6
A. Deyasi and A. Sarkar
R = |r s |
2
(3.15)
T = |t s |
2 n right
n left
(3.16)
If we consider the structure as symmetrical w.r.t incident and propagation of e.m
wave and also w.r.t refractive indices of the interfaces, i.e. n left = n right ≡ n (this
situation is analogous to the cavity-embedded quantum well), then the boundary
conditions lead to:
T
1 + r s
n − nr s
=
t s
nt s
T
t s
−nt s
=
1 + r s
−n + nr s
(3.17)
Solution of Eq. (3.17) leads to the actual composite transfer matrix
T
=
1
2t
t
2
s − r
2
s + 1
−
(1+rs)
2 −t
2
s
n
n
(r s − 1)
2
− t
2
s
t
2
s − r
2
s + 1
(3.18)
Now we have to introduce independently both the polarization conditions. First
we will insert the condition for TE mode for which boundary conditions of field
components will be modified with the following substitutions:
k z = k cos θ
n → n cos θ
(3.19)
where ‘θ ’ is angle of propagation.
Similarly, we can proceed for TM polarization, following the Born and Wolf
approximation, substitutions become
k z = k cos θ
n →
cos θ
n
(3.20)
Introducing suitable suffixes considering propagation from left to right, Eqs. (3.18)
and (3.19) can be reformulated as for TE polarization
n left → n left cos θ left
n right → n right cos θ right
(3.21)
A. Deyasi and A. Sarkar
R = |r s |
2
(3.15)
T = |t s |
2 n right
n left
(3.16)
If we consider the structure as symmetrical w.r.t incident and propagation of e.m
wave and also w.r.t refractive indices of the interfaces, i.e. n left = n right ≡ n (this
situation is analogous to the cavity-embedded quantum well), then the boundary
conditions lead to:
T
1 + r s
n − nr s
=
t s
nt s
T
t s
−nt s
=
1 + r s
−n + nr s
(3.17)
Solution of Eq. (3.17) leads to the actual composite transfer matrix
T
=
1
2t
t
2
s − r
2
s + 1
−
(1+rs)
2 −t
2
s
n
n
(r s − 1)
2
− t
2
s
t
2
s − r
2
s + 1
(3.18)
Now we have to introduce independently both the polarization conditions. First
we will insert the condition for TE mode for which boundary conditions of field
components will be modified with the following substitutions:
k z = k cos θ
n → n cos θ
(3.19)
where ‘θ ’ is angle of propagation.
Similarly, we can proceed for TM polarization, following the Born and Wolf
approximation, substitutions become
k z = k cos θ
n →
cos θ
n
(3.20)
Introducing suitable suffixes considering propagation from left to right, Eqs. (3.18)
and (3.19) can be reformulated as for TE polarization
n left → n left cos θ left
n right → n right cos θ right
(3.21)