Modelling in Optics
Linear and nonlinear Optics
We are used to deal with linear optical materials: a piece of glass is a standard example. It modifies light, it bends it, it absorbs it, but it is not modified by the light itself: its refractive index is not changed by the light, nor its absorption coefficient. Consider now photo-chromic glasses: these are spectacles that become darker in intense sun light. The glass of the lenses is a nonlinear material: its properties, in this case the absorption coefficient, are changed by the light that passes through it. Non linear materials are at the heart of many optical devices, starting from the laser that powers CD and DVD readers. The branch optics that studies optical phenomena based on the nonlinear properties of materials is called nonlinear optics.
What is optical device modelling?
An optical device is an apparatus that uses light to carry out a function. Examples of very 
common optical devices are lasers, optical fibers, liquid crystal displays and, even, simple binoculars. Some of these devices, like binoculars and liquid crystal displays, are "linear" in the sense explained above: the change light, but are not affected by it. Others are inherently non-linear, the laser is the standard example: a laser works because there is a strong interaction between light and matter that enables the device to produce coherent, monochromatic light. Other devices can be considered linear or nonlinear depending on the properties of the material and the intensity of the light shource: optical fibres are examples of such devices.
Modelling an optical device consists in finding a set of ordinary or partial differential equations with appropriate boundary conditions that describe the behaviour of the device: they can be used to analyse its properties, to verify our understanding of its behaviour, to optimise its design to a specific purpose, to test new designs before they are actually built.
The process of modelling a device involves a good understanding of the physics of the device in order to extract the key featires that are essential to a successful model. It also requires a good understanding of mathematics in order to "code" physical understanding in mathematical terms and be able to interpret the resulting equations.
Some examples of our research activity in this area
Giant amplification and control of noise in optics
Noise driven output of an Optical Parametric Oscillator
The study of dynamical systems teaches us that a fixed point is stable if the eigenvalues of the dynamical system linearised around the fixed point have all negative real part. Equivalently, the study of systems of linear first order differential equations with constant coefficients has taught us that a solution will decay asymptotically to zero if all the eigenvalues of the coefficient matrix have negative real parts. This however, is only part of the story. While it is true that the solution will decay asymptotically to zero, it is also possible for it display transient growth, i.e. it initially grows and then decays to zero. This phenomenon is possible if the matrix of the coefficients is non normal, i.e. does not commute with its adjoint.
Non-normal matrices are quite common in physical and engineering systems. In optics they are responsible for giant amplification of noise. For example, we have recently shown that in an Optical Parametric Oscillator noise can be amplified by a factor of 109. It is possible to use the theory of non-normal operators to estimate the magnitude of the amplification and to control it. For more information see the Non-normal effects in optics section of Dr D'Alessandro's web site.
The optics of liquid crystals
Liquid crystals are asymmetric molecules that have some form of long range order, intermediate between the perfect order of a crystal and the complete disorder of a liquid. The most important property from my point of view is that they have a very strong nonlinear response to the light passing through them: this makes them ideal materials to study the strong coupling limit of light and matter. This project has many strands, including beam coupling in photo-refractive liquid crystal cells and the optics of coupled liquid crystal microcavities. For more information see the liquid crystal section of Dr D'Alessandro's web site.
Optical Pattern Formation and Symmetry breaking
As a beam propagates through optical materials its shape can change: for example, 
structures can appear across an originally uniform beam. On the other hand, a predetermined pattern, superimposed on the beam, can be unstable and be destroyed during propagation, causing a loss of information. All these phenomena are described by partial differential equations (mainly of parabolic type) that can be analysed using a variety of techniques, like linear and weakly non-linear stability analysis, group theoretical approaches to bifurcation theory and straightforward numerical integration of the models equations.
For an example of the study of pattern formation in optics using symmetries have a look at the page Anomalous patterns, symmetries and lasers. This work has continued in two directions. On the one hand we have studied the symmetries and bifurcation of the low number of mode dynamics of lasers. This work is summarised in this poster. On the other hand we have studied the many-mode dynamics of wide aperture laser and analysed the behaviour of the complexity of the dynamics (and of the quality of the beam) as a function of the pump power.
For more information please contact Giampaolo D'Alessandro.
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