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Sagnac-Loop-Reflector Fabry-Perot Lattices for Modular 1D Topological Photonics

This paper introduces a modular silicon-photonic Fabry-Perot lattice utilizing cascaded tunable Sagnac loop reflectors to realize the Su-Schrieffer-Heeger model, demonstrating robust topological edge states and disorder resilience through both theoretical derivation and simulation.

Original authors: Siwoo Kim, Yung Kim, Semin Choi, Taeyeon Kim, Seungmin Lee, Kyoungsik Yu, Sangyoon Han, Bumki Min

Published 2026-05-15

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CC BY 4.0

Original authors: Siwoo Kim, Yung Kim, Semin Choi, Taeyeon Kim, Seungmin Lee, Kyoungsik Yu, Sangyoon Han, Bumki Min

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to build a very long, perfect train track for light (photons) to travel on. Usually, scientists build these tracks using tiny loops of wire (microrings). However, making a long line of these loops is tricky. If the factory makes even a tiny mistake in the width of the wire, the light gets confused, and the track stops working as planned. Also, if you want to change how the light jumps from one loop to the next, you often have to redesign the whole thing or tweak it one by one after it's built, which is slow and frustrating.

This paper introduces a new, smarter way to build these light tracks using Sagnac Loop Reflectors (SLRs). Think of an SLR not as a loop you drive through, but as a smart mirror that light bounces off of.

Here is the breakdown of their invention using simple analogies:

1. The "Smart Mirror" vs. The "Loop"

In the old way (microrings), the connection between two stops on the track is like a bridge made of two separate planks. To change how easy it is to cross that bridge, you have to adjust both planks.

In this new way, the connection is controlled by a single "smart mirror" (the SLR).

The Analogy: Imagine a hallway with doors. In the old system, to open a door, you had to turn two different knobs. In this new system, each door has just one single knob.
The Benefit: Because there is only one knob per door, the scientists can easily and independently control how "open" or "closed" each connection is. This makes the whole system much easier to program and tune, like a modular toy set where every piece snaps together perfectly.

2. The "SSH" Dance (The Topological Part)

The scientists arranged these mirrors in a specific pattern: a "strong" mirror, then a "weak" mirror, then a "strong" one, and so on.

The Analogy: Imagine a line of dancers holding hands. Some pairs hold hands very tightly (strong connection), while the pairs in between hold hands loosely (weak connection).
The Result: This creates a "dimerized" chain (a chain of pairs). In the world of physics, this specific pattern is called the Su-Schrieffer-Heeger (SSH) model. It's famous because it creates a special "secret path" for the light.

3. The "Edge State" (The Magic Trick)

When the light travels through this alternating pattern of strong and weak connections, something cool happens:

The Analogy: Imagine a ball rolling down a bumpy hill. Usually, it rolls all the way to the bottom and stops in the middle. But in this specific "SSH" setup, the ball gets stuck only at the very ends of the track. It refuses to go into the middle.
The Paper's Claim: The computer simulations showed that when the light is in the "topological" mode (the special pattern), the energy stays locked at the two ends of the 20-stop track. It creates a "midgap resonance," which is just a fancy way of saying a special frequency where the light lives only at the edges.

4. The "Toughness" Test (Disorder)

Real life is messy. Sometimes the factory makes a mirror slightly too big or too small.

The Analogy: Imagine you have a line of people passing a bucket of water. If one person is a bit clumsy (a "disorder"), does the whole line break?
The Paper's Claim: The scientists tested what happens if they randomly messed up the "knobs" on the mirrors (simulating factory errors). They found that the light at the ends of the track was very tough. Even with the mistakes, the light stayed stuck at the edges. However, the light in the middle of the track got messy and scattered. This proves the "edge" light is robust and reliable, even when the system isn't perfect.

Summary

The paper claims to have built a new type of light circuit using cascaded Sagnac Loop Reflectors.

Modular: Each connection is controlled by a single, tunable knob, making it easier to design and fix than traditional ring-based systems.
Topological: By alternating two types of these mirrors, they created a system that mimics the famous SSH model.
Robust: Simulations show that this system creates special light states that stay locked to the edges of the chip and resist getting messed up by small manufacturing errors.

The authors conclude that this is a great new "complementary platform" for building topological photonics on silicon chips, offering a more direct and programmable way to control how light moves compared to existing methods.

Technical Summary: Sagnac-Loop-Reflector Fabry–P´erot Lattices for Modular 1D Topological Photonics

Problem Statement
Coupled-resonator lattices in silicon photonics, particularly microring-resonator arrays, serve as a versatile platform for band-structure engineering and the emulation of tight-binding Hamiltonians. However, scaling these arrays to larger site counts while maintaining spectrally resolved features presents significant practical challenges. Fabrication-induced variations in waveguide width and etch depth often lead to site-dependent resonance detunings, causing inhomogeneous broadening and unintended shifts in spectral signatures. Furthermore, in conventional ring-resonator lattices, effective hopping amplitudes are typically determined by fixed directional-coupler geometries involving two coupling sections (often mediated by a link ring). Achieving deterministic control over these amplitudes generally requires iterative redesign and careful post-fabrication calibration of individual coupling sections, limiting the flexibility and programmability of the system.

Methodology
The authors introduce a modular one-dimensional Fabry–P´erot (FP) resonator lattice constructed by cascading tunable Sagnac loop reflectors (SLRs). In this architecture, each SLR functions as a compact reflective element where the response is primarily controlled by the cross-coupling coefficient ( $\kappa$ ) of a single directional coupler. This contrasts with ring-based lattices where coupling is mediated by multiple sections.

The study employs a dual-theoretical approach:

Transfer-Matrix Formulation: The authors derive the Bloch dispersion relation for an infinite periodic lattice. By defining a unit cell consisting of alternating SLR types (v and w) separated by waveguide segments, they establish the relationship between the coupler settings and the effective intracell and intercell couplings.
Weak-Coupling Limit Analysis: In the limit where effective SLR transmission amplitudes are small, the discrete round-trip dynamics are approximated by continuous-time coupled-mode equations. This reduction yields an effective tight-binding Hamiltonian of the Su–Schrieffer–Heeger (SSH) form, where the two alternating SLR types dictate the hopping amplitudes.

To validate these theoretical models, the authors perform S-parameter simulations of a 20-site lattice using a circuit-level model and compare these results with temporal coupled-mode theory (TCMT) calculations. They investigate both trivial and topological coupling configurations by varying the cross-coupling coefficients ( $|\kappa_v|$ and $|\kappa_w|$ ). Additionally, they examine the system's robustness by introducing Gaussian disorder to the raw coupler parameters and analyzing the resulting site-resolved power profiles.

Key Contributions and Results

Modular Control Architecture: The paper demonstrates that the SLR-based FP lattice offers a direct, modular handle on effective hopping amplitudes. Unlike ring lattices where coupling is a function of complex geometries, the effective hopping here is controlled by a single coupler element per site, theoretically allowing for independent characterization and tuning (e.g., via MEMS actuation).
SSH Model Realization: By alternating two SLR types within a unit cell, the structure implements a dimerized FP chain. The transfer-matrix analysis confirms that the band gap closes and reopens as the coupling conditions cross the line $|\kappa_v| = |\kappa_w|$ , corresponding to the topological phase transition in the SSH model.
Topological Edge States: S-parameter simulations of a 20-site lattice reveal an isolated midgap resonance in the topological phase ( $|\kappa_v| > |\kappa_w|$ ). The power profiles for this resonance are localized at the lattice boundaries, consistent with topological edge states, whereas bulk-band excitations exhibit extended profiles across the lattice.
Robustness to Disorder: Disorder tests, involving independent Gaussian perturbations to the coupler cross-coupling coefficients, show that the edge-localized response remains robust. The midgap resonance stays concentrated near the boundaries with modest variation across disorder realizations, while bulk modes exhibit significantly larger spatial variation. This confirms that symmetry-preserving hopping disorder does not destroy the topological protection in this regime.
Efficiency: The standing-wave FP implementation reduces the required resonator waveguide length to approximately half that of a traveling-wave ring resonator with the same resonant condition.

Significance and Claims
The paper establishes SLR-based FP lattices as a complementary platform for on-chip topological photonics. The primary significance lies in the platform's ability to provide a more direct link between effective couplings and individual directional couplers compared to conventional evanescently coupled ring arrays. This per-element access is presented as attractive for programmable implementations, particularly those based on MEMS-actuated couplers.

The authors modestly claim that their results validate the effective lattice description in the considered parameter regime and demonstrate the feasibility of realizing SSH-type models with tunable, modular control. They note that while the current analysis assumes lossless waveguides and identical segment lengths, the observed robustness against coupling disorder suggests that the platform could tolerate fabrication imperfections better than systems relying on precise detuning control. The work suggests that this architecture offers a route toward programmable topological photonics on silicon, potentially extendable to longer-range SSH variants, Rice–Mele-type lattices, and non-Hermitian systems, though detailed experimental realization is identified as future work.