# Introduction: Mathematical Modelling

A description of a system using the language of mathematics is called a mathematical model. Many processes occurring in nature can be modelled using partial differential equations together with a set of suitable boundary conditions. As an example we choose the stimulated emission of light in a quantum dot (QD) semiconductor laser. The figure on the right shows the process of stimulated photon emission.

Hovering the mouse cursor on the QD band diagram on the right will start a short animation.

Definitions: The simplified physical system shown in the figure above includes a QD and a photon. The symbols used to visualize the system are listed below:
• The conduction band (CB) on the top can be imagined as a container confining electrons.
• The valence band (VB) is a container with spaces that can be filled by electrons.
• The dot in the CB represents an electron.
• The circles in the VB represent holes (empty states that can be filled up by electrons).
• The wiggly curve on the left side symbolizes an incoming photon (quantum of light).

Process: The incoming photon induces electron-hole recombination leading to the emission of a second photon. At the end of the process we have two photons travelling to the right.
The presence of electrons and holes in the medium can be described by using the notion of material polarisation. Further details of how the material polarisation is used to model semiconductor lasers can be found here.

Why mathematical modelling is useful: Scientist and engineers are keen to obtain accurate models of the phenomena they are studying or technological products they are developing. Mathematical modelling can help us understand and explore complex processes. Mathematical models form the basis of any computer aided simulation package.

The mathematical models found on this site are mainly related to light emitting (photonic) devices like wave-guides, lasers, and light amplifiers.