Subsections
• The Input Data
• Preparing for Transmission
• Digital Communications
• Communicating with Light
• Light Pulses in an Optical Fibre
• Mathematics as the Key Tool

  
Making Messages in a Flash

In this Information Age the increasing pressure on our communication systems is producing a move from electronics to photonics. We shall see that mathematics is a vital tool in the design of optical fibres, which are a key element in modern optical communication systems.

A communications system does the following things.

1.

It takes information or data;

2.

converts it into a form suitable for transmission; and then

3.

strives to efficiently and reliably deliver an accurate version of that information to the receiver.

  
The Input Data

Communication systems must handle inputs of all kinds--text, spoken words and music, video and computer data files. If all of this is to be carried on a single channel it will need to be represented in the same way. Usually an electrical signal is involved.

For example, a telephone converts speech into an electrical signal that is transmitted to a receiver where it drives a microphone to recreate the speech. Mathematicians regularly deal with representations of things. Here then we begin with a quantity continuously varying in time and representing the information or data to be sent to the receivers.

Before we consider how to transmit the signal, there are some other questions, of a basically mathematical nature, that we need to ask. The answers will tell us more about what to send and how to do it.

  
Preparing for Transmission

Assume that the information to be sent is represented by a quantity (`the signal') that varies continuously with time. We first consider how much of that signal we must convey to the receivers, so that they get an accurate version of the information. Then we must ask how best to do that.

Suppose that we had to show someone a circle of a particular size and position in a plane. We could just send them a drawing of the circle. However, if we all know a circle is involved, then it is only necessary to transmit the position of any three points on the circle. These effectively define the whole circle, since only one circle can be drawn through any given three points. Instead of transmitting a whole curve, we need only send three pairs of numbers giving the points specifying the circle.

Then our problem becomes how to best pass on those six numbers specifying the points. That will involve questions of...

• accuracy: how closely can you tell one circle from another; and
• how to represent those numbers: as decimals, in binary notation, or as fractions.

Our signal for transmission presents a similar if more elaborate problem, but it too is essentially mathematical.

• Our signal with all its variations can be expressed as a sum of all possible sinusoidal oscillations.
• However, there will be a limit to these, either naturally or by filtering.
  We can assume that the most rapid ones can be ignored. (Hi-fi enthusiasts know all about the question of high frequency response -- cutting out too much may ruin music, but still leave acceptable speech for telephone conversations.)
• Having done this, we may use an important mathematical result known as the `Sampling Theorem'. This tells us that if we know the values of the signal at regular intervals, then those samples will allow the whole signal to be reconstructed at every instant of time--just like giving three points allowed the whole circle to be reconstructed.
• The required sampling rate depends on the highest frequency variations contained in the signal.
• Having reduced the task to the transmission a series of discrete signal values, we must decide on `accuracy' and `form' for those sample values.
• If we divide up the possible range of values into a number of intervals, then for each sample we need only transmit the relevant interval-number.

  This process is called `quantisation'.

• The fineness of the intervals is chosen so that the receiver of the final signal is not aware that it has happened; that is, any errors are below the level of the inevitable `noise'--which is always present in any system.

  
Digital Communications

Next we recall that all numbers can be expressed in binary form; that is, as a string of zeros or ones. We see that the whole communication process reduces to sending so many binary digits or bits of information per second.

How many bits? Well, ...

• for telephone speech, it is 64,000 bits per second.
• for TV, it is 100 million bits per second.

These are rather large numbers--a point we return to later.

The above argument is general: ``all types of information can be reduced to strings of binary digits or bits''.

If we design a carrier or channel for bits of information then it can deal with everything as long as the coding and unmixing of combined signals is correctly carried out. This capacity to carry a mixture of signals in one data stream is an important property of digital communications.

This idea of a signal being expressed in terms of bits of information is not new. The familiar Morse code is a good example; all letters are expressed in terms of dots and dashes. Morse code also provides an example of a simple use of mathematics in communications. The code design was tackled using probability theory:

• count up the usage of different letters;
• express that as a probability of need; and then
• design the code to most efficiently meet that need; in this case ...
  ... using short strings of bits for the most frequently used letters.

Morse code usage also reminds us that bits of information (dots and dashes) can be represented physically in lots of ways:

• dots and dashes on paper;
• beeps of different lengths; or
• short and long flashes of a signalling light.

... and that brings us back to optical communications.

  
Communicating with Light

The string of bits can be represented by a series of time slots, each of which is empty or contains a flash or pulse of light from a laser. Optical fibre guides these pulses to the receiver.

Describing what happens to optical pulses as they travel along the fibre leads us to Samuel Johnson's old problem:

``we all know what light is, but it is not easy to tell what it is''.

Basically we are compelled to use a mathematical description of light, be it in terms of rays, waves or photons. Einstein ^ wrote:

``All the fifty years of conscious brooding have brought me no closer to the answer to the question, `what are light quanta?'''

For fibre design, we describe light by waves. A pulse of light will be a burst of light waves. Now light waves oscillate at incredibly high frequencies, so a very large number of pulses can be fitted into a second. Despite those great demands stated earlier, the bits required for several TV signals can be fitted onto a single light beam. This is why optical communications are so successful.

  
Light Pulses in an Optical Fibre

An optical fibre is a tiny glass rod with a central `core' region. This core has a slightly higher refractive index that the surround or `cladding'. Light travels along the fibre by bouncing off the core-cladding interface. This is rather like the way sound waves travel along a tube, by reflecting off the inside walls.

As a pulse travels along a fibre two things happen:

• there is some energy loss;
• the pulse gradually spreads out, or `disperses'.

Thus a pulse, originally launched into one time slot, may weaken by losing energy or by spreading into neighbouring time slots, in the process contaminating them.

At the receiver, it is only necessary to decide whether or not there should be a pulse in each time slot--irrespective of its size or distortion.
(This gives digital communications a great advantage over analogue communications.)

The fibre designer makes a mathematical model of pulse propagation.
He/she uses it to see how pulse weakening and distortion can be minimised.

The output of a fibre design will be specifications for:

• size and types of glass for manufacturing the fibre;
• the wavelength of the light to be used.

This design will be optimised to say that: ``if you use these parameters, then for reliable communications the signal can be carried this far (perhaps 20 or even 100 kilometres) before you use it, or check it and regenerate it for the next link.''
That is vital information, because signal `repeaters' are intricate and very costly.

But exactly what does `reliable' mean? Mathematics gives a precise definition.
We can ask that all the noise and interference effects should cause us to expect only rare bit-errors: at most one wrong bit in a thousand million is a common standard.

  
Mathematics as the Key Tool

Mathematics provides the key to designing an optical fibre by:

• specifying precisely the job it has to do;
• describing how it operates; and then
• using that description, to produce optimum designs for reliably communicating the maximum amount of information.