scispace - formally typeset

Journal ArticleDOI

Adiabatic dynamics of edge waves in photonic graphene

13 Apr 2015-Vol. 2, Iss: 2, pp 024003

Abstract: The propagation of localized edge modes in photonic honeycomb lattices, formed from an array of adiabatically varying periodic helical waveguides, is considered. Asymptotic analysis leads to an explicit description of the underlying dynamics. Depending on parameters, edge states can exist over an entire period or only part of a period; in the latter case an edge mode can effectively disintegrate and scatter into the bulk. In the presence of nonlinearity, a 'time'-dependent one-dimensional nonlinear Schrodinger (NLS) equation describes the envelope dynamics of edge modes. When the average of the 'time varying' coefficients yields a focusing NLS equation, soliton propagation is exhibited. For both linear and nonlinear systems, certain long lived traveling modes with minimal backscattering are found; they exhibit properties of topologically protected states.
Topics: Envelope (waves) (52%)

Content maybe subject to copyright    Report


 !"#$%&'(
 ) *"+"!"&*#",##-*.//"#%-$%0-
'.1
2 33* 3$#*$#003"#%-$%0-3"3"3#",##- 433* 3$#*$#003"#%-
$%0-3"3"3#",##-5
6 )   (   
33***333"0,#73
2)')!&
2)'8*' 9 (
')!&3' */ 
(('( )'
(('(:
      )     (  
 )'32  *6
 ''*;''
(((' * 6('
)33***3*
6'<(:)(
)*63(
) (   ) 8 ! 
'=*&

arXiv:1411.6598v1 [physics.optics] 24 Nov 2014
Adiabatic Dynamics of Edge Waves in Photonic
Graphene
Mark J. Ablowitz
1
, Christopher W. Curtis
2
, and Yi-Ping Ma
1
1
Department of Applied Mathematics, University of Colorado, Boulder, Colorado
80309, USA
2
Department of Mathematics and Statistics, San Diego State University, San Diego,
California 92182, USA
Abstract. The propagation of localized edge modes in photonic honeycomb lattices,
formed from an array of adiabatically varying periodic helical waveguides, is considered.
Asymptotic analysis leads to an explicit description of the underlying dynamics.
Depending on parameters , edge states can exist over an entire p eriod or only par t
of a period; in the latter case an edge mode can effectively disinteg rate and scatter
into the bulk. In the presence of nonlinearity, a ‘time’-dependent one-dimens ional
nonlinear Schr¨odinger (NLS) equation describes the envelope dynamics of edge modes.
When the average of the ‘time varying’ coefficie nts yields a focusing NLS equation,
soliton propagation is exhibited. For both linear and nonlinear systems, certain long
lived traveling modes with minimal backscattering are found; they exhibit prope rties
of topologically protec ted states.
PACS numbers: 42.70.Qs, 42.65.Tg, 05.45.Yv

Adiabatic Dynamics of Edge Waves in Photonic Graphene 2
1. Introduction
Recently there has been significant effort directed towards understanding the wave
dynamics in photonic lattices arranged in a honeycomb structure cf. [1, 2, 3, 4, 5, 6, 7].
Due to the extra symmetry of the honeycomb lattice, Dirac points, or conical
intersections between dispersion bands, exist. This is similar to what occurs in carbon-
based graphene [9 ] where the existence of Dirac points is a key reason for many of
its exceptional pro perties. Because o f the correspondence with carbon-based graphene,
the optical analogue is often termed ‘photonic graphene’. In latt ices without edges the
wave dynamics exhibit conical, elliptic, and straight line diffraction cf. [1, 3, 10, 6].
However when edges and a ‘pseudo-field’ are present, remarkable changes occur and
long lived, persistent linear and nonlinear traveling edge waves with little backscatter
appear. These localized waves exhibit the hallmarks of topologically protected states,
thus indicating photonic graphene is a topological insulato r [12, 13, 8].
Substantial attentio n has been paid to the understanding of edge modes in both
condensed matter physics and optics. Interest in such modes goes back to the first
studies of the Quantum Hall Effect (QHE) where it was found that the edge current
was quantized [14, 15, 16]. Investigations related to the existence of edge states and the
geometry of eigenspaces of Schr¨odinger operators has also led to considerable interesting
research [17, 18, 19, 20, 21 , 22]. Suppo r t for the possible existence of linear unidirectional
modes in optical honeycomb lattices was provided in [23, 24]. These unidirectional
modes were found to be related to symmetry breaking perturbations which separated
the Dira c points in the dispersion surface. The modes are a consequence of a nontrivial
integer “topological” charge associated with the separated bands.
Unidirectional electromagnetic edge modes were first found experimentally in the
microwave r egime [25]. These modes were found on a square lattice which have no
associated Dirac points. Recently though, for photonic gra phene, it was shown in [26]
that by intr oducing edges and spatially varying waveguides that unidirectional edge wave
propagation at optical frequencies occurs. The waveguides play the role of a pseudo-
magnetic field, and in certain parameter regimes, the edge waves are found to be nearly
immune to backscattering. The pseudo-magnetic fields used in the experiment s [26] are
created by periodic changes in the index of refraction of the waveguides in the direction
of propagation. The variation in the index of refraction has a well defined helicity and
thus breaks ‘time’-reversal symmetry; here the direction of the wave propagation plays
the role of time.
The a nalytical description begins with the la t tice nonlinear Schr¨odinger (NLS)
equation [26] with cubic Kerr contribution
i∂
z
ψ =
1
2k
0
2
ψ +
k
0
n
n
0
ψ γ |ψ|
2
ψ, (1)
where k
0
is the input wavenumber, n
0
is the ambient refractive index, n/n
0
, referred to
as the potential, is the linear index change relative to n
0
, and γ represents the nonlinear
index contribution. The scalar field ψ is the complex envelope of the electric field, z is

Adiabatic Dynamics of Edge Waves in Photonic Graphene 3
the direction of propagation and takes on the role of time, (x, y) is the transverse plane,
and (
x
,
y
). Below, in Section 3.2, concrete va lues are given for the parameters in
Eq. (1). n is taken to be a 2D lattice potential in the (x, y) -plane which ha s a prescribed
path in the z-direction. This path is characterized by a function a(z) = (a
1
(z), a
2
(z)) ,
such that after the coo rdinate transformation
x
= x a
1
(z), y
= y a
2
(z), z
= z,
the transformed potential n = n(x
, y
) is independent of z
.
Exp erimentally, the path represented by a(z) can be written into the optical
material (e.g. fused silica) [26] via the femtosecond laser writing technique [27]. Since
this technique enables waveguides to be written along general paths, we only require
a(z) to be a smooth function. Introducing a transformed field
ψ =
˜
ψ exp
i
2k
0
Z
z
0
|A(ξ)|
2
,
where A is induced by the path function a via the formula
A(z) = k
0
a
(z), (2)
Eq. (1) is transformed to
i∂
z
˜
ψ =
1
2k
0
(
+ iA(z
))
2
˜
ψ +
k
0
n
n
0
˜
ψ γ
˜
ψ
2
˜
ψ. (3)
In Eq. (3), A appears in the same way as if one had added a magnetic field to Eq. (1);
hence A is referred to as a pseudo-magnetic field.
Taking l to be a typical lattice scale size, employing the dimensionless coordinates
x
= lx, y
= ly, z
= z
z, z
= 2k
0
l
2
,
˜
ψ =
P
ψ
, where P
is input peak power,
defining V (r
) = 2k
2
0
l
2
n/n
0
with r
(x
, y
), rescaling A accordingly, and dropping
the primes, we get the following norma lized lattice NLS equation
i∂
z
ψ = ( + iA(z))
2
ψ + V (r)ψ σ
0
|ψ|
2
ψ. (4)
The potential V (r) is taken to be of ho neycomb (HC) type. The dimensionless coefficient
σ
0
= 2γk
0
l
2
P
is the strength of the nonlinear change in the index of refraction. We also
note that after dropping primes, the dimensionless variables x, y, z, ψ are used; these
dimensionless variables should not be confused with the dimensional variables in Eq.
(1).
In contrast to [28], which in turn wa s motivated by the experiments in [2 6], this
paper examines t he case of periodic pseudo-fields which vary adiabatically, o r slowly,
throughout the photonic graphene la t t ice. We develop a n asymptotic theory which
leads to explicit formulas for isolated curves in the dispersion relation describing how
the structure of edge modes depends on a given pseudo-field A( z). Therefore, we can

Adiabatic Dynamics of Edge Waves in Photonic Graphene 4
theoretically predict for general pseudo-fields when unidirectional traveling waves exist
and the speed with which they propagate.
To exemplify the different classes of dispersion relations allowed in our problem, we
take the pseudo-magnetic field to be the following function (“ Lissajous” curves)
A(z) = (A
1
(z), A
2
(z)) = (κ sin (D
1
z), λ sin (D
2
z + φ)), (5)
where κ, λ, Ω, D
j
, j = 1, 2, and φ a r e constant. Below we will consider two cases
in detail. In the first case we choose D
1
= 1, D
2
= 1, φ = π/2, and κ = λ. This
corresponds to the pseudo-field employed in [26], and in most parts of this paper. In
this case the above function becomes a perfect circle given by
A(z) = (A
1
(z), A
2
(z)) = κ(sin z, cos z), (6)
where κ and are constant. In t he second case we choose D
1
= 2, D
2
= 1, φ = π/2,
and κ = λ. In this case the above Lissajous curve becomes a figure-8 curve given by
A(z) = (A
1
(z), A
2
(z)) = κ(sin (2Ωz), cos (Ωz)), (7)
where κ and are constant. For these classes of pseudo-fields, the numerically computed
dispersion relatio ns and the asymptotic pr ediction of the isolated curves agree very well.
However the dispersion relations also exhibit sensitive behavior in which small gaps in
the spectrum appear when the small parameter ǫ characterizing the slow evolution
increases in size.
Further, we find that when edge modes exist there are two impo r t ant cases. The
first is the case where pure edge modes exist in the entire periodic interval. The second
case is where quasi-edge mo des persist only for part o f the period. Quasi-edge modes
do not exist in the rapidly varying case studied in [28], and so we have shown adiabatic
variation of the pseudo-field allows for new dynamics even at the linear level. We also
present potentia l scaling regimes where these cases mig ht be observed experimentally;
see Section 3.2.
We are also able to analyze the effect of nonlinearity on these slowly varying
traveling edge modes. A nonconstant coefficient (‘time’-dependent) one-dimensional
nonlinear Schr¨odinger (NLS) equation governing the envelope of the edge modes is
derived and is found to be an effective description o f nonlinear traveling edge modes.
In the rapidly varying case [28], the associated NLS equation had constant coefficients,
and so we see adia batic va riation introduces new dynamics into the nonlinear evolution
of edge modes.
Using this new NLS equation, in the focusing case, we find analytically, and confirm
numerically, that unidirectionally propagat ing edge solitons are present in nonlinear
photonic graphene lat t ices. Computation of the NLS equation is compared with direct
simulation of the coupled discrete tight binding mo del with very good agreement
obtained. As with the traditional, constant coefficient, focusing NLS equation,
nonlinearity balances dispersion to produce nonlinear edge solitons. Depending on

Figures (10)
Citations
More filters

Journal ArticleDOI
Abstract: Topological photonics is a rapidly emerging field of research in which geometrical and topological ideas are exploited to design and control the behavior of light. Drawing inspiration from the discovery of the quantum Hall effects and topological insulators in condensed matter, recent advances have shown how to engineer analogous effects also for photons, leading to remarkable phenomena such as the robust unidirectional propagation of light, which hold great promise for applications. Thanks to the flexibility and diversity of photonics systems, this field is also opening up new opportunities to realize exotic topological models and to probe and exploit topological effects in new ways. This article reviews experimental and theoretical developments in topological photonics across a wide range of experimental platforms, including photonic crystals, waveguides, metamaterials, cavities, optomechanics, silicon photonics, and circuit QED. A discussion of how changing the dimensionality and symmetries of photonics systems has allowed for the realization of different topological phases is offered, and progress in understanding the interplay of topology with non-Hermitian effects, such as dissipation, is reviewed. As an exciting perspective, topological photonics can be combined with optical nonlinearities, leading toward new collective phenomena and novel strongly correlated states of light, such as an analog of the fractional quantum Hall effect.

2,132 citations


Journal ArticleDOI
21 Mar 2016
Abstract: Edge states are time-harmonic solutions to energy-conserving wave equations, which are propagating parallel to a line-defect or 'edge' and are localized transverse to it. This paper summarizes and extends the authors' work on the bifurcation of topologically protected edge states in continuous two-dimensional (2D) honeycomb structures. We consider a family of Schrodinger Hamiltonians consisting of a bulk honeycomb potential and a perturbing edge potential. The edge potential interpolates between two different periodic structures via a domain wall. We begin by reviewing our recent bifurcation theory of edge states for continuous 2D honeycomb structures (http://arxiv.org/abs/1506.06111). The topologically protected edge state bifurcation is seeded by the zero-energy eigenstate of a one-dimensional Dirac operator. We contrast these protected bifurcations with (more common) non-protected bifurcations from spectral band edges, which are induced by bound states of an effective Schrodinger operator. Numerical simulations for honeycomb structures of varying contrasts and 'rational edges' (zigzag, armchair and others), support the following scenario: (a) for low contrast, under a sign condition on a distinguished Fourier coefficient of the bulk honeycomb potential, there exist topologically protected edge states localized transverse to zigzag edges. Otherwise, and for general edges, we expect long lived edge quasi-modes which slowly leak energy into the bulk. (b) For an arbitrary rational edge, there is a threshold in the medium-contrast (depending on the choice of edge) above which there exist topologically protected edge states. In the special case of the armchair edge, there are two families of protected edge states; for each parallel quasimomentum (the quantum number associated with translation invariance) there are edge states which propagate in opposite directions along the armchair edge.

31 citations


Journal ArticleDOI
Yu-Liang Tao1, Ning Dai1, Yan-Bin Yang1, Qi-Bo Zeng1  +1 moreInstitutions (1)
Abstract: A second-order topological insulator in three dimensions refers to a topological insulator with gapless states localized on the hinges, which is a generalization of a traditional topological insulator with gapless states localized on the surfaces. Here we theoretically demonstrate the existence of stable solitons localized on the hinges of a second-order topological insulator in three dimensions when nonlinearity is involved. By means of systematic numerical study, we find that the soliton has strong localization in real space and propagates along the hinge unidirectionally without changing its shape. We further construct an electric network to simulate the second-order topological insulator. When a nonlinear inductor is appropriately involved, we find that the system can support a bright soliton for the voltage distribution demonstrated by stable time evolution of a voltage pulse.

11 citations


Journal ArticleDOI
Yu-Liang Tao1, Ning Dai1, Yan-Bin Yang1, Qi-Bo Zeng1  +1 moreInstitutions (1)
Abstract: Higher-order topological insulators have recently witnessed rapid progress in various fields ranging from condensed matter physics to electric circuits. A well-known higher-order state is the second-order topological insulator in three dimensions with gapless states localized on the hinges. A natural question in the context of nonlinearity is whether solitons can exist on the hinges in a second-order topological insulator. Here we theoretically demonstrate the existence of stable solitons localized on the hinges of a second-order topological insulator in three dimensions when nonlinearity is involved. By means of systematic numerical study, we find that the soliton has strong localization in real space and propagates along the hinge unidirectionally without changing its shape. We further construct an electric network to simulate the second-order topological insulator. When a nonlinear inductor is appropriately involved, we find that the system can support a bright soliton for the voltage distribution demonstrated by stable time evolution of a voltage pulse.

8 citations


Journal ArticleDOI
Abstract: For a dissipative variant of the two-dimensional Gross--Pitaevskii equation with a parabolic trap under rotation, we study a symmetry breaking process that leads to the formation of vortices. The first symmetry breaking leads to the formation of many small vortices distributed uniformly near the Thomas--Fermi radius. The instability occurs as a result of a linear instability of a vortex-free steady state as the rotation is increased above a critical threshold. We focus on the second subsequent symmetry breaking, which occurs in the weakly nonlinear regime. At slightly above threshold, we derive a one-dimensional amplitude equation that describes the slow evolution of the envelope of the initial instability. We show that the mechanism responsible for initiating vortex formation is a modulational instability of the amplitude equation. We also illustrate the role of dissipation in the symmetry breaking process. All analyses are confirmed by detailed numerical computations.

4 citations


References
More filters

Journal ArticleDOI
Andre K. Geim1, Kostya S. Novoselov1Institutions (1)
TL;DR: Owing to its unusual electronic spectrum, graphene has led to the emergence of a new paradigm of 'relativistic' condensed-matter physics, where quantum relativistic phenomena can now be mimicked and tested in table-top experiments.
Abstract: Graphene is a rapidly rising star on the horizon of materials science and condensed-matter physics. This strictly two-dimensional material exhibits exceptionally high crystal and electronic quality, and, despite its short history, has already revealed a cornucopia of new physics and potential applications, which are briefly discussed here. Whereas one can be certain of the realness of applications only when commercial products appear, graphene no longer requires any further proof of its importance in terms of fundamental physics. Owing to its unusual electronic spectrum, graphene has led to the emergence of a new paradigm of 'relativistic' condensed-matter physics, where quantum relativistic phenomena, some of which are unobservable in high-energy physics, can now be mimicked and tested in table-top experiments. More generally, graphene represents a conceptually new class of materials that are only one atom thick, and, on this basis, offers new inroads into low-dimensional physics that has never ceased to surprise and continues to provide a fertile ground for applications.

32,822 citations


Journal ArticleDOI
M. Z. Hasan1, Charles L. Kane2Institutions (2)
Abstract: Topological insulators are electronic materials that have a bulk band gap like an ordinary insulator but have protected conducting states on their edge or surface. These states are possible due to the combination of spin-orbit interactions and time-reversal symmetry. The two-dimensional (2D) topological insulator is a quantum spin Hall insulator, which is a close cousin of the integer quantum Hall state. A three-dimensional (3D) topological insulator supports novel spin-polarized 2D Dirac fermions on its surface. In this Colloquium the theoretical foundation for topological insulators and superconductors is reviewed and recent experiments are described in which the signatures of topological insulators have been observed. Transport experiments on $\mathrm{Hg}\mathrm{Te}∕\mathrm{Cd}\mathrm{Te}$ quantum wells are described that demonstrate the existence of the edge states predicted for the quantum spin Hall insulator. Experiments on ${\mathrm{Bi}}_{1\ensuremath{-}x}{\mathrm{Sb}}_{x}$, ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$, ${\mathrm{Bi}}_{2}{\mathrm{Te}}_{3}$, and ${\mathrm{Sb}}_{2}{\mathrm{Te}}_{3}$ are then discussed that establish these materials as 3D topological insulators and directly probe the topology of their surface states. Exotic states are described that can occur at the surface of a 3D topological insulator due to an induced energy gap. A magnetic gap leads to a novel quantum Hall state that gives rise to a topological magnetoelectric effect. A superconducting energy gap leads to a state that supports Majorana fermions and may provide a new venue for realizing proposals for topological quantum computation. Prospects for observing these exotic states are also discussed, as well as other potential device applications of topological insulators.

12,967 citations


Journal ArticleDOI
Abstract: The richness of optical and electronic properties of graphene attracts enormous interest. Graphene has high mobility and optical transparency, in addition to flexibility, robustness and environmental stability. So far, the main focus has been on fundamental physics and electronic devices. However, we believe its true potential lies in photonics and optoelectronics, where the combination of its unique optical and electronic properties can be fully exploited, even in the absence of a bandgap, and the linear dispersion of the Dirac electrons enables ultrawideband tunability. The rise of graphene in photonics and optoelectronics is shown by several recent results, ranging from solar cells and light-emitting devices to touch screens, photodetectors and ultrafast lasers. Here we review the state-of-the-art in this emerging field.

6,298 citations


Journal ArticleDOI
Abstract: Measurements of the Hall voltage of a two-dimensional electron gas, realized with a silicon metal-oxide-semiconductor field-effect transistor, show that the Hall resistance at particular, experimentally well-defined surface carrier concentrations has fixed values which depend only on the fine-structure constant and speed of light, and is insensitive to the geometry of the device. Preliminary data are reported.

4,931 citations


Journal ArticleDOI
Charles L. Kane1, Eugene J. Mele1Institutions (1)
TL;DR: The Z2 order of the QSH phase is established in the two band model of graphene and a generalization of the formalism applicable to multiband and interacting systems is proposed.
Abstract: The quantum spin Hall (QSH) phase is a time reversal invariant electronic state with a bulk electronic band gap that supports the transport of charge and spin in gapless edge states. We show that this phase is associated with a novel Z2 topological invariant, which distinguishes it from an ordinary insulator. The Z2 classification, which is defined for time reversal invariant Hamiltonians, is analogous to the Chern number classification of the quantum Hall effect. We establish the Z2 order of the QSH phase in the two band model of graphene and propose a generalization of the formalism applicable to multiband and interacting systems.

4,328 citations


Network Information
Related Papers (5)
Performance
Metrics
No. of citations received by the Paper in previous years
YearCitations
20212
20202
20171
20162
20151
20141