Polarization coupling in twisted elliptical optical fiber

Yoichi Fujii and Koichi Sano

The polarization of twisted fiber with elliptical cross section was analyzed using the Marcuse mode-couplingtheory. It was concluded that twist coupling is not effective unless the twist is fast (several rads/m). Exper-imental results qualitatively agree with the analysis.

1. IntroductionIn recent applications of single-mode optical fiber,

polarization stabilization of the fundamental mode isa serious problem, for example, in coupling fiber to alaser diode or optical ICs, in optical heterodyne detec-tion, or in the measurement apparatus through whichthe polarization carries information.

Stolen et al. ' first reported a method of polarizationstabilization by a stressed fiber. Many kinds of asym-metrical structures of optical fiber have been pro-posed 2'3 for polarization-stabilized fiber communica-tion.

Yeh,4 Tsuchiya and Sakai,5 and the present authors6

have analyzed the polarization characteristics of fiberswith elliptical cross section and have shown that de-generate HE,, modes in circular fiber are split into two(eHEn1 and oHEnj) and that group delay is different forboth eHE,, and oHEl modes.

The polarization transmitted in circular fiber changesrandomly with intrinsic polarization axes, even whenthe deformation of the cross section is small. When thecross section is elliptical, a small deformation causes nocritical change in the intrinsic polarization axes, so thepolarization transmitted will be stable.

Fiber of precisely circular cross section was made byNorman et al.,7 but a small amount of birefringencecaused by fiber bending8 appears even in this fiber.Recently, Ulrich and Simon9 analyzed elastoopticallyinduced optical activity in twisted single-mode fiber.

In this paper, polarization coupling between thefundamental modes of fibers of elliptical cross section

The authors are with University of Tokyo, Institute of industrialScience, 22-1 Roppongi-7, Minato-ku, Tokyo, 106 Japan.

Received 3 December 1979.0003-6935/80/152602-04$00.50/0.t 1980 Optical Society of America.

is analyzed by considering the geometrical index changedue to the twist at the periphery of the fiber core and isexperimentally verified using a single-mode ellipticalfiber. The effect of twist is calculated as a mode cou-pling between oHE11 and eHEu1 modes, using theMarcuse theory,10 and it is shown that mode couplingis small in optical fiber cable in normal conditions.

II. Coupling Coefficient in Twisted Elliptical FiberConsider two fundamental modes, oHE,, and eHE11,

in an elliptical fiber (see Fig. 1). These modes coupleto each other when the fiber is twisted along its axis.The coupling coefficient is calculated using the Marcuseformulation. 0 Only coupling between forward oHEjand eHE11 waves is considered.

The fields in a twisted elliptical fiber are expandedin terms of series of "local normal modes."' 0 Couplingcoefficient R between forward oHE11 and eHE11 modesis given by

R = 4(jsi an eeedxdy, (1)4 i3,- l) JJ--- Z

where fle and SO are propagation constants;ee and e0 are normalized field distributions of the

eHEI, and oHEI, modes, respectively; andan 2/z denotes the change in the square of the

index distribution along the z axis.We assume a uniform twist with twist angle 0 propor-tional to axial distance z:

0= z. (2)

For small increments of z, the change in the indexdistribution appears only near the periphery of the coreof elliptical cross section t = to, as in Fig. 2. Couplingbetween the oHEni and eHE,, modes for small incre-ments of z can be calculated with Eq. (1), where 0n2/Ozhas a finite value only at the periphery of the ellipticalcore.

After transformation to the elliptical coordinates, Eq.(1) becomes an integration by only at the peripheryof the ellipse.

2602 APPLIED OPTICS / Vol. 19, No. 15 / 1 August 1980

uo = (k'n' - 02)1/

2h coshto,

mode n

OHEi

CLADDING hmode-hcoshiO

=a

Fig. 1. Configuration and normal mode of elliptically cross-sectionedoptical fiber.

Y

Wo = (2 - k2n2)1/2h coshto. (10)

Numerical calculations of R, and Rb are shown in Fig.3.

Ill. Mode Coupling in Twisted FiberSolving the coupling equation between oHE11 and

eHE11 modes with coupling coefficient R, the coupledpower of the eHEu1 mode I Ce (z) 12 at z = z is given ininitial conditions at z = 0 where only the oHE11 modeis incident; i.e., I Ce (0) 12 = 0 and I CO (0) 12 = 1:

ICe(Z) 12 R sin [(3 0 /-e) 1 /24)2 +R2

2+ R2 -.Z)1

2

|C,,(Z)12 = 1 - C,(Z)12.

. H0 *Z

A

z=O

I~

Fig. 2. Slightly twisted elliptical fiber.

(11)

(12)

The maximum coupled power ratio I Ce (z) 1 2/1 C (0) 12is given by Eq. (13), where x is a ratio of the coupledpower between the oHE11 and eHE11 modes permissiblein a fiber system:

ICe(Z)2nax R2IC 2a= <2 -

()|2 ( +o le)+R2

4

(13)

where the birefringence of elliptical fiber f0, - de iscalculated as

f0 - 1e = nlkoA2 (G. - Ge).

For the coupled eHE 11 and oHE11 modes in twistedelliptical fiber (only coupling between forward wavesis considered), coupling coefficient R is obtained:

(Do2/ 2r2(:=lw) J S[(1 + 2A)e,.,e.f + ee .,g

+ e;,.eo,,zt= 0di, (3)

where the integral is calculated only at the periphery ofthe core when

-1 20 sin2-q(sinh.240 + in2n)S= -2

(sinhto cosq)2 + (cosht sinq) 2

n- n2. _

Parameter G - Ge is approximated for small A and

uow 2. wlKo(wo) 1 Go- Ge 22 2 + _ __ 5' (15)

2V4 1o 2 K1(wo) 2

where v is the normalized frequency. Go - Ge 0.5for conventional single-mode fiber (v = 2.4 i 0.2). Thepermissible twist angle for unit length 0 is written withEqs. (6), (7), (8), (14), and (15) as:

(4)

(5) 5

and h is the focal length of the ellipse of the core.Using the analysis of eigenmodes with an expansion

of spheroidal functions by Bessel functions6 and as-suming small ellipticity, coupling coefficient R reducesto

1+ ERb

Ra

4

3

2

1

(6)

where e = 1 - (b/a) is the ellipticity of the core, and

Ra = + ) J(u°) 5Uo 4u) J1(uo) 4

(7)

u_ uOJo(uo) ( w0 Jo(uo)124 Jl(uo) o l(uo)(

are independent of ellipticity E, and uo and wo are de-fined as

F

R = - e [1+Rb]Ra

2a

-Rb

0 0.5 1.0 1.5 2.0 2.5 3.0

NORMALIZED FREQUENCY v k l hcosho T-

Fig. 3. Epsilon-independent terms in coupling coefficient R [Eq.(6)].

1 August 1980 / Vol. 19, No. 15 / APPLIED OPTICS 2603

(14)

I ,, _l , j F Y_ .

_ _ _ _ >

(9)

° -' A *X( 1 rA 2E - o. f3le (rad/m) (16)1-x d- 0.8E) 1~_- (- 0.8e)

as a function of permissible coupled power ratio x. Thisequation means that the permissible twist angle per unitlength is approximately equal to the birefringent phaseangle multiplied by the permissible amplitude ratio ofthe coupled oHEi1 and eHEu1 modes.

Table I. Permissible Twist Angle Per Unit Length (deg/m) for x = 0.1,n = 1.46, A = 0.2%

(Am) E10% = 20% = 30%

0.85 0<45 0<98 0<1601.30 29 63 1091.60 23 52 86

Numerical examples of the permissible twist angleper unit length 0, for permissible coupled power ratiox = 0.1, nj = 1.46, and A = 0.2% are shown in Table I.Photoelastic index change due to mechanical torsion isestimated to be -10-6, which is negligible.

IV. Effect of Bending and Other DeformationIn an elliptical fiber bent in a plane that includes the

short axis, its ellipticity increases asb A 2b2 b

e = -- + a-,a 2p2 a (17)

where a and b are the length of the long and short axes,respectively; A is Poisson's ratio; and p is the radius ofbending curvature. The last term in Eq. (17) is 1.3 X10-11, where a = b = 3 Am, A = 0.17 (SiO2), and p = 10cm. This result has little effect on the ellipticity.

The photoelastic index change is symmetrical forboth axes. So it can be cancelled out in polarizationsplitting." Smith" reported that birefringence ofcurved fiber depends on the inverse square of the cur-vature:

f. - 03e = 0.3/R2 (). (18)

Comparing with Eq. (16), the effect of birefringence dueto bending can be neglected if R >> 0.3 m.

The effects of bulging and deformation of the fiberare found to have zero coupling coefficients.

V. ExperimentFor a single-mode fiber having = 18% ellipticity,

index difference A = 0.2% was used (Fig. 4). The crosssection of the core is a 7.5 X 6.0-um ellipse. The bire-fringence of this fiber is measured at X = 0.8 ,um:

/e - f = 3.5 ± 0.5 rad/m, (19)

Fig. 4. Cross-sectional view of the fiber.

1.0

0.8

S 0.6

0

> 0.4

-J_j

0.2

TWIST ANGLE PER UNIT LENGTH

which agrees with the calculation.6 The result of themode coupling due to twist is shown in Fig. 5.

e [rad/ni

Fig. 5. Coupling of eHEll and oHEI1 modes caused by the twist: fiber length is 589 mm, A = 0.2%, and = 18%. The output analyzer isalso twisted according to the twisted axes of the elliptical fiber, while the input polarizer is kept constant. A X = 0.8 -,4m laser diode was used.

2604 APPLIED OPTICS / Vol. 19, No. 15 / 1 August 1980

polarizzer

perpedcl to axis

. p~~~olarizer\- ~~~~~~~perpendicular to axis

0 2 4 6 8 10 12 14 16 18 20 22

r

I j -I I I

The input polarization is parallel with the long axis.The outputs of components parallel and perpendicularto the long axis of the fiber are measured by twisting thefiber.

Finite extinction at zero twist is due to the noncolli-near lines of electric field strength of the oHE11 andeHE 11 modes and to random imperfections. From thelinear portion of the coupled output, the couplingcoefficient can be estimated. The measured value ofIR/I is 0.21 t 0.17 rad-', which agrees qualitativelywith the calculation,6 IR/0l = 0.14 i 0.07 rad' . Thedifference may be due to the finite extinction of thefiber.

VI. ConclusionPolarization coupling by twisted elliptical fiber has

been analyzed and experimentally verified. Ellipticalfiber was found to have excellent polarization stability,and it was shown that twist has no effect on the polar-ization unless there is large deformation as in Eq.(16).

The polarization stability can be improved by in-creasing index difference A, since I o - Be is propor-tional to A2 but normalized frequency v is proportionalonly to VA.

Elliptical fiber shows great promise in optical com-munications.

The authors are grateful to S. Saito, J. Hamasaki, andT. Okoshi for helpful discussions. The fiber was sup-plied by Sumitomo Electric Company, Ltd.

This research was supported in part by the Fund forOptical Guided Wave Electronics from the Ministry ofEducation, Science and Culture of Japan.

References1. R. H. Stolen, V. Ramaswamy, and P. Kaiser, presented at Topical

Meeting on Integrated and Guided Wave Optics, 16-18 Jan. 1978,Salt Lake City, Utah.

2. V. Ramaswamy and I. P. Kaminow, Laser Focus 14, 48 (Oct.1978).

3. T. Okoshi, Trans. IECE Jpn. OQE 62, 78 (Feb. 1979) (in Japa-nese).

4. C. Yeh, J. Appl. Phys. 33, 2335 (1962).5. H. Tsuchiya and J. Sakai, Trans. IECE Jpn. OQE57, 74 (1974)

(in Japanese).6. Y. Fujii and K. Sano, Trans. IECE Jpn. OQE61, 78 (Oct. 1978)

(in Japanese).7. S. R. Norman, D. N. Payne, and M. J. Adams, presented at Fifth

European Conference on Optical Communications (1979), paper10.1.

8. E. F. Kuester and D. C. Chang, IEEE J. Quantum Electron.QE-11, 903 (1975).

9. R. Ulrich and A. Simon, Appl. Opt. 18, 2241 (1979).10. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic,

New York, 1974).11. A. M. Smith, presented at Fifth European Conference on Optical

Communications (1979), paper 10.2.12. I. P. Kaminow, presented at Fifth European Conference on Op-

tical Communications (1979), paper 10.4.

Twelfth Annual Symposium on OpticalMaterials for High Power Lasers

30 September-1 October 1980National Bureau of Standards

Boulder, Colorado

The Twelfth Annual Symposium on Optical Mate-rials for High Power Lasers will be held at the NationalBureau of Standards in Boulder, Colorado, 30 Sep-tember-1 October 1980. The Symposium is spon-sored by the American Society for Testing and Ma-terials, the Office of Naval Research, Defense ARPA,the Department of Energy, and the National Bureauof Standards.

The Symposium is the principal forum for the ex-change of information on the physics and technologyof laser materials. Topics to be discussed includebulk damage phenomena, surface damage, designconsiderations for high power systems and funda-mental mechanisms of damage.

Co-chairmen for the Symposium will be HaroldBennett, Michelson Lab, Code 38101, Naval WeaponsCenter, China Lake, CA 93555; Alexander J. Glass,L-488, Lawrence Livermore Laboratory, P.O. Box5508, Livermore, CA 94550; Arthur H. Guenther,AFWL/CA, Kirtland AFB, New Mexico 87117; andBrian Newman, Orgn. L-2, Los Alamos ScientificLaboratory, P.O. Box 1663, Los Alamos, New Mexico87545. Abstracts of 150 words should be sent to theConference Secretary, Norma Lear, ElectromagneticsDivision, National Bureau of Standards, 325 W.Broadway, Boulder, CO 80302, by 15 August 1980.

1 August 1980 / Vol. 19, No. 15 / APPLIED OPTICS 2605