488
19 Dielectric Structures
The conservation of energy requires that det M = (1 − R)/T = 1. The eigenvalue equation for M is
μ
2
− 2μ Re(1/t) + 1 = 0 .
(19.12)
The two eigenvalues μ
± are related by μ
+
μ
−
= det M = 1. If we consider an infinite, periodic
structure, we know from Bloch’s theorem (cf. Sect. 6.2.1) that the eigenvector varies between unit cells
only via a phase factor, i.e. |μ| = 1. Therefore, the eigenvalues can be written as
μ
±
= exp(±iβ) ,
(19.13)
where β corresponds to the Bloch phase of a hypothetical infinite periodic structure. This phase β
should not be confused with φ defined earlier, which is associated with the unit cell transmission. We
find the condition
Re(1/t) = cos β
(19.14)
for the Bloch phase. Since every matrix obeys its own eigenvalue equation, we have also (1 being the
unity matrix)
M
2
− 2 M cos β + 1 = 0 .
(19.15)
By induction one can show that the N -period case has the scattering matrix
M
N
= M
sin(N β)
sin β
− 1
sin((N − 1) β)
sin β
.
(19.16)
The solution for the finite period case can be written in terms of the Bloch phase of the infinite potential.
The transmission and reflection of the N -period system are given by
1
t n
=
1
t
sin(N β)
sin β
−
sin((N − 1) β)
sin β
(19.17a)
r n
t n
=
r
t
sin(N β)
sin β
.
(19.17b)
The transmission of intensity can be written as (T = t
∗ t)
1
T N
= 1 +
sin
2
(N β)
sin
2
β
1
T
− 1
.
(19.18)
Again, up to this point no specific distribution of the index of refraction within the unit cell has been
specified.
From (19.17a), a general formula for the mode density ρ N (ω) of the N -stack can be obtained
as [1589]
ρ N =
1
N d
sin(2Nβ)
2 sin β
η
+
η ξ ξ
1−ξ 2
−
N η ξ
1−ξ 2
cos 2 (Nβ) + η 2
sin(Nβ)
sin β
2 ,
(19.19)
where ξ = x/T = cos β and η = y/T .