Birefringence Induced by Stress Ani Sot Ropy and Geometric Deformation

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IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 6, NO. 2, MARCH/APRIL 2000 227

Comparison Between Optical FiberBirefringence Induced by Stress Anisotropy

and Geometric DeformationDipak Chowdhury, Member, IEEE, and David Wilcox

Invited Paper

AbstractA variational formulation-based vector perturbationmodel is used to compare the contribution of stress anisotropyand geometric deformation toward optical fiber birefringence.We show that relative impact of stress on birefringence anddifferential group delay depends significantly on the exact fiberindex profile.

Index TermsAnisotropy, birefringence, optical fiber, polariza-tion mode dispersion (PMD), stress, vector modes.

I. INTRODUCTION

ITH increasing bit rate for optical communication systems,

polarization mode dispersion (PMD) is becoming a major

system impairment [1]. PMD is directly proportional to the

inherent fiber birefringence, i.e.,

[2], [3], where is mean differential group delay (DGD) or

PMD, is the average birefringence (ps/km), is the length

of the fiber, and is the correlation length of perturbation. No

existing direct measurements of the statistical parameter are

found. Direct measurement of fiber birefringence, is time

consuming. Moreover, is most likely a function of length.

So, it is of great interest to be able to evaluate birefringence and

DGD for a fiber from a cross-sectional index measurement,

because these measurements can be obtained in the glass blanks

before drawing them into fibers. Such modeling tools also can

be used for designing fibers with improved PMD performance.In this paper, we describe a vector perturbation analysis

based on the variational expression for the propagation con-

stant, which uses arbitrary two-dimensional (2-D) refractive

index profile to compute the fiber birefringence and DGD

at any given fiber cross section. Birefringence and DGD

are introduced by deviation of the refractive index profilefrom perfect circularity. These perturbations are caused by

imperfections in the manufacturing process. Any perturbation

introduced by the manufacturing process give rise to two

effects: 1) perturbation of the redial index profile because

of the circularly asymmetric distribution of index increasing

dopants, e.g., GeO . In this paper, we refer to it as geometric

Manuscript received July 1, 1999; revised February 11, 2000.The authors are with Corning Incorporated Corning, NY 14830 USA (e-mail:

[email protected]).Publisher Item Identifier S 1077-260X(00)03857-0.

perturbation. 2) Asymmetric and anisotropic stress caused by

the thermal expansion mismatch and fiber draw-induced rapid

quenching. In this paper, we refer to it as stress anisotropy. The

proposed model is general enough to handle both of the effects

mentioned above if the relevant properties can be modeled

or measured. However, in the examples shown in this paper,we chose elliptical perturbations and only consider thermal

expansion mismatch-induced stress anisotropy.

Scalar perturbation theory has been used successfully to in-

vestigate birefringence caused by an elliptic deformation in a

weakly guiding fiber [4][7]. A full vectorial computation of

stress-induced birefringence in a highly elliptic fiber is done by

Fontaine et al., with the use of finite element method to solve

the vector Maxwell equation [8]. Tzolov et al. also presented a

full-vectorial computation of form-induced birefringence based

on a numerical solution of coupled partial differential equations

representing the vector Maxwell equation [9]. Imoto et al. an-

alyzed the stress-induced birefringence in a step index profile

analytically [10]. In all of the above works, the authors eitheranalyzed the problem with the use of a full numerical anal-

ysis, e.g., with the use of the finite element method, or ana-

lyzed simple index profiles, e.g., perfect step index profiles, with

the use of analytical methods. Also, all of the above-mentioned

work is aimed primarily at analytic perturbations, namely, el-

liptic perturbations. In this paper, we address the birefringence

and DGD computation in various communication fibers with ar-

bitrary index profiles having low birefringence. We also include

the stress effect as a perturbation. This formalism allows us to

analyze birefringence and DGD for any 2-D refractive index

profile and corresponding stress field. Although the presented

results are for elliptic perturbation of circular fiber only, the pre-

sented formalism allows for any arbitrary 2-D perturbation for

any waveguide with available vector modes.

In the following section, we first develop the vector pertur-

bation theory for the Maxwell equation to compute modal bire-

fringence of an azimuthally, asymmetric optical fiber waveguide

in the presence of anisotropic stress. Then, we discuss how the

stress field in a fiber for an arbitrary refractive index profile is

computed. Finally, we show some numerical results comparing

relative and absolute impact of stress anisotropy on modal bire-

fringence and DGD for two different types of fibers.

1077260X/00$10.00 2000 IEEE
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CHOWDHURY AND WILCOX: COMPARISON OF STRESS ANISOTROPY AND GEOMETRIC DEFFORMATION 229

where is the Kronecker delta and

mm N and mm N are the

elasto-optic constants for pure silica [13]. The perturbed dielec-

tric tensor is given as . The total stress is

considered as a perturbation to the isotropic dielectric constant.

B. Formulation of Stress Field for 2-D Index Profile

In order to compute the stress field generated by the thermal

expansion mismatch between the silica and GeO , we assume

that the fiber profile is axially invariant. Consequently, we are

left with a 2-D plane strain problem, i.e., the longitudinal

strain . The shear stresses and also are zero

for the plane strain assumption. If we wanted to compute the

stress field fora perfectlyazimuthally symmetric fiber, we could

use a one-dimensional (1-D) formulation. However, the objec-

tive of this work is to study the effect of deviation from perfect

azimuthal symmetry on birefringence. Consequently, a 2-D for-

mulation is required.

With the plane strain assumption, the stress in the plane

is computed by solving the equations of equilibrium [14]

(9)

where

is modulus of elasticity;

is the Poissons ratio;

and are thedisplacementsin and , respectively;

is the input thermal strain.

From the computed displacements, and , we find the stress

components as

(10)

Transformation to the cylindrical coordinates for use in (8) is

simple

(11)

Fig. 1. Radial refractive index profile for (a) an SMF and for a (b) nonzerodispersion shifted fiber (NZDSF).

The thermal expansion coefficient, , and the glass softening

temperature, , are functions of GeO concen-

tration. Because the local index is a function of theGeO concentration, a one-to-one functional relationship exists

between the local index and thermal strain ; i.e.,

, where is

the index delta in percent given by

where is the cladding refractive index for the fiber.

III. RESULTS AND DISCUSSIONS

The two profiles that we will show results for are shown in

Fig. 1. In order to compute the stress field for a 5% elliptic

deformation of the profiles, we used the following coordinate

transformation to obtain an elliptic refractive index profile from

a perfectly circular refractive index profile

i.e.,

(12)

where is the unperturbed, azimuthally, symmetric radial

refractive index profile shown in Fig. 1 and .
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230 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 6, NO. 2, MARCH/APRIL 2000

Fig. 2. for the SMF fiber for a 5% ellipticity.



With the use of the above profile, we computed the amount of

GeO required at a particular location of the profile [15]. Then,

we used the following equation to compute the thermal strain

for a particular location as [16]:

The above values for the thermal strain are used in (9). We used

PDE2D [17] to solve the set of (9) and then use (10) and (11) to

obtain the stress field in the cylindrical coordinate.

Figs. 27 show the computed stress field components for

the two different types of profiles shown in Fig. 1. All

computations are for a 5% ellipticity, i.e., in (12), .

Figs. 27 indicate a couple of features: 1) the overall stress

in the NZDSF fiber [Fig. 1(b)] is almost two times higher
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CHOWDHURY AND WILCOX: COMPARISON OF STRESS ANISOTROPY AND GEOMETRIC DEFFORMATION 231

Fig. 5. for the NZDSF fiber for a 5% ellipticity.



than the same for the SMF fiber [Fig. 1(a)]. This increase

is to be expected because of higher GeO concentration for

higher of the NZDSF fiber. 2) Because of the complicated

index profile, the shear stress, , is more complicated in

NZDSF fiber than the same for the SMF fiber. 3) Although

the peak values of and are about a factor of two

higher in NZDSF fiber than the same for the SMF fiber,

the peak value of is approximately three times higher

(0.15 N/mm in SMF and 0.45 N/mm in NZDSF) in the

NZDSF fiber. These general trends might suggest that impact

of stress in the NZDSF fiber is larger than that in the SMF

fiber. However, the interplay between the form-induced (or

geometrical asymmetry) birefringence and the stress-induced

birefringence is complicated as illustrated below.


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232 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 6, NO. 2, MARCH/APRIL 2000

TABLE IVALUES OF 1 (ps/km) FOR A 5%

ELLIPTIC DEFORMATION OF THE PROFILES SHOWN. THE STRESS TENSOR( ; , AND ) GENERATED BY THERMAL EXPANSION MISMATCH ARE

COMPUTED BY SOLVING THE STRESS EQUATIONS WITH THE USE OFPDE2D [17]

The values of the stress fields are used in (8) to compute the

permittivity tensor. The permittivity tensor then is used in (6)to compute the eigenvalues . Table I shows the computed

values for and for two different profiles, including

corresponding stress fields shown in Figs. 27. Although max-

imumstress levels inNZDSF fiber are twice ashighas thatin the

SMF fiber, inboth fibers,thebirefringence, , increases

by a factor of two in the presence of stress, compared with the

birefringence in the absence of stress. However, contrary to in-

tuition, stress increases the DGD, , by a factor of four in

the SMF fiber and by a factor of 2.5 in the NZDSF fiber when

compared with the values in the absence of stress. If we look

at the absolute values, stress-induced birefringence and DGD

are higher in the NZDSF fiber, as may have been expected. For

the NZDSF fiber, however, form-induced birefringence is muchhigher (factor of six) than that in the SMF fiber. Consequently,

the relative contribution of stress to birefringence and DGD in

the NZDSF fiber is smaller. These results indicate that the rela-

tive impact of stress is strongly dependent on the fiber profile.

IV. CONCLUSION

We presented a comprehensive perturbative model for esti-

mating birefringence and DGD from a 2-D refractive index pro-

file. The model includes the effect of anisotropic stress. Stress

is computed with the use of the finite element method. Despite

the fact that only expansion mismatch-induced stress is consid-

ered, these results illustrate the relative impact of stress for var-ious types of fiber birefringence and group delay and, hence,

PMD. Although we only included expansion mismatch-induced

stress, the model is general enough to handle stresses caused by

quenching and residual glass stresses.

Computer modeling indicates that the relative impact of

stress anisotropy and geometric perturbation on birefringence

and group delay between the two orthogonal modes depend

strongly on the exact fiber profile. So, making a decision on

the relative importance of stress versus geometric perturbation

for a given fiber based on simple analytic models could be

misleading.

ACKNOWLEDGMENT

The authors would like to acknowledge many discussions

with Dr. J. Abbott of Corning and his effort on the initial stress

computation.

REFERENCES

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[9] V. P. Tzolov and M. Fontaine, Theoretical analysis of birefringence andform-induced polarization mode dispersion in birefringent optical fiber:A full-vectorial approach, J. Appl. Phys., vol. 77, pp. 16, 1995.

[10] N. Imoto, N. Yoshizawa, and J.-I. Sakai, Birefringence in single-modeloptical fiber due to elliptic core deformation and stress anisotropy,

IEEE J. Quantum Electron., vol. QE-16, pp. 12671271, 1980.[11] D. Q. Chowdhury and D. A. Nolan, Perturbation model for computing

optical fiber birefringence from two-dimensional refractive-index pro-

file, Opt. Lett., vol. 20, pp. 19731975, 1995.[12] J. G. Dil and H. Blok, Propagation of electromagnetic surface waves in

a radially inhomogeneous optical waveguide, Opto-Electronics , vol. 5,pp. 415428, 1973.

[13] H. Aben and C. Guillemet, Photoelasticity of Glass. Berlin, Germany:Springer-Verlag, 1993.

[14] S. Timoshenko andJ. N. Goodier, Theory of Elasticity. New York:Mc-Graw Hill, 1951.

[15] S. Kobayashi, S. Shibata, and T. Izawa, Refractive-index dispersion ofdoped fused silica, in Proc. IOOC77, vol. B8-3, 1977, pp. 309312.

[16] K. Okamoto,T. Hosaka, andT. Edahiro, Stress analysis ofoptical fibersby finite element method, IEEE J. Quantum Electron, vol. QE-17, pp.21232129, 1981.

[17] G. Sewell, PDE2D: Easy-to-use software for general two dimensionalpartial differential equations, Adv. Eng. Softw., vol. 17, pp. 105112,1993.

Dipak Chowdhury, (S89M89) photograph and biography not available atthe time of publication.

David Wilcox, photograph and biography not available at the time of publica-tion.
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Birefringence Induced by Stress Ani Sot Ropy and Geometric Deformation

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