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RESEARCH | REPORT

pling strengths to the continuum; v g is the group isofrequency contours of the sample. The PhC
velocity describing the dispersion around the DP, sample is illuminated with a tunable continuous-
which for simplicity is here chosen to be the same wave Ti:Sapphire laser that is vertically polar-
along all directions [see (26)for thegeneral case]. ized, while scattered light—arising from natural
The real part of the complex eigenvalues w T char- fabrication imperfections of the sample—is col-
t = 220 nm acterizes the resonance frequency, whereas the lected with a charge-coupled device (CCD) camera
imaginary part represents the linewidth of the placed at the focal plane of a convex lens with
resonance. 10-cm focal length. Because of resonant enhance-
The eigenvalue spectrum exhibits a pair of ment, the scattered light will have strongest
EPs at ðdk x ; dk y Þ¼ð0; Tg=v g Þ, near the original intensityonlyalong directions wheretheunder-
DP, when the square-root term in Eq. 2 vanishes lying resonances share the same frequency as
and the two eigenvectors coalesce (Fig. 1B). The the pump laser, and thus the isofrequency con-
Fig. 2. Fabricated PhC slab and measure- existence of such EPs is topologically robust tours of the sample are directly imaged onto the
ment setup. (A) SEM images of the PhC samples: against continuous changes in the Hamiltonian CCD (26, 28, 29). Compared to previous mea-
side view (top panel) and top view (bottom panel). (8, 26) and does not rely on any symmetries or surement techniques based on the reflection
(B) Schematic of the scattering measurement fine-tuning. Furthermore, this pair of EPs is con- spectrum (13), this scattering method enables
setup. Vertically polarized light from a tunable nected in momentum space by an open-ended fast and direct extraction of band information,
continuous-wave Ti-Sapphire laser (TL) scatters arc—a bulk Fermi arc, along which the real part without fitting to any specific models. To show
off the PhC slab and is collected by a CCD camera of the complex eigenvalues is degenerate at w D the full shrinking-and-reexpanding feature of the
placed at the focal plane of the lens (L).The (Fig. 1C, middle panel). Although sharing fea- isofrequency contours around the bulk Fermi
specular reflection is blocked to ensure only tures similar to previously studied Fermi arcs— arc, the laser wavelength is tuned from 794 nm
scattered light is imaged. POL and QWPare used in both are open-ended isofrequency contours—our down to 788 nm at steps of ~0.2 nm. Furthermore,
polarimetry measurements of the scattered light. bulk Fermi arc resides at one frequency in the the polarization at each point along a given iso-
TL, tunable laser; BS, beam splitter; L, convex bulk dispersion rather than on the surface of a frequency contour is determined through polar-
lens with 10-cm focal length; QWP, quarter-wave 3D Hermitian system and originates from non- imetry measurements, by optionally inserting a Downloaded from
plate; POL, polarizer; BB, beam block. Hermiticity rather than from the presence of quarter-wave plate and a polarizer in front of
Weyl points. As the frequency w decreases from the CCD (26).
Our scheme involves splitting a single DP into above w D , the closed isofrequency contour at w At a few representative wavelengths around
a pair of EPs, which directly leads to the emer- shrinks (Fig. 1C, top panel), eventually turning the Fermi arc, the numerical results of iso-
gence of a bulk Fermi arc. First, consider a 2D- into the open Fermi arc when w ¼ w D (Fig. 1C, frequencycontours(Fig. 3A)obtainedfrom
periodic PhC with a square lattice of circular middle panel), and expands out again into a simulating extracted structural parameters
air holes introduced into a dielectric material. closed contour at even lower frequencies (Fig. 1C, are plotted against the experimental results
In this Hermitian system (no material gain, loss, bottom panel). Taken together, the band structure (Fig. 3B), showing good agreement with each
or radiation loss), the crystalline symmetry (C 4v ) around the EPs forms a double–Riemann sheet other. Here, for better comparison, the numer- http://science.sciencemag.org/
ensures a quadratic band degeneracy at the topology (Fig.1B).Thisoriginatesfromthe com- ical results are offset by 0.5 nm relative to the
center of the Brillouin zone (26). As this C 4v plex square-root term in the dispersion in Eq. 2, experiments. [See (26) about possible reasons
symmetry is broken, e.g., by shearing the struc- which, depending on the sign choice of the for this wavelength offset and the full set of
ture into a rhombic lattice with elliptical holes square root, results in two sheets. The two eigen- isofrequency contours measured at different
(Fig. 1A), the quadratic degeneracy point splits values continuously evolve on each sheet, and wavelengths.] To focus on the bulk Fermi arc,
into a pair of DPs situated at ðTk D ; 0Þ along the their real parts become degenerate along an we highlight the region of interest in both panels,
k x axis. The same splitting behavior is shown in open-ended curve—the bulk Fermi arc. From a where the isofrequency contours clearly demon-
both analytical models and numerical simulations different point of view, this can also be under- strate the shrinking and reexpanding behavior. on March 1, 2018
(21, 26, 27). stood as a nodal arc in the bulk band structure, As shown in Fig. 3, as the wavelength decreases
Next, we consider a non-Hermitian system in analogy to nodal lines in semimetal systems from 794.0 nm, the corresponding isofrequency
consisting of a finite-thickness PhC slab (inset of (2). We further verify the existence of bulk Fermi contour shrinks (top two rows), and eventually
Fig. 1B), where modes near the DP become res- arcs in realistic PhC slab structures via numerical becomes an open-ended arc at 791.0 nm (middle
onances with finite lifetime because of radiative simulations (Fig. 1C, circles), showing a good row), consistent with our previous theoretical
losses toward the top and bottom. Adopting the agreement with analytical results (Fig. 1C, solid predictions in Fig. 1C. As the wavelength is fur-
even and odd y–mirror-symmetric eigenstates lines) (26). The extent of the bulk Fermi arcs ther decreased down to 789.5 nm and 788.7 nm,
at the DP as basis and taking into account the can be tuned by engineering the band structure the arc expands out into closed contours again
radiation losses via non-Hermitian perturbations, and coupling rates to the continuum. (bottom two rows). The bending feature of the
the effective Hamiltonian in the vicinity of the To experimentally demonstrate the bulk Fermi contours is a result of higher-order terms in the
original DP at ðk D ; 0Þ can be written as (8, 21) arc, we use interference lithography to fabricate band dispersion (26). The open contour at 791.0 nm
PhC slabs in Si 3 N 4 (refractive index n ¼ 2:02, (middle row) is a clear, direct observation of the
H eff ¼ w D ig 0 þðv g dk x igÞs z
thickness t ¼ 220 nm) on top of a silica substrate bulk Fermi arc.
þ v g dk y s x ð1Þ
(n ¼ 1:46). The PhC structure consists of rhombic So far, we have shown one direct consequence
with complex eigenvalues of unit cells with side length a ¼ 525 nm, unit cell of the unique double–Riemann sheet topology
angle q ¼ 65:5°, and elliptical air holes with long- near paired EPs—the bulk Fermi arc. Next, we
axis length w ¼ 348 nm and short-axis length demonstrate another consequence: half-integer
w T ¼ w D ig 0
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
T ðv dk g Þ 2igv g dk x ð2Þ h ¼ 257 nm (26). Scanning electron microscope topological charges in the polarization configu-
2
2
2
g
(SEM) images of the fabricated samples are ration, which also serve as a direct experimental
1
Here, s x;z are Pauli matrices, w D is the DP fre- shown in Fig. 2A. The structure is immersed in proof of the n ¼ T 2 = topological index of an EP.
quency,ðdk x ; dk y Þ is the momentum displacement an optical liquid with refractive index matched These topological charges describe the direction
2
2
2
from ðk D ; 0Þ,and dk ¼ dk þ dk .Meanwhile, to that of the silica substrate to create an up- (clockwise or counterclockwise) and number of
x y
g 0 T g are the radiation decay rates of the even and down symmetric environment. times the polarization vector winds around a
odd y–mirror-symmetric modes, taking into We performed angle-resolved scattering mea- point or line singularity in the optical field, and
account that the two modes have different cou- surements (setup shown in Fig. 2B) to image in our particular system, we observe a robust
Zhou et al., Science 359, 1009–1012 (2018) 2 March 2018 2of4