Vol. 2, No. 1/January 1985/J. Opt. Soc. Am. B 173

Instabilities in lasers with an injected signal

J. R. Tredicce, F. T. Arecchi, G. L. Lippi, and G. P. Puccioni

Istituto Nazionale di Ottica, Largo E. Fermi 6, 150125 Firenze, Italy

Received July 10, 1984; accepted October 1, 1984

We consider a laser with an injected signal, in which the polarization can be adiabatically eliminated, we study thestability of the steady-state solutions, and we discuss the time-dependent solutions. For the laser alone, the onlypossible solution is constant intensity. However, the introduction of an external field, with an amplitude that doesnot satisfy the injection-locking condition, destabilizes the system. In such a case, numerical results show the exis-tence of a self-Q-switching process, which induces relaxation oscillations. The frequency of the giant pulses is di-rectly related to the amplitude of the external field, whereas the frequency of the relaxation oscillations dependson the damping rates. We show also that, depending on the value assigned to control parameters, the interactionbetween these frequencies leads to a chaotic behavior through intermittency or period-doubling bifurcations. Fi-nally, topological equivalence between our laser system and a unidimensional circle map is shown for some valuesof control parameters.

1. INTRODUCTION

Shortly after the introduction of the first laser, signal injectionwas considered a suitable method to achieve a stable outputfrom intrinsically unstable lasers.'- 9 This technique wasapplied first to He-Ne lasersl"9"10 to obtain a single mode froma Doppler-broadened gain line and second, to dye lasers inwhich the insertion of an external field from a spectral-bandlaser results in a spectral narrowing of the injected laser.2 8

The method was then applied to C0 2-TEA lasers using dif-ferent arrangements, with two consequences, namely, modeselection and smooth pulses.46 Finally, it was applied tohigh-power conventional CO2 lasers in order to stabilize theoutput intensity of the system.3 More recently, the interestin a laser with an injected signal (LIS) has been centered onsemiconductors, for which the area of application isbroad."1

From a theoretical point of view, locking phenomena werealready well known in electronic oscillators before they werenoted in lasers' 2; a general treatment of the LIS, together witha stability analysis of the steady-state solutions valid for themost-popular atomic gas lasers or dye lasers, is given in Ref.13, where pulsing in the intensity and in the phase is predictedwhen the external field is not strong enough to lock the system.However, the region of interest remains the locked one, anddifferent treatments have been made of homogeneouslyl3 -'5

and inhomogeneously broadened lasers' 5 to establish thegeneral requirements in terms of the detuning and the am-plitude of the external signal in order to reach injectionlocking. A preliminary study of the unstable region is givenin Refs. 10, 11, and 13, based on theoretical and experimentalevidence of stable pulsations of the beating frequencies be-tween the external and internal laser frequencies.' 0 However,a LIS is a good candidate for studying chaotic behavior sinceit can provide the number of degrees of freedom' 6 necessaryfor deterministic chaos; in order to limit ourselves to a smallnumber of dynamical variables, we refer to single-mode lasersin homogeneously broadened media, so that the entire dy-namics is characterized by three dynamical variables, namely,

field amplitude, polarization, and population inversion.' 7

With this in mind, we must distinguish among three classesof homogeneously broadened lasers' 8:

(1) Class A (for example, dye lasers) The loss rate for thefield (K) is much less than the polarization and population-inversion loss rates (yj and -yIl, respectively). In this class,because of the difference among the time scales of the col-lective variables, it is possible to perform an adiabatic elimi-nation of the atomic variables. Therefore the system can bedescribed by means of just one rate equation for the field. Byincluding a constant, coherent, external field, it is possible toincrease the number of degrees of freedom by one. The pos-sible results are stable, locking, or regular pulsations, de-pending both on the detuning between the two fields and onthe amplitude of the external signal. The unstable region isrepresented by the oscillation of the intensity at the beatfrequency.' 0 To achieve higher-order instabilities andeventually chaotic behavior, it is necessary to modulate oneof the control parameters, namely, the external field, thepump rate, or the cavity losses. 9 20

(2) Class B (for example, ruby, Nd:YAG, and CO2 lasers)Here, K and -yII are much smaller than -yI, and adiabaticelimination of the polarization is feasible. The dynamic be-havior of this class of lasers is described by two coupled non-linear equations: one for the field and the other for thepopulation inversion. However, only a stable intensity outputis obtainable in the absence of an external perturbation. Inthese lasers, it is necessary to increase the number of degreesof freedom by at least one in order to reach unstable operation.This can be done in one of the following ways:

(a) Make the system nonautonomous by modulation ofone parameter.21 22

(b) Inject an external field. Such a system has beenbriefly analyzed in the case of semiconductor lasers.23 Reg-ular pulsing with and without damped oscillations was found.However, semiconductor lasers have the peculiarity that theiroperating frequency and gain depend in a nontrivial way on

0740-3224/85/010173-11$02.00 © 1985 Optical Society of America

Tredicce et al.

174 J. Opt. Soc. Am. B/Vol. 2, No. 1/January 1985

the carrier density, and some approximations must be madeto obtain a simple model. We reported previously'8 thetransition to chaos through intermittency for molecular gaslasers.

(c) Increase the number of lasing modes. This situationcan be achieved for a bidirectional ring cavity, and it wasshown that regular and chaotic pulsing were found for a widerange of parameters2 4 (e.g., pressure and excitation). Also,a high value of excitation produces the gain for modes that aredetuned from the central mode to overcome the losses; theninstabilities in those modes are possible even if the predomi-nant mode is stable25; this situation is more easily reached witha Fabry-Perot cavity, in which different modes can be drivensimultaneously by the population inversion.

(3) Class C The three collective variables are damped onthe same time scale (K y yII). Such a case was treatedby Haken,2 6 who showed that instabilities can be reached forthe central mode when K is larger than y + 11 and when theexcitation at threshold becomes larger than (k + y + 11)(yI+ K)/yI(k - y - yII). The first of these conditions requiresa bad cavity (high K) and, considered with the second condi-tion, an extremely high pumping value, is extremely difficultto achieve with most of the usual laser systems. The onlyexample is the far-infrared laser, but no single-mode insta-bility has been reported yet. However, an injected signalincreases the number of degrees of freedom from three to fivebecause field and polarization become complex. This casewas treated theoretically, and evidence of subharmonic bi-furcations and chaos was given27 28 even when the bad-cavitycondition was not satisfied.

This classification system is not applicable to inhomo-geneously broadened lasers (e.g., He-Ne, argon-ion, and Xelasers) in which the coexistence of several gain packets ensuresthat, even with K 0) gives the eigenvalues

X IIZo [ 1zo -2K 11(zo - 1) 12 4

(5)

where

g2ANO

OyIK(1 + 62)

is the unsaturated population inversion normalized to thesteady-state value AN.

Here by inspection we conclude that

(1) The steady-state solutions are always stable abovethreshold (zo > 1) because the second term in Eq. (5) is smallerthan the first one. So the real part of X(XR) is negative for allparameter values. Then the only possible solution is a stablefixed point in the phase space, even if the dynamic behavioris described by two coupled differential equations.

(2) The sum under the square root is positive for all valuesof zo, if y 11 is larger than 2K. In this case, a small perturbationwill induce a monotonic decay to the steady-state values.

(3) If Y

Vol. 2, No. 1/January 1985/J. Opt. Soc. Am. B 175

X = XR + iXI _ - 'y ° +i [2Ky 11(zo - )]1/2. (7)

The existence of an imaginary part in X suggests that thisfrequency appears as a response of the system to perturba-tions. A large external perturbation tuned around that fre-quency can destabilize the system, resulting in large Qswitching for locking of the pulsed output to the modulationfrequency.3 3 For lower amplitude modulation, it was con-jectured that the steady states might be destabilized withoutlocking precisely to the modulation frequency. Following thephilosophy of our previous research,18 we have obtained;chaotic output from a CO 2 laser for a small amplitude mod-ulation of cavity losses near the relaxation-oscillation fre-quency.

3. LASERS WITH INJECTED SIGNALS:STABILITY ANALYSIS OF A CLASS B SYSTEM

Here we introduce an external field with a frequency wl, whichis detuned from the laser frequency L. The rate equationsin a frame rotating with an arbitrary angular frequency Go'are

E = -KE(1 + 0,) + g- EN I+KEoeiaKt,-yj' 1 + i'

= - I (AN - AN o) - 4g jE+ 2AN_y (1 + 3/12)

where EO is the amplitude of the external field and

c- c'

K

O=-0, = co- co'K

=Wo - co'-yJ-

A= I (0 - )j

Fig. 2. Steady-state solution for LIS equations. Output intensityversus intensity of the injected field for constant value of the pumpparameter (zo) and different values of the detuning (1 - WL). Thedashed line shows the unstable region and the solid line shows theregion where the steady-state solutions are stable. For each curve,the critical value of the intensity of the injected field is marked withAj2. For A2 larger than Aj 2, there is no instability.

inversion to its threshold value K-yI(1 + 62)/g2 . In this way,Eqs. (8) can be written as18

= (z - 1)x + (-3z)y + A,

= -(0 - 6z)x + (z - l)y,

K .-z-y1(8)

x 2 + y2

1 + 62(10)

where

2g_ R(L 1) 1/ER,

2g_ EI,

g2AN

yIK(l + 62)'

A= 2gEo(7 l yI)1/2'

(9)

Relations (9) are shown for the sake of clarity in Fig. 1.It is clear that the steady-state solution (E = AN = 0)

represents the situation where a = 0, and thus it can beachieved only by assuming an operating frequency co' equalto the frequency of the injected signal. Hence, without lossof generality, we take as a natural reference frame the externalsignal w' = w1, and we write the complex amplitude E as ER+ iE. For convenience of computation, we normalize thefield to the saturation value (y±Iy )1/2/2g and the population

(11)

and now the dot denotes the derivative with respect to thedimensionless time = Kt.

The nontrivial steady-state solution of Eqs. (10) gives

(12)Zo

z =2 + y2

1 +-1 +2

and

A2

= [(2 - 1)2 + ( - 2)2](g2 + y2)

k

0')CO10

Fig. 1. Qualitative plot of the frequency relations among atomicresonance (homogeneous width y ) centered at w0, cavity resonance(width K) centered at w,, and injected field at wl.

(13)

In Fig. 2, we have plotted the steady-state intensity I = ( 2+ 92)/(1 + 2) as a function of the intensity of the injectedsignal (A2) for different values of the detuning between theexternal signal W and the internal laser frequency L. Thisdetuning in terms of 0 and is given by the relation

(0-) = K (w1-COL)-Ky

(14)

The eigenvalues obtained from the stability analysis ofthe steady-state solution are solutions of the equation

Tredicce et al.

176 J. Opt. Soc. Am. B/Vol. 2, No. 1/January 1985

X3 + Ht" (I + 1) -2(2 - 1)]X2 + f(z - 1)2 + ( - Z)2

-2 'Y" (z - 1)(I + 1) + 2 z X

+ I 1 (I + 1) [(2 - 1)2 + (0 - 52)21+ 2 'Y1 2I[6(0 - 2) -(2-1)]= 0

roots, which, in general, are complex. From the point of viewof these eigenvalues, the stationary state becomes unstablewhen the real part becomes positive. Thus the bifurcationpoint is represented by

X2 ,3 = XR(2,3) ilXI(2,3) = ±iXI(2,3 )-

(15)

(18)

In such a case, the condition for a Hopf bifurcation is

)[(- 1)2 + (0- 62)2] + 2 7K2 I[5(0 - z2) -(2- )]}

- (19)

[' (I + 1)-2(- 1)].

A complete analysis of the solutions of Eq. (15) as a functionof the pumping rate z0 and the detuning (0 - ) is left to afuture paper. We limit ourselves here to the cases in whichzo > 2 and the detunings 0 and are much less than one, whichmeans that the frequency of the external field differs from thecavity frequency by less than 7-I.

Since K

Vol. 2, No. 1/January 1985/J. Opt. Soc. Am. B 177

A2,3 =1 [(-1-K( + 1)]

+ [ 2 (I + 1)2 + I (- 1)(I + 1)

(0- -2

)2 - 12 , + -

2K K ~Asz_ 1, I-zo -1, YllI

178 J. Opt. Soc. Am. B/Vol. 2, No. 1/January 1985

4" ,. ' . '- .

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

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

... ....... . ........ ...............*-------------- .. ......... .............. ............-!----*

..., ....... .... ........ ......... .... . .... ........ ........ .._..., Pt . - -

.B.' _ _ ._ . | .

(a)

X

_

. .1I\

..

(c)

(b)

Z ., , , .. , . -- -- -. - . .

. .. ......... . .. _..-... ............. ........ ..

AM .......... . 1 ... . . . . . . .. . ... i s

...... ....... . ......... ...A n ........... ,............ ...... ...... ................ ... ...................... ..........

(d)

._ ._

Fig. 4. Behavior of the system in the subregion 1 of Fig. 3 [(0 -6) = -0.0214; A = 0.063]. (a) Intensity as a function of time: giant pulses inducedamped relaxation oscillations. The frequency of the giant pulses (1) is much smaller than the relaxation-oscillation frequency, and the peakintensity is regular. (b) Phases (0) as a function of time: the external field locks the frequency of the system during a long period but is notstrong enough to maintain this situation forever. When the system unlocks, a rapid change in the phase by a factor 27r occurs. (c) Imaginarypart of the field (y) versus the real part of the field (x). Each point is taken at a constant time interval; the different densities of points in thefigure indicate the rapid change in phase and the long time during which the systems remains locked. Considering that the intensity is thedistance from these points to the origin, the loop in the upper side shows the existence of relaxation oscillations. (d) Population inversion versustime. This curve shows how the process is similar to Q switching. The population inversion grows until it transfers the energy to the field,giving rise to a giant pulse.

Winst = 1-W 1(31)

where

=arg(x + iy) (32a)

and where

dtj ~~~ x - ~(32b )d(Kt) x 2 +y 2

is approximately equal to w1 . Therefore the laser is ratherclosely locked to the external frequency, and the field is inquadrature with the external field (x = 0; y = ). However,the external signal is not strong enough to maintain this sit-uation permanently, and the system unlocks periodically,changing its phase by an amount 27r and giving out a giantpulse (self-Q-switching) above the stationary value of theintensity. At the same time, this pulse acts as a perturbationof the stationary value, and the intensity relaxes back to thatvalue with damped oscillations with a frequency given by Eqs.(27). The plots of the intensity and the phase of the field asa function of time and the imaginary part versus the real part

of the field are shown in Fig. 4. From Fig. 4(d), it is clear thatthe population inversion grows during the locking and thentransfers the energy into the field. So the effects of the ex-ternal field are, first, locking the frequency of the system toits own frequency over a given time and, second, giving acontinuous growth of the population inversion during thattime until the system unlocks.

It is clear that the pulses do not arise from a Hopf bifurca-tion because the amplitude is proportional to the external-field amplitude, which is larger when it is near the locked re-gi6n.

The second important consideration is that two frequenciesare involved during this process:

(1) The frequency of the giant pulses, which is inverselyproportional to the intensity of the external field and whosemaximum value (for IAl2

Vol. 2, No. 1/January 1985/J. Opt. Soc. Am. B 179

For values of ( 1 - WL) larger than [(2,y1 )/K)(zo-1)11/2Q, the frequency of the giant pulses can be of the order of thedamped oscillations for suitable values of the external am-plitude A. Then if the detuning is larger than or of the orderof £1 the relaxation oscillations become undamped becausethe field cannot relax back onto the steady-state value beforeanother giant pulse arrives.

In Fig. 5(a), we can see five undamped pulses of the relax-ation oscillations between two giant pulses. As the exter-nal-field amplitude is decreased, this number decreases [Fig.5(b)].

In the transition region among situations with differentinteger numbers of relaxation oscillations between giantspikes, a subharmonic bifurcation of the smaller frequency

(a)

. . 0 0 f f . . .. . .. ....: . .. .... ... ................ ''...m _____m______H EW_

(b)

. . . ... .

(c)

Fig. 5. Intensity versus time with ( - 6) =-0.0175 (subregions 2and 4 of Fig. 3). As the amplitude of the external field decreases, thefrequency of the self-Q switching (1) increases, and the relaxationoscillations became undamped. (a) For A = 0.0485, Q1 is around sixtimes larger than the relaxation-oscillation frequency Q2. We observefive undamped pulses between the successive giant pulses. (b) ForA = 0.0475, Q1 increased and is around five times Q2. (c) For 0 =0.0480, %1 and Q2 cannot lock precisely at an integer ratio, and weobserve a subharmonic bifurcation in the lowest frequency.

I ' ' ' ''K ' -'

in

....................... ..................................a.......... ......

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

US .. .... .. . . ... .. .. .. . .. ...........

(a)11%

Y| =oI X

(b)

Fig. 6. ( -) = 0.035 and A = 0.021 (subregion 5 in Fig. 3). (a)Intensity versus time. Owing to the interaction between Q, and Q2,which are of the same order, a third, low-frequency Q3 unequal to Q%appears. (b) The corresponding Poincar6 map, taken for ri = 0 in theplane x - y, shows the inequality of the frequencies by giving a con-tinuous curve indicating the presence of a two-torus as the underlyingattractor.

can be seen [Fig. 5(c)] but only in an extremely small regionof the parameter space. We have described the behavior ofthe system in the regions 1, 2, and 4 of Fig. 3.

When the two frequencies involved are approximatelyequal, we hope to see a resonant destabilization of thesystem.

In fact, for ( - 6) = 0.025 and A _ 0.02, the two frequenciesbecome nearly equal, and a third low frequency appears. Thisthird frequency for small values of A is approximately 17 timessmaller than the high frequency represented by Q, [Fig. 6(a)].For A = 2.1 X 10-2, the low frequency is incommensurate withQ, and we obtain a two-torus, as indicated by the closed loopin a Poincar6 map (x - y) taken for I = 0 [Fig. 6(b)]. WhenA is increased, the system does not go to a chaotic behaviorthrough quasi-periodicity but the two frequencies lock at agiven ratio,3 4 and we can see a discrete number of points in thePoincar6 map (Fig. 7). When the amplitude of the externalfield is increased, the system enters an intermittency regionand eventually approaches a chaotic regime (Figs. 8 and 9).

When the absolute value of ( - 6) is increased, subhar-monic bifurcataions involving the giant pulses and transitionto chaos using period doubling are also possible, as is shownin Fig. 10.

A further increase of ( - ) causes the system to enter aregion where a periodic oscillation with a frequency of about£Q (2K-y11(zo - 1) + (0 - 6)2)1/2 is the most typical solution,but for different values of the initial conditions, results similarto those of the region described in expression (20b) can befound. This phenomenon, which is common in nonlineardynamical systems, is the so-called generalized multistabil-

Tredicce et al.

180 J. Opt. Soc. Am. B/Vol. 2, No. 1/January 1985

," . ., ... ~~~~~~~~~~~~~~~.......................

... .,. ...... ........ ............... .... ......... .. .... ..

............... .... .... ... ..... . ... .... ',..... .........

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

_ . . ... _ se m _ , .in.............. 4 ,

(a)

y

x(b)

Fig. 7. For the same value of (0 - ) as in Fig. 6 and increasing A =0.02295, the two frequencies lock at a ratio Q3/Qj = 14. (a) Intensityversus time, (b) Poincar6 map showing a discrete number of pointers(14).

ity.21 Also, in the other regions, different solutions can coexistfor the same values of the parameters; for example, the two-torus generally coexists with a locked torus and with a periodic

d o, 9

A.0o.02336

3

9

6

3

9

6

3

.9

A= 0.02295

0.5

Ye

Fig. 9. Poincar6 map in the plane x - y for A = 0 when the systementers the chaotic region. The nonregion structure of the strangeattractor can be easily seen.

solution in which only the high frequency is present. In thelatter case, the higher and lower peaks must not be confusedwith a period-two subharmonic bifurcation but must be un-derstood as a locking between the frequency of the giant pulsesand that of the relaxation oscillations.

Thus it is clear that, in presence of generalized multista-bility, the different regions of Fig. 3 do not have a commonboundary, but they are superimposed one upon the othergiving a complicated phase diagram.

It must be also noted that there is an asymmetry in theboundaries of each region on changing the sign of ( - ).This asymmetry arises from the fact that we have considereda detuning different from zero, between the atomic and thecavity frequencies.

_/

/

0.5

- . .

. . * . *..- --

*#**-***-*;*-**E-; * ;**. *-.-.s*.--*vA@@ss *;oveve WF@s Cove yt+**s Ago 00* Cove < *# *>tv *0 * -veX;e a;e He be *; > s %

% . . . . . . . . . .

. . - . . - - * .. . .. * HeI * * . . *.

: . :l :-.:--:- .- -. --- -. --. -.- -a *. ........... |.. *-s

l *-S*--S---- **--- .-S .................. -;-

SS;*se=

. ... A....-- A|.....An-- * lo

- . ; .... :- . .. -. -.-..--.-. ::-.

. .: ........ '.........-....- * $ ; *

.. t ........... t.-9.w. -'-.t . ;'>{ t%'%

. . . . . . . .

........... . ..-..--. -',--.

'I ':;; **........... -- . .- - :.-:

.......... ............... ....... ............ A..*....w §2r,,i. . . . . . . . ...

O .8 16 2.4 m s 7�(a)

V.r

an - In

(b)

Fig. 8. (a) A further increase in A causes the system to enter a region of intermittency. The laminar period decreases as A increases until noperiodicity can be found. For sake of simplicity, we plot only the intensity maxima versus the time for different values of A. (b) Iteration mapof the intensity maxima In+14 versus I. Note the near transparency of the map to the fixed point line (main diagonal), which is a well-knownindication of intermittency.

- --

Tredicce et al.

I

. __ ..... . _

I

I........ � -... .

.i. A� A4�_

Vol. 2, No. 1/January 1985/J. Opt. Soc. Am. B 181

6~~~~~~~~~~~~~~~~~46

Z~~~~ . - 4 6

.. .. . .... .... .... .. ..... . ..... .. ... .. . .. .. .... . .. ... .....A o.oz±

(a)

IT

.

17 ~~~~~~~~~~~~. .. ............. ....... ........ ..............

Z ~ ~ ~ ~ ~ ~ ~ ~~ ~... ...... ..... .... . ..... ....... ........" ~~~~~~~~~~~~. ..... ......... ........ ....l. l .l- .

(c)

i0 , ,.

I ...........,.,,..,.,j,,..........,,,,,,,,x

146

A = o.o27064 mm

;

(e)

Fig. 10. ( -a) = -0.0214 (subregion 6 in Fig. 3). Plotting intensityto chaos through period-doubling bifurcations.

5. DESCRIPTION OF THE LASER WITH ANINJECTED SIGNAL IN TERMS OF THE MAPON THE CIRCLE

It was shown recently that many dynamical systems modeledas a damped driven pendulum (e.g., a Josephson junction ina microwave field and a charge density wave in a periodicelectric field) can be described by one-dimensional discretemaps of the circle onto itself, the so-called circle maps, thatare defined by

Ot+ = f(O.) = fOn + Q + g(0n), (33)

where

g(6n) = g(O,9+ 1)mod 1. (34)

The variable On represents the phase of the oscillatingsystem measured stroboscopically at periodic time intervalstn = 2 rrn/We, using the frequency we of the external force as

"t1- 1 1..1 1 A1'..A1 1° . . 1

* .. ........~~~~~~~~~~~~~~~~~~~~~~~~.......... ............_

A0.022 (b)

4eg . . . , . , . , - . - . - - r 1

r Chaos

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

.:t1:. trill t:1: 1 1....I~~~~~~~~~~~... ....... - l:1Il T am ?O _ 8 sa_ O_ _0 _ m_

A 0.023(d)

, ~ k 12 ,j - .

.. . .. . .. .. .. .. .. .. .. ... ... .... ... .. ... .. ... .. ... . .. .. .. ... ., , ,, ,,,,,,,,,,,,,,,,,,,,,

. . I . . . I . ."F ''I''.'''''''''L''''''' l ' ''''''''''''''''''"1 1 ''I' l l-- lm l l- l

A o.o22 (f)~~~~

as a function of time for different values of A, we observe a transition

a clock. A phase shift O, - 0, + 1 represents a full rotation;hence the periodic property of 0O,, The map has a linear term, and a bias term Q representing the frequency of the system

in the absence of the nonlinear coupling g.The periodicity [Eq. (34)] suggests that the map [Eq. (33)]

be studied for the simplest form of g, that is, that the sinemap should be studied:

n+ = f() = n + Q- - sin(27rn).2 ir(35)

For different values of the two control parameters 2 andK one can build a parameter space showing regions of phaselocking that become larger and larger until they overlap. Thisoverlap coincides with the appearance of chaos because thesystem is uncertain about which value to lock when more thanone locking value is available for the same parameters.

To study the mode locking in the circle map, one considersthe iterations of the map 0, f(O), f 2 (O) .... The iteration is

Tredicce et al.

182 J. Opt. Soc. Am. B/Vol. 2, No. 1/January 1985

..

.-

.. i~~~~~~~~~~~~~~~i

Fig. 11. Poincar~ section in the plane z- nforii, in the region where1 and 23 are unequal.0.

U._-

Fig. 12. Return map for the phase, corresponding to the Poincaresection of Fig. 11. The dashed line shows the return map obtainedby resolving Eqs. (10) numerically. It is clear that the system can bedescribed by a circle map. For comparison, we show by means of acontinuous line the well-known sine map using (4 - COL)/[Ky II (Zo- 1)11/2 and A as parameters.

conveniently described by the winding number

The winding number is the mean number of rotations per it-eration, that is, the frequency of the underlying dynamicalsystem, so W = in the absence of nonlinear coupling. Underiteration, the variable O~ may converge to a series that is pe-riodic:

On+Q = n + P (rational W = P/Q), (37)

quasi-periodic (with irrational W), or chaotic (where the seriesbehaves irregularly).

It can be shown 35 that, for increasing K, the sine map dis-plays phase-locked regions that can widen until, for K = 1,they almost fill the whole interval 0 < Q < 1. For K > 1, thechaotic behavior is possible. At K = 1, the number of residualunlocked regions shrinks with a universal law that seems tobe independent of the exact form of g(O) in Eq. (33).

In our case, we found it convenient to work in the repre-sentation (n, i, z), where n = 2 + y2 is the photon numberand At its time derivative, rather than in the (x, y, z) phasespace. In this representation, for given values of w - coL andA, we have found Poincar6 sections at A = 0 (plots of themaxima of the intensity n), which fill a circle-line regionuniformly (Fig. 11). Shifting the origin at the point n =38.00,z =1.00, we have built aplot of the phase On+1lversus On. Thisis shown in Fig. 12. In the same figure, we have shown the sinemap for comparison.

We see an evident typical equivalence between our LISsystem and the one-dimensional map. This is a remarkableresult, because, in general, our system of Eqs. (10) should beequivalent to a two-dimensional iteration map once we per-form an integration over a period of the external signal. Theone-dimensional character comes from the combination of thedamping rates. We leave a complete analysis of the transitionfrom a two-dimensional to a one-dimensional map, plus ananalysis of the universal behavior near the onset of chaos, toa future paper.

6. CONCLUSIONS

We have presented a detailed stability analysis of a Class Blaser with an injected signal, and we have demonstrated thenecessary and sufficient conditions for obtaining a stableoutput with a frequency locking of the system to the exter-nal-field frequency.

At the same time, in the unstable region, we have shownhow it is possible to reach chaos by two different scenarios:period-doubling cascades and intermittency. We also predictthe appearance of two incommensurate frequencies and thelocking between them. Finally, we have shown that a circlemap describes the behavior of the LIS. However, a detailedanalysis of general solutions for parameter regions that werenot explored here [e.g. (0 - ) 0(1)] and a complete de-scription of the system by means of the circle map are exten-sions of this research currently in progress.

ACKNOWLEDGMENTS

We are grateful to N. B. Abraham, L. M. Narducci, G. L. Oppo,and A. Politi for useful discussions.

F. T. Arecchi is also with Dipartimento di Fisica, Universtadi Firenze, Lango E. Fermi 2, I50125 Firenze, Italy.

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