Dear Editor,

first of all, we would like to thank the referees for their detailed and constructive remarks on our manuscript. We agree that the suggested changes the manuscript required a major revision. We also managed to improve our measurement accuracy by now, so we decided to re-take most of the data presented in the previous manuscript, such that we can also include images of the mode patterns. This should help to better convey some of our points on the importance of the aberrations, as recommended by the referees.

We also added two more figures, and modified most of the existing ones significantly. This should help to address some of the referee's remarks, and re-wrote some paragraphs to both adapt to the new figures and in response to referee comments. 

Our response to individual points raised by the referees in detail:

Comments on points raised by Referee 1:

1) In the introduction, the cited papers (Ref. 4-6 in first version of the manuscript) have been changed with more suitable references, in which high finesse cavity issue is discussed in order to de-emphasize the quantum information applications.

2) The focusing parameter as a function of cavity length is plotted and added to the paper (Fig.2), to make the connection between the cavity length and focusing parameter more clear, as suggested by the referee. The focusing parameter is discussed further in 4 th paragraph of Section 2.

3) For the clarity of the derivation of Equation 2 (equation 1 in previous version), equation 1 is added (basically, the fraction of light reflected of a Gaussian mode, truncated by a finite sized mirror.) The power reflected back from one mirror is squared to get the power left within the cavity mode after one round trip, assuming two identical mirrors.

4) The measured transmission spectrum of the planarconcave cavity is added as new figure 5 as a comparison to the calculated transmission, and in Fig. 4 (former figure 3) we added the estimated transmission spectrum when the fundamental mode of the cavity interacts with a single atom at the center of the cavity. And a brief explanation of the splitted mode figure (Fig. 4, right) is added at the last paragraph of Section 3.

5) The exemplary snapshots of the cavity output at three different frequencies around the transmission peak are shown as well in the new Fig. 5 in order to illustrate how the spatial field profile of the cavity output look like (inverted intensity pattern).

6)Former Figures 2 and 6 are replaced with Fig. 3 and Fig. 8 , respectively. The reason of the large error in the measured linewidth of the previous versions of the figures was the poor mechanical stability of the cavity setup. The measurements are repeated at same focusing parameter values for both graphs and the since the mechanical stability is improved, the error bars are smaller.

7) We changed the text with "(w(z_m)=0.42a)" at the second last paragraph of section 3

8) The new Figure 4 and the last paragraph of section 3 should now explain why the higher order modes prevent the observation of normal mode splitting.

9) The value of 54% (in the previous manuscript) was a measured value, but it was not indicated clearly that it is an experimental value rather than an estimation. The focusing parameter for 54% coupling to a single mode fiber was 0.113, and we repeated the measurement for u=0.36. u=0.36 is the desired value for the suggested anaclastic design as it gives a maximum achievable cooperativity of 150. The coupling value reduced slightly comparing to previous value but as indicated in the last paragraph of section 4, the coupling change depending on the focusing parameter is less than 3%. Ideally, 87% (this is the measured coupling of the cavity input mode into the single mode fiber) of the fundamental mode of the cavity should be coupled to the single mode fiber. However, we don't have an answer of the reduced coupling value yet.

10) We added the missing units in the lens geometry figure (Fig. 6, previously Fig. 4). Similarly, the curvature of the spherical surface is now characterized by a radius (5.5mm), not anymore by a diameter

11) The effect of the higher order modes in Fig. 7 (previously Fig. 5) on the normal mode splitting is discussed at the last paragraph of the section 5.


Our replies to comments by Referee 2:

1) The focusing parameter diverges by getting closer to the exact concentric configuration. The input mode waist at mirror should be infinite in order to match the mode of an exact concentric cavity. Fig. 2 is added for the clarity of the focusing parameter. 

2) Fig 5 (previous manuscript) is now replaced by Fig. 7, and the focusing parameter used for this re-measured graph is 0.36 to make the graphs consistent with the targeted beam geometry. At this focusing parameter the cavity cooperativity has a maximum value of 150.

3) We agree that there are alternative ways to eliminate of the aberrations in the system are discussed in the first paragraph of the section 3 and the related references are added. A brief comment on such methods is added here, also in the conclusion paragraph as well.

4) The higher order modes in Fig. 7 (previously Fig. 5) are the first 8 higher order transverse modes, which have resonance frequencies less than the fundamental mode. For a higher order mode to coincide in frequency with the fundamental mode, the resonance frequency of the higher mode should be (at least) one free spectral range less than the fundamental mode. In order this to happen at a focusing parameter of 0.36 (where the frequency separation of the successive higher order modes is ~60 MHz) the 227th order modes should be excited (one FSR is 13 600 MHz and the frequency separation of successive higher order modes is 60 MHz, 13 600/60=227). In the transmission spectrum shown in Fig. 8 the visible higher order modes are up to 8th order modes, which suggests that there is no higher-order mode falling under the fundamental mode. A brief explanation of the issue is added in the last paragraph of section 5.

5) The abstract is changed accordingly and the wrong expression is corrected. 

Again, we would like to thank our referees for their constructive comments, and hope that with the additional material and a consistent data set in the presentation, we can be considered for publication in the New Journal of Physics.

With Best Regards on behalf of all authors,

Christian Kurtsiefer