8/3/2019 Optical Chaos Character 1

http://slidepdf.com/reader/full/optical-chaos-character-1 1/5

Optical Engineering 50(1), 017005 (January 2011)

Lyapunov exponent of chaos generated byacousto-optic modulators with feedback

Anjan K. GhoshPramode Verma

University of Oklahoma TulsaSchool of Electrical and Computer EngineeringTelecommunication Engineering ProgramTulsa, Oklahoma 74135E-mail: anjn@ieee.org

Abstract. Generation of chaos from acousto-optic modulators with anelectronic feedback has been studied for several years. Such chaotic

signals have an important application in providing secure encryption infree-space optical communication systems. Lyapunov exponent is an im-portant parameter for analysis of chaos generated by a nonlinear system.The Lyapunov exponent of an acousto-optic system is determined andcalculated in this paper to understand the dependence of the chaoticresponse on the system parameters such as bias, feedback gain, inputintensity and initial condition exciting the cell. Analysis of chaos usingLyapunov exponent is consistent with bifurcation analysis and is usefulin encrypting data signals. C2011 Society of Photo-Optical Instrumentation Engineers 

(SPIE). [DOI: 10.1117/1.3530105]

Subject terms: chaos; acousto-optics; optical communication; secure optical com-munication.

Paper 100426R received May 21, 2010; revised manuscript received Nov. 8, 2010;accepted for publication Nov. 29, 2010; published online Jan. 20, 2011.

1 Introduction

Generation of chaos from acousto-optic (AO) modulatorswith an electronic feedback has been studied for severalyears.1–8 It was an interesting field of research for study-ing the behavior of nonlinear dynamical systems using AOcells. Recently, we have shown that such chaotic signals havean important application in providing secure encryption infree-space optical communication systems.9, 10 In general,chaos encryption of data is known to provide higher levelsof security than what is available by standard cryptographictechniques. To realize a secure optical communication link using AO generated chaos we need to know how to obtaina chaotic response from an AO system and the properties of the chaotic signal in detail. We found that an understandingof the Lyapunov exponent (LE)11–13 of the nonlinear AO sys-tem with feedback is essential in building a chaos-encryptedoptical communication system.

In this paper, we calculate the LE of an AO Bragg modula-tor with feedback to characterizeand understand the behaviorof the chaos generated. It is well known that if the LE of anonlinear system is greater than unity the system generateschaotic signals; otherwise the time series obtained from thenonlinear system may be a stable periodic sequence.11, 12 Weshow how to determine the LE of the acousto-optic systemanalytically in terms of the AO system parameters.

2 Theory

A typical AO modulator set-up with feedback is shownschematically in Fig. 1. We assume that a coherent planewave from a laser is incident on the AO cell in Fig. 1 andundergoes a Bragg interaction. The intensity of the + 1 orderdiffracted beam is detected by a photodetector. The detectedsignal is fed back to the electronic processor driving the AOcell. A bias voltage of value α is added to the amplified signalfrom the photodetector and the total signal is fed to the AOcell driver. Assuming that the sound pressure remains con-

0091-3286/2011/$25.00 C 2011 SPIE

stant during the interaction and the interaction time is muchless than the feedback delay time, the intensity Y (n) of thefirst-order diffracted beam at the time t = nτ  canbe describedas a nonlinear one-dimensional iterative process

Y (n) = I in sin2[α + βY (n − 1)], (1)

where β is the net feedback gain, I in is the intensity of theinput light beam, τ  is the time delay in the feedback loop, andn = 1,2, 3, . . . (Ref. 6 and 7). We assume that the parametersα and β are real numbers. Since the parameter I in > 0 it iseasy to see from Eq. (1) that

0 ≤ Y (n) ≤ I in , (2)

for all values of n and the initial condition Y (0). We can alsorepresent the difference equation or map in Eq. (1) as

Y (n) = f [Y (n − 1)]. (3)

It is clear that the function f  in Eq. (3) depends on threeparameters α, β and I in.

If the initial condition is Y (0) then the trajectory or theorbit of the map in Eq. (3) is given by12

Y (n) = f (n)[Y (0)] = f [ f [. . . f [Y (0)] . . .]]. (4)

When the initial condition is perturbed to Y (0) = Y (0)+ ε, the sequence generated by Eq. (3) is given by

Y (n) = f (n)[Y (0) + ε]. (5)

If the perturbation is small, as n increases the separationbetween the orbits of  Y (n) and Y ’(n) evolves approximatelyas13

|Y (n) − Y (n)| ≈ |ε| exp(λn), (6)

where the parameter λ is the Lyapunov exponent of chaoticsystem.11, 12 From Eq. (6) we obtain

λ ≈1

nln

 f (n)[Y (0) + ε] − f (n)[Y (0)]

ε

. (7)

Optical Engineering January 2011/Vol. 50(1)017005-1

8/3/2019 Optical Chaos Character 1

http://slidepdf.com/reader/full/optical-chaos-character-1 2/5

Ghosh and Verma: Lyapunov exponent of chaos generated by acousto-optic modulators with feedback

Fig. 1 A schematic diagram of an acousto-optic system withfeedback.

As we are interested in the effects of very small perturba-tions, the limit of Eq. (7) is taken as ε → 0. Then the terminside the logarithm in Eq. (7) is expanded using the chainrule of differentiation:

limε→0

f (n)[Y (0) + ε] − f (n)[Y (0)]

ε

=

d f (n)[ y]

d y

 y=Y (0)

=

n−1k =0

d f [ y]

d y

 y=Y (k )

, (8)

and substituting the expression in Eq. (8) in Eq. (7) weobtain

λ ≈1

n ln

n−1k =0

d f [ y]

d y y=Y (k )

=1

n

n−1k =0

ln

d f [ y]

dy

 y=Y (k )

.

(9)

Finally, the limit of the expression in Eq. (9) is taken asn → ∞ to obtain the LE λ of the nonlinear one-dimensionalmap in Eq. (3) as11, 12

λ = limn→∞

1

n

n−1k =0

ln

d f  [ y]

d y

 y=Y (k )

. (10)

From Eq. (6) and the subsequent derivation, we can con-clude that if the LE λ > 0, the orbits of  Y (n) and Y (n) will

diverge away from one another and the sequence Y (n) willexhibit chaotic behavior, whereas if λ < 0 we expect that theorbits will converge to a steady oscillation.11–13 Along withthe initial condition Y (0) the LE in Eq. (10) is a function of all the parameters of the nonlinear system.

3 LE for AO System

From Eqs. (1) and (10) we obtain the following expressionfor the LE of the AO system shown in Fig. 1.

λAO = limn→∞

λAO(n). (11)

100

101

102

103

104

105

−0.55

−0.5

−0.45

−0.4

−0.35

−0.3

n

   L  y  a  p  u  n  o  v   E  x  p .

α = 2.1, β = 1.25, Iin

= Y(0) = 2.199

Fig. 2 Convergence of the Lyapunov exponent as more terms areused in calculating the sum in Eq. (11).

where

λAO(n) = ln |β I in | + Sn , (12)

Sn =1

n

n−1k =0

ln |sin[2α + 2βY (k )]|. (13)

The LE λAO depends on four parameters Y (0), α, β and I in

of the AO system. To compute the LE we evaluate the termλAO(n) for a large value of n until we obtain

|λAO(n + 1) − λAO(n)| ≤ 10−6.

In Fig. 2 we plot the convergence of  λAO(n) calculatedfor a system with α = 2.1, β = 1.25, Y (0) = 2.199 and

 I in

= 2.199. These values of the parameters were chosenarbitrarily and we notice that the calculated value of  λAO

tends to − 0.55. This system does not produce chaos. Theconvergence is attained for n > 2000. For other values of the system parameters the convergence is attained for n inthe range of 1000 to 8000.

0 20 40 60 80 100

2.16

2.165

2.17

2.175

2.18

2.185

2.19

2.195

2.2

Time step n

   I  n   t  e  n  s   i   t  y

   Y   (  n   )

α = 2.1, β = 1.25, Iin

= Y(0) = 2.199

Fig. 3 Y (n ) converges to a steady state when LE < 0.

Optical Engineering January 2011/Vol. 50(1)017005-2

8/3/2019 Optical Chaos Character 1

http://slidepdf.com/reader/full/optical-chaos-character-1 3/5

Ghosh and Verma: Lyapunov exponent of chaos generated by acousto-optic modulators with feedback

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Time step n

   I  n   t  e  n  s   i   t  y   Y   (  n   )

α = 2.1, β = 1.25, Iin

= 2.199, Y(0) = 0.5

Fig. 4 Y (n ) undergoes a chaotic oscillation when LE > 0.

In Fig. 3 we depict the output sequence Y (n) versusn for the parameter values given above. It is clear fromFig. 3 that the AO system’s output converges to a steady-state as indicated by its λAO < 0. In Fig. 4 we showthe output sequence Y (n) versus n for an AO system withα = 1.0, β = 1.3, Y (0) = 2.0 and I in = 2.0 so that λAO

= 0.18. Chaos is evident in the oscillations of  Y (n) in Fig. 4.Since 0 ≤ |sin[θ ]| ≤ 1 wehave −∞ ≤ ln |sin[θ]| ≤ 0for

all values of  θ . Therefore, the term Sn in Eq. (12) or (13)is always less than zero, for all values of  α, β and Y (0). If we want chaos in the output of the AO system, that is, if wewant λAO > 0, we must make the first term in the right-handside of the Eq. (11), namely, the term ln |β I in | greater thanzero, or |β I in | greater than unity. This condition is clearly anecessary condition for chaos. It is not a sufficient condition

as we see in the case depicted in Figs. 2 and 3 with β = 1.25and I in = 2.199. In Figs. 2 and 3 we have ln |β I in | = 1.01 butwith λAO = − 0.55 the chaos is not produced. We can thusstate the following theorem.

0 0.5 1 1.5 2 2.5 3 3.5 4−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

Parameter β

   L  y  a  p  u  n  o  v   E  x  p  o  n  e  n   t

α = 2, Iin

= Y(0) = 0.55

Fig. 5 Variation of the Lyapunov exponent with the feedback gain β.

0 0.5 1 1.5 2 2.5 3 3.5 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Parameter β

   I  n   t  e

  n  s   i   t  y   Y

α = 2, Iin

= Y(0) = 0.55

Fig. 6 Bifurcation of the orbit of Y (n ) with the feedback gain param-eter β.

Theorem 1: A necessary condition for chaos in the output

of the AO system with feedback is that the effective feedback gain β and the input light intensity I in satisfy

|β I in | > 1. (14)

To illustrate the behavior of the LE with respect to theparameters of the AO system we performed numerical ex-periments in which one of the parameters is varied over arange (0,4) while other three parameters were kept constant.In Fig. 5 we depict the variation of λAO as a function of thefeedback gain β for α = 2, I in = 0.55 and Y (0) = 0.55. InFig. 5 we notice that for smaller values of β the LE increaseslogarithmically till ln |β I in | is approximately equal to unity.For larger values of  β the value of the LE depends on thesummation term Sn in Eq. (13).

There are disjointed ranges of  β for which LE > 0 andwe observe chaos in the output of the nonlinear AO system.If the values of any of the other three parameters change theranges of  β for chaos production would also change. The

0 0.5 1 1.5 2 2.5 3 3.5 4−5

−4

−3

−2

−1

0

1

Parameterα

   L  y  a  p  u  n  o  v   E  x  p  o  n  e  n   t

β = 2, Iin

= 2, Y(0) = 2

Fig. 7 Dependence of the Lyapunov exponent on the feedback biasfactor α.

Optical Engineering January 2011/Vol. 50(1)017005-3

8/3/2019 Optical Chaos Character 1

http://slidepdf.com/reader/full/optical-chaos-character-1 4/5

Ghosh and Verma: Lyapunov exponent of chaos generated by acousto-optic modulators with feedback

Fig. 8 Deterministic output when the bias parameter α varies such that LE < 0 for both ‘1’ and ‘0’ levels.

Fig. 9 Chaotic output when the bias parameter α varies such that LE > 0 for both ‘1’ and ‘0’ levels.

Optical Engineering January 2011/Vol. 50(1)017005-4

8/3/2019 Optical Chaos Character 1

http://slidepdf.com/reader/full/optical-chaos-character-1 5/5

Ghosh and Verma: Lyapunov exponent of chaos generated by acousto-optic modulators with feedback

ideas obtained from Fig. 5 can also be determined from tra-ditional bifurcation diagrams in Fig. 6 for α = 2, I in = 0.55and Y (0) = 0.55. From Fig. 5 we find that the LE > 0 forβ approximately greater than 3.5. In the bifurcation diagramof Fig. 6 we also notice that chaos start for β approximatelygreater than 3.5. Thus, the method of calculating LE forcharacterization of chaos in a nonlinear system is entirelyconsistent with other methods of analyzing chaos.14 How-

ever, unlike Fig. 5, it is not easy to calculate the exact rangeof values of the feedback gain from Fig. 6 so that chaoticoscillations are produced by the AO system.

In Fig. 7 we show how the LE depends on the feedback bias factor α when the other parameters were chosen ar-bitrarily as β = I in = Y (0) = 2. Since the term ln |β I in |

= 1.38 the AO system can become chaotic if S < − 1.38. InRef. 6 and 9 it was proposed that a signal can be encodedwith chaos by feeding the signal as a time varying bias α(n)to the AO cell. From Fig. 7 we notice that care must be takenin selecting the amplitude levels of the signal or α(n) so thatchaotic oscillations are generated by the AO system. For ex-ample, if we feed binary data with a ‘1’ level correspondingto α = 2.7 and a ‘0’ level to α = 0.7, from Fig. 7 we noticethat for both levels LE < 0 and hence, chaos is not be presentin the output. This case is depicted in Fig. 8. We notice thatthe output Y (n) is basically an inverted version of the inputα(n). In Fig. 9 we show the chaos generated when a binarywaveform (‘0’ level is 0.5 and the ‘1’ level is 2.5) is suppliedto the AO cell as a time-varying α(n) so that both ‘1’ and ‘0’levels of α(n) produce LE > 0. Fig. 5 and Fig. 7 of LE versusthe system parameters are thus useful tools for designing achaos based communication system. We need to develop andconsult such plots to make sure that our nonlinear optical sys-tem with feedback produces chaotic waveforms to encrypt ordecrypt the data signals.

4 Conclusion

In this paper we discussed the concept of the Lypunov ex-ponent of a nonlinear system for determining whether thesystem output would show chaos or not. We determined andcalculated the LE of a nonlinear AO system with feedback and characterized its chaotic behavior with respect to thesystem parameters such as bias, feedback gain, input inten-sity and initial condition. We derived a necessary conditionfor the AO system to show feedback. Our results are usefulin designing AO systems for modulating or demodulatingmessages with chaos.

Acknowledgments 

The authors aregrateful to Mr. S. Gangadhar for his help with

calculations and to Professor M. R. Chatterjee and ProfessorP. P. Banerjee of the Department of Electrical and ComputerEngineering in the University of Dayton for fruitful discus-sions.

References 

1. J. Chrostowski, “Noisy bifurcations in acoustooptic bistability,” Phys. Rev. A 26, 3023–3025 (1982).

2. R. Vallee and C. Delisle, “Mode Description of the Dynamical Evo-lution of an acousto-optic bistable device,” IEEE Journal of Quantum Electronics QE-21, 1423–1428 (1985).

3. R. Vallee and C. Delisle, “Route to chaos in an acousto-optic bistabledevice,” Phys. Rev. A 31, 2390–2396 (1985).

4. J. P. Goedgebauer, et al., “Demonstration of bistability and multistabil-ity in wavelength with a hybrid acoustooptic device,” IEEE J. Quant.

 Electron. QE-23, 153–157 (1987).5. T. -C.Poon andS. K. Cheung, “Performanceof a hybridbistabledeviceusing an acoustooptic modulator,” Appl. Opt. 28, 4787 (1989).

6. P. P. Banerjee, et al., “Response of an acousto-optic device with feed-back to time-varying inputs,” Appl. Opt . 31, 1842–1852 (1992).

7. M. R. Chatterjee and J. -J. Huang, “Demonstration of acousto-opticbistability and chaos by direct nonlinear circuit modeling,” Appl. Opt.31, 2506 (1992).

8. S.-T. Chen and M. R. Chatterjee, “Dual-input hybrid acousto-optic set– reset flip-flop and its nonlinear dynamics,” Appl. Opt. 36, 3147–3154(1997).

9. M. Chatterjee and M. Al-Saedi, “Examination of Chaotic Signal En-cryption, Synchronization and Retrieval Using Hybrid Acousto-OpticFeedback,” in OSA FiO/LS/META/OF&T , 6624 (2008).

10. A. K. Ghosh, et al., “Design of acousto-optic chaos based securefree-space optical communication links,” in Proceedings of SPIE , pp.74640L–74640L-6 (2009).

11. T. Bountis, “Fundamental concepts of the theory of chaos and fractals,”in Chaos Applications in Telecommunications, P. Stavroulakis, Ed.,CRC Press, Boca Raton, FL (2006).

12. W. Kinsner, “Characterizing chaos through Lyapunov metrics,” IEEE Transactions on Systems, Man and Cybernetics, Part C (Applicationsand Reviews), 36, 141–151 (2006).

13. I. Procaccia, “The Static and Dynamic Invariants that CharacterizeChaos and the Relations Between Them in Theory and Experiments,”Phys. Scr. T9, 40–46 (1985).

14. D. M. Heffernan, et al., “Characterizationof chaos,” International Jour-nal of Theoretical Phys. 31, 1345–1362 (1992).

Anjan K. Ghosh hasmore than twenty yearsof research and teaching experience in theareas of optical information processing, op-tical communications and photonic sensorsand instrumentation. He obtained his doctor-ate in Electrical Engineering from Carnegie-Mellon University, Pittsburgh. Dr. Ghosh wasa memberof technicalstaff in AT&T Bell Lab-oratories for two years. He served as a fac-

ulty member in the University of Iowa, IowaCity, IIT Kanpur, India and Nanyang Techno-

logical University, Singapore. He was the Head of the Departmentof Electrical Engineering, Adv. Center for Electronic Sciences, LaserTechnology Program and the Center for Laser Technology, all at IITKanpur. He published over 150 papers in journals and conferenceproceedings. He is a Senior Member of IEEE, a Fellow of the In-stitute of Electronics and Telecomm. Eng. (India) and members ofSPIE, Optical Society of America, Optical Soc. of India, Eta KappaNu and Sigma Xi.

Pramode Verma is Director—Telecommunications Engineering Programand holds the Williams Chair in Telecom-munications Networking at the Universityof Oklahoma-Tulsa. Prior to joining theUniversity of Oklahoma, he was with AT&T

Bell Laboratories and Lucent Technologiesfor over 20 years. His research areasencompass all areas in telecommunicationsnetworking including network security. Heobtained his PhD from Concordia University.

Optical Engineering January 2011/Vol. 50(1)017005-5