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### Design of an optomagnonic crystal: Towards optimal magnon-photon mode matching at the microscale

#### Abstract

We put forward the concept of an optomagnonic crystal: a periodically patterned structure at the microscale based on a magnetic dielectric, which can co-localize magnon and photon modes. The co-localization in small volumes can result in large values of the photon-magnon coupling at the single quanta level, which opens perspectives for quantum information processing and quantum conversion schemes with these systems. We study theoretically a simple geometry consisting of a one-dimensional array of holes with an abrupt defect, considering the ferrimagnet yttrium iron garnet (YIG) as the basis material. We show that both magnon and photon modes can be localized at the defect, and use symmetry arguments to select an optimal pair of modes in order to maximize the coupling. We show that an optomagnonic coupling in the kHz range is achievable in this geometry, and discuss possible optimization routes in order to improve both coupling strengths and optical losses.

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• Received 8 December 2020
• Revised 11 February 2021
• Accepted 3 March 2021

DOI:https://doi.org/10.1103/PhysRevResearch.3.013277

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

#### Physics Subject Headings (PhySH)

1. Research Areas
1. Physical Systems
Condensed Matter & Materials Physics

#### Authors & Affiliations

Jasmin Graf1,2, Sanchar Sharma1, Hans Huebl3,4,5, and Silvia Viola Kusminskiy1,2

• 1Max Planck Institute for the Science of Light, Staudtstraße 2, 91058 Erlangen, Germany
• 2Department of Physics, University Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany
• 3Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, Walther-Meissner-Straße 8, 85748 Garching, Germany
• 4Physik-Department, Technische Universität München, James-Franck-Straße 1, 85748 Garching, Germany
• 5Munich Center for Quantum Science and Technology (MCQST), Schellingstraße 4, 80799 München, Germany

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#### References

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##### Issue

Vol. 3, Iss. 1 — March - May 2021

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#### Images

• ###### Figure 1

Investigated geometry: (a) Optomagnonic crystal with an abrupt defect at its center for localizing an optical and a magnon mode at the same spot in the defect area. (b) Optomagnonic crystal from the side representing a heterostructure. (c) Mode profiles of the localized optical and magnon mode discussed in the main text. Note: All mode shape plots are normalized to their corresponding maximum value.

• ###### Figure 2

General structure of a 1D photonic crystal and mode localization at a defect: (a) 1D photonic crystal consisting of periodic layers alternated by the lattice constant $a$ with different dielectric constants ${\varepsilon }_{1}>{\varepsilon }_{2}$ and widths ${d}_{1}$ and ${d}_{2}$. (b) A defect breaks the symmetry and can pull a band-edge mode into the photonic band gap. Since a mode in the band gap cannot propagate into the structure, the light is Bragg reflected and is thus localized (see, e.g., [47]).

• ###### Figure 3

Symmetries of the optical modes in a periodic waveguide: (a) Symmetry planes of the investigated 1D photonic crystal shown in Fig. 1. (b) Symmetry of a transverse electric (TE)-like and a transverse magnetic (TM)-like optical mode in a thin 3D structure. The red arrows indicate the electric field vector $\mathbit{E}$ which for $z=0$ (middle of the crystal along the height) lie in-plane for TE-like modes and point out of plane for TM-like modes. For $z\ne 0$ this is not fulfilled anymore (see, e.g., [47]).

• ###### Figure 4

Optical [(a) and (b)] and magnetic modes (c): (a) Band diagram (obtained with MEEP) for TE-like modes within the irreducible BZ with a state that was pulled into the gap from the upper band-edge state by the insertion of a defect (note that the gap state was not obtained by band diagram simulations). The bands in the green shaded area representing the light cone are leaky modes which couple with radiating states inside the light cone [112]. From the shape of the localized defect mode (obtained with Comsol) with a frequency of ${\omega }_{\text{opt}}=2\pi ×246\phantom{\rule{0.16em}{0ex}}\mathrm{T}\mathrm{Hz}$ (middle layer in the $xy$ plane) we see that this mode is odd with respect to $x=0$ and $y=0$ (and even with respect to $z=0$). (b) Optical spin density (middle layer in the $xy$ plane) of the localized mode, fulfilling the same symmetries, see main text. (c) Band diagram of backward volume waves within the irreducible BZ showing magnetic modes with extended $\mathbit{k}$ values but preferring wave vectors at the edge of the BZ. The highest excited localized mode has a frequency of ${\omega }_{\text{mag}}=2\pi ×13.12\phantom{\rule{0.16em}{0ex}}\mathrm{G}\mathrm{Hz}$ and is odd along the mirror symmetry planes for $x=0$ and $y=0$ (and additionally even with respect to the plane for $z=0$). The dashed line in the middle inset shows the mode spectrum in case of no defect. Note: All mode shape plots are normalized to their corresponding maximum value.

• ###### Figure 5

Dipolar spin wave types and symmetries in the magnonic crystal: (a) Dipolar spin waves can be divided into three types: backward volume waves (BWVW) with their wave vector parallel to the external field which both lie in the plane of the structure ($\mathbit{k}\phantom{\rule{0.16em}{0ex}}\parallel \phantom{\rule{0.16em}{0ex}}{\mathbit{m}}_{0}\phantom{\rule{0.16em}{0ex}}\parallel \phantom{\rule{0.16em}{0ex}}{\mathbit{H}}_{\mathbf{ext}}$). Forward volume waves (FWVW) with their wave vector in-plane and perpendicular to the external field which lies normal to the structure's plane ($\mathbit{k}\phantom{\rule{0.16em}{0ex}}\perp \phantom{\rule{0.16em}{0ex}}{\mathbit{m}}_{0}\phantom{\rule{0.16em}{0ex}}\parallel \phantom{\rule{0.16em}{0ex}}{\mathbit{H}}_{\mathbf{ext}}$). Surface waves are also forward volume waves but they have their wave vector in-plane and perpendicular to the external field which also lies in-plane of the structure ($\mathbit{k}\phantom{\rule{0.16em}{0ex}}\perp \phantom{\rule{0.16em}{0ex}}{\mathbit{m}}_{0}\phantom{\rule{0.16em}{0ex}}\parallel \phantom{\rule{0.16em}{0ex}}{\mathbit{H}}_{\mathbf{ext}}$). (b) Symmetries of the investigated 1D magnonic crystal shown in Fig. 1. Since the external magnetic field breaks two mirror symmetry planes only the mirror symmetry plane normal to the saturation direction remains. Additionally a $\pi$-rotation symmetry around the saturation axis is present.

• ###### Figure 6

Spatial shape of the coupling and different symmetries of the optical and the magnetic mode: (a) Spatial shape of the coupling similar to the magnon mode shape. (b) Different symmetries along the crystal's height of the optical spin density and the magnon mode. Due to the external magnetic field the mirror symmetry along the height is broken and only a $\pi$-rotation symmetry remains, resulting in different mode shapes along the height of the crystal. For thin films this difference is rather small. Note: All mode shape plots are normalized to their corresponding maximum value.

• ###### Figure 7

Fine structure of the optical spin density and the magnon mode along the length of the crystal for a fixed height and width.

• ###### Figure 8

Height dependence of the Faraday component of the optomagnonic coupling: The coupling shows a $\sqrt{{V}_{\text{mag}}}$ dependence since the optical mode volume in the YIG and the ${\text{Si}}_{3}{\text{N}}_{4}$ slab is constant. The decrease with larger height can be explained by the shrinking directionality measure [see Eq. (26)] between the optical and the magnetic mode.

• ###### Figure 9

Optimization of the geometry: Through increasing the parameters along the width of the crystal we create more space for the modes without touching the optical optimization of the original crystal (dashed line). We note that we also increased the defect size, not shown here.

• ###### Figure 10

Optical [(a) and (b)] and magnetic modes (c) of the optimized crystal: (a) Band diagram for TE-like modes within the irreducible BZ with a defect mode in the photonic band gap which was pulled from the upper band-edge state into the gap by the insertion of a defect. From the mode shape of the localized mode with a frequency of ${\omega }_{\text{opt}}/2\pi =279\phantom{\rule{0.16em}{0ex}}\mathrm{T}\mathrm{Hz}$ (middle layer in the $xy$ plane) we see that this mode is odd with respect to $x=0$ and $y=0$ [and even with respect to ($z=0$)]. (b) Optical spin density of the localized mode (middle layer in the $xy$ plane) which is odd with respect to $x=0$ and $y=0$ (and even with respect to $z=0$). (c) Band diagram of backward volume waves within the irreducible BZ showing magnetic modes with extended $\mathbit{k}$ values but preferring wave vectors at the edge of the BZ. The highest excited localized mode has a frequency of ${\omega }_{\text{mag}}=2\pi ×13.17\phantom{\rule{0.16em}{0ex}}\mathrm{G}\mathrm{Hz}$ and is odd along the mirror symmetry planes for $x=0$ and $y=0$ (and additionally even with respect to the plane for $z=0$). The dashed line in the middle inset shows the mode spectrum in case of no defect. Note: All mode shape plots are normalized to their corresponding maximum value.

• ###### Figure 11

Fine structure of the optical spin density and the magnon mode along the length of the optimized crystal for a fixed height and width.

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