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Appendix I
2 x 2 Matrix formulation for Isotropic Layered Media
[Yeh 88]
tel-00916300, version 1 - 10 Dec 2013
Let us consider the case of a plane wave incident on an interface between two media
with refractive indices n 1 and n 2 respectively. It is know that a part of the incident
wave will be transmitted into medium 2 and a part reected back at medium
1. If E i exp [i (ωt − k i .r)], E r exp [i (ωt − k r .r)] and E t exp [i (ωt − k t .r)] are the eld
amplitudes of the incident plane wave, reected and transmitted waves respectively,
following the boundary conditions all the three propagation vectors k i ,k r and k t must
lie in a plane. This plane where the propagation vectors of incident, reected and
transmitted wave lie is known as the plane of incidence. In such a plane, the tangential
components θ i ,θ r and θ t of the wave vectors with regard to the normal of the
plane interface satisfy the following condition,
n 1 sinθ i = n 1 sinθ r = n 2 sinθ t
Eqn (i)
These kinematic properties hold true for many types of propagation vectors such
as light waves, sound waves, matter waves etc.,. But, the dynamical properties
such as the intensity of the reected and transmitted waves, phase changes and
the polarization eect depend on the specic nature of wave propagation and their
boundary conditions.
Extending the case above to TE polarization in a thin lm with three interfaces
at air, thin lm and susbstrate, −→ E (x) for the s wave is a continuous function of
x. However when decomposed into the right and left travelling components (Ref.
Fig.5.1), they are no longer continuous at the interfaces. The amplitude of the eld
at the left and right side of the interface at x = 0 can be represented as column
vectors and linked by
(
(
A ′ 2
B ′ 2
(
) (
A 1
= D1 −1 D 2
B 1
)=P 2
(
) (
A 2
=
B 2
) (
A 2
=D2 −1 D 3
B 2
A ′ 2
B ′ 2
A ′ 3
B ′ 3
)≡ D 12
(
e iφ 2
0
0 e −iφ 2
)≡D 23
(
A ′ 2
B ′ 2
)(
A ′ 3
B ′ 3
)
)
A 2
B 2
)
Eqn (ii)
Eqn (iii)
Eqn (iii)
where D 1 ,D 2 and D 3 are the dynamical matrix which relates the refractive index
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