Topology-optimized dual-polarization Dirac cones

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Topology-optimized dual-polarization Dirac cones

Zin Lin, Lysander Christakis, Yang Li, Eric Mazur, Alejandro W. Rodriguez, and Marko Lončar

Phys. Rev. B 97, 081408(R) – Published 23 February 2018

Abstract

We apply a large-scale computational technique, known as topology optimization, to the inverse design of photonic Dirac cones. In particular, we report on a variety of photonic crystal geometries, realizable in simple isotropic dielectric materials, which exhibit dual-polarization Dirac cones. We present photonic crystals of different symmetry types, such as fourfold and sixfold rotational symmetries, with Dirac cones at different points within the Brillouin zone. The demonstrated and related optimization techniques open avenues to band-structure engineering and manipulating the propagation of light in periodic media, with possible applications to exotic optical phenomena such as effective zero-index media and topological photonics.

Received 29 May 2017
Revised 28 January 2018

DOI:https://doi.org/10.1103/PhysRevB.97.081408

Physics Subject Headings (PhySH)

Optical & microwave phenomena Photonics Refraction

Polarization

Topology

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Zin Lin¹, Lysander Christakis², Yang Li¹, Eric Mazur¹, Alejandro W. Rodriguez³, and Marko Lončar^1,*

¹John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
²Department of Physics, Yale University, New Haven, Connecticut 06511, USA
³Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA

^*loncar@seas.harvard.edu

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Issue

Vol. 97, Iss. 8 — 15 February 2018

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Images

Figure 1
(a) Topology-optimized unit cell. Black (white) regions have relative permittivity $ε_{r} \approx 5.5$ ( $ε_{r} = 1$ ). Gray regions with intermediate permittivities are also seen. Note that the structure obeys $C_{4 v}$ symmetry. (b) The band structure reveals two overlapping Dirac cones, one for TM polarization (solid lines) and the other for TE polarization (dashed lines). Transverse magnetic Dirac bands (dark red lines) are formed by the degeneracy of one monopolar (M) and two dipolar (D) modes manifested by the $E_{z}$ component, whereas transverse electric Dirac bands (light-blue lines) are formed by the degeneracy of two dipolar (D) modes and one quadrupolar (Q) mode manifested by the $H_{z}$ component (see figure inset). (c) Configuration of the prism test designed to illustrate effective zero-index behavior for designs with dual-polarization Dirac cones (DPDCs). (d) FDTD analysis of the DPDC structures and their far-field patterns through the prism test show orthogonally emerging beams at the prism facet ( $θ = 0$ ), validating the effective zero-index behavior for both TM- and TE-polarized waves incident on the nonbinary design. Also shown are the retrieved TM and TE effective indices for the nonbinary design (e).
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Figure 2
(a) Binary regularized version of a DPDC PhC unit cell, with the corresponding band structure (b) showing TM (solid lines) and TE (dashed lines) Dirac cones. (c) FDTD analysis and the far-field patterns through the prism test show orthogonally emerging beams at the prism facet $(θ = 0)$ , validating the effective zero-index behavior for both TM- and TE-polarized incident waves. (d) Also shown are the TM and TE effective indices retrieved from scattering coefficients.
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Figure 3
(a) Binary regularized DPDC PhC with fabrication-friendly features. (b) The corresponding band structure shows two overlapping TM (solid lines) and TE (dashed lines) Dirac cones. (c) FDTD analysis and the far-field patterns through the prism test [Fig. 1] show orthogonally emerging beams at the prism facet $(θ = 0)$ , validating the effective zero index behavior for both TM- and TE-polarized incident waves. (d) Also shown are the TM and TE retrieved effective indices.
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Figure 4
Detailed image of a (a) low-index ( $ε_{r} = 3.3$ ) and a (b) high-index ( $ε_{r} = 9$ ) topology-optimized hexagonal unit cell. (c) Band structure of the low-index design exhibiting overlaid TE (dashed line) and TM (solid lines) Dirac bands (red and blue). The degenerate modes (insets) transform according to the $E$ irreducible representation of the $C_{3 v}$ group.
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