Redshifting light in an expanding universe It's evident and well known that light traveling across an expanding FLRW universe is redshifted via an equation:
$$\frac{\lambda_{arriving}}{\lambda_{emitted}}=\frac{a_{now}}{a_{then}}$$
Where $a$ is the cosmological scale factor when the light is emitted and observed (denoted then and now respectively).
Let's say the light was traveling through a waveguide over that same distance. Calculations shouldn't be effected, and the redshift would follow the same equation.
If we now take that same waveguide and make it a large circle of the same total length, would that effect the redshift equation? I don't see how, but maybe someone here knows better.
If light is still redshifted the same it seems we can shrink the size of the waveguide arbitrarily down to a small local system. Does cosmological redshift happen locally? I've found arguments that energy isn't lost to bound systems lacking.
 A: The cosmological expansion can be seen only with very large structures. Its effective "force" is so weak that even galaxies are not affected, gravity keeps them bound and invariant .

Thus, the Andromeda galaxy, which is bound to the Milky Way galaxy, is actually falling towards us and is not expanding away. Within the Local Group, the gravitational interactions have changed the inertial patterns of objects such that there is no cosmological expansion taking place. Once one goes beyond the Local Group, the inertial expansion is measurable, though systematic gravitational effects imply that larger and larger parts of space will eventually fall out of the "Hubble Flow" and end up as bound, non-expanding objects up to the scales of superclusters of galaxies. 

Structures bound by the stronger interactions like electromagnetic and strong are of course not affected. The raisin bread analogue helps understand this:


Animation of an expanding raisin bread model. As the bread doubles in width (depth and length), the distances between raisins also double.

the dough is expanding, but the raisins are stable in size because the electromagnetic bindings are not affected by the  yeast in the dough.
The waveguide you are envisaging, is bound together with the electromagnetic force, and any interactions with electromagnetic waves will be within the "raisin".
A: Let's simplify: Instead of using a (possibly dielectric) waveguide (the picture your description summoned in my mind was a ring of optical fibre), just have a photon bounce between mirrors in a vacuum.
At first glance, you should indeed get the same effect from bouncing the photon a few times between mirrors that are far apart and bouncing the photon many times between mirrors close together: In Friedmann cosmology, the total redshift the photon 'accumulates' during its travel through curved spacetime will depend only on the time of emission and absorption.
But you also have to take into account what happens at the mirrors, and ask yourself if it will make a difference if the mirrors are, say, both comoving with the Hubble flow (eg mounted in different galaxies), or if they are kept at constant proper distance and thus decreasing comoving distance (eg through use of a rigid frame).
When deriving cosmological redshift, your starting point are comoving source and observer. Relative to this situation, a mirror at fixed proper distance will be moving towards the observer, and the photon should pick up some energy on its bounce. I suspect this might compensate the redshift, though I have yet to verify this by calculation (or even better, by coming up with a convincing argument).
