Dear Editor, first, we like to thank all reviewers for their careful reading of the manuscript, and the constructive remarks to improve it. In the following, we address the points raised by the reviewers. Comments from the first reviewer: "1. I couldn't find certain crucial parameters, like the radius of curvature of the mirrors and the measured finesse of the cavity. Please check." Our reply: Noted - we have included the radius of curvature in Section 3.1 and measured finesse in Section 3.1.3. "2. I was intrigued by the argument in section 2.2 on identical effective mode volumes for all radial tranverse modes. Can the authors formulate a physical explanation for this, on top of the mathematical explanation mentioned in the manuscript?" Our reply: We think that the physical explanation is that the beam waists for all radial transverse modes are the same, just that it appears larger due to ring structure, but this is not yet clear to us. To hint at this conjecture, we have included the following sentence in Section 2.2 -- "This relation also implies that even though higher order radial modes appear to be “larger”, their beam waists are the same." "3. The curves in Fig. 4 are not Lorentzian but suggest the presence of a sidemode. Do the authors understand the origin of the observed side peaks? Is this worth mentioning?" Our reply: We suspect indeed that the small sidemodes near the Lorentzian LG peaks are other transverse modes of similar near-degenerate frequencies, which get coupled to some degree due to the imperfection of the cavity cylindrical symmetry -- the sidemodes become more prominent as the cavity either gets more misaligned in the transverse direction, or moves closer to the critical point. However, we don't have corroborating evidence for it, and prefer not to further comment. Comments from the second reviewer: "1. Similar work has been demonstrated by using DMD and other types of SLMs. It is unclear to me that how the transmission coefficient is calculated. I suspect that the input power refers to the beam power after the beam is modulated by the SLM, but perhaps the real benefit of using phase-only SLM as compared to DMD based devices is the efficiency as compared to the input beam power out of the fiber. Some clarification and comparison would be recommended." Our reply: We agree that clarifying losses and a reference to DMDs is in order. We added a paragraph in Section 3.1.1. to better explain the different contributions to the overall transmission, and to highlight the distinction with the DMD-based devices. "2. In Fig. 1, it is illustrated that higher order LG modes would result in smaller peak transmission in resonance. Can the authors explain what causes the reduction in peak transmission, especially in theory?" Our reply: The plot in Fig. 1 is an example of a cavity transmission spectrum obtained with a Gaussian input mode from a single mode fiber output, before performing any optimization or mode matching with SLM. The purpose is to define the variable $\delta v_{tr}$ (frequency spacing of the transverse modes), and to illustrate the transmission profiles of the transverse modes and the camera-observed spatial profile. "3. The authors states “In addition, all the radial transverse modes (LG modes with l = 0) at a particular critical distance d have identical effective mode volumes”. It is unclear to me how the math comes to such conclusion. Can the authors explain a bit more on this issue?" Our reply: We modify the explanation following this statement to make the mathematical argument clearer. In summary, we relate the normalization condition to the definition of the mode volume, to show that the mode volumes are indeed the same for all radial tranverse modes. We have also provided a physical argument to why this is the case, following the suggestion from the first reviewer. "4. The definition of mode overlap after Eq.(9) needs more clarification as the mode profile U1 and U2 are not defined here. Since the LG modes are orthogonal to each other, why the mode matching efficiency is high between an input Gaussian beam and a high-order LG mode?" Our reply: We have clarified the definition of the mode profiles U1 and U2 in the paragraph after Eq. (9). In Table 1, the mode matching efficiency is calculated between the SLM-modulated output of an input Gaussian beam and a high-order LG mode. By using the SLM modulation technique described in Section 2.4, we could create an SLM output mode that matches the high-order LG mode very well. "5. Some grammatical improvement is needed in Paragraph 2, section 3.1.2: “The detuning from the cavity resonance is expressed is corresponding units of light frequency”." Our reply: We agree and have modified the sentence to read as follows: “The detuning from the cavity resonance is expressed in corresponding units of light frequency”. "6. When the SLM is set to impose a super-position of multiple LG modes, what is the overall modulation efficiency as compared to the input power?" Our Reply: The overall modulation efficiency of the SLM, which we understand to be the overall diffraction efficiency of the SLM, is roughly constant (~60%) over any LG modes, including any super-position of multiple LG modes. This is because the SLM phase transformation imposed by Eq. (9) has only a minimal effect on the power output, but a large effect on the phase output which allows for high mode matching efficiency. We suspect that the main contribution to the SLM diffraction losses are due to the SLM design, i.e. fill factors, LCD absorption, etc, which can be improved with better SLM design. Along with the first point, we have included another paragraph in Section 3.1.1. to explain this observation. "7. Fig. 5 shows that the mode matching efficiency varies as a function of phi, the relative phase between LG00 and LG10 modes. Can the authors explain a bit more here as I would assume the efficiency should remain constant for a rotationally-symmetric system? I am also not following the statement “in which case the in-phase component of the beam encodes the LG00 mode, while the quadrature component of the beam encodes the LG10 mode.” What is the in-phase and quadrature components of the beam? And which beam are we talking about here?" Our reply: We thank the reviewer for pointing this out - we accidentally overload the variable phi to mean both the azimuth angle of the coordinate system and the relative phase between the LG00 and LG10 modes. In the revised manuscript, we have changed the relative phase variable to xi and clarify the statement regarding the in-phase and quadrature component of the beams. In summary, the beam refers to the SLM output, and the in-phase (quadrature) component here refers to the zero (or pi/2) phase shift of the LG modes with respect to the SLM output. "8. Fig. 7 reveals the impact of limited aperture of the mirrors to higher order LG modes. Is this limitation fundamental or related to particular mirrors used in the experiment?" Our reply: This is a very interesting question. For our implementation, we supect that this limitation is mainly due to the mirror imperfections, i.e. sphericity, mechanical stresses, and alignment of the mirrors. On the other hand, we estimated an effective aperture of ~1.4 mm, which corresponds to ~15 degrees of divergence angle. At this angle, it is not very straightforward to conclude whether the paraxial approximation has broken down for higher-order LG modes, and may require mirrors with different shapes. It is certainly interesting to pursue this question in a future research. Comments from the third reviewer: "How specific are these results to the specific mirror shape used by the authors, which integrates "anaclastic" lenses? Can we expect similar higher-order coupling efficiencies with a more conventional design? I would suggest to address this question more explicitely at a suitable place in the paper." Our reply: The main purpose of the "anaclastic" lenses is to match the strongly diverging mode of the near-concentric cavity to nearly-collimated mode with a single (ellipsoidal) surface, which reduces the number of optical components and surfaces in the vacuum system we intend to use them in. A more conventional design, comprising of a group of several spherical surfaces to convert the curved wavefronts would probably have similar higher-order couploing efficiencies to higher order modes, as it can be as close to an ideal lens action as desired. If at all, such a multi-lens arrangement even may simplify alignment, as the elliptical surface we use is quite sensitive to off-axis incidence of the collimated beam. However, we don't have any experimental evidence for this, and also think that the type of collimation surfaces are not essential to the mode matching to higher order cavity modes, and thus would prefer not to make explicit comments in this direction in the paper. "When mentioning small mode volumes in the near-concentric region and their potential for strong interaction between light and atoms [refs. 21-23], I would suggest to add a reference to work with near-concentric cavities in the Schleier-Smith group, such as Davis et al, PRL 122, 010405 (2019)." Our reply: We thank the reviewer for the suggestion and include the aforementioned reference in the paper. "p. 10, "which on the same order as the mirror transmission and scattering losses": verb missing." Our reply: We have modified the phrase to read as follows: “which is on the same order as the mirror transmission and scattering losses”. List of other changes: 1. We clarified the formulation of the cavity finesse in Section 2.3 2. Throughout the text, we replaced the index for the radial modes from formerly m to now p, because the index combination {m,l} is too close to the angular momentum indices commonly used to be confusing. So the LG mode indices are now {p,l} for the radial and azimuthal aspect. 3. We flip the ordering of the prefactor $A_{l,m}$ to $A_{p,l}$ in equation (1) to be consistent with the indices ordering of other variables and other relevant locations. 3. We have changed the variable on the x-axis label on Figure 5 (left) from phi to xi. With this, we hope to have addressed all point indicated by the referees. For convenience, we added a differential pdf version highlighting the changes of the revised manuscript compared to the original manuscript, and look forward to your reply. With Best Regards on behalf of all authors, Christian Kurtsiefer