Biomedical Engineering Reference
In-Depth Information
a right-handed spiral
R
m
corresponds to
m
>
0 and a left-handed
spiral
L
m
to
m
<
0.Aswesummarizebelow,thereisancloserelation
betweenthespiralpitch
m
andthetopologicalcharge
ofthevortex
generated in the near field.
Near-field generation of OAM at the front side (
z
0
+
)ofthe
structured membrane can be understood by considering that each
point
=
ρ
n
of the groove illuminated by the incoming field is an SP
point source, launching an SP wave perpendicular to the groove.
Withgroovewidthsmuchsmallerthantheilluminationwavelength,
the in-plane component of the generated SP field in the vicinity of
the center of the structure is
E
SP
(
0
+
)
ρ
0
,
z
=
∝
G
·
[
E
in
(
ρ
n
,
z
=
0
+
)
e
ik
SP
|
ρ
0
−
ρ
n
|
/
)
1
/
2
is the Huygens-
·
n
n
]
n
n
where
G
=
|
ρ
−
ρ
|
(
0
n
Fresnelplasmonicpropagatorand
n
n
=
κ
−
1
(
d
2
d
s
2
)thelocalunit
normal vector determined from the curvature
κ
and the arc length
s
of the groove. The resultant SP field is the integral of elementary
point sources over the whole groove structure. As indicated by a
full evaluation, we can conveniently limit the integration to radial
regions
ρ
n
ρ
0
where the grooves become practically annular.
This leads to
n
n
∼−
ρ
and therefore to a simple expression of the
integrated SP field
E
SP
ρ
n
/
C
in
·
E
in
, connected to the incoming field
=
by an in-couplingmatrix
2
π
C
in
(
m
)
∝
e
im
ϕ
0
ϕ
e
im
ϕ
e
−
ik
SP
ρ
0
cos
ϕ
ρ
⊗
ρ
,
d
(2.63)
0
⊗
the
symbol denoting a dyadic product.
Contrasting with the recent studies that have been confined to
the near field, our suspended membrane opens the possibility to
decouple the singular near field into the far field, with an additional
structure on the back side of the membrane connected to the front
sidebythecentralhole.Bysymmetry(assumingloss-freeunitarity),
the out-coupling matrix is simply given as the hermitian conjugate
of the in-coupling matrix, that is,
C
out
C
in
, corresponding to
a surface field that propagates away from the central hole on the
back side. The inout coupling sequence corresponds to the product
T
=
C
†
(
m
out
)
·
=
C
(
m
in
)which,inthecircularpolarizationbasis,writes
explicitly as
∝
e
i
(
m
out
−
m
in
)
ϕ
t
+−
e
2
i
ϕ
t
++
T
(2.64)
t
−+
e
−
2
i
ϕ
t
−−
