Post on 06-Apr-2018
8/3/2019 Birefringence Induced by Stress Ani Sot Ropy and Geometric Deformation
1/6
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:
dipak@corning.com).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
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-8/3/2019 Birefringence Induced by Stress Ani Sot Ropy and Geometric Deformation
2/6
http://-/?-http://-/?-http://-/?-8/3/2019 Birefringence Induced by Stress Ani Sot Ropy and Geometric Deformation
3/6
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 .
http://-/?-http://-/?-http://-/?-http://-/?-8/3/2019 Birefringence Induced by Stress Ani Sot Ropy and Geometric Deformation
4/6
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.
Fig. 3. for the SMF fiber for a 5% ellipticity.
Fig. 4. 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
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-8/3/2019 Birefringence Induced by Stress Ani Sot Ropy and Geometric Deformation
5/6
CHOWDHURY AND WILCOX: COMPARISON OF STRESS ANISOTROPY AND GEOMETRIC DEFFORMATION 231
Fig. 5. for the NZDSF fiber for a 5% ellipticity.
Fig. 6. for the NZDSF fiber for a 5% ellipticity.
Fig. 7. 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.
8/3/2019 Birefringence Induced by Stress Ani Sot Ropy and Geometric Deformation
6/6
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
[1] C. D. Poole and J. Nagel, Polarization effect in lightwave systems, inOptical Fiber Telecommunications IIIA, I. P. Kaminow and T. L. Koch,Eds. New York: Academic, 1997.
[2] F. Curti, B. Daino, G. De Marchis, and D. Matera, Statistical treatmentof the evolutionof principal statesof polarizationin single-mode fibers,
J. Lightwave Technol., vol. 8, pp. 11621165, 1990.[3] G. J. Foschini and C. D. Poole, Statistical theory of polarization
dispersion in single mode fibers, J. Lightwave Technol., vol. 9, pp.14391456, 1991.
[4] D. L. A. Tjaden, Birefringence in single-mode opticalfibers dueto coreellipticity, Phillips J. Res., vol. 33, pp. 254263, 1978.
[5] M. J. Adams, D. N. Payne, and C. M. Ragdale, Electron. Lett., vol. 15,
pp. 298299, 1979.[6] R. A. Sammut, C. D. Hussey, J. D. Love,and A.W. Snyder,Modal anal-
ysis of polarization effects in weakly-guiding fibers, Proc. Inst. Elect.Eng., vol. 128, pp. 173187, 1981.
[7] C. Vassallo, Optical Waveguide Concepts, Amsterdam: Elsevier, 1991.[8] M. Fontaine, B. Wu, V. P. Tzolov, W. J. Bock, and W. Urbanczyk, The-
oretical and experimental analysis of thermal stress effects on modal po-larization properties of highly birefringent optical fibers, J. LightwaveTechnol., vol. 14, pp. 585591, 1996.
[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.
http://-/?-http://-/?-