Ellipsometry: finding n, k
or thicknesses, roughness…
3
starts linear,
angle 
linear polarizer,
angle 
1
2
Wikipedia
does
something
 cos2 

 sin  cos 
sin  cos   rp

2
sin   0
http://www.gaertnerscientific.com
/ellipsometers/l116sf.htm
0  cos  


rs  sin  
Superposition of Waves
Single frequency light has infinite extent in time
(mathematical, not physical)
Any light with information (e.g. pulsing) has to
contain a spread in frequencies.
Any light created in the universe has more than one
frequency (it had a beginning).
Superposition of Waves
Addition of two waves, equal magnitude:
cos(k1 x  1t )  cos(k2 x  2t )
 a b 
 ab 
cos(a)  cos(b)  2 cos 
cos



 2 
 2 
Superposition of Waves
Phase velocity
c

vp  f  
=
n( ) k ( )
Group velocity
speed of phase oscillations

d
vg 

k
dk
(this definition not valid when absorption is strong)
speed of “envelope”, pulse, signal, image, energy,
information
Dispersion due to n()
Wave speed depends on 
Dispersion is when vphase
is not constant: k() or
(k) are not linear:
vphase =c/n()
kRe ()=n() /c
k/ 1/c
n


Does group velocity exceed c?
kRe ()=n() /c
k

kvac/ 1/c

1
vp
Look at slope for vg :
dk
1

d  vg
Where our definition is
valid (low absorption),

vg  c even when v p  c
k4
  3c 3
ko
If
in some strange system,
then the group velocity is
k3
a) vg  c k 3
o
b)
c)
d)
k3
v p  12c 3
k
3 o
k
v p  3c 3
ko
k4
vp  c 3
ko
2
If k 
o c
a)
b)
c)
d)
in some strange system, the group velocity is
3
vg  c 3
o
2
vg  c
2o
o
vg  c
2

vg  2c
o
Energy and superimposed plane waves
Practicing simple addition of (quasi-parallel) waves
(Phscs 123)
n o c
I
| E |2
2
You add two 1-D waves, of amplitude E1 and 2E1. In
some places they add constructively, and some places
destructively. E1 by itself corresponds to intensity I1 .
The brightest intensities seen in the interference pattern
are:
a)
b)
c)
d)
e)
3I1
4I1
6I1
9I1
25I1
You add two coherent 1-D waves, of amplitude I1 and
9I1. In some places they add constructively, and some
places destructively.
The dimmest intensities seen in the interference pattern
are:
a)
b)
c)
d)
e)
2I1
3I1
4I1
6I1
8I1
Use
k
n
c
to write
Find f ( ) in terms of n( )
n
a)  k
n
kvac
b)

c)  2 n
k
2 vac

vg 1  v p 1  f ( )