Page 75 - Science

RESEARCH

◥ point (DP) with nontrivial Berry phase can split
REPORT into a pair of EPs (20–22) when radiation loss—
a form of non-Hermiticity—is added to a two-
dimensional (2D)–periodic photonic crystal (PhC)
OPTICS structure. The EP-pair generates a distinct double–
Riemann sheet topology in the complex band struc-
Observation of bulk Fermi arc and ture, which leads to two notable consequences:
bulk Fermi arcs and polarization half topo-
polarization half charge from paired logical charges. First, we demonstrate that this
pair of EPs is connected by an open-ended iso-
frequency contour—a bulk Fermi arc—in direct
exceptional points contrast to the common intuition that isofre-
quency contours are necessarily closed loops.
The bulk Fermi arc here is a special topolog-
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1
3
Hengyun Zhou, *† Chao Peng, 1,2 * Yoseob Yoon, Chia Wei Hsu, Keith A. Nelson, 3 ical signature of non-Hermitian effects in paired
1
1
1
5
Liang Fu, John D. Joannopoulos, Marin Soljačić, Bo Zhen ‡ EPs and resides in the bulk dispersion of a 2D
system. This is fundamentally different from the
The ideas of topology have found tremendous success in closed physical systems, but even previously known surface Fermi arcs that arise
richer properties exist in the more general open or dissipative framework. We theoretically from the 2D projection of Weyl points in 3D
propose and experimentally demonstrate a bulk Fermi arc that develops from non-Hermitian Hermitian systems. Moreover, we find experi-
radiativelossesinanopensystemofphotonic crystal slabs. Moreover, we discover half-integer mentally that around the Fermi arc, the far-field
topological charges in the polarization of far-field radiation around the bulk Fermi arc. Both polarization of the system exhibits a robust half-
phenomena are shown to be direct consequences of the non-Hermitian topological properties integer winding number (23–25), analogous to
of exceptional points, where resonances coincide in their frequencies and linewidths. Our work the orientation reversal on a Möbius strip. We
connects the fields of topological photonics, non-Hermitian physics, and singular optics, show that this is a direct consequence of the Downloaded from
providing a framework to explore more complex non-Hermitian topological systems. topological band-switching properties across the
Fermi arc connecting the EP pair and is direct
1
experimental proof of the n ¼ T 2 = topological
n recent years, topological effects have been We theoretically design and experimentally index associated with an EP (8). With compre-
widely explored in closed and lossless sys- realize a new configuration of isolated EP pairs in hensive comparisons between analytical models,
tems, where the physics is characterized by momentum space, which allows us to reveal topo- numerical simulations, and experimental mea-
a Hermitian operator that ensures a real en- logical signatures of EPs in the band structure surements, our results are a direct validation
I ergy spectrum and a complete, orthogonal set and far-field polarization, and to extend topolog- of non-Hermitian topological band theory and
of eigenfunctions. This has revealed a number of ical band theory into the realm of non-Hermitian present its novel application to the field of sin- http://science.sciencemag.org/
previously unknown phenomena such as topo- systems. Specifically, we demonstrate that a Dirac gular optics.
logically nontrivial band structures (1, 2), and
promising applications including backscattering-
immune transport (3–5). However, most systems, Dirac point Paired exceptional points Bulk Fermi arc
particularly in photonics, are generically non-
Analytics Numerics
Hermitian because of radiation into open space
or material gain or loss. Non-Hermiticity enables Radiation
k y k y
even richer topological properties, often with no Loss on March 1, 2018
counterpart in Hermitian frameworks (6–8). One
such example is the emergence of a new class of
k x k x
degeneracies, commonly referred to as excep-
tional points (EPs), where two or more resonances 0.6658
0.5946
of a system coalesce in both eigenvalues and
eigenfunctions (9). So far, isolated EPs in pa-
rameter space (10–12) and continuous rings of
0.6655
EPs in momentum space (13) have been studied 0.5942
across different wave systems because of their
intriguing properties, such as unconventional 2 2
transmission or reflection (14) and relations to 0.5938 ×10 -3 0.6652 ×10 -3 ×10 2 -3
parity-time symmetry (15), as well as their unique 0 0 y
applications in sensing (16, 17)and single-mode y 2.85 y
3.93 -2 2.85 -2
lasing (18, 19). ×10 -2 x 4.13 -2 ×10 x 3.05 -2 ×10 x 3.05 -2
1 Department of Physics, Massachusetts Institute of Technology, Fig. 1. Bulk Fermi arc arising from paired exceptional points split from a single Dirac point.
2
Cambridge, MA 02139, USA. State Key Laboratory of Advanced (A and B) Illustration of photonic crystal (PhC) structures, isofrequency contours, and band structures.
Optical Communication Systems and Networks, Department of (A) Band structure of a 2D-periodic PhC consisting of a rhombic lattice of elliptical air holes, featuring
Electronics, Peking University, Beijing 100871, China.
3 Department of Chemistry, Massachusetts Institute of a single Dirac point on the positive k x axis. (B) The real part of the eigenvalues of an open system
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Technology,Cambridge,MA 02139,USA. Department of consisting of a 2D-periodic PhC slab with finite thickness, where resonances experience radiation loss.
Applied Physics, Yale University, New Haven, CT 06520, USA. The Dirac point splits into a pair of exceptional points (EPs).The real part of the eigenvalues is
5 Department of Physics and Astronomy, University of degenerate along an open-ended contour—the bulk Fermi arc (blue line)—connecting the pair of EPs.
Pennsylvania, Philadelphia, PA 19104, USA. (C) Examples of the isofrequency contours in this system, including the open bulk Fermi arc at the EP
*These authors contributed equally to this work. †Present address:
Department of Physics, Harvard University, Cambridge, MA 02138, USA. frequency (middle panel), and closed contours at higher (upper panel) or lower (lower panel)
‡Corresponding author. Email: bozhen@sas.upenn.edu frequencies. Solid lines are from the analytical model, and circles are from numerical simulations.
Zhou et al., Science 359, 1009–1012 (2018) 2 March 2018 1of4