Lee's Formulation

Introduction to recent research topics 

Revealing new topological effects in optical metasurfaces

Out-of-plane radiation from leaky topological states is a unique feature of planar photonic resonant systems, which has no counterpart in condensed matter systems. Our study starts from this fundamental difference and tries to find new ways to explore theoretical predictions as well as useful functionalities for free-space optical/metasurface applications.

Ki Young Lee

Topology: Mathematical object that is invariant under the continuous deformation can be new degrees of freedom for controlling waves
Recent attempts to understand quantum materials in terms of topology have led to the discoveries of new phases of matters such as topological insulator and topological semimetals, changing the paradigm of the exploration of functional materials. The topological degrees of freedom are not exclusive to condensed matter systems and are being expanded to various classical wave platforms such as artificially constructed photonic or phononic metamaterials, broadening the scope of various academic and practical investigations.

The term 'topological' refers to mathematical quantities that are invariant under continuous parametric changes. In physics, the overall distribution characteristics of eigenstates in the energy band of a periodic system are an example of topological invariants, called the band topology (Zak or Berry phase and Chern number). This is a global feature of the system, and the manifestation of states based on it has anomalous features such as self-localization and topological protection that is insensitive to local parametric changes. 

Nanophotonic systems such as waveguide arrays, coupled ring resonators, photonic crystals, and metamaterials are being recognized as a useful platform for exploring hypothetical topological phenomena that cannot be observed in condensed matter system thanks to their fabrication flexibility and material diversity. Furthermore, the additional topological degrees of freedom open up promising functionalities for practical photonic applications with topologically induced unprecedented physical natures. Examples include Pancharatnam-Berry phase based flat optical devices and robust single mode lasing from topological interface states. 

The question
Many studies in a photonic platform that mimic the topological effects of electronic systems have naturally focused on the Hermitian axiom and the in-plane dynmamics of the bound states. In the case of electrons, out-of-plane emission from the material surface is forbidden because their energy must be conserved under the time evolution, whereas photonic states in the certain condition allow leakage out of the confinement plane. This leakage means an open system nature in which the confined state and the external environment strongly interact. Therefore, the leaky resonance of the quasi-bound state existing on the plane has the characteristics of a non-Hermitian band topology, and these have recently been reported in the thin-film photonic crystal slab also called metasurfaces. 

Metasurfaces are thin-film flat optical resonators with subwavelength-scaled pattern in horizontal dimension for trapping lightwaves. These platforms have shown a range of fascinating optical functions in a flat and compact manner by spatially controlling the leakage interaction between a photonic quasi bound state and free-space propagating waves in a subwavelength scale. However, the topological effects, particularly band topology, in metasurfaces has been rare to be investigated due to the inherent open system nature, which necessitate a careful non-Hermitian extension of the Hermitian topological quantities.

What are the physical implications of this fundamental difference (non-Hermiticity) on the existing topological phenomenon? Could it potentially serve as a promising avenue for discovering new topological phenomena or suggesting intriguing functionalities for practical metasurface applications? It is crucial to explore the potential additional functionalities that can be achieved by utilizing the photonic version of topological states or manipulating invariants in a leaky resonance system, particularly in the context of flat-optical elements, color filters, beam steering devices, or near-eye displays, which are the main targets of metasurface technologies. We are placing emphasis on these issues.

The solutions
"What is the significance of studying the topological degrees of freedom for light that is trapped and moves along the surface (in-plane) of a patterned thin film photonic medium, i.e. a metasurface, especially when light can easily leak out of the surface (out-of-plane)?" This question has guided our recent works, through which we have uncovered several underlying principles and potential applications.

• Principles: Interpreting the band structure of a metasurface from the perspective of the Dirac equation can provide a new framework for understanding the boundary topological physical phenomena between different metasurfaces.
  - Guided-mode resonance platform as analogy with 1D Dirac domain wall problem: Nanophotonics (2021).
  - Interlayer interaction platform & its Dirac-semimetal analogy: Physical Review Letters (2022).
  - Surface plasmon polariton platform: Current Applied Physics (2023).

• Applications: Boundary physics and band topology in metasurfaces offer unprecedented control over light, which can help overcome limitations of existing applications or serve as a foundation for discovering entirely new ones.
  - Topological beam emitters with compact, efficient, and tailored beams:
  Science Advances (2022), Advanced Functional Materials (2024),  Light: Science & Applications (2025).
  - Enabling compact coherent perfect absorber: Scientific Reports (2024).
  - The concept of interlayer interaction driven Dirac semimetal phase and its associated photonic band topology enables the emergence of a continuum of unidirectional guided resonances (UGR continnua): Nature Nanotechnology (2025).
  - Enabling broadband transmissive phase-only modulation with UGR continuum: Physical Review Research (2025).

So far, we have shown that interpreting the band topology of metasurfaces not only offers new principles for beam manipulation via boundary states, but also allows for a systematic classification of the collective behavior of global photonic modes, enabling broadband and robust control of leakage radiation.

Highlights: "Unidirectional guided resonance continua", Nature Nanotechnology (2025).

Unidirectional guided-resonance (UGR) has been known as an extreme interference effect observed in vertically asymmetric metasurfaces. Because of this, it was believed to appear as a singularity—a “point” or “discretuum”—in momentum space. However, we predicted and demonstrated that when two identical metasurfaces are slightly misaligned with glide symmetry, such UGRs can form a “continuum”.

This finding breaks the conventional belief that UGR is confined to isolated conditions, and opens up exciting opportunities for broadband optical couplers and directional emitters.

The key lies in the anomalous Dirac band structure of the glide-symmetric bilayer metasurface. At specific misalignments, oppositely propagating internal guided-modes remain orthogonal across the entire band (complex Dirac mass is completely zero), despite the inherently non-Hermitian nature of the system. This enables persistent unidirectional coupling — for example, left-going modes leak upward, while right-going modes only leak downward.

Considering that the band structure of metasurfaces is inherently non-Hermitian, such orthogonality is quite anomalous. The misalignment also makes the structure resemble an array of tilted mirrors. So, this can be understood as a synergistic effect between anomalous orthogonality of the modes and critical vertical asymmetry of the structure.

The visualization below shows how a single point source inside such a bilayer metasurface leads to spatially separated leakage directions for oppositely propagating guided-modes. What’s remarkable is that this behavior will occur across a broadband wavelength range and, theoretically, across the entire band.

This is just one of the intriguing aspects of bilayer metasurfaces. For a deeper dive into this underlying physics and its wavefront-shaping applications, check out our another theoretical paper: Physical Review Research (2025).

And our PRL paper also explains the topological analogy between glide symmetric bilayer metasurfaces and Dirac-semimetals: Physical Review Letters (2022).