The authors describe the operation of a near-concentric cavity close
to instability, where the mode waist in the center becomes microscopic
and alignment requirements are extraordinarily stringent. Such
cavities have interesting applications in the field of trapped ion
experiments and multimode cavity QED.

The investigation is performed with care and is well described in the
manuscript. The authors report the remarkable result that the finesse
does not significantly change for the last stable mode, which was not
achieved so far to my knowledge.

As a criticism, I need to add that the study, while being sound and
probably tedious to conduct, is of technical nature and does not
convey or discuss novel physical insight. One thus may ask whether PRA
is the right journal.

Below I add some more detailed comments:

1) The motivation to study near-concentric cavities is not fully
convincing / uses inconcise arguments:

i. Intro: The authors motivate their cavity to be advantageous for
scaling up cavity experiments, but the challenging operation (active
3D stabilization!) poses questions about benefits along this line

ii. The authors state that near-concentric cavities exhibit the
smallest mode waist; this should be specified more explicitly. The
mode waist can become similarly small for smallest mirror separation.
The waist size is symmetric with respect to the mirror separation and
gets small for both short and long cavities. It is only by fine tuning
the cavity length of the last mode closer to the stability than
lambda/2 (e.g. by wavelength tuning) that one can get smaller than for
l_cav = lambda/2.

iii. The figure of merit for light matter coupling scales with Finesse
/ mode cross section (~Cooperativity), such that a microcavity and a
near-concentric cavity with same mode cross section also require the
same finesse (i.e. mirror reflectivity) for a given Cooperativity. So
the argument is not quite correct to state that near-concentric
cavities require less demanding coatings. On the other hand, the ratio
of coupling strength over cavity decay rate does improve for a long
cavity, and as long as the rates remain larger than the atomic decay,
one can enter the strong coupling regime more easily. This should be
described more clearly.

iv. An important argument is missing in my view: For ion trap
experiments, large distance from the trapped ion to the dielectric
mirrors is required to avoid trap deformation and heating. This is
also a type of experiment where such rather unpractical cavities are
used (e.g. in Innsbruck).

v. Multimode cavity qed appears to be the most interesting direction
where this could lead to; this could be emphasized and detailed a bit
more.

2) Space between words / sentences is sometimes missing

3) The finesse of the cavity should be mentioned

4) The mode size at the mirrors should be mentioned for the last
stable mode