Problem set #4, Due Apr 5th
1. Maxwell Equations and Operators:
For plane waves of the form,
 
 i ( kr t )
E (r , t )  E0e
,
show the following relationships apply:
 

  E  ik  E
 

  E  ik  E


E
 i  E
t
Please use these spatial and temporal derivatives to simplify Maxwell’s equation below for
the case of plane waves
B
t
E
  B  
t
E   /
 E  
B  0
2. Fresnel Formulas Fresnel was the first to do the calculation on the fraction of a light wave
reflected and transmitted by a flat interface between two media with different refractive
indices. We would like to derive such formulas
First we consider a beam geometry for light with its electric field perpendicular to the plane of
incidence (i.e., out of the page). The definitions for all the relevant parameters are given in the left
panel of the figure below.
(1) Write down the boundary condition for the electric field at the two interfaces below.
(2) Given the boundary condition above, confirm the Fresnel’s equations for the polarized
light parallel to the incident plane (right panel)
r||  E0 r / E0i   ni cos(t )  nt cos(i ) /  ni cos(t )  nt cos(i )
t||  E0t / E0i  2ni cos(i ) /  ni cos(t )  nt cos(i )
ki
Er
Ei
Bi
ki
kr
ni
Bi
Br
i r
kr
Ei
Br
ni
i r
Er
Interface
Interface
t
t
Et
Bt
nt
kt
Et
Bt
kt
nt
3. We consider a beam geometry for light with its electric field parallel to the plane of
incidence (i.e., in the page). The definitions for all the relevant parameters are given in the
right panel of the figure above.
(1) Calculate the reflection & transmission coefficients.
(2) Let’s define ni = 1(air) and nt = 1.5 (glass), plot the above reflection & transmission
coefficients as function of incident angle θi from 0 to 90 degrees for an Air-to-Glass
Interface. It will be ideal to use a graphic software to do the plotting (any software
is fine). If you decide to do it with hand, you are required to at least sample 20
points for θi from 0 to 90 degrees.
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