Stability of Nonlinear Te Surface Waves along the Boundary of Left-Handed Material

This paper is concerned with the stability characteristics of nonlinear surface waves propagating along a left-handed substrate (LHM) and a non-linear dielectric cover. These characteristics have been simulated numerically by using the perturbation method. The growth rate of perturbation is computed by solving the dispersion equation of perturbation. I found that the stability of nonlinear surface waves is affected by the frequency dependence of the electric permittivity εh and magnetic permeability μh of the LHM. The spatial evolution of the steady state field amplitude is determined by using computer simulation method. The calculations show that with increasing the effective refractive index nx at fixed saturation parameter μp, the field distribution is sharpened and concentrated in the nonlinear medium. The waves are stable of forward and backward behavior. At higher values of nx, attenuated backward waves are observed.


Introduction
Recently, there has been great interest in new type of electromagnetic materials called left-handed media [1].Over fifty years ago, Veselago was the first to consider the left-handed meta-material (LHM) which he defined as media with simultaneously negative and almost real electric permittivity and magnetic permeability in some frequency range [2].The electric and magnetic fields form a left-handed set of vectors with the wave vector [3].These materials have been shown to exhibit unique properties, such as Snell law and Doppler shift.Smith, et al. [4] have built these materials by using two dimensional arrays of splitting resonators and wires and are operating the microwave range.Nonlinear surface waves propagating along the interface of linear and nonlinear media have a number of novel extraordinary properties which attracted attention of many investigators [5][6][7][8].Understanding the stability of nonlinear surface waves is essential for the exploitation of these waves in various devices.There are numbers of approaches to the problem both using numerical simulations methods by Akhmediev et al. [8] and Moloney et al. [5] and analytical methods by Tran [6] which has been based on steady-state solutions to a nonlinear wave equation which contains an intensity dependent refractive index.The question is whether these wave solutions are stable on propagation of waves.Akhmediev et al. [8] had shown when the growth rate of perturbation of waves  is real, the sur-face waves are unstable and when  is imaginary, the waves are stable.Akhmediev et al. [7] explained the stability behavior of antisymmetric and symmetric solutions of a linear core sandwiched between two nonlinear media.They showed that the antisymmetric wave is stable at high values of the propagation constant, in contrast to the symmetric wave.Hasegawa [9] studied the soliton effects in various fibers, he reported that, optical soliton is formed by a balance between the dispersion velocity of the waves and the Kerr nonlinearity of the fiber.Sukhorukov et al. investigated the Spatial optical solitons in waveguide arrays, they predicted, two-dimensional (2D) networks of nonlinear waveguides which allow a possibility of realizing useful functional operations with discrete solitons such as signal switching, blocking, routing, and time gating [10,11].Setzpfandt et al. described discrete solitons in quadratic waveguide arrays [12].Their results demonstrated that a power threshold may appear for soliton formation, leading to a suppression of beam self-focusing which explains recent experimental observations.Shabat and Mousa have studied the stability of nonlinear surface waves along the boundary of linear semiconductor [13] and along the boundary of lateral antiferromagnetic/nonmagnetic superlattice (LANS) [14].These studies were carried out in a media with positive refractive index.Such media are called right handed materials.
This paper is concerned with the stability of nonlinear surface waves propagating along the boundary of left-H.M. MOUSA 124 handed media [1] (LHM).
To study the stability of the corresponding surface waves, it is necessary to select a particular form of the frequency dependence of the electric permittivity h  and magnetic permeability h  of the LHM, I solve this problem by using computer simulation method [15].
The geometry is shown in Figure 1.It consists of a non-linear semi-infinite cladding contact everywhere to a linear, semi-infinite LHM substrate at planar interface.The coordinate system is such that, the y axis is normal to the interface and the wave vector is directed along the 0 y  x axis.

Theoretical Analysis
Since the wave propagation is in x-direction then, the Maxwell equations for S-polarized wave (TE) are reduced to the following Equation [8]   The dielectric constant of the linear medium in the re- Assuming that the nonlinear medium is self-focusing, the solution of the wave equation which is polarized along the z-axis is: where   , A x y is a slowly varying field envelope, x n is the effective refractive index.
By substituting Equation (3) into Equation (1), the equation for the slowly varying amplitude   where is the decay constant of the nonlinear medium, 3  is the linear part dielectric function of the non linear medium, the coordinates x and are normalized by the factor y c  , and the fields are normalized by the factor 1 2 0  , where  is the wave angular frequency, c is the light velocity in free space, and 0  is the non-linearity coefficient.
The investigation of the stability of nonlinear surface wave (NSW) propagation along the interface between the linear and non linear medium has been focused in looking for the steady-state solution 2 , 0, for linear medium 2 sec , 0, for nonlinear medium -At the interface between the two media we as su 0 y  , me the condition that the dielectric constant of the linear medium where is the decay constant of the linear medium and permeab .Both a negative dielectric permittivity ility are written as [3]: with plasma frequency and resonance frequency 0  p  .To determine th m e stability criterion for NSWs, I -nu erically stimulated the steady-state solution of Equation (4a) with small perturbation as [8]: where   , f x y i lution, s a perturbation function of the steadystate so p  is the saturation parameter.
Substituting uation (7) into Equation (4a), we can Eq obtain: We shall consider the dependence of the perturbatio form [8]: where and are functions of only.We take case 1 , , where where , c c cond ect to are constants to be determined from the boundary ition, and primes denote vatives with resp the deri y .In a linear medium, the solutions are decaying as , where where y which implies and Equation (12a) may be solved analy ing each of the two expressions under the absolute value in k tically by expandterms of  up to the fourth order and by calculating the absolute values of these expressions, one obtains that [8]   , , , c c A A t y th hrough appl tio l e dete ican of the boundary conditions at y = 0 as [5]: (1) It is found by substituting Equations ( 5) & (7) into Equation (3), which results in (2) znl zl Since the wave function u vanishes at the boundary, say y =10 then (3) 0 at 10 At the initial pe rturbation where , it is convenient to take , , , c c A A .By numerical lation method it is easy to stu of the steady-state field amplitude

 
, , , at 0, 2.9, and 3 The variation of the energy integral of the nonlinear surface waves with x n h is also calculated analytically for different values of t e wave frequency through the integral of square perturbed field amplitude in linear and nonlinear medium as [8]    

Computer Simulation and Discussion
Some numerical calculations are presented for the lation of the stability Equation ( 7) of the proposed ture, which consists of LHM substrate and a nonlinear di electric cover.Computer simulation software (Maple) [15] is used in our computation, where the run takes a reasonable usage time.The parameters are [3] as follows: x n  , the energy becomes negative, where the waves can be switched to the backward propagation as an effect of the LHM.
These results are different from that obtained for the magnetic medium such as lateral antiferromagnetic/nonmagnetic superlattice (LANS) [14] and gyrodielectric medium as a semiconductor [13].The existence of the magnetic matter causes the growth rate to be always real and the waves are always unstable.For a semiconductor substrate, the waves are stable of forward traveling.

Conclusions
The stability characteristics of nonlinear surface waves  propagating along a left-handed substrate(LHM) and a non-linear dielectric cover the stability of the waves in LHM can be controlled by the frequency dependence of the electric permittivity and magnetic permeability of the LHM.By increasing the effective refractive index at fixed saturation parameter, the field distribution is sharpened which is implying the possibility of optical switching and the field concentrated in the nonlinear medium (optical soliton) which is useful for practical ultrahigh-speed communications.At higher values of x n , attenuated backward waves are observed.I believe that the stability which has been investigated and reported here may provide new opportunities for the design of future microwave-photonic devices.

Figure 1 .
Figure 1.Configuration of a single interface nonlinear cover/LHM substrate structure.
z n function, so that the function can be written in the Copyright © 2012 SciRes.OPJ H. M. MOUSA 125 quation(9) into Eq ation (8), we obtain the s ifferential equations which have solutions decay as y   for self focused waves in nonlinear medium of the form: wave  is either real or im y, thus by aginar a bit of algebra we can obtain a dispersion relation for determining  of the form [8]:

Figure 3 .
Figure 3.The field distribution of the nonlinear su waves A(y, z) for (a) n x = 3, th rate  = 0.8266; (b) n x =