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A waveguide is any of several kinds of structure intended to direct the propagation of highfrequency electromagnetic energy along specific paths. While the analysis of bends and terminations in such a system is an essentially threedimensional problem, the propagation in long straight segments of the guide can be reduced to a twodimensional analysis. In this case, we assume that the guide is of uniform crosssection in the (X,Y) plane, unvarying in the Zdimension of the propagation direction. In this configuration, we can make the assumption that the fields inside the guide may be represented as a sinusoidal oscillation in time and space, and write
(3.1)
It is easy to see that these expressions describe a traveling wave, since the imaginary exponential generates sines and cosines, and the value of the exponential will be the same wherever . A purely real implies an unattenuated propagating mode with wavelength along the z direction.
We start from the timedependent form of Maxwell’s equations
(3.2)
Assume then that and , and apply (3.1) :
(3.3)
Taking the curl of each curl equation in (3.3) and substituting gives
(3.4)
In view of (3.1), we can write
(3.5)
with denoting the operator in the transverse plane.