Bragg cell nonlinear dynamics and bifurcation properties under first-order feedback with variable aperture are examined. Chaotic encryption and recovery of low-bandwidth signals, and optimal performance are evaluated for fixed and variable apertures

Chaparala, Suman

Chatterjee, Monish Ranjan

English

University of Dayton

University of DaytoneCommonsElectrical and Computer Engineering FacultyPublicationsDepartment of Electrical and ComputerEngineering10-2016Nonlinear Dynamics, Bifurcation Maps, SignalEncryption and Decryption using Acousto-OpticChaos under a Variable Aperture IlluminationMonish Ranjan ChatterjeeUniversity of Dayton, mchatterjee1@udayton.eduSuman ChaparalaUniversity of DaytonFollow this and additional works at: http://ecommons.udayton.edu/ece_fac_pubPart of the Optics Commons, Signal Processing Commons, and the Systems andCommunications CommonsThis Conference Paper is brought to you for free and open access by the Department of Electrical and Computer Engineering at eCommons. It hasbeen accepted for inclusion in Electrical and Computer Engineering Faculty Publications by an authorized administrator of eCommons. For moreinformation, please contact frice1@udayton.edu, mschlangen1@udayton.edu.eCommons CitationChatterjee, Monish Ranjan and Chaparala, Suman, "Nonlinear Dynamics, Bifurcation Maps, Signal Encryption and Decryption usingAcousto-Optic Chaos under a Variable Aperture Illumination" (2016). Electrical and Computer Engineering Faculty Publications. Paper399.http://ecommons.udayton.edu/ece_fac_pub/399Nonlinear Dynamics, Bifurcation Maps and Signal Encryption and Decryption Using Acousto-Optic Chaos Under a Variable Aperture Illumination  Monish Chatterjee1*  and Suman Krishna Chaparala1      1University of Dayton, Department of Electrical and Computer Engineering, 300 College Park, Dayton, Ohio 45469, USA *Corresponding author e-mail address: mchatterjee1@udayton.edu  Abstract: The nonlinear dynamics (NLD) and bifurcation characteristics of a Bragg cell under first-order feedback are examined for a variable aperture.  Corresponding chaotic encryption and recovery of low-bandwidth signals are examined for fixed (optimal) photodetector aperture (at about 73%), and subsequent sub-optimal aperture variation.  Device performance and tolerances are evaluated and compared relative to the optimal fixed aperture, and the dynamical impact of the encryption system is evaluated in terms of possible variations in the photodetector illumination area.  OCIS codes: (070.1060) acousto-optic signal processing; (060.4785) optical security & encryption; (190.3100) instabilities and chaos   1.  Introduction   The acousto-optic Bragg cell is a piezoelectric oscillator in which acoustic vibrations lead to a diffraction grating thereby causing an input light beam to undergo scattering into two well-defined orders.  The device behavior may be studied using the so-called Klein-Cook parameter (Q) [1].  Under first-order feedback, the device exhibits interesting dynamical behavior including mono-, bi- and multistability leading to various degrees of chaos.  The chaos in the feedback loop has been utilized for a variety of signal encryption and recovery applications [2,3].  The key parameters such as acoustic bias 0ˆ , feedback gain  , time delay (𝑇𝐷) and incident light intensity (𝐼𝑖𝑛𝑐) enable encrypted information to be communicated securely over arbitrary distances [3].  Recent work has included examining the dynamical system for conditions of variable feedback gain, whereby it is found that chaotic passbands may be extended and consequently higher signal dynamic ranges may be possible.  This paper examines a corresponding problem relative to a photodetector with variable illumination aperture.  To that end, the overall dynamical behavior for a variable aperture (with openings from 0-100%) is first studied, and it is found that optimal chaotic passband occurs at around 73% preferably at higher 0ˆ -centered passbands.  The problem is then numerically studied at the 73% aperture point (whereby encrypted signals are recovered with reasonable fidelity), and  - tolerances are calculated.  A variable aperture is then introduced around of the optimal opening, and the system is examined both in terms of the resulting NLD and pass/stopband bifurcations, and also the signal encryption properties of the system.   2.  Band Selection For simplicity the input light beam is assumed as a uniform plane wave; the first-order output at the photo detector is given by [2]:           2 01 1ˆ( ) sin2 2incI t I I t TD      .                                           (1) The amount of light incident on the photo detector (∝ 𝐼𝑖𝑛𝑐) would vary in practical applications. This can be viewed as a variable aperture illuminating the photo detector. From this perspective, it may be shown that the incident light intensity itself varies with time, since the power incident is related to the integral of the Poynting vector.  Thus,  𝐼𝑖𝑛𝑐  may be treated as  ?̌?(𝑡). To identify and choose better parameters for signal transmission in the chaotic region bifurcation maps are developed for a fixed feedback gain and varying bias. In this case the Eq. (1) may be re-written as            2 01 1ˆ( ) ( ) sin2 2I t A t I t TD      .                                                           (2)   From these maps the pass band and stop bands for the transmission of the message signal can be inferred. ( )A t is varied from 0 to 3, and there  is  an optimally large pass band ( 0ˆ ) from .0287- 2.246 when ?̃? is 2.18  (i.e., 73% aperture opening).  The  blue  regions and  gaps  in  the Figs.1 and 2 indicate the pass bands (PB) and stop bands with 𝑇𝐷 = 1𝜇𝑠,  = 4, 0ˆ = 4 𝑡𝑜 0.  Thus, the dynamic range varies from  0,maxˆ − 0,minˆ  in the chosen PB.  3.  Encryption and Decryption  For modulation or encryption the message signal (s(t)) is added to a dc value, which is center of the pass band and fed as an input bias to the Bragg cell is set at about as ˆdc = 1.123.  A heterodyne receiver is designed for the decryption of the modulated message signal which would be matched to the transmitter keys , generating an unmodulated chaos.  At the receiver the transmitted signal (I_Tx) is multiplied with the generated chaos wave at the receiver (IRR) and filtered through a low pass filter for the recovery of the signal. The input (red) and output (blue) wave forms are normalized for comparison in Figs.4 and 5.  4.  References  [1] A.Korpel, “Acousto-Optics,” 2nd edition (Marcel Dekker, New York, 1997). [2] M.R. Chatterjee and M. Al-Saedi, “Examination of chaos- based encryption and retrieval in a hybrid acousto- optic device,” Frontiers in Optics, OSA Technical Digest(CD), paper # FCW3, Rochester, NY(2008). Fig.2 Bifurcation map for ?̌? = 2.18 Fig.1 Bifurcation map for ?̌? =1  Fig.3. Transmitter and receiver block diagram Fig 4. Encrypted 10 KHz trailing sawtooth recovery at 73% aperture.   Fig.5. Encrypted 10 KHz rising sawtooth recovery at 73% aperture. [3] M.R. Chatterjee and F.S. Almehmadi, “Information encryption, transmission, and retrieval via chaotic modulation in a hybrid acousto-optic Bragg cell under profiled beam illumination,”Proc.SPIE 9216,92160S (2014). 

Nonlinear Dynamics, Bifurcation Maps, Signal Encryption and Decryption using Acousto-Optic Chaos under a Variable Aperture Illumination

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Nonlinear Dynamics, Bifurcation Maps, Signal Encryption and Decryption using Acousto-Optic Chaos under a Variable Aperture Illumination

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