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This paper tackles the wave attenuation along with a cylindrical waveguides composed of a left Handed material (LHM), surrounded by a superconducting or metal wall. I used the transcendental equations for both TE and TM waves. I found out that the waveguide supports backward TE and backward TM waves since both permittivity and magnetic permeability of LHM are negative. I also illustrated the dependence of the TE and TM wave attenuation on the wave frequency and the reduced temperature of the superconducting wall (T/T
_{c}). Attenuation constant increases by increasing the wave frequency and it shows higher values at higher T/T
_{c}. Lowest wave attenuation and the best confinement are achieved for the thickest TE waveguide. LHM-superconductor waveguide shows lower wave attenuation than LHM-metal waveguide.

Recently, there has been a great interest in new type of electromagnetic materials called left-handed media [

A superconductor like Yttrium barium copper oxide (YBCO), is a famous “high-temperature superconductor”, achieves prominence because it is the first material to achieve superconductivity above (77 K), the boiling point of liquid nitrogen. It is associated with the formula YBa_{2}Cu_{3}O_{7−x} The superconducting properties of YBa_{2} Cu_{3}O_{7−x} are sensitive to the value of x, its oxygen content with 0 ≤ x ≤ 0.65. R.D Black et al. [_{2}Cu_{3}O_{7−x} ceramics normally display superconductivity and metallic conductivity below and above the critical temperature of the superconductor respectively. Mohazzab, et al. described the synthesis of epoxy-modified YBCO ceramics and evaluates the semiconducting properties at temperature above [_{2}Cu_{3}O_{7−x} has been developed in thin-film form to the point of practical applications, and several devices are available. Intensive materials research have resulted in techniques, notably laser-ablation and radio-frequency sputtering.

In this analysis, a theoretical study of the propagation characteristics of TE and TM waves guided by an optical structure is presented. This structure consists of a left Handed material (LHM) cylinder with superconducting walls like YBa_{2}Cu_{3}O_{7−x}. and are electric permittivity and magnetic permeability of LHM respectively.

In this context, the structure geometry of the problem considered is shown in

The longitudinal electric and magnetic fields and respectively, propagating in the waveguide can be derived by aid of Helmholtz’s wave equation [

Decomposing Helmholtz’s wave equation into a radial and a longitudinal part in cylindrical coordinates for the electric field yields:

A plane wave solution for the electric field of the form

is substituted into Equation (1) as:

with where represents the wave angular frequency, is the wave number in free space

, and are the dielectric permittivity and magnetic permeability of free space respectively. is the propagation constant. Both a negative dielectric permittivity and permeability are written as [

Equation (2a) can be split into two equations by a separation of variables with the form:

Equation (3b) describes a simple harmonic oscillator and Equation (3a) is one form of the Bessel equations, whose solutions are the Bessel functions [

where and denote the coefficients of the longitudinal fields, is the radial distance , is called the Bessel function of the first kind and is the order of the Bessel function.

The propagation constant is a complex variable which constitutes a phase constant and an attenuation constant as:

By Maxwell’s curl equations, the transverse field components can be written as:

Substituting Equation (4a) and Equation (4b) into Equation (5a) and Equation (5b), yields

and

At the wall, the tangential electric and magnetic fields and respectively are related to a surface impedance by [24,25]:

With

By substituting Equation (6d) into Equation (6c) one obtains:

and

can be expressed in terms of electrical properties of the wall material (superconductor) as:

is the permittivity of the superconductor. It is complex with the form [

where

is the field penetration depth at temperature T = 0 K, is the conductivity of the superconductor and is the critical temperature of the superconductor. is called the reduced temperature of the superconductor. At the boundary of the wall (), by substituting Equation (4a), Equation (6a) and Equation (8) into Equation (7a), and dividing by, one obtains:

By substituting Equation (4b), Equation (6b) and Equation (8) into Equation (7b), and dividing it by, one gets:

Equation (10) and Equation (11) constitute a homogeneous system which admits a non trivial solution only in case its determinant is zero. Solving the determinants of the coefficients and in Equation (10) and Equation (11) results in the following transcendental equation:

The roots of Equation (12) are the allowed values of the propagation constant. Thus, it determine the characteristics modes of propagation. For each value of there is infinity of roots, any one of which can be denoted by the subscript. Any root of Equation (12) can then be designated by. In Equation (12), since TE modes are determined by roots of [

An alternate form of the Equation (12) is required for TM modes by substituting Equation (4a), Equation (6a) and Equation (8) into Equation (7a), and dividing it by, then the result is :

By substituting Equation (4b), Equation (6b) and Equation (8) into Equation (7b), and dividing it by, the result is :

In same way, the transcendental equation of TM modes is:

Since TM modes are determined by roots of

, the attenuation constant of TM modes ()

can be obtained [

In this paper, the numerical calculations for LHM cylinder with a superconducting wall like YBa_{2}Cu_{3}O_{7−x}, are taken with the following parameters: and [

and, [

tively observed in (5.8, 5.7 and 5.6 GHz). By Equation (8), Equation (9a) and Equation (9b), increasing by increasing will decreases. As a result, high and then high attenuation is achieved. The solution for the attenuation constant of TM waves is found by solving Equation (15).

in the frequency range 4 to 6 GHz and at. As a comparison between the results of Figures 3(a) and (b), the TE waveguide with a = 3 mm has lower attenuation than the TM wave-guide.

By increasing the band’s order to the values (1, 2, 3, 4, 5, 6, 7 and 8), roots of are increasing to the values (−1.84, −3, −4.2, −5.3, −6.4, −7.5 and −8.577) respectively. As a result, high attenuation of waves is realized at a higher band’s order.

In Figures 4(a) and (b), the wave frequency has been plotted against the attenuation constant for the first five

TM bands. By increasing the band's order to the values (1, 2, 3, 4 and 5), roots of are (−3.83, 30.56, −6.38, 49.3 and −8.7) respectively. For the odd band's order, (i.e., (1, 3, 5)) as the frequency decreases further to cutoff, the attenuation rises to high negative values, signals propagation become almost impossible. The attenuation is decreased to high negative values as compared to attenuation of even band's order which is increased to small negative values by increasing frequency.

The effect of the waveguide thickness on attenuation of the second band of TE waves is noticed in

This means that, in this waveguide, the lowest wave attenuation and the best confinement are achieved for the thickest TE waveguide. _{2}Cu_{3}O_{7−x} is replaced by a metal like ferrite. Both and are replaced by and respectively. According to Lichtenecker’s formula and in frequency range (4 to 5.8 GHz) [

wave frequency for different values of LHM-Ferrite radius. By decreasing the radius to the values 10 mm, 7 mm, 3 mm, the attenuation increases to the values of (−200, −280, −660) respectively at frequency 5.6 GHz. For the same range of a, the TM second band’s attenuation increases to the values of (160, 260, 660) respectively at frequency 5.6 GHz as displayed by

As a comparison between Figures 5 and 6, the implementation of YBa_{2}Cu_{3}O_{7−x} has reduced the TE and TM wave attenuation ratio by a factor of 10^{5} and 10 respectively as compared to that achievable with LHM-Ferrite structure.

The attenuation characteristics of both TE and TM waves in a waveguide structure containing LHM-superconductor or LHM-metal are dicussed. I found out that, LHM stimulate the modes to be backward of large propagation lengths. The lowest wave attenuation and the best confinement are achieved for the thickest TE waveguide. I compared the loss of LHM-superconductor waveguide with that of LHM-metal waveguide. The LHMsuperconductor waveguide is able to reduce the propagation losses, and to increase the mode’s propagation lengths which are very promising results in designing some future microwave devices.