The path of the wave packets in the FP resonator is limited to one direction (a one-dimensional structure). The ring resonator is two-dimensional and topologically distinguishable. A cavity is three-dimensional. In all three structures you have standing waves with places of constructive interference.

Ring cavity without an optical valve (Faraday cell or other) stay also the  standing wave cavity. Optical valve is device  that pass light  just in clockwise direction and  dose not  pass in anticlockwise one (or vice verse) . 

Hi Vincent,

But in the case of two counter-traveling beams inside the ring cavity, if we take the resultant intensity, then also we can find standing wave pattern of nodes and anti-nodes.

Oh!!! That clears a bit of confusion.

It means that ring cavities can also support standing wave modes.

Are standing wave eigenmodes and traveling wave eigenmodes two different representation techniques of optical fields inside a resonator?

Waves travel always. If you have at least two waves (with the same frequency) in opposite direction, you have superposition and you will find nodes, where the amplitude ist always low and some other points, where the amplitude is much higher. Looks as if the waves are "standing", but that describes only the average over a long time.

(Normal) modes of any resonator are only standing waves. Traveling waves are not resonator modes; they are solutions of the Helmholtz equation obtained as the sum of two degenerated standing waves shifted for Pi/2.

                  cos(wt-kx) =cos(wt) cos(kx) + sin(wt) sin(kx)

If mode degeneracy is somehow removed, there is no traveling wave, even in ring resonators.

Hi Branislav

If I remove the mode degeneracy by rotating the resonator (Sagnac Effect), will the traveling wave nature will be lost ?

If this rotation removes degeneracy, there would be no traveling wave anymore.

Rotation removes frequency degeneracy but not  standing wave laser operation absolutely. Exists some minimal  velocity rotation below which we have standing wave  mode operation and just above some  velocity value we  recreate  travelling wave laser operation. This affect stipulated by  frequency  mode locking  in given laser and depends from many reasons. To great regret  similar laser behaviour limits measurement of small rotation velocity  by laser gyroscopes  based on Sagnac effect.

To Vincent. You are absolutely right! Thank You for understanding of the problem. Eugene Tikhonov