FULL WAVE ANALYSIS OF DIELECTRIC RECTANGULAR WAVE GUIDE WITH HARMONICS

Abstract

The concept of guiding electromagnetic waves either along a single conducting wire with finite surface impedance or along a dielectric rod/slab has been known for a long time. There has been a lot of study on microwave waveguides and various methods have been evolved to find propagation constant and other parameters of a wave guide. Most of studies are being done by assuming cosine and sinusoidal fields as incidence field but very less people has assumed square field or harmonics field as incidence. Propagation modes of rectangular dielectric waveguide based on the expansion of electromagnetic field in terms of a series of circular harmonics (Bessel and modified Bessel multiplied by trigonometric function) has been done by Goell [10]. In this thesis, a non-sinusoidal signal has been considered as incidence wave to a Rectangular Dielectric waveguide. The wave function has been represented by a square wave which is distributed in two dimensions. These square waves have been represented by harmonics of sine and cosine function and their 3-D plots had been drawn using Matlab 7.0. The 3-D graphical representations of Eymn mode wave function for various harmonics (square wave) are similar as that obtained for simple cosine and sine functions [1]. These 3-D graph plots, give better visualization and understanding of field distribution in x and y directions for even and odd functions. Modes consisting of linear combinations of the established TE, TM, or HE modes can be constructed in a wave guide. Eymn and Exmn as well as hybrid modes are supported by the waveguide. The wave guidance takes place by the total internal refection at the side walls. The field components for the Eynm modes are Ex, Ey, Ez, Hx, and Hz, with Hy = 0 and the independent set of fields for the Exnm modes are Ex, Ez, Hx, Hy, and Hz, with Ey = 0. The complete set of fields is the sum of the Eynm and Exnm modal fields. The solutions to the rectangular dielectric guide problem have been derived by assuming guided mode propagation along the dielectric, and exponential decay of fields transverse to the dielectric surface. Thus, in the region of confinement, (inside the guide) due to reflections there is standing wave patterns and when the field goes out of the boundary of the guide, in the absence of reflection, the field moves away from the guide exponentially i.e. there is an exponential decay of fields, transverse to the dielectric surface. The fields are -xiiassumed to be approximately square wave distributed inside the waveguide and decaying exponentially outside. Wave function based on harmonics for even and odd functions have been derived for Eymn mode. Using Marcatili‟s, approximation method, the approximation of fields has been applied for inside and outside fields. Field at the extreme outside corner of the waveguide has been neglected as field strength is very weak at corners. Applying mode matching technique, the transverse plane of the waveguide has been divided into different regions, such that in each region canonical Eigen functions represent the electromagnetic fields. The Eigen value problem has been constructed, by enforcing the boundary conditions at the interface of each region. Assuming air dielectric interface and square wave form distribution of field inside the waveguide characteristic equations has been derived. Solution of characteristics equations for Ey11 modes assuming three harmonics of even and odd function is calculated graphically by MathCad Tool. Calculation of transverse propagation constants for inside (u and u1) and outside (v and v1) in x and y directions of waveguide has been done by taking a particular value to the ratio c1 and c2 (u/u1=c1 and v/v1=c2). The value of c1 is optimized to F/60 and c1 is taken equal to c2. Where, F is the operating frequency. Relative dielectric constant inside the waveguide is taken as three. Comparison of results of normalized propagation constant kz/k0 using this graphical method to that of Marcatili's and Goell's methods, for a silicon dielectric waveguide with a=0.5mm and b=1mm cross section, Ey11 mode has been done. This method works quite well for frequencies at the lower and middle range, when the wave is well guided, the results agree very well with the Marcatili's and Goell's method. At higher frequencies above cut-off, because of presence of harmonics, the normalized propagation constant differ from experimental or direct methods. Accurate calculations are more complicated at higher values of harmonics so it is done, only up to three harmonics. The result of three harmonic functions is very much consistent at lower frequencies and differs at higher frequencies.