19.1 Photonic Band Gap Materials
487
Fig. 19.2 1D scattering
problem: a General
scattering of an index of
refraction distribution, b
N -period stack, c two-layer
(quarter-wave) stack
(a)
x
0
n
d
t
0
r
1
(b)
x
0
n
d
t N
0
r N
...
2d
3d
Nd
(N-1)d
1
(c)
x
0
n
d
t N
0
r N
...
2d
3d
Nd
(N-1)d
1
n 1
d 2
d 1
n 2
Evaluating the derivative we find
d
cos 2 (k d)
dk
dω
=
y
x − x
y
x 2
,
(19.8)
where the prime denotes derivation with respect to ω. Using the relation cos
2
θ = (1 + tan
2
θ)
−1 , we
obtain the general expression
ρ(ω) =
dk
dω
=
1
d
y
x − x
y
x 2 + y 2 .
(19.9)
19.1.3 Transmission of an N-Period Potential
Now, the behavior of N periods of a given index distribution n(x) within a thickness d of one period
(Fig. 19.2b) is investigated. The scattering matrix M connects the intensity at x = 0 with that at x = d.
We use the column vector u = (u
+
, u
−
)
T containing the right- and left-going waves (labeled ‘+’ and
‘−’, respectively), u
±
= f
± exp(±ik x),
u(0) = M u(d) .
(19.10)
Using the boundary conditions u(0) = (1, r ) and u(d) = (t, 0), we find that M has the structure
M =
1/t r
∗
/t
∗
r/t 1/t
∗
.
(19.11)