**footnotetext<EMAIL_ADDRESS>
# Point-Defect-Localized Bound States in the Continuum in Photonic Crystals
and Structured Fibers

Sachin Vaidya,1,∗ Wladimir A. Benalcazar,1 Alexander Cerjan,1,2,3 and Mikael
C. Rechtsman1 1Department of Physics, The Pennsylvania State University,
University Park, Pennsylvania 16802, USA
2Sandia National Laboratories, Albuquerque, New Mexico 87123, USA
3Center for Integrated Nanotechnologies, Sandia National Laboratories,
Albuquerque 87123, New Mexico, USA

###### Abstract

We show that point defects in two-dimensional photonic crystals can support
bound states in the continuum (BICs). The mechanism of confinement is a
symmetry mismatch between the defect mode and the Bloch modes of the photonic
crystal. These BICs occur in the absence of bandgaps and therefore provide an
alternative mechanism to confine light. Furthermore, we show that such BICs
can propagate in a fiber geometry and exhibit arbitrarily small group velocity
which could serve as a platform for enhancing non-linear effects and light-
matter interactions in structured fibers.

Over the last three decades, photonic crystals (PhCs) have been shown to
exhibit exceptional confinement and transport properties that exploit the
existence of a photonic bandgap, a band of frequencies where no
electromagnetic waves may propagate Joannopoulos  _et al._ (2008); Sakoda
(2005); Joannopoulos  _et al._ (1997); John (1987). Photonic bandgaps can
inhibit spontaneous emission of embedded quantum emitters Yablonovitch (1987);
Noda  _et al._ (2007); Fujita  _et al._ (2005); Singh (2007), facilitate slow-
light through band-edge operation Baba (2008) or host localized defect modes
that can serve as high-Q resonators or waveguides. Confined defect modes form
the basis of many devices such as PhC fibers Knight (2003); Russell (2003),
spectral filters, and lasers Imada  _et al._ (1999); Kurosaka  _et al._ (2010)
and to achieve near-perfect confinement, defect modes are constructed to lie
within photonic bandgaps so as to spectrally isolate them from the extended
states of the PhC. However, this necessitates the use of materials with a
sufficiently high refractive index to open complete gaps. An alternative
mechanism for confinement could circumvent the need for bandgaps, enabling the
use of many low-refractive index materials such as glasses and polymers as
well as increasing design flexibility for the realization of PhC-based
devices.

One possible way to achieve this is by using bound states in the continuum
(BICs). BICs are eigenmodes of a system that, despite being degenerate with a
continuum of extended states, stay confined – this confinement may result from
a variety of mechanisms Hsu  _et al._ (2016). For example, modes of a PhC slab
that lie above the light line of vacuum and therefore could radiate, can
remain perfectly bound to the slab Pankin  _et al._ (2020); Taghizadeh and
Chung (2017); Cerjan  _et al._ (2019); Hsu  _et al._ (2013); Lee  _et al._
(2012); Liu  _et al._ (2009). Previous designs with BICs have mostly shown
confinement of a mode in one dimension lower than that of the environment.
Recently, corner-localized BICs were predicted and observed in two-dimensional
chiral-symmetric systems with higher-order topology Benalcazar and Cerjan
(2020); Cerjan  _et al._ (2020). However, chiral (sub-lattice) symmetry is, in
general, strongly broken in all-dielectric PhCs. Indeed, confinement in the
continuum has to this point not yet been achieved in point defects embedded
inside multi-dimensional PhCs.

In this work, we predict the existence of BICs that are exponentially confined
to point defects in a two-dimensional PhC environment. The defect cavity and
bulk PhC are designed such that radiation leakage is prohibited due to a
symmetry mismatch between the defect mode and the ambient continuum states.
The BICs proposed here are protected by the simultaneous presence of time-
reversal symmetry (TRS) and the point group of the lattice and as such are
robust as long as these symmetries are maintained. As an application for these
BICs, we also show how they can circumvent bandgap requirements and be used as
propagating fiber modes with arbitrarily small group velocity in a low-
contrast slow-light PhC fiber.

Figure 1: (a) The unit cell of a two-dimensional PhC consisting of circular
discs. The symmetry operators of the $C_{4v}$ point group are labelled. (b)
The Brillouin zone of the PhC showing its HSPs and the little groups under
which the HSPs are invariant. The solid color consists of all momenta that lie
within the irreducible Brillouin zone.

We draw a distinction between our BICs and the previously reported defect
modes degenerate with Dirac points in 2D PhCs Xie  _et al._ (2014); Mao  _et
al._ (2016, 2017); Hu  _et al._ (2018); Yang  _et al._ (2019); Xie  _et al._
(2015). In the latter case, the confinement of light to a defect site is due
to a vanishing density of states at the Dirac point, which is where that
confined mode’s frequency lies. Characteristically, such defect modes exhibit
weak confinement due to the algebraic mode profile away from the defect site.
In contrast, the defect modes presented here are bona fide symmetry protected
BICs that are exponentially localized to the defect site.

Consider a two-dimensional PhC consisting of a square lattice of discs with
dielectric constant $\varepsilon$ and radius $r$ embedded in vacuum. This PhC,
as shown in Fig. 1 (a) is invariant under 90∘ rotations ($C_{4}$, $C_{4}^{2}$,
$C_{4}^{-1}$), and reflections along the $x$, $y$ axes and two diagonals
($\sigma_{x}$, $\sigma_{y}$, $\sigma_{d_{1}}$, $\sigma_{d_{2}}$). These
symmetry operations constitute the $C_{4v}$ point group. The irreducible
Brillouin zone of this lattice contains three inequivalent high symmetry
points (HSPs), namely, $\mathbf{\Gamma}=(0,0)$,
$\mathrm{\mathbf{X}}=(\pi/a,0)$ and $\mathrm{\mathbf{M}}=(\pi/a,\pi/a)$, as
shown in Fig. 1 (b). The HSPs $\mathbf{\Gamma}$ and $\mathrm{\mathbf{M}}$ are
invariant under the full $C_{4v}$ group, while $\mathrm{\mathbf{X}}$ is
invariant only under the little group, $C_{2v}$. Eigenmodes of the PhC at a
HSP transform according to the irreducible symmetry representations (irrep) of
the group under which the HSP is invariant. The $\mathrm{\mathbf{X}}$ point
has four possible one-dimensional irreps $(a_{1},a_{2},b_{1},b_{2})$ with
character table as shown in Table I. Similarly, the $\mathbf{\Gamma}$ and
$\mathrm{\mathbf{M}}$ points have four one-dimensional irreps
$(A_{1},A_{2},B_{1},B_{2})$ and one two-dimensional irrep $(E)$ with character
table as shown in Table II Sakoda (2005). The eigenmodes of a $C_{4v}$
symmetric PhC that transform according to the two-dimensional irrep $(E)$ of
the $C_{4v}$ point group, commonly manifest as quadratic two-fold degeneracies
at $\mathbf{\Gamma}$ and $\mathrm{\mathbf{M}}$ in the presence of TRS. When
$C_{4v}$ is broken, this degeneracy splits into two Dirac points as long as
inversion and TRS are retained. However, breaking TRS can lift the degeneracy
completely Chong  _et al._ (2008).

Figure 2: (a) The TM bands and photonic DoS of a square lattice of dielectric
discs of $\varepsilon=4$ and $r/a=0.275$ calculated using MPB Johnson and
Joannopoulos (2001). The spectrally-isolated two-fold degeneracy is marked
with an arrow. (b) Quality factor (Q) of the defect mode as a function of
defect radius $(r_{d})$. The sharp divergence in Q indicates the existence of
a BIC at $r_{d}/a=0.224$. The inset shows the dependence of the defect mode
frequency on $r_{d}$. (c) The $\mathbf{E}$-field intensity envelope of the BIC
showing exponential localization as a function of distance (along the
$y$-axis) from the defect site. The inset shows the $z$-component of the
$\mathbf{E}$-field of the BIC, extracted from FDTD simulations. (d) The
$\mathbf{E}$-field intensity envelope of the resonance when the symmetry of
the supercell is reduced from $C_{4v}$ to $C_{2v}$. The inset shows the
$z$-component of the $\mathbf{E}$-field of the resonance, extracted from FDTD
simulations.

$C_{2v}$ | $\mathrm{I}$ | $C_{2}$ | $\sigma_{x}$ | $\sigma_{y}$
---|---|---|---|---
$a_{1}$ |  1 |  1 |  1 |  1
$a_{2}$ |  1 |  1 | $-1$ | $-1$
$b_{1}$ |  1 | $-1$ |  1 | $-1$
$b_{2}$ |  1 | $-1$ | $-1$ |  1

Table 1: Character table for the $C_{2v}$ point group.

$C_{4v}$ | $\mathrm{I}$ | $2C_{4}$ | $C_{2}$ | $2\sigma_{x,y}$ | $2\sigma_{d_{1},d_{2}}$
---|---|---|---|---|---
$A_{1}$ |  1 |  1 |  1 |  1 |  1
$A_{2}$ |  1 |  1 |  1 | $-1$ | $-1$
$B_{1}$ |  1 | $-1$ |  1 |  1 | $-1$
$B_{2}$ |  1 | $-1$ |  1 | $-1$ |  1
$E$ |  2 |  0 | $-2$ |  0 |  0

Table 2: Character table for the $C_{4v}$ point group.

We now describe the general mechanism for creating defect-localized BICs. By
changing the geometric parameters of the lattice, the band dispersion of a
$C_{4v}$ and TRS-symmetric PhC can be designed such that the two-fold
degeneracy at either $\mathbf{\Gamma}$ or $\mathrm{\mathbf{M}}$ is spectrally
isolated from other bands. In a large system consisting of many unit cells of
such a PhC (a supercell), a single defect site with radius $r_{d}\neq r$ is
introduced at the center. This creates modes with a significant support on the
defect site that generally radiate by hybridizing with the bulk states of the
PhC, forming leaky resonances that are characterized by a complex frequency
with a negative imaginary part. The frequency of such modes can be tuned by
changing the parameters of the defect site such as size or dielectric
constant. When the real part of the frequency of the defect mode exactly
matches that of the spectrally-isolated two-fold degeneracy of the bulk, it
becomes a perfectly confined BIC provided that the defect mode transforms
according to a one-dimensional irrep that is orthogonal to the two-dimensional
irrep of the bulk. The presence of this BIC can be inferred from the vanishing
of the imaginary part of the frequency and hence a diverging quality factor,
$Q=-\mathrm{Re}(\omega)/2\mathrm{Im}(\omega)$, of the defect mode.

To demonstrate this, we simulate this system using finite-difference time
domain method (FDTD) as implemented in MEEP Oskooi  _et al._ (2010). The bulk
band requirements are easily met in a simple square lattice of discs with
dielectric constant $\varepsilon=4$ and radius $r/a=0.275$, where $a$ is the
lattice constant in both $x$ and $y$ directions. The chosen values of
$\varepsilon$ and $r/a$ allow the spectrally-isolated two-fold degeneracy to
occur between TM bands 10 and 11 at the $\mathbf{M}$ point as shown in Fig. 2
(a). The photonic density of states (DoS), also shown in the same figure, is
given by
$\mathrm{DoS}(\omega)=\sum_{n}\int_{\mathbf{k}\in\mathrm{BZ}}\delta[\omega-\omega_{n}(\mathbf{k})]\mathrm{d}\mathbf{k}$,
where $\omega_{n}(\mathbf{k})$ is the frequency eigenvalue at the momentum
$\mathbf{k}$ and band index $n$. Since each band undergoes an extremum at the
degeneracy, the DoS exhibits a jump-discontinuity-type Van Hove singularity
between two finite and non-zero values. The non-vanishing set of states at the
degeneracy forms the continuum within which a BIC can be created.

In a large supercell, we now introduce a defect by changing the radius
($r_{d}\neq r$) of a single disc in the center of the supercell. As we scan
the values of $r_{d}$, a BIC emerges for the specific value of the defect
radius that corresponding to a mode with the exact frequency of the bulk
degeneracy. This is seen from the sharp divergence of the Q-factor of the
defect mode as shown in Fig. 2 (b). Examining the mode profile shown in the
inset of Fig. 2 (c) reveals that the defect mode transforms according to the
irrep $A_{1}$ which is prevented from mixing with the basis modes of the
orthogonal two-dimensional irrep, $E$, of the bulk. Moreover, the mode shows
very strong exponential localization to the defect site which can be seen by
plotting the intensity envelope as shown in Fig. 2 (c). Another important
feature of this BIC is its occurrence above $\omega a/2\pi c=1$. This implies
that the lattice constant of the bulk PhC is larger than the wavelength of the
BIC mode, a property which could prove useful for fabrication, because
features sizes would need not be subwavelength.

Figure 3: (a) The $k_{\parallel}$-band structure of the defect-free PhC fiber
at $k_{z}=0.18$ $(2\pi/a_{\parallel})$. The spectrally-isolated two-fold
degeneracy is marked with an arrow. (b) $\mathbf{D}$-field intensity profile
of a solid-core fiber BIC mode that occurs at $k_{z}=0.18$
$(2\pi/a_{\parallel})$. (c) $\mathbf{D}$-field intensity profile of a hollow-
core-like fiber BIC mode.

To conclusively show that this BIC is indeed symmetry protected, we change the
defect site from a disc to a filled ellipse, which reduces the symmetry of the
supercell from $C_{4v}$ to $C_{2v}$. Due to this deformation, the degeneracy
between the two modes that formed the two-dimensional irrep, $E$, of $C_{4v}$
is very slightly lifted and the resultant non-degenerate modes have the one-
dimensional irreps $b_{1}$ and $b_{2}$ of $C_{2v}$. As before, we vary the
defect size to tune the frequency of the defect mode and find a maximum $Q\sim
10^{4}$ indicating that the mode is not a BIC but a resonance. Indeed, the
field pattern of the defect mode as shown in the inset of Fig 2 (d),
transforms according to $b_{2}$, which coincides with one of the irreps of the
bulk enabling the defect and bulk modes to couple and create a leaky resonance
with a finite Q (see Supplementary Material). This is also evident from the
intensity envelope of the resonance as shown in Fig 2 (d) that markedly
demonstrates the lack of exponential confinement to the defect site.
Displacing the defect site away from the center also breaks the $C_{4v}$
symmetry of the supercell and has a similar effect of degrading the Q-factor
of the mode (see Supplementary Material).

The symmetry mismatch between the defect mode and bulk bands requires the
existence of a spectrally-isolated two-fold degeneracy in the bulk PhC so the
question naturally arises: how easy is it to design this bulk band
requirement? It is clear from our findings that even simple PhC designs are
able to satisfy the requirements for reasonably low dielectric contrast and in
fact, the feature in the TM bands of the PhC discussed in Fig. 2 (a) persists
down to $\varepsilon=3$ for a slightly smaller value of $r/a$. Furthermore,
such quadratic degeneracies can also occur at the $\mathbf{\Gamma}$ point in
$C_{3v}$ and $C_{6v}$ symmetric lattices, forming two-dimensional irreps of
the respective point groups. In the Supplementary Material, we outline a
method for finding optimized structures with tunable parameters that exhibit
such degeneracies.

For traditional defect modes in 2D PhCs, it suffices to have a bandgap for one
polarization, either TE or TM, since they constitute orthogonal subspaces that
do not mix. However, for applications such as PhC fibers, (i.e., where the 2D
pattern described above is extruded in the third direction, $z$, and
$k_{z}\neq 0$ generally), the distinction between TE and TM is lost and one
requires an overlapping bandgap for both polarizations to confine defect
modes. In particular, slow-light PhC fibers rely on the existence of a
complete bandgap at $k_{z}=0$ which persists for a small range of $k_{z}$
Grgić  _et al._ (2010); Lægsgaard (2011); Oskooi  _et al._ (2009). The
arbitrarily small group velocity of the propagating modes in such fibers is
achieved by operating near the $k_{z}=0$ band edge. These slowly-propagating
modes can then be used to strongly enhance interactions of light with either
the dielectric material itself or an infiltrated material Donvalkar  _et al._
(2015); Sparks  _et al._ (2011), depending on whether the fiber hosts a solid
or hollow core. Thus, the design of these fibers requires a high dielectric
contrast to open a complete bandgap at $k_{z}=0$. To the best of our
knowledge, the smallest contrast for which a complete bandgap exists for 2D
PhCs is for $\varepsilon=4.41$ Cerjan and Fan (2017). We now extend the idea
of point-defect-localized BICs to propagating slow-light fiber modes,
circumventing the requirement for a complete bandgap.

The fiber design that we propose is identical to an extruded version of the 2D
PhC discussed before, now consisting of cylinders extended along the direction
of propagation in the fiber. However, since the distinction between TE and TM
polarizations is lost, the spectrally-isolated two-fold degeneracy of the bulk
must occur in the full band structure in order to create a BIC. This is easily
achieved in our structure for a range of $k_{z}$ values around $0$. For
instance, Fig. 3 (a) shows the band structure of the fiber with
$\varepsilon=4$, $r/a=0.2755$ at $k_{z}=0.18$ $(2\pi/a_{\parallel})$, where
$a_{\parallel}$ is the lattice constant in the $x,y$ plane. As before, we
introduce a defect site and tune the radius $r_{d}$ and find a BIC at
$r_{d}/a=0.230$ for this particular value of $k_{z}$. The field profile of the
BIC is plotted in Fig. 3 (b), forming a solid-core mode and displaying strong
confinement to the defect site. Since the spectrally-isolated two-fold
degeneracy persists down to $k_{z}=0$, the group velocity of this BIC along
the length of the fiber $(v_{g_{z}}=\mathrm{d}\omega/\mathrm{d}k_{z})$ can be
made arbitrarily small with an appropriate choice of $r_{d}$. It is also
possible to create a hollow-core-like fiber mode where the BIC has reasonable
support in the air region. To achieve this, we omit the central defect site
and instead tune the radius of the nearest eight sites uniformly so as to
maintain $C_{4v}$ and find a BIC as shown in Fig. 3 (c).

The BICs presented here could be experimentally realized in a variety of
systems. For example, these principles could be applied to create high-Q
nanocavities in gapless PhC slabs where some vertical leakage is unavoidable
but in-plane leakage could be suppressed through the symmetry mismatch
mechanism. Functionally, these modes would behave similarly to run-of-the-mill
PhC slab-based cavities that rely on a bandgap but could be realizable in
alternative structures with potentially lower dielectric contrast. Similarly,
the PhC fiber design discussed here could be implemented straightforwardly by
complex fiber drawing techniques Russell (2003). Furthermore, such isolated
degeneracies are also known to occur in 3D PhCs which could lead to true
gapless confinement of light in all directions such as in structures that are
precursors to ones with Weyl points Lu  _et al._ (2013); Wang  _et al._
(2016); Yang  _et al._ (2017); Vaidya  _et al._ (2020). Evidently, these BICs
rely solely on symmetry considerations and can also be readily realized using
other periodic systems such as acoustic crystals, waveguides Plotnik  _et al._
(2011); Cerjan  _et al._ (2020) and coupled resonator arrays.

In conclusion, we have proposed BICs that are exponentially localized to
defects beyond bandgaps in both 2D PhCs and structured fibers. The PhC slow-
light fiber implementation relaxes the need for bandgaps at $k_{z}=0$ and thus
allows for a wider range of materials to be used for their implementation. The
results presented here have consequences for the general design of PhC-based
devices since the requirement for finding bandgaps could potentially be
replaced with finding isolated degeneracies at HSPs, which occur more
commonly, at lower dielectric contrast and at higher frequencies in the band
structure. Furthermore, it may be possible to use the BIC mechanism to realize
hinge modes in higher order photonic topological insulators Noh  _et al._
(2018); Peterson  _et al._ (2018); Benalcazar  _et al._ (2017); Benalcazar and
Cerjan (2020); Cerjan  _et al._ (2020) due to their structural similarity with
PhC fiber modes.

###### Acknowledgements.

M. C. R. acknowledges support from the Office of Naval Research (ONR)
Multidisciplinary University Research Initiative (MURI) grant N00014-20-1-2325
on Robust Photonic Matertials with High-Order Topological Protection as well
as the Packard Foundation under fellowship number 2017-66821. W.A.B. is
grateful for the support of the Eberly Postdoctoral Fellowship at the
Pennsylvania State University. This work was performed, in part, at the Center
for Integrated Nanotechnologies, an Office of Science User Facility operated
for the U.S. Department of Energy (DOE) Office of Science. Sandia National
Laboratories is a multimission laboratory managed and operated by National
Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary
of Honeywell International, Inc., for the U.S. DOE’s National Nuclear Security
Administration under contract DE-NA-0003525. The views expressed in the
article do not necessarily represent the views of the U.S. DOE or the United
States Government. Computations for this research were performed on the
Pennsylvania State University’s Institute for Computational and Data Sciences
Roar supercomputer.

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