Adiabatic dynamics of edge waves in photonic graphene

TL;DR: In this paper, the propagation of localized edge modes in photonic honeycomb lattices, formed from an array of adiabatically varying periodic helical waveguides, is considered.

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.

Summary

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].
• The pseudo-magnetic fields used in the experiments [26] are created by periodic changes in the index of refraction of the waveguides in the direction of propagation.
• 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, the authors find that when edge modes exist there are two important cases.
• The results in this and previous papers show that nonlinear photonic graphene can also be thought of as a topological insulator.

2. Discrete Equations

• This generalizes [33] where purely stationary edge modes are obtained.
• The function C(Z, ω) is determined below.

3. Linear system

• The authors first return to investigate the properties of the edge solution (20-21).
• In the case where the topological index is nontrivial, i.e.
• Interestingly, the dispersion relation of edge modes in Fig. 2(a) breaks up into multiple segments near ω = 0, π. Also, in Fig. 2(b) there are scattered eigenvalues near α = 0 for ω immediately outside Ip = (ω−, ω+).
• To illustrate the possibility of an I = 0 (topologically trivial) dispersion relation in Case (II), the authors take the pseudo-field A(Z) to be Eq. (7).
• Thus the initial condition (28) with δ = 1 remains localized for part of the period before disintegrating into the bulk with power distributed into both an and bn.

3.1. Numerical Computation of Dispersion Relations

• It is interesting to see where the dispersion relations of pure edge modes as shown in Fig. 2 lie in the full Floquet spectra.
• The Floquet spectrum is computed using Eqs. (12– 13) with a finite number of lattice sites, taken to be d/2, for each vector a and b and zig-zag boundary conditions on both ends.
• From this, the authors can determine each eigenvalue λj up to an integer multiple of 2π/T ; this represents an ambiguity that cannot be resolved by studying Q(T ) alone.
• Let us consider the particular choice f = Vj(z0) where z0 is arbitrary.
• For either ρ < 1/2 or ρ > 1/2, the overall structure of the spectrum is similar to the case where the pseudo-field A is absent [33], though in their case the bulk spectrum no longer consists of regular bands.

3.2. Length Scales Associated with Quasi-Edge States

• The scales of the problem suggest that both pure and quasi- edge modes might be observable in experiments.
• A discussion of potential scales for such an observation follows.
• In the introduction it was shown that the typical length scale in the longitudinal direction is z∗ = 2k0l 2, where k0 = 2πn0 λ , λ being the input wavelength.
• It is useful to compare this length scale with the fast evolution problem discussed in [26, 28].
• As in the slow evolution case, a perfect HC lattice with ρ = 1 was taken, while the strength of the pseudo-field was κ = 1.4.

4. Nonlinear Two-Dimensional Localized Edge Modes

• Importantly, nonlinear edge modes can also be constructed via the same asymptotic analysis used in Sec. 3.
• In all cases, it can be seen that the localized mode is eventually destroyed by dispersion after sufficient evolution.
• Figure 10 shows the nonlinear evolution at the same parameters as Fig. 8 but with σ 6= 0. As shown in Fig. 10(a,b), when the NLS equation has third order dispersion due to ᾱ′′(ω0) = 0, weak nonlinearity enhances dispersion somewhat.
• As in the linear case, the 2D evolution becomes somewhat weaker than predicted by the NLS equation beyond the time scale z ∼ 1/(ǫν3) for panels (a,b) and z ∼ 1/(ǫν2) for panels (c–f).
• This absence of backscattering in the presence of nonlinearity suggests that the edge soliton is indeed topologically protected in the same parameter regime as topologically protected linear modes.

5. Conclusion

• A method is developed which describes the propagation of edge modes in a semi-infinite honeycomb lattice in the presence of a periodically and relatively slowly varying pseudo-field with weak nonlinearity.
• In the linear case, the dispersion relations of pure edge modes indicate that some modes may exhibit topological protection.
• With weak nonlinearity included, it is shown that in the narrow band approximation, a time-dependent NLS equation is obtained.
• These 1D NLS solitons correspond to true edge solitons propagating on the edge of the semi-infinite honeycomb lattice.
• On the other hand when the NLS equation is defocusing significant dispersion occurs.

Figures

Figure 1: The width |Ip| of the existence interval Ip of pure edge states as a function of the deformation parameter ρ and the pseudo-field parameter κ. The color scale interpolates between white for 0 and black for π. The dispersion relations at the labeled points (a)–(b) are shown in Fig. 2.

Figure 2: The dispersion relation of pure edge modes for points (a) and (b) in Fig. 1 computed using the circular pseudo-field Eq. (6) with κ = 0.3 and (a) ρ = 0.4; (b) ρ = 1. Panel (c) is computed using the Lissajous pseudo-field Eq. (5) with κ = λ = 0.3, D1 = 2, D2 = 1, φ = π/2, and ρ = 1. The number of lattice sites is 40 and the small parameter is ǫ = 2π/60. The blue curve represents pure edge modes computed numerically from Eqs. (12–13). The black curve shows the asymptotic prediction Eq. (24).

Figure 6: The full Floquet spectra computed at the same parameters as in Fig. 2.

Figure 10: Plot of |bm0(z)| (at the edge) at the same parameters as in Fig. 8 but with σ: (a,b) 5 × 10−4; (c–f) 2 × 10−3. Remarkably, in panel (e) the edge soliton persists until at least z = 2× 104 and has the signature of a nonlinear topological edge state.

Figure 7: Plot of the time evolution of the 2D discrete system Eqs. (10–11) with σ = 0 (panels (a)) and the 1D LS equation (39) with σ̃ = 0 (panel (b)) in terms of |bm0(z)| (at the edge). The parameters are (ρ, κ) = (1, 0.3), ǫ = 2π/60, and ω0 = 2π/3 as in Fig. 5(b), and the envelope width is ν = 0.1. Periodic boundary conditions in m are used. Note in panel (b) the computation of the 1D LS equation stops at z ≈ z+ = 15 where the edge state delocalizes.

Figure 5: Time evolution of Eq. (12–13) with parameters (ρ, κ) = (1, 0.3) and ǫ = 2π/60 as in Fig. 2(b). The initial condition is given by Eq. (28) with (ω, δ): (a) (π/2, 1); (b) (2π/3, 1); (c) (π/3, 0.7). The number of lattice sites is 80, but only the leftmost 10 sites are shown.

Figure 8: Plot of the time evolution of the 2D discrete system Eqs. (10–11) with σ = 0 (panels (a,c,e)) and the 1D LS equation (39) with σ̃ = 0 (panels (b,d,f)), shown in terms of |bm0(z)| (at the edge). The parameters for panels (a–b) agree with Fig. 5(a) with ν = 0.1. The parameters for panels (c–f) agree with Fig. 9(a) with (ω0, ν): (c,d) (3π/8, 0.1); (e,f) (5π/8, 0.1). The edge state persists over a long distance. Periodic boundary conditions in m are used.

Figure 4: The shaded region shows the region in the (ω, Z)-plane determined by the localization criterion Eq. (22) at the same parameters as in Fig. 2(b). The two solid white lines bound the existence interval Ip of pure edge modes. The dashed lines show the values of ω used in the panels of Fig. 5.

Figure 3: Plots of the pseudo-field A(Z) = (A1(Z), A2(Z)) corresponding to the parameters used in (a) Fig. 2(a,b); (b) Fig. 2(c).

Figure 9: The dispersion relation of pure edge modes (panel (a)) and the full Floquet spectrum (panel (b)) computed at κ = 0.5 and ρ = 0.51, using 40 lattice sites and ǫ = 2π/100. The blue curve represents pure edge modes computed numerically from Eqs. (12–13). The black curve shows the asymptotic prediction Eq. (24).

Full text

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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

Frequently asked questions

Q1. What are the contributions in "Adiabatic dynamics of edge waves in photonic graphene" ?

In this paper, the propagation of localized edge modes in photonic honeycomb lattices, formed from an array of adiabatically varying periodic helical waveguides, is considered. 

Q2. Why does the edge soliton exhibit a slow modulation in its amplitude and?

Due to the z-dependence of the coefficients of the NLS equation, the edge soliton exhibits slow modulation in its amplitude and width. 

Q3. What is the effect of nonlinearity on the evolution of edge modes?

As with the traditional, constant coefficient, focusing NLS equation, nonlinearity balances dispersion to produce nonlinear edge solitons. 

Q4. What is the role of the waveguides in the optical field?

The waveguides play the role of a pseudomagnetic field, and in certain parameter regimes, the edge waves are found to be nearly immune to backscattering. 

Q5. What are the dimensions of the variables used in Eq. 1?

The authors 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). 

Q6. What is the effect of nonlinearity on the envelope of the edge modes?

A nonconstant coefficient (‘time’-dependent) one-dimensional nonlinear Schrödinger (NLS) equation governing the envelope of the edge modes is derived and is found to be an effective description of nonlinear traveling edge modes. 

Q7. What are the characteristics of the edge waves?

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. 

Q8. What is the localization interval of the region in the (, Z)-plane?

4. Thus the localization interval IZ(ω) corresponds to the vertical slice through the shaded region at fixed ω, and the existence interval Ip = (ω−, ω+) of pure edge modes is bounded by the two solid white lines. 

Q9. What is the length scale for the pseudo-field Eq. (6)A?

For the pseudo-field Eq. (6)A = (A1(z), A2(z)) = κ(sinΩz,− cosΩz) = κ(sinZ,− cosZ), Z = ǫz,taking Ωz∗ = ǫ where ǫ = 0.1 leads to Ω = 14.8 rad/m and the period T = 2π Ω = 42 cm. 

Q10. What is the simplest solution of Eqs?

Taking a discrete Fourier transform in m, i.e. letting amn = ane imω and bmn = bne imω, yields the simplified systemi∂zan + e id·AL−bn + σ|an|2an = 0, (12) i∂zbn + e −id·AL+an + σ|bn|2bn = 0, (13)whereL−bn = (bn + ργ∗(z;ω)bn−1) (14) L+an = (an + ργ(z;ω)an+1) (15)γ(z;ω) = 2eiϕ+(z) cos(ϕ−(z)− ω) andϕ+(z) = (θ2(z) + θ1(z))/2, ϕ−(z) = (θ2(z)− θ1(z))/2.Since γ(z;ω + π) = −γ(z;ω), if (an(z), bn(z)) is a solution of Eqs. (12–13) at any given ω, then (−1)n(an(z), bn(z)) is a solution of Eqs. (12–13) at ω + π. 

Q11. What is the dispersion relation of edge modes in Fig. 2?

the dispersion relation of edge modes in Fig. 2(a) breaks up into multiple segments near ω = 0, π. Also, in Fig. 2(b) there are scattered eigenvalues near α = 0 for ω immediately outside Ip = (ω−, ω+). 

Q12. What is the second case where quasi-edge modes persist only for part of the period?

In the rapidly varying case [28], the associated NLS equation had constant coefficients, and so the authors see adiabatic variation introduces new dynamics into the nonlinear evolution of edge modes. 

Q13. What is the generalization of pure edge modes?

The authors note that since no pure edge mode exists for any range of frequencies ω in Case (III), this case will be omitted in the following discussion of pure edge modes. 

Q14. What is the second case where quasi-edge modes exist?

Quasi-edge modes do not exist in the rapidly varying case studied in [28], and so the authors have shown adiabatic variation of the pseudo-field allows for new dynamics even at the linear level.