Dear Editor,

first, we would like to thank the referee for the careful reading of our
work, and the constructive comments for preparing a revised manuscript. Our
replies to the referee comments point by point:

1) "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."

Reply: The near-concentric cavity presented here exhibits the smallest
cavity waist with a cavity geometry that is favorable for atom trapping
experiments. In addition, the presented work carried out the mode
analysis and demonstrated that the Gaussian beam theory remains valid
in such strong focusing regime. This can provide a benchmark for
theoretical work to investigate the breakdown of paraxial approximation in
describing resonant modes of optical cavities. 
Hence, we believe that our work is interesting to the audience of PRA.


2) "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."

Reply: The cavity alignment is automatically checked and the realignment
procedure is also automatically done and takes only few seconds. Thus
this does not affect the experimental duty cycle significantly.

In addition, the need for transverse realignment is mainly caused by
temperature changes. This work is only a first step of this approach. It
takes the high-finesse cavity community more than 20 years to achieve the
current state of the art (see Kimble, Physica Scripta T76, 127-137, 1998).
We believe that the 3D stabilization is not a fundamental limit and 
there are feasible improvement which can be done in future designs to
improve the passive stabilization. For example, the two cavity mirror holders
can be made of materials with different thermal expansion coefficients with
different lengths to minimize transverse misalignment. We added the reference
above to the manuscript in the introductory section as well.


3) "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."

Reply: We agree with the referee's comment and removed the statement about
the tightest focus of the cavity mode.
We would like to note that theoretically, both very short cavities and
near-concentric cavities can obtain small cavity mode waists.  However, the former can only achieve a diffraction-limited waist (~lambda/2)
with an almost zero cavity length, which is technically difficult to
achieve due to the penetration of the cavity field into the mirror coating,
especially when the coating must be thick to obtain a high finesse (Hood et
al., 2001 PhysRevA.64.033804, our reference ). Such an attempt would probably
be closer to field geometries in a photonic bandgap structure, while we
explore a more traditional cavity geometry.

In this work, from the measured cavity length we determined that the cavity
mode waist is 2.4 micron, which is, to the best of our knowledge, the smallest
waist obtained so far for a cavity geometry that is favorable for atom
trapping experiments. For a comparison with fiber-based cavities, 
Brandstatter et al. give a nice overview in Review of Scientific Instruments
84, 123104 (2013). We include the above statement with the reference in the
second paragraph in section IV in the manuscript.



4) "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."

Reply: It is true that the Cooperativity does not depend on the
cavity length given a cavity mode cross section. However, the
Cooperativity being larger than one is only a necessary condition for the
strong atom-cavity coupling. We are more interested in achieving the
sufficient condition that g>>kappa and g>>gamma (gamma is the cavity
decay rate). The ratio of g over gamma is proportional to F(L^1/2)/w, which
is more favorable for long cavities at a given Finesse and a cavity mode cross
section.

To clarify this, we include a corresponding statement at the second paragraph
of the introduction in the revised manuscript:
"The ratio of coupling strength over cavity decay rate is proportional to 
$F\sqrt{l_{cav}}/w$, which is more favorable for longer cavity lengths at a 
given cavity finesse $F$ and a cavity mode waist $w$."


5) "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)."

Reply: The second paragraph of the introduction included already a similar
explanation of the advantage of using near-concentric cavities in ion
trap experiments in the manuscript:
"For use with trapped ions, the large separation between two mirrors 
provides the ability to avoid charging problems with dielectric surfaces, 
which has been a major hindrance to the development of ion traps in 
optical cavities."
In order to make this part of the introduction more accessible, we broke it
down in two paragraphs in the revised manuscript.


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

Reply: We are still working on a better understanding of this topic and will
present the detailed studies of applications of multimode cavity QED once
ready.


7) "Space between words / sentences is sometimes missing"

Reply: We corrected the missing spaces.


8) "The finesse of the cavity should be mentioned. The mode size at the
mirrors should be mentioned for the last stable mode."

Reply:
We added the cavity finesse of F=603 and an inferred cavity mode
waist at the mirror of 0.56 mm at the second paragraph of page 4 in the
manuscript. 


With this, we hope to have addressed the issues pointed out by the referee,
and look forward for consideration of publication in Physical Review A.

With Best Regards on behalf of all authors,

Christian Kurtsiefer