Simultaneous photonic and phononic bandgaps in a hexagonal lattice geometry with gradually transforming circular-to-triangular air gap holes

TL;DR

This work demonstrates simultaneous photonic and phononic bandgaps in a scalable 2D silicon phoxonic crystal by engineering a hexagonal lattice with air-gap holes that smoothly transform from circular to triangular via independent controls of $R$ and $l$. Using COMSOL Multiphysics, Bloch-periodic eigenproblems for both photonics and phononics are computed along the irreducible Brillouin path in the unit cell, enabling precise band-structure mapping. By varying $R$ and $l$, the authors achieve up to 49.7% photonic tunability and 24.8% phononic tunability, with photonic gaps in $f_{norm}$ between $[0.078,0.102]$ and phononic gaps in $f_{norm}$ between $[0.089,0.102]$; TE polarization sustains wide photonic gaps while phononic gaps exhibit polarization- and orientation-dependent behavior. This geometrically programmable framework offers a fabrication-friendly route to integrated Bragg filters, sensors, and co-localized photonic–phononic devices for on-chip optomechanical and acousto-optic applications.

Abstract

The integration of photonic and phononic bandgaps within a single scalable architecture promises transformative advances in optomechanical and acousto-optic devices. Here, we design and simulate a two-dimensional hexagonal lattice in silicon with air-gap holes that transition smoothly from circular to triangular via tuneable geometrical parameters including air-gap hole radius (R) and tether length (l). By independently varying these two parameters, we systematically explore diverse honeycomb lattice geometries and their bandgap properties. This transformation from circular to triangular air-gap holes enables suppression of both electromagnetic and elastic wave modes through Bragg scattering and symmetry modulation. We demonstrate that systematic variation of R and l allows tuning of photonic and phononic bandgaps upto 49.7% and 24.8% respectively. This possibility of geometrically tuning bandgaps provide a strong foundation for applications in Bragg filters, sensors etc. without the need for complex defects or exotic materials.

Figures (10)

• Figure 1: Construction of the unit cell. (a) A regular hexagon of side length a; (b) Circular sectors of radius R are centered at each vertex of the hexagon; (c) Tethers of length l/2 are attached to the hexagon; (d) Removal of the sectors to obtain a geometry element; (e) Tiling the elementary patterns to produce a lattice; (f) The unit cell of the lattice.
• Figure 2: (a-e) Change in geometry observed by increasing l value and keeping the R value constant. The air gap holes gradually transform from circular to triangular; (f-j) Change in geometry observed by decreasing the R value. The air gap holes are observed to reduce in size with decrease in R value and gradually transform from triangular holes of filleted vertices to sharp vertices.
• Figure 3: Electric field plots showing the z-component of the initial seven eigenmodes of the unit cell with parameters $\textit{R} = 0.38\,\mu m$ and $\textit{l} = 0.5\,\mu m$. Each subplot (a–g) corresponds to a distinct normalized eigenfrequency as indicated above each plot, visualized using a blue-red colormap where red denotes regions of maximum field intensity and blue denotes minimum intensity. Within the analyzed seven eigenmodes, plots (d-e) correspond to a pair of degenerate mode with $f_{norm} \approx 0.155$. Note: The normalized frequency $f_{norm}$ is calculated by multiplying the eigenfrequency by a normalization factor $1\,\mu m/c$, where $c$ is the speed of light in vacuum ($3 \times 10^{8}\,m/s$).
• Figure 4: Solid displacement plots showing the z-component of the initial seven eigenmodes of the unit cell with parameters $\textit{R} = 0.38\,\mu m$ and $\textit{l} = 0.5\,\mu m$. Each subplot (a–g) corresponds to a distinct normalized eigenfrequency as indicated above each plot, visualized using a blue-red colormap where red denotes regions of maximum field intensity and blue denotes minimum intensity. Within the analyzed seven eigenmodes, plots (e-f) correspond to a pair of degenerate mode with $f_{norm} \approx 0.103$. Note: The normalized frequency $f_{norm}$ is calculated by multiplying the eigenfrequency by a normalization factor $1\,\mu m/c$, where $c$ is the speed of sound in Silicon ($8.518.4\,m/s$).
• Figure 5: (a) Normalized frequency of each of the initial photonic modes vs. $k$-vector for the geometry $R = 0.38\,\mu m$, $l = 0.5\,\mu m$, with the photonic bandgap highlighted in grey; (b) Normalized frequency of each of the initial phononic modes vs. $k$-vector for the geometry $R = 0.38\,\mu m$, $l = 0.5\,\mu m$, with the phononic bandgap highlighted in grey. Here, the normalized frequency is the eigenfrequency multiplied by a normalization factor $1\,\mu m/c$, where $c$ is the speed of light ($3 \times 10^{8}\,m/s$) for photonic wave propagation and the speed of sound in Silicon ($8518.4\,m/s$) for phononic wave propagation.