Chaotic Oscillator in Wavelength - IEEE Xplore

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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 34, NO. 4, APRIL 1998

Chaotic Oscillator in Wavelength: A New Setup for Investigating Differential Difference Equations Describing Nonlinear Dynamics Laurent Larger, Jean-Pierre Goedgebuer, and Jean-Marc Merolla

Abstract—A generator of chaos in wavelength is reported. It is formed by a wavelength-tunable laser diode with a time delayed feedback loop in which a wavelength nonlinearlinear device is introduced. The dynamical regime of wavelength emission thus obtained is ruled by a differential difference equation. Experimental results are compared with numerical simulations and with previous theoretical and experimental results. Index Terms—Chaos, delay systems, distributed Bragg reflector lasers, dynamics, nonlinear difference equations, nonlinear differential equations, optical frequency conversion, optical oscillators.

I. INTRODUCTION

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INCE Ikeda et al. [1] predicted chaotic behavior in optical ring cavity, much work has been devoted to feedback is ruled by a systems [2]–[14] whose dynamical state first-order differential difference equation expressed as (1) in which is the time constant of the device, is the delay introduced by the feedback loop, is a nonlinear function and is the so-called bifurcation parameter. The general scheme for such oscillators is depicted in Fig. 1. It essentially consists of a first-order low-pass filter with a response time , a nonlinear medium in which the amplitude of the nonlinearity can be adjusted through the parameter , and a feedback loop with a delay line . In these systems, the dynamics can be theoretically studied from (1) or, in some cases, using a discrete model (also termed adiabatic approximation) which can be expressed as (2) at time is related to the state of the where device at previous time . Equation (2) holds as the variations of are slow, i.e., as the first time derivative in (1) is negligibly small. The discrete model has been extensively studied in mathematics, especially as the function is quadratic, yielding the so-called logistic map Manuscript received June 23, 1997; revised December 11, 1997. This work was supported by France Telecom. The authors are with GTL-CNRS Telecom, UMR CNRS 6603, Georgia Tech Lorraine, 57070 Metz, France, and also with Laboratoire d’Optique P.M. Duffieux, UMR CNRS 6603, Université de Franche-Comté, 25030 Besan¸con Cedex, France. Publisher Item Identifier S 0018-9197(98)02388-4.

Fig. 1. Oscillator ruled by a differential difference equation.

[15] and the well-known perioddoubling route to chaos. The devices ruled by (1) have also been investigated using optoelectronic hybrid systems because of their experimental simplicity. In order to validate the configuration described in the following, the transition between the two models is experimentally revisited. The results are compared with previous experimental and theoretical studies [4]–[8]. In this paper, we report a new experimental configuration which allows the study of a number of dynamics of the general form (1), in which the nonlinear function can be of any type. Another originality of the work concerns the bifurcation parameter which can be experimentally tuned over a much greater range than in the previous comparable setups. This should allow the study of experimentally unexplored dynamical regimes. Whereas most of the optical systems reported up to the present use optical power [4]–[9], [16] or polarization as the operating variable [17], the oscillator reported in the following works with a nonlinearity which deals with optical wavelength. The potential advantage of such a system is the experimental easiness to use the wavelength to induce a nonlinear behavior. We report the first results obtained, using a -function as the function to demonstrate the principle of operation and to compare with results reported previously. II. EXPERIMENTAL SETUP The experimental setup is represented in Fig. 2. The chaotic oscillator consists of: • a tunable laser source (distributed Bragg reflector (DBR) double electrode laser diode) whose wavelength can be tuned continuously, i.e., without mode hopping, by a DBR-section injection current, . The wavelength tun0.2 nm/mA around the center wavelength ability is

0018–9197/98$10.00  1998 IEEE

LARGER et al.: CHAOTIC OSCILLATOR IN WAVELENGTH

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1550 nm. The continuous tuning range is 1.5 nm. is wavelength-independent and is The optical power adjustable up to 20 mW by means of an injection current . • a wavelength nonlinear element formed by a birefringent and . plate (BP) set between two crossed polarizers The fast and slow axes of BP are oriented at 45 to the polarizing directions of and . The power spectrum is a channeled density of light at the output of spectrum, i.e., a function of wavelength , which can be expressed as Fig. 2. Experimental setup.

(3) 11 mm is the optical path-difference (OPD) where of BP. The tuning range of the laser diode being much smaller than the bias wavelength , (3) can be approximated as (4) , and where is with the wavelength deviation from the bias wavelength as the laser diode is tuned with a current around the bias current . Equations (3) and (4) also indicate that the function exhibits sinusoidal lobes inside the tuning range of the laser diode, spaced in wavelength by a free 0.2 nm. spectral range FSR • a photodetector providing a linear conversion of the optical power into a photocurrent with a conversion factor . The latter is electronically adjustable. which introduces a time delay much • a delay line longer than the response time of the loop, in order to obtain the Ikeda instabilities and the chaotic regimes ( much larger than unity). The delay line is formed by an analog CCD memory component (first in first out), of 256 memories (RD 5106 Reticon). The delay value ( 0.51 ms) is fixed by an external sampling clock. filter, which determines the • a first-order low-pass response time of the loop, denoted in (1). It was adjusted to be slow ( 8.6 s) using a simple filter, to be sure that the dynamical process is of first order (the time constants in the laser diode and the detector are much shorter, typically 1 and 10 ns, respectively). Then the output wavelength of the system is ruled by a differential difference equation of the same type of (1)

center wavelength of the tunable DBR laser, ad; justable through adjustable photodetector gain; optical power of the tunable DBR laser; wavelength/DBR-current tuning rate of the laser. The working regime of the device is set by the value of the bifurcation parameter as will be explained later. When working as a generator of chaos, the wavelength emitted by the laser chip exhibits chaotic fluctuations whose properties are -nonlinearity induced by the wavelength determined by the nonlinear element. The first advantage of such a system lies in the nonlinearity which is extrinsic to the laser chip and which is in wavelength instead of in density of power. This allows a very easy and accurate control of the nonlinear response of the system and, hence, of the chaotic fluctuations of the wavelength emitted by the laser diode. The second advantage is in the potential possibility to operate with other types of nonlinear functions, by using spectral filters with suitable transmission curves. This should offer a much broader range of applications than other optical intrinsic or hybrid systems reported up to the present, which operated with nonlinear -functions in most cases. In the functions restricted to following, we limit the discussion to the setup shown in Fig. 2, -function, i.e., as the nonlinear function is a wavelength in order to compare our results with those reported by other authors. III. NUMERICAL SIMULATIONS OF THE DYNAMICS AND IDENTIFICATION OF A CHAOTIC REGIME In this section, we report numerical simulations which were used to determine the working conditions of the system. To make comparison easy with previous results published in the literature, we rewrite (5) in the form of (1) using the normalized variable

(5) (6) with

, , and . Usually the dynamics of a system ruled by (6) can be simulated numerically directly from this equation. In some cases, a simplified version of (6) can be used, which holds as the time derivative and the response time are small.

with ; ; wavelength deviation from the bias wavelength bifurcation parameter in -unit; optical path-difference of the birefringent plate;

;

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(a)

(a)

(b) Fig. 4. Lyapounov exponent (discrete model) for 80 = 0.3; the fractal structure can be noticed. (a) Lyapounov exponent versus bifurcation parameter . (b) Lyapounov exponents for period-4, -8, -16 cycles, and fractal structure of the period doubling. (b) Fig. 3. Bifurcation diagram, obtained from the discrete model (80 = 0.3): (a) bifurcation diagram, discrete or mathematical model (iterative) and (b) magnified region.

Under these conditions, (6) can be written in the form of (2) as (7) In the following, the model associated to (6) is termed the “continuous” model, in contrast with the “discrete” model (also termed adiabatic model) attached to (7). We review hereafter some qualitative considerations on the differences between the dynamical behavior attached to the discrete model and to the continuous one. A more precise discussion concerning these discrete and continuous models can be found in the literature (see [10]–[13], [18], [19]). A convenient method to describe the dynamics of chaotic systems is to plot the bifurcation diagram for different values of the bifurcation parameter , as shown in Fig. 3. To do this, we used the rigorous (6) as well as the case of the adiabatic approximation (7).

A. Dynamics Attached to the Adiabatic Model Fig. 3 represents the bifurcation diagram calculated after (7) for 0.3. It is obtained by calculating the statistics of the normalized variable (which is related to the wavelength ) versus . a typical period-doubling cascade occurs, For similar to the well-known period-doubling cascade of the logistic map. The rapid geometric convergence of the bifurca-

tion values in the period-doubling cascade was independently demonstrated by Feigenbaum [20] and Tresser and Coullet , a so-called accumulation point is [21]. For obtained; it corresponds to the asymptotic end of the cascade. , an inverse cascade is obtained and For the chaotic regimes start. The inverse cascade is a sequence of bifurcations of chaotic regimes termed period- chaos where the values evolve chaotically inside intervals. When passing through a bifurcation in the inverse cascade, chaos. the period- chaos evolves in a periodFor , the variable takes values inside a single interval, yielding the so-called fully developed chaos. One should note that some periodic windows (e.g., at 2.35, period 3) can be seen inside the chaotic regimes. Each of these periodic windows evolve to chaotic regimes when increasing , through a period-doubling cascade. Another approach to describe the behavior of the system is to analyze the stability which is described by the Lyapounov exponent. The inverse Lyapounov exponent, which is related to the number of iterations, describes how fast the system converges to a periodic regime (for negative values) or diverges in a chaotic regime (for positive values). Fig. 4 represents the versus the bifurcation parameter . Lyapounov exponent For 2.1, Fig. 4(a) shows that the Lyapounov exponent feature negative values. This corresponds to periodic regimes. Their stability decreases when increasing , except near the bifurcation points, where the system is critically stable ( 0). B. Dynamics Attached to the Continuous Model Fig. 5 shows the bifurcation diagram calculated from Euler numerical integration of (6). It can be seen that , (i.e., the


(a)

(b) Fig. 5. Simulated bifurcation diagram, obtained with the continuous model (80 = 0.3): (a) bifurcation diagram, continuous or physical model (differential) and (b) magnified region.

wavelength emitted by the laser diode) takes values that are more uniformly distributed than in the discrete model. No windows attached to periodic regimes are now seen inside the , chaotic domain. Before the accumulation point the first periodic regimes (fixed point, period-2 also named , period-4 or ) are in good agreement with those of the discrete model. However it can be noticed that the bifurcation diagram differs from the adiabatic model as one goes from the period-doubling sequence to the direction of chaos, especially . This can be explained from considerations on the for dynamics of the system and the Lyapounov exponent, as it is discussed more deeply in the following. , the bifurcation diagrams As already mentioned, for of the discrete and continuous models are very similar. Hence, the stability analysis given previously can be used to describe the time evolution of the system, especially for the fixed point, period-2, period-4 [Fig. 6(a)], etc., regimes. Physically, the system oscillates between states that are stable over a time . When switching from a state to the next, the time (which is not neglected in the continuous derivative model) induces perturbations. These perturbations are damped rapidly from one state to the next as the Lyapounov exponent is strongly negative. This explains why the discrete model can be regarded as a good approximation of the continuous . model for In contrast with the previous situation, for , the perturbations produced by the time derivative are amplified when the system switches from one state to the next, since the Lyapounov exponent takes positive values. Then, the cannot be neglected in (6); this explains time derivative

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why the continuous bifurcation diagram for differs significantly from that of the discrete model. , the general form One should note that for of the arborescence of the inverse cascade is similar for the continuous and discrete models. This is due to the so-called , as repulsive cycles which still exist for explained in [3]. However, the dynamical properties of the period- chaos [period-4 chaos in Fig. 6(b)] thus obtained with the continuous model are completely different from those calculated with the discrete model. For 2.166, so-called higher harmonic synchronization regimes occur [3]. The latter are characterized by periodic and which exhibit fine ososcillations whose period is . Such higher harmonic cillations with a frequency around synchronizations do not exist in the adiabatic approximation and are typical of the continuous model [see Fig. 6(c)]. Mathematically, they can be explained by a linearized approach of (6) as described in [3], [6], and [18]. They were also observed and discussed experimentally by Hopf et al. [4], [5], [7]. 2.166, full chaos is obtained [Fig. 6(d)]. The For latter is characterized by a white noise spectrum with a cut-off . The dimension of the chaos thus obtained frequency of ) [10]–[14]. is related to (

IV. EXPERIMENTAL RESULTS Experimental verifications were conducted using the device 0.51 ms in the feedback shown in Fig. 2, with a time delay loop. The laser diode was a DBR laser whose wavelength could be tuned continuously over a range of 1.5 nm (200 GHz) centered at 1550 nm. The birefringent plate was 6 cm, yielding an optical path a calcite slab of thickness difference 11 mm between its fast and slow axes, where 1.477 and 1.634 are the extraordinary and ordinary refractive indices, respectively. Then its spectral function expressed transmission curve, which is also the by (3), exhibits seven sinusoidal peaks centered at 1550 nm inside the 1.5-nm tuning range of the laser diode. The linewidth of the laser diode was 10 MHz. The wavelength emitted by the laser diode was measured 0.8 mm, set using a birefringent plate with an OPD between two crossed polarizers and operating at the inflection point of an edge of its spectral transmission curve. Under these conditions, the chaotic fluctuations of the wavelength were detected using a photodiode. The wavelength resolution thus obtained is related to the detection noise at the photodetector. It was estimated to be about 5 pm (0.6 GHz) for the detector used. Fig. 7 is the experimental plot of the bifurcation diagram obtained as 0.3 (the vertical axis is the wavelength, the horizontal axis is the bifurcation parameter). The bifurcation parameter was tuned by changing electronically the photodetector gain in the feedback loop. The experimental bifurcation diagram is in quite good agreement with the calculated diagram in Fig. 5(a). The period-doubling cascade (period-2 and period-4 ) route to chaos is obtained as the bifurcation parameter is increased from 0.7 to 2.1. Assessing experimentally higher period cycles (such as

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(a)

(b)

(c)

(d)

Fig. 6. Time evolution simulated with the continuous model. (a) Fully developed chaos.

Fig. 7. Experimental bifurcation diagram for

T4

periodic regime. (b)

80 = 0.3.

and ) requires both a very fine tuning of and a high spectral resolution to resolve each of the wavelength states. This was not possible with the equipment available at the laboratory. This probably explains why we did not observe regime. When comparing with period cycles higher than the in other experimental results, periodic regimes higher than the doubling cascade were also very rare and difficult to be observed. This fact is usually attributed to mode competition, experimental noise, signal resolution, and to the low stability (and higher period) in the discrete model analysis of the [see Lyapounov exponent values of in Fig. 4(b)]. When considering the general shape of the doubling cascade, it is also in good agreement with previous experimental results from Vallée et al. [7], [8] and Hopf et al. [4], [5]. Fig. 8 shows experimental dynamical regimes for different values of the bifurcation parameter and the corresponding fast Fourier transform (FFT) spectrum. Period-2 in Fig. 8(a)

T4

chaotic regime. (c) Higher harmonic synchronization. (d)

is a typical square waveform. The wavelength emitted by the laser diode oscillates periodically between two states spaced by 41 pm (6.3 GHz). As expected the spectrum in Fig. 8(b) shows odd harmonics whose amplitude decreases with increasing the odd number. The fundamental frequency is 970 Hz. 59 pm, Four wavelength states (located at 66 pm, 110 pm, 114 pm) are obtained in the period-4 regime, as shown in Fig. 8(c). The spectrum of period-4 [Fig. 8(d)] exhibits subharmonic peaks at frequencies , where is a positive integer, located between the previous peaks of the period-2 spectrum. The dc background in the spectrum, i.e., the noise level, is slightly increased compared with that of the period-2 regime; this noise is attached to the small oscillations that can be seen on the period-4 levels in Fig. 8(c). As exceeds 2.11 [Fig. 8(e) and (f)], the peak-to-peak amplitude of these small oscillations increases, yielding a high noise level in the spectrum. These oscillations may be regarded as a starting chaotic regime at each of the levels of the period-4 cycle. The time evolution of Fig. 8(e) is termed period-4 chaos. When slightly increasing from 2.12 to 2.18, the device enters a period-2 chaos, as shown in Fig. 8(g) and (h). The noise level in the spectrum is relatively high, but still features harmonic peaks. The two successive regimes period-4 chaos and period-2 chaos illustrate the inverse cascade in the bifurcation diagram. These nonperiodic but chaotic regimes with a growing complexity are also in very good agreement with the behaviors observed in previous experimental investigations [4]–[9].


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(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Fig. 8. Experimental regimes 80 = 0.3) in the left column: time evolution of the wavelength for different values of the bifurcation parameter . In the right column, the FFT spectrum for: (a), (b): = 1.93, period-2 cycle, (c), (d) = 2.04, period-4 cycle, (e), (f) = 2.12, period-4 chaos, (g), (h) = 2.18, period-2 chaos, (i), (j) = 2.26, higher harmonic synchronization, and (k), (l) = 2.52, fully developed chaos.

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Another example of periodic regime is illustrated in Fig. 8(i) and (j); a complicated high-frequency periodic oscillation is observed, previously named higher harmonic synchronization. This is a typical regime that cannot be interpreted using the discrete model. Its description and observation is given in [3], [4], [6], and [7]. For values of over 2.5 [Fig. 8(k) and (l)], fully developed chaos is clearly obtained, without any remaining influence of the higher harmonic synchronization. The spectrum is similar to that of a white noise in the bandwidth of the dynamical system. The dimension can be estimated here to be about 70, according to the theoretical and numerical investigations of Dorizzi et al. [11]. It should be noticed that the values of involved in this numerical analysis could not be experimentally tested with the devices reported so far, based on time-delayed differential equations. In the experiment reported here, the is related to the continuous tuning normalized value of range of the laser diode and also to the periodicity of the wavelength nonlinear function. The first is currently limited technically to a few nanometers for DBR laser diodes, but the second experimental parameter can be easily increased, for example, using a fiber interferometer with a large OPD. is fixed by the In such a configuration, the limitation of linewidth of the laser, i.e., by its coherence length which cannot be greater than the OPD in order to obtain interference. When using a 10-MHz linewidth laser, the maximum value of is approximately of the order of 1000. This value is to be compared with the maximum value achieved with setups previously published. The maximum reached in the present setup (OPD 11 mm) was about 15.

V. CONCLUSION A new chaotic optical oscillator was reported that uses -wavelength wavelength as the dynamical variable. A nonlinearity was used to test the experimental setup and to compare its behavior with intensity chaotic oscillators (Ikeda systems), in which the nonlinearity is well known to be restricted to a -function in most cases. Results are in fairly good agreement with the theory and with the other experi). These values mental realizations for low values ( correspond to those previously investigated experimentally by other authors. The setup also allows experimental investigations of dynamical regimes corresponding to much higher values, such as those studied numerically in [10]–[14]. The shape of the nonlinear delayed driving force involved in the dynamics can also be changed easily, simply by using another spectral profile for the wavelength spectral filter. The advantage of the system reported here is in the high flexibility produced by the use of wavelength to induce chaos. Work is still in progress to demonstrate routes to chaos using other nonlinear functions in wavelength. ACKNOWLEDGMENT The authors would like to thank F. Delorme from CNET Bagneux for the DBR multielectrode tunable laser diode used in the experiments.

REFERENCES [1] K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun., vol. 30, no. 2, pp. 257–261, Aug. 1979. [2] K. Ikeda and K. Matsumoto, “High-dimensional chaotic behavior in systems with time-delayed feedback,” Physica D, vol. 29, pp. 223–235, 1987. [3] K. Ikeda, K. Kondo, and O. Akimoto, “Successive higher-harmonic bifurcations in systems with delayed feedback,” Phys. Rev. Lett., vol. 49, no. 20, pp. 1467–1470, Nov. 1982. [4] F. A. Hopf, D. L. Kaplan, H. M. Gibbs, and R. L. Schoemacker, “Bifurcation to chaos in optical bistability,” Phys. Rev. A, vol. 25, no. 4, pp. 2172–2182, Apr. 1982. [5] H. M. Gibbs, F. A. Hopf, D. L. Kaplan, and R. L. Schoemacker, “Observation of chaos in optical bistability,” Phys. Rev. Lett., vol. 46, no. 7, pp. 474–477, Feb. 1981. [6] H. M. Gibbs, Optical Bistability: Controlling Light With Light. New York: Academic, 1985. [7] R. Vallée and C. Delisle, “Mode description of the dynamical evolution of an acousto-optic bistable device,” IEEE J. Quantum Electron., vol. QE-21, pp. 1423–1428, Sept. 1985. [8] , “Route to chaos in an acousto-optic bistable device,” Phys. Rev. A, vol. 31, no. 4, pp. 2390–2394, Apr. 1985. [9] J. Chrostowski, R. Vallée, and C. Delisle, “Self-pulsing and chaos in acoustooptic bistability,” Can. J. Phys., vol. 61, pp. 1143–1148, 1983. [10] J. D. Farmer, “Chaotic attractor of an infinite-dimensional dynamical system,” Physica D, vol. 4, pp. 366–370, 1982. [11] B. Dorizzy, B. Grammaticos, M. Le Berre, Y. Pomeau, E. Ressayre, and A. Tallet, “Statistics and dimension of chaos in differential delay systems,” Phys. Rev. A, vol. 35, pp. 328–339, Jan. 1986. [12] M. Le Berre, E. Ressayre, A. Tallet, and Y. Pomeau, “Dynamic system driven by a retarded force acting as colored noise,” Phys. Rev. A, vol. 41, no. 12, pp. 6635–6646, June 1990. [13] M. Le Berre, E. Ressayre, A. Tallet, and H. M. Gibbs, “High-dimension chaotic attractor of a nonlinear ring cavity,” Phys. Rev. Lett., vol. 56, no. 4, pp. 274–277, Jan. 1986. [14] M. Le Berre, E. Ressayre, A. Tallet, H. M. Gibbs, D. L. Kaplan, and M. H. Rose, “Conjecture on the dimension of chaotic attractors of delayed-feedback dynamical systems,” Phys. Rev. A, vol. 35, no. 9, pp. 4020–4022, May 1987. [15] R. May, “Simple mathematical models with very complicated dynamics,” Nature, vol. 261, pp. 459–467, June 1976. [16] R. Roy, T. W. Murphy, T. D. Maier, Z. Gills, and E. R. Hunt, “Dynamical control of a chaotic laser: Experimental stabilization of a globally coupled system vectorial nonlinear dynamics in lasers with one or two stable eigenstates,” Phys. Rev. Lett., vol. 68, no. 9, pp. 1259–1262, Mar. 1992. [17] J.-C. Corvette, F. Bretenaker, A. Le Floch, and P. Glorieux, “Vectorial nonlinear dynamics in lasers with one or two stable eigenstates,” Phys. Rev. A, vol. 49, no. 4, pp. 2868–2880, Apr. 1994. [18] P. Manneville, Dissipative Structure and Weak Turbulence. New York: Academic, 1990. [19] Chow and Mallet-Parret, Coupled Nonlinear Oscillators. Amsterdam, The Netherlands: North-Holland. [20] M. J. Feigenbaum, “Quantitative universality for a class of nonlinear transformation,” J. Stat. Phys., vol. 19, p. 25, 1978. [21] Tresser and Coullet, “Instabilités et chaos,” C. R. Acad. Sci., vol. 278A, pp. 577–585, 1978.

Laurent Larger was born in Colmar, France, in 1968. He received the degree in electronic engineering from the University of Paris XI, Orsay, France, in 1990 and the Ph.D. degree in optical engineering from the University of Franche-Comté, Besan¸con, France, in 1997. Since 1994, he has been working at the Optics Laboratory of Besan¸con, University of FrancheComté, in the field of chaos in optical systems for secure communications. His research interests include nonlinear dynamics, laser diodes, and otpical telecommunication systems.


Jean-Pierre Goedgebuer was born on November 19, 1950. He received the Doctorat de 3eme cycle and the Doctorat d’Etat degree from the University of Franche-Comté, Besan¸con, France, in 1975 and 1978, respectively. In 1974, he joined the Optics Laboratory, University of Franche-Comté, where he worked on dye lasers and on acousto- and electrooptic systems. In 1985 and 1986, he was on sabbatical at the Centre National d’Etudes des Télécommunications, Lannion, France, where he has worked in the area of fiber communications systems. He presently holds the position of Professor at the University of Franche-Comté. His current research activities deal with optoelectronic systems for applications in transmission systems, signal processing, optical computing, and optical cryptography.

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Jean-Marc Merolla was born in Montbéliard, France, in 1971. He received the DEA degree of optoelectronics from the University of FrancheComté, Besan¸con, France. He is currently working toward the Ph.D. degree at the Optics Laboratory P.M. Duffieux, Besan¸con, France. His research concerns optical systems for secure communication, nonlinear optics, and quantum communication.