Biomedical Engineering Reference
In-Depth Information
Equating the denominator
in (4.21) to zero, we obtain the
dispersion equation for the wavenumbers
q
of the chiral-plasmon
modes excitedon the surface the chiral half-space
(
S
(
k
L
)
/
Z
+
S
(
k
0
))(
S
(
k
R
)
+
S
(
k
0
)
/
Z
)
+
(
S
(
k
R
)
/
Z
+
S
(
k
0
))(
S
(
k
L
)
+
S
(
k
0
)
/
Z
)
=
0.
(4.23)
Ass
uming
χ
=
0, from (4
.23), we get t
wo inde
pen
dent equat
ions:
k
0
−
k
0
εμ
−
k
0
−
k
0
εμ
−
ε
0,
which define wavenumbers of the TM and TE modes, respectively,
[3]. In the case
q
2
+
q
2
=
0and
μ
q
2
+
q
2
=
μ
=
1, the first equation is reduced to
the usual
dispersion equation for the surface plasmon
q
k
0
√
ε/
=
ε
+
(
1)
ε<
−
(where
1) [43, 59].
In the case of a small chirality (
χ
→
χ>
0), one can find an
asymptotic solution of (4.23). In the most interesting cases
μ
=±
1
and
ε
=−|
ε
|
,onecanobtain
0,
1
+
,
2
|
ε
|
|
ε
| −
1
|
ε
|
χ
q
=
k
0
(
|
ε
| +
1)
2
(
|
ε
| −
1)
μ
=
1,
|
ε
|
>
1,
q
≥
k
0
,
q
=
k
0
|
ε
| −
1
|
ε
|
>
1,
q
≥
k
0
χ
√
2
|
ε
|
,
μ
=−
1,
|
ε
|
,
q
=
k
0
1
− |
ε
|
χ
√
2
|
ε
|
,
μ
=−
1, 0
<
|
ε
|
<
1,
q
≥
k
0
. (4.24)
In Fig. 4.5, the dispersion curves corresponding to (4.23) and (4.24)
are shown. From this figure, one can see that in the case of a metal
half-space, which has a small addition of chirality (
ε<
0,
μ
=
1,
χ
=
0.1), the wavenumbers of the surface chiral-plasmon modes
almost coincide with the wavenumbers of the surface plasmon
modesinanonchiralmetal(seeFig.4.5a).InthecaseofchiralDNG-
metamaterial (
ε<
0,
μ
=−
1,
χ
=
0.1), there are two branches
of surface modes (see Fig. 4.5b). For nonchiral DNG half-space (
μ
=
−
1,
χ
=
0), these branches coincide and transformed to
ε
=−
1for
all
q
≥
k
0
(see Fig. 4.5b). It is this degenerate plasmon mode that
leads to a number of paradoxical properties of the dipole radiation
neartheDNGhalf-space[39,41,56,58].FromFig.4.5,italsofollows
that the asymptotic expressions (4.24) describe the solutions of the
dispersion equation well.
