Biomedical Engineering Reference
In-Depth Information
Figure 5.2
Photonic band structures of helix crystals along the helix axis
for three lattice constants (a)
d
=
20 mm, (b)
d
=
11 mm. The pitch
p
=
4.4
mm, the radius
a
=
3.3 mm.
for ourhelical system
k
z
a
2
y
n
I
n
(
n
2
k
2
τ
−
k
2
a
2
cot
2
+
ψ
τ
n
a
)
n
n
τ
n
a
)
R
n
ψ
z
n
I
n
(
+
k
2
a
2
cot
2
x
n
τ
n
=
0.
(5.16)
We can see that the Eq. (5.16) regresses back to the eigen-equation
of a singletape helix if weremove the lattice term in
x
n
,
y
n
,and
z
n
.
5.2.3
Band Structure of Metallic Helix Array
The computed photonic band dispersions give us an intuitive
understanding of the optical property of helix arrays. Figure 5.2a,b
show the band structure of the helix array along the helix axis for
different lattice constants. We label the eigenmodes in Fig. 5.2 by
their dominant term in Eq. (5.2). For example, the (
−
1,S) modes
(blue stars in Fig. 5.2) have
n
=−
1 term as the dominant term
and “S” stands for a “slow mode” below the light line, and we use
the subscript “F” for a mode inside the light cone. Equations (5.3,
5.4) show that there is a
±
/2 phase difference between the radial
andtheangularcomponentsforboththeelectricfieldandmagnetic
field,implyingthattheeigenmodesareLHorRHcircularlypolarized
(LCP or RCP). Eigenmode analysis indicates that the
n
=±
1modes
are indeed either LCP or RCP. The lowest frequency branch (red),
