Why TEM doesnot exist in waveguide while TE and TM are possible

This post again discusses the propagation in a waveguide. A waveguide is a single conductor with a dielectric inside it (Air is also a dielectric). TEM mode is characterised by electric fields and magnetic fields perpendicular to one another and perpendicular to the direction of propagation. Inorder to have such a configuration, there has to be a source of electric field at the centre from where E fields originate and terminate on the outer conductor and there has to be a current source which can generate magnetic fields. For TEM mode the current flow should be along the axis of the waveguide, which creates rotating magnetic fields which are normal to the electric fields generated due to the (moving) charges in the current carrying conductor. Since such a current source is absent and waveguide being a single conductor configuration, TEM mode cannot exist inside a waveguide. Also it is evident from the above explanation that for TEM mode to exist, presence of atleast two conductors is compulsory.

Now the question is what are the other field configurations possible?

We so far talked about conduction currents along the axis of the waveguide whose absence prevented TEM mode. But there are another class of currents called as Displacement Currents, whose existence is proposed to explain flow of currents in a capacitor. Displacement current is formed by time varying Displacement Field (D). Now imagine time varying Electric Field to be present along the axis of the wave guide. This causes a time varying Displacement Field in the same direction. This Displacement field in turn results in Displacement Currents which can serve as the source of magnetic fields. The direction of Magnetic fields can be found from amperes law and deduced to be in the transverse plane. Thus "Electric field along the axis and Magnetic field in the transverse plane" is a non-vanishing combination inside a waveguide. Such a mode is called Transverse Magnetic(TM) mode.

If a axial time varying magnetic field is present in the waveguide, it results in Electric Fields in the transverse plane from Faraday's law. Thus "Magnetic field in the axial direction and Electric Fields in the transverse direction" is another possible non-vanishing combination. This mode is called Transverse Electric(TE) mode.