Sending Signals

You have already learned about traveling waves and how they can interfere to produce maxima and minima, and how total internal reflection can trap light in an optical fiber.  Now let's look at how to describe the light traveling through a fiber and how those traveling waves can combine to produce signals.
 

More about optical fibers and light guidance is found in Part A of your reading assignment:  Chapter 3 of William Grant's book Understanding Lightwave Transmission:  Applications of Fiber Optics.  This is the first chapter of the handout you received in class.

One important difference between Grant's work and the information about optical fibers in the on-line tutorial is the definition of numerical aperture.  Grant's book assumes light enters the fiber from air, so the index of refraction outside of the fiber is 1.0.  The tutorial is more general, allowing the medium outside of the fiber to be anything.  Thus the definition of numerical aperture in the tutorial is more general as well:

where n₀ is the index of refraction outside the fiber, and q₀ is the angle of acceptance for light to be completely trapped inside the fiber.  In the special case where n₀=1.0 (fiber is in air), this definition of NA simplifies to the one given in Grant's book.  An important concept to understand is that NA is completely determined by the indices of the cladding and the core rather than the medium by which the fiber is surrounded.  The definition above may appear to depend on n₀, but a change in n0 will result in a corresponding change in the angle of acceptance so NA does not change.  NA is a property of the fiber, not of how it is used.
 

The next section on Fourier Analysis may seem like a non-sequitor to fiber optics, but it is really very closely related.  You have undoubtedly heard how superior fiber optic connections are to older coaxial cable connections.  You might even have heard that one of the advantages is the large bandwidth fiber optics can carry.  Carrying a large bandwidth means fibers can transmit large quantities of data.  But have you ever wondered what the connection between bandwidth and quantity of information is?  Or why you need to transmit a large range of frequencies to send information quickly?  The connection lies in a mathematical relationship first derived by Jean Baptiste Fourier.
 

Many naturally occurring waves (such as traveling electromagnetic waves) oscillate sinusoidally. Many others, however, do not. Consider the waves at the beach, or a wave created by someone's voice. These waves are erratic and non-uniform - certainly not sinusoidal. Trying to reproduce these waves and transmit them without huge distortions could be nearly impossible if it were not for the mathematical relationship proven by Jean Baptiste Joseph Fourier.

Fourier proved that every periodic wave can be expressed as a sum of sine waves. The frequencies of the component sine waves are integer multiples of the frequency of the original wave. For example, a square wave (a value of 1.0 for T/2, -1.0 for the next T/2, then repeating, see below) can be created by the Fourier series

The above sum and square wave it produces are shown in the figure below. Because the square wave is very discontinuous, and sine functions are very continuous and smooth, many terms of the series must be added before the desired squareness is achieved. An important note to remember is that Fourier series assume a repeating function (otherwise, the frequency is not defined). Even though the figure only shows one cycle, this square wave (and the sine waves adding to produce it) continues indefinitely.  In the figure, the blue line is the desired square wave, the fuchsia line represents the term from the above equation being added, and the red line represents the sum up to and including the fushcia line.

Most signals, however, are not periodic. No information could be sent if they were. A periodic square wave, for example would just indicate 101010101010 . . . . In order for a signal to contain meaningful information, it must change, and the frequency of the signal is no longer well-defined. Fortunately, Fourier addressed this case too. In essence, a non-periodic signal is just a signal with infinite period (it takes forever before it repeats). So the frequency of the signal is the inverse of a very large number, or zero. Thus the frequencies of the different terms in the sum differ by a very small amount (zero). If I take the limit of a sum for which the difference between successive terms becomes zero, that's an integral! So a non-periodic function can be represented as an integral over frequency of sine functions with coefficients that depend on the frequency and so are inside the integral. Such an integral is known as a Fourier transform.

This section contains some upper-level mathematics concepts that are not the focus of this course. The main thing you should understand is that any periodic function can be expressed as a sum of sines, and any non-periodic function can be expressed as an integral of sines. For an exact solution, the sums and integrals contain frequencies up to infinity. In practice, the sums and integrals can be truncated at some maximum frequency. The more frequencies you include in the sum or the integral, the more exact the expression will be.

Fourier series and transforms are important when considering data transfer since different media have different ranges of frequency, or bandwidths, for which they are reliable. For example, some frequencies of electromagnetic radiation will be absorbed more by the atmosphere than other frequencies will. A signal traveling through the atmosphere will thus be distorted if it contains those frequencies which are absorbed. So we want to transmit signals through the atmosphere only in bandwidths that do not contain easily-absorbed frequencies.
 

Any mathematical physics or mathematical engineering text such as Mathematical Methods in the Physical Sciences by Mary L. Boas..