Chemistry Reference
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where nonlocal contributions to the polarization have been ignored. Useful quan-
tities are the average dielectric constant
1
2
ðe
k
þ e
?
Þ
e ¼
(24)
and the dielectric anisotropy
1
2
ðe
k
e
?
Þ:
d ¼
(25)
Solving Maxwell's equations in a helical cholesteric structure gives the secular
equation for the eigenvalues
q
d
2
2
2
n
2
¼ e þ b
þ
4
eb
;
(26)
where
b ¼ l
o
=p ¼ q
o
l
o
=
2
p
, and the optical eigenmodes
d
ðn þ bÞ
2
p
l
o
n
2
p
ÞzotÞ
þ
2
p
l
o
nþ
2
p
ÞzotÞ
g;
e
i
ðð
e
i
ðð
E
¼ E
o
fð
^
x
þ i
^
y
Þ
ð
^
x
i
^
y
Þ
(27)
2
e
where x and y are unit vectors in the
x-y
plane (plane of the director), and light
branches,
the branch corresponding to the negative radical
is negative for
e
p
, where the index
n
is imaginary. In this regime, there is no light
propagation; the modes are exponentially decaying evanescent waves. This region
therefore corresponds to the photonic band gap, with the low energy band edges at
l
o
¼ pn
k
¼ p
p
e
k
< b <
e
p
. Outside the band,
the eigenmodes are combinations of left-and right-circularly polarized waves,
propagating with different velocities. Inside the band, the mode corresponding to
imaginary
n
is evanescent, with counter-propagating circularly polarized waves. At
the band edges,
n ¼
l
o
¼ pn
?
¼ p
p
e
k
and the high-energy edge at
2
, and the modes are standingwaves, with E parallel
to the director at the low energy edge, and perpendicular at the high-energy edge.
Ignoring finite size effects, the DOS is
0,
e b
¼d
d
n
2
d
d
k
d
d
d
2
pn
l
o
¼
n
c
l
o
r ¼
o
¼
l
o
:
(28)
o
2
nc
Since
n
2
is a smoothly varying function of
diverges at the band edges as 1
0. Photon emission by excited atoms
and molecules and distributed feedback lasing is therefore expected at the band
edges. Alternately, for wavelengths at the band edges (where the refractive index
is zero), the material simply acts as a Fabry-Perot cavity for circularly polarized
waves. If excited dye or LC molecules are present, there is gain and the material,
acting both as the cavity and the gain medium, can lase.
=n
, where
n !
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