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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