Engineering Equifrequency Contours of Metasurfaces for Self-Collimated Surface Wave Steering Sara M. Kandil1 Diaaaldin J. Bisharat2and Daniel F. Sievenpiper1

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Engineering Equifrequency Contours of Metasurfaces for Self-Collimated Surface

Wave Steering

Sara M. Kandil1, Dia’aaldin J. Bisharat2and Daniel F. Sievenpiper1

1University of California San Diego, Department of Electrical and Computer Engineering

La Jolla, CA 92093, U.S.A

2City University of New York, Graduate Center

New York, New York 10016, USA

(Dated: October 18, 2022)

Metasurfaces provide unique capability in guiding surface waves and controlling their polariza-

tion and dispersion properties. One way to do that is by analyzing their equifrequency contours.

Equifrequency contours are the 2D projection of the 3D dispersion diagram. Since they are a k-

space map representation of the surface, many of the wave properties can be understood through

the Equifrequency contours. In this paper, we investigate numerically and experimentally the engi-

neering of equifrequency contours using C-shape metasurface design. We show the ability to provide

high self-collimation as well as spin-dependent wave splitting for the same metasurface by tuning the

frequency of operation. We also show the ability to steer the wave along a deﬁned curved path by

rotating the C-shape which results in rotating its equifrequency contours. This work demonstrates

how engineering equifrequency contours can be used as a powerful tool for controlling the surface

wave propagation properties.

I. INTRODUCTION

During the last decade, there has been a great interest

in making conventional optics such as lenses, waveguides,

couplers and polarization-based devices using ﬂat sur-

faces [1–3]. These surface platforms provide the advan-

tage of being scalable and easily integrated on chips. Ad-

ditionally, they provide additional degrees of freedom for

controlling the wave propagation as well as eliminating

the accumulated changes in phase and amplitude when

propagating over distances as the case in conventional

optical systems [4].

Metasurfaces which are 2D surfaces patterned with

subwavelength scatterers are extensively studied as plat-

forms for ﬂat optics. They provide unique capabilities

in controlling the dispersion properties, phase and po-

larization of the propagating wave [5, 6]. This can be

done by carefully choosing the unit cell design to engi-

neer the interaction between the wave and the surface.

Several research has been done to study the wide capa-

bilities of metasurfaces from guiding [7–9], beam focusing

[10, 11], splitting [12–14], steering [15], and lensing [16].

One way to control the interaction between the wave and

the surface is by engineering the equifrequency contour

of the unit cell design. Equifrequency contour (EFC),

also called isofrequency contour (IFC), is the 2D projec-

tion of the 3D dispersion diagram at diﬀerent frequencies

[17, 18]. It represents a k-space map of the wave possible

trajectories at diﬀerent frequencies. The direction of the

wave is determined by the direction of its group velocity

where the group velocity of the wave is deﬁned as δω/δk.

This can be determined through the EFCs. Anisotropic

shapes where some symmetries are broken have interest-

ing, nonconventional EFCs where the contours can vary

from elliptical to ﬂat allowing the wave to propagate in

one direction (normal to the ﬂat contour) with high self-

collimation [19].

Another capability of metasurfaces is that they can

control the spin-orbit coupling of surface waves. It was

recently shown that surface waves with evanescent tails

possess a transverse spin that is locked to the wave mo-

mentum. A property termed as Spin-Hall Eﬀect (SHE)

where opposite spins propagate in opposite directions

[20–22]. This is analogous to the SHE phenomenon ini-

tially discovered in electronic systems [23]. It is also re-

ferred to as spin-momentum locking where spin repre-

sents the circular polarization of the electric and/or mag-

netic ﬁeld of the surface wave. Several studies showed

the ability to achieve spin-dependent unidirectional prop-

agation using metasurfaces with engineered anisotropy

[24–26], bandgap materials [27, 28], gradient metasur-

faces [29], and near-ﬁeld interference with asymmetrically

placed dipole sources [30–32].

In this paper, we investigate the diﬀerent ways the

EFC of a metasurface can be engineered to control the

surface wave propagation and spin-dependent direction-

ality. We show various wave properties achieved through

the same metasurface design such as self-collimation,

polarization-based beam splitting and wave steering. The

paper is organized as follows: in Section II, we discuss

the homogeneous metasurface design formed of metallic

C-shape unit cell, its self-collimation and spin-dependent

wave propagation. In Section III, we present the inho-

mogeneous design formed of the same metallic C-shaped

unit cells. We discuss how its EFCs change with the ro-

tation angle resulting in surface wave steering along two

predeﬁned paths. We study these phenomena using nu-

merical simulations as well as experimental results which

are presented in Section IV. The paper is concluded in

Section V.

arXiv:2210.08999v1 [physics.class-ph] 13 Oct 2022

FIG. 1. (a) Schematic showing the C-shape metasurface

design. (b) The dimensions of the C-shape unit cell consisting

of (d) a C-shape metallic post of 0.143mm thickness placed

on a Roger’s substrate.

II. HOMOGENEOUS METASURFACE

Fig.1 presents a schematic showing the C-shaped meta-

surface design we will study throughout this section. As

depicted in Fig.1(b), the unit cell consists of a metallic

C-shaped of 0.143mm thickness placed on a Roger’s 5880

substrate (r= 2.2). The C-shape metallic post has a

width of 0.715mm and radius of 2.145mm where its edge

is extended from the center by 0.429mm. The whole sur-

face is 250mm×250mm. The C-shape metasurface de-

sign was studied and optimized numerically using Ansys

HFSS. In this section, we will explore diﬀerent surface

wave properties supported by the C-shaped metasurface

through studying its equifrequency contours.

A. Self-Collimation

Isoropic shapes with small or no asymmetry have EFC

close to a circle where the wave propagates equally in

all direction. As the asymmetry of the shape increases,

the contour becomes ﬂatter with higher wave directional-

ity and hence, higher self-collimation [33]. The C-shape

design has broken rotational symmetry along the z- and

x-axes which makes it a low-symmetry shape and hence,

have high self-collimation. This can be observed from

the calculated EFC of the C-shape unit cell shown in

Fig.2(a). It can be shown that the C-shape design pos-

sesses diﬀerent contour shapes which dictate various wave

propagation properties. At 14 GHz, the EFC is elliptical

which then becomes ﬂatter at higher frequencies where

high self-collimation takes place. The E-ﬁeld proﬁles at

diﬀerent frequency contours are shown in Fig.2(b) where

the surface is excited with an Eydipole at the bottom

center. The magnitude of Exmaps are calculated for

each EFC. It can be observed that the wave is split where

the split angle decreases with the increase of frequency.

At 19 GHz, the surface wave is highly collimated as well

as spin-independent. This means that any polarization

will excite the wave to propagate with high collimation

and zero split angle.

B. Spin-dependent Propagation

Fig.2(c) shows the spin-dependent behavior of the sup-

ported surface wave. The EFC calculated for the C-shape

shows a spin-based wave splitting property where the

wave is split into left-handed and right-handed circularly

polarized waves. By excitation of the surface with an

Ey+iEzdipole source, the wave propagates along the

left arm where it propagates along the right arm when

excited with an Ey−iEzdipole source. The spin density

is deﬁned as a vector quantity whose direction is normal

to the plane of the ﬁeld circular rotation. The following

equation can be used to calculate the spin density of the

propagating surface wave [34]:

S= Im (E∗×E+H∗×H

E2+H2),(1)

where Sis the spin density vector in Gaussian units nor-

malized per one photon in units of ~= 1. E and H are the

electric and magnetic ﬁelds of the surface wave. Fig.3(a)

shows a schematic representation of the two components

of the transverse spin with respect to the two wave propa-

gation directions where the Sxcomponent ﬂips sign while

the Symaintains the same sign for both wave propaga-

tion directions. Fig.3(b) shows the numerically calcu-

lated spin maps for the two split waves for Sxand Sy.

III. INHOMOGENEOUS METASURFACE

As demonstrated, the EFC is closely related to the

shape of the unit cell; engineering it can be done through

diﬀerent ways. For example, some studies showed that

the change of the unit cell from square to rectangular or

parallelogram can increase the ﬂatness of the EFCs and

hence increase the collimation [35, 36]. The symmetry of

the shape itself can also result in changing the contour

shape. For example, as described earlier, a broken rota-

tional symmetry can produce ﬂatter contours along the

x- or y-axis. 450mirror symmetry of a shape can result

in tilted contours [37].

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EngineeringEquifrequencyContoursofMetasurfacesforSelf-CollimatedSurfaceWaveSteeringSaraM.Kandil1,Dia'aaldinJ.Bisharat2andDanielF.Sievenpiper11UniversityofCaliforniaSanDiego,DepartmentofElectricalandComputerEngineeringLaJolla,CA92093,U.S.A2CityUniversityofNewYork,GraduateCenterNewYork,NewYork10016,US...

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Engineering Equifrequency Contours of Metasurfaces for Self-Collimated Surface Wave Steering Sara M. Kandil1 Diaaaldin J. Bisharat2and Daniel F. Sievenpiper1

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