Total reflection
Let’s go back to the phenomenon of total reflection, as in the figure
The condition for total reflection is:
sin  l 
n2
n1
(EM52)
and can be fulfilled only if n2  n1 . What happens if  i   l ? The refraction condition gives
n2 sin  t  n1 sin  i  n1 
n2
 n2
n1
hence sin  t >1. The transmission angle becomes imaginary. We have to use in our
computations only real angles, therefore we replace  t with some function of  i , which is
definitely a real angle. Consider the figure used for Snell laws:
The electric field in the 2nd material is
1




Et  E0t exp i t  k t r

 
n
Here kt  r  k t x sin  t  k t z cos t and kt  2 . Replace  t by  i :
c
n
sin  t  1 sin  i ,
n2
2
n

cos t   1   1 sin  i   i
 n2

2
 n1

 sin  i   1
 n2

where the square root is real. The electric field of the transmitted wave is
2






n1
n2  n1




Et  E0t exp i t 
x sin  i  exp 
z  sin  i   1
 c

c


 n2



Discussion during the lectures.
2