The Harry Dym equation, which is related to the classical string problem, is derived in three different ways. An implicit cusp solitary-wave solution is constructed via a simple direct method. The existing connections between the Harry Dym and the Kortwegde Vries equations are uniformised and simplified, and transformations between their respective solutions are carried out explicitly. Whenever possible, physical insights are provided.

Generation of chaos from acousto-optic (A-O)Bragg cell modulators with an electronic feedback has been studied for over 3 decades. Since an acousto-optic Bragg cell with zeroth-and first-order feedback exhibits chaotic behavior past the threshold for bistability, such a system was recently examined for possible chaotic encryption of simple messages (such as a low-amplitude sinusoidal signal) applied via the bias input of the sound cell driver. Subsequent recovery of the message signal was carried out via a heterodyne-type strategy employing a locally generated chaotic carrier, with threshold parameters matched to the transmitting Bragg cell. In this paper, we present numerical results and detailed interpretations for signal encryption and recovery under hybrid A-O electronic feedback using a heterodyne strategy. Important features of this setup, such as the system robustness in terms of parameter matching (feedback gain, dc bias, and time delay) are also examined in some detail. C 2011 Society of Photo-Optical Instrumentation Engineers (SPIE). IntroductionIn an acousto-optic (A-O) modulator, an RF signal applied to a piezo-electric transducer, bonded to a suitable crystal, will generate an acoustic wave. It is well known that the acoustic wave acts like a "phase grating" that diffracts any incident laser beam into a number of diffracted orders. The Raman-Nath regime is characterized by multiple scattered orders while in the Bragg regime there are typically only two scattered orders (zeroth-and first-orders). 1 Around 1978, it was reported that A-O devices with positive feedback gain exhibit bistability characteristics. 2 In an A-O device, the amplitude of the diffracted fields that operate in the Bragg regime, i.e., the zeroth-and first-orders, which appear at the output of the Bragg cell, are related through a set of coupled differential equations. In Refs. 3 and 4, Chrostowski and co-workers present experimental results for A-O bistability and chaos using an equivalent circuit model of the Bragg cell with feedback. In a standard setup, a Bragg cell is driven by an ultrasonic sound wave from an RF generator at 40 MHz, and the resulting sound grating diffracts an incident He-Ne laser beam into the first Bragg order under Bragg condition. The first-order is then picked up by a linear photodetector, fed to an amplifier, and then returned to the bias input of the RF generator. The arrangement is shown in Fig. 1. Nominally, the scattered light beam is intrinsically frequency or phase modulated (with the acoustic frequency). In typical waveform sources, the external bias input amplitude modulates the RF waveform. A plot of the first order intensity (I 1 ) versus the bias inputα 0 yields the well known bistable and hysteretic behavior. 3,4 The bistability and hysteresis characteristics depend strongly on the feedback gain (β), the feedback time delay (TD), and the amplitude (I inc ) of the incident

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A novel technique involving nonlinear dependent sources expressed as convergent power series expansions of one or more variables representing the output fields of a Bragg-domain hybrid acousto-optic device with feedback is presented. Using some straightforward modifications of a basic circuit prototype, we show how optical bistability, possible multistability, and chaos may be generated for three fundamental types of tuning effects, i.e., feedback gain, bias voltage, and input amplitude. The results obtained are shown to compare favorably with the existing theory and experiments.

Gaussian beam propagation through a turbulent layer has been studied using a split-step methodology. A modified von Karman spectrum (MVKS) model is used to describe the random behavior of the turbulent media. Accordingly, the beam is alternately propagated (i) through a thin Fresnel layer, and hence subjected to diffraction; and (ii) across a thin modified von Karman phase screen which is generated using the power spectral density (PSD) of the random phase obtained via the corresponding PSD of the medium refractive index for MVKS turbulence. The random phase screen in the transverse plane is generated from the phase PSD by incorporating (Gaussian) random numbers representing phase noise. In this paper, numerical simulation results are presented using a single phase screen whereby the phase screen is located at an arbitrary position along the propagation path. Specifically, we examine the propagated Gaussian beam in terms of several parameters: turbulence strength, beam waist, propagation distance, and the incremental distance for Fresnel diffraction for the case of extended turbulence. Finally, on-axis temporal statistics (such as the mean and variance) of the amplitude and phase of the propagated field are also derived.