Barrier transmission map of one-dimensional nonlinear split-ring-resonator-based metamaterials: Bright, dark, and gray soliton resonances

摘要

The barrier transmission characteristics of a one-dimensional chain of optically linear split-ring resonators (SRRs) containing a barrier composed of optically nonlinear split-ring resonators are studied. (This is an analogy to the quantum mechanical problem of the resonant transmission of a particle through a finite barrier potential.) The SRRs are idealized as inductor-resistor-capacitor-equivalent resonator circuits where the capacitance is either from a linear dielectric medium (optically linear SRRs) or from a Kerr-type nonlinear dielectric medium (optically nonlinear SRRs). The SRRs are arrayed in a one-dimensional chain and interact with one another through weak nearest-neighbor mutually inductive couplings. The transmission maxima of the SRR barrier problem are studied as they are located in a two-dimensional parameter space characterizing the linear mutually inductive coupling and the nonlinear Kerr dielectric of the SRRs of the barrier. The result is a two-dimensional map giving the conditions for the existence of the resonant-barrier modes that are excited in the transmission process. The various lines of transmission maxima in the two-dimensional plot are associated with different types of resonant excitations in the barrier. The map is similar to one recently made in McGurn [Phys. Rev. B 77, 115105 (2008)] for the resonant-transmission modes of a nonlinear barrier in a photonic crystal waveguide. The SRR problem, however, is quite different from the photonic crystal problem as the nonlinear difference equations of the two systems are different in the nature of their nonlinear interactions. Consequently, the results for the two systems are briefly compared. The transmission maxima of the SRR system occur along lines in the two-dimensional plot, which are associated with modes resonantly excited in the barrier. These lines of resonant modes either originate as a simple evolution from the resonant modes of the linear barrier limit or as lines of soliton breather modes of the bright, dark, or gray types. The soliton modes arise either spontaneously at a lower critical value of the nonlinearity of the barrier media or from the breakup of a barrier mode in the linear media limit as the barrier media become increasingly nonlinear. The soliton resonances are identified by comparing the wave functions and energies of the barrier excitations with those of solitons in an infinite chain composed of barrier material. Results for the infinite chain in Giri, Choudhary, Gupta, Bandyopadhyay, and McGurn [Phys. Rev. B 84, 155429 (2011)] for bright solitons are used for this identification, and in the Appendix we present additional solutions for dark and gray solitons that were not included in our previous treatments.