WOLFRAM|DEMONSTRATIONS PROJECT

Constant Angle of Incidence Contours for an Air-SiO2-Si System

​
incidence angle
metric thickness
The ellipsometric function
ρ
r
for an ambient-film-substrate system is defined as the ratio of complex-amplitude reflection coefficients for the
p
and
s
polarization:
ρ
r
=
Rp
R
s
=
A+BX+C
2
X
D+EX+F
2
X
,
where
A,B,C,D,E,andF
are functions of Fresnel interface reflection coefficients
r
01p
,
r
12p
,
r
01s
, and
r
12s
. The pairs
01
and
12
signify the ambient-film and film-substrate interfaces. Here
X=exp-j2πd
D
ϕ

is an exponential function of metric film thickness
d
and
D
ϕ
is the thickness period. If the ambient and film media are transparent, the point representing
X
moves uniformly in a clockwise direction around the unit circle in the complex
X
plane as
d
increases from 0 to
D
ϕ
. Because
ρ
r
is related to
X
by an analytic rational function, the point representing
ρ
r
must trace a closed contour in the complex
ρ
r
plane as
X
traces the unit circle in the complex
X
plane.
The figure shows such a family of non-intersecting, constant angle of incidence contours (CAIC) for
ϕ=0
(normal incidence) to
ϕ=90°
(grazing incidence) for the
air-
SiO
2
-Si
system at
λ=632.8nm
.