Scattering of the discrete solitons on the PT -symmetric defects

We study the propagation of linear waves and solitons in an array of optical waveguides with an embedded defect created by a pair of waveguides with gain and loss satisfying the socalled parity-time (PT ) symmetry condition. We demonstrate that in the case of small soliton amplitudes, the linear theory describes the scattering of solitons with a good accuracy. We find that the incident high-amplitude solitons can excite the mode localized at the PT -symmetric defect. We also show that by exciting the localized mode of a large amplitude, it is possible to perform phase-sensitive control of soliton scattering and amplification or damping of the localized mode. Copyright c © EPLA, 2012 Introduction. – Photonic structures consisting of coupled waveguides with regions of gain and loss offer many novel possibilities for shaping optical beams in comparison with traditional conservative or low-loss structures [1–3]. Such structures can be constructed as optical analogues of the complex space-time potentials possessing the so-called parity-time (PT )-symmetry, which can have an entirely real eigenvalue spectrum, corresponding to the energy conservation of optical eigenmodes. The beam dynamics in this case may demonstrate the properties qualitatively different from those usually observed in conservative systems [4–6]. Among them are the PT -symmetry breaking that occurs for sufficiently large powers [7], power oscillations [4,7–9], nonmonotonous dependence of the transmission on absorbtion [10], unidirectional invisibility [7,11], conical diffraction [12], a new type of Fano resonance [13]. Such systems may support solitons [14–16], demonstrating the amplification of extended waves or solitons [13,17,18], and the nonlocal response that manifests itself through the nontrivial effect of the boundary conditions [19]. Recently, PT -symmetric properties of couplers composed of two waveguides have been demonstrated experimentally [8,10]. Various schemes have been suggested to tailor the beam shaping and switching using PT -symmetric structures, including introduced PT -defects in periodic lattices [20] and self-induced refractive index change in nonlinear PT -symmetric structures [1,21,22]. It was found, in particular, that linear waves and small-amplitude solitons can be damped or amplified as they are scattered on PT -symmetric couplers embedded into conservative waveguide arrays [17,18,23]. In this paper, we study the interaction of solitons with a PT -symmetric defect with balanced gain and loss (the so-called PT -symmetric coupler, or PT -coupler), and show that stronger nonlinear interactions may lead to the excitation of a large-amplitude defect mode localized at the PT -coupler. Additionally, we show that by specially exciting a large-amplitude PT -coupler mode, it becomes possible to realize phase-controlled soliton scattering, which is accompanied by damping or amplification of the localized modes. The letter is organized as follows. First, we introduce our model. Then we summarize our recent results [17] on the reflection and transmission coefficients describing linear waves scattered by the PT -symmetric coupler, and then identify new effects in the process of highamplitude soliton scattering. After that we analyze the soliton interaction with a large-amplitude mode localized at the PT -symmetric coupler. Finally, we present several conclusions stemming from our work. Model. – We consider a chain of coupled optical waveguides with an embedded pair of PT -symmetric waveguides, where one waveguide experiences gain, and the other one loss, as shown schematically in fig. 1. Light propagating through the waveguide array can be described by a set of the discrete nonlinear Schrödinger