The general approach to study the wave guiding systemsSuppose the wave guiding system is infinitely long, and let it beplaced along the z-axis and the propagating direction be along thepositive z-direction. Then the electric and the magnetic fieldintensities can beexpressedasE(x, y,z)= E(x, y)e-k.:H(x, y,z)= H,(x, y)e-ik:where k,is the propagation constantin the z-direction, and they satisfythefollowing vectorHelmholtzequation:a?EaEa?E+kE=0ayO22ax?a?Hα?Ha"H+k"H=0dy202?ax?U
The general approach to study the wave guiding systems Suppose the wave guiding system is infinitely long, and let it be placed along the z-axis and the propagating direction be along the positive z-direction. Then the electric and the magnetic field intensities can be expressed as k zz x y z x y j 0 ( , , ) ( , )e − E = E k zz x y z x y j 0 ( , , ) ( , )e − H = H + = + + + = + + 0 0 2 2 2 2 2 2 2 2 H H H H E E E E 2 2 2 2 2 2 k x y z k x y z where kz is the propagation constant in the z-direction, and they satisfy the following vectorHelmholtz equation:
The above equation includes six components, E, E,E, andH.,H.,H,in rectangular coordinate system,and they satisfythescalarHelmhotzequationBased on the boundary conditions of the wave guiding systemand by using the method of separation ofvariables, we can findthesolutions fortheseequationsFrom Maxwell's eguations, we can find the relationshipsbetweenthe x-componentor they-componentand the z-componentasaEaH.aH.aEHjk-jkHj0sjouaxayyah.oH.OE.-jkE1jkH+ joμj0skeayayk2axaxWhere k? = k?-k?. These relationships are called the representationof the transverse components by the longitudinal components.u
The above equation includes six components, and , in rectangular coordinate system, and they satisfy the scalar Helmhotz equation. Ex Ey Ez , , H x H y H z , , From Maxwell’s equations, we can find the relationships between the x-component or the y-component and the z-component as − = − y H x E k k E z z x j z j 1 2 c + = − x H y E k k E z z y j z j 1 2 c − = x H k y E k H z z z x j j 1 2 c − = − y H k x E k H z z z y j j 1 2 c Where . 2 2 2 c z k = k − k Based on the boundary conditions of the wave guiding system and by using the method of separation of variables, we can find the solutions for these equations. These relationships are called the representation of the transverse components by the longitudinal components
We onlyneed to solvethe scalarHelmholtzequationforthelongitudinal components, and then from the relationships between thetransverse components and the longitudinalcomponents all transversecomponentscanbederived.In the same way,in cylindricalcoordinatesthe z-componentcanbe expressed in terms of the r-component and -component asE.ouaEjkkearadaH.aE合jopadaroH.aE08jkHapk.arOEaHk.HJOek2OradU
We only need to solve the scalar Helmholtz equation for the longitudinal components, and then from the relationships between the transverse components and the longitudinal components all transverse components can be derived. In the same way, in cylindrical coordinates the z-component can be expressed in terms of the r-component and –component as + = − z z r z H r r E k k E j j 1 2 c + = − r E H r k k E z z z j j 1 2 c − = r H k E k r H z z z r j j 1 2 c + = − z z Hz r k r E k H j j 1 2 c
2.Equationsfor ElectromagneticWavesin RectangularWaveguidesSelect the rectangularcoordinate system and let the broad sidebe placed along the x-axis, the narrow side along the y-axis,and thepropagating directionbe along the z-axis.For TM waves, H, = o , andaccordingto the method oflongitudinal fields,the componentE,shouldfirstbe solved,and fromhe,uwhichthe other components can0be derived.The z-component of the electric field intensity can be written asE. = E.o(x, y)e-ik:UV
2. Equations for Electromagnetic Waves in Rectangular Waveguides Select the rectangular coordinate system and let the broad side be placed along the x-axis, the narrow side along the y-axis, and the propagating direction be along the z-axis. a z y x b , For TM waves, Hz = 0 , and according to the method of longitudinal fields, the component Ez should first be solved, and from which the other components can be derived. The z-component of the electric field intensity can be written as k z z z z E E x y j 0 ( , )e − =
It satisfies the following scalarHelmholtzeguation,i.eo'E."E.+kE.=0oy?Ox?And the amplitude is found to satisfy the same scalar Helmholtzequation, given byOE00'E.0+kE.-0ax?OyIn order to solve the above equation, the method of separation ofvariablesis used. LetE.0 (x、y)= X(x)Y(y)XV--kWeobtainXVwhere X" denotes the second derivative of X with respect to x, and y"denotes the second derivativeof Y with respectto y.UV
It satisfiesthe following scalarHelmholtz equation, i.e. 0 2 2 c 2 2 2 + = + z z z k E y E x E And the amplitude is found to satisfy the same scalar Helmholtz equation, given by 0 0 2 2 c 0 2 2 0 2 + = + z z z k E y E x E In order to solve the above equation, the method of separation of variablesis used. Let ( ) ( ) ( ) 0 E x y X x Y y z 、 = We obtain 2 c k Y Y X X = − + where X" denotes the second derivative of X with respect to x, and Y" denotes the second derivative of Y with respect to y