19.1 Photonic Band Gap Materials
489
19.1.4 The Quarter-Wave Stack
A quarter-wave stack, also known as a Bragg mirror, exhibits a one-dimensional photonic band gap. One
period consists of two regions with thickness and index of refraction (d 1 , n 1 ) and (d 2 , n 2 ), respectively
(Fig. 19.2c). In the quarter-wave stack each region has an optical thickness of λ/4 (the wave accumulates
in each region a phase of π/2) for a particular wavelength λ 0 or (midgap) frequency ω 0 . Thus, the
condition reads
n 1 d 1 = n 2 d 2 =
λ 0
4
=
π
2
c
ω 0
.
(19.20)
Using the Fresnel formulas, the transmission of an arbitrary two-layer cell is
t =
T 12 exp(i ( p + q))
1 + R 12 exp(2i q)
,
(19.21)
where p = n 1 d 1 ω/c and q = n 2 d 2 ω/c are the phases accumulated in the two layers, respectively. The
values of T 12 and R 12 are given as
T 12 =
4 n 1 n 2
(n 1 + n 2 ) 2
(19.22)
R 12 =
(n 1 − n 2 )
2
(n 1 + n 2 ) 2 .
(19.23)
For the quarter-wave stack ( p = q = π/2), we obtain for (19.21)
t =
T 12 exp(iπ ˜
ω)
1 + R 12 exp(iπ ˜
ω)
,
(19.24)
where ˜
ω = ω/ω 0 is the frequency scaled to the midgap value.
The transmission of a single two-layer cell is
T =
T
2
12
1 − 2 R 12 cos(π ˜
ω) + R
2
12
,
(19.25)
and the Bloch phase is given by
cos β = ξ =
cos(π ˜
ω) − R 12
T 12
(19.26)
η =
sin(π ˜
ω)
T 12
.
(19.27)
For the N -period quarter-wave stack the transmission is given by
T N =
1 + cos β
1 + cos β + 2 (R 12 /T 12 ) sin
2
(N β)
.
(19.28)
A band gap forms. Within the band gap, the density of modes is lowered, at the edges it is enhanced
(Figs. 19.3 and 19.4). The transmission at midgap decreases ∝ (n i /n j )
2N , where n i < n j .