Geometrical fiber is the simplest type of fiber optics, has cylindric form with core and cladding layers. Cladding refractive index is lower that core refractive index.

These types of fibres differentiates with the way how refracting index is changing between cladding and the core:

  • step-index fiber;
  • graded-index fiber.

graded and step index fibres

In case of step-index fiber the refractive index changes significantly. And in case of graded-index fiber, refractive index changes gradually.

This type of fiber optics works when the wavelength is much smaller than the core radius.

Rays in fiber

Let’s consider geometry of step-index fiber (below). Here n0sinθ0=n1sinθ1. If we will follow the ray way – the initial ray is refracting on the border air-core, and refracting on the border core-cladding. The refraction on the border core-cladding is possible if the angle sinθ<n2n1.

And n0sinθ0=n12n22, where n0sinθ0 is a numerical aperture NA.

These fibers are not very useful for communication purposes because of modal dispersion, when all the rays are spreading differently (with different angle in the fiber resulting different time dispersion at the output).

When multiple rays are traveling across the fiber, they can go through multiple lengths. Rays will end at the same time at the end of the fiber if they were travelled with the same speed.

Let’s consider an impulse through the fiber. We can calculate the time gap between the shortest and longest pathways of the dispersed rays through the fiber. The shortest pathway will be in case of θ0=0, and the longest – for φ=arcsin(n2n1). If L is a length of the fiber, then time difference between shortest and longest ray will be T=Lcn21n2n1n2n1 at the end of the fiber cable. This time difference is a measure of the signal broadening. This time difference should be less than a time slot 1B. So BL<n2cn1(n1n2). This relation is the limitation for step-index fibres.

Most fibres are designed so the ratio n1n2n1<0.01.

The gradual-index fibres are different from step-index ones. Here the refractive index is changing gradually from n1 to n2 value. Most graded-index fibers are designed towpath the α-profile:

n(ρ)=n1[1n1n2n1(ρa)α], ρ<an1(1n1n2n1)=n2, ρa, here a is a core radius.

The most frequent fiber profile is graded-index profile, corresponding to α=2, and is called parabolic-index profile.

The minimal dispersion corresponds to α=2(1n1n2n1) so TL=n128c and BL<8cn12, where =n1n2n1. The product of BL shows the quantitative characteristics of fiber, so it can be prepared the way to carry data with high bit rate to the long distances.