ECE4853
Physical optics
version 1.1
Exercise F-1: For two parallel reflecting surfaces, physical optics requires that waves travelling different
paths will destructively interfere unless the difference between the phase shifts 1 and 2 of the two paths
 = 2m, where m is an integer.
For the geometry of two reflecting surfaces, as shown by figure F-1 (a) :
Figure F-1(a): The dashed lines represent wavefronts. The angle  is the grazing angle of an optical ray
path. Total internal reflection will occur for  < 90 – C, whereC is the critical angle
sin C 
( defined by)
n2
n1
Phase-shift due to path length is
S 
2d sin 
1
where  0/n1, for which n1 = refractive index of the core
Phase-shift due to reflection (Fresnel laws) is
1
 TE   N  2  tan 1  
 n
 TM
n 2 cos 2   1 

sin 


n 2 cos 2   1 
  p  2  tan n 

sin 


1
In these relationships n is the relative index of refraction = n1/n2 .
These are phase shifts due to perpendicular (N) and parallel (P) polarizations, respectively (of the Efield)
The ray represented by path AB includes two reflections before coming back to the same direction it
started with. So when the difference between S and 2N (or between S and 2P ) = 2x m, then nondestructive interference will result. This condition selects a particular grazing angle m .
The angle m can only be found by iteration (or by graphical assessment). We will use the graphical
means using Excel by executing a plot of N (or P) vs a plot of
S 
2d sin 
1
 m
The result should look something like that shown by Figure F-1(b). The intersection identifies the value
of m corresponding to the value of m given in the header.
Figure F-1(b). The intersection shows that m corresponding to the value of m= 1 is m = 3.66o (which
can probably be determined more easily by comparing adjacent column values than by plot intersection).
Assignment: Set up a plot of  vs N similar to that shown as a means to determine the (1) grazing angle
and (2) the acceptance angle(s) at which constructive interference will occur. Keep in mind that the
intersection condition may be satisfied for several different values of m. Show your construct as your
Figure F-1 as set up for extraction of info for task (a) below.
(a) For an optical fiber with n1 = 1.525 and n2 = 1.520 and core diameter d = 10m determine the
allowed grazing angle(s)  m and (2) acceptance angle(s) m for lightwaves having  = 850nm.
Express angles in terms of degrees rather than radians.
(b) Repeat for lightwaves having  = 1255nm
(c) For (1) d = 10m and (2) d = 5m
use your construct to determine the cutoff wavelengths for m = 1 and 2.
List the extracted values (and include the specifications) for these three requirements in a table = your
Table F-1
Submit electronically (pdf files only. Snapshots of figures and tables with supporting text explanations)