Controllable Angular Scattering with a Bianisotropic Metasurface

0.23 MB·

by  Karim Achouri

Controllable Angular Scattering
with a Bianisotropic Metasurface
Karim Achouri, and Christophe Caloz
Dept. of Electrical Engineering, Polytechnique Montre´al, Montre´al, QC H2T 1J3, Canada
Email: see http://www.calozgroup.org/
Abstract—We propose the concept of a bianisotropic meta- susceptibility tensors as
7 surface with controllable angular scattering. We illustrate this
1 concept with the synthesis and the analysis of a metasurface P “(cid:15)0χee¨Eav`χem¨Hav{c0, (2a)
0 exhibiting controllable absorption and transmission phase as M χ H χ E η , (2b)
2 function of the incidence angle. “ mm¨ av` me¨ av{ 0
whereE andH aretheaverageelectricandmagneticfields
n av av
a I. INTRODUCTION on both sides of the metasurface.
J x
Over the past few years, metasurfaces have proven to be
5
2 impressivelypowerfulinmanipulatingelectromagneticwaves. z
However, most studies have been restricted to metasurfaces Ei,1 Et,3

] performing electromagnetic transformations for a unique set
h θi,1 θt,3
p of incident, reflected and transmitted waves. If the incidence
- angle would change, the scattered waves would experience Ei,2 Et,2
ss major and uncontrollable changes compared to the specified θi,3 θt,1
a ones. Only a few studies have attempted to analyze or syn-
l
c thesis metasurfaces with angle-independent scattering as, for Ei,3 Et,1
. instance, in [1]–[3].
s
c In this work, we propose a new technique to synthesize Fig.1:Multiplescatteringfromauniformbianisotropicmeta-
i
s a metasurface with controllable angular scattering. For sim- surface.
y
plicity, we consider the case of a uniform metasurface, only
h Let us now consider the electromagnetic transformations
transforming the phase and the amplitude of the scattered
p
depicted in Fig. 1 where p-polarized incident plane waves are
[ waves.Themetasurfaceissynthesizedbyspecifyingthereflec-
tion and transmission coefficients for three different incidence scattered, without rotation of polarization, by a bianisotropic
1 metasurface. In this transformation, the only electromagnetic
angles which, by continuity, allows a relative smooth control
v field components that are not zero are E ,E and H and
of the angular scattering as function of the incidence angle. x z y
5
thereforeonlyafewsusceptibilitycomponentswillbyexcited
4 The synthesis of a metasurface performing three transforma-
bysuchfields.Consideringthateachofthefoursusceptibility
8 tions requires a number of degrees of freedom which are
8 here obtained by leveraging bianisotropy and making use of tensors in (2) contains 3 3 components, the only suscepti-
ˆ
0 bilities that are relevant to the problem of Fig. 1 are
longitudinal susceptibilities [4], [5]. ¨ ˛ ¨ ˛
.
1 χxx 0 χxz 0 χxy 0
0 ˝ ee ee‚ ˝ em ‚
7 II. METASURFACEDESIGN χee “ χ0zx 00 χ0zz , χem “ 00 χ0zy 00 , (3a)
1 A. Metasurface Synthesis and Analysis ¨ ee ee˛ ¨ em ˛
:
v 0 0 0 0 0 0
i A bianisotropic metasurface may be described by χ ˝χyx 0 χyz‚, χ ˝0 χyy 0‚, (3b)
X zero-thickness continuity conditions conventionally called me “ me me mm “ mm
0 0 0 0 0 0
r GSTCs [6], [7]. For a metasurface lying in the xy-plane at
a z 0, the GSTCs read where all the susceptibilities that are not excited by the
“ fields have been set to zero for simplicity. Note that this
zˆ ∆H jωP zˆ ∇ M , (1a) metasurface does not induce rotation of polarization. The
ˆ “ (cid:107)´ ˆ (cid:107)ˆ z ˙
susceptibility tensors in (3) contain a total number of 9
P
∆E zˆ jωµ0M(cid:107) ∇(cid:107) z zˆ, (1b) unknown components. However, in this work, we wish to
ˆ “ ´ (cid:15) ˆ
0
design a reciprocal metasurface, which reduces the number of
where ∆E and ∆H are the differences of the electric and unknowns to 6 since, by reciprocity, χxz χzx, χxy χyx
magneticfieldsonbothsidesofthemetasurfaceandwhereP and χzy χyz. ee “ ee em “´ me
em me
“´
andM are,respectively,theelectricandmagneticpolarization In order to simplify the synthesis and the analysis, we
densities, which may be expressed in terms of bianisotropic specify that the metasurface is uniform in the xy-plane. Then
the susceptibilities are not function of x and y and hence the and transmission coefficients are plotted in Figs.2a and 2b,
spatial derivatives on the right-hand side of (1) only apply respectively. As may be seen in these graphs, the metasurface
to the fields and not to the susceptibilities through (2). This exhibits the specified response in terms of both coefficients at
restriction means that the reflection and transmission angles thethreespecifiedangles.Moreover,thetransmissionexhibits
follow conventional Snell’s law, i.e. θ θ and θ θ. acontinuousamplitudedecreaseasθ increasesbeyond 50 .
r i t i i ˝
“´ “ ´
Letusnowsubstitutethesusceptibilities(3)into(1)with(2)
1
and enforce reciprocity. This operation reduces (1) to the two
following equations:
0.8 T 0.75
| |“
∆H jω(cid:15) χxxE χxzE jk χxyH (4a)
y “´ 0p ee x,av` ee z,avq´ 0 em y,av de0.6 T 0.5
∆Ex “´jωµ0χymymHy,av`jk0pχxemyEx,av`χzemyEz,avq(4b) plitu | |“
χxz E χzz E η χzy H m0.4 T 0.25
´ eeBx x,av´ eeBx z,av´ 0 emBx y,av A | |“
where x is the partial derivative along x. The synthesis 0.2
B
technique consists in solving (4) for the susceptibilities in (3). R 0 R 0 R 0
However, as previously mentioned, there are 6 unknown sus- 0 “ “ “
−90 −45 0 45 90
ceptibilities,andthesystem(4)containsonly2equations.This θi ˝
p q
means that, to be determined, the system (4) may be solved (a)
for three independent sets of incident, reflected and trans-
mitted waves [4], [5]. Thus, the reflection and transmission
135
coefficients of the metasurface in Fig. 1 may be specified for
threedifferentanglesofincidence.Byspecifyingthereflection 90
andtransmissioncoefficientsforthreespecificangles,onecan 45 =T “45˝
q
˝
athcehierevsepoconnsetroolflatbhlee qmueatsais-ucorfnatcineufooursnaonng-uslpaercsificaetdterainngglessindcee asep 0=T “0˝ =T “0˝
h
facto corresponds to an interpolation of the three specified P −45
responses. −90
Once the synthesis has been completed, following the
−135
aforementioned procedure, the response of the metasurface −90 −45 0 45 90
versusincidenceangleforthesynthesizedsusceptibilitiesmay θi ˝
p q
be performed by analysis, which consists in solving (4) to (b)
determinethereflection(R)andtransmission(T)coefficients. Fig. 2: Reflection (dashed red line) and transmission (solid
blue line) amplitude (a) and phase (b) as function of the inci-
B. Illustrative Example
dence angle for a metasurface synthesize for the transmission
We now illustrate by an example the synthesis and analysis coefficients T 0.75;0.5ej45˝;0.25 (and R 0) at the
“ t u “
of a bianisotropic metasurface with controllable angular scat- respective incidence angles θ 45 ;0 ; 45 .
i ˝ ˝ ˝
“t´ ` u
tering.Letusconsiderareflection-lesstransformation(R 0)
“
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