Chaos Induction in a Laser Diode

Mauricio Reyes and Efrain Solarte

Grupo cle Optica Cudntica, Departamento cle Fisica, Universidad del Valle, A. A. 25360 Santiago de Cali, Colombia

Abstract. Chaos is a class of complex behavior that can emerge from nonlinear dynamical systems, and it exists in both the natural and the technological world. Many biological systems such as the human heart and invertebrate neurons naturally exhibit chaotic behavior. Furthermore, digital computing has made feasible the creation of fractal patterns based on chaos. The beauty of chaos, however, lies not in the aesthetic of fractals, but in the simplicity of the system from which such complex, unpredictable behavior can emerge. The chaotic behavior of a non-linear R-L-Diode circuit has been studied. The non-linear behavior of the diode was modeled to compare the measured curves with the predicted ones. Period unfolding was observed by feeding the R-L-Diode circuit with a sinusoidal signal, varying the frequency and holding constant the amplitude of the input signal. The output voltage was measured on the resistor and bifurcation phenomenon was observed for 2, 4 and 8 periods and for a wide range of frequencies, before reaching the chaotic behavior. The voltage drop on the resistor was used as a source, and coupled to a diode laser excitation circuit. This signal has been coupled to the laser circuit using operational amplifiers to secure the extinction of noises, to provide the adequate signal level and to couple the circuit impedances. The laser response was studied as a function of output signal of the R-L-Diode circuit, especially where it becomes chaotic.

Keywords: Chaos, bifurcation, laser diode, non linear circuits, R-L-Diode, signal conditioning.

INTRODUCTION

The chaotic dynamics in optical systems takes considerable attention in the last decade because of their potential apphcations in telecommunications: the optical chaos can be used to transmit encrypted messages. The study of chaotic phenomena is a wide part of the theory of nonlinear systems dynamics. The birth of this science goes back to the second half of the 17th century when Newton invented the differential calculus, the motion laws and the gravitation theory. Physicists have believed that periodic solutions were sufficient to describe the oscillatory behavior of dynamic systems. Henri Poincare, at the end of the 19th century, showed that simple dynamic laws can drive to very complex behaviors that we call chaotic behaviors. There is no a unique definition for a dynamic chaotic system, but in any case the complex behavior of the system is reflected in the existence of periodic solution points in any small portion of the space in which the dynamic variables take the values. Moreover, solutions with very near initial conditions, will exhibit very different behavior, and will take very different values along the system evolution. And finally, for given initial conditions, as the time flows the system will assume values very near to any of the values that the variable can take. The electronic RLD circuit is one example of a simple system that can exhibit chaotic behavior. Building a chaotic circuit is a useful aid for understanding the mathematics and apphcations of this pervasive phenomenon, this project consists of building a chaotic circuit, analyzing the circuit itself, analyzing chaotic data from the circuit, and an attempt at building of a secure communication system.

BEHAVIOR OF A RL-DIODE CIRCUIT

Let us consider the circuit of figure la, whose nonlinear element is a diode. Considering the simphcity of this model circuit, one should be able to prove rigorously some properties that are associated with chaotic behavior, as the high sensitivity and the strong dependence on the initial conditions. In this model circuit, the resistance and the inductance were choose to be: R = (220 + 0,05) £2, L = (2 + 0.01) mH., and a diode 1N4001 was used.

CP992, RIAO/OPTILAS 2007, edited by N. U. Wetter and J. Frejlich © 2008 American Institute of Physics 978-0-7354-0511-0/08/$23.00

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0

L1

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a)

D1

L-1 D1N4004 \ v

$ R1

i*

6t

A / . ' tj

b)

- ^

}

A • ^ . .

j | •

FIGURE 1. (a) Circuit R-L-Diode used to produce a chaotic voltage source, (b) Current and Charge characteristics of a diode model.

Theoretical considerations

The diode can be simulated as a capacitor in parallel with a resistance, whose characteristic behaviors are shown in the figure 1(b). For tensions greater than Uo, the diode is conducting, with a capacity C2 and a resistance R2=l/G2; and for polarization voltages under Uo, or for inverse polarization, the circuit current is very small or zero, and the diode works as a parallel RC circuit, with a capacity CI and a resistance R1=1/G1, with G1=0, or Rl = «. From Kirchhoffs laws for the equivalent circuit RL-Diode, the following differential equations are obtained:

dt C. (q-q^) + i

di \ , s. R . (Ecos(cot)-Ur,) iq-qo)--i + dt LC L L

The index k, depending on the voltage region, stays for 1 or 2. Defining the vector x (t) as:

x(t) = (q(t)-q„i(t))

(1)

(2)

(3)

The system (1) and (2) can be written of the form: dx — (t) = -A,x(t) + b,(t) dt

With f-G,,

A„ C,

yC,L

1

-R L

(4)

(5)

And:

(6)

The solution of equations (4) has the following form:

x{t) = x^,{t) + e'^''-'^\x{t,)-x^M) With:

:̂* (0 = d^ cos(cot) + f^ sm(cot) + g^

(7)

(8)

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Complex and chaotic behavior observation

In order to observe the period unfolding in the output signal of this circuit, the input signal amplitude was holding on a fixed voltage value, and the signal frequency was swept. The voltage across the resistance was used as output signal. The phase diagrams were observed on the oscilloscope screen, taking the circuit output in the vertical axis and the input signal on the horizontal one. The data were stored in a PC using a measurement interfacing circuit. Every periodic solution produces a close path in the phase space, resulting in a close figure on the oscilloscope screen. Period unfolding has been observed with the input signal frequency as parameter, up to 6 periods could be clearly distinguished before chaos emerged, as is shown in the following figures 2 and 3.

3

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FIGURE 2. Measured phase diagrams: Voltage drop across the resistance Vs voltage input (from signal generator), the signal frequency is used as parameter, (a) Frequency 67.05 KHz, one period; (b) Frequency 76.39 KFLz, two periods; (c) Frequency

122.83 KFLz, three periods, and (d) Frequency 133.90 KFLz, four periods.

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FIGURE 3. Measured phase diagrams: Voltage drop across the resistance Vs voltage input (from signal generator), the signal frequency is used as parameter. Emergence of chaos: Single periodic solutions can not be distinguished (a) Frequency

203.1KHz, near the chaos threshold (b) Frequency 206.5KF[z, above the chaos threshold.

By Fourier transforming the electrical output signals, the occurrence of different periodic solutions (finite distinguishable frequencies) has been corroborated.

LASER LIGHT CHAOS INDUCTION

The chaotic output signal of the RL-Diode circuit was used as a source, to modulate the injection current of a diode laser. Because of the power requirements and the impedance differences, a conditioning stage was designed to match both circuits coupling the circuit impedances, sinking the source circuit load, and amplifying the output signal from the chaotic RL-Diode circuit. The design takes into account other parameters as the input and output signals and the frequency bandwidth. The chaos induction concept is schematic presented in the figure 4.

Circuit Cliaotic R-L-Diode

J

—> Sig nal Processing

l̂ ^ f >

Chaos Induction in a Laser Diode

FIGURE 4. Schematic block diagram of the laser pumping system used to induce a laser chaotic behavior.

Electrically pumped laser diode, using a conditioned signal, obtained from a RL-Diode circuit, produces chaotic laser hght. This behavior was observed by measuring the output light from the laser diode. Light signal from a semiconductor laser was detected using a high speed photodiode, which generates an electrical signal that is measured using a Tektronix TDS 200b oscilloscope. To observe the laser signal behavior, the light detector signal was feed to the oscilloscope vertical plates and the signal generator to the horizontal ones. Some typical results are shown in the figures 5, 6 and 7.

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FIGURE 5. Measured diagrams. Voltage output from the light detector vs. Input signal voltage from signal generator. Signal frequency is used as parameter, (a) 109.3KHz, (b) 120.6Khz,

2.5

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FIGURE 6. Measured diagrams. Voltage output from the light detector vs. Input signal voltage from signal generator. Signal frequency is used as parameter (a) 127.9KHz, (b) 135.3K, (c) 171.3KHz, (d) 186.2KHz.

Period unfolding has been observed with the input signal frequency as parameter, periods were not so clearly distinguishable as in the RL-Diode Circuit (see figure 5 and 6), and chaos emerged at different frequencies as in the case of the simple RL-Diode circuit as is shown in figures 5 and 6. A chaotic input signal produces a chaotic laser signal too, as is to see in figure 6, for a frequency above the RL-Diode Chaos threshold.

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FormatoXY

^:5

FIGURE 7. Measured diagrams. Voltage output from the light detector vs. Input signal voltage from signal generator. Signal frequency is used as parameter. Input signal frequency: 206.6 KHz.

CONCLUSIONS

Chaos in a quite diverse rank of frequencies was observed, although always for the order of kHz. The bifurcation phenomenon for 2 was observed, 4 and 8 periods. A very simple nonlinear electronic system, the RL-Diode circuit, allowed inducing chaos effects in a laser diode by current injection. Bifurcations and period unfolding phenomena has been observed and corroborated.

ACKNOWLEDGMENTS

We gratefully acknowledge Universidad del Valle and the Department of Physics for supporting this project.

REFERENCES

1. M Richard A. Holmgren, "A First Course in Discrete Dynamical System" second edition - springer 2. M A. Azzouz, M. Hasler, "Orbits of the R-L-Diode Circuit". On Circuits and Sys. Vol 37, pp. 1330-1339, Nov. 1990 3. K. Murali, M. Lakshmanan, L. O. Chua, "The Simplest Dissipative Nonautonomous Chaotic Circuit". IEEE Trans. On

Circuits and Sys. Vol 41, n° 6, Jun. 1994, pp. 462-463. 4. T. Matsumoto, L. Chua, S. Tanaka, "Simplest Chaotic Nonautonomous Circuit". Phys. Rev. A, Vol. 30, pp. 1155-1157, Aug.

1984 5. R. W. Rollins, E.R Hunt, "Exactly Solvable Model of a Physical System Exhibiting Universal Chaotic Behavior".Phys. Rev.

Letters, Vol. 49, n° 18, pp. 1295-1298, Nov. 1982 6. A. Azzouz, M. Hasler, IEEE. 37 (11), (1990). 7. B. Prusha, The College of Wooster, Measuring Feigenbaum's 5 in a bifurcation electric circuit. 8. R. Lua, University of Minnesota, Period doubling in a simple diode circuit, (2002).

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