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The surface wave dispersion relations of surface Plasmon at the interface of a left-handed material and a non-linear Kerr medium of arbitrary nonlinearity are derived based on a generalized first integral approach. The normalized power flow is also investigated for various values of frequency. The above study is conducted for both cases: self-focusing (α≺0) and de-focusing (α≻0) nonlinear Kerr coefficient.

Recently there are great interest and investigation of plasmonics. This is due to the increasing of transmission in layered thin films composed of metals and their experimental applications. Surface Plasmons are charges oscillations occurring at the interface between metal and dielectric layers. Plasmonics concerns with the surface Plasmons and the light interaction with metals.

The interaction of light and surface Plasmon has increased many applications and investigation studies such as developed spectroscopy, high resolution microscopy and sensing, development of light sources and cloaking left-handed materials.

Surface Plasmon excitation concerns with the free electrons oscillation of the interface between metal and dielectric layers leading to the resultant excitation which depends of the optical properties of the two layers and the interface geometry between the two layers. The surface Plasmon excitation is also investigated at simple geometry interface where the left-handed material and the dielectric layers have an interface which is infinite planar [

In our study, a Kerr-nonlinear type dielectric has been investigated where the dielectric function of the dielectric media depends on the electric field intensity. The surface Plasmons dispersion equation at a planar interface between a metal layer and a linear optical layer (where the wave number is k and the angular frequency is ω) can be expressed as the following:

where c is the speed of light in vacuum and the

where ε is the linear part and α is a nonlinear dielectric coefficients, which frequency-dependent.

In recent years, there has been an increasing growing interest in new artificial metamaterials. One of the most important reasons is due to the unusual characteristics and behaviors. Some of new interesting application of metamaterial is to use the left handed material or metamaterial in construction optical wave guide sensors [

Instead of a semi-infinite metallic region, we study here the surface Plasmon dispersion relation of a left- handed material (LHM). LHM is a medium with negative permittivity, permeability, and refractive index, which was initially discussed by Veselago [

ollowing the theory and approach of a TM-polarized wave which is considered to be propagated at the interface of a Kerr-type medium and a LHM, and by ignoring any loss in both media [

where c.c. is complex conjugate constant, and the relative phase of the two components

The solution of Maxwell Equations (4) in the LHM (

ponent electric field amplitude will have the form:

The losses are neglected. Here

where D_{z} is the z component of the electric induction vector D. Equation (6) hold in a linear medium with replaced ε_{d} by ε_{eff} .

The continuity of

with

A standard treatment of the nonlinear region 1 invokes a “first integral” to get at an equation for

An integration with respect to z gives

where C is an integration constant. The key step is to recognize the identity

The Equation (9) can be rewritten as

Applying the boundary conditions,

Applying the boundary conditions at Equation (11)

where

Furthermore, using Equation (7) together with the relation

where

Now let us apply the above results to derive an explicit and exact dispersion relation for the surface Plasmon at a LHM-Kerr dielectric interface. Substituting Equation (14) into Equation (13), the result can finally be reduced to a quadratic equation in the wave number k which, leading to the following relation:

Putting the limits

which are unacceptable since these will make k = 0 in Equation (15). Thus, we obtain the following implicit expression for

Then, the surface plasmon frequency can be solved from Equation (16) to give the following relation:

This result is compared and contrasted with the one from the inexact approach [

Note that even in the weak field limit, Equation (17) implies that:

which contains an extra factor of 1/2 compared within Equation(18).

The power flow in the LHM linear medium is given by:

And the power flow in the nonlinear medium is written as:

It has been noticed that the dispersion curves are obviously changed by the effect of the nonlinearity for both self-focusing (

In

real solution of

The normalized power

It is noticed from our above treatment that the dispersion curves are strongly dependent on the intensity of the electric field. The most important conclusions are that for

creases monotonically with

versal and bistability cases have been clearly observed. The two interesting behaviors could lead to new design of future application in Optoelectronic-Microwave technology.