19.1 Photonic Band Gap Materials
503
Fig. 19.20 Dispersion of
cavity photon mode and
electronic resonance at
E el = 3.0 eV (dashed
lines) and coupled modes
(solid lines) for
2V = 40 meV
Fig. 19.21 Equifrequency
surfaces (19.54) for a
‘normal’ isotropic
( = ⊥ > 0) and uniaxial
( > 0, ⊥ > 0, here
shown for ⊥ > > ) optical
medium and type I ( < 0,
⊥ > 0) and type II
( > 0, ⊥ < 0)
hyberbolic metamaterials
=
⎛
⎝
0
0
0 0
0 0 ⊥
⎞
⎠
(19.53)
with ⊥ < 0. A negative dielectric function is known from metals below the plasma frequency
(cf. Sect. 9.9.1). From the conventional formula ω
2
= k
2 c
2
/n
2 in a isotropic medium, in an uniaxial
medium the isofrequency surface is given by
ω
2
c 2 =
k
2
x + k
2
y
+
k
2
z
⊥
(19.54)
and for ’normal’ uniaxial materials an ellipsoid (or a sphere for isotropic materials, cmp. to band
dispersions Fig. 6.35a, b). If one of the tensor elements is negative, two possible types of ’hyperbolic’
metamaterials develop with < 0 (HMM type I) and with ⊥ < 0 (HMM type II). Their isofrequency
surfaces (for TM waves) for the two types are depicted in Fig. 19.21.
Apart from homogeneous media, various geometries for HMM have been proposed and investigated
[1639, 1640]. HMM can transport high-k waves and may enable devices for sub-wavelength resolution
imaging. Also, the enhanced density of photonic states (within a restricted wavelength range) can be
used for enhancement of spontaneous recombination rates (Purcell effect, cf. Sect. 19.2.2) [1641]. A
possible epitaxial, almost perfectly lattice matched superlattice HMM system of alternating dielectric
and metallic materials is MgO/TiN [1642]. Also HMM involving the anisotropic magnetic permittivity
tensor have been considered [1643].