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

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

The ellipsometric function for an ambient-film-substrate system is defined as the ratio of complex-amplitude reflection coefficients for the and polarization:

ρ

r

p

s

ρ

r

Rp

R

s

A+BX+C

2

X

D+EX+F

2

X

where are functions of Fresnel interface reflection coefficients , , , and . The pairs and signify the ambient-film and film-substrate interfaces. Here is an exponential function of metric film thickness and is the thickness period. If the ambient and film media are transparent, the point representing moves uniformly in a clockwise direction around the unit circle in the complex plane as increases from 0 to . Because is related to by an analytic rational function, the point representing must trace a closed contour in the complex plane as traces the unit circle in the complex plane.

A,B,C,D,E,andF

r

01p

r

12p

r

01s

r

12s

01

12

X=exp-j2πd

D

ϕ

d

D

ϕ

X

X

d

D

ϕ

ρ

r

X

ρ

r

ρ

r

X

X

The figure shows such a family of non-intersecting, constant angle of incidence contours (CAIC) for (normal incidence) to (grazing incidence) for the system at .

ϕ=0

ϕ=90°

air--Si

SiO

2

λ=632.8nm