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Homework Problem

Pb. 8.31 Use the decomposition given by Eq. (8.103) and the results of
Pb. 8.15 to prove the Sylvester theorem for the unimodular matrix, which
states that:

 sin[( n + ) ]1 θ − Dsin( n )θ Bsin( n )θ 
a  b n  
M =   =  sin( )θ n )θ n ) sin[(θ sin( )θ n − ) ]1 θ 
n
−
c 
d
 C sin( Dsin( 
 sin( )θ sin( )θ 
where θ is defined in Equation 8.97.

Application: Dynamics of the Trapping of an Optical Ray
in an Optical Fiber
Optical fibers, the main waveguides of land-based optical broadband net-
works are hair-thin glass fibers that transmit light pulses over very long dis-
tances with very small losses. Their waveguiding property is due to a
quadratic index of refraction radial profile built into the fiber. This profile is
implemented in the fiber manufacturing process, through doping the glass
with different concentrations of impurities at different radial distances.
The purpose of this application is to explain how waveguiding can be
achieved if the index of refraction inside the fiber has the following profile:

 n 2 
n = n 1 − 2 r 2 (8.104)
0   2  

22
where r is the radial distance from the fiber axis and nr is a number smaller
2
than 0.01 everywhere inside the fiber.
This problem can, of course, be solved by finding the solution of Maxwell
equations, or the differential equation of geometrical optics for ray propaga-
tion in a non-uniform medium. However, we will not do this in this applica-
tion. Here, we use only Snell’s law of refraction (see Figure 8.4), which states
that at the boundary between two transparent materials with two different
indices of refraction, light refracts such that the product of the index of refrac-
tion of each medium multiplied by the sine of the angle that the ray makes
with the normal to the interface in each medium is constant, and Sylvester’s
theorem derived in Pb. 8.31.
Let us describe a light ray going through the fiber at any point z along its
length, by the distance r that the ray is displaced from the fiber axis, and by
the small angle α that the ray’s direction makes with the fiber axis. Now con-
sider two points on the fiber axis separated by the small distance δz. We want

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