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In this work, we derived the modal dispersion relation for TE_{m} modes for a symmetric slab waveguide constructed from SiO_{2} dielectric guiding core material with lossy left-handed material (LHM) as cladding and substrate, and the power confinement factor. The dispersion relations and the power confinement factor were numerically solved for a given set of parameters: allowed frequency range; core’s thicknesses; and TE_{m} mode order. We found that the real part of the effective refractive index decreased with thickness and frequency increase. Moreover, the imaginary part (extinction coefficient) of the effective refractive index has very small values for all thickness in the frequency ranges, which means the waveguide structure is transparent for the used frequencies. The waveguide structure offers good guiding power for all thickness in the frequency range with low power attenuation. The real part of the effective refractive index increases with the increase of mode order, and the power confinement factor decreases with the increase of mode order.

Metamaterials are unlike conventional materials, which gain their properties from their inherent composition of atoms and molecules. Metamaterial changed our perspective to this concept by replacing the molecules with manmade structures that might have dimensions of nanometers for visible light (nanorods) or in the case of GHz radiation that may be as large as a few millimeters (Split Rings Resonator SRR), but still much less than the wavelength [_{m} modal dispersion relation in lossy LHM-Dielectric structure and the power confinement factor; Section 3 is devoted for discusses the numerical results; Section 4 is solely devoted to the conclusion.

We briefly outline the derivation of the dispersion relation for TE_{m} mode in the proposed waveguide structure [14,15]. The dispersion relation for TE_{m} modes propagation in the with complex propagation wave constant is represented in the form, where, k is the effective wave index in each layer, and is the free space wave number which equals, where is the velocity of light, and is the applied angular frequency.

Substitution of Equations (1) and (2) into Maxwell’s equation yields the following linear differential equations for the core and cladding respectively are given by

and

Where and are the wavenumbers in the core and cladding respectively. Where is the electric field in the guiding core, and is the frequency dependent electric permittivity, which can be obtained from Sellmeier dispersion relationship, which is given by [

Where is a constant, and are called Sellmeier coefficients, and have the following values for fused silica:

.

Where, and are the effective dielectric permittivity and permeability respectively, and they are given by the well-known Drude model [17-21]

and

Where is the plasma angular frequency of the wires, is the operating angular frequency, is the structural factor (sometimes is called the filling fraction of the material) which depends on the characteristics of the embedded split rings resonators in host material, is the damping frequency (is the collision frequency of the electrons), and is the resonant frequency of the split ring resonators. Besides that, the magnetic field components can be written as

and

We consider the slab waveguide with uniform refractive-index profile in the core. We use the fact that the guided electromagnetic fields are confined in the core and exponentially decay in the cladding, the electric field distribution in the core and cladding is

The electric field components in Equation (10) is continuous at the interface between the core and the cladding at. Neglecting the terms independent of, the boundary condition for is treated by the continuity condition of as

From the condition that the are continuous at, we obtain the following

Eliminating A from Equation (12), and rearranging we have the dispersion relation for modes, that is

Where, , , and is the order of the mode.

The power flow is the real part of the integral of the complex Poynting vector over the waveguide cross section, that is

For wave we rewrite Equation (14) using Equations (8) and (9), as

Substituting Equation (10) in Equation (15), we have the power flow in each layer, that is

Where and are the power in the core, cladding, core, and substrate. The power confinement factor in the core is defined and the power flow in the core to the total power flow in the waveguide. Thus, the power confinement factor can be calculated using Equation (18), that is

Where is the total power flow in the waveguide structure.

The dispersion relation, Equation (13), numerically solved to find the complex effective wave index as a function of the angular frequency in the allowed frequency spectrum for different dielectric constants, for dielectric film thicknesses, and mode order. The power confinement factor for the structure under investigation has been investigated for the film thicknesses and mode order. The parameters of the lossy LHM have been theoretically adjusted to have negative permittivity and negative permeability in the frequency range which lies between 10.5 ~ 15.5 GHz. The parameters were used in carrying out the numerical calculations are: the plasma frequency ω_{p}/2π = 10.95 GHz, is the damping frequency = 0.5 GHz, and the electrons resonant frequency ω_{0} = 8 GHz, and the structure factor F = 0.8. In _{2} core thickness. It is noticed that the real part of the effective refractive index attains small index values, which means the phase front variation upon propagation through this waveguide structure is small. Besides that, the real part of the effective refractive index decreases smoothly with frequency increase with negative slope. The negative slope indicates that the overall effect of structure behaves like left handed material (LHM), since the slope of the dispersion relation represents the group velocity [22,23].

In _{0} mode versus the operating frequency range at different dielectric SiO_{2} core thickness. It is noticed that the extinction coefficient attains very small negative values, which means the structure is transparent for the allowed frequency spectrum. The value of the extinction coefficient increases with core’s thickness increase. However, the extinction coefficient decreases with frequency increase.

In _{0} mode (Equation (17)) versus the allowed frequency range for different core’s thicknesses.

flips at approximately ω = 12 GHz and the structure guides the power better for the thicker core than the slim one. But the overall performance of the waveguide structure is good enough to be used in any possible application. This implies the structure guides the power through the core more than wasting it in the cladding, and this conclusion can also be seen from

In

In

In

We derived the modal dispersion relation for TE_{m} modes for symmetric slab waveguide constructed from dielectric guiding core material with cladding and substrate made of lossy left-handed material (LHM), and the power confinement factor. The numerical solutions showed that the real part of the effective refractive index decreased with thickness and frequency increase. Moreover, the imaginary part (extinction or attenuation coefficient) of the effective refractive index has very small values for all thickness in the frequency ranges, which means the waveguide structure is transparent for the used frequencies. The waveguide structure offers good guiding for all thickness in the frequency range with low power attenuation. The real part of the effective refractive index increases with mode order increase, and the power confinement factor decreases with mode order increase. This waveguide structure is a good candidate for coupling or guiding electromagnetic waves.