In order to define the usual modes of EM waves in a confined space, TEM, TE and TM, one must have a well defined notion of "transverse" and "longitudinal" in the system. In the case of cylindrical waveguides, there is the axis of translational symmetry and in the case of spherical cavities, there is the radial direction.

Now imagine arbitrarily shaped cavity. Can the cavity modes in it be classified as TEM, TE and TM in the absence of any special direction which we call longitudinal?

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    $\begingroup$ What's wrong with defining "longitudinal" as along the direction of propagation of the mode? $\endgroup$ – probably_someone Jul 17 at 22:02
  • $\begingroup$ your question is not clear because you are asking about radiation modes and cavity modes at the same time; the two are not the same and are not directly related for they satisfy very different boundary conditions. please clarify your question.. $\endgroup$ – hyportnex Jul 18 at 13:34
  • $\begingroup$ @probably_someone: ok, so how do you define a direction of propagation in an arbitrarily shaped cavity? Is there a canonical choice for this? $\endgroup$ – Fizikus Jul 19 at 21:33
  • $\begingroup$ @hyportnex: corrected, thanks. $\endgroup$ – Fizikus Jul 19 at 21:33
  • $\begingroup$ @Fizikus Solve for the modes of the cavity first, and determine the wavevector by applying a Fourier transform to the solutions. $\endgroup$ – probably_someone Jul 19 at 21:35

In general, neither the cavity modes nor the waveguide modes can be classified as TE or TM.

For example. if the waveguide is uniform along an axis, say, $z$, but it is partially filled with dielectric, so in the $xy$ plane it is inhomogeneous, no propagating TE or TM modes exist. Note the emphasis on propagating for the cutoff modes are still either TE or TM (or TEM if the conducting surface is multiple connected). In practice, this means that near cutoff one has propagating quasi-TE or quasi-TM modes such that the longitudinal components are much smaller than the transversal ones. If you make a cavity out of such partially filled guide by terminating its ends with PEC walls the "quasiness" will carry over to the cavity but again no true TE or TM modes can resonate it.

If you take an arbitrarily shaped cavity whose inside is empty (vacuum) then, in general, again no TE or TM modes exist. Just recall that even if you start with a rectangular cavity oscillating in its $TE_{111}$ mode and slightly deform its end wall then the "TE" nature will surely be violated there. In some special symmetrical geometries you can have TE or TM modes, for example, a spherical resonator has modes that are TE or TM with respect to the $r$ coordinate. These modes can be derived from electric or magnetic vector potentials. A clean derivation is in chapters 3 and 6 of [1].

[1]: Harrington: Time-Harmonic Electromagnetic Fields

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