Multisolitons in SRR based Metamaterials in Klein-Gordon Lattice

Abstract:

             Solitons are the solutions of nonlinear partial differential equations whose modal behavior can be described by a nonlinear Klein-Gordon (NLKG) equation that is derived through variational principle based on a generalized Hamiltonian. It is known that solitons gives robust solutions that can retain their shape to a longer extent without any distortion and hence also treated as particles. In the realm of their application in nonlinear optical fibre communications as well as in various nano-structured devices, NLKG equation seems to be a better option. It is a PDE with 2^nd order spatial and temporal terms that describe the nonlinear modes in the communication systems involving both ferroelectrics and metamaterials that are well-known nonlinear optical materials. Here, we derive NLKG for metamaterials as split-ring-resonators (SRR) for the applications in antenna. Various parameters involved in the Hamiltonian indicate interesting behaviour against amplitude of magneto-inductive waves that also shows multi-soliton behaviour in the metamaterials system.

Keywords: Multi-solitons, Metamaterials, Variational Principle, K-G Equation.

            The prefix ‘meta’ in Greek means ‘beyond’ or ‘after’. This suggests that the "metamaterial" exhibits characteristics surpassing those in nature. The ‘electrodynamics of substances with both negative dielectric constant (e) and magnetic permeability (m) is very interesting indeed. They are predicted to possess a negative index of refraction, and consequently they exhibit a variety of optical properties that are not found in positive indexed materials. About fifty years ago, Veselago (Sov. Phys. Uspekhi 1968, 10, 509-514) was the first to create a theoretical foundation of whether the refractive index of materials can possess negative values, and predicted a number of unusual optical properties associated with such negative index materials (NIM). A lucid account of transforming this theory into reality, i.e. metamaterials, was later given by Litchinitser and Shalaev (J. Opt. Soc. Am. 2009, B26, 161-169) and elaborated in many other important investigations.

The fascinating world of optics deals with ‘light’ that is the ultimate means of sending information to and from the interior structure of materials. It packages data in a signal of zero mass and unmatched speed. However, light in a sense is ’one-handed’ when interacting with atoms of conventional materials. This is because from the two field components of light, i.e. electric and magnetic, only the electric-field probes the atoms of a material, whereas the magnetic-field component of light is normally weak. Metamaterials, i.e. artificial materials with rationally designed properties, can allow both field components of light to be coupled to the ‘metaatoms’, enabling entirely new optical properties and exciting applications with such ‘two-handed’ light. Among the interesting properties is a ‘negative refractive index’. The refractive index is one of the most fundamental characteristics of light–propagation in materials. With negative refractive index, metamaterials may lead to the development of a ‘superlens’ that is capable of imaging objects and fine structures, which are much smaller than the wavelength of light. Other exciting applications of metamaterials include ‘antennae’ with superior transmission properties, optical nanolithography and nanocircuits, and also ‘metacoatings’ that can make the objects invisible.        

In the field of applied physics, a fertile ground was created with the tremendous surge of research activity on nonlinear optical materials and devices involving ferroelectrics and metamaterials. For such materials, the use of non-integrable Klein-Gordon (K-G) equation with 2^nd order spatial and temporal terms is already described (see later for Bandyopadhyay et al in Phys. Rev. B) for modal behavior in various device applications including those in the nano-structured systems in terms of shape and velocity of the soliton. The main focus of this article is on the application of nonlinear K-G equation that could describe the modal behavior of soliton and multisoliton waves in metamaterials.

It may be pointed out that discrete solitons or discrete breathers (DBs) (also called ‘intrinsic localized mode’ or ILM) are observed both in the integrable (viz. sine-Gordon equation) and non-integrable systems (viz. Klein-Gordon equation). However, the integrability imposes a criterion for obtaining DBs analytically that is possible only in the integrable systems, while for the non-integrable systems it is obtained by various numerical methods viz. a) spectral collocation method, b) finite-difference method, c) finite element method, d) Floquet analysis, etc. As evident from many numerical experiments, DBs mobility is achieved by an appropriate perturbation (Tsironis et al, Phys. Rev. Lett. 1996, 77, 4776-4779). Within the ambit of integrable systems, although the above nonlinear differential equations, such as KdV equation, sine-Gordon equation and nonlinear Schrodinger equation (NLSE) give rise to soliton solutions, these equations are not discussed here in order to remain within our main focus on the derivation via Non-linear Klein-Gordon Equation, i.e. NLKG (Bandyopadhyay et al, Phys. Rev. 2010, B81, 064104-111; Phys. Rev. 2011, B84, 155429-437; Phys. Rev. 2012, B86, 184101-108)

In Book (September, 2017): "Computational Chemistry Methodology in Structural Biology & Materials Sciences", (Part II: Computational Chemistry Methodology in Materials Science) Publisher: Apple Academic Press, New Jersey (USA), Editors: T. Chakraborty, et al., pp.273-308.