8
A. Deyasi and A. Sarkar
Therefore, we can modify Eq. (3.26) as
T
TE
1
κ
left
z
κ 0
= A
1
κ
right
z
κ 0
(3.28)
where κ
left
z and κ
right
z
are wave vectors along Z-axis.
The transcendental equation for the eigenmodes of the structure:
t
TE
11
κ
right
z
κ 0
+ t
T E
12
κ
left
z κ
right
z
κ
2
0
− t
TE
21 −
κ
left
z
κ 0
t
TE
22 = 0
(3.29)
where t i j have usual meanings as matrix elements of T
TE . Solution of Eq. (3.29) leads
to complex values of frequencies. Henceforth, we have to consider only those values
as real solutions which consist of positive real values and zero imaginary values. So
solution numbers will be limited, and they can only carry the practical aspects.
Similar procedure can be taken for TM mode also. Again, we have to choose
respective Cartesian system along with existing field components as given by
E =
E x 0 E z
(3.30)
B =
0 B y 0
(3.31)
The transfer matrix under that polarization concerned T
TM can be correlated with
the existing field vectors
B y
−E x
at both the boundaries
T
TM
B
left
y
−E
left
x
=
B
right
y
−E
right
x
(3.32)
The Maxwell equation gives
−
κ z
n 2 κ 0
B y = E x
(3.33)
Therefore, modifying (3.32) as
T
TM
1
κ
left
z
n
2
left κ 0
= A
1
κ
right
z
n
2
right κ 0
(3.34)
Corresponding transcendental equation is