-ization cross talk < -30 dB) will not be seriouslydegraded by this small misalignment.2 However, abuffer offset of 20 jim will more than triple the effect,degrading the coupler performance. Therefore it isimportant to ensure that fibers used in the produc-tion of PM couplers have good buffer concentricity ifthe elasto-optic alignment method is to be used.

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