MODAL DISPERSION CURVES OF A METAL COATED OPTICAL WAVEGUIDE WITH TREFOIL CROSS-SECTION UNDER WEAK GUIDANCE APPROXIMATION

In this paper the author investigated a special type of core cross-section bounded by metallic boundary. In this structure three cores embedded in a common metallic boundary. Characteristic equation in derived for the guided modes under weak guidance approximation by choosing an appropriate coordinate system. This study in done by using analytical method. From the characteristic equation, dispersion curves are obtained and interpreted.

In this paper the author investigated a special type of core cross-section bounded by metallic boundary. In this structure three cores embedded in a common metallic boundary. Characteristic equation in derived for the guided modes under weak guidance approximation by choosing an appropriate coordinate system. This study in done by using analytical method. From the characteristic equation, dispersion curves are obtained and interpreted.

…………………………………………………………………………………………………….... Introduction:-
In the beginning, only two type of waveguides were used in optical range -waveguides with circular cross-section [1][2] and planner waveguides [3][4]. Now, optical waveguides with various non-circular cross-sections like rectangular and elliptical, triangular, have been studied by many researchers. The analysis of waveguides with non circular cross-sections is generally difficult, and usually approximate or numerical methods are employed [5][6][7][8] in such investigations. In the present paper author treat a very special type of cross-section of the shape of a trefoil. If three cores with such a cross-section is embedded in a common boundary, the structure is similar to a waveguide with three circular cores are embedded in a common boundary, each circular loop represents the cross-section of one core. Due to the presence of neighbouring circular loops, these circular loops are distorted into flatten shape. In this proposed waveguide the boundary in highly conducting. In order to obtain the number of modes, which can propagate through such a fiber for a given 'V' parameter, characteristics equation for the guided modes in derived. We obtain the dispersion curves for some lower order modes by the investigating of this characteristic equation. It is found that the dispersion curves are of the expected shapes, and these dispersion curves can be used to determine the number of modes which a fiber of this type can sustain at a known operating frequency for a given size parameter and fixed value of care refractive index.

Theory
We consider an optical fiber with trefoil like cross-section bounded by highly conducting material, represent by given equation in polar coordinates, as  which is appropriate for this geometry.
The scalar factors can easily be derived.
Now assuming a harmonic time dependence of the electric field, the vector wave equation for the electric field Using the weak guidance approximation, equation (6) can be reduced to a scalar wave equation.
Where 0  and  are permeability of free-space and the permittivity of the medium.
Where  is unknown constant.To separate and ,   we consider two special cases. Firstly, we assume   and we get following two equations.  This equation (19) is known as modal characteristic equation of the waveguide under consideration.

Numerical Computation
We now make some numerical estimates of the modal properties of a fiber with trefoil cross-section having metallic boundary. The obvious step is to determine the value of propagation constant  by solving the equation (19). We now choose the refractive index of core 1 1.48. n  The wavelength of wave is fixed at 0 1.55 .

 
In order to obtain the dispersion curves we choose a given value of 'a' and then plot the L.H.S of eq. Here 'V' parameter is plotted along the abscissa and the normalised propagation constant 'b' is plotted along the ordinate. The possible number of modes can be determined by these curves for the proposed waveguide. From these curves we see that when V 5.  There is a only a single mode sustain by the guide and when V7  there are two guided modes. When the value of V parameter increases, the number of modes also increase, an expected. We have five modes when V 19.  The dispersion curve for the lowest mode in fig.(3) does not begin at V 0,  this anomaly appears because of the singularity of curve for   0 V 0 a  . For all the dielectric fiber, however, the fundamental mode always exists.