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

.