Hanle-Zeeman redistribution matrix. I. Classical theory expressions in the laboratory frame

Polarized scattering in spectral lines is governed by a 4; 4 matrix that describes how the Stokes vector is scattered and redistributed in frequency and direction. Here we develop the theory for this redistribution matrix in the presence of magnetic fields of arbitrary strength and direction. This general magnetic field case is called the Hanle- Zeeman regime, since it covers both of the partially overlapping weak- and strong- field regimes in which the Hanle and Zeeman effects dominate the scattering polarization. In this general regime, the angle-frequency correlations that describe the so-called partial frequency redistribution (PRD) are intimately coupled to the polarization properties. We develop the theory for the PRD redistribution matrix in this general case and explore its detailed mathematical properties and symmetries for the case of a J = 0 -> 1 -> 0 scattering transition, which can be treated in terms of time-dependent classical oscillator theory. It is shown how the redistribution matrix can be expressed as a linear superposition of coherent and noncoherent parts, each of which contain the magnetic redistribution functions that resemble the well- known Hummer- type functions. We also show how the classical theory can be extended to treat atomic and molecular scattering transitions for any combinations of quantum numbers.

Geometric polarimetry − part I: spinors and wave states

A new formal approach for representation of polarization states of coherent and partially coherent electromagnetic plane waves is presented. Its basis is a purely geometric construction for the normalised complex-analytic coherent wave as a generating line in the sphere of wave directions, and whose Stokes vector is determined by the intersection with the conjugate generating line. The Poincare sphere is now located in physical space, simply a coordination of the wave sphere, its axis aligned with the wave vector. Algebraically, the generators representing coherent states are represented by spinors, and this is made consistent with the spinor-tensor representation of electromagnetic theory by means of an explicit reference spinor we call the phase flag. As a faithful unified geometric representation, the new model provides improved formal tools for resolving many of the geometric difficulties and ambiguities that arise in the traditional formalism.

Optical interface based on a nanofiber atom-trap

In this thesis, I present the realization of a fiber-optical interface using optically trapped cesium atoms, which is an efficient tool for coupling light and atoms. The basic principle of the presented scheme relies on the trapping of neutral cesium atoms in a two-color evanescent field surrounding a nanofiber. The strong confinement of the fiber guided light, which also protrudes outside the nanofiber, provides strong confinement of the atoms as well as efficient coupling to near-resonant light propagating through the fiber. In chapter 1, the necessary physical and mathematical background describing the propagation of light in an optical fiber is presented. The exact solution of Maxwell’s equations allows us to model fiber-guided light fields which give rise to the trapping potentials and the atom-light coupling in the close vicinity of a nanofiber. Chapter 2 gives the theoretical background of light-atom interaction. A quantum mechanical model of the light-induced shifts of the relevant atomic levels is reviewed, which allows us to quantify the perturbation of the atomic states due to the presence of the trapping light-fields. The experimental realization of the fiber-based atom trap is the focus of chapter 3. Here, I analyze the properties of the fiber-based trap in terms of the confinement of the atoms and the impact of several heating mechanisms. Furthermore, I demonstrate the transportation of the trapped atoms, as a first step towards a deterministic delivery of individual atoms. In chapter 4, I present the successful interfacing of the trapped atomic ensemble and fiber-guided light. Three different approaches are discussed, i.e., those involving the measurement of either near-resonant scattering in absorption or the emission into the guided mode of the nanofiber. In the analysis of the spectroscopic properties of the trapped ensemble we find good agreement with the prediction of theoretical model discussed in chapter 2. In addition, I introduce a non-destructive scheme for the interrogation of the atoms states, which is sensitive to phase shifts of far-detuned fiber-guided light interacting with the trapped atoms. The inherent birefringence in our system, induced by the atoms, changes the state of polarization of the probe light and can be thus detected via a Stokes vector measurement.

Identification of geostationary satellites using polarization data from unresolved images

In order to protect critical military and commercial space assets, the United States Space Surveillance Network must have the ability to positively identify and characterize all space objects. Unfortunately, positive identification and characterization of space objects is a manual and labor intensive process today since even large telescopes cannot provide resolved images of most space objects. Since resolved images of geosynchronous satellites are not technically feasible with current technology, another method of distinguishing space objects was explored that exploits the polarization signature from unresolved images. The objective of this study was to collect and analyze visible-spectrum polarization data from unresolved images of geosynchronous satellites taken over various solar phase angles. Different collection geometries were used to evaluate the polarization contribution of solar arrays, thermal control materials, antennas, and the satellite bus as the solar phase angle changed. Since materials on space objects age due to the space environment, it was postulated that their polarization signature may change enough to allow discrimination of identical satellites launched at different times. The instrumentation used in this experiment was a United States Air Force Academy (USAFA) Department of Physics system that consists of a 20-inch Ritchey-Chrétien telescope and a dual focal plane optical train fed with a polarizing beam splitter. A rigorous calibration of the system was performed that included corrections for pixel bias, dark current, and response. Additionally, the two channel polarimeter was calibrated by experimentally determining the Mueller matrix for the system and relating image intensity at the two cameras to Stokes parameters S0 and S1. After the system calibration, polarization data was collected during three nights on eight geosynchronous satellites built by various manufacturers and launched several years apart. Three pairs of the eight satellites were identical buses to determine if identical buses could be correctly differentiated. When Stokes parameters were plotted against time and solar phase angle, the data indicates that there were distinguishing features in S0 (total intensity) and S1 (linear polarization) that may lead to positive identification or classification of each satellite.

Approximate Approximations and a Boundary Point Method for the Linearized Stokes System

The method of approximate approximations, introduced by Maz'ya [1], can also be used for the numerical solution of boundary integral equations. In this case, the matrix of the resulting algebraic system to compute an approximate source density depends only on the position of a finite number of boundary points and on the direction of the normal vector in these points (Boundary Point Method). We investigate this approach for the Stokes problem in the whole space and for the Stokes boundary value problem in a bounded convex domain G subset R^2, where the second part consists of three steps: In a first step the unknown potential density is replaced by a linear combination of exponentially decreasing basis functions concentrated near the boundary points. In a second step, integration over the boundary partial G is replaced by integration over the tangents at the boundary points such that even analytical expressions for the potential approximations can be obtained. In a third step, finally, the linear algebraic system is solved to determine an approximate density function and the resulting solution of the Stokes boundary value problem. Even not convergent the method leads to an efficient approximation of the form O(h^2) + epsilon, where epsilon can be chosen arbitrarily small.

Recombinant protein production using a Tobacco yellow dwarf virus-based episomal expression vector : control of Rep activity

Over the past decade, plants have been used as expression hosts for the production of pharmaceutically important and commercially valuable proteins. Plants offer many advantages over other expression systems such as lower production costs, rapid scale up of production, similar post-translational modification as animals and the low likelihood of contamination with animal pathogens, microbial toxins or oncogenic sequences. However, improving recombinant protein yield remains one of the greatest challenges to molecular farming. In-Plant Activation (InPAct) is a newly developed technology that offers activatable and high-level expression of heterologous proteins in plants. InPAct vectors contain the geminivirus cis elements essential for rolling circle replication (RCR) and are arranged such that the gene of interest is only expressed in the presence of the cognate viral replication-associated protein (Rep). The expression of Rep in planta may be controlled by a tissue-specific, developmentally regulated or chemically inducible promoter such that heterologous protein accumulation can be spatially and temporally controlled. One of the challenges for the successful exploitation of InPAct technology is the control of Rep expression as even very low levels of this protein can reduce transformation efficiency, cause abnormal phenotypes and premature activation of the InPAct vector in regenerated plants. Tight regulation over transgene expression is also essential if expressing cytotoxic products. Unfortunately, many tissue-specific and inducible promoters are unsuitable for controlling expression of Rep due to low basal activity in the absence of inducer or in tissues other than the target tissue. This PhD aimed to control Rep activity through the production of single chain variable fragments (scFvs) specific to the motif III of Tobacco yellow dwarf virus (TbYDV) Rep. Due to the important role played by the conserved motif III in the RCR, it was postulated that such scFvs can be used to neutralise the activity of the low amount of Rep expressed from a “leaky” inducible promoter, thus preventing activation of the TbYDV-based InPAct vector until intentional induction. Such scFvs could also offer the potential to confer partial or complete resistance to TbYDV, and possibly heterologous viruses as motif III is conserved between geminiviruses. Studies were first undertaken to determine the levels of TbYDV Rep and TbYDV replication-associated protein A (RepA) required for optimal transgene expression from a TbYDV-based InPAct vector. Transient assays in a non-regenerable Nicotiana tabacum (NT-1) cell line were undertaken using a TbYDV-based InPAct vector containing the uidA reporter gene (encoding GUS) in combination with TbYDV Rep and RepA under the control of promoters with high (CaMV 35S) or low (Banana bunchy top virus DNA-R, BT1) activity. The replication enhancer protein of Tomato leaf curl begomovirus (ToLCV), REn, was also used in some co-bombardment experiments to examine whether RepA could be substituted by a replication enhancer from another geminivirus genus. GUS expression was observed both quantitatively and qualitatively by fluorometric and histochemical assays, respectively. GUS expression from the TbYDV-based InPAct vector was found to be greater when Rep was expected to be expressed at low levels (BT1 promoter) rather than high levels (35S promoter). GUS expression was further enhanced when Rep and RepA were co-bombarded with a low ratio of Rep to RepA. Substituting TbYDV RepA with ToLCV REn also enhanced GUS expression but more importantly highest GUS expression was observed when cells were co-transformed with expression vectors directing low levels of Rep and high levels of RepA irrespective of the level of REn. In this case, GUS expression was approximately 74-fold higher than that from a non-replicating vector. The use of different terminators, namely CaMV 35S and Nos terminators, in InPAct vectors was found to influence GUS expression. In the presence of Rep, GUS expression was greater using pInPActGUS-Nos rather than pInPActGUS-35S. The only instance of GUS expression being greater from vectors containing the 35S terminator was when comparing expression from cells transformed with Rep, RepA and REnexpressing vectors and either non-replicating vectors, p35SGS-Nos or p35SGS-35S. This difference was most likely caused by an interaction of viral replication proteins with each other and the terminators. These results indicated that (i) the level of replication associated proteins is critical to high transgene expression, (ii) the choice of terminator within the InPAct vector may affect expression levels and (iii) very low levels of Rep can activate InPAct vectors hence controlling its activity is critical. Prior to generating recombinant scFvs, a recombinant TbYDV Rep was produced in E. coli to act as a control to enable the screening for Rep-specific antibodies. A bacterial expression vector was constructed to express recombinant TbYDV Rep with an Nterminal His-tag (N-His-Rep). Despite investigating several purification techniques including Ni-NTA, anion exchange, hydrophobic interaction and size exclusion chromatography, N-His-Rep could only be partially purified using a Ni-NTA column under native conditions. Although it was not certain that this recombinant N-His-Rep had the same conformation as the native TbYDV Rep and was functional, results from an electromobility shift assay (EMSA) showed that N-His-Rep was able to interact with the TbYDV LIR and was, therefore, possibly functional. Two hybridoma cell lines from mice, immunised with a synthetic peptide containing the TbYDV Rep motif III amino acid sequence, were generated by GenScript (USA). Monoclonal antibodies secreted by the two hybridoma cell lines were first screened against denatured N-His-Rep in Western analysis. After demonstrating their ability to bind N-His-Rep, two scFvs (scFv1 and scFv2) were generated using a PCR-based approach. Whereas the variable heavy chain (VH) from both cell lines could be amplified, only the variable light chain (VL) from cell line 2 was amplified. As a result, scFv1 contained VH and VL from cell line 1, whereas scFv2 contained VH from cell line 2 and VL from cell line 1. Both scFvs were first expressed in E. coli in order to evaluate their affinity to the recombinant TbYDV N-His-Rep. The preliminary results demonstrated that both scFvs were able to bind to the denatured N-His-Rep. However, EMSAs revealed that only scFv2 was able to bind to native N-His-Rep and prevent it from interacting with the TbYDV LIR. Each scFv was cloned into plant expression vectors and co-bombarded into NT-1 cells with the TbYDV-based InPAct GUS expression vector and pBT1-Rep to examine whether the scFvs could prevent Rep from mediating RCR. Although it was expected that the addition of the scFvs would result in decreased GUS expression, GUS expression was found to slightly increase. This increase was even more pronounced when the scFvs were targeted to the cell nucleus by the inclusion of the Simian virus 40 large T antigen (SV40) nuclear localisation signal (NLS). It was postulated that the scFvs were binding to a proportion of Rep, leaving a small amount available to mediate RCR. The outcomes of this project provide evidence that very high levels of recombinant protein can theoretically be expressed using InPAct vectors with judicious selection and control of viral replication proteins. However, the question of whether the scFvs generated in this project have sufficient affinity for TbYDV Rep to prevent its activity in a stably transformed plant remains unknown. It may be that other scFvs with different combinations of VH and VL may have greater affinity for TbYDV Rep. Such scFvs, when expressed at high levels in planta, might also confer resistance to TbYDV and possibly heterologous geminiviruses.

Accurate series solutions for gravity-driven Stokes waves

In the past, high order series expansion techniques have been used to study the nonlinear equations that govern the form of periodic Stokes waves moving steadily on the surface of an inviscid fluid. In the present study, two such series solutions are recomputed using exact arithmetic, eliminating any loss of accuracy due to accumulation of round-off error, allowing a much greater number of terms to be found with confidence. It is shown that higher order behaviour of series generated by the solution casts doubt over arguments that rely on estimating the series’ radius of convergence. Further, the exact nature of the series is used to shed light on the unusual nature of convergence of higher order Pade approximants near the highest wave. Finally, it is concluded that, provided exact values are used in the series, these Pade approximants prove very effective in successfully predicting three turning points in both the dispersion relation and the total energy.

Temporal-based support vector machines for facial expression recognition

When classifying a signal, ideally we want our classifier to trigger a large response when it encounters a positive example and have little to no response for all other examples. Unfortunately in practice this does not occur with responses fluctuating, often causing false alarms. There exists a myriad of reasons why this is the case, most notably not incorporating the dynamics of the signal into the classification. In facial expression recognition, this has been highlighted as one major research question. In this paper we present a novel technique which incorporates the dynamics of the signal which can produce a strong response when the peak expression is found and essentially suppresses all other responses as much as possible. We conducted preliminary experiments on the extended Cohn-Kanade (CK+) database which shows its benefits. The ability to automatically and accurately recognize facial expressions of drivers is highly relevant to the automobile. For example, the early recognition of “surprise” could indicate that an accident is about to occur; and various safeguards could immediately be deployed to avoid or minimize injury and damage. In this paper, we conducted initial experiments on the extended Cohn-Kanade (CK+) database which shows its benefits.

Exponential asymptotics with multiple stokes lines

When asymptotic series methods are applied in order to solve problems that arise in applied mathematics in the limit that some parameter becomes small, they are unable to demonstrate behaviour that occurs on a scale that is exponentially small compared to the algebraic terms of the asymptotic series. There are many examples of physical systems where behaviour on this scale has important effects and, as such, a range of techniques known as exponential asymptotic techniques were developed that may be used to examinine behaviour on this exponentially small scale. Many problems in applied mathematics may be represented by behaviour within the complex plane, which may subsequently be examined using asymptotic methods. These problems frequently demonstrate behaviour known as Stokes phenomenon, which involves the rapid switches of behaviour on an exponentially small scale in the neighbourhood of some curve known as a Stokes line. Exponential asymptotic techniques have been applied in order to obtain an expression for this exponentially small switching behaviour in the solutions to orginary and partial differential equations. The problem of potential flow over a submerged obstacle has been previously considered in this manner by Chapman & Vanden-Broeck (2006). By representing the problem in the complex plane and applying an exponential asymptotic technique, they were able to detect the switching, and subsequent behaviour, of exponentially small waves on the free surface of the flow in the limit of small Froude number, specifically considering the case of flow over a step with one Stokes line present in the complex plane. We consider an extension of this work to flow configurations with multiple Stokes lines, such as flow over an inclined step, or flow over a bump or trench. The resultant expressions are analysed, and demonstrate interesting implications, such as the presence of exponentially sub-subdominant intermediate waves and the possibility of trapped surface waves for flow over a bump or trench. We then consider the effect of multiple Stokes lines in higher order equations, particu- larly investigating the behaviour of higher-order Stokes lines in the solutions to partial differential equations. These higher-order Stokes lines switch off the ordinary Stokes lines themselves, adding a layer of complexity to the overall Stokes structure of the solution. Specifically, we consider the different approaches taken by Howls et al. (2004) and Chap- man & Mortimer (2005) in applying exponential asymptotic techniques to determine the higher-order Stokes phenomenon behaviour in the solution to a particular partial differ- ential equation.