486
19 Dielectric Structures
Fig. 19.1 Right:
electromagnetic dispersion
with a forbidden gap at the
wavevector of the
periodicity. Left:
Electron-wave dispersion
typical of a direct-gap
semiconductor. When the
photonic band gap
straddles the electronic
band gap, electron–hole
recombination into photons
is inhibited since the
photons have no place to go
(zero final density of states)
19.1.2 General 1D Scattering Theory
The formation of a photonic band gap in a one-dimensional dielectric can be calculated to a large extent
analytically and thus with direct insight. Let n(x) be the spatially varying index of refraction (no losses
or nonlinear optical effects). The one-dimensional wave equation (Helmholtz equation) reads for the
electric field E
∂
2 E(x)
∂ x 2 + n
2
(x)
ω
2
c 2 E(x) = 0 .
(19.3)
A comparison with a one-dimensional Schrödinger equation
∂
2
(x)
∂ x 2 +
2m
2 [E − V (x)] (x) = 0
(19.4)
shows that the Helmholtz equation corresponds to the quantum-mechanical wave equation of zero
external potential V and a spatially modulated mass, i.e. a case that is usually not considered.
Let us consider now the amplitude a k of the k eigenvector. The eigenvalue is then ω k . The onedimensional mode density ρ(ω) (per energy and per unit length) is
ρ(ω) =
dk
dω
,
(19.5)
which is the inverse of the group velocity.
We follow one-dimensional scattering theory as presented in [1589]. At this point we do not rely
on any specific form of n(x) (Fig. 19.2a). The (complex) transmission coefficient t for any index
structure is
t = x + iy =
√
T exp(iφ) ,
(19.6)
where tan φ = y/x. φ is the total phase accumulated during propagation through the structure. It can
be written as the product of the physical thickness of the structure d and the effective wave number k.
Hence we have the dispersion relation
d
dω
tan(k d) =
d
dω
y
x
.
(19.7)
19 Dielectric Structures
Fig. 19.1 Right:
electromagnetic dispersion
with a forbidden gap at the
wavevector of the
periodicity. Left:
Electron-wave dispersion
typical of a direct-gap
semiconductor. When the
photonic band gap
straddles the electronic
band gap, electron–hole
recombination into photons
is inhibited since the
photons have no place to go
(zero final density of states)
19.1.2 General 1D Scattering Theory
The formation of a photonic band gap in a one-dimensional dielectric can be calculated to a large extent
analytically and thus with direct insight. Let n(x) be the spatially varying index of refraction (no losses
or nonlinear optical effects). The one-dimensional wave equation (Helmholtz equation) reads for the
electric field E
∂
2 E(x)
∂ x 2 + n
2
(x)
ω
2
c 2 E(x) = 0 .
(19.3)
A comparison with a one-dimensional Schrödinger equation
∂
2
(x)
∂ x 2 +
2m
2 [E − V (x)] (x) = 0
(19.4)
shows that the Helmholtz equation corresponds to the quantum-mechanical wave equation of zero
external potential V and a spatially modulated mass, i.e. a case that is usually not considered.
Let us consider now the amplitude a k of the k eigenvector. The eigenvalue is then ω k . The onedimensional mode density ρ(ω) (per energy and per unit length) is
ρ(ω) =
dk
dω
,
(19.5)
which is the inverse of the group velocity.
We follow one-dimensional scattering theory as presented in [1589]. At this point we do not rely
on any specific form of n(x) (Fig. 19.2a). The (complex) transmission coefficient t for any index
structure is
t = x + iy =
√
T exp(iφ) ,
(19.6)
where tan φ = y/x. φ is the total phase accumulated during propagation through the structure. It can
be written as the product of the physical thickness of the structure d and the effective wave number k.
Hence we have the dispersion relation
d
dω
tan(k d) =
d
dω
y
x
.
(19.7)