Completed research projects

This pages contains a list of research projects I am no longer working on.

Multiscale models of liquid crystals suspensions of nanoparticles

Very dilute suspensions of nanoparticles (e.g. gold) in liquid crystals seem to have remarkable effects on the liquid crystal properties. Dr Keith Daly (Engineering Sciences) and I are running a long term project on using homogenisation theory to develop macroscopic models of these suspensions. This project has involved my ex-PhD student Dr Tom Bennett and my current PhD, Jordan Gill.

Suspensions of nano particles in liquid crystals have been studied by means of molecular simulations and macroscopic models. Simulations of rod like molecules are computationally expensive and are limited to small systems. On the other hand macroscopic models contain unknown phenomenological parameters that attempt to capture the coupling between liquid crystal and its dopant. Unfortunately there is often no way to self consistently compute the value of these parameters. The aim of this project is to use homogenisation theory obtain a macroscopic description of a doped nematic system that allows a consistent computation of the new dopant dependent terms.

The outcome of the homogenisation process is that all of the complicated geometric information encoded in the original problem has been moved into effective material parameters. This is the crucial advantage over more phenomenological models. The microscopic detail is retained in the form of a cell problem, which is used to compute the effective material parameters. The end results of the process is two-fold: (i) we obtain equations that are less computationally intensive to solve and (ii) we can compute the new material parameters from the underlying geometry of the system.
The first results of this project focused on nematic liquid crystals with fixed inclusions. They have appeared in Physical Review E:

T.P. Bennett, G. D'Alessandro and K.R. Daly, Multiscale models of colloidal dispersion of particles in nematic liquid crystals, Phys. Rev. E 90(6), 062505 (2014)

We have extended this work to rotating inclusions in two dimensions. The derivation of the model has appeared in the SIAM Journal on Applied Mathematics:

T.P. Bennett, G. D'Alessandro and K.R. Daly, Multiscale models of metallic nanoparticles in in nematic liquid crystals, SIAM J. Appl. Math. 78, 1228-1255 (2018)

Fast Q-tensor algorithms for liquid crystal alignment away from defects

This project was started with my ex-PhD student, Dr Keith Daly and was continued with my EPSRC funded post-doc, Dr Yogesh Murugesan, and Dr Giovanni de Matteis.
Modelling liquid crystals in optical devices requires us to determine the alignment of the liquid crystal molecules as a function of applied external fields. The equations for the liquid crystal alignment can be "stiff", i.e. they may contain terms that evolve on vastly different time scales. This property makes them very hard to integrate numerically. To overcome this problem we have exploited its causes and have used the vastly different time scales to develop an approximation to the liquid crystal equations that is non-stiff and very accurate. The draw-back of this approximation is that it is valid only in regions away from defects. The approximation, called Defect Free Q-Tensor Approximation (DFQTA), is described fully in

K.R. Daly, G. D'Alessandro and M. Kaczmarek, An efficient Q-tensor based algorithm for liquid crystal alignment away from defects SIAM Journal on Applied Mathematics 70(8), 2844-2860 (2010)
An introductory explanation and a Matlab code of the equations is available from the DFQTA Homepage.

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.
I am interested in studying models and numerical techniques to determine the alignment of liquid crystals in cells and microcavities, the effects of micro- and nano-particles on the properties of liquid crystals, the dynamics of light beams (including plasmons) in liquid crystals cells and cavities.
Below are some projects I have been involved with, with the most recent nearer the top.

Thermal nonlinearities in liquid crystals

A first phase of this project was in collaboration with Prof Malgosia Kaczmarek and with Andrew Acreman, at the time a PhD student in the Functional materials group. Gold nanoparticles suspended in liquid crystals may heat up when exposed to light and alter the (temperature-dependent) material properties of the liquid crystals. We have extended work by Ouskova et al (2011) and Lysenko et al (2012) to describe the effect of voltage and particle shape on these effect. The results are described in a paper published in Phys Rev E:

A. Acreman, M. Kaczmarek and G. D'Alessandro, Gold nanoparticle liquid crystal composites as a tunable nonlinear medium, Phys. Rev. E 90(1), 012504(8) (2014)
A second phase of this project involved Prof Malgosia Kaczmarek and Dr Yogesh Murugesan. In this phase we focussed on the effects that a finite size beam width has on the thermal nonlinearity of the suspension. Its results and comparison with experimental data are in
O. Kurochkin, Y.K. Murugesan, T.P. Bennett, G. D'Alessandro, Y. Reznikov, G.H. Mehl, B.J. Tang and M. Kaczmarek, Thermal optical non-linearity of nematic mesophase enhanced by gold nanoparticles - an experimental and numerical investigation, Phys. Chem. Chem. Phys. 18, 11503-11512 (2016) - Doi: 10.1039/C6CP00116E [PDF]

Photorefraction and beam coupling in liquid crystal cells

This project started as a collaboration with my ex-PhD student, Keith Daly, and with Professor Malgosia Kaczmarek (Physics, Southampton).
Light beams propagating through a slab of material with periodically modulated refractive index, e.g. a diffraction grating, are deflected by an angle that depends on the wave length of the modulation and of the light. This effect can be used to transfer energy from one beam to the other: this phenomenon is called beam coupling.
We are interested in devices where the light itself induces the refractive index grating that will then deflect it: this variation on the beam coupling theme is called photorefractive beam coupling. The system we are studying consists of a planar liquid crystal cell where the entry glass plate has been replaced by a photo-sensitive layer. When two light beams interfere on the photo-sensitive layer they induce a periodic modulation of the conductance of the layer that can then be used to create a periodic modulation of the liquid crystal alignment in the cell. We are developing models of the liquid crystals and of the propagation of the light beams in the cell. Some preliminary results are contained in this poster. The beam coupling model is discussed in this paper published in Applied Physics B,

K. R. Daly, G. D'Alessandro and M. Kaczmarek, Regime independent coupled-wave equations in anisotropic photorefractive media, Appl. Phys. B 95(3), 589-596 (2009)
Some preliminary experimental and theoretical results on beam coupling in photorefractive cells drives with a low frequency AC voltage have appeared in European Physics Letters (EPL),
M. Herrington, K. Daly, O. Buchnev, G. D'Alessandro and M. Kaczmarek, AC field enhanced beam coupling in photorefractive, hybrid liquid crystals, EPL 95, 14003(5) (2011)
while a comprehensive review of the topic has appeared in the Journal of the Optical Society of America:
M. Proctor, J. Bateman, K.R. Daly, M. Herrington, O. Buchnev, N. Podoliak, G. D'Alessandro and M. Kaczmarek, Light-activated modulation and coupling in integrated polymer-liquid crystal systems, J. Opt. Soc. Am. B 31, 3144-3152 (2014)

Ferronematic liquid crystals

This project is in collaboration with Professor Malgosia Kaczmarek's group in Physics, in particular her PhD student Nina Podoliak, and Professor Tim Sluckin (Maths).
The inclusion of even a very low fraction of nanoparticles in liquid crystals can have a large effect on their properties. In this project we study the effects that the inclusion of ferromagnetic or super-paramagnetic nanoparticles has on the magnetic properties of liquid crystals. These colloidal mixtures are called "ferronematics". We have developed models of ferronematics and successfully fitted them to experimental data. Some of our results have appeared in

N. Podoliak, O. Buchnev, O. Buluy, G. D'Alessandro, M. Kaczmarek, Y. Reznikov and T.J. Sluckin, Macroscopic optical effects in low concentration ferronematics, Soft Matter 7(10), 4742-4749 (2011)
Interestingly, while studying the effect of nanoparticles on liquid crystals we have also noticed that a small bias field can also significantly affect the magnetic Frederiks transition, i.e. the value of the externally applied magnetic field at which the liquid crystal shows significant re-orientation. These results have appeared in:
N. Podoliak, O. Buchnev, G. D'Alessandro, M. Kaczmarek and T.J. Sluckin, Large effect of a small bias field in liquid-crystal magnetic transitions, Phys. Rev. E 82(3), 030701 (4) (2010)

Plasmons and liquid crystals

Schematic diagram of a plasmonic photorefractive liquid crystal cell. The plasmon sits at the interface between the gold and the PVK layer. This is sufficiently thin so that the plasmon are also sensitive to the alignment of the liquid crystal molecules. The liquid crystals are controlled by the applied voltage V and by steering beams (not shown) that enter the system through the hemispherical prism on the left and modulate the conductance of the PVK layer.
This project was in collaboration with my ex-PhD student, Keith Daly, with Dr David Smith and his ex-PhD student Stephen Abbott in Physics and with Professor Malgosia Kaczmarek (Physics, Southampton).

A surface plasmon is a particular type of electromagnetic wave: it exists at the interface between a metal and a dielectric and propagates along the boundary between the two. Surface plasmons have attracted a lot of interest in recent years for a variety of reasons. One of them is that they are very sensitive to the refractive index of the metal and dielectric very close to the interface. This makes them ideal probes for optical sensors: a slight change in the metal or dielectric layer, caused for example by some impurities that we wish to detect, can cause a significant change in the optical properties of the surface plasmon.
Surface plasmons are very sensitive to the refractive index of the dielectric adjacent to the gold layer. This raises the possibility of controlling their behaviour using liquid crystals: a device of this type could be reconfigured in real time either optically or electrically and seems an ideal tool to study the physics of plasmons and their interaction. A schematic diagram of how a photorefractive liquid crystal cell can be used to manipulate plasmons is shown on the right.
The aim of this project was to model analytically and numerically the interaction of surface plasmons with the liquid crystals. Ultimately, we used the liquid crystals to study plasmon-plasmon interactions and to transfer energy from one to the other.

We have developed an analytical model of the effect of the liquid crystal orientation on the dispersion of surface plasmons,
K. R. Daly, S. Abbott, G. D'Alessandro, D. C. Smith and M. Kaczmarek, Theory of hybrid photorefractive plasmonic liquid crystal cells, J. Opt. Soc. Am. B 28(8), 1874-1881 (2011).
Preliminary results on plasmon diffraction have appeared in Optics Letters
S.B. Abbott, K.R. Daly, G. D'Alessandro, M. Kaczmarek and D.C. Smith, Photorefractive Control of Surface Plasmon Polaritons in a Hybrid Liquid Crystal Cell, Opt. Lett. 27(13), 2436-2438 (2012)
while a more comprehensive analysis has appeared in J. Opt. Soc. Am. B:
S.B. Abbott, K.R. Daly, G. D'Alessandro, M. Kaczmarek and D.C. Smith, A Hybrid Liquid Crystal Photorefractive System for the Photorefractive Coupling of Surface Plasmon Polaritons, J. Opt. Soc. Am. B 29(8), 1947-1958 (2012)
and
K.R. Daly, S.B. Abbott, D.C. Smith and G. D'Alessandro, Optimization of plasmon-plasmon coupling in photorefractive layered media, J. Opt. Soc. Am. B 30(8), 2090-2099 (2013)

Liquid crystals in hemispherical microcavities

A project (now ended) on liquid crystals in optical devices consisted in modelling the configuration of nematic liquid crystals in arrays of microcavities with variable geometry and in planar cells. The results of this project have been presented in this seminar given at EQEC in 2005 and have been published in

G. Vijaya Prakash, M. Kaczmarek, A. Dyadyusha, J.J. Baumberg and G. D'Alessandro, Control of topological defects in microstructured liquid crystal cells, Opt. Expr. 13(6), 2201-2209 (2005)
The second stage of this project consisted in studying the optical spectrum of the empty microcavities. Experimental results show that the modes of these cavities are very similar to the Gauss-Laguerre modes of a macroscopic cavity, but with fewer frequency degeneracies. We have used a simple paraxial theory to get a key to interpret the experimental results. The first results of this analysis were published in Optics Letters
R. C. Pennington, G. D'Alessandro, J. J. Baumberg, and M. Kaczmarek, Tracking spatial modes in nearly hemispherical microcavities, Opt. Lett. 32(21), 3131-3133 (2007)
A full analysis has appeared in Physical Review A:
R. C. Pennington, G. D'Alessandro, J. J. Baumberg, and M. Kaczmarek, Spectral properties and modes of surface microcavities, Phys. Rev. A 79(4), 043822 (2009)

Bistability in planar cavities

A first project in this programme of study was motivated by the experimental observation of bistability in planar cells filled with nematic liquid crystals. We have developed a model of this optical device and studies its properties, thus confirming and explaining the experimental observations. The results of this project were published in

G. D'Alessandro and A. A. Wheeler, Bistability of liquid crystal microcavities, Phys. Rev. A 67(2), 023816(12) (2003).
and were presented as a talk at the British Applied Mathematics Colloquium in 2003 and as a seminar at ITAM also in 2003.

Non-normal effects in optics

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. For a very comprehensive introduction to this system have a look at the Pseudospectra gateway. Non-normal effects have been known in optics for some quite time, even though not by this name. They are often lumped under the category of excess noise and are associated to optical devices with non-orthogonal cavity modes. We (and others) have shown that this point of view is rather restrictive. For example, in the paper

F. Papoff, G. D'Alessandro and G.-L. Oppo, State Dependent Pseudoresonances and Excess Noise, Phys. Rev. Lett. 100(12), 123905 (2008)
we have shown that a simple model of semiconductor laser displays non-normal effects, even though its cavity modes are perfectly orthogonal.

We have extended this work to synchronously pumped optical parametric oscillators. These devices split one high frequency photon (called pump) in two lower frequency ones (called signal and idler). They are used, for example, as a tunable source of laser light. Our study shows that these devices are able to sustain transient amplification of the order of 109. This amplification factor is so large than noise sustained signal and idler fields can exist even though the optical parametric oscillator is nominally below threshold. A summary preliminary results with a simple example on non-normal effects is contained in this seminar given at the Southampton Photonics Day in September 2008. These were also published in

G. D'Alessandro and C. Brent-Laforet, Giant noise amplification in synchronously pumped optical parametric oscillators, Opt. Lett. 34(5), 614-6 (2009)
A more complete set of results has appeared in
G. D'Alessandro and F. Papoff, Giant sub-threshold amplification in synchronously pumped optical parametric oscillators Phys. Rev. A 80(2), 023804(8) (2009)

Waveguide sensors

The aim of this project was to develop mathematical and numerical models of optical waveguide sensors. This project was in collaboration with my ex-PhD student, Keith Daly, and with Professor Peter Smith and his group at the Opto-Electronics Research Centre (ORC), Southampton.

By precision modification of the properties of glass (e.g. etching very fine ridges) it is possible to build waveguides (e.g. the equivalent of a water channel for light) that bring light in contact with very small quantities of materials to be analysed (analyte): the close and accurately controlled contact between light and analyte allows us to determine its properties, for example if there are contaminants or specific biological molecules. In other words, we have built a waveguide based optical sensor. These sensors have many advantages: they are small and require only very small quantities of analyte, their geometry is very flexible and can be tailored to specific applications, they can be attached to the end of a long optical fibre and be safely placed in harsh environments. As part of this more general project, we mainly focussed on the basic sceince description of plasmons and waveguides.

Plasmons and waveguides

A surface plasmon is a particular type of electromagnetic wave: it exists at the interface between a metal and a dielectric and propagates along the boundary between the two. Surface plasmons have attracted a lot of interest in recent years for a variety of reasons. One of them is that they are very sensitive to the refractive index of the metal and dielectric very close to the interface. This makes them ideal probes for optical sensors: a slight change in the metal or dielectric layer, caused for example by some impurities that we wish to detect, can cause a significant change in the optical properties of the surface plasmon.

Schematic diagram of a trench optical waveguide with gold cladding. The light travels mainly in the core (small green square), but it is affected by variations in the refractive index of the material in the trenches (usually a liquid, indicated by the light-blue colour) in contact with the gold layers.
In this project we model sensors that couple gold-coated optical ridge waveguides with surface plasmons to obtain remotely accessible, highly sensitive sensors. This project aims to marry the ability of wave guides to carry information in the form of a light wave to the exquisite sensing ability of surface plasmons by studying optical wave guides coated with a metallic layer. In these, the property of the waves propagating along the wave guide will be influenced by the surface plasmons at the metal-glass interface and will hopefully allow remote sensing using light signals.
We have developed a comprehensive numerical model of the modes of the waveguide and their interaction with the surface plasmons. The theoretical analysis of a ridge waveguide was published in
K.R. Daly, C. Holmes, J.C. Gates, P.G.R. Smith and G. D'Alessandro, Complete Mode Structure Analysis of Tilted Bragg Grating Refractometers in Planar Waveguides Toward Absolute Index Measurement, IEEE Photonics J. 3(5), 861-871 (2011).
while the experimental results with a gold coated ridge waveguide appeared in
C. Holmes, K.R. Daly, I.J.C. Sparrow, H.L. Rogers, J.C. Gates, G. D'Alessandro and P.G.R. Smith, Excitation of Surface Plasmons using tilted planar-waveguide Bragg gratings, IEEE Photonics J. 3(5), 777-788 (2011).

Pattern formation in nonlinear optics

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 page is the summary of two papers:

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 was published in
D. R. J. Chillingworth, G. D'Alessandro and F. Papoff, Explicit centre manifold reduction and full bifurcation analysis for an astigmatic Maxwell-Bloch laser, Physica D 177, 175-202 (2003)
and 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. This work was published in Part of the results of the first paper were presented as a poster at Eqec 2003.

I am no longer active in the field of pattern formation, but I am still very much interested in it. Please get in touch if you are interested in working in this area.

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