PHYSICS 415/416 SUPPLEMENTAL PROBLEMS
Part 7
7A. (Fresnel Equations)
Derive the Fresnel equations for TM polarized light. I suggest you follow the outline below. This attack
closely follows what we did in class for the TE case.
Begin with the following geometry. Note that the positive x-direction is to the left. Also note that the reflected
magnetic field points in the negative y-direction. This direction was chosen so that the reflected electric field in
the diagram is not inverted with respect to the incident and transmitted electric fields (the x-components all
point in the same direction). If the actual reflected wave is inverted upon reflection then this will make the
reflection coefficient negative. This is what we want in our coefficient since this will be consistent with the
meaning of a negative reflection coefficient in the TE case. (Some books will flip the reflected fields. This
makes all of the magnetic fields point in the same direction, which makes for a simpler set-up of the problem.
However, then a positive TM reflection coefficient means an inversion upon reflection, which is in direct
contrast with the TE case. I think it is better to go with this geometry.)
E
^z
Br
B
Er
^x
^y
Bt
Et
Apply the boundary conditions that the components of the magnetic and electric fields tangent to the surface
must be continuous. Again, be aware that the reflected magnetic field points in the negative y-direction and that
the positive x-axis points to the left.
Once you have these equations, follow the steps that we did for the TE case. Specifically, relate the magnetic
field amplitudes to the electric field amplitudes, express cost in terms of sin, and use the definitions of the
index ratio and the reflection and transmission coefficients to finally arrive at the following equations,
rTM 
tTM 
 n 2 cos   n 2  sin 2 
n 2 cos   n 2  sin 2 
2n cos 
n 2 cos   n 2  sin 2 
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7B. (Fresnel Equations)
Use the Fresnel equations to demonstrate the following:
 1  n
(a) Show that for normal incidence, the reflectance of both polarizations is given by R  
 . How much
 1  n
light energy is reflected if light traveling in air is normally incident on a piece of glass with an index of 1.5?
(b) Show that Brewtser’s Law follows from the Fresnel equations. Recall that at the Brewster angle (or
polarization angle), the reflected light is TE polarized. Thus, at i = p , the TM reflection coefficient vanishes.
[You may find the following trig identity helpful: sec2 = 1 + tan2 ]
(c) Show that the expression for the critical angle derived from Snell’s Law,  c  sin 1 (n2 / n1 ) , can also be
derived from the Fresnel equations for the reflection coefficients. Recall that all of the light is internally
reflected so that |r| = 1. Show that the expression is valid for both polarizations.
WARNING: Do not try to derive the expression by requiring that t=0 and working with the transmission
equations. This will not work! We will say more about this in class.
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7C. (Fresnel Equations)
TE polarized light hits side 1 of the 45°-45°-90° glass prism shown below with an incident power of 50 W at
normal incidence. The index of the glass is 1.65. Find the power of the light that leaves side 3.
Side 3
Side 1
Side 2
7D. (Fresnel Equations)
Consider another 45°-45°-90° prism. Again, TE polarized light hits Side 1 at normal incidence as shown in the
previous problem.
(a) What phase shift does the light transmitted through side 1 experience?
(b) What phase shift does the light transmitted through side 3 experience?
(c) Suppose that you wanted the prism to act as a quarter-wave plate, i.e. you would like to give the light a
phase shift of /2 when it reflects off of side 2. Show that this cannot be done by finding the required refractive
index of the glass.
(d) Could you give the light a phase shift of /4 when it reflects off of side 2? Answer this by finding the
required refractive index of the glass.
(e) Sketch how you could use two prisms to make a quarter-wave plate.