Adiabatic topological photonic interfaces

Topological phases of matter have been attracting significant attention across diverse fields, from inherently quantum systems to classical photonic and acoustic metamaterials. In photonics, topological phases offer resilience and bring novel opportunities to control light with pseudo-spins. However, topological photonic systems can suffer from limitations, such as breakdown of topological properties due to their symmetry-protected origin and radiative leakage. Here we introduce adiabatic topological photonic interfaces, which help to overcome these issues. We predict and experimentally confirm that topological metasurfaces with slowly varying synthetic gauge fields significantly improve the guiding features of spin-Hall and valley-Hall topological structures commonly used in the design of topological photonic devices. Adiabatic variation in the domain wall profiles leads to the delocalization of topological boundary modes, making them less sensitive to details of the lattice, perceiving the structure as an effectively homogeneous Dirac metasurface. As a result, the modes showcase improved bandgap crossing, longer radiative lifetimes and propagation distances.

However, the authors did not mention the advances in the introduction at all. The authors need to do systematical investigation, introduce some important developments in this field, and discuss the differences and relations between their results and the existing findings.
#2 The authors did some case studies to show how the adiabatic topological interface is superior to the step and parabolic topological interfaces. However, as the width of linear topological interface changes from 0 to infinity, the topological interface also changes from abrupt to adiabatic types. To better understand the crossover mechanics, the authors should show how the width of linear interface affect the behaviors of topological boundary states.
#3 Related to the second comment, I do not believe the more adiabatic connection between two different topological structures, the linear topological interface has better behaviors in most circumstances. There may be an optimal width of linear topological interface that the topological boundary states have features of longer lifetime, propagation distances, and better topological protection again disorder. Consider an extreme case where the width of linear interface is infinite, the spatial width of topological boundary state is also infinite. I believe that any tiny disorder could break the topological boundary states. However, the step and parabolic interfaces are still immune to disorder to some extent. The authors should objectively evaluate the pros and cons of the adiabatic topological interface.
#4 The authors mainly discuss the topological interfaces in symmetry-protected topological systems. It is very interesting to consider the topological interfaces in photonic analogues of quantum-Hall and Chern insulators which are more robust. Could the authors give some intuitive discussion and comment on the advantages of adiabatic topological interface in these systems? #5 The expression $\Delta f/f \approx (0.4 − 0.6) \times 10^{-4}$ in the line 94 is ambiguous.
Reviewer #3 (Remarks to the Author): In their manuscript "Adiabatic topological photonic interfaces", the authors introduce the concept of adiabatic topological photonic interfaces. Their aim it to demonstrate theoretically and confirm experimentally that adiabatic topological metasurfaces with slowly varying synthetic gauge fields significantly improve the guiding features of spin-Hall and valley-Hall topological photonic structures, which are commonly used in the design of symmetry-protected topological devices. This is a very interesting manuscript; it will enrich the field as it has a very good story and very good experiments. I recommend this work for publication in Nature Communications.
For the sake of clarity, there are a few points the authors may answer/optimize before publication: -Why is the SNR seemingly worse for these new types of interfaces (se Fig. 2b linear) -How much light is left in the states? -The authors may add a numerical analysis about how much more robust the states at the novel interfaces are against symmetry breaking (as this was part of the motivation) -Figs. 4g,h are hard to decipher… -Why are these interfaces called "adiabatic" and which role does it play?
-Why does this idea do not work for electronic TI and, more generally, for all SPT phases?
In conclusion, while I think the paper is already fine for publication, addressing the points above may serve for even more readability.

Authors' Response to Reviewers
Reviewer #1 (Remarks to the Author): I enjoyed reading this paper and the results achieved here, and would like to recommend the publication of this manuscript because it does provide a sound solution to the practical problems in topological photonics. As shown by the convincing experimental results here, the quality factor and the propagation length are significantly improved by introducing linearly modulated domain walls. However, there is still a question I would like to raise and wish to see the authors give some answer in this direction. The question is: whether the approach raised here is general (with some theoretical proof from possibly synthetic dimensions or gauge fields) or case by case?

Authors' response to Reviewer #1:
We thank Reviewer #1 for emphasizing importance of our results in demonstrating practical solutions of existing problems in topological photonic structures. We truly appreciate their recommendation to publish our work in Nature Communications.
As for their question about whether the approach used by us is general, the answer is yes. The proof is as follows. The localization of the modes to the topological boundary is defined by the gauge field profile in the metasurfaces. Thus, for the mass term of a generic profile ( ), where is the direction perpendicular to the domain wall, the edge mode localization to the wall ( ( = 0) = 0) is given by the  Fig. R1 below) and experiment also confirm this prediction that the field becomes less localized for smoother profiles, with the linear case yielding the least localized boundary mode among the tested profiles. Importantly, responding to the Referees' question, this simple analytical form of the mode profile is generic and applies to any topological system, e.g., valley, spin-Hall, or QHE-like systems, and therefore the mass term modulation does in general allow to control properties of the boundary modes, such as their localization to the topological boundary.
In the context of the Wu and Hu structure specifically, the observed increase of the quality factor is the consequence of fact that the modes become less localized to the interface for smoother mass term profiles. Indeed, it is known that radiative lifetime of topological edge modes increases for less localized modes, which has been demonstrated for the step-like profiles with smaller mass term by Hafezi's group in Refs.  (2023) https://doi.org/10.48550/arXiv.2206.07056]. In our systems, however, this is achieved by different means the mass-term profile modulation, but the result is the same, although with the benefit of larger topological band gap.
The mechanism for increased radiative lifetime (and quality factor) by itself is explained as a consequence of cancellation of the far fields radiated by two (trivial and topological) domains. Thus, because the respective fields are out-of-phase, this leads to better cancellation of the far-field radiation of the boundary mode when the radiating mode is less localized to the domain wall. The mode that is more spread out spatially into the bulk gives rise to more directional far field radiation. Therefore, for smoother mass term profiles, the far fields from the two domains are better aligned with one another, which, considering their out-of-phase character and equal amplitudes, gives rise to stronger cancellation and overall suppression of the radiative leakage into far field. Interestingly, this out-of-phase character of the far-fields of topological and trivial domains always manifests itself in experiments as the dark (nodal) line in the images of the edge modes at the center of the domain wall (also seen in Refs. [S. Arora, et al. Phys. Rev. Lett. 128, 203903 (2022), andS. Peng, et al. Phys. Rev. Lett. 122, 117401 (2019)]). Figure R1. Simulation results for far field distributions of electrical field Ex of the edge mode for domain walls with different profiles. Field distribution is plotted in the xy-plane 1.5 μm above the metasurface.
We have reflected the above discussion to the revised version of the manuscript (in the main text and in Supplementary Note 1) in response to the Reviewer #1 comment.

Reviewer #2 (general comment):
The authors theoretically and experimentally studied three different interfaces (step, parabolic, and linear types) in spin-Hall and valley-Hall topological photonic structures which are protected by geometrical symmetry. Because of fragility of symmetry-protected topology and the radiative and bosonic natures of photonic systems, topological boundary states at the abrupt (step) interface are less robust and may be partially broken. However, the authors found that adiabatic (linear) interfaces are superior to the abrupt and parabolic interfaces in the aspects of less sensitivity to details of the lattice, narrower linewidth, longer radiative lifetime and propagation distance. The adiabatic interface may be used to improve photonic devices, such as topological lasers and topological polaritons.
Considering the unique properties of photonics, to the best of my knowledge it is the first time that the authors compare different topological interfaces in topological photonics and clearly point out the advantages of adiabatic topological interface. In general, the authors have provided some interesting, fresh, novel and concrete results, and the manuscript is well-written and easy to follow. I believe that this manuscript could potentially benefit the practical applications of topological photonics and trigger more theoretical and experimental studies. However, at this stage, I could not convincingly recommend the acceptance of this manuscript in the current form, for the following reasons.

Authors' response to Reviewer #2 general comment
We would like to thank Reviewer #2 for carefully reading our manuscript and for highlighting novelty of our work and the advantages of the approach used for topological photonics. We also thank Reviewer #2 for pointing out ways to further improve our work by addressing their specific remarks raised, which all were addressed in the revised manuscript (detailed in our response below).

Reviewer #2 specific remark #1:
Topological boundary states, as key features of topological state of matter, have attracted broad research interest. In the community of condensed matter physics, many studies have considered and compared different type of topological interfaces including abrupt and smooth types; see some references [J. Liu, et al., Phys. Rev. B 88, 241303(R) (2013) However, the authors did not mention the advances in the introduction at all. The authors need to do systematical investigation, introduce some important developments in this field, and discuss the differences and relations between their results and the existing findings.

Authors' response to Reviewer #2 remark #1
We would like to thank Reviewer #2 for pointing out all these important developments in condensed matter physics, which escaped our attention, but are indeed directly relevant to our own work in photonics. Specifically, the choice of boundary, which is truly vital for crystalline topological phases, was investigated in Phys. Rev. B 88, 241303(R). Similarly, interfacing magnetic and topological insulators was studied in Nano Lett. 18, 6521-6529 (2018), and an advantage of smooth connections between magnetic insulators and topological insulators over sharp interfaces was discovered as a trivial boundary state appeared to be suppressed. While different from what we report in our work, the idea of interface engineering for improved performance of topological systems is strikingly similar to our own. Even more relevant is the work [Materials 13, 4481(2020)], where a smooth interface between topological insulator and topological crystalline insulator is created by epitaxial immersion growth, which leads to improved topological boundary modes. Finally, another very interesting discovery reported in [Phys. Rev. B 107,075129] is that smooth heterojunctions of trivial and topological insulators lead to emergence of coexistence of massless Weyl and massive Dirac fermions, which further evidences importance of the topological profile engineering in both condensed matter and photonics.
Following Reviewer #2 suggestion, the respective advances brought by the topological interface engineering in condensed matter physics were mentioned in our revised manuscript and properly cited.

Reviewer #2 specific remark #2:
The authors did some case studies to show how the adiabatic topological interface is superior to the step and parabolic topological interfaces. However, as the width of linear topological interface changes from 0 to infinity, the topological interface also changes from abrupt to adiabatic types. To better understand the crossover mechanics, the authors should show how the width of linear interface affect the behaviors of topological boundary states.

Authors' response to Reviewer #2 remark #2
This is an interesting point which indeed deserves an additional elaboration in the manuscript. There are two main consequences of the transition from abrupt to adiabatic interface. Firstly, the boundary modes become progressively less localized. Thus, for the specific case of linear profile, with the mass term = , the Dirac equation admits an exact edge mode solution, which shows that the modes are not exponentially localized to the interface but exhibit the Gaussian profile ( , )~exp {− 1 2 2 }. Thus, for the case of infinitely wide linear region, → ∞, the modes too become completely delocalized, i.e., effectively become one of the bulk modes. Secondly, some of the bulk modes also start to localize to the linear transition region, and, as its width approaches infinity, there is an infinite number of such modes which continuously populate the former gap region. Thus, the physics in the limit → ∞ becomes indistinguishable from the case of massless = 0 and gapless Dirac equation. This behavior was confirmed in our numerical modelling by directly solving Maxwell's equations with the use of the finite element method, as shown in Fig. R2.
To summarize, when optimizing the smoothness of the mass-term profile, one should consider tradeoffs and aim for the scenario when the edge modes are still well-defined, i.e., exist within a reasonably wide topological bandgap, but are still sufficiently localized to the interface as required by a particular application.
The above discussion and results investigating dependence on the width of the linear profile were added to the revised Supplementary Information as a new Note 3. Fig. R2. Photonic band structure of topological boundary modes for linear mass-term profile with gradually increasing width of transition (a-d) from 0 unit cells up to 37 unit cells. Color-coded diagrams of the degree of perturbation and edge mode near field distribution for each interface are shown on the top panels. Radiative quality factor of the edge modes and guided bulk modes are shown in color in these band diagrams.

Reviewer #2 specific remark #3:
Related to the second comment, I do not believe the more adiabatic connection between two different topological structures, the linear topological interface has better behaviors in most circumstances. There may be an optimal width of linear topological interface that the topological boundary states have features of longer lifetime, propagation distances, and better topological protection again disorder. Consider an extreme case where the width of linear interface is infinite, the spatial width of topological boundary state is also infinite. I believe that any tiny disorder could break the topological boundary states. However, the step and parabolic interfaces are still immune to disorder to some extent. The authors should objectively evaluate the pros and cons of the adiabatic topological interface.