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TM01 mode doesn't exist because If anyone of the two (m or n)

becomes 0 then the z component of electric field that is Ezs becomes zero which in turn makes all the other components (Exs, Eys, Hxs, Hys) vanish.

you don't have any component of field for this configuration.

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Q: Sketches of field patterns of dominant modes in rectangular waveguides?
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