Stationary localized states due to a nonlinear dimeric impurity embedded in a perfect one-dimensional chain

Abstract

The formation of stationary localized states due to a power law nonlinear dimeric impurity embedded in a perfect one-dimensional chain is studied here using the appropriate discrete nonlinear Schrödinger equation. A proper ansatz is introduced in the appropriate Hamiltonian to obtain the reduced effective Hamiltonian. The Hamiltonian contains a parameter, ␤ϭ 1 / 0 , which is the ratio of stationary amplitudes at impurity sites. Relevant equations for localized states are obtained from the fixed point of the reduced dynamical system. The complete phase diagram in the (,) plane for all permissible values of ͉␤͉р1 is obtained. Furthermore, the maximum number of localized states is found to be 6. ͓S0163-1829͑97͒11417-5͔The discrete nonlinear Schrödinger equation ͑DNLSE͒ is a set of n coupled differential equationswhere V m,mϩ1 ϭV mϩ1,m à and mϭ1,2,3, . . . ,n.

Symmetry breaking for transmission in a photonic waveguide coupled with two off-channel nonlinear defects

“…Assuming that the defect rods are very thin and that the nonlinear contribution in Eq. (16) is small compared to 0 , we obtain for the matrix elements (14) m|V |n ≈− 3 16 σ χ (3) …”

“…15 and 20), the resonance spectra in the PhC waveguide with coupled off-channel defects are located in a rather narrow frequency domain. Therefore, we neglect the frequency dependence in the nonlinear susceptibility χ (3) (ω) in the following. Assuming that the defect rods are very thin and that the nonlinear contribution in Eq.…”

“…The system has attracted interest over the past two decades because of analytical treatment and its generality. [2][3][4][5][6][7][8][9][10][11][12] On the other hand, the system can be realized in the two-dimensional photonic crystals (PhC).…”

Bound states in photonic Fabry-Perot resonator with nonlinear off-channel defects

“…The system attracts interest over decade because of analytical treatment and its generality. [2][3][4][5][6][7][8][9][10][11][12] On the other hand, the system can be realized in the twodimensional photonic crystals ͑PhC͒.…”

“…Then the corresponding eigenfunction becomes localized, i.e., the in-channel nonlinear defect can give rise to the bound state. 2,3,9 One may ask does exist the localized bound state with discrete eigenfrequency inside the propagation band. First, this question was formulated and positively answered by von Neumann and Wigner in 1929.…”