Vol. 10, No. 7/July 1993/J. Opt. Soc. Am. B 1191

Polarization instability in a twisted birefringent optical fiber

Sandra F. Feldman,* Doreen A. Weinberger,' and Herbert G. Winful

Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, Michigan 48109

Received December 4, 1991; revised manuscript received January 19, 1993The fast-axis polarization instability arises in a weakly birefringent fiber as a result of competition betweenthe natural fiber birefringence and the nonlinear ellipse rotation. Direct observation of the fast-axis polariza-tion instability is reported. A full theoretical development of the polarization instability in a twisted, birefrin-gent optical fiber is presented. The theory includes the derivation of and full solutions for the evolution of lightin a twisted fiber as well as stability analysis and phase-plane representation of the solutions. The experimentis described in detail; good agreement is obtained between theory and experiment. As a result of the insta-bility, very small variations in either the input power or the input polarization to the fiber result in largechanges in the output polarization. A crossed polarizer at the fiber end converts the polarization variation intointensity information. Thus the modulation depth of an input pulse has been increased from 15% to 100%.Modulation gains of as much as 10 times are theoretically possible.

INTRODUCTIONThe interaction between natural, linear fiber birefrin-gence and birefringence induced through the nonlinearindex of refraction has aroused interest for both pulse-shaping and switching applications. Pulse shapers maybe used to clip the wings or winglike pedestals oftenpresent in ultrashort pulses.'13 Intensity- or polarization-dependent switches may be used in fiber-optic logic gates,4shutters,5 or modulators.6 Each of these effects is basedon the same basic phenomenon. As a result of the nonlin-ear refractive index, more intense light experiences a dif-ferent net birefringence than less intense light. Lightwaves of different intensities therefore evolve through dif-ferent polarization states as they propagate in a fiber. Ifone appropriately orients a polarizer at the fiber output,the high- and low-power components may be effectivelyseparated.

In a weakly birefringent fiber, the interplay betweennatural birefringence and nonlinear effects, including thenonlinear refractive index and the nonlinear ellipse rota-tion, leads to a fast-axis polarization instability.6 9 Un-avoidable twisting of the fiber in the laboratory inducesadditional circular birefringence,' 0 which complicates theanalysis but does not change the basic manifestation of thepolarization instability."" 2 In the study reported here wehave used long pulses from a Q-switched ND:YAG laser tomake a direct observation of asymmetry between the fastand the slow axes of a birefringent fiber arising owing topolarization instability.'3" 4 Good qualitative agreement isobtained between experiment and theory. Additionally, asixfold increase in the depth of a periodic modulation onthe input beam is observed in the neighborhood of thepolarization instability. Previous researchers used circu-larly polarized input to demonstrate dramatic pulse shap-ing that is consistent with a polarization instability.' 5However, the researchers did not directly probe the fiberprincipal axes, nor did they observe modulational gain.

The polarization instability could be used to enhance theoperation of nonlinear switches and pulse shapers as wellas to construct a novel amplifier to increase the modula-

tion depth of a signal. On the other hand, in a fiber de-vice that relies on the preservation of linear polarizationto function correctly, such a polarization instability couldbe detrimental. It is necessary to understand the polar-ization instability both to determine how the instabilitymay be used in nonlinear devices and to determine thecritical parameters for the instability so that it may beavoided if necessary.

NONLINEAR POLARIZATION EVOLUTION INTWISTED FIBER

Derivation of Propagation EquationsThe equations governing polarization evolution in the fi-ber are derived in the cw approximation, so that time-dependent effects are ignored. This approximation isvalid as long as the fiber length is short and the lightpulses are long enough that dispersion may be neglected.The equations including dispersion have been consideredboth for strongly birefringent fibers, in which case it isvalid to neglect nonlinear ellipse rotation, 6 and for weaklybirefringent fibers (n < 10-6), in which case one obtainsa soliton polarization instability that is similar to the cwpolarization instability.'7 The equations are not inte-grable when time-dependent effects are included andmust be investigated numerically. When the time depen-dence may be neglected, the equations may be solvedexactly.

Since the equations governing the evolution of the polar-ization state of a light wave in an untwisted fiber are con-tained as a special case of the equations for the twistedfiber, and since small amounts of twist are nearly impos-sible to avoid in the laboratory, the theory is developedhere for a twisted fiber." When one twists the opticalfiber there is a twofold effect: the first is a strictly geo-metric effect that is due to the precession of the fiber prin-cipal axes'" and the second is shear-strain-induced circularbirefringence. 0 A fiber twist rate of q rad per unit lengthresults in circular birefringence a = hq, where h 0.13-0.16 for silica fiber. We define a normalized birefringence

0740-3224/93/071191-11$06.00 1993 Optical Society of America

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1192 J. Opt. Soc. Am. B/Vol. 10, No. 7/July 1993

by -q = Aa/r. We take the z direction as the direction ofpropagation. In a coordinate frame that twists alongwith the fiber the dielectric permittivity tensor becomes

1.= l 2 (1)

n1. 62

The real diagonal terms account for the natural linear fi-ber birefringence, while the imaginary off-diagonal termsrepresent the twist-induced circular birefringence. In anisotropic fiber the diagonal terms el and 62 would be equal.Since Maxwell's equations are simpler in an untwistedframe, we transform the dielectric tensor into the labora-tory coordinate system, using the rotation matrix

[ cos(qz) sin(qz) 1-sin(qz) cos(qz)

In the laboratory coordinate system the dielectric tensortransforms to

e = R(-qz)etwj1 tR(qz)Eo + Ea cos(2qz) ca sin(2qz) + iq I, (3)ea sin(2qz) -iq c - ea cos(2qz) J

where eo = 1/2(el + 62) = n2 is the average permittivityand ea = 1/2(el - 62) is proportional to the linear fiberbirefringence.

We follow the derivation of Winfull9 and write theelectric-field vector as a superposition of the two ortho-gonally linearly polarized modes, such that E = [LEj(z) +SEy(z)]f(x,y)exp(-icot). E and Ey are the complexlinearly polarized mode amplitudes, and f(x,y) isthe transverse-mode pattern normalized so thatf f2 (x, y)dxdy = 1. The mode pattern does not changesignificantly as the light propagates down the fiber. Thefield components propagating in the fiber must satisfyMaxwell's wave equation,

d2E, + () = E=o2p.NL (4)where i,j = x, y. The nonlinear polarization pNL leadsto the familiar phenomenon of the intensity-dependentrefractive index and nonlinear ellipse rotation.7 2 0 Thefirst term in the expansion of the nonlinear polarizationin an optical fiber is the third-order term2 0

pNL = (xeo/2)[(E E*)E + 1/2(E E)E*], (5)where X = 8nn2/3. In silica fiber the nonlinear index co-efficient n2 3.2 X 10- cm2/W and the fiber core indexn 1.5.

We resolve the field into complex circularly polarizedcomponents by explicitly writing the field amplitude andphase as

c(z)exp(ikoz) = 1/V2[E(z) iEy(z)]. (6)The dependence on distance that is due to the linear propa-gation term is explicitly written as exp(ikoz), where ko =wn/c. We now expand Eqs. (4) and (5), using the circularly

polarized basis fields of Eq. (6). We consider the slowlyvarying envelope approximation, which applies when thefield changes gradually as it propagates, and so the termsthat involve second derivatives with respect to distancemay be neglected. After much algebra, a pair of coupleddifferential equations is obtained for the two circularlypolarized modes, c+ and c_:

d a i(C1 2Cd c+ = i c+ + i exp(i2qz)c_ + iP(Ic+2 + 21c j 2)c+,dz 2n(7a)

d -c = iK exp(-i2qz)c+ - i-c_ + i3(Ic_12 + 21c+12)c_,dz 2n

(7b)where ,B = XcW/4ncAeff= 2 cfn2/ 3 cAeff and the fibereffective-modal area Aeff is the inverse of the inte-gral f If(x, y)14dxdy. The normalized birefringence K =ln/A = koan/2n. The fiber beat length is defined asLb = 7/K and corresponds to the fiber length required inthe linear regime for the polarization state to undergo onefull evolution and to return to its initial polarization.

Finally, we further simplify the equations by explicitlyrewriting the modes in terms of amplitudes and phases sothat c(z) = Ic(z)Iexp[i45(z)]. It is also convenient todefine T =+ - 0 - 2qz.; Equating the real parts ofthe resulting equations yields differential equations forthe amplitudes of c+ and c_. Equating the imaginaryparts of the equations yields differential equations for thephases + and 0-, which in turn yields a single compactequation for T:

d Ie+ = cc-Isintr,d =d-Ie-I =. -iIc.IsinP,dz

(8a)

(8b)

d a. q+KCsI)(cII+d =-- 2q + K(cos/) -1 - II + (Ic-I2 - Ic+12).dz IC+I Ic-I(

(8c)

The polarization of the light is completely determinedonce solutions for c+I, Ic-I, and T are obtained.

It is relevant to note that the theory presented abovecorresponds to the limiting case of a vanishing modulationfrequency in the theory presented by Wabnitz for modula-tional polarization instability, which includes the effect ofchromatic dispersion.2'

Derivation of Conservation EquationsEquations for conservation of power and momentum arederived from Eqs. (8). It will prove to be convenient tonormalize the powers in the circularly polarized compo-nents by /3/ 2K so that u = IC+12 /3/2K and v = ICI|2 3/2K.The power conservation equation is easily obtained fromEqs. (8a) and (8b). The equation d/dz(lc+12 + IcI 2) = isreadily apparent since Ic+IdIcI/dz (1/2)dlc+12/dz, and asimilar equation holds for Ic-1. With the normalized pow-ers u and v it is clear that

p = u + v, (9)

Feldman et al.

Feldman et al.

where p is a constant that expresses the normalized, totalpower in the system. The equation for the conservationof momentum is derived from the identity

d (1c+l Ic-lcosP) = cosPd (lc+l Ic-I) - Ic+I Ic-IsinT dP(10)

We rewrite the derivative d(Ic I Ic-I1)/dz, using Eqs. (8a)and (8b), and we recast the expression for Ic+I IcJdP/dz,using Eq. (8c). After these substitutions and additionalalgebra, a second equation for the conservation of momen-tum is obtained:

r = Vu vco st - u ( + , - p) , (where jL = q/K(1 h/2n). The two conservation equa-tions are used to solve the set of coupled differential equa-tions describing the evolution of the light polarization in atwisted fiber.

Solutions for Evolution of Light in Twisted FiberWe use the normalized power variables u and v as well asAz to recast Eqs. (8) as follows:

du-= 2KN/U sinP,dz dv -= -2K sin,dz

d = K , - TACOS T - 2Kt + 2K(V - U) .

Vol. 0, No. 7/July 1993/J. Opt. Soc. Am. B 1193

Once u(z) is found, v(z) is immediately obtained fromthe conservation of power. The relative phase differenceI is determined from the conservation of momentum.However, is not unambiguously defined by the inversecosine operation implied in Eq. (11). To determine thecorrect value for T, refer to Eqs. (12). If du/dz 2 0, thensin 0 and P is assigned in the range 0 ' ' -,. Ifdu/dz < 0, then sinP < 0 and T is in the range - ir 5T < 0. Once u, v, and T are known, the output light po-larization is calculated. The azimuth of the polarizationellipse in laboratory coordinates is 6 lab = T/2 + qz. How-ever, it is more useful to consider the orientation of thepolarization ellipse relative to the principal fiber axes, inwhich case 0 = /2P The angle 0 = 00 corresponds to theslow axis, while 0 = 90 corresponds to light polarizedalong the fast axis. The ellipticity is given by e = (r - 1)/(r + 1), where r = I.Determination of Eigenmodes and Stability Analysisof SolutionsThe eigenmodes of this system are those polarizationstates that propagate along the fiber without changing.Once the eigenmodes are known, a stability analysis isperformed and the results are diplayed in bifurcation dia-grams. To find the modes that propagate without chang-ing, one rewrites Eqs. (8) and sets the derivatives with

(12a)

(12b)

(12c)

Equation (11), which expresses the conservation of momen-tum, may be used to reduce Eq. (12a) to the quadrature,

f|U0 f = + 2Kdz' = 2Kz, (13)

where the quartic Q(u) = u(p - u) - [F + u(u - p + IL)]2and uo is the input power in the right-circularly polarizedcomponent. One chooses the sign on the right-hand sideof Eq. (13) by evaluating sin P at the fiber input. Ifsin T > 0, then the positive sign is chosen; otherwise thenegative sign is used.

Equation (13) may be inverted to solve for u in terms ofelliptic functions.22 The exact solutions depend on thenature of the quartic Q(u) = -(u - U1)(U - U 2 )(U - U3)(U - U4 ). Q(u) may have four real roots or two real rootsand two complex-conjugate roots. In case 1, Q(u) has fourreal roots, ul 12 2 U3 2 U4, as seen in Fig. 1(a), where-Q(u) is plotted versus u. If one considers -Q(u) as apotential, it becomes clear that the allowable values of uare confined to either (a) ul U 2 U2 or (b) U3 U 11 U4.In case 2 [Fig. 1(b)], Q(u) has two real roots, ul U 12, andtwo complex roots, U3 = U4*. In this case u is constrainedby ul - U 2 U2 . Separate solutions exist for each of thesecases [Figs. 1(a), 1(b), and 2] and are summarized inTable 1. The solutions of Table 1 were checked in se-lected cases by direct numerical integration of the coupleddifferential equations.

Right-Circularly Polarized Power Component

(a)

0.5

-0.51

-1 -0.5 0 0.5 1 1.5 2 2.5Right - Circularly Polarized Power Component

3

(b)Fig. 1. Roots and allowed values of the quartic Q(u). The quar-tics are calculated for circularly polarized light in the same fiber(a) with no twist and (b) with a twist rate of one-half twist perbeat length. The normalized power P = 2.2 in both cases. Theroots are indicated by filled circles. The value of u is constrainedto lie within one of the regions bounded by arrows.

1194 J. Opt. Soc. Am. B/Vol. 10, No. 7/July 1993

Table 1. Solutions for the Nonlinear Polarization Evolution in a Twisted Optical Fiber

Four Real RootsU1 U > U2 > U3 > U4

Four Real RootsU1 U2 2 U3 2 U > U4

Two Real Roots U1 2 U > U2Two Complex Roots U3 U4* = a + ib

A2 = ( - a)2 + b2; B 2 (122 - a)2 + b2

m (Ul - U2)(U3 - U4 )/(U, - U3 )(U2 - U4) (u - U2)(U3 - U4 )/(Ul - U3)(U2 - U4) [(U1 - U2 )2 - (A - B)2]/4ABg 2/\/(ul - U3)(U2 - U4) 2/\/(ul - U3 )(U2 - U4) 1/VA7_

'ko sin'l [(iU U3)(Uo - U2) 1/2L (U - U2)(UO - U3)uWz) U3(U1 - U2) - sn2(x I m)U 2 (u -U3).,

(1 - U2) - sn2(x I m)(u, - U3)

sin [(U - U3)(UO - 4)11/2(3 - U4)(U1 - UO)U1(U3 - U4) + sn 2(xIm))U4 (Ul - U3)

(U3 - U4) + sn2 (x I m)(ul - U3)

cos-1 (, - uo)B - (UO - 2)A1L(u, - uo)B + (WO - U2 )AJ(u2A - uB)cn(x m) + (u2 A + uB)

(A - B)cn(x m) + (A + B)ax = 2z/g + F0o1m); F is an elliptic integral of the first kind.bsn and cn are elliptic functions.

respect to distance (d/dz) equal to 0 as follows:

du= f,(u',v',P) = KV' sinP n 0,

dz'dz= f2(U',V',>P) = -/CU'sinP=> 0,

dz = f3(U', V',P) = K( - - )(cos ) - 2L+ 2K(V'2 - U'2) => 0,

state of the input is close to that of an unstable eigenmode(but does not exactly reproduce the SOP of the eigenmode),

(14a) then the light may evolve in a complicated way in the fiber,and the output SOP need not remain close to the inputpolarization state. One determines the stability of the

(14b) eigenmodes by forming the Jacobian matrix J and examin-ing its eigenvalues.2 3 The Jacobian matrix is defined as

(14c)

where u' = = V|7Ic+| and v' = = \"17/Kc-j.To satisfy Eqs. (14a) and (14b) simultaneously, we requiresin P = 0, corresponding to P = 0 or T = 1800. Thus theazimuth of the polarization ellipse in the fiber frame,

= /2% is equal to 0 or 90. The fiber eigenmodes will

of, of,au' Ov'

af2 af2au' av'

af3 af3au' av'

Of,aPaf2

af3oP

where fl, f2, and f are defined above in Eqs. (14).derivatives are evaluated, and one obtains

00

00

L- Kv'(cos'I) - 4K' KU'5((cos ) + 4 KV'

always be aligned with the principal axes. The ellipticityof the eigenmodes may be determined from Eq. (14c).Since T is known to be either 0 or 1800, cos T = +1 foreigenmodes aligned along the slow and fast axes, respec-tively. We use the power conservation equation, p =ui + v2, square both sides, and collect terms to obtain thefollowing quartic for u'2:

16(U,2)4 + 16(,u - 2p)(u'2) 3 + 4(1 - 6 p + 5p2 + 2)(U2)2+ 4(-p - p2 + 2p2A - p3)(U'2) + p2 = 0. (15)

To determine whether the solutions to Eq. (15) correspondto the fast- or slow-axis eigenmodes, the solutions mustbe substituted back into Eq. (14c). As already stated, ifthe positive or negative sign holds, the solution corre-sponds to an eigenmode aligned with the fast or slow axis,respectively.

Once the fiber eigenmodes are known, one completesthe analysis by determining which of the eigenmodes arestable and which are unstable. If the polarization state ofthe input light is close to that of a stable eigenmode, thenit will not evolve far from that state of polarization (SOP)as it propagates. On the other hand, if the polarization

Kv' cos 1-KU'cosP1 ~,0] (17)

where 6 = (1/1IU' 2 + 1/Iv'I2 ). One evaluates the matrix J,using the values for u' v' and P for each polarization ei-genmode. For each eigenmode, three eigenvalues, j(j = 1, 2,3), of J are determined. If for all the eigenval-ues Re(uji) < 0, then the eigenmode being evaluated isstable. If for any one of the eigenvalues Re(,.j) > 0, thenthe eigenmode is unstable.

The bifurcation diagrams in Fig. 2 show the fiber eigen-modes as a function of input power. Figure 2(a) shows thefast- and slow-axis eigenmodes for the untwisted fiber.The slow axis remains a stable guiding center for linearlypolarized light at any power. Below the critical power(P < 1) the fast axis maintains stable guiding for linearlypolarized light. At the critical power (P = 1) the fastaxis undergoes a pitchfork bifurcation. Although linearlypolarized light remains an eigenmode, the eigenmode isno longer stable. Two new, stable, elliptically polarizedeigenmodes emerge, which correspond to those polariza-tions at which the nonlinear ellipse rotation exactly bal-ances the effect of the natural, linear fiber birefringence.Figure 2(b) shows a similar diagram for a fiber with atwist rate of only 0.3 twist per natural beat length, cor-responding to a small induced circular birefringence.

Parameters

(16)

The

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Vol. 10, No. 7/July 1993/J. Opt. Soc. Am. B 1195

SLOW- AXIS EIGENMODE1.

U 1-1

0 0.5 1 1.5 2 2.5 3FAST- AXIS EIGENMODES

0

-l10 0.5 1 1.5

Normalized Power

(a)

SLOW- AXIS EIGENM

0

X1 -1o0 0 0.5 Iv2. FASm Ir

1.5T- AXIS EIGENMI

0 If

-0I 0.5 1 1.5Normalized Powe

(b)Fig. 2. Bifurcation diagrams showing thEand unstable (dotted curves) eigenmodestwisted fiber and (b) a fiber with a twist ratlength (after Ref. 14).

The slow axis again maintains stablesomewhat elliptically polarized light.power the fast-axis eigenmode is alsocally polarized. The fast axis again1eigenmode that is most nearly linearlyunstable. Although the nature of thchanged, the onset of instability is dpower because of the need to use theance both the natural linear birefringecircular birefringence.

The effect of twist on the polarizatilight in the fiber depends strongly on tftwist-induced birefringence and the nagence. In a slightly twisted but hifringent fiber, light input along eitheroriented along that axis, twisting slowframe along with the fiber axes. Ina fiber with low linear birefringencestrongly twisted (many twists per beatcircular eigenmodes. To propagate vthe SOP, the light must be input withtion. This fiber is also stable againststability, since it requires extremelynonlinear ellipse rotation to countercircular birefringence. In the intermcular birefringence induced by the fibejof magnitude comparable with that cfringence, resulting in an elliptical bi

input with the correct elliptical polarization will propa-gate without changing, while all other states of polariza-tion will evolve periodically as they propagate. Theperiod of this evolution is given by the modified beat lengthLb = VI/K(1 + p)1 2. In general, however, it is always pos-sible to find a configuration in which a pair of orthogonal,linearly polarized input states leads to linearly polarizedoutput states. These states are referred to as the nomi-nal axes." In practice it is often convenient to use thesenominal axes as a laboratory reference.

2 2.5 3 Phase-Plane Presentation of Polarization EvolutionOne uses phase portraits to visualize the solutions to setsof coupled differential equations such as Eqs. (8). Thephase portrait plots ellipticity (e) versus azimuth () as a

[ODE function of propagation distance at a given light intensity.In the phase plane a rectilinear coordinate system is used.Ellipticity is indicated along the y axis, so that linearpolarization (e = 0) corresponds to a horizontal line acrossthe center of the plot. The upper half of the phase plane

2 2.5 3 represents right-elliptically polarized light with the right-DDES circular polarization along the top edge, and the lower half

represents left-elliptically polarized light with the left-circular polarization along the bottom. The polarizationangle is plotted along the x axis. The angle 0 = 00 corre-sponds to the slow axis, while 0 = 900 corresponds to the

2 2.5 3 fast axis. Therefore the phase plane may be thought of aser having been wrapped around the surface of a cylinder.

Phase portraits are entirely equivalent in function to plotson the Poincar6 sphere.24 In the Poincar6-sphere repre-

estable (solid curves) sentation, the Stokes parameters are plotted on the sur-for both (a) an un- face of a sphere. Linear polarization is represented along

.e of 0.3 twist per beat the equator, while the circular polarizations occur at thenorth and the south poles. The phase plane and thePoincar6 sphere are used in the same way: an input SOP

guiding, albeit for is chosen and located on the plot. The trajectories on theSimilarly, at low phase plane or sphere indicate how the light will evolve as

* stable and ellipti- it propagates in the fiber.bifurcates, and the Phase portraits are included in Fig. 3 for untwisted andI polarized becomes twisted fibers at powers less than and greater than thee instability is un- critical power. The low-power phase portrait for an un-delayed to a higher twisted fiber [Fig. 3(a)] consists of two sets of concentricnonlinearity to bal- ovals, centered on linearly polarized light oriented alongace and the induced the fast and slow axes. The central point of these ovals

corresponds to a fiber eigenmode. The circular paths[on evolution of the surrounding the eigenmode indicate that the light polar-ie ratio between the ization undergoes oscillatory behavior; the azimuth oscil-tural fiber birefrin- lates back and forth between the two angles 6 and passesghly linearly bire- through both right- and left-elliptical polarizations in eachaxis should remain period. This oscillatory nature is clarified if one notesly in the laboratory the polarization evolution explicitly drawn beneath thethe other extreme, phase portrait. As the power is increased to three times(8n < 10-6) that is the critical power [Fig. 3(b)], the phase plane is dividedlength) will develop into two distinct regions by a separatrix orbit. The sepa-ithout a change in ratrix orbit passes through an unstable saddle point corre-a circular polariza- sponding to linearly polarized light oriented along the fastthe polarization in- axis. Oscillatory behavior is again observed about thehigh power for the slow axis for light input near that axis. However, nearthe strong induced the fast axis the evolution is drastically changed by theediate case the cir- presence of the saddle point. Linearly polarized light in-r twist is of an order put near the fast axis but slightly off axis (inside the sepa-of the natural bire- ratrix) will now oscillate about the slow axis. Light thatrefringence. Light is precisely aligned with the axis but is slightly elliptically

BU

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I

.................................................

I

1196 J. Opt. Soc. Am. B/Vol. 10, No. 7/July 1993

FAST AXIS SLOW AXISAzimuth

( ) C\() 1 2 3 4 5 6 7 8 I11

9 10 11 12 13 14 15 16 9(a)

-90 -80 -60 -40 -20 0 20FAST AXIS SLOW AXIS

RCP

LIN

-&I LCP80 90

FAST AXISAzimuth

~ e:) ':/L1 2 3 4 5 6 7 8 l

9 10 1 1 12 9(b)

polarized (outside the separatrix) executes rotatory mo-tion: the light orientation traverses all the possible an-gles (-90 to +900) and always maintains the samehandedness. Again, the polarization evolution is explic-itly displayed below the phase portrait to illustrate the dif-

0Azimuth(d)

Fig. 3. Phase portraits displaying the state of polarization of thelight as it propagates in both untwisted and twisted fibers atpowers less than and greater than the critical power (afterRef. 14). (a) Low-power phase portrait for an untwisted fiber(P = 0.5). Note the two regions of oscillatory motion centeredon linearly polarized light aligned with the principal axes.(b) High-power phase plane for an untwisted fiber (P = 3.0).Note the separatrix orbit (drawn as a heavy curve) separating theregions of oscillatory and rotatory motion. (c) Low-power phaseplane for a weakly twisted fiber (P = 0.01, twist rate = 0.3).The phase plane now consists of two regions of oscillatory motionas well as a region of rotatory motion. (d) High-power phaseplane for a weakly twisted fiber (P = 4.0, twist rate = 0.3). Thephase plane is once again divided by a separatrix orbit. Notethat the saddle point has moved from the linear polarization to aslightly elliptical polarization.

ferent nature of oscillatory and rotatory motion. Notealso the two regions of oscillatory motion for ellipticallypolarized light aligned along the fast axis; these corre-spond to the two additional stable eigenmodes seen in thebifurcation diagram of Fig. 2(a).

I RCP

121

LIN

14

-aI.LCP80 90FAST AXIS Azimuth

(C)

i

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Vol. 10, No. 7/July 1993/J. Opt. Soc. Am. B 1197

When the fiber is twisted, the eigenmodes become ellip-tical, as already observed from the bifurcation diagrams.Thus at low power the phase plane consists of regions ofoscillatory motion centered on elliptical polarizationsalong each axis [Fig. 3(c)]. A comparison of Figs. 3(a) and3(c) shows the profound effect of fiber twist on the evolu-tion of a weak light beam. The phase plane for thetwisted fiber contains not only the two regions of oscilla-tory motion about the eigenmodes as before but also alarge region of rotatory motion. The circular birefrin-gence induced by the twist has an important part in deter-mining how the polarization evolves in the fiber. Whenthe light intensity is increased above the bifurcation pointshown in Fig. 2(b) the phase plane is again separated intoregions of oscillatory and rotatory motion by a separatrixorbit [Fig. 3(d)]. Comparison of Figs. 3(b) and 3(d) showsthat the location of the saddle point is shifted slightly toan elliptical polarization because of the fiber twist. Forthe instability in a twisted fiber to be reached with lin-early polarized light, the azimuth of the light must be afew degrees from the fast axis. The similar evolution ofthe polarization state at high power for the twisted anduntwisted fiber indicates that even when the fiber isslightly twisted the instability will manifest itself in aform similar to that predicted for the untwisted case.This occurs because the twist-induced birefringence be-comes a fairly small perturbation on the system comparedwith the magnitude of the nonlinear effect.

OBSERVATION OF POLARIZATIONINSTABILITY

Estimation of Fiber Beat LengthBecause of the long beat length of the fiber used in thisexperiment, standard measurement techniques, such asobservation of the Rayleigh scattering in the fiber,2 5 can-not be used. To estimate the beat length, the followingprocedure was developed that is particularly suited forfibers the beat lengths of which range from a few hundredcentimeters to a few meters. The fiber is supported invacuum chucks in a flat position to eliminate any effectscaused by external stress or bending. First, one locatesone of the principal axes of the fiber by rotating linearlypolarized input light (using a half-wave plate) and simulta-neously adjusting an analyzing polarizer until the lightout of the fiber is also highly linearly polarized. Next, thelight input to the fiber is rotated in small steps. At eachstep, one determines the orientation of the output light bylocating the major and minor axes of the elliptical polar-ization, and the ellipticity is calculated from the intensityof the light along the ellipse axes. Since both the inputand the output polarizations are known, one may estimatethe fiber birefringence by treating the fiber as a wave plateof unknown retardance in the Jones matrix formulation.

P, and P11 are the partial powers transmitted throughcrossed and uncrossed polarizers corresponding to thepowers along the minor and major axes of the polarizationellipse (Fig. 4). PTOT is the total power of the light beambeing analyzed. Rin is the rotation angle of the light intothe fiber relative to a fiber axis, and Rout is the orientationof the light leaving the fiber relative to the same axis.Then in the fiber coordinate system the Jones matricesrepresenting the system2 6 are

,,p-I rcos Rout sin Rout1-Tsin Rout cos Rout

0 1 cos Rin -sin Rinl['1exp(iF/2) [sin Rin cos Rin L 0]'

(18)

where exp(iO) represents an arbitrary phase factor intro-duced in traveling through the fiber and F is the relativephase delay between the parallel and perpendicular lightcomponents in the fiber. The allows for right- or left-elliptically polarized light. To solve Eq. (18) it proves con-venient to take the magnitude of each side and to workwith powers rather than fields, so the sign of the on theleft-hand side of the equation does not enter the finalequation for F. The above equation is straightforward tosolve for F, yielding

2 F P1/PTOT - sin2 (Rin + Rout)2 sin2(Rin - Rout) - sin2(Rin + Rout) (19)

From r, the beat length Lb may be calculated from

fiber length X 2ir fiber length X 360F (rad) r (deg)

(20)

When one takes the inverse cosine to find F, the first sev-eral possible values should be retained, yielding severalvalues for the beat length. Physically, this is a directconsequence of the ability to estimate the beat length onlyto within modulo 27mr For example, this method cannotdistinguish between fibers 0.25 beat length long and those1.025 beat lengths long. Repeating the procedure with adifferent length of fiber will establish which value for thebeat length is correct. We used this method repeatedlyto obtain a beat length estimate of 4 m for the fiber usedin this experiment. Some representative data that wereused to calculate this value for the beat length are pre-sented in Table 2. Our calculation of the beat length is inreasonable agreement with the estimation of a 10-m beatlength by Corning Glass Works, which supplied the fiber.The critical power, P, = 2K//3, yields P, = 250 W for a 4-mbeat length.

Experiment DescriptionIn this experiment 200-ns (FWHM)from a Nd:YAG laser (A = 1.064 ,um)

Q-switched pulseswere used. These

Fig. 4. Determination of the beat length of a weakly birefrin-gent fiber by means of a generalized Jones matrix technique.The measurements required in Eq. (19) are indicated.

Feldman et al.

exp(iO) _, N/P_11-=-i VP,

X exp(-iF/2)0

1198 J. Opt. Soc. Am. B/Vol. 10, No. 7/July 1993

Table 2. Representative Data Used to CalculateFiber Beat Length'

Rin (deg) Rout (deg) P., E/2 (deg) Beat Length (in)10 8 0.075 36 420 12 0.24 36 440 48 0.51 28 3.850 64 0.41 24 3.7

a In all cases the fiber length L = 3.2 m and the total power into the fiberP = 2.2 (relative units). Approximately 20 mW of power was transmittedthrough the fiber during the measurement.

Pi1

ND X/2 lox

Fiber lies in aluminumsupport

IZ~Folding

l. ... ... P.Mirrorlox /2 Polarizer

MO MO

in

Nd:YAG 1.06pm 0 . /odnAttenuator Collimating 90-10 Mirror

AteutrLenses Bean splitterFig. 5. Schematic of the experimental layout. The attenuator iscomposed of a half-wave plate followed by a polarizer. The waveplate and the polarizer marked with an asterisk form a second at-tenuator, permitting more accurate control over the pulse power.The fiber is held in vacuum chucks and supported in an alu-minum bar to minimize stress-induced birefringence. Neutral-density (ND) filters are used as needed to prevent damage to thephotodiodes. lox microscope objectives (MO's) are used tocouple light into and out of the fiber.

pulses were long enough and the fiber was short enoughthat the effects of dispersion were negligible and thecw approximations made throughout the preceding calcu-lations were valid. The pulse repetition rate was kept to500 Hz to avoid fiber damage. Since damage to the fiberend face is affected both by the pulse peak power and theaverage power, we used a low repetition rate to obtainhigher peak power. Even at only 500 Hz, catastrophicfiber damage becomes a problem at peak powers exceeding1 kW. The pulses were modulated at 76 MHz because oflongitudinal mode beating in the laser. We included asinusoidal approximation to the modulation in the model,rather than using injection seeding or an intracavity eta-lon to attempt to eliminate the modulation. A schematicof the experiment is shown in Fig. 5. We controlled thelaser power by using a pair of attenuators consisting ofhalf-wave plates followed by polarizers to adjust the inten-sity precisely while still maintaining the polarization ofthe light. Two attenuators were needed because the laseritself was polarized only slightly better than 10:1. Asingle attenuator did not permit precise control over thelight intensity, particularly at low powers. Collimatinglenses were necessary to prevent excessive diffraction ofthe beam before we launched the beam into the fiber. Abeam splitter placed in front of the final set of polarizingoptics split 10% of the light into a detector, which al-lowed us to monitor the input pulse shape during theexperiment. The final folding mirror was followed by apolarizer so that the light entering the fiber was linearlypolarized at better than 500:1. We could orient the polar-ization at any angle to the principal axes by using a half-

wave plate. Light was coupled into and out of the fiberwith 1OX microscope objectives. Typically 50-60% of theincident light was successfully coupled into the fiber.The analyzer consisted of a half-wave plate followed by apolarizing beam-splitter cube. We set the analyzer byaligning the linearly polarized light with the fiber axes atlow power. The polarizer was then crossed to the fiberoutput. It proved convenient to null the output, which wedid by rotating an output half-wave plate while keepingthe orientation of the polarizer fixed. This way, the po-larizing cube permitted simultaneous monitoring of bothP-L, the power transmitted when the polarizer is in thecrossed position, and PI1, the power rejected by the crossedpolarizer. EG&G FND-100 photodiodes in a simple cir-cuit with no preamplifier were used to detect the pulses.The 200-ns Q-switched pulses were long in comparison tothe photodiode rise time (< 1 ns), so that we could directlydetect the output without power averaging. Neutral-density filters were used as needed to prevent photodiodedamage.

The silica fiber used was equivalent to a step-index fi-ber with a core radius of 2.6 Axm and A = 4.8 X 10' andwas 3.2 m in length. The fiber was held at either end bymeans of low-vacuum chucks and was allowed to rest in along, V-shaped piece of aluminum to minimize birefrin-gence effects caused by clamping, bending, and twisting.Although a more highly birefringent fiber would be moreresistant to external effects and therefore easier to handle,the critical power would also increase. The informationused for calculating the fiber beat length was obtained ata cw power level of -100 mW to minimize nonlinear ef-fects. At the same time, a low-power polarization extinc-tion ratio of better than 500:1 was measured. The beatlength (Lb) was estimated to be -4 m, so that the fiberlength L 0.8 Lb and the corresponding critical powerP, 250 W. At peak-pulse powers approaching 1 kW,stimulated Raman and Brillouin scattering as well assignificant pump depletion resulting from the Brillouinscattering became problems. Therefore the peak powerwas always kept below this experimentally determinedstimulated-scattering threshold.

Results and Comparison with TheoryThe results of the coupled-mode theory for a weaklytwisted fiber were used to predict the transmissionthrough the exit analyzer. Exact quantitative agreementbetween theory and experiment could not be expected be-cause of the difficulty of either mounting a fiber with ab-solutely no twist or precisely measuring the twist rate.However, we found that we could obtain good agreementbetween theory and experiment by adjusting the variousparameters in the theory.

To compare the experiment with the theory, we approxi-mated the input pulses as Gaussian pulses with anexponentially decaying tail and with 15% (peak-to-peak)sinusoidal modulation riding on the pulse that was due tolongitudinal mode beating in the laser [Fig. 6(a)]. Weused the theory for a fiber with a twist rate of 0.1 twistper beat length to calculate the expected output polariza-tion state for each point of the (model) input to predict theoutput pulse shapes. Simulations showing the predictedoutput through the crossed polarizer (Pj), together withthe corresponding experimental oscilloscope traces, areshown in Figs. 6(b)-6(f). Note that the peak power given

Feldman et al.

Vol. 10, No. 7/July 1993/J. Opt. Soc. Am. B 1199

0 0 30(TIME (ns) -200 0TIME (ns)

(a) (c)

PEAK POWER = 1.00.02

0.01 - ~~~~~~~~~~0.010.01~ ~~~~~~~~-U

-200 0TIME (ns)

PEAK POWER = 0.45

0.2

1 -200 0TIME (ns)

(b) (d) (f)Fig. 6. Comparison of the oscilloscope traces from the experiment with the theoretical calculations for a weakly twisted fiber. Theupper half of each part of the figure shows the theoretically predicted pulse shape transmitted through the crossed polarizer. The uppertrace in the oscilloscope photographs corresponds to P11, while the lower trace corresponds to P 1. The lower oscilloscope trace should becompared with the theoretical prediction. Note that the vertical scale on the oscilloscope plots is in arbitrary units, and therefore thetransmittance near the slow and fast axes may not be directly compared. In addition, after data were taken for (e), the neutral-densityfilters in front of the photodiode were changed from 1.9 to 3.1 for (f). (a) Theoretical approximation and oscilloscope trace of theQ-switched pulse input into the fiber. The modulation is due to longitudinal mode beating in the laser. (b) Light at the critical powerinput at =3 to the slow axis. Little light leaks through the crossed polarizer. (c) Intense light (P = 2.5) input to the fiber at =30 to theslow axis. Again, little pulse shaping is observed. (d) Low power light (P = 0.45) is input to the fiber at -10' to the fast axis. Nonlin-ear pulse-shaping effects begin to appear at a light intensity of only one half of the critical power. (e) Light at the critical power is inputto the fiber at =10 from the fast axis. Significant pulse shaping is observed at the critical power. The fiber-crossed polarizer systemis far more transmitting for the central region of the pulse than in the wings. (f) Intense light (P = 2.5) input at -10 from the fastaxis. Significant pulse shaping is observed. The pulse appears narrower since the power in the wings is so strongly suppressed relativeto the peak.

for each plot as well as the values along the power axis isin units normalized to P,. On the oscilloscope traces, thelower trace corresponds to P1 and the upper trace corre-sponds to PI1. The theoretical predictions should be com-pared with the lower oscilloscope traces. Typically only asmall percentage (maximum 10%) of the light is passedthrough the crossed polarizer; hence pulse-shaping effectsare most easily observed in P,.

For linearly polarized light input near the slow axis, afairly constant, low-transmission rate is expected throughthe analyzing polarizer; the output pulse undergoes littleshaping and should be similar to the input pulse [Figs. 6(b)and 6(c)]. Linearly polarized light input near the fastaxis undergoes pulse shaping that is highly power depen-dent. At power well below the critical power, the fast axis

is a stable eigenmode, the behavior is similar to that nearthe slow axis, and a small fraction of the pulse intensity istransmitted essentially unshaped. At a slightly higherpower, the fiber-and-analyzer combination results in aflat-topped output pulse [Fig. 6(d)] as a result of power-dependent pulse shaping. As the peak power is increasedto the critical power, switching occurs, and the central re-gion of the pulse is transmitted more strongly than thewings [Fig. 6(e)]. Finally, as the power continues toincrease above the critical power, this nonlinear transmis-sion becomes more pronounced, and the modulation depthof the pulse also increases [Fig. 6(f)]. The pulse appearsnarrower, since the wings are strongly suppressed relativeto the central peak. In addition, the peak of P11 tends tobecome slightly flatter as more power is transferred to P,.

2.5

1.5

0.5

-2C

PEAK POWER = 2.5

300 TIME (ns)

(e)

PEAK POWER = 2.5

300 TIME (ns)300

Feldman et al.

_

0

-1 300

1200 J. Opt. Soc. Am. B/Vol. 10, No. 7/July 1993

r_.a0

-

2

I . ._ _ - I I

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Normalized Input Power

Fig. 7. AM gain as a function of normalized input power. NoAM gain is expected below the critical power. AM gain as highas 10 may be realized.

Fig. 8. Oscilloscope trace showing a nearly fully modulated out-put pulse for an AM gain of -6. This expanded view was takenat the same time as the pulses in Figs. 6(d)-6(f) (after Ref. 13).

The inclusion of a small amount of twist in the theorygreatly improves the fit of the experiment to the theoreti-cal curves, as shown by a comparison of Fig. 1 of Ref. 13and Fig. 6 herein. Not only is the qualitative agreementbetter, as evidenced by the better match of the curveshapes, but the quantitative agreement is also improved.In Fig. 6(e) the ratio of the peak power to the power on theshoulder is approximately three for both theory and ex-periment. The ratio between the peak powers forFigs. 6(e) and 6(f) is 50 in the experiment and-30 in the theory if twist is included but only 8 if thetheory neglects fiber twist.

In the alignment of the analyzing optics, the assump-tion was made that if linearly polarized light into the fiberyields linearly polarized light out of the fiber at low power,then the light is traveling along a principal axis of thefiber. However, in a twisted fiber this condition is metalong the fiber nominal axes. Only for negligible or smalltwist do the nominal and principal axes coincide. Theexperimental assumption that the nominal axes and theprincipal axes are coincident is justified in light of the im-proved agreement between the theory and the experimentwhen a small twist is incorporated into the theory.

Amplitude Modulation GainAn important feature of nonlinear dynamical systems istheir ability to amplify fluctuations when they are drivennear an instability. For an input power in the vicinity ofthe critical power for the polarization instability, a smallrelative modulation of the input intensity can result in amuch larger modulation in the power transmitted in theorthogonal polarization.+ If we define the modulationdepth as AP/P, then the amplitude-modulation (AM) gainis defined as

IP _ 1AP 1 _ 1 DIgIAPi.J/Pin T

(21)

where gD = APa/APin reduces to the differential gain andthe transmissivity T = Pl/P1n. Figure 7 plots gAM versusthe normalized input power and shows that AM gains ashigh as 10 can be expected from this system. The experi-mental results shown in an expanded view in Fig. 8 con-firm the predicted AM gains. A 15% modulation in theinput intensity is transformed into a nearly 100% modula-tion in the output P,. The modulation in P11 remains thesame as that in the input. Essentially, the input to thisamplifier is a large amplitude pulse with a small modula-tion, while the output is a smaller amplitude pulse carry-ing a large modulation.

CONCLUSIONWhen the intensity of light incident upon an untwisted,birefringent optical fiber is low enough that nonlinear ef-fects are negligible, then linearly polarized light that isoriented along either of the fiber principal axes maintainsits state of polarization as it propagates. With increasinglight intensity a nonlinear birefringence is induced bymeans of the intensity-dependent refractive index. Whenthe induced birefringence becomes comparable with thenatural fiber birefringence, asymmetry results betweenthe fast- and slow-fiber modes. Along the slow axis theinduced birefringence adds to the natural birefringence;the slow axis therefore maintains stable guiding. Alongthe fast axis the induced birefringence opposes the natu-ral birefringence. The two birefringences exactly canceleach other at the critical power. For linearly polarizedlight above the critical power the fast axis becomes an un-stable saddle point, and even a slight deviation from per-fect linear polarization or from perfect alignment alongthe fast axis leads to completely different output polariza-tion states. Additionally, even at a given input polariza-tion small changes in the input intensity may lead todifferent output polarization states. Thus a signature ofthe polarization instability is that small changes in inputintensity may lead to large changes in the intensity trans-mitted through a crossed polarizer located at the fiberexit. If a weakly modulated beam is input into the fiberwith sufficient intensity to excite the polarization insta-bility, the output beam transmitted through the crossedpolarizer may have a significantly greater modulationdepth. This increase is referred to as AM gain.

The perturbation caused when the fiber is slightlytwisted does not significantly alter the dynamics de-scribed above. When the fiber is twisted, circular bire-fringence is induced that, in turn, leads to elliptically po-larized eigenmodes. At any input power light must belaunched with an appropriate elliptical polarization to

Feldman et al.

Vol. 10, No. 7/July 1993/J. Opt. Soc. Am. B 1201

maintain the state of polarization as the light propagates.These elliptically polarized eigenmodes are aligned alongthe principal axes of the fiber. In a twisted fiber it is nolonger appropriate to speak of the presence of fast or slowaxes because linearly polarized light launched along theseaxes does not maintain its state of polarization. How-ever, the principal axes for the linear, natural fiber bire-fringence remain a convenient reference for measuringthe orientation of the light polarization as it evolves in thefiber. In this case, the birefringence induced by intenseelliptically polarized light may cancel the combined natu-ral and twist-induced birefringence.

When long lengths of low-birefringence fiber are usedin nonlinear switching applications, the effect of twist canbe quite important. A number of recent papers have de-scribed polarization-switching experiments involving bire-fringent fibers that are multiple beat lengths long.27 28 Inthe absence of twist, theory predicts that the switchingpower P, will depend on the polarization angle of the inputfield with respect to the birefringence 9 In particular,near the slow axis (0 0), P, scales with the fiber lengthas 1L, while at 45 (or for circularly polarized input) itscales as 1\/E. Near the fast axis, the switching power isindependent of length and depends only on fiber birefrin-gence and nonlinearity. Unavoidable twists in a long fiberwill lead to some averaging over these length dependencesof the switching power. This would explain some dis-crepancies noted in Ref. 27 between experiments and thetwist-free theory.9

In conclusion, we have presented direct observations ofAM gain and of asymmetry between the fast and slowmodes of a birefringent fiber resulting from polarizationinstabilities. Good qualitative agreement is obtained be-tween the experiment and the theory. We observed a six-fold increase in the modulation depth of the input beam byprobing the instability. Such AM gain has potential usein a novel amplifier with moderate power and fiber lengthrequirements.

ACKNOWLEDGMENTSThe authors thank Corning Glass Works for the donationof the low-birefringence fiber necessary to conduct theexperiment. This work is based on dissertation researchsupported under a National Science Foundation GraduateFellowship awarded to S. R Feldman. The authors ac-knowledge National Science Foundation support undergrants EET-8552520 (to D. A. Weinberger) and EET-8712877 (to H. G. Winful).

*Present address, General Electric Corporate Researchand Development, Schenectady, New York 12309.

'Present address, Department of Physics, Smith College,Northampton, Massachusetts 01063.

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