Coherent control of topological states in an integrated waveguide lattice Alexey MikhinyzViktoriia RutckaiaxzRoman S. SavelevyzIvan S. Sinevy

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Coherent control of topological states

in an integrated waveguide lattice

Alexey Mikhin,†,‡Viktoriia Rutckaia,¶,§,‡Roman S. Savelev,†,‡Ivan S. Sinev,†

Andrea Al`u,¶,kand Maxim A. Gorlach∗,†

†School of Physics and Engineering, ITMO University, Saint Petersburg 197101, Russia

‡Authors who contributed equally to this work

¶Photonics Initiative, Advanced Science Research Center, The City University of New

York, New York 10031, USA

§Centre for Innovation Competence SiLi-nano, Martin-Luther-University,

Halle-Wittenberg, 06120 Halle (Saale), Germany

kPhysics Program, Graduate Center, The City University of New York, New York 10016,

USA

E-mail: m.gorlach@metalab.ifmo.ru

Abstract

Topological photonics holds the promise for enhanced robustness of light localiza-

tion and propagation enabled by the global symmetries of the system. While traditional

designs of topological structures rely on lattice symmetries, there is an alternative

strategy based on accidentally degenerate modes of the individual meta-atoms. Using

this concept, we experimentally realize topological edge state in an integrated optical

nanostructure based on the array of silicon nano-waveguides, each hosting a pair of de-

generate modes at telecom wavelengths. Exploiting the hybrid nature of the topological

mode formed by the superposition of waveguide modes with diﬀerent symmetry, we im-

plement coherent control of the topological edge state by adjusting the phase between

arXiv:2210.01648v1 [physics.optics] 4 Oct 2022

the degenerate modes and demonstrating selective excitation of bulk or edge states.

The resulting ﬁeld distribution is imaged via third harmonic generation allowing us

to quantify the localization of topological modes as a function of the relative phase of

the excitations. Our results highlight the impact of engineered accidental degeneracies

on the formation of topological phases, extending the opportunities stemming from

topological nanophotonic systems.

Keywords

Topological photonics, topological edge states, subwavelength grating waveguides, integrated

photonic circuits, coherent control

Introduction

Topological photonics provides a promising avenue to manipulate light in engineered nanos-

tructures by creating disorder-robust edge or interface states immune to backscattering at

sharp bends and defects1–3. The ﬁrst approaches to tailor such states have been relying on

broken time-reversal symmetry by applying an external magnetic bias in magneto-optical

materials4–6. This strategy however faces practical limitations, which have been inspiring

a search for alternative time-reversal-invariant platforms7–11, such as crystalline topological

metamaterials11–19, particularly appealing for its experimental accessibility.

In crystalline topological systems, the nontrivial topology of the bands and the associated

edge or interface states originate due to the special choice of lattice symmetries that ensure

topological degeneracies (e.g. Dirac points), as well as the opening of topological bandgaps

for suitable lattice parameters. Prominent examples are one-dimensional (1D) zigzag ar-

rays20–25, 2D breathing honeycomb11,13 and breathing kagome26–28 lattices. However, since

lattice geometries are challenging to modify dynamically, the topological properties of such

systems, e.g., the existence of topological states and their localization length, are diﬃcult to

control in real time.

To overcome this limitation, it has recently been suggested to utilize the accidental de-

generacy of modes in the individual meta-atoms of a simple lattice29,30. In such case, the

interference of the near ﬁelds of the degenerate modes can regulate the coupling between

neighboring meta-atoms, while the detuning between these modes controls the topological

transitions29–31. Experimental demonstrations of this strategy to create topological struc-

tures at the nanoscale have remained elusive so far.

In this Article, we implement this strategy experimentally realizing an integrated optical

structure based on an array of multimode nanostructured waveguides that supports topolog-

ical edge state in the telecom range. By optimizing the design of the waveguides, we ensure

that they host a pair of modes with the same propagation constants but diﬀerent symmetry

of the near ﬁeld proﬁles [Figure 1a]. Such accidental degeneracy is crucial for the formation

of a Floquet topological phase7and the emergence of topological edge states. In contrast to

the traditional implementations of topological physics, these modes have hybrid origin being

a superposition of two waveguide modes with diﬀerent symmetry of the near ﬁeld. As we

demonstrate, this feature enables a coherent control of the topological edge state.

Generally, the goal of coherent control phenomena is to tailor the relative phases of mul-

tiple excitation signals to control the response of the system in real-time32–34. In photonics,

this is often achieved through the interference of excitations from multiple ports35,36. In our

case, coherent control is ensured by manipulating the relative phase between two waveguide

modes, which allows us to switch between the excitation of edge and bulk modes of the

lattice. To image the modes of the fabricated waveguide nanostructure, we excite it with

short laser pulses and collect the third harmonic signal, which provides a sensitive tool for

detection and localization.

edge mode

excitation

bulk modes excitation

selective excitation

of bulk/edge modes

(a)

-γ

iΔ

≈0

≈κ+γ

(b)

Figure 1: (a) Scanning electron microscopy image (SEM) of the fabricated structure with

schematic proﬁles of the near ﬁeld distributions for two quasi-degenerate modes of the struc-

tured waveguide (yellow curves). Depending on the relative phase diﬀerence with which

these modes are initiated through the directional couplers, either a topological edge mode

(red curve) or bulk modes (green curves) of the waveguide array are excited. (b) Schematic

of the |Ey|2ﬁeld distribution for the two modes of the neighboring waveguides. The coupling

between them gives rise to the hybrid modes with asymmetric ﬁeld proﬁles.

Design of the Waveguide Lattice

The key ingredient necessary to obtain the topological features in our setup is the degeneracy

of two modes with distinct symmetries of their near ﬁeld distributions as sketched in Fig. 1b.

If two degenerate modes Ey1and Ey2exhibit diﬀerent parity of the ﬁeld relative to reﬂection

in Oyz plane, their linear combinations become strongly asymmetric relative to the waveguide

axis. This immediately aﬀects the magnitude of the evanescent coupling between the adjacent

waveguides [Fig. 1b].

In an array of waveguides, hybridization of their modes gives rise to bulk and edge states.

If two degenerate modes of an edge waveguide are excited in phase, their linear combination

Ey1+Ey2will feature the hot-spot at the edge of the structure decoupled from the rest

of the lattice. On the contrary, if the same modes are excited out of phase, their linear

combination Ey1−Ey2will feature the hot-spot on the other side of the waveguide ensuring

good coupling to the bulk of the structure [Fig. 1b]. This reasoning suggests the emergence

of an edge-localized state with a hybrid nature.

To achieve the desired functionality and support our qualitative reasoning, we design

subwavelength structured waveguides etched from crystalline silicon on sapphire substrate

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CoherentcontroloftopologicalstatesinanintegratedwaveguidelatticeAlexeyMikhin,y,zViktoriiaRutckaia,{,x,zRomanS.Savelev,y,zIvanS.Sinev,yAndreaAlu,{,kandMaximA.Gorlach,yySchoolofPhysicsandEngineering,ITMOUniversity,SaintPetersburg197101,RussiazAuthorswhocontributedequallytothiswork{PhotonicsInitiativ...

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Coherent control of topological states in an integrated waveguide lattice Alexey MikhinyzViktoriia RutckaiaxzRoman S. SavelevyzIvan S. Sinevy

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