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Modulation instability in nonlinear positive-negative index waveguide array

Modulation instability in a nonlinear optical waveguide array with alternating positive and negative refractive indices is investigated analytically. Particular solutions of a system of coupled nonlinear equations are found. These solutions describe propagation of the continuum backward and forward waves in longitudinal direction of considered system. Instability of these waves is investigated with a linear stability analysis. A gain spectrum depending on different parameters of considered system: electromagnetic field power and ratio of the backward to forward wave amplitude, is studied analytically and numerically. It was shown that in case of Bragg resonance the continuum waves are stable. If nonlinearity assists at one kind of waveguides only and another one is linear one then the modulation instability does not occur. Out of Bragg resonance in defocusing waveguides the instability of the continuum waves is reduced.