Instabilities in lasers with an injected signal

Transcription

1 Tredicce et al. 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, Firenze, Italy Received July 10, 198; accepted October 1, 198 We consider a laser with an injected signal, in which the polarization can be adiabatically eliminated, we study the stability of the steady-state solutions, and we discuss the time-dependent solutions. For the laser alone, the only possible solution is constant intensity. However, the introduction of an external field, with an amplitude that does not satisfy the injection-locking condition, destabilizes the system. In such a case, numerical results show the existence of a self-q-switching process, which induces relaxation oscillations. The frequency of the giant pulses is directly related to the amplitude of the external field, whereas the frequency of the relaxation oscillations depends on the damping rates. We show also that, depending on the value assigned to control parameters, the interaction between these frequencies leads to a chaotic behavior through intermittency or period-doubling bifurcations. Finally, topological equivalence between our laser system and a unidimensional circle map is shown for some values of control parameters. 1. INTRODUCTION Shortly after the introduction of the first laser, signal injection was considered a suitable method to achieve a stable output from intrinsically unstable lasers.'- 9 This technique was applied first to He-Ne lasersl" 9 "1 0 to obtain a single mode from a Doppler-broadened gain line and second, to dye lasers in which the insertion of an external field from a spectral-band laser results in a spectral narrowing of the injected laser. 2 8 The method was then applied to C0 2 -TEA lasers using different arrangements, with two consequences, namely, mode selection and smooth pulses. 6 Finally, it was applied to high-power conventional CO 2 lasers in order to stabilize the output intensity of the system. 3 More recently, the interest in a laser with an injected signal (LIS) has been centered on semiconductors, for which the area of application is broad."1 From a theoretical point of view, locking phenomena were already well known in electronic oscillators before they were noted in lasers' 2 ; a general treatment of the LIS, together with a stability analysis of the steady-state solutions valid for the most-popular atomic gas lasers or dye lasers, is given in Ref. 13, where pulsing in the intensity and in the phase is predicted when the external field is not strong enough to lock the system. However, the region of interest remains the locked one, and different treatments have been made of homogeneouslyl 3 -' 5 and inhomogeneously broadened lasers' 5 to establish the general requirements in terms of the detuning and the amplitude of the external signal in order to reach injection locking. A preliminary study of the unstable region is given in Refs. 10, 11, and 13, based on theoretical and experimental evidence of stable pulsations of the beating frequencies between the external and internal laser frequencies.' 0 However, a LIS is a good candidate for studying chaotic behavior since it can provide the number of degrees of freedom' 6 necessary for deterministic chaos; in order to limit ourselves to a small number of dynamical variables, we refer to single-mode lasers in homogeneously broadened media, so that the entire dynamics is characterized by three dynamical variables, namely, field amplitude, polarization, and population inversion.' 7 With this in mind, we must distinguish among three classes of homogeneously broadened lasers' 8 : (1) Class A (for example, dye lasers) The loss rate for the field () is much less than the polarization and populationinversion loss rates (yj and -yil, respectively). In this class, because of the difference among the time scales of the collective variables, it is possible to perform an adiabatic elimination of the atomic variables. Therefore the system can be described by means of just one rate equation for the field. By including a constant, coherent, external field, it is possible to increase the number of degrees of freedom by one. The possible results are stable, locking, or regular pulsations, depending both on the detuning between the two fields and on the amplitude of the external signal. The unstable region is represented by the oscillation of the intensity at the beat frequency.' 0 To achieve higher-order instabilities and eventually chaotic behavior, it is necessary to modulate one of the control parameters, namely, the external field, the pump rate, or the cavity losses (2) Class B (for example, ruby, Nd:YAG, and CO 2 lasers) Here, and -yii are much smaller than -yi, and adiabatic elimination of the polarization is feasible. The dynamic behavior of this class of lasers is described by two coupled nonlinear equations: one for the field and the other for the population inversion. However, only a stable intensity output is obtainable in the absence of an external perturbation. In these lasers, it is necessary to increase the number of degrees of 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 of one parameter. Inject an external field. Such a system has been briefly analyzed in the case of semiconductor lasers. 23 Regular pulsing with and without damped oscillations was found. However, semiconductor lasers have the peculiarity that their operating frequency and gain depend in a nontrivial way on /85/ $ Optical Society of America

2 17 J. Opt. Soc. Am. B/Vol. 2, No. 1/January 1985 the carrier density, and some approximations must be made to obtain a simple model. We reported previously' 8 the transition to chaos through intermittency for molecular gas lasers. (c) Increase the number of lasing modes. This situation can be achieved for a bidirectional ring cavity, and it was shown that regular and chaotic pulsing were found for a wide range of parameters 2 (e.g., pressure and excitation). Also, a high value of excitation produces the gain for modes that are detuned from the central mode to overcome the losses; then instabilities in those modes are possible even if the predominant mode is stable 25 ; this situation is more easily reached with a Fabry-Perot cavity, in which different modes can be driven simultaneously by the population inversion. (3) Class C The three collective variables are damped on the same time scale ( y yii). Such a case was treated by Haken, 2 6 who showed that instabilities can be reached for the central mode when is larger than y + 11 and when the excitation at threshold becomes larger than (k + y + 11)(yI + )/yi(k - y - yii). The first of these conditions requires a bad cavity (high ) and, considered with the second condition, an extremely high pumping value, is extremely difficult to achieve with most of the usual laser systems. The only example is the far-infrared laser, but no single-mode instability has been reported yet. However, an injected signal increases the number of degrees of freedom from three to five because field and polarization become complex. This case was treated theoretically, and evidence of subharmonic bifurcations and chaos was given even when the bad-cavity condition was not satisfied. This classification system is not applicable to inhomogeneously broadened lasers (e.g., He-Ne, argon-ion, and Xe lasers) in which the coexistence of several gain packets ensures that, even with << yi, 7yI, the number of degrees of freedom is large enough to find instabilities and the bad-cavity condition is not necessary If an inhomogeneously broadened laser is set to operate in a single-mode frequency, then a bad cavity is necessary; however, the conditions are not so restrictive as in the homogeneously broadened case We present a detailed analysis of LIS instabilities for Class B lasers below. 2. STABILITY ANALYSIS OF A CLASS B LASER With the assumption that, 7y11 <<1, the Maxwell-Bloch equations describing N homogeneously broadened two-level atoms coupled to a single-mode field' 6 can be reduced to two rate equations after adiabatic elimination of the polarization: E =-E(1 +i)+ g 2 EAN yi(1 + i) g 2 El 2 AN'1 AN = - 11(AN - ANo) (1) where E is the field amplitude, AN is the population inversion, g is the coupling constant, ANo is the unsaturated population inversion, and 0 and 6 are given by 0 = (c - WL)/, 6 = (o -L)l (2),wc, wo, and CtL being the cavity, atomic, and laser operating frequencies, respectively. If we normalize E 2 in units of the photon number inside the cavity, the coupling constant in MS units is given by g2 = COA 2 /heov. (2) Solutions of Eqs. (1) at steady state (E = AN = 0) yield with the condition or _AI2= 7±711 [g 2 ANo ](+62), N= (1 + 2), (3) 0 = -6 (a) Wac WL COL WO y±l This is the frequency-pulling formula for a homogeneously broadened laser. A linear stability analysis of the nontrivial steady-state solution (El2 > 0) gives the eigenvalues where X IIZo [ 1zo -2 11(zo - 1) 1 2 g 2 ANO OyI(1 + 62) is the unsaturated population inversion normalized to the steady-state value AN. Here by inspection we conclude that (1) The steady-state solutions are always stable above threshold (zo > 1) because the second term in Eq. (5) is smaller than the first one. So the real part of X(XR) is negative for all parameter values. Then the only possible solution is a stable fixed point in the phase space, even if the dynamic behavior is described by two coupled differential equations. (2) The sum under the square root is positive for all values of zo, if y 11 is larger than 2. In this case, a small perturbation will induce a monotonic decay to the steady-state values. (3) If Y <2 in the presence of a perturbation, the laser can react with a damped oscillation lasting 2/7y1z0 under the condition of the excitation z 0 that [8 (2_1)/2 7Y Z <-+ [( - 1/ 'y11 rl11 Tredicce et al. In the limiting case 71 << (common to many Class B lasers (e.g., ruby or C0 2 lasers), oscillations can be observed in the range of the possible experimental values of zo. In this case, the eigenvalue X becomes (5) (6)

3 Tredicce et al. Vol. 2, No. 1/January 1985/J. Opt. Soc. Am. B 175 X = XR + ixi _ - 'y +i [2y 11 (zo - )]1/2. (7) 3. LASERS WITH INJECTED SIGNALS: STABILITY ANALYSIS OF A CLASS B SYSTEM Here we introduce an external field with a frequency wl, which is detuned from the laser frequency L. The rate equations in a frame rotating with an arbitrary angular frequency Go' are E = -E(1 + 0,) + g- EN I+Eoeiat, -yj' 1 + i' = - I (AN - AN o ) - g je+ 2 AN _y (1 + 3/12) where EO is the amplitude of the external field and c- c' The existence of an imaginary part in X suggests that this frequency appears as a response of the system to perturbations. A large external perturbation tuned around that frequency can destabilize the system, resulting in large Q switching for locking of the pulsed output to the modulation frequency. 3 3 For lower amplitude modulation, it was conjectured that the steady states might be destabilized without locking precisely to the modulation frequency. Following the philosophy of our previous research, 18 we have obtained; chaotic output from a CO 2 laser for a small amplitude modulation of cavity losses near the relaxation-oscillation frequency. O=- 0, = co- co' =Wo - co' -yj- 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 be achieved only by assuming an operating frequency co' equal to the frequency of the injected signal. Hence, without loss of generality, we take as a natural reference frame the external signal w' = w1, and we write the complex amplitude E as ER + ie. For convenience of computation, we normalize the field to the saturation value (y±iy ) 1 / 2 /2g and the population (8) (9) A= I (0 - )j Fig. 2. Steady-state solution for LIS equations. Output intensity versus intensity of the injected field for constant value of the pump parameter (zo) and different values of the detuning (1 - WL). The dashed line shows the unstable region and the solid line shows the region where the steady-state solutions are stable. For each curve, the critical value of the intensity of the injected field is marked with Aj2. For A 2 larger than Aj 2, there is no instability. inversion to its threshold value -yi( )/g 2. In this way, Eqs. (8) can be written as 18 where. -z -y1 = (z - 1)x + (-3z)y + A, = -(0-6z)x + (z - l)y, x 2 + y g_ R (L 1) 1/ER, 2g_ EI, g 2 AN yi(l + 62)' A= 2gEo (7 l yi)1/2' (10) (11) and now the dot denotes the derivative with respect to the dimensionless time = t. The nontrivial steady-state solution of Eqs. (10) gives z = Zo 2 + y (12) and k A 2 = [(2-1)2 + ( - 2)2](g2 + y2) (13) In Fig. 2, we have plotted the steady-state intensity I = ( )/(1 + 2) as a function of the intensity of the injected signal (A 2 ) for different values of the detuning between the external signal W and the internal laser frequency L. This detuning in terms of 0 and is given by the relation 0') CO10 Fig. 1. Qualitative plot of the frequency relations among atomic resonance (homogeneous width y ) centered at w 0, cavity resonance (width ) centered at w,, and injected field at wl. (0-) = (w1-col)- y (1) The eigenvalues obtained from the stability analysis of the steady-state solution are solutions of the equation

4 176 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) 'Y 1 2I[6(0-2) -(2-1)]= 0 (15) roots, which, in general, are complex. From the point of view of these eigenvalues, the stationary state becomes unstable when the real part becomes positive. Thus the bifurcation point is represented by X 2, 3 = XR(2,3) ilxi(2,3) = ±ixi(2, 3 )- Tredicce et al. In such a case, the condition for a Hopf bifurcation is (18) 711 U+ (2-1)2+ (0-62)2-2'Y'l ( - 1)(I+ 1) 2 y 112IJ-1 )[(- 1)2 + (0-62)2] I[5(0 - z2) -(2- )]} [' (I + 1)-2(- 1)]. - (19) A complete analysis of the solutions of Eq. (15) as a function of the pumping rate z 0 and the detuning (0 - ) is left to a future paper. We limit ourselves here to the cases in which zo > 2 and the detunings 0 and are much less than one, which means that the frequency of the external field differs from the cavity frequency by less than 7-I. Since << My, this necessarily limits us to small detunings (in units of the cavity frequency) from the atomic frequency [in units of -yi, (o, - coo)/y << 1]; however, some results can be easily extrapolated to large detunings. With these assumptions, we can obtain the points where the real root of Eq. (15) becomes zero. If XA = 0, 7ll (I + )[(z - 1)2 + (0 52) 'Y I[6(0-62) - ( - 1)] = 0. (16) Equation (16) corresponds to the condition necessary to have da 2 /dj = 0; therefore XA = 0 at the two points of infinite slope in Fig. 2. This result is also valid for large detunings. Thus, taking into account the approximations considered above, we obtain the 2 and I values: Z1 ; I, -= ZO -1, The point 23, I3 has no physical sense for a laser because I3 is negative for z 0 larger than zero. The two possible solutions are (2i, I) and (2, I2). The first one is equivalent to the steady-state values of the laser without the external signal and corresponds to point 1 in Fig. 2, whereas the second one corresponds to point 2, where the intensity value is less than one. Between these two points, the value of XA is positive and the steady-state solutions are unstable with respect to small perturbations of the system. Now we will study the two other We consider three different regions in the values for 2, I: 0 < ;f ; 1, IO = ZO - I ;! I, [1 + (1 + 8/zo) 1 / 2 < 2 X 1 < 0, Z -< '[1 + (1 + 8/zo)] 1 /2 2 <z 0, (1 +1I+ 8/zo) 1 / 2 1 (20c) (20a) (20b) Independently of the region of interest, as =2_0 0 <1 the condition can be written as -2 I (I )(z-1) I 2I + (-1)2 U - 1) 1'Y (I + 1) ( - 1) + 2 l 2( - 1) 71 U'I( + 1) In the region described in expression (20a) no Hopf bifurcation is possible because Eq. (21) cannot be satisfied Z2 - [1 + (1 + 8/zo)'/ 2 ], when (2-1) is less than zero. In such a case, the intensity or the In -1, population inversion should be negative, which makes no 1 + (1 + 8/zo) 1 / 2 physical sense. As a conclusion for values of the intensity larger than (zo - 1), the system is stable and locked to the Z3 - [1 - (1 + 8/zo) 1 2], external signal. It is easy to evaluate X 2, 3 when XA changes sign, with the I3 1. ~~~~~~~(17) condition that _ 1 and I z - 1. In this case, Eq. (15) is 3 1-(1 + 8/ZO)1/2 (7 reduced to X2 + f1 (I + 1) - 2(2-1)X + 1)2 + (0-62)2-2 z ( - 1)(I+ 1) for which the solutions are (21) , (22)

5 Tredicce et a. A2,3 =1 [(-1-( + 1)] Vol. 2, No. 1/January 1985/J. Opt. Soc. Am. B 177 X2,3 = [1 + (1 + 8/zo)1/ 2 ] - J + [ 2 (I + 1)2 + I (- 1)(I + 1) ±i{0-6z [ + (1 + 8/zo)1/2]}. (28) ( )2-12, + - Asz_ 1, I-zo -1, YllI <<, 2 ~ 2 1 i[2 't Z- 1) + (0-)2 1/2; (23) It is evident that A 2, 3 has a real part that is positive for z > (23) 1; so the system reacts with stable oscillations at a frequency Q2 given by (2) hence, at point 1 in Fig. 2, the system relaxes to the steadystate value with a damped oscillation having a frequency (in units of ) of the order of (25) In the region described in expression (20c), where 2 is bounded by zo/2 <2f < zo and the intensity by 0 < I < 1, depending on the value of zo, the only possible solution for zo > 2 is 2 _ 1. However, this value belongs not to this region but to the region described in expression (20b). So Hopf bifurcations are not possible in this region for z > 2. To find the stability of the system, we must evaluate at least the real part Of X2,3 in this region. We know that Xi is less than zero and that the three roots are connected by the relations 1\X 2 X 3 = (2-1)[-( - 1)(I + 1) + 2I], X 2 X 3 + X 1 (X 2 + X 3 ) = (E- 1)2 + (0 62)2-2 zi [(z1)(i + 1) + 2I], X3 = 2(2-1)- -(I + 1), (26) which are general relations for the three regions described in expressions (20a)-(20c). In the region described in expression (20c), and with the condition that zo > 2, relations (26) become Xl + X 2 + X 3 = 2(2-1), X 2 X 3 + X 1 (X 2 + 3) = ( - 1)2, and 1/2 Q, = 2 'I' (ZO - 1) + ( - )2 - I I X 1 X 2 X 3 = (2-1 )[71 (2-1)(I + 1) + 2 A 3 = X2*- 2IJzl, (27) As a consequence, Xi = 2[(z - 1) - a], with a = Re(X 2, 3 ), and, if Xi < 0, it follows that a > +(2-1), which is larger than zero; then the system will be unstable in the region described in expression (20c), and no Hopf bifurcation is possible for zo > 2. We can evaluate 2, 3 in point 2 of Fig. 2, i.e., for the values 02 = (1 + 8/zo)1/ 2. (29) The last element to take into account in this stability analysis is the value of the external field necessary to reach a stable output, which is the value of A for which 2 1 and I zo - 1. From Eq. (13) it is easy to obtain Ao 2 (0-6) 2 (zo- 1) Ao _ /~ (0-6). (30) Then, for an external field larger than Ao, we obtain a stable output locked in frequency to the external signal. This result is important for laser stabilization by signal injection because it is clear that the amplitude of the external field necessary for locking is proportional to the square root of the intensity of the laser without an injected signal and to the detuning between the external and internal laser frequencies. For high-power lasers it is important to reduce the difference (0 - ) to the minimum possible value. This is done by reducing the cavity losses and controlling the cavity frequency of the laser (e.g., the distance between mirrors). When the external-field amplitude is less than AO, the system becomes unstable, and oscillations must grow. In fact, we have obtained different kinds of instabilities for small values of A.. NUMERICAL RESULTS On integrating Eqs. (10) numerically for an external-field amplitude less than AO, we have obtained different results that are summarized in the phase diagram (0-6) as a function of A in Fig. 3. For A < AO and co, - L, the output intensity remains constant for a long time at about the steady-state value I zo - 1. During this time, the instantaneous frequency 1 3 (8- ).- (- 6) 2 =-(1 + (1 + 8/zo)1/ 2 Using Eq. (23), it follows that 1= ( + 8/zo)1/2 Fig. 3. Phase diagram in the parameter space; detuning versus amplitude of the external field. The limits in the unstable region indicate the different kinds of solutions as shown in Figs. -10.

6 178 J. Opt. Soc. Am. B/Vol. 2, No. 1/January 1985 Tredicce et al. I ",. '. ' * !----*.B.'._....., _..., Pt. - - (a),~._ 2_ 1_. X _ I Z.,,,.., I\..1.. I _ a AM i s A n..., _._ (c) Fig.. Behavior of the system in the subregion 1 of Fig. 3 [(0-6) = ; A = 0.063]. (a) Intensity as a function of time: giant pulses induce damped relaxation oscillations. The frequency of the giant pulses (1) is much smaller than the relaxation-oscillation frequency, and the peak intensity is regular. Phases (0) as a function of time: the external field locks the frequency of the system during a long period but is not strong enough to maintain this situation forever. When the system unlocks, a rapid change in the phase by a factor 27r occurs. (c) Imaginary part 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 the figure indicate the rapid change in phase and the long time during which the systems remains locked. Considering that the intensity is the distance from these points to the origin, the loop in the upper side shows the existence of relaxation oscillations. (d) Population inversion versus time. 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. (d) where and where Winst = 1-W 1(31) =arg(x + iy) (32a) dtj x - ~~~ ~(32b ) d(t) x 2 +y 2 is approximately equal to w 1. Therefore the laser is rather closely locked to the external frequency, and the field is in quadrature with the external field (x = 0; y = ). However, the external signal is not strong enough to maintain this situation permanently, and the system unlocks periodically, changing its phase by an amount 27r and giving out a giant pulse (self-q-switching) above the stationary value of the intensity. At the same time, this pulse acts as a perturbation of the stationary value, and the intensity relaxes back to that value with damped oscillations with a frequency given by Eqs. (27). The plots of the intensity and the phase of the field as a function of time and the imaginary part versus the real part of the field are shown in Fig.. From Fig. (d), it is clear that the population inversion grows during the locking and then transfers the energy into the field. So the effects of the external field are, first, locking the frequency of the system to its own frequency over a given time and, second, giving a continuous growth of the population inversion during that time until the system unlocks. It is clear that the pulses do not arise from a Hopf bifurcation because the amplitude is proportional to the externalfield amplitude, which is larger when it is near the locked regi6n. The second important consideration is that two frequencies are involved during this process: (1) The frequency of the giant pulses, which is inversely proportional to the intensity of the external field and whose maximum value (for IAl 2 << IAol 2) is proportional to the detuning between the external-field' frequency w 1 and the laser frequency WL-. (2) The damped relaxation oscillations generated by the presence of the giant pulses.

7 Tredicce et al. For values of ( 1 - WL) larger than [(2,y 1 )/)(zo-1)11/2 Q, the frequency of the giant pulses can be of the order of the damped oscillations for suitable values of the external amplitude A. Then if the detuning is larger than or of the order of 1 the relaxation oscillations become undamped because the field cannot relax back onto the steady-state value before another giant pulse arrives. In Fig. 5(a), we can see five undamped pulses of the relaxation oscillations between two giant pulses. As the external-field amplitude is decreased, this number decreases [Fig. 5]. In the transition region among situations with different integer numbers of relaxation oscillations between giant spikes, a subharmonic bifurcation of the smaller frequency in Vol. 2, No. 1/January 1985/J. Opt. Soc. Am. B 179 I ' ' ' '' ' ' a US (a) 11% Y =o I X. 0.0 (a).m m H EW_ f f : ''.. Fig. 5. Intensity versus time with ( - 6) = (subregions 2 and of Fig. 3). As the amplitude of the external field decreases, the frequency of the self-q switching (1) increases, and the relaxation oscillations became undamped. (a) For A = 0.085, Q 1 is around six times larger than the relaxation-oscillation frequency Q 2. We observe five undamped pulses between the successive giant pulses. For A = 0.075, Q1 increased and is around five times Q2. (c) For 0 = 0.080, %1 and Q 2 cannot lock precisely at an integer ratio, and we observe a subharmonic bifurcation in the lowest frequency. (c) Fig. 6. ( -) = and A = (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. The corresponding Poincar6 map, taken for ri = 0 in the plane x - y, shows the inequality of the frequencies by giving a continuous curve indicating the presence of a two-torus as the underlying attractor. can be seen [Fig. 5(c)] but only in an extremely small region of the parameter space. We have described the behavior of the system in the regions 1, 2, and of Fig. 3. When the two frequencies involved are approximately equal, we hope to see a resonant destabilization of the system. In fact, for ( - 6) = and A _ 0.02, the two frequencies become nearly equal, and a third low frequency appears. This third frequency for small values of A is approximately 17 times smaller than the high frequency represented by Q, [Fig. 6(a)]. For A = 2.1 X 10-2, the low frequency is incommensurate with Q, and we obtain a two-torus, as indicated by the closed loop in a Poincar6 map (x - y) taken for I = 0 [Fig. 6]. When A is increased, the system does not go to a chaotic behavior through quasi-periodicity but the two frequencies lock at a given ratio, 3 and we can see a discrete number of points in the Poincar6 map (Fig. 7). When the amplitude of the external field is increased, the system enters an intermittency region and eventually approaches a chaotic regime (Figs. 8 and 9). When the absolute value of ( - 6) is increased, subharmonic bifurcataions involving the giant pulses and transition to chaos using period doubling are also possible, as is shown in Fig. 10. A further increase of ( - ) causes the system to enter a region where a periodic oscillation with a frequency of about Q (2-y 11 (zo- 1) + (0-6)2)1/2 is the most typical solution, but for different values of the initial conditions, results similar to those of the region described in expression (20b) can be found. This phenomenon, which is common in nonlinear dynamical systems, is the so-called generalized multistabil-

8 - -- % J. Opt. Soc. Am. B/Vol. 2, No. 1/January 1985 Tredicce et al. I,"..,... ~~~~~~~~~~~~~~~......, ', I i. A A_ m _ se _,.....in..., (a) Ye Fig. 9. Poincar6 map in the plane x - y for A = 0 when the system enters the chaotic region. The nonregion structure of the strange attractor can be easily seen. y x Fig. 7. For the same value of (0 - ) as in Fig. 6 and increasing A = , the two frequencies lock at a ratio Q3/Qj = 1. (a) Intensity versus time, Poincar6 map showing a discrete number of pointers (1). ity. 21 Also, in the other regions, different solutions can coexist for the same values of the parameters; for example, the twotorus generally coexists with a locked torus and with a periodic solution in which only the high frequency is present. In the latter case, the higher and lower peaks must not be confused with a period-two subharmonic bifurcation but must be understood as a locking between the frequency of the giant pulses and that of the relaxation oscillations. Thus it is clear that, in presence of generalized multistability, the different regions of Fig. 3 do not have a common boundary, but they are superimposed one upon the other giving a complicated phase diagram. It must be also noted that there is an asymmetry in the boundaries of each region on changing the sign of ( - ). This asymmetry arises from the fact that we have considered a detuning different from zero, between the atomic and the cavity frequencies. d o, 9 A.0o *. * A= *#**-***-*;*-**E-; 9 :. : 6 l :-.:--: a*..... *-s l *-S*--S---- **---.-S... -;- 3 SS;*se= ;... : :: * ;**. *-.-.s*.- -*va@@ss *;oveve WF@s Cove yt+**s Ago 00* Cove < *# *>tv *0* -vex;e a;e He be *; > s % *..... * He I * *.. *..... A...-- A... An-- * lo ' $..:... * ; *.. t... t.-9.w. -'-.t. ;'>{ t%'% 'I ':;; ** ', :.-: A..*... w 2r,,i. O m s (a) 0.5 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 no periodicity can be found. For sake of simplicity, we plot only the intensity maxima versus the time for different values of A. Iteration map of the intensity maxima In+1 versus I. Note the near transparency of the map to the fixed point line (main diagonal), which is a well-known indication of intermittency. _/ / I 0.5 an - In V. r

9 Tredicce et al. Vol. 2, No. 1/January 1985/J. Opt. Soc. Am. B 181 ; Z~~~~. - 6 A o.oz± 6~~~~~~~~~~~~~~~~~6 (a) *. "t A1'..A A ~~~~~~~~~~~~~~~~~~~~~~~~......_ eg...,.,., r 1 r Chaos. IT 17 ~~~~~~~~~~~~ :t1:. t:1: trill 11...I~~~~~~~~~~~ l:1il (c) ~ ~ ~ ~ ~ ~~ ~ Z ~ ~ " ~~~~~~~~~~~~ l. l.l-. i0,,. I...,.,,..,.,j,,...,,,,,,,,x T am?o _ 8 sa_ O 0 _ m_ A (d), ~ k 12,j-. "F ''I''.'''''''''L l ' '''''''' ,,,,,,,,,,,,,,,,,,,,,,,,,.. I... I.. 16 A = o.o2 706 mm (e) Fig. 10. ( -a) = (subregion 6 in Fig. 3). Plotting intensity to chaos through period-doubling bifurcations. "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 5. DESCRIPTION OF THE LASER WITH AN INJECTED SIGNAL IN TERMS OF THE MAP ON THE CIRCLE It was shown recently that many dynamical systems modeled as a damped driven pendulum (e.g., a Josephson junction in a microwave field and a charge density wave in a periodic electric field) can be described by one-dimensional discrete maps of the circle onto itself, the so-called circle maps, that are defined by where Ot+ = f(o.) = fon + Q + g(0n), (33) g(6n) = g(o,9+ 1 )mod 1. (3) The variable On represents the phase of the oscillating system measured stroboscopically at periodic time intervals tn = 2 rrn/we, using the frequency we of the external force as 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. (3)] suggests that the map [Eq. (33)] be studied for the simplest form of g, that is, that the sine map should be studied: n+ = f() = n + Q- - sin(27rn). 2 ir (35) For different values of the two control parameters 2 and one can build a parameter space showing regions of phase locking that become larger and larger until they overlap. This overlap coincides with the appearance of chaos because the system is uncertain about which value to lock when more than one locking value is available for the same parameters. To study the mode locking in the circle map, one considers the iterations of the map 0, f(o), f 2 (O)... The iteration is

10 182 J. Opt. Soc. Am. B/Vol. 2, No. 1/January 1985 Tredicce et al i~~~~~~~~~~~~~~~i We see an evident typical equivalence between our LIS system and the one-dimensional map. This is a remarkable result, because, in general, our system of Eqs. (10) should be equivalent to a two-dimensional iteration map once we perform an integration over a period of the external signal. The one-dimensional character comes from the combination of the damping rates. We leave a complete analysis of the transition from a two-dimensional to a one-dimensional map, plus an analysis of the universal behavior near the onset of chaos, to a future paper. Fig Poincar~ section in the plane z- nforii, in the region where 1 and 23 are unequal. U._- Fig. 12. Return map for the phase, corresponding to the Poincare section of Fig. 11. The dashed line shows the return map obtained by resolving Eqs. (10) numerically. It is clear that the system can be described by a circle map. For comparison, we show by means of a continuous line the well-known sine map using ( - COL)/[y 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 iteration, that is, the frequency of the underlying dynamical system, so W = in the absence of nonlinear coupling. Under iteration, the variable O~ may converge to a series that is periodic: On+Q = n + P (rational W = P/Q), (37) quasi-periodic (with irrational W), or chaotic (where the series behaves irregularly). It can be shown 35 that, for increasing, the sine map displays phase-locked regions that can widen until, for = 1, they almost fill the whole interval 0 < Q < 1. For > 1, the chaotic behavior is possible. At = 1, the number of residual unlocked regions shrinks with a universal law that seems to be independent of the exact form of g(o) in Eq. (33). In our case, we found it convenient to work in the representation (n, i, z), where n = 2 + y 2 is the photon number and At its time derivative, rather than in the (x, y, z) phase space. In this representation, for given values of w - col and A, we have found Poincar6 sections at A = 0 (plots of the maxima of the intensity n), which fill a circle-line region uniformly (Fig. 11). Shifting the origin at the point n =38.00, z =1.00, we have built aplot of the phase On+1lversus On. This is shown in Fig. 12. In the same figure, we have shown the sine map for comparison. 6. CONCLUSIONS We have presented a detailed stability analysis of a Class B laser with an injected signal, and we have demonstrated the necessary and sufficient conditions for obtaining a stable output with a frequency locking of the system to the external-field frequency. At the same time, in the unstable region, we have shown how it is possible to reach chaos by two different scenarios: period-doubling cascades and intermittency. We also predict the appearance of two incommensurate frequencies and the locking between them. Finally, we have shown that a circle map describes the behavior of the LIS. However, a detailed analysis of general solutions for parameter regions that were not explored here [e.g. (0 - ) 0(1)] and a complete description of the system by means of the circle map are extensions of this research currently in progress. ACNOWLEDGMENTS 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, Universta di Firenze, Lango E. Fermi 2, I50125 Firenze, Italy. REFERENCES 1. H. L. Stover and W. H. Steier, Appl. Phys. Lett. 8, 91 (1966). 2. L. E. Erickson and A. Szabo, Appl. Phys. Lett. 18, 33 (1981). 3. C. J. Buczek and R. J. Freiberg, IEEE J. Quantum Electron. QE-8, 63 (1972).. A. Girard, Opt. Commun. 11, 36 (197). 5. J. L. Lachambre, P. Lavigne, G. Otis, and M. Noel, IEEE J. Quantum Electron. QE-12, 756 (1976). 6. R. Daigle and P. Belanger, Opt. Commun. 23,165 (1977). 7. Y.. Park, G. Giuliani, and R. L. Byer, Opt. Lett. 5, 96 (1980). 8. P. Burlamacchi and R. Salimbeni, Opt. Commun. 17, 6 (1976). 9. R. W. Dunn, S. T. Hendow, W. W. Chow, and J. Small, Opt. Lett. 8, 319 (1983). 10. W. Annovazzi and S. Donati, IEEE J. Quantum Electron. QE-16, 859 (1980). 11. R. Lang, IEEE J. Quantum Electron. QE-18, 979 (1982), and references therein. 12. R. Adler, Proc. IRE 3, 351 (196). 13. M. Spencer and W. Lamb, Jr., Phys. Rev. A 5, 88 (1972). 1. C. J. Buczek, R. J. Freiberg, and M. Skolnick, Proc. IEEE 61,111 (1973). 15. W. W. Chow, IEEE J. Quantum Electron. QE-19, 23 (1983). 16. H. Haken, in Handbuch der Physik, L. Genzel, ed. (Springer- Verlag, New York, 1970), Vol. 25/2c.

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