Abstract
Modulation instability (MI) in a coupled resonator optical waveguide (CROW) and photonic-crystal waveguide (PCW) with nonlinear Kerr media was studied by using the tight-binding theory. By considering the coupling between the defects, we obtained a discrete nonlinear evolution equation and termed it the extended discrete nonlinear Schrödinger (EDNLS) equation. By solving this equation for CROWs and PCWs, we obtained the MI region and the MI gains, G(p,q), for different wavevectors of the incident plane wave (p) and perturbation (q) analytically. In CROWs, the MI region, in which solitons can be formed, can only occur for pa being located either before or after pi/2, where a is the separation of the cavities. The location of the MI region is determined by the number of the separation rods between defects and the sign of the Kerr coefficient. However, in the PCWs, pa in the MI region can exceed the pi/2. For those wavevectors close to pi/2, the MI profile, G(q), can possess two gain maxima at fixed pa. It is quite different from the results of the nonlinear CROWs and optical fibers. By numerically solving the EDNLS equation using the 4th order Runge-Kutta method to observe exponential growth of small perturbation in the MI region, we found it is consistent with our analytic solutions.
Highlights
Photonic crystals (PCs) are artificial structures in which the refractive index is periodically distributed at a length scale comparable to the operating wavelength [1, 2]
The first method is to design a proper structure to create a linear dispersion curve in the range of operating frequency; the second method is to add nonlinear Kerr media to provide solitons propagation [8,9,10,11]. In the latter case, the criteria of forming a soliton is that the wavevector of the incident wave must be located within the modulation instability (MI) regions [12,13,14], where the MI refers to a process in which small perturbations upon a uniform intensity beam would grow exponentially [14]
We have successfully used the tight binding theory (TBT) to investigate MI in both coupled resonator optical waveguide (CROW) and photonic crystal waveguide (PCW) by considering growth of a small perturbation superimposed on a plane wave
Summary
Photonic crystals (PCs) are artificial structures in which the refractive index is periodically distributed at a length scale comparable to the operating wavelength [1, 2]. The first method is to design a proper structure to create a linear dispersion curve in the range of operating frequency; the second method is to add nonlinear Kerr media to provide solitons propagation [8,9,10,11] In the latter case, the criteria of forming a soliton is that the wavevector of the incident wave must be located within the modulation instability (MI) regions [12,13,14], where the MI refers to a process in which small perturbations upon a uniform intensity beam would grow exponentially [14].
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