Department of Electronics, Weizmann Institute of Science, Rehovot, Israel
Received March 5, 1984; accepted April 18, 1984
Instabilities and self-oscillation in systems containing an optical Kerr medium are studied in detail. Nonlinearinteractions, both within and without a cavity, are discussed. Instability thresholds and frequencies of self-oscilla-tion are treated as a gain-feedback process. A detailed investigation of the instabilities in a high-finesse nonlinearring cavity shows that both Ikeda instabilities and bistability are obtained in a rather limited regime of detuning.In most of the detuning range, the system exhibits period-i- oscillations. Counterpropagating waves in a Kerr medi-um are shown to become unstable above a certain threshold intensity. This results in a system that is equivalentto a Raman-like laser that is being excited in the distributed-feedback structure generated by the two waves. Aqualitative description of harmonic generation and period doubling, based on wave-mixing processes, is also pre-sented.
1. INTRODUCTION
Nonlinear optical systems, like nonlinear systems in otherfields, exhibit a rich spectrum of temporal behavior. Abovea certain threshold their output intensity undergoes periodicand chaotic self-oscillations. During the last few years therehas been a growing interest in the temporal stability of opticalbistable systems. Ikeda et al. were the first to point to theconnection between optical chaos in bistable systems andturbulence.' They have studied a system of a ring cavitycontaining a nonlinear Kerr medium irradiated by light, andthey discovered instabilities and chaos at the output-lightintensity. It was shown later that the same phenomena occurin nonlinear Fabry-Perot resonators 2 as well as in nonlineardistributed-feedback resonators.3 Others have studied bi-stable systems that contain two-level atoms and have foundthe existence of instabilities and self-oscillations as well.4
Recently we showed that even a simple system, consisting oftwo counterpropagating beams in a Kerr medium, may self-oscillate without an external resonator.5 Some of the theo-retical predictions, including the appearance of period dou-bling and chaotic oscillations, were verified experimentallyin hybrid bistable systems 6 as well as in an all-opticalsystem. 7
Being simple to model, a Kerr medium is attractive foranalytical as well as numerical studies. It can be character-ized by just two parameters, namely, the nonlinear refractiveindex n2 and the response time T, and it can serve as a modelfor both transparent media and two-level media far fromresonance. Indeed, one of the most thoroughly investigatedsystems is a ring cavity containing a Kerr medium. Twolimiting cases are usually discussed, depending on the relationbetween T, the response time of the medium, and tr, the transittime in the resonator. The first case is that of a fast Kerrmedium, i.e., r << t,. Here, two different regimes of self-oscillation were found. The first, known as Ikeda instability,'is characterized by an oscillation period of 2tr, whereas in thesecond regime, which was predicted by us,8 the oscillation ismuch faster, with frequency of order T1. In the second case,that of slow nonlinearity, >> tr, self-oscillation with a periodof order was predicted. 9
Not much was said about the physical mechanisms that ledto the various temporal effects. It was proposed by us,8 and
independently by Firth and Abraham,'0 that four-wavemixing interactions can be the reason for the appearance ofinstabilities. In this paper we investigate in detail the in-teraction of light waves in a Kerr medium and present acomprehensive analysis of the various oscillation regimes.Briefly stated, we show that optical instabilities can be at-tributed to gain-feedback processes, or, in other words, thatself-oscillation is one form of a parametric oscillator. Theoscillating output intensity is then interpreted as the beatingbetween the original injected field and the fields generatedby the parametric oscillator. We analyze and explain thebehavior of two specific systems; the first is a nonlinear ringcavity, and the second is interfering beams in a Kerr me-dium.
In Section 2 we analyze a simple interaction of two nonco-propagating light waves in a Kerr medium and show the ex-istence of optical gain in one of them. The interaction of alight wave with two sidebands is discussed in Section 3. It isshown that the sidebands are amplified through a four-wavemixing process. Section 4 is devoted to a detailed investiga-tion of a high-finesse nonlinear ring cavity, and the variousoscillating regimes are explained. In particular, we show thatthere is a continuous transition from Ikeda instability to pe-riod-T oscillations and that the system may oscillate at anyfrequency between the two. It is also known that the ap-pearance of instabilities leaves a limited range for the imple-mentation of bistability in a high-finesse cavity. In Section5 we treat the case of two counterpropagating waves in a Kerrmedium. We show that the grating induced in the mediumby the standing-wave pattern generates a distributed feed-back. The self-oscillation is interpreted as a distributed-feedback parametric oscillator. In Section 6 we propose amechanism for the behavior of the system beyond the stabilitythreshold. In particular, we claim that period doubling andthe generation of harmonics can be also attributed to wave-mixing processes.
2. ENERGY TRANSFER BETWEEN LIGHTWAVES
Consider the situation of Fig. la, in which two plane waves,E0 and El, interfere in a nonlinear Kerr-like medium. The
Fig. 1. Nonlinear interaction in a Kerr medium: a, interaction oftwo plane waves; b, geometry for forward four-wave mixing.
fields are represented by
E = Ao exp[i(wot - ko r)] + c.c.,
E = A1 exp[i(colt - kiz)] + c.c. (1)
Assume first that the two fields have the same frequency, coo= wl. The fields interfere to form a phase hologram throughthe nonlinearity. One may suspect that this hologram, havingthe right spacing and orientation, will diffract light from onewave into the other. The nonlinear part of the refractiveindex nNL is
The first two terms on the right-hand side of Eq. (2) de-scribe a uniform bias of the nonlinear index, and the other twoterms may be called holographic terms. Equation (2) issubstituted into the Maxwell wave equation. By using theslowly varying amplitude approximation, and by collectingall the terms that oscillate as exp[i(co1 t - k1z)], we get anequation for A 1(z):
= -i(AoAo* + A1Al*)Al - i(AiAo*)Ao, (3)dV
where v = n2kz. Equation (3) was written so that the originof the different terms is apparent (two from the bias term andone from the holographic term). Investigation of I =AAl*
yields
d = Al* d + A d = 'd1 1 d A1 d ~i (4)
X exp[-i(wo - co1 )t + i(ko - k1) r]}. (7)
Again we recognize two bias terms and two holographic terms.Note that the holographic terms represent the moving gratingbut that grating is shifted in phase compared with the lightinterference pattern because of the complex multiplyingfactor. This shift is caused by the finite response time of themedium, and it disappears when r = 0. The wave equationnow yields
d = _i(AoAo* + AiAi*)A- - Ao*AlAo,d~~ I _~1iA
(8)
where A = (coo - col)j. The intensity I, changes accordingto
dIl 2A IoId~ +A 2 (9)
For n2 > 0, Eq. (9) predicts gain for Il if wl < coo and lossotherwise. The source of this gain is power transfer from Ioto I, as a result of diffraction. The gain is maximized for A= 1, i.e., when coo - co, = r-1. When A = 0 there is no powertransfer, as we showed before. Figure 2 depicts the depen-dence of the gain factor on the frequency difference. Notethat if Io >> I,, so that pump depletion can be neglected, I,grows exponentially with z.
It should be pointed out that this gain process is by nomeans new. It would be named differently in differentphysical situations. In particular, if the nonlinearity sourceis electrostriction, and r is the optical-phonon lifetime, theprocess is a model for Raman gain. Usually Raman gain ischaracterized by a spectrum that is more complicated thanthat of Fig. 2. This means only that the assumption of aDebye-relaxation equation with a single decay time r is onlyan approximation of the real physical process. The point ofview presented here stresses the intimate relation between
1.5
1.0
X0
zE-,which means that no power is being transfered between thetwo waves. The presence of a nonlinear medium influencesonly the phase of the interacting fields.
Consider next the case of different frequencies. The in-terference pattern is then dynamic, i.e., a moving grating. Inorder to investigate the interaction, it is necessary to state thedynamical behavior of the nonlinearity. We assume aDebye-relaxation relation,
T NL + nNL = n 2 (E 2 (t)),dt
which is solved by
(5)
0.5
0.0
-0.5
-1.0
-1.5-3 -2 -1 0 1 2 3
FREQUENCY DIFFERENCE A
Fig. 2. Exponential gain factor for I, as a function of the normalizedfrequency difference between pump and probe, A = (coo - co)r.
E (
NONLINEAR
KERR MEDIUM
(6)
I . . . . I . . . I . . . 1I 1
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664 J. Opt. Soc. Am. B/Vol. 1, No. 4/August 1984
Kerr-type nonlinearities and gain processes. These variousprocesses can thus be represented by a single physicalmodel.
3. PARAMETRIC FOUR-WAVE MIXING GAIN
Consider the same situation as before, but now with two co-propagating waves in the +z direction, as shown in Fig. lb.For simplicity we assume that A >> jA,. A field at W2 = 2wo- wl, which we denote as E 2, will be generated through thenonlinear interaction. This frequency was neglected beforebecause it was not phase matched. In a collinear propagation,neglecting material dispersion, it is phased matched and mustbe included. Note that the frequencies wl and w2 are sym-metrically located around wo. Maxwell's equations then yieldthe following coupled-wave equations for the field ampli-tudes:
dA _iAoAo*Ao,
dA,
= iAoAo*Al - i(l - iA)lAoAo*A
-i(l - iA)AoAoA2 *,dA2 Aoo2
= _iAoAo*A2 - i(l + iA)lAoAo*A 2
- i(l + iA)-lAoAoA,*, (10)where we have neglected terms proportional to two or moreof the weak amplitudes A, and A2. Juxtaposing Eqs. (8) and(10), we find that the main difference is the presence of thewave mixing terms.
Defining a = Aj exp(iIo0), where I = AoAo* = constant,yields
Note that the collinear geometry results in significantly dif-ferent interaction from the noncollinear case because of thewave-mixing process. First, the field growth is not expo-nential but linear with . Second, the nature of the processdepends strongly on the phase relation between the inputfields. Figure 3 depicts the intensity amplification factor forIa ()I = a2(0) with different relative phases. The two fieldsEl and E2 may be the sidebands that result from modulationof the central field Eo. It can then be deduced that in the caseof frequency modulation, when a(0) = -a 2 *(0), there is nointeraction between the two sidebands. This is not surprising,since a frequency-modulated field has a constant time-aver-aged intensity, and, therefore, its effect on the medium is onlyto increase the bias refractive index. Only when the field hasan amplitude-modulated component does the intensity vary,and the oscillating (although sluggishly so) refractive indexcouples among the various frequency components.
The four-wave mixing process modeled here depends onfulfillment of phase matching. As the frequency differencebetween o and co1 increases, dispersion of the linear refractive
1.3
1.20
tr/21.1
1.0
0.9 3T/2
0.8
0.7-3 -2 -1 0 1 2 3
FREQUENCY DIFFERENCE A
Fig. 3. Relative gain for a probe field in forward four-wave mixingprocess as a function of the normalized frequency difference betweenpump and probe for several values of relative phases between theinput probe fields. Equal input amplitudes for the two sidebandsare assumed.
index prevents phase matching. In this case A2 is not gen-erated efficiently, and one will have, for example, pureRaman-like gain, as modeled in Section 2. There are cases,such as in optical fibers, in which phase matching can beachieved for even a large frequency difference, and stimulatedgeneration of both fields has been observed.10
4. SELF-OSCILLATION IN A NONLINEARRING CAVITY
In Sections 2 and 3 we determined that a nonlinear Kerr me-dium pumped by a wave at coo may induce gain at otherfrequencies. Consider now the nonlinear ring cavity of Fig.4. An input field at co0 is injected from the outside. It is wellknown that this system may exhibit bistability for certaindetuning ranges. Here we disregard for the moment thepossible bistable nature of the system, and we investigate thesteady-state solution. Since gain is induced by the pump,oscillations may be achieved once the gain is strong enough.If phase matching for four-wave mixing is not possible, am-plification at co = W - 1 is expected to be dominant,therefore creating a Raman parametric oscillator or simplya Raman laser. If, on the other hand, phase-matching con-ditions are fulfilled, we have to take into account the morecomplicated process described by Eqs. (10).
Let B be the round-trip feedback and the 00 be the initialdetuning from resonance. Using Eqs. (12) and the boundaryconditions A,(L) = A(0) and A2(L) = A2(0), we obtain acondition for oscillation at w1 and 2:
where 0 = o - ( - w0 )tr, P = n2k101, I is the intracavityintensity of the input field, and I is the length of the nonlinearmedium. Requiring a vanishing determinant, we get
Equation (14) is exactly identical with Ikeda's characteristicequation, which was derived by linear-stability analysis.1
This identity proves that the four-wave mixing parametricprocess of Eqs. (10) describes the entire dynamical behaviorof the system. In other words, the gain-feedback model is notan approximation but an exact description of the physicalprocess.
The main advantage of this model is its intuitive appeal.Indeed, it does not yield any new results that cannot be de-rived by linear-stability analysis, being exactly analogous toit. However, a model that is based on a threshold for oscil-
Fig. 4. A nonlinear ring cavity.
5
4
3
2
i
0
1.50
zW 1.000D
X 0.75z0
E-
1 0.50i5
o 0.25
0.000 7T 27
DETUNING PHASE ,
Fig. 5. Instability threshold intensity and oscillation frequency ina high-finesse ring cavity containing a fast nonlinear medium as afunction of initial detuning 0o. The lines in a describe the boundaryof the unstable regimes, and in b they show the frequency of self-oscillation at these boundaries. In most of the detuning range thethreshold intensity is about 2e, and the oscillation frequency is aboutT-1, as expected in a pure-Raman-like laser. Deviation from thesevalues is in small detuning ranges around ko = 0 and ko = Tr.
W2
shifted mode pattern
low power mode pattern
CO
Fig. 6. The relation between the input-light frequency and the cavitymodes in the case of Ikeda instability. The lower line describes thesituation at low pump intensity; the upper line shows the shifted modepattern that is due to the nonlinear interaction. Ikeda instability isobtained when the pump frequency is located exactly between the twocavity modes. The output intensity then oscillates at the beat periodof 2tr.
lations and a resonance condition is most familiar to physi-cists, and, more importantly, is useful for predictions of newphenomena.
We now examine, as an illustration, the stability of a high-finesse nonlinear ring cavity, i.e., B = 1 - E. We assume firsta fast nonlinearity, T << tr. Expanding Eq. (14) up to a secondorder in e and searching for the lowest threshold-intensitysolution for each value of 00 reveal a complicated instabilitystructure. Figure 5 depicts schematically the threshold in-tensity and oscillation frequency as a function of the detuning.Note that the intensities are the intracavity intensities, whichcan be easily calculated from the input value. We find thatIkeda instabilities, i.e., oscillations with a period of 2t, occurat a rather limited detuning range of \/3e < 00 - r </ 3 /e.As 00 is increased, oscillations at higher frequencies have alower threshold intensity. The oscillation frequency and thethreshold intensity grow as 00 is increased, and in most of thedetuning range A 1 and P 2E.
The Ikeda instability can be explained by the excitation oftwo cavity modes adjacent to the input pump frequency co.Since r << t, those modes are excited with equal amplitudes,as can be derived from Eq. (12). The pump frequency mustfall exactly between the two modes; hence the basic beatingperiod is 2tr. This situation is depicted in Fig. 6. The furtherthe initial detuning is from 7r, the larger the intensity that isrequired to induce this oscillation. However, this intensityhas to be above a certain value in order to meet the gain-feedback condition. This value corresponds to a detuning of
/3e from 7r.As the detuning is increased above 7r + \/3 /e, higher modes
are successively excited: first the next-nearest-neighbormodes, then the two modes beyond them, and so on. As thefrequency of these modes shifts away from wo, the lower-fre-quency mode of the two has a larger amplitude, because it hasa higher gain, as can be seen from Fig. 3. Eventually, whenthis mode is located around w0 - -r1, it becomes dominant.This is the peak of the Raman-like gain. From Eq. (9) we caneasily derive the threshold intensity for this oscillation as 2E.
Hence in most of the detuning range we have essentially aRaman-like oscillator. The transition from Ikeda oscillationto Raman-like oscillation is shown in Fig. 7, which is a mag-nified view of Fig. 5 in the detuning range around 00 = 7r.
Figure 8 depicts the behavior of the system for detuningrange around 00 = 0. The threshold line in the range V-e <00 < /3\/3E describes here the bistable nature of the systemrather than an oscillatory behavior. The shaded region is not
I III II II II I
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666 J. Opt. Soc. Am. B/Vol. 1, No. 4/August 1984
5
0~
4
3
2
1
0
2.0
z
0'W
0D
0r
1.5
1.0
0.5
0.00 1
5
W
N-
P.
0
4
3
2
1
0
2.0
cv
0
0
2 3 4 5
DETUNING PHASE (o-7r)/c
Fig. 7. Threshold intensity and oscillation frequency for a high-finesse ring cavity containing a fast nonlinear medium for the de-tuning range around 7r. Ikeda instability is obtained only in the rangeof V3 < 00 - r < %v'3c. The dashed line is the theoreticalthreshold for the Ikeda instability. For 00 above this range, highermodes are excited first; in most of the detuning range, modes shiftedby Tr' oscillate, as shown in Fig. 5.
accessible, and there are no steady-state values of circulatingintensity in this region. For 00 > 5/V3e, the lower branch ofthe bistability curve becomes unstable and exhibits oscilla-tions. It can thus be concluded that bistability in a high-finesse cavity exist only in a limited detuning range.
Another limiting case, which is discussed in the literature,is that of a slow nonlinear medium, i.e., T >> t. The analysisof Eq. (14) in this case reveals instabilities, provided that T isnot too large. Figure 9 shows the threshold of instability asa function of the initial detuning phase for various values oftIT when the detuning range is around 00 = 0. The oscilla-tion frequencies are of order T-1. It is evident from Figs. 2and 3 that fields with frequencies shifted from wo by muchmore than a few T- 1 are only slightly amplified, and thereforethey will not oscillate. Hence only the cavity mode adjacentto o0 can be excited. As the response time r is increased, thefrequency range of amplification decreases, and the instabilityrange is more limited. Note that only the upper branch of thebistability curve becomes unstable in this case.
1.5
1.0
0.5
0.00 1 2 3 4 5
DETUNING PHASE f0 /e
Fig. 8. Threshold intensity and oscillation frequency for a high-finesse ring cavity, containing a fast nonlinear medium, for the de-tuning range around 0. The shaded region is inaccessible becauseof bistability. Bistability is obtained only in the range of V/3 < o< 5 /3V"3. For 00 above this range, the lower branch of the bistabilitycurve turns unstable. The upper branch is always unstable beyonda certain intensity value.
5. ON THE STABILITY OF INTERFERINGBEAMS
In Section 4 we showed how parametric gain that is due to afour-wave mixing process leads to self-oscillation. Thefeedback necessary for the oscillation was supplied by theexternal resonator in which the interaction takes place. Inthis section we treat the case of two monochromatic wavesinteracting in a Kerr-like medium. We show that in thissystem self-oscillation and chaos are obtained, even withoutany external feedback. We still maintain that the phenom-enon can be explained by gain-feedback arguments; yet thefeedback in this case stems from the interaction itself. Thestanding-wave pattern that is generated induces refractive-index modulation with a period of half a wavelength. Sucha construction may lead to laser oscillation, in which case thelaser is a distributed-feedback laser.12 The instability ofcounterpropagating beams originates from the same gainmechanisms discussed before, but it supplies its own (dis-
b
- Ikedainstability
_,,I ,I,, I . I .,..-I
l
Y. Silberberg and1. Bar-Joseph
Vol. 1, No. 4/August 1984/J. Opt. Soc. Am. B 667
10
W
"I
z0
z
:U2Z
8
6
4
2
0
to
8
6
4
2
0-4 -2 0 2 4
DETUNING PHASE 00/e
Fig. 9. Threshold intensity and oscillation frequency in a high-fi-nesse ring cavity, containing a slow nonlinear medium, for several tT
values. The shaded region is inaccessible because of bistability. Onlythe upper branch of the bistability curve is unstable.
tributed) feedback. Note, however, that the steady-statebehavior of this system is not bistable. Moreover, thesteady-state output power of the two beams is not affected atall by the nonlinear interaction. Thus it can be concludedthat the phenomena of self-oscillation and chaos are notnecessarily connected with bistability.
We consider the geometry of Fig. 10(a), although ourtreatment will trivially hold for the geometry of Fig. 10(b) by
redefining z/cos 0 for z. The two fields are taken to be
El = Al(z, t)expi(wt - kz)] + c.c,
E2 = A2 (z, t)exp[i(cot + kz)] + c.c., (15)
with constant input amplitudes of VII and v\7I, respectively.We assume a Kerr-like medium, characterized by the
Debye-relaxation relation of Eq. (5). When the slowly varyingamplitude approximation is used Maxwell's equations yield
dz+ no- = -i(Hil + H 22)A 1 - iH 1 2 A2 ,
n0c a 2 if(H 1 l + H 22)A2 - H21A1, (16)az C at
with j = n2k and
Hij = - f Ai(z, t')Aj*(z, t)expt ) dt'. (17)
The Hij are the holographic terms discussed in Section 2.Assuming a steady state, i.e., vanishing time derivatives, theholographic terms are independent of time:
Hi = Ai (z)Aj*(z),
and the solutions for Eqs. (16) are
A10 (z) = \/-I exp[-if(hl + 212)z + i ],
A20(z) = \ /2exp[+i(I2 + 2I)z + i2],
(18)
(19)
where 01 and 02 are determined by the initial conditions. Theintensities Ii = AiAi* do not change along the medium, andthe two waves do not interchange energy, as we proved inSection 2. Note, however, that the phase shifts that the wavesacquire while traveling through the medium are not neces-sarily equal. The nonlinear interaction induces some non-reciprocity, although it does not affect the intensities. Thiseffect has been investigated experimentally in a fiber-opticrotation sensor, in which an induced phase difference mayaffect its output. 13 Application of the nonreciprocity has beenproposed for enhancing the sensitivity of Sagnac interfer-ometers.1 4
We now wish to investigate whether the steady-state solu-tions of Eqs. (16) are stable, i.e., whether they represent realphysical solutions. This is done by using linear-stabilityanalysis. We examine a solution of the form
which is a perturbed steady-state solution. We look for agrowing perturbation, i.e., Re(X) > 0. The form of Eq. (20)is necessary because the interaction mixes conjugate terms.The resulting linear system of equations for Fj and G is
2P2
-A'- Pi + 2P2
-2P2
P2 ) (21)
P 1 + P2 G
Here X' = Xno/c, P1 = iOI, and P2 = P1 /(1 + X-). Equalinput intensities I, = I2 = I were assumed. We are lookingfor a solution of the system given by Eq. (21), which fulfills theboundary conditions F1 (0) = G1(0) = F2 (L) = G2(L) = 0.These perturbations will then grow (or decay) according toexp[Re(Xt)]. The boundary conditions will yield four ho-mogeneous equations for Fj and Gj. If we are to get nontrivial
Y. Silberberg and L Bar-Joseph
668 J. Opt. Soc. Am. B/Vol. 1, No. 4/August 1984
solutions, a determinant has to vanish. This will give a valuefor X for each set of parameters and determine whether the
a
NONLINEAR
KERR MEDIUM E2
b I
Fig. 10. Geometries for interfering beams in a Kerr medium.
5
4
3
2
0
5
4
3
2
1
01 10 100
TRANSIT TIME tr/T
Fig. 11. Threshold intensity and oscillation frequency for interferingbeams in a Kerr medium versus the transit time in the medium. Thedashed lines are different solutions, which correspond to differentlongitudinal modes of the distributed-feedback resonators. Only thefirst eight modes are shown. The system is unstable above the solidline, where at least one solution exists. As the transit time increases,higher-order modes are excited first.
system is stable. We sought the boundaries of the instabilityregions by setting Re(X) = 0 and solving for Im(X) and I.
Figure 11 depicts the threshold intensity for instability asa function of tlr, t = nL/c being the transit time in themedium. In the region above the solid line in Fig. 11 there isat least one solution for a growing perturbation. Note that,for each value of t,, there is a series of solutions, which lie onthe different dashed lines. These solutions correspond toexcitation of different longitudinal modes of the distrib-uted-feedback resonator, which is generated by the stand-ing-wave pattern through the nonlinearity. The modestructure of a distributed-feedback resonator was investigatedby Kogelnik and Shank.' 2 These modes have eigenfrequen-cies separated by approximately (2t,)-', and they are arrangedsymmetrically around wo, which is the frequency that fulfillsthe Bragg condition. The amount of feedback for a modedecreases with its distance from wo. Indeed, we find that inour case for t < r, the lowest-order mode (the one nearest wo)is the one that is excited first. Note that, besides having ahigher feedback, it also has a higher gain, since the gain de-creases for frequencies beyond wo -1, as is shown in Figs.3 and 4. For t > r, a higher mode, which is around wo +-1,will be the one to be excited first, as can be seen in Fig. 11, yetit requires higher intensity.
The gain-feedback model, then, explains qualitatively theresults of the stability analysis in an interfering-beams system.This system is unique in the sense that both the gain and thefeedback stem from the nonlinear interaction. Its steady-state solution is simple and does not hint of the complextime-dependent solutions that are obtained in the regions ofstability. Some of these solutions are described in Ref. 4.
6. HARMONICS AND PERIOD DOUBLING
One major advantage of our model is its ability to explain, atleast qualitatively, phenomena that occur beyond the insta-bility threshold when the system is in the oscillatory regime.In this section we discuss the generation of harmonics and thephenomenon of period doubling.
Consider a bistable system pumped by a field Eo at coo. Wehave shown that, once the intensity is strong enough, a fieldEl at wl may be excited, usually accompanied by E 2 at 2 oo -wl. The output intensity will then oscillate at a basic fre-quency of xl - c0 . The fields Eo and El will now generate,through the nonlinearity, a term of the form EEo*El. Thisis a source for oscillations at coo + 2(w, - coo). This situationis depicted in Fig. 12a. The new field Eh will beat with E, andE 2 at twice the basic frequency. That means that a harmonicof the oscillation frequency has been generated. Its effect onthe output intensity is to change the oscillation pattern froma pure sinusoidal pattern into a more complex one withoutchanging the basic period. Note that a source term of thiskind is always present once E and El are present, andtherefore the harmonic field Eh always exists. Moreover,higher harmonics will also be generated by the mixing of lowerharmonics, in a similar way. In the case of Ikeda instability,we may expect that harmonics of odd order will be dominant,because they match the cavity modes, whereas even-orderharmonics will not be resonant. This is in agreement with therecent numerical investigation of Ikeda et al. 15
The phenomenon of period doubling, on the other hand,occurs only at a certain threshold intensity. It occurs when
K}
zW
N
0A:
6
r
0
Y. Silberberg and 1. Bar-Joseph
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Vol. 1, No. 4/August 1984/J. Opt. Soc. Am. B 669
Eh
II
Ii
a
El Eo E2
Edt 1 t E
b
Fig. 12. The relation between the oscillation frequencies:eration of harmonics; b, period doubling.
a, gen-
a field Ed with a frequency Wd = (°o + wl/2 oscillates. Thisfield will beat with E0 or El at half the basic frequency, i.e.,period doubling. This situation is shown in Fig. 12b. How-ever, the field Ed is amplified by interacting with E0 and Elthrough the wave-mixing process EOEEd*. Note that thisterm is a gain term, i.e., is proportional to Ed, rather than asource term. Only when this gain is sufficiently strong willoscillation at Ed start. This explains why bifurcation throughperiod doubling is characterized by thresholds.
Obviously, resonance conditions also have to be met. Evenif the field Ed is initially far from resonance, the nonlineardispersion can shift the cavity modes enough to permit os-cillations. For example, in the case of Ikeda instability, as inFig. 7, the field Ed will be located at a quarter of the distancebetween the oscillating modes. Yet, because of nonlineardispersion, this field may have the same wave vector as El andhence be a cavity mode. This phenomenon is known as modesplitting, and it is often discussed in relation to instabilitiesin lasers.16
7. CONCLUSIONS
In this paper we have presented a physical model that explainsthe phenomenon of instabilities in nonlinear optical systems.We have shown that self-oscillation can be attributed to again-feedback mechanism, in which the gain originates fromfour-wave mixing interactions, and the feedback can be eitherexternal or inherent in the nonlinear interaction. It shouldbe pointed out that instabilities do not occur only in bistablesystems. A nonlinear cavity exhibits instabilities even whenthe initial detuning is such that the system is not bistable. Inthe interfering-beams configuration, the appearance of self-oscillation is more dramatic, since below the threshold forinstabilities the behavior of the system is rather simple, andthe output intensity is equal to the input intensity. The factthat a simple system like this exhibits self-oscillation hintsthat instabilities in nonlinear optics are much more funda-mental than is usually assumed. In fact, although there aresome differences in the details, instabilities in nonlinear op-tical systems and instabilities in lasers appear to originatefrom a similar mechanism.
As was mentioned earlier, a Kerr medium can serve as amodel for a two-level medium far from resonance. The ap-
proach presented here can be generated to account also for aresonant medium. Indeed, sideband amplification resultingfrom four-wave mixing in a two-level medium was suggestedas a source for instabilities.' 7 The phenomenon of instabili-ties in various nonlinear optical systems can thus be viewedin a single frame.
We have also shown that Ikeda instability, i.e., self-oscil-lation with a period of 2tr, occurs only in a limited detuningrange in a high-finesse cavity containing fast nonlinear me-dium. In most of the detuning range, the system oscillateswith a period of order r, and there is a continuous transitionbetween the two types of oscillation. This result can begeneralized to a lower-finesse cavity; the lower the finesse, thelarger is the detuning range where Ikeda instability appears.The appearance of self-oscillations of period r also limits therange for bistability in a high-finesse cavity. As we haveshown, for Oo > 5
/3E, the lower branch of the bistabilitycurve turns unstable; hence bistability occurs only for d\/e <¢O < /3-6.
The model presented here can yield quantitative resultsonly near the threshold for instabilities but may provide aqualitative description of nonlinear optical systems beyondthat threshold. Indeed, we have suggested a model for thegeneration of harmonics and period doubling and have usedwave-mixing processes to explain some earlier numerical ob-servations.
ACKNOWLEDGMENT
*We wish to acknowledge the support and encouragementgiven by A. A. Friesem.
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