Kerr effect in all-fiber cavities of optical gyros

Brigitte Lamouroux, Bernard Prade, and Jean-Yves Vinet

The transmission spectrum of an all-fiber cavity is theoretically analyzed, in order to evaluate the perturba-tion induced in the eigenfrequency measurement by the Kerr effect. Existence of a threshold separating thebistable operation region from the monostable is shown. In the weakly nonlinear operation, the asymmetry ofthe line is evaluated and the resulting error in the frequency measurement is analytically derived.

1. IntroductionRecall briefly that some fiber resonant gyros involve

all-fiber cavities,1 2 the eigenfrequencies of which aresplit by the rotation induced Sagnac effect. The rota-tion rate is directly related to the frequency lag be-tween two counterpropagating eigenmodes.

We intend to study the perturbation in the measure-ment process induced by the Kerr effect when thesemodes are alternatively launched into the cavity, thusdiscarding wave mixing effects. First, special atten-tion will be devoted to the Kerr effect in optical fiber toget the effective index perturbation.

11. Nonlinear Effective Index of a FiberIn what follows we shall consider only weakly guid-

ing monomode fibers; therefore, we neglect polariza-tion effects and use a transverse approximation of thefields. Within this frame, we consider the simplestsecond order expansion of the refractive index of anisotropic dielectric medium:

n = n(°) + ,IEI 2, (1)

where n(O) is the ordinary linear refractive index and Ethe amplitude of the optical field.

It is well known that the effective index of a dielec-tric guide obeys the following relation:

S f(E*curl H - H*curl E)dS + ik f(HH* + n2EE*)dSe k f (E X H* + E* X H)edS

(2)

where E and H are the transverse amplitudes of theEM field. The integrations are taken over the wholetransverse cross section (r,O) of the guide.

By substituting Eq. (1) into Eq. (2) and consideringthe nonlinear terms as a perturbation, we can write:

nfff = n(f) + n(l); E = E(°) + E(l); H = H(0 ) + H(l),

where n(l), E(, HM1 ) are perturbation terms. By usingthe variational3 properties of Eq. (2), we obtain:

n() - 2 f n(0o)ylE(0)12 . EO) 12 dSeff (E(°) X H(°)* + E(°) X H(o))edS

If we let P be the total power carried by the mode, weget:

ce n(°)en ci)= f ynO)fFO) 14 dS.neff 2P I.eff~~ 4

Taking into account the weak guiding approxima-tion, we have replaced n(O) by n(°) in Eq. (3) and, more-over, supposed y to be uniform all over the cross sec-tion of the guide. Consider a circular core monomodefiber; it has been shown that the fundamental modecan be fairly approximated by a Gaussian function ofthe form:

E = A exp(-r2/w 2), (5)

where A is a constant related to the carried power P by:P = I 2 x~~~~P = con(O)A2 exp(-2r2/w') r dr dO,

o of ' (6)

where w is the waist of the beam related to the normal-ized frequency V and the core radius a, according toMarcuse4 by:

w = a(0.65 + 1.619V-31/2 + 2.879V-6). (7)

By substituting Eq. (5) into Eq. (4), we finally obtain

n() = (0) Peff~i - yneff -

1rw2(8)

The authors are with Ecole Polytechnique-ENSTA, AppliedOptics Laboratory, Centre de l'Yvette, 91120 Palaiseau, France.

Received 1 August 1989.0003-6935/90/060750-04$02.00/0.© 1990 Optical Society of America.

Note that the circle of radius w may be viewed as theeffective cross section of the guide.

For a step-index silica fiber with a core radius of a =2 ,um and An = 4 X 10-3 illuminated at wavelength X =0.6328 ,m, we get n(l) = 8 X 10-13P (P is expressed inmW).

750 APPLIED OPTICS / Vol. 29, No. 6 / 20 February 1990

(3)

(4)

Ill. Study of the Resonance Line Shape of an All-FiberCavity

An all-fiber cavity may consist of a single fiber tailfolded onto itself by means of a strongly coupling di-rectional coupler (see Fig. 1), which will be supposed tohave a linear index, owing to its short effective length.

Let a,, a2, a3, a4 be the optical amplitudes at the fourgates of the coupler and let us set T = a4 /a112, I = a3 12

and Io = la12. Thus Io is the power launched in theinput fiber tail, whereas I is the intracavity powereventually enhanced by the resonance. The followingrelations have been established5:

T=1- 11 + 4S 2 sin2'{

I1= SIO

1 + 4S2 sin 2 I

'I = kneffL/2 + 7r/4,

(9)

(10)

COUPLEUR

a 1

Fig. 1. Sketch of the all-fiber cavity closed by means of a direction-al coupler.

1

(11)

where S is a dimensionless parameter proportional tothe finesse which is assumed much larger than unity, Lis the length of the fiber loop, and k = wic. Moreover,in Eqs. (9) and (10), propagation losses were neglected,and the coupling coefficient was adjusted so as to givezero transmission of the cavity at resonance. We canintroduce further the time constant of the linear cavi-ty:

r = n(°)LS/C

and the nonlinear perturbation coefficient:

A = n(°f) 2(° Ar- e- Ic 2 n~ 0'

where

(12)

0.5

0-4 -2 0 2 4

Fig. 2. Resonance curve of a nonlinear cavity, monostable region:S = 100; A = 0; 2 X 10-5; 5 X 10-5.

1

(13)

0.5

The following dimensionless variables will be used,w(°) being a given eigenfrequency of the linear cavity:

{X = (-- W(0))T

ty = I/Io

The phase becomes:

X 1 (2)*=I-+Ay+Bxy with B = ( IO.

2S 28 ~oneff

By substituting I into Eq. (10), we obtain:

S

1 +4S2 sin 2(xI2S + Ay + Bxy)

(14)0 v' . , . I * . . . . J

-16 -8 0 8 16Fig. 3. Resonance curve of a nonlinear cavity, bistable region:

S = 100; A = 0; 1 X 10-4; 2 X 1O-4; 5 X 10-4; 10-3.

(15)

It seems interesting to evaluate the threshold be-tween the monostable and the bistable regime. The

(16) -solution of Eq. (16) may be thought of as the solution ofthe system:

Inversion of the preceding formula makes possible toplot the relation:

= B4- sin,1 S )- 1)AY} -

S= 1 + 4S2 sin2

qI

P = + Ay28

(17)

We obtain characteristic diagrams as shown on Figs.2 and 3. In Fig. 2, a small nonlinear perturbation wasassumed and the resulting curve exhibits a monostablebehavior, whereas in Figure 3, a stronger value of A wastaken, leading to bistability.

(18a)

(18b)

Depending on the parameter A, Eq. (18) is seen to haveone solution (monostable operation) or up to three(bistable operation). Consider an approximate solu-tion T which differs from an exact one TO°) by a smallamount e:

20 February 1990 / Vol. 29, No. 6 / APPLIED OPTICS 751

= (O) + C. (19)

By substituting Eq. (19) into Eq. (18a) and keepingthe first order result, then substituting again that re-sult into Eq. (19), we get the new approximate solution:

where

e'/e = - 8AS 3 sinI(O) cosq(O)[1 + 4S2 sin 2 (0 )] 2 . (21)

The solution I(O) is stable if IE'/el < 1. The discus-sion is made simpler by noting that y nearly vanisheswhen ' differs significantly from zero. We can there-fore use the Lorentzian approximation and write:

101 < 1(1 + 02)2 4AS 2 (22)

(25)2 T) f d

'T=-m T(t) sinwmtdt.

In the preceding integration, values of t small com-pared to the period will not contribute significantly;now, if t is not neighboring zero, we can write at firstorder in x0:

T = x0T1 + 71T2 (26)

with

=1 A sin2 a A(I + 2-/T, =J o (.1 +A2si 2a,)2 d

T = 1 2 A sin 2a da = A(1 + A2

3/2

2ir Jo (1 + A2 sin 2 )3a 8 )52

(27)

(28)

with = 0/2S. The left hand side of inequality (22)has a maximum of 9/16 3 and, thus, the condition (22)is fulfilled as long as 4AS2 < 163)/9.

The parameter 7 = 4AS 2 appears, thus, to be therelevant nonlinear characteristic number and we seeby example that the bistable regime arises whenX > ac= 3.08. The S2 dependence of X results from the lineardependence on S of both the intracavity stored powerand the effective optical path length.

we find

= A2 32[XO + 3 A2] (29)

In the linear case ( = 0), it is known6 that theoptimal A, which maximizes the shot noise limitedsignal to noise ratio, is A = 2-1/2. Therefore, if theservo loop is tuned to make T vanishing, it results abias of

Ax = - /8.

IV. Eigenfrequency MeasurementsIn resonant ring cavity gyros, the angular velocity

measurement is reduced to a frequency shift measure-ment between two counterpropagating optical eigen-modes. To measure each eigenfrequency, the lineshape is periodically scanned at a frequency win, givinga modulated output power. By detecting the firstharmonic term of that signal, one usually obtains afeedback command which is used to serve the laserfrequency at the line center. In the linear regime, theline shape is symmetrical with respect to the line cen-ter and when the offset of the modulation is zero, thefirst harmonic vanishes and may be regarded as anunbiased feedback signal. Now, in the nonlinear re-gime, even far beyond the bistability threshold, theline shape was seen above to be asymmetrical, a corre-sponding bias is expected to arise and must be evaluat-ed.

In the Lorentzian approximation, the transmissionfactor T is of the form:

0 = o2 (23)1 + 02

with 0 = x + 1/2X7(l - T). By solving Eq. (23) in T, weget at first order with respect to 7:

x2X (24)1 + X2

+ (1+ X2)3

The line scanning will be represented by x = xo + Asin wt, where xo is the small offset we want to correctand A a scanning amplitude. We shall assume Ixol <<Aand A < 1.

The first harmonic amplitude of T is given by

(30)

Constant bias can always be removed, but in thepreceding expression contains the power of thesource which although somewhat averaged by the lock-in detection, is nevertheless yet subject to fluctuationsmuch more difficult to compensate. These fluctua-tions appear as a noise on the measurement of theSagnac frequency difference.

Recall that for a mode propagating in a rotatingfiber, the apparent trip duration undergoes the sameSagnac perturbation as a plane wave circulating in thevacuum along the same rotating geometrical path,7namely, for a cavity, each eigenfrequency o is splitinto two new eigenfrequencies wocw and woccw corre-sponding to clockwise and counterclockwise propagat-ing eigenmodes. We have for a circular fiber loop oflength L:

2L6WS = OC-ZOC = (0 .(31)

n~fX

where Q denotes the angular velocity and X the opticalwavelength. The noise on the frequency bias due tononlinearities is otherwise:

aWN = Ax = 1 anr 8 r

(32)

where bn results of the long term power drift of thelaser source.

Consider, for example, a 1-m long cavity with S =100 and a source power of 10 ,gW; to detect 10% of theearth's rotation rate, the fluctuations of the relativeintensities of the two counterpropagating waves mustbe kept <4%.

752 APPLIED OPTICS / Vol. 29, No. 6 / 20 February 1990

= (0) + e, (20)

V. ConclusionThe influence of the Kerr effect in dielectric wave-

guides upon the measurement of the Sagnac frequencyshift in a rotating all fiber cavity was evaluated. Athreshold for bistable operation was given. Even inthe weakly nonlinear regime, much smaller than thatthreshold, a noise arises in the frequency measure-ment, due to the differential intensity fluctuationsbetween the two successive counterpropagating eigen-modes.

The authors would like to thank the Director of theDirection Recherches Etudes Techniques for permis-sion to publish this paper.

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Oster, and W. Schroder, "Passive Fiber Resonator Gyro," inProceedings Symposium on Gyro Techology (Stuttgart, Germa-ny, September 22-23, 1986) pp. 8.0-8.19.

2. M. Ohtsu and S. Araki, "Using a 1.5 um DFB InGaAsP Laser in aPassive Ring Cavity-Type Fiber Gyroscope," Appl. Opt. 26,464-470 (1987).

3. D. Nelson, Electric, Optic and Acoustic Interactions in Dielec-trics (Wiley, New York, 1979).

4. D. Marcuse, "Loss Analysis of Single-Mode Fiber Splices," BellSyst. Tech. J. 56, 703-717 (1977).

5. L. F. Stokes, M. Chodorow, and H. J. Shaw, "All-Single-ModeFiber Resonator," Opt. Lett. 7, 288-290 (1982).

6. G. A. Sanders, M. G. Prentiss, and S. Ezekiel, "Passive RingResonator Method for Sensitive Inertial Rotation Measurementin Geophysics and Relativity," Opt. Lett. 16, 569-571 (1981).

7. B. Prade and J. Y. Vinet, "Guided Optics in Rotating DielectricMedia," II Nuovo Cimento 10iB, 323-334 (1988).

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