Teaching Modern Physics - QuarkNet - Fermilab

Teacher’s Key
Index of Refraction Problems
1. Assume that you have a long optical fiber with core index of refraction ncore = 1.59 and
cladding nclad = 1.49. The fiber is 2 m long with an inner-core diameter of 1 mm. Light is
generated on the axis of the fiber at one end and bounces along it to the other end. Due to
small imperfections in manufacturing, the light is not quite perfectly reflected at each
bounce. Calculate what fraction of light is reflected at each bounce so that 50 percent of
the light makes it out of the far end of the fiber. Assume that the reflection fraction is
identical for each bounce and consider only light transmitted at the critical angle.
θc
t1
t2
First calculate the critical angle, then the amount of distance traveled by the light
(along the fiber-axis direction) for each bounce. From that, you can calculate the
number of bounces (n). Since each bounce reflects a factor r of the light, the light
transmitted will be r n .
sin θc = 1.49/1.59 → θc = 69.6°
tan θc = t1/(1 mm) → t1 = 2.69 mm
tan θc = t2/(2 mm) → t1 = 5.38 mm
How many bounces?
t1 + n t2 = 2000 mm → n = 371 bounces.
Since each reflection reflects r fraction of the light,
0.5 = r n ,
which means that r = 0.998 or 99.8%.
2. Standard materials used in scintillating fiber in particle physics experiments are
polystyrene (n = 1.59), acrylic (PMMA, n = 1.49) and fluorinated acrylic (fPMMA, n =
1.42).
Calculate the critical angle (i.e., the angle at the core-cladding interface) for light trapped:
a. In a fiber with a polystyrene core and a PMMA cladding.
sin θc = 1.49/1.59 → θc = 69.6°
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