A. Frequency Animation

When interactively modifying or automatically animating the source frequency, the displayed propagating mode is by default the highest order mode. This is very usefull to intuitively understand the order of succession of the cutoo frequencies of the various modes. When taking a very rectangular waveguide { e.g. with a high a=b ratio { (see Fig. 3), it becomes evident, that the cutoo frequencies of mode m; 0 are proportional to m, and that the cutoo frequencies of mode 0; n are proportional to n. Once increasing the source frequency one immediately understands that mode 0; 1 appears after mode m; 0 as soon as the \wave width" of mode m+1; 0 is greater than b. This is an intuitive understanding of the fact that the cutoo frequencies of modes m; 0 are proportional to m=a and that the cutoo frequencies of mode 0; n are proportional to n=b (See (1)). For example, in Fig. 3 cutoo frequency of mode 0; 1 is higher than cutoo frequency of mode 4; 0 and lower than cutoo frequency of mode 5; 0 for an a=b ratio of 4:4 B. Coordinate system for circular waveguides Since the eld topology in circular waveguides is determined using polar coordinates, these topologies are always represented using the (; ') coordinate system. However, such a system { although very usefull for calculations { is absolutely not intuitive, as shown on Fig. 4. Looking at Fig. 4.a and Fig. 4.b one rst expects a sin-gularity at the wave guide center. However, after deeply analyzing these results, one may nally understand that there is no singularity at all, that the eld is vertical at the waveguide origin, and that the discontinuity is simply due to the polar coordinate system. On the other hand, Fig. 4.c and Fig. 4.d show the horizontal and vertical eld components, which may be immediately and correctly interpreted. Fig. 3. mWave showing electric eld amplitude for mode TE40 propagating in rectangular waveguide with high a=b ratio Fig. 4. Electric eld components of mode TE11 propagating in a circular waveguide. a) Radial component E (upper left), b) tangential component E' (upper right), c) horizontal component Ex (lower left) and d) vertical component Ey (lower right). \Three dimensional nite element analysis of high power microwave devices," IEEE Trans.