# Adiabatic dynamics of edge waves in photonic graphene

## Summary (2 min read)

### 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.

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##### Frequently Asked Questions (14)

###### 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.