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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
dξ
,
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
Ωz), λ sin (D
2
Ωz + 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 Ωz, −cos Ωz), (6)
where κ and Ω 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 Ω 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
