Parametric waves in birefringent fiber oscillators

s.

15 March 1996

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IfiB +

OPTICS COMMUNICATIONS

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Optics Communications

124 (1996) 628-641

Full length article

Parametric waves in birefringent fiber oscillators S. Longhi Centro di Elettronica

Quantistica e Strumentazione Elettronica de1 CNR, Dipartimento P.za L. da Vinci 32, 20133 Milan, Italy Received 29 June 1995; accepted

di Fisica de1 Politecnico,

12 October 1995

Abstract

Parametric generation of waves in a weakly damped nonlinear-optical system is theoretically investigated. We consider a birefringent, nonlinear and dispersive fiber ring that is driven to oscillation via a three-wave interaction in a nonlinear ,I)‘) medium. Stability of the parametric waves beyond threshold is investigated by use of standard linear stability analysis and by amplitude equations and secondary instabilities that are typical of nonlinear spatially extended systems are discussed. We also investigate the existence of stable optical solitons in the weakly dissipative limit, pointing out the close analogy between these solitons and the parametrically excited solitary waves studied in hydrodynamics.

1. Introduction

Propagation of the electromagnetic field in nonlinear optical fibers offers the possibility of exploring a rich variety of nonlinear effects and instabilities that may be described in terms of universal dynamical models widely encountered in different physical contexts. A well-known example is provided by the modulational (or Benjamin-Feir) instability, which is a general phenomenon of wave propagation in nonlinear dispersive media [ 1] . One of the most interesting features of nonlinear effects in optical fibers is certainly the possibility of propagating optical solitons. The existence of solitary waves is closely related to the possibility of reducing the Maxwell equations to the nonlinear Schroedinger equation, which is known to possess localized propagating pulse solutions in the ideal dissipationless case. Solitary waves in birefringent fibers have also been investigated to include the vectorial nature of the electric field. In the dissipationless case, polarization dynamics of vector solitons may give rise to a rich variety of nonlinear effects that are of particular 0030.4018/96/$12.00 0 1996 Elsevier Science B.V. All rights reserved SSDIOO30-4018(95)00641-9

interest in the context of soliton switching [ 2,3]. The soliton-formation capability of optical fibers has also been successfully exploited to develop new kinds of lasers for producing ultrashort pulses. Since the first demonstration of the soliton laser [ 41, much effort has been expended on the generation of stable soliton pulses in fiber lasers [ 51. These lasers consist of a fiber ring including typically an erbium-doped fiber amplifier [ 61 which provides the gain of the oscillator. Pulse propagation in fiber lasers is governed by a complex Ginzburg-Landau equation [ 71 and stabilization of the soliton pulses against continuous-wave oscillation requires the use of some external techniques, such as active or passive mode-locking of the cavity modes [ 81 or continuous sliding of the laser frequency [ 91. From a fundamental point of view, the possibility of generating ultrashort optical pulses in fiber lasers is closely related to the existence of structurally stable solitary waves of the complex Ginzburg-Landau equation [ l& 121. This equation is universal in the sense that it represents a general model for pattern formation in various fields of physics, such as plasma physics, hydrodynam-

S. Longhi/Opics

Communications

its and nonlinear optics [ lo]. The excitation of parametric waves in nonlinear processes provides another class of pattern-forming systems widely observed in different fields of physics, particularly in hydrodynamics [ lo]. Parametric waves are nonlinear waves that arise quite generally in weakly damped nonlinear systems driven far from equilibrium by an external force oscillating at a frequency twice their natural frequency. An important example of parametric waves in hydrodynamics is the excitation of waves on a surface of a vibrating liquid layer, known as Farady waves [ 131. In this paper we present a relevant example of parametric waves that is taken from nonlinear optics. We discuss the nonlinear dynamics of a birefringent fiber ring which employs a parametric amplifier. Excitation of parametric waves arises in this case from three-wave interaction in a nonlinear x(‘) medium. This means that the gain is provided by the conversion of a pump field into idler and signal fields in the nonlinear xc2) medium. Signal and idler waves are assumed to have the same frequency and to be polarized along the two polarization axes of the fiber. The coupled equations that govern the dynamics of the two polarization modes (derived in Section 2) are expressed by two nonlinear Schroedinger equations, modified to include the effects of cavity losses, detuning and parametric gain. The oscillation condition for the generation of parametric waves is investigated in Section 3, where it is shown that the sign of the detuning parameter plays a fundamental role in defining the nature of the bifurcation of the nonlasing solution and in determining the patterns of the two polarization modes. It is predicted that the onset of parametric waves can be either supercritical (smooth) or subcritical (hysteretic). The stability of these waves beyond threshold is investigated both by deriving amplitude equations (in the supercritical case) and by direct linear stability analysis of the nonlinear modes. The emergence of phase and amplitude instabilities beyond threshold is discussed in Section 4. Above the neutral stability curve the parametric waves may become unstable to the growth of low-frequency perturbations (Eckhaus instability) and the stability boundary in the parameter space is derived by a direct analysis of the low-frequency behavior of the matrix eigenvalues in the linearized problem. Amplitude instabilities have also been observed in numerical simulations but only in the subcritical case and below the neutral stability curve. In Section 5 the existence of

124 (1996) 628-641

629

solitary waves is investigated in the weakly dissipative limit by applying standard perturbation analysis. This section extends the results of recent studies on the effects of a phase-sensitive gain on optical soliton generation and transmission [ 15,161 to the case of a birefringent fiber, pointing out the hydrodynamical counterpart of this phenomenon. Existence of stable parametrically excited solitary waves is in fact a wellknown phenomenon in hydrodynamics [ 12,141 and is related to the presence of a subcritical instability, so that in a suitable range of the bifurcation parameter the trivial homogeneous state is stable against infinitesimal disturbances, but can be triggered into an oscillatory state by a finite disturbance [ 121.

2. Description of the model and derivation of the coupled field equations Let us consider the cavity configuration containing a degenerate parametric amplifier, e.g. a nonlinear xc2) medium which converts coherent pump radiation at frequency 2w0 into radiation at frequency w,. The electric field at frequency w, resonates in the ring and its expression along the propagation direction z of the fiber can be written as

00

Output Coupler h Fig. 1. Schematic of the nonlinear, dispersive and birefringent fiber ring cavity with degenerate parametric amplification. The pump field at frequency 204, is converted into degenerate signal and idler fields at frequency o, in the nonlinear x”’ medium. Signal and idler waves are assumed to be polarized along the two axes of the birefringent fiber, and phase-matching of type II is assumed in the parametric conversion process. L is the length of the ring and h is the cavity loss at frequency o, for both polarization states.

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S. Longhi / Optics Communications

wot>le^,

E(z, t) = U(z, t)exp[i(kg+V(z,

t)exp[i(Zoz-w,t)]e^,+c.c.,

(1)

where e^i.2represent the polarization states, U and V are the slowly-varying complex amplitudes of the modes, k = k(w) and I = Z(W) are the dispersion relations corresponding to the two eigenmodes and k,, = k( CC+,), 1, = I( q,) . In writing Eq. ( 1), we assume the planewave model and do not consider the transverse field distribution, which we suppose to be the same for both polarization modes and not substantially modified by nonlinearity [ 171. Moreover, we assume that both modes are close to resonance with two resonances of the ring cavity, e.g. we assume that there are two integers n and m such that the detunings vi = k,J - 2nrr and v2 = Z,,L- 2mn are much smaller than 2rr. Parametric interaction in the nonlinear xC2’ medium is modeled assuming collinear phase-matching of type II between pump, signal and idler fields [ 181, the signal and the idler fields being polarized along the two fiber polarization states &r.2. In this way, both polarization modes in Eq. ( 1) are simultaneously involved in the parametric conversion process of pump photons. If we assume that the field envelope suffers small changes after propagation in one round-trip, the effects of losses and parametric gain can be considered distributed in distance. In the slowly-varying envelope approximation, the mode amplitudes U and V satisfy the coupled nonlinear equations [ 171: ill,U+ik’~,U--~k”~~U+(2a+bco~~6)~U~~U

Xexp[i(k,-Z,)z]

g,=p-

(a02fJv,

(3)

where u is the nonlinear conversion parameter, ,u = aZE,, 1 is the length of the xC2’ medium and Ep is the amplitude of the pump field. Eqs. (2) are associated with the boundary conditions = U(0, 0 ,

U(L t)exp(W) V(L, t)exp(iZ&)

(da)

=V(O, t) ,

Cab)

which express the closure of the ring. Introducing the new variables v=z, r=t[(Z’+k’)/2]z, Eqs. (2) assume the form ia,U+i&,U-&Y’@J+(2u+bcos26)~U~*U

exp[ - 2i( k, - I,) q] + b cosd sin6{ U’V*

+(21u12+ IV12)V

Xexp[-i(ko-b)gll +i(h-g,)UIL-i(g,lL)V*=O,

(24

+i(A-g,)UIL-i(g,lL)V*=O, ia,V+ iZ’a,V- iZ”afV+ (2a + b

cos26)

I VI

*V

&Z”@+ (2~ + b cos26) 1VI *V

Xexp[ -i(k,-Z,)v]

lU12)U

+bcos8sin6{V2U* +(21Vj2+

IUl’)U

xexp[i(ko-4drll)

xexp[i(k”-kdzll +i(h-g,)VIL-i(g,/L)U*=O.

iSa,V-

Xexp[2i(k,-Zo)n]

+b cos4sin6{V2U* +(2jVj2+

ia,V-

(Sal

+ (2~ + 2b sin28) ( U \ *V+ b cos26U2V*

+ (2a + 2b sin24) 1U I 2V+ b cos26U2V*

Xexp[ -i(k,-Z,)z]

3

xew[i(k,-kd771

IV12)V

Xexp[-i(k,,--Z,,)zl]

xexp[2i(k,-Z,)z]

g,=p=i2

X

+bcostYsin9{U*V* +(21U12+

In Eqs. (2)) a and b are material coefficients which depend on the specific form of the tensorial third-order nonlinear susceptibility (a = b in silica fibers), 8 is the ellipticity angle of the polarization eigenmodes ( 8 = 0 for a linearly polarized fiber and 6= ~12 for a circularly birefringent fiber), h is the loss per round-trip (including coupling losses), g, and g, describe parametric interaction in the nonlinear x’*’ medium, k’, I’, k” and Z”are the first and second derivatives with respect to frequency of the dispersion functions k= k(w) and 1 = Z(w), evaluated at w = 0,. The expressions of the coefficients g, and g, can be obtained by treating the amplification of the idler and signal waves in the nonlinear crystal perturbatively, assuming the lower pump conversion limit. In the case of perfect phase-matching and up to second order, we obtain (see Appendix A) :

+ (2~ + 2b sin28) I VI 2U+ b cos29V2U*

+(2~+2bsin*9-)~V~~U+bcos~9V~U* Xexp[ -2i(k,-Z,)z]

I24 (1996) 628-641

+i(h-g,)V/L-i(g,/L)U*=O,

(2b)

and the boundary conditions

(5b) (Eqs. (4) ) yield

S. Longlzi/Optics

U(0,

~)=(l+iv*)U(L,

7-TR),

V(0, 7)=(1+ivz)V(L,

7-TR),

Communications

(6)

where S = (k’ - I’) /2 is the group velocity mismatch between the two polarization states and TR= L( k’ + I’) /2 is the average cavity round-trip time. The parameter g2 given by Eq. (3) depends on the field envelopes U and V at second order, and therefore its effect is to modify the nonlinear fiber parameters a and b so as to include saturation of the pump photons at leading order. Note that, in deriving Eq. (6), we have used the near-resonance condition 1ul,* 1-=K27~ and we have expanded the exponential terms in Eqs. (4) to first order. Eqs. (5,6) allow one to determine the temporal evolution of the field envelopes U and V at successive round-trips for a given initial field distribution. In particular, integration of Eqs. (5) permits to determine the field envelopes for a fixed value of the time r at 71=L starting from q=O, and therefore Eqs. (6) represent a map. Assuming that the field envelopes suffer small changes per round-trip, we can derive a continuation of the map. Integrating both sides of Eqs. (5) with respect to 7~ over the interval [0, L] and assuming that U and V are almost constant (mean-field limit), we get the following equations from Eq. (6) after continuation: T,&lJ=

-L6a,U-$ik”La~U+(g,--h+iv,)U

+g,V*+i[(2a+b +(2a+2b

sin26)LIV1*U],

T,a,V=L&3,V-$iZ”L@+ +g,U*+i[(2a+b + (2a+2b

cos2@)L] U12U

sin26)Ll

(74

(g,-h+iv2)V cos*6)LIV/*V U12V] ,

(7b)

where T is the slow-time variable describing the evolution of the field envelopes at successive round-trips. In deriving Eqs. (7), we have assumed that the intrinsic birefringent length of the fiber (defined as 2rr/ I k, - lo I ) is much larger than the fiber length L, so that the terms in which exponential factors appear in Eqs. (5) are rapidly oscillating and may be neglected in the mean-field limit. This assumption is certainly satisfied in the case of relatively large birebirefringent fibers, where typical intrinsic fringent lengths at 1.5 p,rn are concentrated in the range IO-100 cm [ 191. In order to reduce Eqs. (7) to

124 (1996) 628-641

631

normalized form, we next assume that light is propagating in the anomalous dispersion regime where I?, Z”< 0 and that, as is appropriate for optical fibers, the small difference between k” and Z”may be neglected. In this case, introducing the scaled variables n = ( - 2/ Lk”)“*r, y=TIT,, A= [L(2u+b cos6)]“*U exp { -i[(vl-v2)T/2TR] -i[&/k”]} and B=[L(2u + b cos28) ] 1’2V* exp{i[(v,-v2)T/2TR+i[&/ k”] }, we obtain from Eqs. (7) and Eq. (3) the following dimensionless coupled nonlinear equations

+i( IAl*+

IB12)A+~JB/2A,

@a>

ayB=( -A+4p2-iv)B+fi-iazB -i(

IA12+ IB12)B+~*IA12B,

(8b)

where we have set Y = ( ui + v2 - LS2/k”) I2 and E= i[(2b sin26-b cos29-)/(2u+b cos*6)] -{(al)‘/ [ L( 2u + b cos24) ] }. The physical meaning of all terms in Eqs. (8) can be summarized as follow: h and p represent the cavity losses and the parametric gain, respectively; v is an effective detuning parameter, the terms involving derivatives stand for fiber dispersion and the nonlinear terms describe self-phase modulation, cross-phase modulation and saturation effects. In particular, saturation of the pumping photons is measured by the real part of E, which is always negative. Note that, without loss of generality, the parametric gain may be assumed real and positive. Eqs. (8) govern the nonlinear dynamics between the two polarization modes of the fiber and represent the starting point of our analysis. We note that coupling between modes is due both to cross-phase modulation induced by third-order nonlinear susceptibility of the fiber and to parametric interaction caused by the nonlinear x(‘) medium. In order to specify the order of magnitude of the various terms which enter our calculation, we assume a high-finesse cavity and we treat the parameter A as a small parameter. We also assume that the detuning parameter v and the parametric gain ZJ in Eqs. (8) are of the same order of magnitude as A. In this case, from Eqs. (8) it follows that A and B are of the order of 0(h1’2) and the terms $A/2 and $B/2 in Eqs. (8) may be neglected. Note however that the order of magnitude of the real part of E may be 0( 1) and therefore saturation of the pump photons can not be neglected in general. For instance, for a linearly

632

S. Longhi /Optics Communications I24 (1996) 628-641

polarized SiOp glass fiber at &, = 1.5 p,m of length L = 5 m and a KTP nonlinear xc” crystal of length 4Y= lcm, we can estimate E= - 1 - 0.33i. Let us note that when neglecting cavity losses, parametric interaction and detuning effects, Eqs. (8) reduce to two coupled conservative nonlinear Schroedinger equations which have been extensively analyzed in the context of nonlinear fiber couplers [ 2,3]. In particular, cross-phase modulation has been demonstrated to play an important role in determining polarization dynamics and switching properties of vector solitons. The presence of losses and parametric gain in Eqs. (7) breaks the conservative nature of the equations and introduces new features in the dynamics of the polarization modes. A system of equations similar to Eqs. (8) is well-known, however, in other fields of physics; they describe quite generally parametric waves in weakly-damped oscillatory systems which are forced at twice their natural frequency [ 10,20,21].

3. Oscillation condition and parametric

waves

The coupled equations (8) have the trivial equilibrium zero solution A = B = 0 which corresponds to the fiber oscillator being below threshold. As usual, we expect the trivial solution to lose its stability as the pumping parameter p in Eqs. (8)) which plays the role of bifurcation parameter, is increased. It is interesting to note that, depending on the value of the pumping parameter, the nonlinear Eqs. (8) have also the following family of stationary (i.e. independent of y) nonlinear modes

the reference frequency w,. As a matter of fact, the frequency q should assume a discrete series of values in order to satisfy the boundary conditions; however, as in the case of fiber lasers, the cavity length is unusually large, leading to a correspondingly small free spectral range, and therefore we may treat the parameter q as a continuous variable. All the solutions given by Eq. (9) are exact solutions of the coupled nonlinear equations, and correspond to polarization modes with the same intensity (independent of time), but with different carrier frequencies equally detuned from w, by the amount a= ( -2/Ll?)“*q. The fact that the two polarization modes have different frequencies equally detuned from w, is a consequence of energy conservation in the parametric conversion process, so that each pump photon at frequency 2~0, generates a pair of Stokes- anti-Stokes photons at frequencies we - 0 and o,, + 0, respectively. Because the Stokes photons may be generated in either one of the two polarization states of the fiber, Eq. (9) is also an exact solution of Eqs. ( 8) under the transformation q -+ - q. In order to determine which of the solutions (9) the fiber oscillator selects above threshold, linear stability analysis of the trivial zero solution is needed. Therefore, as usual in the linear stability analysis, we add small perturbations to the zero solution and look for exponential growth of such perturbations. We find the following spectral problem: AA=(

AB= (-h-iv)B+@-i@,

A B

(h-E~)*+[q2-v-(2+Ei)X]*=~*, and the phase difference

(10)

( $-- 9) is given by

exp[i($-cp)]=[iq*+h-iv-(%+/3i)X]lp. (11) In Eq. ( lo), .srand ei are the real and the imaginary parts of E, respectively; /3 = 2 + ei and 4 is an arbitrary real parameter representing the frequency offset from

aexp(iqx)

(124 t 12b)

where A is growth rate of the perturbation. have solutions of the form

0 which are the analogous of parametric waves in hydrodynamics. In Eq. (9)) the field intensity X satisfies the second-order algebraic equation

-h+iv)A+@+i@A,

Eqs. (12)

,

where q is the frequency of the perturbation corresponding eigenvalues are given by A(q) = --A+ [$-

(q2- zy]“2.

and the (13)

Therefore, the trivial zero solution loses its stability via a steady-state bifurcation. The threshold for oscillation corresponds to the value of the pumping parameter p such that an eigenvalue vanishes, and is given by pn(q) = [P+

(q2- v)2]l’2.

(14)

Eq. ( 14) defines the neutral stability curve and shows at once that its behavior basically depends on the sign

S. Longhi / Optics Communications

0.00 -0.50

0.50

0.00

Frequency

q

Fig. 2. Sketch of the neutral stability cume for (a) a negative detuning (V = - 0.05) and (b) for a positive detuning (V = 0.05). The values of the other parameters are: A= 0.05, •~= - 1 and ei = - 0.5.

of the detuning parameter. In fact, when u < 0 the lower threshold is given by pti= ( hZ+ v”) “’ and corresponds to an homogeneous perturbation (with frequency q = 0)) which has the maximum growth rate. On the other hand, when v > 0, the lower threshold, given by pti = h, becomes independent of detuning, and the maximum growth rate corresponds to perturbations with frequencies + 6, as indicated in Fig. 2. It should be noted that in the latter case the validity of the model represented by Eqs. (8), which assumes a large separation between the fast timescale x and the slow one y, requires strictly that the period of the perturbation be much shorter than the cavity round-trip time T, (or, equivalently, that the detuning parameter Y be greater than - 27r2Lk”lTg). Therefore, for a cavity configuration very close to resonance, the validity of the averaged Eqs. ( 8) is broken and the map defined by Eqs. (5) and (6) should be considered. In the pos-

124 (1996) 628-641

633

itive detuning case, at the linear instability both modes with frequencies f & are equally good solutions, and so is any superposition of these. The linear stability analysis can not predict the linear combination of the two unstable modes which is actually selected, and weakly nonlinear analysis is needed to answer this question [ lo]. In Appendix B, we show that only one of the two perturbations may be stable above threshold. Therefore, when v > 0, field oscillation is characterized by two polarization modes having the same intensity (independent of time), but with different carrier frequencies equally detuned from w, by the amount ( - 2vILV)“2 . We note that the physical phenomenon which makes the threshold pth independent of detuning when v > 0 is quite similar to the origin of roll patterns in degenerate optical parametric oscillators [22] and of traveling-wave patterns in lasers [ 231, where diffraction plays the same role as fiber dispersion in our model. In particular, frequency shift of the carrier frequencies associated with fiber dispersion can compensate for off-resonance operation whenever Y is positive. As a final remark, we note that the threshold for oscillation of the mode with frequency q, given by the neutral stability curve Eq. (14)) coincides with the threshold for existence of the nonlinear mode given by Eq. (9) whenht;+p(q’V) 0, the threshold for existence of the nonlinear mode is less than that for oscillation, and is given by /G(4) =Jk+(qG+[h~~++(q2-V)]v(~+~*)

. (15)

In this case, the bifurcation is subcritical and there are two acceptable solutions of Eq. ( 10) when p, < p < Pi. However, the lower branch is always unstable, as it will be shown in the next section.

4. Stability analysis of polarization modes beyond threshold: secondary instabilities When the pump parameter p is increased above the threshold value ,u~, the fiber oscillator undergoes a steady-state bifurcation to polarization modes with the

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S. Longhi / Optics Communications

same carrier frequency for v < 0 or to polarization modes with different frequencies equally detuned from the reference frequency we for v > 0. In the former case, moreover, the bifurcation may be subcritical. An inspection of the neutral stability curves given in Fig. 2 for both signs of detuning shows that a continuous band of active modes may be excited above threshold, and nonlinear terms govern the competition among these modes. The question of mode selection and mode stability near threshold operation may be addressed in a general way by applying weakly nonlinear analysis in the vicinity of the bifurcation point, deriving an amplitude equation which is universal in the sense that it describes pattern-forming instabilities in widely different physical contexts [ 10,231. This is done in Appendix B for the positive detuning case, where the bifurcation is always supercritical. The validity of such an amplitude equation is limited, however, near threshold operation and can not correctly predict mode instabilities well beyond threshold. Nevertheless, in our case exact solutions of the nonlinear equations are available, and therefore a direct linear stability analysis of these solutions can be done. To this aim, we assume A=A,(l+u),

B=B,(l+h),

(16)

where A,, B, are the stationary solutions given by Eq. (9) and u and h are small perturbations. By substituting Eq. ( 16) into Eqs. (8)) we find the following linearized equations for the perturbations: $u=

(-h+iY)n+@zexp(i@)

+ (i+e)X(U+h+h*) t$h=(

-h-iv)h+pz4exp(

+iX(2u+u*) +i(&+iq)*u, -iO)

+(-i+e*)X(h+u+u*)-i(&+iq)*h,

(174

-iX(2h+h*) (17b)

where O= $J-- cp. The most general solution of Eqs. ( 17) is given by a linear combination of solutions of the form

exp ( Ay + iQx) .

In Eq. ( 18)) Q is the frequency of the perturbation A(Q) are the four eigenvalues of the matrix

(18)

and

124 (1996) 628-l

(4 MC

a2

-* $ a1 4

i a4

a3

a3

a4\

y*

2; ,

al a2

al

_

I

where a,‘= -h+iv+(3(i+e)X--(q&Q)*, a2= ix, a,=pexp(iO) +(i+e)X and a4=(i+E)X. The stationary solution is unstable to growth of perturbations with frequency Q if the matrix M has at least one eigenvalue with positive real part. Let us first consider stability against homogeneous perturbations, corresponding to Q = 0. In this case, a: = a;, and any eigenvalue of the matrix M is an eigenvalue of either M, or IV,, where M,=

M2= anda, =a: =a; . The eigenvalues of M2 are 0 and - h and therefore they do not lead to instability. Note that the existence of a neutral mode, corresponding to the eigenvalue with vanishing real part, is a consequence of the translational symmetry of Eqs. (8). On the other hand, the eigenvalues associated with Ml have two sources of instability. The first one is expressed by the condition - A + 2eJ> 0 and coincides with the secondary instability previously predicted and observed in Ref. [ 201 in the context of parametric waves generated in a horizontal layer of fluid submitted to vertical vibrations. In our case, however, this type of instability is absent, because or is always negative. The second source of instability arises when the bifurcation is subcritical and indicates that the lower branch of the stationary solution is always unstable. Consider now the most general case of nonhomogeneous perturbations; the condition for the rise of an instability could be determined analytically by use of the Routh-Hurwitz criterion, which allows one to check, directly from the properties of the matrix M, whether Re( A) < 0. However, due to the presence of a free parameter in the matrix coefficients, this method appears too involved for a full analysis and it seems more practical to numerically evaluate the frequency of the perturbations with the highest growth rate for each value of the pumping parameter and then look for the zero points of this function. Fig. 3 shows typical stability boundaries in

S. Lmghi/

Optics Communications

124 (1996) 628-641

635

to considering the stability against low-frequency perturbations (Q + 0)) instability may arise when the zero eigenvalue become positive. Let us indicate with A4 + c1A3 + c2A2 + c,A + c4 = 0 the characteristic polynomial of the matrixM, where the explicit expressions of the coefficients Ci are given in Appendix C. Expanding the zero eigenvalue in series around Q = 0 to second order and making use of the identity (&,&IQ) o=a = 0, it is straightforward to show that

-u.50

0.00

u.5u

Therefore stability of the stationary solution against low-frequency perturbations is ensured provided that 5_ @

0.20

g

which, using Eqs. (C3) and (C4) and after some algebra, reads explicitly as

EL

z

2

0.10 + ( 4pq2 0.00 -0.50

0.00

0.50

Frequency q Fig. 3. Stability diagrams in the plane (q, p) for parameter values as in Fig. 2. The thick solid lines represent the neutral stability curves, while the thick dashed lines (partially obscured by the solid curve in (b) ) correspond to the boundaries of existence of stationary solutions. In the portions of the neutral stability curves which lie above the dashed lines, parametric waves bifurcate subcritically. The shaded areas represent the stability domains as predicted by numerical analysis of the matrix eigenvalues in the linearized problem and coincide with the Eckhaus stability domains as given by Eq. (19). In (b) , the inset is an enlargement around the minima of the neutral stability curve, as indicated by the dashed circle. Note that very close to threshold, parametric waves are stable in a frequency band which has the usual parabolic shape as predicted by the amplitude equation.

the (q, ,u) plane obtained by numerical analysis for both positive and negative detuning cases. We can identify one source of the instability observed in the numerical analysis as phase instability arising from low-frequency unstable growth bands, known as Eckhaus instability [ lo]. In fact, the four eigenvalues A,(Q) (k= 1, 2, 3, 4) of the matrix M are analytic functions of Q and at Q = 0 one eigenvalue is zero and the other three are negative (not considering the lower branch in the subcritical case). If we limit our analysis

- p1 VfJ- 2p2)XZ + 2p1X3} < 0 )

(19)

where uo=q2- v, p1=p2+G and p2=heIf p( q2 - v) . In the positive detuning case and very close to threshold, the Euckhaus stability boundary (as indicated in the inset of Fig. 3 (b) ) has a parabolic shape as predicted by stability analysis of the amplitude equations (see Appendix B) . Beyond threshold, the exact boundary of the Eckhaus stability domain in the plane (q, p), defined by Eq. ( 19), shows, for parameter values of physical interest (h > 0, or < 0, p > 0)) the emergence of two lateral sub-bands which are separated by a region of unstable frequencies, as indicated in Fig. 3 for both signs of the detuning parameter. We note that, for the various parameter values we have investigated, numerical simulations indicate that the Eckhaus stability domain given by Eq. ( 19) coincides with the entire stability domain determined by direct numerical evaluation of the matrix eigenvalues, at least above the neutral stability curve. Therefore no high-frequency instabilities have been observed above the neutral stability curve. These features are in agreement with the results of Ref. [21], where stability of parametric waves was investigated in a different way by deriving a phase diffusion equation. The existence of two stable sub-bands separated by aregion of unstable frequencies can lead to formation of stable wave-number kinks

636

S. Longhi /Optics

Communications

2.

3

0.20

8 g I

0.10

O.O!OL

0.50 Frequency

q

Fig. 4. Emergence of amplitude instabilities below the neutral stability curve in the subcritical case. The solid line is the neutral stability curve, the dashed line is the boundary of existence of the stationary solutions and the dark-shaded area indicates the stability domain as predicted by numerical analysis of the matrix eigenvahtes. The lightly shaded area, partially obscured by the dark-shaded area, indicates the Eckhaus stability domain given by Eq. ( 19). The values ofvarious parametersare: V= -O.Ol,h=O.OS, eC= -0.5 and li= 1. [ 2 I]. In the present context, this means that by increasing the pumping parameter beyond the split-up point of the Eckhaus stability domain, the frequencies of the two polarization modes may change on the slow timestale between two stable frequencies which lie in the two stable subbands. High-frequency instabilities were observed in the numerical analysis below the neutral stability curve in the case of subcritical bifurcations, as shown in Fig. 4. In this case, the upper branch of the stationary solution, which is stable against low-frequency perturbations, may become unstable against perturbations of finite frequency Q. This type of instability is clearly illustrated in Fig. 5, where the real part of the eigenvalue with maximum growth rate is plotted as a function of the perturbing frequency Q.

124 (1996) 628-I

parameter. Stable solitary waves of Eqs. (8) are therefore expected to exist in the subcritical case. From an experimental point of view, these nonlinear solutions are of particular interest because they correspond to ultrashort pulses of light. The possibility of generating ultrashort pulses in fiber lasers has generated a vast literature and various mode-locking techniques have been proposed and experimentally demonstrated to produce stable short pulses. In fiber lasers, the interplay between group-velocity dispersion and self-phasemodulation leads to soliton shaping and temporal compression of the output pulses [4,5]. However, the gain element itself is not able to suppress continuous wave laser oscillation, and stability of the soliton pulse requires some other external actions, such as active or passive periodic modulation of the cavity losses [ 81 or continuous shifting of the frequency of the light that circulates inside the ring laser [ 91. On the other hand, the use of a parametric gain may spontaneously suppress continuous wave operation. This interesting feature has recently been investigated both in the fields of soliton transmission [ 151 and of mode-locking in degenerate optical parametric oscillators [ 161. We guess that this different behavior is closely related to the existence of a subcritical instability when the gain element becomes phase-sensitive. To investigate the existence of stable solitary waves of Eqs. (8)) we note that, in the case A= p= v = e=O, i.e. in absence of losses, parametric gain and detuning effects, the coupled equations (8) reduce to the Manakov equations,

5. Solitary waves Fauve and Thual [ 121 have recently shown that the possible origin of stable solitary waves widely observed in dissipative systems driven far from equilibrium is associated to the existence of a subcritical instability, which implies that two stable homogeneous states can coexist in a suitable range of the bifurcation

Frequency Q Fig. 5. The largest real part of any eigenvahre as a function of the perturbing frequency Q for the stationary solution corresponding to q = - 0.5 and y= 0.15. The values of the other parameters are the same as in Fig. 4.

S. Longhi/ Optics Communications 124 (1996) 628-641

which are integrable by using the inverse scattering transform [ 31. In this case, they posses a family of exact solitons, called vector solitons, given by [ 2,3] :

where 7 is an arbitrg parameter which defines the soliton width, y1 = 42~~ q and yZ = &v are the soliton amplitudes, p I f p, = 1, and the soliton’ s phases pi, (pz evolve according to q,(y) = $y + c,cJ,,,p*(y) = - $y + qO, qc, being an arbitrary soliton’s internal phase. To study the effects of losses, parametric gain and detuning on the soliton dynamics, we limit our analysis to the case of small values of h, p, E, and V, so that the adiabatic variations of the soliton parameters 77, ~i,~, (P,,~ under the action of these terms can be investigated using perturbation analysis. Note that the soliton parameters yi, yZ and 7 are not completely independent, but 3 + r’, = 27’ as a consequence of the condition pi +p2 = 1. The central point of the perturbation method is to assume that the vector soliton shape remains close to that given by the unperturbed problem (Eq. (20))) allowing the soliton’s parameters to become slowly-varying functions of y under the action of the perturbation terms. A standard procedure to determine the slow evolution of the soliton’s parameters is to derive the dynamical equations of a few loworder conserved quantities of the unperturbed problem, calculated by making the ansatz (20) [ 21. The adiabatic evolution of the soliton’s parameters can be derived here in a simpler way by multiplying both sides of Eq. (8a) and (8b) by A* and B* respectively, and integrating over the variablex. Making the ansatz (20)) we obtain

Wa)

(21b)

+;

- !j [3+(1+Ei)yT1.

Note that, introducing the quantity jYmdx (IA]‘JB]2)=($-~)/2~,whichisaconstant of motion in the absence of perturbations, Eqs. (21a, b) it is straightforward to show that

(214 I_ = from

aJ-=-2hl_. Therefore, after transient I- =O, which means that yi = y2, and so 7)= y, where y denotes either yi or y2. The physical explanation of this fact is obvious: the gain of each polarization mode is provided by the other polarization mode, and therefore the two modes must grow together. Introducing the phase difference the dynamics of the soliton’s parameters P=P2-~1? can be reduced to that of y and cp, which is governed by the following equations: a,y=

-2hy+2/_Ly

cosrp+ jE,Y )

a,cp=-2v-2~Sinrp-2~(1+~Ei).

(22a) (22b)

Depending on the values of the various parameters, Eqs. (22) may have no more than two fixed points. It is straightforward then to investigate the stability of these stationary solutions against infinitesimal disturbances by using standard linear stability analysis. If we assume that the smallness of the parameters A, p, E, and u is of the same order of magnitude, we can drop (at first order) the terms containing er and Ei in Eqs. (22a, b) . The stability condition gives sinrp< 0, and there is one stable stationary solution, given by coscp= Al/_&)

y=(-v+@T>.

(23)

The perturbation analysis does not, however, consider the problem of energy transfer from the solitary wave into dispersive waves. Stability of the solitary wave against the growth of radiation modes can be investigated by considering the continuous eigenvalue spectrum of the linearized Eqs. (8) around the solitary

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Communications

solution [ 161. This spectrum coincides with that given by Eq. ( 13). Therefore the solitary wave is stable against the growth of continuous modes only in the case of a subcritical bifurcation ( Y < 0 at first order) when the pumping parameter p varies in the range h
6. Conclusion Hydrodynamical aspects of the dynamics of a birefringent fiber ring driven to oscillation by parametric interaction in a nonlinear x”’ medium have been investigated. The two polarization modes of the fiber, which are coupled by means of cross-phase modulation and parametric conversion of the pump field, organize themselves in nonlinear waves characterized by modes with the same intensity but with different frequencies, equally detuned form the reference frequency to satisfy the energy conservation law in the parametric conversion process. Pattern selection and threshold condition at the onset of oscillation have been investigated and linear stability analysis of the nonlinear waves beyond threshold has shown the emergence of phase and amplitude instabilities which are commonly encountered in spatially extended systems driven far from equilibrium. As with parametrically excited solitary waves observed in hydrodynamics, stable optical solitons may exist when oscillation occurs via a subcritical instability. We envisage that this phenomenon is of particular interest in nonlinear optics and may have important applications to the generation and storage of ultrashort soliton trains.

Appendix A Let us consider collinear propagation of pump, signal and idler waves in a nonlinear ,$‘) medium of length +?’and assume phase matching of type II. In this case, although signal and idler fields are degenerate in frequency, they have different polarization and are considered to be different waves. Assuming perfect phase-matching and neglecting group-velocity mismatch inside the medium, the three-wave equations which describe parametric interaction are [ 181

124 (1996) 628-641

(a,+ ilca,)

u= av*z,

(A.11

(a,+ l/ca,)v=aU*z,

(A.21

(a,+ llc$)Z=

(A.3)

-2aUV,

where Z is the envelope of the pump field at frequency 2~0, and CTis the nonlinear cross section of the parametric process. Introducing the new variables 17= z/t, t= t-z/c, Eqs. (A.l-A.3) yield

a,u=dv*z,

(A.4)

a,v=

a-elJ*z )

(A.3

a&=

-2a~UV.

(A.6)

In the low pump-conversion limit, i.e. for ue + 0, we may assume that the changes suffered by the envelopes U and Vdue to nonlinear interaction are small, and we can solve Eqs. (A&A.6) perturbatively by expanding all functions as series in the smallness parameter (~4. Up to second order, we find U(v) = u(O) +E”v’O’*r] -(,,)*1V(0)1*C7(0)~*+~~*U(O)~*,

(A-7)

V(v) = V’O’ + /_&u’o’*~ -(,~)*1u(o)1*V(o)r1*+~~*V(o)r12,

(A.8)

where U(O), V (‘) are the signal and idler envelopes at the entrance of the nonlinear medium, p= alE, and Ep is the undepleted pump envelope. The changes of the signal and idler envelopes after propagation in the nonlinear medium are obtained from Eqs. (A.7, A.8) by making 77= 1, and are given by 6U=~v~-((,e>*Iv*Iu+~~*U,

(A.9)

sv=~u*-((T~)*~u*Iv+~~2v,

(A.lO)

where we have dropped all suffixes on the right-hand sides. From Eqs. (A.9) and (A. 10) we get the expression fo the parameters g, and g, given by Eq. (3) in the text.

Appendix B An inspection of the neutral positive detuning case ( v > 0) uous bands of active modes can old, leading to the expectation

stability curve in the shows that two continbe excited near threshthat a standing wave

S. Longhi / Optics Communications

124 (1996) 628-641

639

pattern could be excited. In this appendix we show that the standing wave solution is always unstable, and on1 a single traveling wave state (at either fi or - s’ v ) may be a stable solution of Eqs. (8). To this aim, we apply standard weakly nonlinear analysis close to threshold [ lo], expanding the solution of Eqs. (8) as an asymptotic series in the smallness parameter cy= Al.- h. We write therefore A

UE

=cu”*~‘o’+(yy(l)+cy3’2y(*)+...)

0B (B-1) where uCk) = (‘(“) L(Ck1 , an d we introduce a multiple scale expansion for y, y = Y, + &‘*Yi + (YY~+ . . ., so that a,=a,+a

112 2 a,,+&*+...

.

03.2)

Fig. 2(b) indicates that, above threshold, there exists a finite bandwidth of modes around q = k 6 which become unstable. A usual way to take into account the effects of these modes is to introduce a multiple scale expansion for x. The behavior of the neutral stability curve given in Fig. 2(b) around its minima suggests to set x=X, + (u”*X,, so that (B.3) Putting Eqs. (B.l), (B.2) and (B.3) intoEqs. (8), a hierarchy of equations for successive corrections to v is obtained. At the order 0( c&‘*), we recover the linearized problem

ay,do)= b(O)

,

where the linear operator

(B.4)

L is given by

(B.13) Eqs. (B.12) and (B.13) are two coupled GinzburgLandau equations which have two types of stationary solutions: (a) traveling waves (A, = 0 or A, = 0) and (b) standing waves (A, = &A*). The standing-wave solution corresponds to a conversion of the pump photons into Stokes and anti-Stokes photons in the same polarization mode of the fiber, and therefore such a solution gives rise to a time-modulated interference

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S. Longhi /Optics Communications

pattern in the field intensity. Stability of standing wave versus traveling wave solutions is a general question in the context of nonlinear systems driven far from equilibrium, and depends strongly on the coefficients of the nonlinear terms in the amplitude equations [ IO]. In our case, it can easily be shown by linearizing the amplitude equations about the standing wave solution that this solution is always unstable. Therefore the nonlinear terms select the traveling wave solution. By making, for instance, A2 = 0 in Eq. (B. 12)) we obtain the amplitude equation for the traveling wave solution, and stability analysis of this solution gives the usual parabolic Eckhaus stability boundary [IO], as shown in the inset of Fig. 3(b).

Appendix C The coefficients of the characteristic the matrix A4 are given by

polynomial

of

c, =4(A-5X)

c~=

-4~iQ”X-64~iQ’X-i + 32vq4Q’-

+ 8Vq2Q4+ 32EiQ4X2

64vQ2X2 - 16gQ2X3 +4$Q4X2

- 16q2v’Q2-4.Qb+4y2Q4-

12Q6X

+ 4Q4q’ - 4q’Q” - 16q6Q2 + 32vQ4X+48q4Q’X+

16q2Q4X+Q8+

16v’Q’X

12veiQ4X

-~~v~*Q~X-~V~E~Q~X+~E~Q~~~X+~E~~’Q~X - 64Q’X3 + 48Q4X2 - 16e,2Q2x3 + 4$Q4X2 + 8hve,.Q*X-24herq’Q2X-44ht;Q4X+8$q’Q2X’+

16+Q2X2-48eivQ2X2

8u2~Q2X2

124 (I 996) 628-641

+ 16Eiq2Q2X2-8~2VQ2X2+

S~q'Q*X'.

(C.4)

We note explicitly that all the coefficients of the characteristic polynomial are functions of Q*, and therefore (~c,/~Q)~=~=O identically for eachj= 1, 2, 3,4.

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