Dispersion

1. One of the homework problems was designed to lead you to recognize an important physical limitation on the transmission rate possible through optical fiber.

   • I'm referring to the problem that asked you to think about the difference in propagation speed between the light rays that travel "straight" through a fiber and those that bounce back and forth between the boundaries of the core and cladding.

   • The problem was intended to make you realize that the light pulses sent through the fiber get wider as they travel. This is called dispersion.

     At the point where the pulses have spread so much that the pulses on either side of a given pulse completely overlap the given pulse, it will become basically impossible to decode the intended signal.

   • As a result, in such fiber there is an upper bound on the product of the data rate and the length of the fiber.

   • Graded-index fiber and single mode fiber are used to reduce this effect.

2. In addition to spreading the pulse out, the different paths light rays can follow in fiber will distort the "squareness" of the light pulses.

   • The intensity at the leading edge will be reduced as only the fastest rays of the leading edge will remain "leading".

3. By investigating the source of this effect a bit more closely, we will realize that a form of dispersion will also effect electrical signals and optical signals in single mode fiber.

4. The problem I assigned involved an angle that depended on the indices of refraction of the core and cladding of the fiber. Let's do a little physics to make sure we understand what the index of refraction of a material really is.

   • The diagram below represents two parallel rays of light being refracted at the boundary between two materials with different indices of refraction, eta₁ and eta₂.

     

   • Assuming that ₁ is the index of refraction of the material through which the light passes first, then the angles can be determined using Snell's Law:

     (₂)/(₁) = (cos)/(cos)

   • To appreciate this formula, one must understand what the constant known as the index of refraction really is.

     • When we speak of the speed of light, "c", we are really talking about the speed of light in a vacuum.

     • When light (or other electromagnetic signals) travel through other materials they travel at speeds less than c.

     • The index of refraction of a material is the number by which one must divide "c" to obtain the speed of light in a particular material.

   • In the diagram, the point C represents the location of the leading edge of one of the rays at the point that the other ray reaches the boundary (at A). The points B and C similarly represent the positions of the two rays at a slightly later point in time.

   • From the diagram it is clear that

     CD = AD cos

     and

     AB = AD cos

   • Therefore:

     (CD)/(AB) = (cos)/(cos)

   • If t is the time it takes one ray to travel from A to B, then AB must equal t times the speed at which the ray is traveling. This speed is c/₂. Similarly, the speed of the ray going from C to D is c/₁.

   • Replacing the lengths of CD and AB then yields:

     (₂)/(₁) = (cos)/(cos)

5. Now that you understand how refraction behaves in a fiber, let's get back to an example of refraction with which you are probably more familiar: the behavior of white light passing through a prism.

   • As you know, passing white light through a prism separates the light showing the colors of the spectrum.

   • This occurs because different frequencies of light are refracted differently.

   • This implies that the index of a refraction is not really a constant associated with a material, but a function of both the material and the frequency of the light passing through.

   • Finally, since the index of refraction is just the ratio of the speed of light in a material to its speed in a vacuum, this implies that different frequencies of light travel at different speeds (unless in a vacuum).

6. Now, we can put this together with Fourier analysis to realize the horrible truth.

   • When we send anything other than a pure sine wave, we are actually sending a wave that is composed of many different frequencies.

   • These different sine wave will travel at different speeds.

   • As a result, they will spread apart as they travel just like the rays of light taking different paths through a fiber.

7. The horrible truth: All baseband and modulated signals suffer from dispersion.