REFEREE REPORT(S):
Referee: 1

COMMENTS TO THE AUTHOR(S)
In their paper „Diffraction-limited Fabry-Perot Cavity in the Near Concentric Regime“ the authors compare two different cavity designs in the near concentric regime. First, they experimentally measure the linewidth of a cavity with plano-concave mirrors and show that the linewidth grows much faster with the so-called focusing parameter (ratio of mode waist over half the cavity length) than it could be explained by diffraction losses only. In a numerical model in which they include aberrations, and coupling to higher order modes of the cavity, they can reproduce the experimental data much better, concluding that aberrations play a major role in the increase of the linewidth. In contrast, their newly designed cavity in an anaclastic design with aspherical entrance surface avoids these additional aberrations and shows diffraction-limited behaviour of the linewidth when changing the focusing parameter. It therefore has a major advantage over plano-concave cavity designs in the near concentric regime when aiming for strong atom–cavity coupling. 

The paper is very well written, and it comprises a nice study of the origins of linewidth broadening for plano-concave cavities. The authors demonstrate experimentally the diffraction-limit for their anaclastic cavity design and estimate huge atom-cavity cooperativity for a single Rubidium atom held in the centre of the cavity, giving a promising perspective to future experiments. However, as the authors claim themselves, the idea of such a cavity design was presented before and is not new. Therefore, I would consider the experimental demonstration of the diffraction limited design and the investigation of the aberration induced coupling to higher order modes as a rather technical study with minor advancements not suited for publication in New Journal of Physics.

Before submitting elsewhere, I would advice the authors to further discuss, change or clarify the following points in their next version:

(1)	in the introduction, they cite papers [Refs 4-6] from the world leading groups in optical cavity QED (Kimble & Rempe), however, the papers they have picked do not discuss the “struggle for higher coupling”. These papers fit better under the topic of quantum information processing. The authors should pick more appropriate publications here.

(2)	in section 2, it might not be obvious for every reader that changing the focussing parameter actually involves a change of the cavity length. The authors should state that more clearly, and then also the sentence “The linewidth increases dramatically as the cavity length approaches the concentric limit…” would be easier to understand.  

(3)	in section 2, it is not obvious for me how the correction factor per round trip in Eq. (1) was determined. 

(4)	in Figure 3 a calculated spectrum is shown. It would be interesting to see how well this calculated cavity spectrum matches the experimental observation. The calculated spectrum has a quite large pedestal. Will it only be the increased cavity linewidth or as well this pedestal that will make it difficult to observe the normal mode splitting for a single atom?

(5)	in section 3, can one observe the excitation of higher order modes with a camera behind the cavity? It would be instructive to show such a picture.

(6)	in section 3, the error bars in Fig. 2 & 6 look very large if one considers an averaging over 100 spectra as it is indicated in the text. Is there a reason for this large uncertainty in determining the width of the cavity transmission line?

(7)	in section 3, second last paragraph, make it clearer that: w(z_m)=0.42a. 

(8)	in section 3, last paragraph, it is not clear to a general reader why the higher order modes prevent to observe the normal mode splitting. This point should be discussed in more detail. 

(9)	in section 4, how do the authors conclude that they will be able to couple 54% of the light exiting the cavity into a single mode fibre? Is this true for all focusing parameters? And what limits the coupling efficiency?

(10)	Figure 4 needs improvement, there are no units, and it should be made more clear, that 11 is twice the radius of curvature of the spherical inside of the cavity mirrors.

(11)	in Figure 5 higher order modes are visible. How will these influence the coupling strength of atom to the cavity or make an observation of normal modes more difficult?