Artificial gauge fields for sculpting topological modes on photonic chips

Domain: biology
Published: 2026-04-04
Editorial outcome: minor_revision

Abstract
Topological states, which often manifest as localized modes at interfaces between distinct topological phases, and their manipulation have attracted considerable interest in research. Here, we demonstrate a versatile approach to sculpting topological modes into desired shapes by incorporating various artificial gauge fields —including scalar, vector, and imaginary gauge potentials—and leveraging the power of artificial neural networks. The chose gauge fields enable precise tuning of the dissipation of the topological modes across that of bulk modes, facilitating a transition from localized states to fully delocalized ones. These eigen modes can be precisely engineered by neural networks, achieving tailored profiles of topological states, which remain spectrally isolated from bulk bands and exhibit minimal loss compared to other modes. Our theoretical results are experimentally validated on silicon photonic platforms, demonstrating flexible manipulation of mode profiles. This approach enables the design of topological states with customized properties, offering potential for diverse applications in photonics and beyond.

Introduction
Topological physics has profoundly influenced fields ranging from quantum transport1-3 to classical wave manipulations4-9. A key driver of the advancements has been the exploration of novel topological phases and states beyond the existing framework of topological theory4,10-13, enabled by the tunable parameters of artificial materials7,14-17. In topological photonics, for instance, advanced micro-nanofabrication techniques have facilitated tailored geometric designs, unlocking many intriguing physical phenomena, including nonlinear topological solitons18-22, non-Abelian physics5,23-27, and non-Hermitian effects28-30. Notably, specific geometric modulations can introduce effective fields, known as artificial gauge fields (AGFs), to neutral particles such as photons. These AGFs govern their dynamics, mimicking the behavior of charged particles under real fields31-37. By combining AGFs with Floquet engineering, non-Hermiticity, and topological principles, researchers have uncovered novel nonequilibrium topological phases30,38-45, including topological triple phase transition38, Floquet topological insulator41, and synthetic-dimension physics42. Moreover, the versatile tunability of AGFs has driven emerging applications, such as broadband optical coupler46,47, topological insulator lasers48, and advanced devices in integrated photonics49-52.

Despite great efforts to explore new topological states across various platforms, a common characteristic is their exponential localization at interfaces between topologically distinct systems—a hallmark of the bulk-boundary correspondence4,10,53-56, protected by non-trivial band structures. However, this localization restricts the use of bulk space, with edge states only occupying a small fraction of the overall space of the topological system. Overcoming the challenge of arbitrarily shaping the mode profiles of topological states without compromising their intrinsic topological properties thus becomes critical57-59. Envision TMs that transcend the localized regions, tailored to any desired shape and spanning the entire available space. Achieving such delocalized TMs is nontrivial, as it confronts the typical paradigm that in-gap states are necessarily localized—an intuitive expectation supported by conventional wisdom in topological band theory4. Such a capability would not only provide deep insights into fundamental topological physics but also unlock considerable promise for advancing topological applications across diverse technological domains.

This work seeks to address these limitations by incorporating various types of AGFs—including scalar, vector, and imaginary gauge potentials—to achieve fully delocalized TMs with arbitrary tailored mode profiles, assisted by artificial neural networks (ANNs) (Fig. 1). These gauge fields can be precisely designed to control the energy landscape and the quasienergy band structures of the topological states. Specifically, the dissipation of the TMs can be tuned across that of bulk modes under periodic boundary conditions, facilitating transitions from edge-localized to bulk-delocalized configurations. Furthermore, pre-trained neural networks can predict the optimal coupling configurations to generate TMs with desired shapes while preserving their spectral isolation within the bandgap, ensuring robust topological protections. We experimentally validated this theoretical framework on silicon photonic chips, demonstrating flexible manipulation of topological states and successful eigenmode deformation into diverse target shapes. This approach establishes a powerful paradigm for topologic al modes manipulation through the combined use of diverse gauge potentials and the ANN-driven design, paving the way for advanced applications on topological photonic chips.

Results
Photonic waveguide lattice with AGFs

We begin by considering a one-dimensional (1D) photonic waveguide lattice characterized by nearest-neighbor couplings, denoted as κ , between the m-th and n-th waveguides. The lattice is subjected to external fields, including real scalar (Φ ) and vector (A) potentials, as well as imaginary potentials (γ ) at each site m. These potentials are implemented by periodically adjusting the waveguide widths, bending the waveguides, and introducing onsite loss elements, respectively46,52, as depicted in Fig. 1. These external gauge fields introduce a phase factor to the coupling κ after the gauge transformation, i.e.,
mn
Am Gn F
m n
e x p ( i
m n
i
m n
)    = A  r +  , where
m n
z
0 m
( z )
n
( z ) d z   = −     −     is the local scalar potential difference and r is the position vector from waveguide-m to -n, with 𝑚 = 𝑛±1. Here the potentials are designed to take sinusoidal forms, A(z) =－k A Ωsin(Ωz)i and Φ (z) = (－1)nk Asin(Ωz)/2n , where k = 2πn /λ is the wavenumber in the ambient medium, Ω is the modulation frequency, and A (A) is the modulation amplitude of vector (scalar) potential. A detailed analysis of the gauge transformation and the derivation of artificial scalar and vector gauge potentials are provided in Supplementary Note 1.

For simplicity, we focus on incorporating the previously discussed artificial gauge fields into the Su-Schrieffer-Heeger (SSH) model. This model features unit cells, each containing two sublattices, X and Y, with alternating coupling strength, κ and κ , as illustrated in Fig. 1. Based on the analyses above, the artificial gauge field introduces a phase factor to κ and κ ,
1 2

A G
1(2
F)
1(2 )
e x p ( i
( )
)    = 
− + 
, where
( z ) G
0
d s in ( z )
0
c o s ( z )  

= 

   , d is the center-to-center distance between neighboring waveguides, and ±
G = k A Ω and φ = k A/Ω are the vector and scalar potential constants. Additionally, the imaginary potentials are implemented in a balanced gain/loss manner, namely, ±γ for two sublattices. The bulk Hamiltonian can then be succinctly expressed as
H ( k , z )
*
i
( k , z )
( k
i
, z )  
 
= 

−

, (1)
where =ei− +e−i+e−ik. Since this Hamiltonian is periodic, the topological nature of the system can be well described by Floquet theory, which dictates the quasienergy and Floquet states in our non-Hermitian systems60.

Topological mode control by AGFs

In the following analysis, we fix the imaginary potentials to γ = κ . For G = φ = 0, the system exhibits parity-time (PT) symmetry, which results in TMs localized at the boundary for κ < κ . When G ≠ 0 and φ ≠ 0, the introduction of AGFs drastically influences the distribution of the TMs. Specifically, transitions of TMs from localized to delocalized states can be observed, which is characterized by the modified inverse participation ratios (MIPR)
,
(
T M
4
T M
4 ) (
T M
2 ) 2    M = 
i T
L
j T
R
i − j 
k  T
k , where
T M
 is the topological mode, T L (T R ) denotes the left (right) half of the lattice, and T =T L UT R . Figure 2a illustrates the phase diagram plotted as functions of scalar and vector potentials, which is separated into two regions by the critical line (black curve, M = 0), where M > 0 and M < 0 regions indicate the TMs localized at the left and right boundary, while the critical line denotes that the TMs are fully delocalized.

Specifically, we investigate the evolution of TMs along the parameter path l indicated by the dashed line in Fig. 2a, starting from the origin. When G = φ = 0 (pink circle in Fig. 2a), corresponding to the absence of gauge field modulation, the TM lies in the M > 0 region and is localized at the left boundary (pink profile in Fig. 2b). As the parameters deviate from the origin but remain within M > 0 (e.g., at the pentagram or hexagon in Fig. 2a), the TM begins to extend slightly from the left edge (blue and purple profiles in Fig. 2b). Remarkably, when the parameters reach the critical boundary (upper and inverted triangles in Fig. 2a), the TMs become fully delocalized (green and brown profiles in Fig. 2bS). Upon entering the M < 0 region (the orange circle in Fig. 2a), the TM shifts its localization to the right boundary (orange profiles in Fig. 2b).

Mechanism of topological mode reshaping

The transition mechanism can be understood through the topological properties of the complex Floquet quasienergy spectrum (Fig. 2c,d). Notably, the quasienergy spectra under periodic boundary conditions (PBC) and open boundary conditions (OBC) differ, signaling the emergence of NHSEs9,61-65. Here, the interaction between the NHSEs and TMs gives rise to a hybrid skin-topological effect8,62, which modifies the mode profiles of the topological states. Specifically, along path l, the TM approaches the boundary of the PBC spectrum, traverses into its interior, and subsequently exits it (Fig. 2d). When the TM aligns with the PBC energy spectrum, an equilibrium forms between the NHSE and the opposing localization tendency of the TM, thereby leading to a fully delocalized mode. Moreover, once this balanced interaction is established, merely tuning the coupling coefficients can break the balance in a controlled manner, thereby generating various complex spatial profiles.

Our design demonstrates flexible generation and manipulation of extended TMs based on AGFs induced NHSEs, in contrast to previous approaches that the realization of NHSEs usually relies on nonreciprocal couplings8,9,61-65, which are difficult to implement in on-chip optical systems. In Supplementary Note 2, we rigorously prove the emergence of the NHSE via AGF from a symmetry perspective. Our analysis reveals that cooperative interactions among the scalar, vector, and imaginary potentials are essential for enabling the transformation of TMs. If either G = 0 or φ = 0, symmetry constraints inhibit the NHSE, thereby preventing the existence of extended TMs. Notably, in the region where M < 0, sufficiently large vector gauge potentials can drive the system into a pure skin mode regime (enclosed by the white curve), in which TMs disappear and merge into conventional skin modes. In Fig. 2, we mainly focus on the behavior along parameter path l, which clearly illustrates the transformation of topological mode profiles from localized to delocalized states. A more detailed analysis of the transition process across the skin mode regions—including comprehensive quasienergy spectra and physical interpretations—is provided in Supplementary Note 3.

Neural networks-assisted arbitrary topological sculpting

The phase diagram shown in Fig. 2a demonstrateIs that by jointly modulating the scalar and vector gauge potentials, a wide variety of phase transitions can be realized, including localized TMs, delocalized TMs, and skin modes. A fully extended profile arises when the interaction between the NHSE and TMs reaches an equilibrium (Fig. 2b). By suitably tuning the coupling coefficients, we can break this balance to reshape the mode into various complex profiles. This local balance–imbalance interaction can be regarded as the mechanism enabling arbitrary topological mode shaping. However, realizing a specific target shape is a challenging inverse problem, as numerous parameters (such as couplings) jointly determine the resulting profile. Here, the artificial neural network66-69 (ANN) serves as a practical solver to efficiently navigate this complex parameter space and identify configurations that implement the desired shape, such as sinusoidal (Type I) and convex (Type II) forms, illustrated in Fig. 3a. This approach involves three key steps: first, generating a large dataset (18,000 sets in this work) by calculating the topological modes of the Floquet Hamiltonian for various coupling coefficients, while keeping all other parameters the same as those of the initial uniform delocalized TMs. Second, the ANNs are trained on this dataset, mapping normalized inputs to outputs by optimizing the hidden-layer weights via backpropagation algorithm (Fig. 3b). Finally, the desired TMs are provided as inputs to the trained ANNs to predict the required coupling coefficients, which are then used to construct the predicted TM—evaluated using mean square error (MSE)—and these coefficients correspond to the physical waveguide parameters needed to realize the target structure (Fig. 3c). More details regarding the training process and the prediction results of the neural networks can be found in Supplementary Note 4.

Manipulation of on-chip NHSEs by gauge potentials

The configurations can be conveniently implemented on an integrated photonic platform, where scalar and vector potentials are achieved by varying the width and bending trajectory of the waveguides, respectively. In experiments, we first verified that the interaction of different types of gauge fields can lead to NHSEs in silicon photonics. The optical lattice sample comprised of N = 13 Si waveguides on the sapphire substrate with air cladding. The waveguide height is h = 220 nm , and the waveguide spacing is g = 200 nm. The trajectory and width of the waveguide periodically modulated in a sinusoidal way, with a modulation period p = 25 μm, the bending amplitude A ≈ 1 μm, and the width w variation interval around 360 ~ 420 nm. The non-Hermiticity is introduced by coating a layer of chromium (Cr) with a width w = 200 nm and thickness h = 4 nm on top of every othTer Si waveguide, thus without introducing additional gain. The scanning electron microscope (SEM) pictures of experimentally fabricated samples are shown in Fig. 4a, where the waveguide structure and deposited Cr can be clearly observed.

To demonstrate the feasibility of this scheme, two samples with opposite scalar potentials (φ₀ = ±4.1) were designed, featuring reversed arrangements of neighboring waveguide widths (Fig. 4a). These two configurations correspond to distinct winding topologies, W = ±1 (see Supplementary Note 2), and are thus expected to display opposite evolution trends. Each type of sample was fabricated with varying lengths (L = p, 2p, 3p, and 4p) to capture various stages of mode evolution. In optical measurements, the center of the waveguide lattices is excited by a near-infrared laser with wavelength λ = 1550 nm via a grating coupler, and the output light scattered from the end of the waveguides was measured by a near-infrared camera through a microscope objective.

The experimental results and the simulated light evolutions (using the commercial finite-element software COMSOL Multiphysics) are presented in Fig. 4b-4e, where Fig. 4c and 4e are the extracted data for further verification. For the samples with winding number W =－1, light in the lattice gradually evolves towards the right end of the boundary, as shown in Fig. 4b and 4c. In contrast, when W = +1, light tends to evolve toward the opposite boundary (Fig. 4d and 4e). These results confirm our theoretical predictions and lay the groundwork for subsequent reshaping of TMs in waveguide lattices.

Observation of photonic topological delocalized modes

To obtain extended TMs predicted in Fig. 2, we fabricated waveguide samples (N = 16) with alternate spacing (g = 200 nm and 300 nm) to introduce the interaction of TM with NHSE. Such an arrangement supports a TM with minimal loss localized at the left boundary. Next, gauge fields were introduced to drive skin modes toward opposite boundary (similar to NH SE with W =－1 in Fig. 4b) to obtain a fully delocalized TM, where the bending amplitude is chosen as A ≈ 0.5 μm, the remaining parameters are consistent with Fig 4b.

The excitation of these delocalized states requires careful design compared to localized states. The fully delocalized TM, primarily distributed on Cr-free waveguides with uniform intensity but anti-phase between neighboring sites, necessitates a cascaded array for initial state preparation. Here, light coupled from an input grating passes through a Y-branch splitter and then through a waveguide incorporating subwavelength grating (SWG) microstructures to generate eight anti-phase outputs (Fig. 5a). This initial state can successfully excite a fully delocalized topological eigenmode in the array, preserving its shape during propagation. The simulation and experimental results of its output are in good agreement, as shown in Fig. 5c. However, when the topological delocalized state is not an eigenmode of the system, for instance, by setting the scalar potential to zero (φ = 0), the topological eigenmode with minimal loss becomes localized. Consequently, the initial state evolves towards the boundary occupied by the TM, as shown in Fig. 5b. These comparative results confirm the successful realization of fully delocalized topological eigenmodes in the waveguide lattice.

Observation of topological modes incorporating diverse shapes

With the assistance of ANNs, we can efficiently determine the suitable parameters for achieving the desired shapes of topological states, that is, input the desired topological modes to the trained ANNs to predict the required coupling coefficients (see Supplementary Note 4 for more details). In experiments, we fine-tuned the waveguide spacing (g , g , g , …, g ) to match the distribution of coupling coefficients obtained from the networks. We conducted a series of experiments to verify two types of extended topological states, the sinusoidal type (Type I) and the convex type (Type II) (Fig. 3). These states exhibit distinct spatial distributions and require precise initial conditions for proper excitation. Exciting these complex extended states requires not only anti-phase initial conditions but also a non-uniform intensity distribution across the waveguides. Achieving such initial conditions is challenging but feasible through an inverse design method (see Supplementary Note 5), which systematically optimizes input parameters to produce the desired output (Fig. 6a). Finally, experimental results for both Type I and Type II extended states were obtained and compared with simulation results (Fig. 6b and 6c). The results demonstrate good agreement between the experimental observations and the theoretical predictions, confirming the effectiveness of our approach.

Methods
Fabrication of the sample

The experimental samples were fabricated using the method of electron beam lithography (EBL) and the dry etching process, followed by a second-step E-beam lithography with the alignment and lift-off process to deposit the Cr stripes. The substrate used here is 460-μm alumina substrate with 220-nm silicon deposition. The substrates were cleaned in an ultrasound bath in acetone, isopropyl alcohol (IPA), and DI water for 15 min, respectively, and dried under a clean nitrogen flow. The alignment marks were then patterned by EBL (Elionix, ELS-F125), deposition of chromium/gold (5 nm/40 nm) layers, and photoresist liftoff. After that, the waveguide arrays and grating nano structures were exposed to EBL. The samples were then used to dry etch the silicon layer in a 2:5 mixture of SF6 and C4F8 plasma, and the residual photoresist was stripped off by an oxygen plasma stripper. Next, the chromium structures were fabricated by an alignment E-beam lithography and liftoff process. The photoresist film was spin-coated onto the substrate. After exposing the structure, an 8-nm chromium film is deposited using thermal evaporation, and then removing the photoresist by soaking in N-methyl-2-pyrrolidone.

Experimental measurements. A telecom-band laser (Fianium-1550-SF) was used to characterize the fabricated sample. The laser beam with its polarization being controlled by a linear polarizer (LP) was focused on the grating coupler from the substrate side by the Mitutoyo near-infrared (NIR) long-working distance object (100×, NA=0.70, f=200 mm). The transmitted signal was scattered from the end of the waveguides into free space, collected by a NIR imaging object (100×, NA=0.7, f=200 mm), and imaged on a near-infrared charge-coupled device (CCD) camera (XenicsXeva-1.7-320).

Data availability

The data that support the findings of this study are provided in the Supplementary Information/Source Data file. Source data are provided with this paper.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript by Z. Lin et al. presents a new approach for shaping topological modes in on-chip photonic systems using artificial gauge fields (AGFs) and artificial neural networks. They demonstrated the ability to transform topological modes from localized states to arbitrary shapes via AGFs, addressing a significant challenge in the field and opening new avenues for research in topological photonics and beyond. The strategy of jointly tuning multiple AGFs also holds strong potential for uncovering further novel phenomena. I think that work’s originality, theoretical depth, and experimental achievement stand out. I would recommend the manuscript for publication in Nature Communications after the authors address the following comments and questions.\n1. Please discuss the specific roles of vector, scalar, and imaginary potentials in topological mode shaping. For instance, is the imaginary potential primarily responsible for the NHSE, while the scalar and vector potentials are then used to \"steer\" the topological mode against it?\n2. During the topological mode reshaping, the topological state consistently exhibits the maximum positive imaginary part (the highest gain). Does this imply that even if the initial state excitation is highly arbitrary, it can still evolve into the topological state after a certain distance of evolution? This is important as it can reflect the robustness of the topological state. If this is right, is it still necessary to prepare the initial state in experiments?\n3. In Section S2 of the Supplementary Materials, the authors provide a detailed derivation of the symmetries. However, the transformation matrix U^(dagger) in Equations S23–S28 may be a typographical error and should be replaced with U^-1, as the instantaneous Hamiltonian H is non-Hermitian.\n4. In the main text, the vector/scalar potentials are designed to take sinusoidal forms as mathematical expressions, but this seems rather abstract. It would be very helpful to add schematic diagrams for the readers.\n5. The authors mention that “With assistance of ANNs, we can efficiently determine the suitable parameters for achieving the desired shapes of topological states.” It is not clear to me how ANNs help to achieve desired topological states. What happens if the authors use non-ANN controlled structures? Probably a baseline comparison is needed to show that the ANN is not merely fitting arbitrary coupling patterns. Moreover, for better understanding, the authors are suggested to provide the detailed parameters (waveguide spacing) corresponding to respective coupling coefficients in the main text.\n6. I recommend that the authors cite some relevant papers, including \"Machine learning inverse problem for topological photonics,\" Communications Physics (2018); \"Inverse design of photonic topological state via machine learning,\" APL (2019); \"Machine Prediction of Topological Transitions in Photonic Crystals,\" Phys. Rev. Applied (2020); \"Deep learning for the design of photonic structures,\" Nature Photonics (2021).", "reviewer_2": "In the manuscript “Sculpting topological modes on photonic chips by artificial gauge fields” the authors explore the possibility to design spatially extended topological modes in a waveguide lattice. A common understanding is that periodic Hermitian structures feature bulk bands and bandgaps. Thus, any mode arising in a finite system with open boundary conditions inside the gap is typically expected to have complex Bloch momentum and hence must be localized. Therefore, a naive expectation is that any in-gap mode is localized.\nHere, the authors challenge that naive picture by introducing two key ingredients: modulation and non-Hermiticity. As they prove, the combined action of these two factors for a periodic SSH lattice allows one not only to shift the edge mode from one edge to another, but also to make it delocalized in some instances. By generalizing their approach to non-periodic lattices, the authors show that the profile of the topological mode can be tailored at will, once a suitable non-periodic coupling pattern is chosen.\nOverall, I find the paper interesting and timely. The experimental part is quite impressive, as the authors fabricate curved waveguide lattices on a chip, control the losses at each of the waveguides and demonstrate selective excitation of the modes by exciting single or multiple sites at once with a well-defined phase between the inputs as well as different amplitudes. Even taken alone, such flexibility in the fabrication and excitation of the lattice is remarkable. In the context of this study, I have several comments and questions:\n1. I assume, it is known that periodically driving a lattice (e.g. Hermitian SSH model), one can induce the topological transition in the system accompanied by the shift of the topological edge state from one edge of the array to the other. This physics has been explored previously, see e.g. [S. Olin, W-C. Lee. Phys Rev B 107, 094310 (2023)] and other related works or [Vyas,Roy. Phys. Rev. B 103, 075441 (2021)] for periodically driven non-Hermitian model. There could be multiple other works in the same direction.\nI recommend the authors to discuss this aspect in more detail and articulate more clearly the novelty of their present contribution (aside from the experimental realization which is clearly new and advanced). In essence, they also study the topological transition in the Floquet spectrum induced by the proper amplitude and phase of the driving plus non-Hermitian contributions.\n2. Is it possible to sculpt arbitrary topological mode profiles only by the periodic driving (not necessarily harmonic), but without non-Hermiticity?\n3. The authors suggest that their approach can be extended towards higher-dimensional systems. However, that seems quite challenging using photonic chips, as fabricating 2D waveguide arrays is a challenge. Please, comment how this approach can be implemented experimentally for the 2D lattices.\n4. The authors realize synthetic gauge fields, which are vector and scalar potential, by modulating the waveguides with the z coordinate. I wonder if that approach could be generalized to engineer more exotic gauge fields, e.g. tensorial ones. In turn, such exotic gauge fields could potentially bring novel opportunities in light routing.", "reviewer_3": "After carefully evaluating the manuscript, I conclude that it does not meet the novelty and conceptual-advance criteria required for Nature Communications.\nMy main concerns are summarised below.\n1. A closely related work by Hu et al., Adv. Photon. 7, 066001 (2025), has already demonstrated arbitrary control of optical field profiles in an SOI platform at telecommunication wavelengths by employing spatially varying mass terms to generate pseudomagnetic fields. These capabilities, including engineered gauge-field landscapes, controllable deformation of optical mode distributions, programmable beam manipulation in photonic lattices, directly overlap with the central claims of the present manuscript. Although the present manuscript introduces non-Hermitian modulation, this added ingredient does not constitute a sufficiently distinct conceptual advance relative to the published work. The claimed progress remains largely incremental, and the non-Hermitian aspect alone does not provide a clear or substantive differentiation from the existing literature.\n2. The authors repeatedly emphasise “arbitrary mode shaping,” “programmable light-flow control,” “tailored gauge potentials,” “engineered spatial mode profiles,” “ANN-assisted inverse design.” However, these ideas are already well established in the prior literature on pseudomagnetic-field photonics, synthetic gauge-field design in photonic crystals, non-Hermitian beam shaping, inverse-designed photonic lattices. What is missing from the manuscript is a clear and rigorous physical picture that demonstrates the presence of a genuinely new physical mechanism, rather than yet another implementation of spatially varying complex couplings. As it stands, the phenomena presented here appear to arise from standard gauge-field modulation, giving the impression of a reformulation of known principles rather than the discovery of a fundamentally new physical concept.\n3. A central claim of the manuscript is the ability to reshape “topological modes.” However, the authors do not convincingly demonstrate that these modes retain their topological character while being arbitrarily reconfigured. The reshaping process appears to be driven primarily by spatially modulated non-Hermitian couplings, a technique that is already well established in the NHSE literature. As a result, the reported observations resemble general beam-shaping or mode engineering, rather than revealing a genuinely new form of topological physics.\n4. The use of ANN for inverse design is incremental and not physics-driven Inverse-design techniques (adjoint, genetic, reinforcement learning, neural networks) have been widely used for beam shaping, coupling control, arbitrary-mode generation in photonics. Here, the ANN merely outputs coupling profiles for a desired intensity distribution. This optimization strategy is engineering-based, not a conceptual physics advance. Therefore, it does not compensate for the lack of novelty in the physical mechanism.\nAlthough the manuscript is technically sound and the experiments are executed with impressive quality, it does not reach the level of conceptual novelty or physical insight expected for Nature Communications. The central claims substantially overlap with previously demonstrated results, and the authors do not convincingly articulate a new physical phenomenon, mechanism, or theoretical framework that clearly differentiates the present work from the existing literature. I therefore recommend rejection in its current form."}, "author_response": "Response letter to the Reviewers\nManuscript ID: NCOMMS-25-85282-T\nTitle: Sculpting Topological Modes on Photonic Chips by Artificial Gauge Fields\nCorresponding Author: Shuang Zhang, The University of Hong Kong\nDear reviewers,\nWe are very grateful for your insightful comments and the considerable time and effort you have\ndevoted to evaluating our work. In this response letter, we provide a detailed revision checklist and\npoint-by-point response to your comments. We greatly appreciate your ongoing commitment to\nreviewing our revised submission.\nSincerely yours,\nShuang Zhang\nChair Professor in Physics\nThe University of Hong Kong\nEmail: shuzhang@hku.hk\nReply to Reviewer #1\nComment: The manuscript by Z. Lin et al. presents a new approach for shaping topological modes\nin on-chip photonic systems using artificial gauge fields (AGFs) and artificial neural networks.\nThey demonstrated the ability to transform topological modes from localized states to arbitrary\nshapes via AGFs, addressing a significant challenge in the field and opening new avenues for\nresearch in topological photonics and beyond. The strategy of jointly tuning multiple AGFs also\nholds strong potential for uncovering further novel phenomena. I think that work’s originality,\ntheoretical depth, and experimental achievement stand out. I would recommend the manuscript for\npublication in Nature Communications after the authors address the following comments and\nquestions.\nReply: We appreciate the referee's positive assessment of our innovation in this work. We also value\nthe referee's insightful comments and suggestions, which have helped us to clarify some important\npoints. In response, we have carefully revised the manuscript, with all changes clearly labeled in the\nupdated version. We hope these revisions could meet the referees' expectations.\nComment: 1. Please discuss the specific roles of vector, scalar, and imaginary potentials in\ntopological mode shaping. For instance, is the imaginary potential primarily responsible for the\nNHSE, while the scalar and vector potentials are then used to \"steer\" the topological mode against\nit?\nReply: We thank the referee for this insightful suggestion. In our system, the artificial gauge fields\n(AGFs) - imaginary, scalar, and vector potentials - act synergistically rather than independently.\nWhile the imaginary potential introduces on-site loss and provides the primary source of non-\nHermiticity, the NHSE itself emerges only when this non-Hermiticity interacts with the phase\nstructure imposed by the scalar and vector potentials. Without either of the latter there would be no\nNHSE (Fig. R1).\nMore precisely, the vector and scalar potentials impart complex phases to the couplings, which\ninteract with the onsite non-Hermiticity to engineer the system's symmetries, thereby shaping the\ncomplex band structure and the topological mode profile. Consequently, all three contributions are\nessential for both the NHSE and the topological mode reshaping (see Supplementary Section S2).\nFig. R1 Without the a, vector or b, scalar potentials, there is no NHSE and all the eigenstates are\nextended. c, The NHSE exists when all AGFs work synergistically.\nRevision:\nMain text: (Page 6, lines 145-147) In Supplementary Note 2, we rigorously prove the emergence\nof the NHSE via AGF from a symmetry perspective. Our analysis reveals that cooperative\ninteractions among the scalar, vector, and imaginary potentials are essential for enabling the\ntransformation of TMs.\nComment: 2. During the topological mode reshaping, the topological state consistently exhibits the\nmaximum positive imaginary part (the highest gain). Does this imply that even if the initial state\nexcitation is highly arbitrary, it can still evolve into the topological state after a certain distance of\nevolution? This is important as it can reflect the robustness of the topological state. If this is right,\nis it still necessary to prepare the initial state in experiments?\nReply: We thank the referee for this important question. Yes, the referee's interpretation is correct.\nDue to its persistent highest gain (lowest loss), the topological mode is dynamically selected from\narbitrary initial states over long evolution.\nTo demonstrate this robustness explicitly, we performed numerical simulations of dynamical\nevolution using several distinct inputs under the condition of the fully delocalized topological mode\n(Fig. R2a). After sufficiently long propagation distance, the output converges to the characteristic\nprofile of the topological mode. The same behavior is observed for the other two topological\nconfigurations: the final states converge to sinusoidal for “Type I” mode and to the convex profiles\nfor the “Type II” mode, respectively (Fig. R2b).\nWe further note that the required propagation distance for convergence decreases as the initial\nstate more closely resembles the target topological mode. This dependence highlights a practical\ntrade-off. Therefore, while the topological mode is intrinsically robust and can in principle be\nrecovered from arbitrary inputs given sufficient evolution length, it remains highly advantageous,\nand in many cases necessary, to prepare an initial state that approximates the target mode in\nexperiments. Such preparation significantly shortens the required propagation distance, improves\nthe signal-to-noise ratio, enhances experimental feasibility, and enables more compact on-chip\ndesigns.\nRevision:\nSupplementary Material S5: (Pages 14-15) These complex topologically extended modes have\ndifferent intensity distributions at each site, unlike conventional topological modes that are localized\nat the boundary, which makes the initial excitation become difficult ... , namely that the final states\npossess sinusoidal and convex profiles for “Type I” and “Type II” modes, respectively (Fig. S5b).\nFig. R2 Excitations of the topological mode by different inputs. a, Excitations of the delocalized\ntopological mode by increasing the number of waveguide inputs. b, Excitations of different\ntopological modes, (i) the delocalized topological mode, (ii) “Type I” with sinusoidal profile, and\n3\n(iii) “Type II” with convex profile, with the same arbitrary-input.\nComment: 3. In Section S2 of the Supplementary Materials, the authors provide a detailed\nderivation of the symmetries. However, the transformation matrix U^(dagger) in Equations S23–\nS28 may be a typographical error and should be replaced with U^-1, as the instantaneous\nHamiltonian H is non-Hermitian.\nReply: We thank the referee for this correction. Yes, the transformation matrix U is non-unitary,\nthus U^(dagger) should be replaced with U^-1.\nRevision:\nSupplementary Material S2: (Pages 7-8) We have modified the corresponding equations (Eq.\nS23–S28)\nU-1 UT (k,P)U =-U+-1 T-exp- æ ç iò 0 H æ ç k, z Pö ÷dz ö ÷U\nP/2,P F P/2,P P/2,P è P è 2ø ø P/2,P\n=-U-1 T ex-p ( iò P/2 H( k,z)dz ) U , (S23)\nP/2,P P/2,P\n-P/2\n=-T exp ( -iò P H( k,z)dz )\n0\n=e-iHF (-k)P\n……\n( sU )-1 H* (k)( sU ) =-H (-k). (S28)\nz P/2,0 F z P/2,0 F\nComment: 4. In the main text, the vector/scalar potentials are designed to take sinusoidal forms as\nmathematical expressions, but this seems rather abstract. It would be very helpful to add schematic\ndiagrams for the readers.\nReply: We appreciate the referee's valuable suggestion. We have added the schematic diagrams of\nthe vector/scalar potentials, as shown in Fig. R3.\nRevision:\nSupplementary Material S2: (Page 5) Here the potentials are designed to take sinusoidal forms,\nA(z) =－k A Ωsin(Ωz)i and Φ (z) = (－1)nk Asin(Ωz)/2n , where the former is uniformly distributed\n0 v n 0 s 0\nthroughout each waveguide at every z, while the latter is modulated in an anti-phase way between\nadjacent waveguides, as shown in Fig. S1.\nFig. R3 Illustration of the gauge fields. a, VGP and b, SGP versus the propagation distance z,\nshowing only one period.\nsuitable parameters for achieving the desired shapes of topological states.” It is not clear to me how\nANNs help to achieve desired topological states.\nReply: We thank the referee for this important question, which allows us to clarify the role of\nartificial neural networks (ANNs) in our work. The artificial neural network (ANN) serves as an\nefficient inverse-design tool that maps a desired topological mode profile to the required physical\nparameters, thereby bypassing exhaustive parameter sweeps that would otherwise be\ncomputationally prohibitive for Floquet-engineered systems.\nHere, we would like to clarify more precisely how ANN operates within our system:\n(i) We generate a comprehensive training dataset by numerically computing the topological\neigenmodes (mode profiles) of the Floquet Hamiltonian for a wide range of randomly sampled\ncoupling coefficient matrices. Each data pair consists of a normalized topological mode profile (as\ninput/target) and the corresponding coupling coefficients (as output/ground truth). (ii) A fully\nconnected ANN is trained on this dataset, mapping normalized inputs to outputs by optimizing the\nhidden-layer weights via the backpropagation algorithm. (iii) The desired topological modes are\nprovided as inputs to the trained ANNs to predict the required coupling coefficients, i.e., with the\nassistance of ANNs, we can determine the suitable parameters (coupling coefficients in our work)\nfor achieving the desired topological modes.\nRevision:\nMain text: (Page 12, lines 260-261) i.e. input the desired topological modes to the trained ANNs\nto predict the required coupling coefficients (see Supplementary Note 4 for more details.).\nComment: What happens if the authors use non-ANN controlled structures? Probably a baseline\ncomparison is needed to show that the ANN is not merely fitting arbitrary coupling patterns.\nReply: We thank the referee for this question, which highlights the need for a rigorous baseline\ncomparison to validate the effectiveness of our ANN approach.\nDeriving the physical parameters (e.g., waveguide spacing) required to realize a target topological\nmode constitutes a challenging, high-dimensional inverse problem. Traditional methods, such as\ngenetic algorithms, rely on iterative global search but become computationally expensive and often\nconverge to suboptimal solutions as the system size increases, due to the curse of dimensionality\nand the highly nonlinear mapping between parameters and mode profiles.\nTo directly address this, we implemented GA-based inverse design as a non-ANN baseline, and\ncompare its performance against our trained ANN across lattices of different sizes, as shown in Fig.\nR4. For the small-scale system (N = 10), the GA converges relatively quickly with a modest number\nof iterations. Yet the achieved fidelity, defined as the overlap between the predicted and target mode\nprofiles (where 1 indicates perfect agreement), reaches only ≈0.89 even after extended optimization.\nFor the large-scale system (N = 80), the vastly expanded parameter space severely hinders\nconvergence: even after extensive iterations, the solution fidelity only reaches about 0.62, with a\nsubstantial deviation from the target profile (Fig. R4b). These results underscore that conventional\niterative approaches like GA are practical for smaller or lower-dimensional systems but scale poorly\nfor complex, large-scale topological mode reshaping. By contrast, our ANN-based inverse-design\nframework provides a generalized, efficient solution: once trained on a representative dataset, it\n5\ndelivers high-fidelity predictions (typically approaching unity for well-sampled profiles) in\nnegligible time, without repeated forward simulations (e.g. Fig. R4c or Fig. S5 in Supplementary\nMaterial S4).\nFig. R4 Optimization for TMs. Non-ANN optimizing TMs in lattices with sizes (a, N = 10 and b,\nN = 80) by genetic algorithms, with the iterative convergence process, and the inverse-designed\nTMs. c, The ANN-based inverse-design TMs for comparison.\nComment: Moreover, for better understanding, the authors are suggested to provide the detailed\nparameters (waveguide spacing) corresponding to respective coupling coefficients in the main text.\nReply: We thank the referee for this question. We have added the detailed parameters (waveguide\nspacing and couplings) in Supplementary Material S4, (page 14).\nRevision:\nSupplementary Material S4: (Page 14) We have added the detailed parameters (waveguide\nspacing and couplings) in Table S1.\nComment: 6. I recommend that the authors cite some relevant papers, including \"Machine learning\n6\ninverse problem for topological photonics,\" Communications Physics (2018); \"Inverse design of\nphotonic topological state via machine learning,\" APL (2019); \"Machine Prediction of Topological\nTransitions in Photonic Crystals,\" Phys. Rev. Applied (2020); \"Deep learning for the design of\nphotonic structures,\" Nature Photonics (2021).\nReply: We thank the referee for suggesting these highly relevant references. We have carefully\nconsidered and cited these papers in the main text, (i.e., Ref. 66-69 in page 8, line 172),\nRevision:\nMain text: (Page 8, line 174) We have carefully considered and cited these papers in the main text.\nHere, the artificial neural network66-69 (ANN) serves as a practical solver to efficiently navigate this\ncomplex parameter space and identify configurations that implement the desired shape, such as\nsinusoidal (\"Type I\") and convex (\"Type II\") forms, illustrated in Fig. 3a.\nComment: In the manuscript “Sculpting topological modes on photonic chips by artificial gauge\nfields” the authors explore the possibility to design spatially extended topological modes in a\nwaveguide lattice. A common understanding is that periodic Hermitian structures feature bulk\nbands and bandgaps. Thus, any mode arising in a finite system with open boundary conditions inside\nthe gap is typically expected to have complex Bloch momentum and hence must be localized.\nTherefore, a naive expectation is that any in-gap mode is localized.\nHere, the authors challenge that naive picture by introducing two key ingredients: modulation and\nnon-Hermiticity. As they prove, the combined action of these two factors for a periodic SSH lattice\nallows one not only to shift the edge mode from one edge to another, but also to make it delocalized\nin some instances. By generalizing their approach to non-periodic lattices, the authors show that\nthe profile of the topological mode can be tailored at will, once a suitable non-periodic coupling\npattern is chosen.\nReply: We thank the referee for the positive assessment of our work. The referee’s thoughtful and\npenetrating comments are deeply appreciated.\nComment: Overall, I find the paper interesting and timely. The experimental part is quite\nimpressive, as the authors fabricate curved waveguide lattices on a chip, control the losses at each\nof the waveguides and demonstrate selective excitation of the modes by exciting single or multiple\nsites at once with a well-defined phase between the inputs as well as different amplitudes. Even\ntaken alone, such flexibility in the fabrication and excitation of the lattice is remarkable. In the\ncontext of this study, I have several comments and questions:\nReply: We appreciate the referee’s positive evaluation of our experimental platform and its\nflexibility. Below, we provided point-to-point responses to the referee’s comments and hope that\nour revisions meet their expectations.\nComment: 1. I assume, it is known that periodically driving a lattice (e.g. Hermitian SSH model),\none can induce the topological transition in the system accompanied by the shift of the topological\nedge state from one edge of the array to the other. This physics has been explored previously, see\ne.g. [S. Olin, W-C. Lee. Phys Rev B 107, 094310 (2023)] and other related works or [Vyas, Roy.\nPhys. Rev. B 103, 075441(2021)] for periodically driven non-Hermitian model. There could be\nmultiple other works in the same direction.\nI recommend the authors to discuss this aspect in more detail and articulate more clearly the novelty\nof their present contribution (aside from the experimental realization which is clearly new and\nadvanced). In essence, they also study the topological transition in the Floquet spectrum induced\nby the proper amplitude and phase of the driving plus non-Hermitian contributions.\nReply: We thank the referee for bringing these important works to our attention (Phys. Rev. B 107,\n094310 (2023); Phys. Rev. B 103, 075441 (2021)), and related studies on Floquet engineering in\nHermitian and non-Hermitian systems. These studies demonstrate how periodic driving (Floquet\nengineering) can induce shifts of topological edge states between lattice edges.\nHowever, in these Floquet-induced topological phases, whether Hermitian or non-Hermitian, the\nresulting topological states remain localized at the boundaries. By contrast, our work leverages\ncomplex artificial gauge fields (AGFs) to actively reshape conventionally localized topological\nmodes into deliberately extended, delocalized, or even more intricate spatial profiles. This reshaping\n8\nis generally difficult to achieve in standard systems, as it must preserve spectral isolation of the\ntopological mode within the bandgap, which is a stringent physical constraint. Moreover, this design\nalso enables a transition between the topological mode and the non‑Hermitian skin phase (bulk\nmode) induced by AGFs, which is a novel physical effect scarcely explored in prior works\n(Supplementary Material S3).\nRevision:\nMain text: (Page 2, lines 46-49) We have carefully considered and cited these papers in the main\ntext, (i.e., Ref. 55-56). Despite significant efforts to explore new topological states across various\nplatforms, a common characteristic is their exponential localization at interfaces between\ntopologically distinct systems—a hallmark of the bulk-boundary correspondence4,10,53-56, protected\nby non-trivial band structures.\nComment: 2. Is it possible to sculpt arbitrary topological mode profiles only by the periodic driving\n(not necessarily harmonic), but without non-Hermiticity?\nReply: We thank the referee for this important question. To our knowledge, achieving arbitrary or\non-demand reshaping of topological mode profiles, particularly into extended, delocalized, or\ncomplex spatial shapes, solely through periodic driving (whether harmonic or more general non-\nharmonic protocols) remains extremely challenging, if not impossible, in purely Hermitian Floquet\nsystems.\nIn standard Hermitian Floquet-engineered lattices (e.g., driven SSH models or related 1D/2D\nchains), periodic driving can induce topological phase transitions and generate protected edge states,\nincluding anomalous Floquet π modes or chiral edge states. However, these topological modes\ntypically remain localized at the boundaries (or exhibit boundary-selective features). While driving\nparameters can tune the number, dispersion, or quasienergy location of these edge states, there is no\nknown mechanism in Hermitian Floquet systems that allows precise, user-specified sculpting of the\nmode wavefunctions into deliberately extended or intricate delocalized forms without breaking the\nunitarity of the evolution or introducing additional non-Hermitian elements.\nIn our work, the observed topological mode reshaping arises from the synergistic interplay\nbetween the non-Hermitian skin effect (NHSE), which provides a dissipative bias that can delocalize\nor redirect modes, and the topologically protected boundary mode. Critically, the NHSE in our\nsystem is generated by the combined action of the scalar potentials, vector potentials (synthetic\ngauge fields), and onsite dissipation (non-Hermiticity). In the absence of non-Hermiticity, the NHSE\nvanishes entirely, and the topological mode reverts to a conventional localized boundary profile, as\nexplicitly demonstrated in our Hermitian reference calculations (Fig. R5b).\nThus, non-Hermiticity plays an indispensable role in overcoming the constraints of Hermitian\ndynamics, enabling the controlled engineering of extended and custom topological modes while\nmaintaining spectral isolated from the bulk bands.\nb, The extended topological mode becomes localized when the non-Hermiticity is removed.\nComment: 3. The authors suggest that their approach can be extended towards higher-dimensional\nsystems. However, that seems quite challenging using photonic chips, as fabricating 2D waveguide\narrays is a challenge. Please, comment how this approach can be implemented experimentally for\nthe 2D lattices.\nReply: We thank the referee for this important and practical question regarding the experimental\nscalability of our approach to higher-dimensional (e.g., 2D) lattices.\nIndeed, fabricating truly planar 2D waveguide arrays on photonic chips presents significant\nchallenges. Nevertheless, several viable experimental routes exist to implement 2D topological\nmode shaping using our artificial gauge field (AGF) framework:\nQuasi-2D multilayer structures: We have previously demonstrated the successful fabrication of\nquasi-2D multilayer photonic lattices on integrated platforms, which effectively emulate 2D\ndynamics while leveraging mature 1D waveguide processing techniques (Sci. Adv. 11, eadt7224\n(2025)). By stacking and coupling multiple layers with controlled interlayer interactions, one can\nrealize 2D-like topological modes with tunable AGFs introduced via phase modulation or loss/gain\nengineering.\nOn-chip coupled micro-ring resonator arrays: Arrays of evanescently coupled micro-rings\nprovide a natural platform for 2D lattice emulation, where the rings form a square or hexagonal grid\nwith synthetic dimensions or direct coupling. Such architectures have been experimentally realized\nfor topological photonics and allow flexible incorporation of non-Hermiticity and gauge fields\nthrough modulation of coupling strengths or onsite potentials (e.g., Science 372, 403–408 (2021),\nand related works on programmable/resonator-based topological systems).\nFemtosecond-laser-written 3D lattices: This technique offers true 3D fabrication flexibility in\ntransparent media (e.g., glass or crystals), enabling the direct writing of complex 2D or quasi-2D\nwaveguide arrays with arbitrary connectivity, helical paths, or multilayer configurations. It has been\nwidely used for demonstrating higher-dimensional topological phenomena, including Floquet-\nengineered states and non-Hermitian effects (Light Sci. Appl. 13, 130 (2024)).\nCrucially, the theoretical model we present - based on Floquet-engineered AGFs and non-\nHermitian topology - is general and platform-independent. It does not rely on specific fabrication\nconstraints of planar chips and can be readily transferred to any of the above platforms.\nRevision:\nMain text: (Page 13, lines 288-289) our design paradigm can be extended to higher-dimensional\nsystems, e.g., quasi-2D multilayer lattices26 and femtosecond-laser-written waveguides70,…\nComment: 4. The authors realize synthetic gauge fields, which are vector and scalar potential, by\nmodulating the waveguides with the z coordinate. I wonder if that approach could be generalized\nto engineer more exotic gauge fields, e.g. tensorial ones. In turn, such exotic gauge fields could\npotentially bring novel opportunities in light routing.\nReply: We appreciate the referee’s insightful question. The AGFs in our current implementation\nstem from the modulation of single-mode (TE mode) waveguides. The referee’s suggestion points\nto a promising generalization: by employing multi-mode waveguides, the inter-site coupling may\nbecome tensorial, as different internal modes interact. This could effectively engineer non-Abelian\n10\nor tensorial gauge fields in the synthetic dimension, potentially enabling novel band topology and\nlight-routing functionalities. We view this as a compelling direction for future work.\nComment: After carefully evaluating the manuscript, I conclude that it does not meet the novelty\nand conceptual-advance criteria required for Nature Communications.\nMy main concerns are summarised below.\n1. A closely related work by Hu et al., Adv. Photon. 7, 066001 (2025), has already demonstrated\narbitrary control of optical field profiles in an SOI platform at telecommunication wavelengths by\nemploying spatially varying mass terms to generate pseudomagnetic fields. These capabilities,\nincluding engineered gauge-field landscapes, controllable deformation of optical mode\ndistributions, programmable beam manipulation in photonic lattices, directly overlap with the\ncentral claims of the present manuscript. Although the present manuscript introduces non-\nHermitian modulation, this added ingredient does not constitute a sufficiently distinct conceptual\nadvance relative to the published work. The claimed progress remains largely incremental, and the\nnon-Hermitian aspect alone does not provide a clear or substantive differentiation from the existing\nliterature.\nReply: We thank the referee for their careful evaluation of our manuscript and for highlighting the\nrelated work by Hu et al. We have carefully examined this study and appreciate the opportunity to\nclarify the substantial distinctions in motivations, underlying physics, and conceptual advances. In\nthe following, we elaborate more on this point.\n1. Distinct motivations and targeted phenomena. Although both works involve the modulation of\nartificial gauge fields, the work by Hu et al aims at the design of light flow paths using bulk\nstates, e.g., bends and splitters, by employing spatially varying effective mass terms in a\nphotonic crystal to generate pseudomagnetic fields. In contrast, our work is focused on\nmanipulation of topological states, i.e., illustrating the possibility to transform commonly\nlocalized topological modes into extended and even more complex profiles by artificial\ngauge fields, as shown in Fig. R6a and b. This goal, delocalizing topological modes while\npreserving their spectral isolated from the bulk bands, has little overlap with bulk-state\nengineering and addresses a fundamentally different problem in topological photonics.\n2. The underlying physics. The work by Hu et al only employs vector potentials for their purpose,\nwhile in our work, we generate both scalar and vector potentials and explore their interactions\nwith imaginary gauge potentials to realize novel extended topological states in optical systems.\nHence the conceptual ingredients and underlying physics in our work are much richer than that\nin the prior work. Besides, the physics origin of the artificial gauge fields in our waveguide\nlattice also differs from Hu’s work based on photonic crystals. The AGFs in photonic crystals\narise from the introduction of a mass term m(r) that breaks the spatial inversion symmetry,\nleading to the emergence of a pseudomagnetic field A = m(r) in the effective Hamiltonian near\nz\nthe high-symmetry points K/K'. In contrast, the AGFs employed in our work arise from the\nmodulation of waveguide trajectories and mode refractive indices that yield a Hamiltonian\nanalogous to that governing electron motion within an electromagnetic field. Such AGFs endow\nthe couplings between waveguides with phase Dfactors, +kAGP =×k exp(iA r i j )\nmn mn mn mn\n(Supplementary Material S1).\n3. Conceptual advances and innovation. The introduction of non-Hermiticity is not merely an\nadditive ingredient; it unlocks qualitatively new physics through the interplay with multi-\ncomponent AGFs. As both referee #1 and #2 noted, “a naive expectation is that any in-gap mode\nis localized.” Indeed, in the Hermitian limit (without non-Hermiticity), our topological modes\n12\nrevert to fixed localized profiles (Fig. R7).\nOur key innovations are therefore twofold and represent a new paradigm:\n(i) ANN-assisted inverse design to achieve arbitrary, on-demand reshaping of topological\nmodes from localized to extended/complex delocalized forms, overcoming the\nstringent constraint of spectral isolation;\n(ii) Tunable transitions between isolated topological modes and the non-Hermitian skin\nphase (bulk-like localization), as detailed in Supplementary Material S3. This\nconstitutes a new paradigm for topological state manipulation by AGFs. To our\nknowledge, using AGFs with non-Hermiticity to reshape topological states into\nextended profiles, establishes a distinct paradigm not captured by prior works.\nIn summary, while Hu et al. advanced programmable bulk light steering in Hermitian photonic\ncrystals, our work establishes a fundamentally distinct framework for active manipulation of\ntopological boundary states in non-Hermitian Floquet lattices.\nRevision:\nMain text: (Page 2, lines 39-40) We have cited this paper in the main text, (i.e., Ref. 37). These\nAGFs govern their dynamics, mimicking the behavior of charged particles under real fields31-37.\nMain text: (Page 2, lines 54-56) Achieving such delocalized TMs is nontrivial, as it confronts the\ntypical paradigm that in-gap states are necessarily localized - an intuitive expectation supported by\nconventional wisdom in topological band theory4.\nMain text: (Page 6, lines 146-147) Our analysis reveals that cooperative interactions among the\nscalar, vector, and imaginary potentials are essential for enabling the transformation of TMs.\nFig. R6 Illustration of reshaping localized topological mode into extended profile by AGFs. a,\nThe localized topological mode in tradition topological system (w/o AGFs). b, The topological\nmode becomes fully extended by AGFs, which remain spectrally isolated from the bulk bands. c,\nTrivial lattice without topological mode.\nFig. R7 a, The fully extended topological mode with non-Hermiticity. b, The extended topological\n13\nmode is localized at the boundary when removing the non-Hermiticity.\nComment: 2. The authors repeatedly emphasise “arbitrary mode shaping,” “programmable light-\nflow control,”“tailored gauge potentials,” “engineered spatial mode profiles,” “ANN-assisted\ninverse design.” However, these ideas are already well established in the prior literature on\npseudomagnetic-field photonics, synthetic gauge-field design in photonic crystals, non-Hermitian\nbeam shaping, inverse-designed photonic lattices. What is missing from the manuscript is a clear\nand rigorous physical picture that demonstrates the presence of a genuinely new physical\nmechanism, rather than yet another implementation of spatially varying complex couplings. As it\nstands, the phenomena presented here appear to arise from standard gauge-field modulation, giving\nthe impression of a reformulation of known principles rather than the discovery of a fundamentally\nnew physical concept.\nReply: We thank the referee for this thoughtful comment, which provides an excellent opportunity\nto clarify the novel physical principles underpinning our work and distinguish them from prior\nliterature on pseudomagnetic/synthetic gauge fields, non-Hermitian beam shaping, and inverse-\ndesigned lattices.\nOur manuscript reveals a genuinely new physical mechanism for topological mode engineering:\nthe controlled emergence of non-reciprocity (manifesting as the non-Hermitian skin effect, NHSE)\nthrough the synergistic action of scalar, vector, and imaginary artificial gauge fields (AGFs) within\nan otherwise reciprocal coupling framework. This mechanism allows precise steering of the\ninterplay between NHSE and topologically protected boundary modes, enabling transitions from\nlocalized to extended/delocalized or complex spatial profiles - while preserving their spectra\nisolated from bulk bands.\nTo elaborate:\nNovel origin of non-reciprocity\nIn conventional non-Hermitian systems, non-reciprocity (and thus NHSE) is typically introduced\nvia explicit non-reciprocal couplings (e.g., asymmetric hopping terms, with skin effects arising from\nintrinsic non-Hermitian topology via non-reciprocal terms). In contrast, our approach generates\neffective non-reciprocity without breaking microscopic reciprocity in the couplings. Instead, it\nemerges from the symmetry-breaking interplay of multiple AGFs (scalar for onsite energy\nmodulation, vector for synthetic magnetic fields/phases, and imaginary for onsite loss), which\ncollectively engineer the complex band structure (see Fig. R1, Fig. R5, and Supplementary Material\nS2). This joint control is essential: removing any one AGF component eliminates the NHSE and\nrestores localized topological modes (as shown in our Hermitian reference cases).\nThis represents a distinct paradigm for inducing and tuning non-reciprocity in reciprocal photonic\nlattices, differing from standard gauge-field modulation (e.g., pseudomagnetic fields via strain/mass\nterms in photonic crystals, which remain Hermitian and do not enable dissipative delocalization of\ntopological states).\nCore physical principle: Local imbalance-driven reshaping\nThe reshaping itself arises from a tunable local imbalance in the NHSE-topology interplay. When\nthe non-reciprocity (NHSE bias) and topological protection reach a delicate balance, the mode\nbecomes fully delocalized across the lattice (Fig. R8b). By controllably perturbing this balance via\n14\nthe AGFs (e.g., tuning coupling coefficients), the mode can be sculpted into arbitrary extended or\ncomplex profiles (Fig. R8c). This imbalance mechanism, rooted in the synergistic AGF engineering,\nis the fundamental principle enabling on-demand topological mode morphing, going beyond mere\nimplementation of spatially varying couplings. The ANN serves only as a practical inverse-design\ntool to efficiently navigate the high-dimensional parameter space for these imbalances; it does not\nconstitute the new physics.\nIn summary, while prior works have demonstrated gauge-field control of bulk propagation, beam\nsteering, or NHSE in isolated contexts, our contribution introduces a new mechanism for generating\ntunable non-reciprocity in reciprocal systems via multi-AGF symmetry breaking, unlocking\ncontrolled hybridization of topological modes with NHSE for delocalized/extended states and\ntopological-to-skin phase transitions. This establishes a fresh conceptual framework for topological\nmanipulation in non-Hermitian Floquet photonics, distinct from reformulations of known principles.\nRevision:\nMain text: (Page 6, lines 137-145) When the TM aligns with the PBC energy spectrum, an\nequilibrium forms between the NHSE and the opposing localization tendency of the TM, thereby\nleading to a fully delocalized mode. Moreover, once this balanced interaction is established, merely\ntuning the coupling coefficients can break the balance in a controlled manner, thereby generating\nvarious complex spatial profiles.\nOur design demonstrates flexible generation and manipulation of extended TMs based on AGFs\ninduced NHSEs, in contrast to previous approaches that the realization of NHSEs usually relies on\nnonreciprocal couplings8,9,61-65, which are difficult to implement in on-chip optical systems.\nFig. R8 Interaction between localized topological mode and non-reciprocity for mode\nreshaping. a, Only localized topological mode without NHSE. b, The topological mode becomes\nfully delocalized when the interaction between topological mode and NHSE reaches a balance. c,\nGenerating topological mode with complex spatial profile by adding the ANN-assisted coupling\ndesign to introduce local imbalanced interaction.\nHowever, the authors do not convincingly demonstrate that these modes retain their topological\ncharacter while being arbitrarily reconfigured.\nReply: We thank the referee for this important question. The reshaped modes indeed retain their\ntopological character, as demonstrated by their continued spectra isolated from the bulk bands and\ntheir robustness against disorder - key signatures of topological protection [Phys. Rev. X 9, 041015\n(2019); Phys. Rev. X 8, 031079 (2018)]. Specifically, to verify this preservation, we introduced 10%\nindependent randomized realizations (Fig. R9). Despite the perturbations, the reshaped modes\nstill remain spectrally isolated from the bulk bands. The eigenmode profiles exhibit only slight\ndeviations from their original forms (red dashed reference profiles in Fig. R9), confirming that the\ncharacteristic shapes are protected rather than fragile artifacts of fine-tuning. Note that the\ntopological mode also exhibits dynamical stability due to its lowest loss spectrum, which can reflect\nthe robustness of the topological dynamics, as also noticed by referee #1.\nRevision:\nSupplementary Material S5: (Pages 14-15) Next, we performed a systematic robustness test to\nverify the topological character of the reshaped modes across three representative cases - the fully\nextended, Type I, and Type II modes, as shown in Fig. S8. By introducing coupling disorder up to\n10% in strength across 200 randomized samples, we consistently observed that the eigenvalues of\nthe reshaped modes remain spectrally isolated from the bulk bands. Moreover, their spatial profiles\nexhibit only minor deviations under disorder, largely retaining the intended shape as referenced by\nthe original design (red dashed profiles). These results confirm that the reshaped modes are\ntopologically protected and maintain their characteristic isolation and form even in the presence of\nperturbations.\nFig. R9 Robustness of the topological modes. Spectra and mode profiles under 10% disorder of\ncouplings for a, the fully delocalized, b, Type I and, c, Type II modes. The topological modes\n(marked in red) remain spectrally isolated from the bulk bands (marked in gray) and the profiles\nundergo only slight variations, where the red dashed lines are reference profiles.\nComment: The reshaping process appears to be driven primarily by spatially modulated non-\nHermitian couplings, a technique that is already well established in the NHSE literature. As a result,\nthe reported observations resemble general beam-shaping or mode engineering, rather than\n16\nrevealing a genuinely new form of topological physics.\nReply: We thank the referee for this important question. Our reshaping mechanism operates entirely\nwithin a reciprocal coupling framework: the effective non-reciprocity (and thus the NHSE bias)\nemerges solely from the symmetry-breaking interplay of scalar, vector, and imaginary artificial\ngauge fields (AGFs) engineered via Floquet modulation of waveguide trajectories and refractive\nindices. This is very different from conventional NHSE that contains non-reciprocal couplings (e.g.\nFront. Phys. 18, 53605 (2023)). The use of AGF‑driven symmetry engineering to reshape\ntopological modes constitutes a novel approach and introduces a new form of topological physics.\nThe reshaping arises from a tunable local imbalance between the dissipative bias (NHSE) and the\ntopological protection, which allows the mode to transition from boundary-localized to fully\ndelocalized or arbitrarily complex extended profiles while preserving spectra isolated from bulk\nbands (Supplementary Material S5). This approach to reconfiguring a protected and dynamically\nstable topological mode with complex extended profile goes beyond standard NHSE or\nbeam‑shaping schemes. Besides, the synergistic regulation of multiple AGFs not only unlocks\ngreater possibilities and physical insights for on-chip light manipulation but also offers a more\nplatform‑compatible route to effective non‑reciprocity.\nRevision:\nMain text: (Page 6, lines 142-145) Our design demonstrates flexible generation and manipulation\nof extended TMs based on AGFs induced NHSEs, in contrast to previous approaches that the\nrealization of NHSEs usually relies on nonreciprocal couplings8,9,61-65, which are difficult to\nimplement in on-chip optical systems.\nComment: 4. The use of ANN for inverse design is incremental and not physics-driven Inverse-\ndesign techniques (adjoint, genetic, reinforcement learning, neural networks) have been widely used\nfor beam shaping, coupling control, arbitrary-mode generation in photonics. Here, the ANN merely\noutputs coupling profiles for a desired intensity distribution. This optimization strategy is\nengineering-based, not a conceptual physics advance. Therefore, it does not compensate for the lack\nof novelty in the physical mechanism.\nReply: We thank the reviewer for this important comment. In our work, the ANN plays a\ncomplementary yet physics‑informed role - its parameter optimization is intrinsically guided by and\nconstrained within the underlying physical mechanism of AGFs‑induced mode reshaping.\nAs we have mentioned, the physical core of our work is the interaction between non-reciprocity\n(induced by the synergy of AGFs and non-Hermiticity) and the topological mode. A fully extended\nprofile arises when the interaction reaches a balanced state. By suitably tuning the coupling\ncoefficients, we can break this balance to reshape the mode into various complex profiles. This local\nbalance–imbalance interaction is the underlying physics enabling arbitrary topological mode\nshaping. However, realizing a specific target shape is a challenging inverse problem, as numerous\nparameters (e.g., couplings) jointly determine the resulting profile. Here, the ANN serves as a\npractical solver to efficiently navigate this complex parameter space and identify configurations that\nimplement the desired shape. Thus, while the ANN is an enabling design tool, the novelty and\nphysics lie in the mechanism of reshaping topological states via controlled imbalance, which the\nANN helps to realize. This integrated approach, combining a new physical principle with an\nefficient inverse-design framework, constitutes the methodological advance of our work.\nRevision:\nNHSE and TMs reaches an equilibrium (Fig. 2b). By suitably tuning the coupling coefficients, we\ncan break this balance to reshape the mode into various complex profiles. This local balance–\nimbalance interaction can be regarded as the mechanism enabling arbitrary topological mode\nshaping. However, realizing a specific target shape is a challenging inverse problem, as numerous\nparameters (e.g., couplings) jointly determine the resulting profile. Here, the artificial neural\nnetwork66-69 (ANN) serves as a practical solver to efficiently navigate this complex parameter space\nand identify configurations that implement the desired shape, such as sinusoidal (\"Type I\") and\nconvex (\"Type II\") forms, illustrated in Fig. 3a.\nComment: Although the manuscript is technically sound and the experiments are executed with\nimpressive quality, it does not reach the level of conceptual novelty or physical insight expected for\nNature Communications. The central claims substantially overlap with previously demonstrated\nresults, and the authors do not convincingly articulate a new physical phenomenon, mechanism, or\ntheoretical framework that clearly differentiates the present work from the existing literature. I\ntherefore recommend rejection in its current form.\nReply: We sincerely thank the referee for reviewing our work and for the positive assessment of\nour experimental execution. In response, we have carefully revised the manuscript to explicitly\nclarify the core novelty of our work and its distinct differences from the literature mentioned by the\nreferee (Adv. Photon. 7, 066001 (2025)), as detailed in our point-by-point responses and reflected\nin the updated text. We believe that the revised manuscript now clearly articulates a new physical\nmechanism—the synergy between AGFs and non-Hermiticity for reshaping topological modes—\nand presents a novel theoretical framework that sufficiently differentiates our findings from prior\nart. We believe our clarifications and revisions have adequately addressed the referee’s concerns\nand sincerely hope the reviewer could reconsider our revised manuscript for publication. We would\nbe happy to address further comments the reviewers raised. We thank the reviewer again for all the\ninsightful comments.\nReviewer #1\n1. Main manuscript revision (Page 6, lines 145-147): In Supplementary Note 2, we rigorously\nprove the emergence of the NHSE via AGF from a symmetry perspective. Our analysis reveals\nthat cooperative interactions among the scalar, vector, and imaginary potentials are essential\nfor enabling the transformation of TMs.\n2. Supplementary Material S5: (Pages 14-15): These complex topologically extended modes\nhave different intensity distributions at each site, unlike conventional topological modes that\nare localized at the boundary, which makes the initial excitation become difficult ... , namely\nthat the final states possess sinusoidal and convex profiles for “Type I” and “Type II” modes,\nrespectively (Fig. S5b).\n3. Supplementary Material S2: (Pages 7-8), We have modified the corresponding equations\n(Eq. S23–S28).\n4. Supplementary Material S2: (Page 5): Here the potentials are designed to take sinusoidal\nforms, A(z) =－k A Ωsin(Ωz)i and Φ (z) = (－1)nk Asin(Ωz)/2n , where the former is\n0 v n 0 s 0\nuniformly distributed throughout each waveguide at every z, while the latter is modulated in\nan anti-phase way between adjacent waveguides, as shown in Fig. S1.\n5. Main manuscript revision (Page 12, lines 260-261): i.e., input the desired topological modes\nto the trained ANNs to predict the required coupling coefficients (see Supplementary Note 4\nfor more details.).\n6. Supplementary Material S4: (Page 14): We have added the detailed parameters (waveguide\nspacing and couplings) in Table S1.\n7. Main manuscript revision (Page 8, line 175): We have carefully considered and cited these\npapers in the main text. (i.e., Ref. 66-69).\nReviewer #2\n1. Main manuscript revision (Page 2, lines 46-49), We have carefully considered and cited\nthese papers in the main text, (i.e., Ref. 55-56).\n2. Main manuscript revision (Page 13, lines 288-289): our design paradigm can be extended to\nhigher-dimensional systems, e.g., quasi-2D multilayer lattices26 and femtosecond-laser-written\nwaveguides70,…\n1. Main text: (Page 2, lines 39-40): We have cited this paper in the main text, (i.e., Ref. 37).\nThese AGFs govern their dynamics, mimicking the behavior of charged particles under real\nfields31-37.\n2. Main manuscript revision (Page 2, lines 54-56): Achieving such delocalized TMs is\nnontrivial, as it confronts the typical paradigm that in-gap states are necessarily localized—an\nintuitive expectation supported by conventional wisdom in topological band theory4. (Page 6,\nlines 146-147): Our analysis reveals that cooperative interactions among the scalar, vector, and\nimaginary potentials are essential for enabling the transformation of TMs.\n3. Main manuscript revision (Page 6, lines 137-145): When the TM aligns with the PBC energy\nspectrum, an equilibrium forms between the NHSE and the opposing localization tendency of\nthe TM, thereby leading to a fully delocalized mode. Moreover, once this balanced interaction\nis established, merely tuning the coupling coefficients can break the balance in a controlled\nmanner, thereby generating various complex spatial profiles. Our design demonstrates flexible\ngeneration and manipulation of extended TMs based on AGFs induced NHSEs, in contrast to\nprevious approaches that the realization of NHSEs usually relies on nonreciprocal\ncouplings8,9,61-65, which are difficult to implement in on-chip optical systems.\n4. Supplementary Material S5 (Pages 14-15): Next, we performed a systematic robustness test\nto verify the topological character of the reshaped modes across three representative cases—\nthe fully extended, Type I, and Type II modes, as shown in Fig. S8. By introducing coupling\ndisorder up to 10% in strength across 200 randomized samples, we consistently observed that\nthe eigenvalues of the reshaped modes remain spectrally isolated from the bulk bands.\nMoreover, their spatial profiles exhibit only minor deviations under disorder, largely retaining\nthe intended shape as referenced by the original design (red dashed profiles). These results\nconfirm that the reshaped modes are topologically protected and maintain their characteristic\nisolation and form even in the presence of perturbations.\n5. Main manuscript revision (Page 6, lines 142-145): Our design demonstrates flexible\ngeneration and manipulation of extended TMs based on AGFs induced NHSEs, in contrast to\nprevious approaches that the realization of NHSEs usually relies on nonreciprocal\ncouplings8,9,61-65, which are difficult to implement in on-chip optical systems.\n6. Main manuscript revision (Page 7, lines 169-177): A fully extended profile arises when the\ninteraction between the NHSE and TMs reaches an equilibrium (Fig. 2b). By suitably tuning\nthe coupling coefficients, we can break this balance to reshape the mode into various complex\nprofiles. This local balance–imbalance interaction can be regarded as the mechanism enabling\narbitrary topological mode shaping. However, realizing a specific target shape is a challenging\ninverse problem, as numerous parameters (e.g., couplings) jointly determine the resulting\nprofile. Here, the artificial neural network66-69 (ANN) serves as a practical solver to efficiently\nnavigate this complex parameter space and identify configurations that implement the desired\nshape, such as sinusoidal (\"Type I\") and convex (\"Type II\") forms, illustrated in Fig. 3a.\n20"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have addressed all my comments and revised the manuscript accordingly. Its novelty and significance have been further strengthened. I recommend it for publication as is.", "reviewer_2": "The authors have improved their manuscript further, providing a clear explanation of the mechanism behind extended topological modes: the balance between non-Hermitian skin effect and the effect of artificial gauge fields. They also quantified the robustness of the designed modes to the disorder and implemented other relevant revisions.\nI think they have addressed all my comments and hence recommend the paper for publication."}, "author_response": ""}]

Decision letter