Chiral Effects in Plasmonic Metasurfaces and Twisted Metamaterials - Singular and Chiral Nanoplasmonics - page 102

coupling among the inclusions [59, 60]:
p 00 /ε 0
η 0 m 00 /μ 0

α ee α em
α me α mm

− 1 − 1 C tt C tz

C zt C zz

E inc × z
η 0 H inc · z

= − ·

E inc × z
η 0 H inc · z

.

= α s ·

(3.1)

Inthisformula, α ee isthetransverseelectricpolarizabilitytensor
(2 × 2), α mm is the normal magnetic polarizability coe cient,
and α em , α me are the magnetoelectric polarizability vectors, which
take into account the cross-coupling between transverse electric
and normal magnetic effects due to artificial magnetism and
polarization effects. In addition, the C elements are the Green's
dyadic coe cients taking into account the array coup ling [5 9], z is

the unit normal vector to the array, ε 0 , μ 0 ,and η 0 = √ μ 0 /ε 0 are the
free-space permittivity, permeability, and characteristic impedance,
respectively.ThereciprocityofthemetasurfaceimposestheOnsager
constraints[61,62]ontheseelements,whichmaybecombinedwith
the reciprocal properties of the coupling dyads:
α ee = α

em , C tt = C tt , C zt =− C tz , (3.2)
where the superscript T indicates the transpose operation, to en-
sure that the generalized polarizability tensor also has a reciprocal
form:

ee ,

α me =− α

⎛ ⎞

α

e xx e xy e xz

α α

ee
s

em
s

α α

⎝ ⎠ =

e yx e yy e yz

− α

e s T

α s =

α α α

. (3.3)

mm
s

α

me
zx

me
zy

mm
zz

α α α

In general, the elements of the generalized polarizability tensor
are dependent on the incidence angle, as a symptom of spatial
dispersion in the array, due to the influence of the coupling dyadics.
For smaller periods, however, these effects are usually small. The
induced dipole moments on all the other inclusions are related
to p 00 , m 00 through a simple phase shift associated with the
momentum of the impingingexcitation, which is conserved.
It follows that, within the dipolar limit, the tensor α s compactly
describes the metasurface interaction with an arbitrary impinging
plane wave, which effectively induces an averaged electric surface
current density tangential to the plane
K e =

j p 00

d x d y

ω

[ A

/

m ] (3.4)