THz Bandpass Filter Design Using Metamaterial-Based Defected 1D …
9
t
TM
11
κ
right
z
n
2
right κ 0
+ t
TM
12
κ
left
z κ
right
z
n
2
left n
2
right κ
2
0
− t
TM
21 −
κ
left
z
n
2
left κ 0
t
TM
22 = 0
(3.35)
Progressing towards the condition of normal incidence, we find that both
Eqs. (3.29) and (3.35) are identical. This also ensures the validity of our mathematical
formulation.
Using appropriate transformations, both equations can be reduced to
t 11 n right + n left t 22 − t 12 n right n left − t 21 = 0
(3.36)
The condition for convergence for reflection and transmission coefficients
assuming both the refractive indices at left and right boundaries may be written
as
|r s |
2
+
n right
n left
|t s |
2
≤ 1
(3.37)
Now we will consider the unit cell for calculation for reflection and transmission
coefficients.
3.3 Transmission Coefficient
We have now considered the smallest unit of one-dimensional photonic crystal
structure, where propagating waves can be represented in generalized form
p j = t ji p i + r i j q j
(3.38)
q i = t i j q j + r ji p i
(3.39)
where t ij and r ij are transmissivity and reflectivity in passing from layer i to layer j
(Fig. 1).
From the coupled wave equations, transfer matrix at the interface for ideal (defectfree) structure can be obtained as [31]
Fig. 1 Unit block of
one-dimensional photonic
crystal