490
19 Dielectric Structures
(a)
(b)
Fig. 19.3 Quarter-wave stack with indices of refraction a n 1 , n 2 = 1.0, 1.5 and b 1.0, 3.0. Solid lines: dimensionless
density of modes ρ N (19.19), dashed lines: transmission T N (19.28) for two different numbers of pairs N = 5 (left
panels) and 10 (right panels) versus the dimensionless frequency ˜
ω
(a)
(b)
Fig. 19.4 Quarter-wave stack with indices of refraction n 1 , n 2 = 1.0, 1.5: a Transmission T N at midgap ( ˜
ω = 1, down
triangles) and at the band edge ( ˜
ω = 1 − ˜
ω/2, up triangles) versus number of pairs N . b Dimensionless density of
modes ρ N at maximum near the band edge and at midgap versus number of pairs N
In the limit of large N the complete width ˜
ω of the band gap is implicitly given by
cos
π
2
˜
ω
= 1 − 2
n 1 − n 2
n 1 + n 2
2
.
(19.29)
If |n 1 − n 2 | | n 1 + n 2 , we find
˜
ω ≈
4
π
|n 1 − n 2 |
n 1 + n 2
.
(19.30)
19 Dielectric Structures
(a)
(b)
Fig. 19.3 Quarter-wave stack with indices of refraction a n 1 , n 2 = 1.0, 1.5 and b 1.0, 3.0. Solid lines: dimensionless
density of modes ρ N (19.19), dashed lines: transmission T N (19.28) for two different numbers of pairs N = 5 (left
panels) and 10 (right panels) versus the dimensionless frequency ˜
ω
(a)
(b)
Fig. 19.4 Quarter-wave stack with indices of refraction n 1 , n 2 = 1.0, 1.5: a Transmission T N at midgap ( ˜
ω = 1, down
triangles) and at the band edge ( ˜
ω = 1 − ˜
ω/2, up triangles) versus number of pairs N . b Dimensionless density of
modes ρ N at maximum near the band edge and at midgap versus number of pairs N
In the limit of large N the complete width ˜
ω of the band gap is implicitly given by
cos
π
2
˜
ω
= 1 − 2
n 1 − n 2
n 1 + n 2
2
.
(19.29)
If |n 1 − n 2 | | n 1 + n 2 , we find
˜
ω ≈
4
π
|n 1 − n 2 |
n 1 + n 2
.
(19.30)