Instabilities in a hybrid bistable system without delayed feedback

Volume 75, number 2

OPTICS COMMUNICATIONS

15 February 1990

INSTABILITIES IN A HYBRID BISTABLE SYSTEM WITHOUT DELAYED FEEDBACK F.A. N A R D U C C I ', D.W. BROMLEY, G.-L. O P P O 2 and J.R. T R E D I C C E Department of Physics and Atmospheric Science, Drexel University, Philadelphia, PA 19104, USA

Received 1 September 1989

We show that a hybrid bistable system can display oscillatory or even chaotic behavior when an amplifier is introduced in the feedback loop, despite the absence of a delay. We study the effects of bias and the amplifier'sgain and bandwidth on the dynamics of the system.

1. Introduction

simulations o b t a i n e d from a simple model with only three degrees o f freedom.

During the past several years, hybrid bistable systems have been widely studied, both theoretically and experimentally [1]. Self-pulsing and chaos via period-doubling bifurcations have been observed when the feedback loop was affected by a delay time greater than the response time o f the system [2,3]. These devices generally consist o f a laser b e a m passing through an acousto-optic ( A O M ) or electro-optic ( E O M ) modulator. A detector then transform the output intensity into a voltage which is used to control the losses introduced by the A O M [4] or EOM [ 5 ] driver. Usually, the output signal o f the detector is in the range o f hundreds o f millivolts, while the required voltage of the nonlinear device is tens o f volts for the A O M to h u n d r e d s o f volts for the EOM. An amplifier introduced in the feedback loop is then absolutely necessary to observe bistability. Here, we show both theoretically and experimentally that the presence o f the amplifier drastically changes the d y n a m ical behavior o f the system, enlarging its phase space dimensionality so that oscillations and chaos can be observed without a delayed feedback. The experimental results agree qualitatively with numerical Present address: Department of Physics, University of Rochester, Rochester, NY 14627, USA. Present address: Department of Physics and Applied Physics, University of Strathclyde, John Anderson Building, Glasgow G4 0NG, Scotland, UK. 184

2. Experimental results The experimental setup is schematically shown in fig. I. The intensity o f the H e N e polarized laser is controlled by an acousto-optic m o d u l a t o r ( A O M l ). Input and output intensities on the A O M 2 are measured by two p h o t o d i o d e s (P 1 and P2, respectively). The corresponding output voltage is amplified and sent back into the A O M 2, closing the feedback loop. There are no extra delays except those introduced by the length o f the wires ( a p p r o x i m a t e l y 30 c m ) .

~ pl

Ch. I.

Ch. 2.

PA

-

A0M I

I He NeLaser /

Fig. 1. Experimental setup. The first order diffraction lines from each AOM are used. P1 and P2 are detectors of 40 MHz bandwidth, PA are pre-amplifier units and AMP corresponds to the different amplifiers used.

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OPTICS COMMUNICATIONS

We tried different amplifiers in the feedback loop. One of them consisted of a two stage operational amplifier with a maximum gain of 20 and a bandwidth on the order of 50 kHz. The second one was a commercial ORTEC model 410 amplifier and differentiator with gains in the range from 0.1 up to 100, and a 125 kHz bandwidth with a resonance frequency at around 130 kHz. Finally, we also used the amplified output of a series 7000 Tektronix oscilloscope. This output has a bandwidth of 300 kHz and a variable gain from 0.1 up to 50. We obtained qualitatively similar results for all these cases. A dc bias voltage added to the input of the driver of the AOM 2 was an optional feature to control the range of parameters in which we performed the experiments. An oscilloscope recorded the output signal of the amplifier and the input intensity provided by P 1. This scheme was already used and discussed in ref. [4]. For low amplifier gain ( < 5 ) and no bias, the behavior of the system shows a bistable cycle with stable upper and lower branches, for the whole range of input intensities available to our laser (~< 4 mW). However, increasing the gain (above 5 ) and for negative dc bias, an unstable region appears in the upper branch of the bistable loop at a critical value of the input intensity. In fig. 2, we display the output intensity (lout) as a function of/in (fig. 2a) and lou t as a function of time for a swept input intensity (fig. 2b). From the figures, it becomes clear that the appearance of oscillations grow from the critical point with a non-zero frequency, as we might expect from a Hopf bifurcation. For values of/in near the critical threshold, the oscillation frequency is of the order of 33 kHz and increases up to 50 kHz before disappearing in a reverse Hopf bifurcation. The critical point shows a strong dependence on the gain and bandwidth of the amplifier. Increasing gain and decreasing bandwidth enlarge the unstable region to cover the entire upper branch of the bistable cycle. By using the ORTEC amplifier, which gives a higher amplification factor and which has a higher resonance frequency, a more complex dynamical behavior is observed. Period-two oscillations and chaotic behavior were seen for many different values of the input intensity. The output intensity shows periodic oscillations of low amplitude for small values of the input intensity, Ii,. Increasing Iin, a first period-two appears followed by a sudden jump into an-

15 February 1990

Input I n t e ~ | t y (e~u;)

1

I ¢i v

Q

i

i

ii

iiiiii

i

T i m e (a.u.)

I

I

Fig. 2. Experimental results. (a) Output intensity versus input intensity; (b) time evolution of the output intensity. These pictures have been obtained by using the ORTEC410 amplifier. other period-two of a much bigger amplitude. A further increase in Ii, sets the system into a chaotic region, probably originated by a period-doubling cascade. However, we were able to observe only an incipient period-four because of the amount of noise in the system (fig. 3). The above experimental results have been analyzed in terms of a simple model that is described in the next section.

3. Theoretical results The dynamical equations describing the output voltage (V) of the detector in a hybrid bistable de185

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15 February 1990

d V / d t = - V+ y[ 1 - k c o s ( 0 + Z) ] ,

(5a)

dZ / dt = R,

(5b)

d R / d t = [ ( 7 - 1 ) sin 0+O~o cos 0117o V + Fo sin O y [ 1 - k cos( O+ Z) ] -27R-

(72+~o~)Z

(5c)

is obtained. The steady state solution of this system is l~t = y [ l - k

cos(0+Zst)] ,

Zst = [/'0(7 sin 0+O)o cos 0 ) / ( 7 2 + 0 9 2 ) ] Vst, Time (a.u.)

I Rst -- 0 .

Fig. 3. Time evolution of an incipient period-two. The large amount of noise of the system prevents the observation of any higher periodicity. vice can be written in dimensionless variables as [ 2 ]

d V / d t = - V+ y [ 1 - k cos(O+ Z) ] ,

(1)

where y is proportional to the input intensity, k is the effective modulation depth of the AOM, 0 is the bias voltage and Z is the voltage driving the AOM, The relation between V and Z is given by the convolution integral

The dependence of Vst o n c o s ( 0 - - ~ Z s t ) makes the system a multistable one, with the possibility of controlling the size of the multistable region with the bias 0. Since Zst is proportional to V,t, the steady state solutions of these equations do not differ much from those obtained from the usual model [ 1 ]. A linear stability analysis of eq. (5) around the steady state solutions gives two conditions for the real part of at least one eigenvalue to be positive,

Saddle node condition /'0(7 sin 0+COo cos O)yk s i n ( 0 + Z~t) -- ( O)2 "t- 72) 2> 0 ,

Z= i F ( t - Y )

V(t')dt',

(2)

(6)

Hopf bifurcation condition

0

Fo[ (7+ 1 ) sin 0-OJo cos 0] yk sin (0+Zs~) where F ( t - t ' ) is the transfer function of the amplifier. I f F ( t - t ' ) is a delta function in t which corresponds to an amplifier of infinite bandwidth and unity gain, the first order differential equation used in previous works is recovered [ 1-5 ]. In the following, we consider the simplest case of a linear amplifier, then

F(t)=Fo e x p ( - 7 t ) sin(cOot+0),

(3)

where Fo is the amplification factor, 7 is the bandwidth, mo is the resonance frequency, and 0 is the phase factor. Defining

R=dZ/dt

(4)

and taking the second derivative of Z, a closed set of three ordinary first-order differential equations 186

- 27(o,~2 + 7 2 + 27+ 1 ) > 0 .

(7)

If condition (6) is satisfied, a real eigenvalue is positive. Eq. (6) is satisfied between the two turning points of the bistable cycle which are located by setting the right hand side ofeq. (6) equal to zero. The point at which eq. (7) is equal to zero is the position of a H o p f bifurcation (the real part of a complex eigenvalue crossing the axis). By simple inspection, it is evident that higher gain, lower bandwidth and higher resonance frequency of the amplifier, as well as higher input intensity and efficiency of the modulator will increase the probability of finding a H o p f bifurcation. In fig. 4, we show a typical bistable cycle as well as the points that satisfy conditions (6) and (7), for the given values of Fo, k, 7, 0, 0, and ~oo. We notice

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OPTICS COMMUNICATIONS 5.0

L

X X ~ X ~ x x x

I

out

~

//.oooooOOOO°°°5~

0.0

I

0.0

15 February 1990

I

I

I

I

y

I

I

I

I

130.0

I out

0.0 0.0

Fig. 4. A typical bistable curve obtained from the linear stability analysis of eqs. (5) for/'0=2.0, y=l.0, k=0.9, 0=0.0, ~=0.0 and ~Oo= 0.1. The circles and the crosses correspond to instabilities generated by Hopf bifurcation and saddle-node bifurcation respectively.

5.0

that two H o p f bifurcations are present in the upper branch, while the lower branch remains stable. This situation can drastically change if the factor ~ (phase o f the amplifier response to a step signal) is changed. In fact, by varying ~, it is possible to move the H o p f bifurcation points along the bistable cycle, and therefore obtain self oscillation in the lower branch also. Numerical integration o f eq. ( 5 ) (slowly sweeping the input intensity y ) is shown in fig. 5.

out

4. Comparison between experimental and theoretical results The agreement between fig. 2b and 5c is striking. However, before claiming a general agreement between experiments and eq. ( 5 ) , we need to make some i m p o r t a n t considerations. F r o m the theoretical analysis, it is evident that the transfer function o f the amplifier plays an essential role in the a p p e a r a n c e and relative position o f the H o p f bifurcation in the bistable cycle. We measured these p a r a m e t e r s for our amplifiers so that we could use realistic values in the numerical simulations. We obtained O=n/5, ~Oo= 10 kHz, and 7 = 1 kHz. T i m e units in the equations are normalized to the RC value o f the detector which is 40 M H z for R = 50 fL The time signal in fig. 5c gives the impression o f not having a H o p f bifurcation (at least a supercritical one) upon the appearance o f the oscillations. However, this effect is due to a delayed bifurcation

y --)

30.0

0.0 0.0

~ y

30.0

19.0

l out

0.0 6x103

Time

17x103

Fig. 5. Sweeping diagram of the output intensity versus the input intensity y from the numerical integration eq. ( 5 ). (a) Forward sweeping, (b) backward sweeping, (c) time evolution of the output intensity during the forward sweeping. induced by the sweeping o f the input intensity. Similar results have been observed experimentally when the input intensity is swept (fig. 6). This effect is much larger in the numerical results (fig. 5a,b) due to the fact that noise is not included in the equations while it is u n a v o i d a b l e in the experiment (fig. 6). Furthermore, chaos was not observed numerically when 0 = 0 (even if a more accurate search is per187

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OPTICS COMMUNICATIONS

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ior. An accurate design o f an amplifier needs to be done if stable operation is desired. Small gain and a b a n d w i d t h which is much larger than that o f the m o d u l a t o r and detector are required. However, technical restrictions make it difficult to accomplish in real systems, especially when an EOM is used because its response time is fast and it requires high driving voltages. On the other hand, this system is another tool to study complex d y n a m i c a l b e h a v i o r while maintaining a relatively simple experimental scheme and theoretical model.

I

¢l v

w-

eI

e~

C~

Input Intensity (a.u.)

I

Acknowledgements Fig. 6. Experimental forward and backward sweeping around the instability threshold. f o r m e d ) but it appears easily if we consider ~ = g / 2 . The transition from periodic to chaotic solutions needs to be studied further, both theoretically and experimentally.

5. Conclusions We showed here that the introduction o f a " r e a l " amplifier in the feedback loop o f a hybrid bistable system has an essential role on its dynamical behav-

188

We thank Dr. L.M. Narducci for useful discussions. J.R.T. acknowledges a "Joseph H. D e F r e e s " grant from Research Corporation.

References [ 1] H.M. Gibbs, D.L. Kaplan and R.L. Shoemaker, Phys. Rev. Lett. 46 (1981) 474. [ 2 ] K. Ikeda, Optics Comm. 30 ( 1979 ) 251; K. Ikeda and O. Akimoti, Phys. Rev. Len. 41 ( 1982 ) 617. [ 3 ] R. Val6e and C. Delisle, IEEE J. Quantum Electron. QE-21 (1985) 1423. [4] R. Val6e and C. Delisle, Phys. Rev. A 31 ( 1985 ) 2390. [ 5 ] M. Okada and K. Takizawa, IEEE J. Quantum Electron. QE17 (1981) 2135.