Most people who have a passion for maths, who are charmed by its universal language, fall in love with this utopian but imaginary world, an unbelievably beautiful world filled with never-ending variety of phenomena. This world is interesting also because it represents many aspects of the real world, helping us to understand some physical or biological processes.

My research deals with problems in communications engineering. For a mathematician, even the day-to-day life poses interesting and challenging questions. For example, it is possible to exploit maths for every estimate or forecast in any given sector. Over the last few years, I have been working with mathematical models that describe the propagation of light waves through optical fibres. Every impulse constitutes one bit, the fundamental unit of information. From a mathematical point of view, an impulse is only a particular function with a fixed shape. The cables used in communication systems are made of special fibres to transmit voluminous data over long distances: telecommunications engineering tries to improve this technology by finding ways to transmit information faster and more accurately.

Unfortunately, these are conflicting goals: the faster the transmission, the greater the distortion and the weaker the signal. In fact, the impulses tend to interact during their passage through the fibres, changing and dispersing the information being transmitted. This problem is solved using a particular function for the impulse as it speeds along the cable. The function preserves the shape of the impulse and removes any small distortions as they are introduced. Also, the impulses are sent at slightly longer intervals—which, of course, slows down the transmission but keeps the successive impulses intact by preventing collision.

The mathematical models treat optical fibres as a straight line and assume that the material of the fibre has a refractive index proportional to the square of the amplitude of the impulse. Both the assumptions are valid for many optical materials and for impulses with smaller wavelengths.

Now think of a sheet that we are reading as consisting of two series of lines, one comprising vertical lines and another comprising horizontal lines. The idea is to send the impulses or light waves in the two orientations simultaneously, thereby doubling the speed of transmission. Such simultaneous transmission in two orientations is called bimodal transmission because the data are sent in two different modes. Mathematical models for bimodal transmission are, naturally, more complex than those for unimodal transmission because the two modes tend to interact. The inevitable result of the interaction is distortion of signals. It is therefore necessary to find a suitable shape for each of the two impulses that can prevent – or at least minimize – such distortion.

In the last months of 2009, my collaborators and I proved the existence of at least three such shapes, which seem good candidates for improving bimodal data transmission. We now have to study the properties of these elusive shapes in greater detail to show that the loss of information is minimal. This work also has some interesting practical applications; for example, it can make your internet connection faster and more efficient!

Eugenio Montefusco

University of Rome La Sapienza

www.atomiumculture.eu