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FIGURE 8.4
Parameters of Snell’s law of refraction.

to find r(z + δz) and α(z + δz), knowing r(z) and α(z). We are looking for the
iteration relation that successive applications will permit us to find the ray
displacement r and α slope at any point inside the fiber if we knew their val-
ues at the fiber entrance plane.
We solve the problem in two steps. We first assume that there was no bend-
ing in the ray, and then find the ray transverse displacement following a
small displacement. This is straightforward from the definition of the slope
of the ray:

δr = α(z)δz (8.105)

Because the angle α is small, we approximated the tangent of the angle by the
value of the angle in radians.
Therefore, if we represent the position and slope of the ray as a column
matrix, Eq. (8.105) can be represented by the following matrix representation:

 rz( + δ z) 1 δ z  rz()
  =     (8.106)
α z ( + δ z) 0 1  α z ()

Next, we want to find the bending experienced by the ray in advancing
through the distance δz. Because the angles that should be used in Snell’s law
are the complementary angles to those that the ray forms with the axis of the
fiber, and recalling that the glass index of refraction is changing only in the
radial direction, we deduce from Snell’s law that:

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