Development of a Uniform Theory of Diffraction for Scattered and Surface Electromagnetic Field Analysis on a Cylinder

Con esta tesis ”Desarrollo de una Teoría Uniforme de la Difracción para el Análisis de los Campos Electromagnéticos Dispersados y Superficiales sobre un Cilindro” hemos iniciado una nueva línea de investigación que trata de responder a la siguiente pregunta: ¿cuál es la impedancia de superficie que describe una estructura de conductor eléctrico perfecto (PEC) convexa recubierta por un material no conductor? Este tipo de estudios tienen interés hoy en día porque ayudan a predecir el campo electromagnético incidente, radiado o que se propaga sobre estructuras metálicas y localmente convexas que se encuentran recubiertas de algún material dieléctrico, o sobre estructuras metálicas con pérdidas, como por ejemplo se necesita en determinadas aplicaciones aeroespaciales, marítimas o automovilísticas. Además, desde un punto de vista teórico, la caracterización de la impedancia de superficie de una estructura PEC recubierta o no por un dieléctrico es una generalización de varias soluciones que tratan ambos tipos de problemas por separado. En esta tesis se desarrolla una teoría uniforme de la difracción (UTD) para analizar el problema canónico del campo electromagnético dispersado y superficial en un cilindro circular eléctricamente grande con una condición de contorno de impedancia (IBC) para frecuencias altas. Construir una solución basada en UTD para este problema canónico es crucial en el desarrollo de un método UTD para el caso más general de una superficie arbitrariamente convexa, mediante el uso del principio de localización de los campos electromagnéticos a altas frecuencias. Esta tesis doctoral se ha llevado a cabo a través de una serie de hitos que se enumeran a continuación, enfatizando las contribuciones a las que ha dado lugar. Inicialmente se realiza una revisión en profundidad del estado del arte de los métodos asintóticos con numerosas referencias. As í, cualquier lector novel puede llegar a conocer la historia de la óptica geométrica (GO) y la teoría geométrica de la difracción (GTD), que dieron lugar al desarrollo de la UTD. Después, se investiga ampliamente la UTD y los trabajos más importantes que pueden encontrarse en la literatura. As í, este capítulo, nos coloca en la posición de afirmar que, hasta donde nosotros conocemos, nadie ha intentado antes llevar a cabo una investigación rigurosa sobre la caracterización de la impedancia de superficie de una estructura PEC recubierta por un material dieléctrico, utilizando para ello la UTD. Primero, se desarrolla una UTD para el problema canónico de la dispersión electromagnética de un cilindro circular eléctricamente grande con una IBC uniforme, cuando es iluminado por una onda plana con incidencia oblicua a frecuencias altas. La solución a este problema canónico se construye a partir de una solución exacta mediante una expansión de autofunciones de propagación radial. Entonces, ésta se convierte en una nueva expansión de autofunciones de propagación circunferencial muy apropiada para cilindros grandes, a través de la transformación de Watson. De esta forma, la expresión del campo se reduce a una integral que se evalúa asintóticamente, para altas frecuencias, de manera uniforme. El resultado se expresa según el trazado de rayos descrito en la UTD. La solución es uniforme porque tiene la importante propiedad de mantenerse continua a lo largo de la región de transición, a ambos lados de la superficie del contorno de sombra. Fuera de la región de transición la solución se reduce al campo incidente y reflejado puramente ópticos en la región iluminada del cilindro, y al campo superficial difractado en la región de sombra. Debido a la IBC el campo dispersado contiene una componente contrapolar a causa de un acoplamiento entre las ondas TEz y TMz (donde z es el eje del cilindro). Esta componente contrapolar desaparece cuando la incidencia es normal al cilindro, y también en la región iluminada cuando la incidencia es oblicua donde el campo se reduce a la solución de GO. La solución UTD presenta una muy buena exactitud cuando se compara numéricamente con una solución de referencia exacta. A continuación, se desarrolla una IBC efectiva para el cálculo del campo electromagnético dispersado en un cilindro circular PEC recubierto por un dieléctrico e iluminado por una onda plana incidiendo oblicuamente. Para ello se derivan dos impedancias de superficie en relación directa con las ondas creeping y de superficie TM y TE que se excitan en un cilindro recubierto por un material no conductor. Las impedancias de superficie TM y TE están acopladas cuando la incidencia es oblicua, y dependen de la geometría del problema y de los números de onda. Además, se ha derivado una impedancia de superficie constante, aunque con diferente valor cuando el observador se encuentra en la zona iluminada o en la zona de sombra. Después, se presenta una solución UTD para el cálculo de la dispersión de una onda plana con incidencia oblicua sobre un cilindro eléctricamente grande y convexo, mediante la generalización del problema canónico correspondiente al cilindro circular. La solución asintótica es uniforme porque se mantiene continua a lo largo de la región de transición, en las inmediaciones del contorno de sombra, y se reduce a la solución de rayos ópticos en la zona iluminada y a la contribución de las ondas de superficie dentro de la zona de sombra, lejos de la región de transición. Cuando se usa cualquier material no conductor se excita una componente contrapolar que tiende a desaparecer cuando la incidencia es normal al cilindro y en la región iluminada. Se discuten ampliamente las limitaciones de las fórmulas para la impedancia de superficie efectiva, y se compara la solución UTD con otras soluciones de referencia, donde se observa una muy buena concordancia. Y en tercer lugar, se presenta una aproximación para una impedancia de superficie efectiva para el cálculo de los campos superficiales en un cilindro circular conductor recubierto por un dieléctrico. Se discuten las principales diferencias que existen entre un cilindro PEC recubierto por un dieléctrico desde un punto de vista riguroso y un cilindro con una IBC. Mientras para un cilindro de impedancia se considera una impedancia de superficie constante o uniforme, para un cilindro conductor recubierto por un dieléctrico se derivan dos impedancias de superficie. Estas impedancias de superficie están asociadas a los modos de ondas creeping TM y TE excitadas en un cilindro, y dependen de la posición y de la orientación del observador y de la fuente. Con esto en mente, se deriva una solución UTD con IBC para los campos superficiales teniendo en cuenta las dependencias de la impedancia de superficie. La expansión asintótica se realiza, mediante la transformación de Watson, sobre la representación en serie de las funciones de Green correspondientes, evitando as í calcular las derivadas de orden superior de las integrales de tipo Fock, y dando lugar a una solución rápida y precisa. En los ejemplos numéricos realizados se observa una muy buena precisión cuando el cilindro y la separación entre el observador y la fuente son grandes. Esta solución, junto con el método de los momentos (MoM), se puede aplicar para el cálculo eficiente del acoplamiento mutuo de grandes arrays conformados de antenas de parches. Los métodos propuestos basados en UTD para el cálculo del campo electromagnético dispersado y superficial sobre un cilindro PEC recubierto de dieléctrico con una IBC efectiva suponen un primer paso hacia la generalización de una solución UTD para superficies metálicas convexas arbitrarias cubiertas por un material no conductor e iluminadas por una fuente electromagnética arbitraria. ABSTRACT With this thesis ”Development of a Uniform Theory of Diffraction for Scattered and Surface Electromagnetic Field Analysis on a Cylinder” we have initiated a line of investigation whose goal is to answer the following question: what is the surface impedance which describes a perfect electric conductor (PEC) convex structure covered by a material coating? These studies are of current and future interest for predicting the electromagnetic (EM) fields incident, radiating or propagating on locally smooth convex parts of highly metallic structures with a material coating, or by a lossy metallic surfaces, as for example in aerospace, maritime and automotive applications. Moreover, from a theoretical point of view, the surface impedance characterization of PEC surfaces with or without a material coating represents a generalization of independent solutions for both type of problems. A uniform geometrical theory of diffraction (UTD) is developed in this thesis for analyzing the canonical problem of EM scattered and surface field by an electrically large circular cylinder with an impedance boundary condition (IBC) in the high frequency regime, by means of a surface impedance characterization. The construction of a UTD solution for this canonical problem is crucial for the development of the corresponding UTD solution for the more general case of an arbitrary smooth convex surface, via the principle of the localization of high frequency EM fields. The development of the present doctoral thesis has been carried out through a series of landmarks that are enumerated as follows, emphasizing the main contributions that this work has given rise to. Initially, a profound revision is made in the state of art of asymptotic methods where numerous references are given. Thus, any reader may know the history of geometrical optics (GO) and geometrical theory of diffraction (GTD), which led to the development of UTD. Then, the UTD is deeply investigated and the main studies which are found in the literature are shown. This chapter situates us in the position to state that, as far as we know, nobody has attempted before to perform a rigorous research about the surface impedance characterization for material-coated PEC convex structures via UTD. First, a UTD solution is developed for the canonical problem of the EM scattering by an electrically large circular cylinder with a uniform IBC, when it is illuminated by an obliquely incident high frequency plane wave. A solution to this canonical problem is first constructed in terms of an exact formulation involving a radially propagating eigenfunction expansion. The latter is converted into a circumferentially propagating eigenfunction expansion suited for large cylinders, via the Watson transformation, which is expressed as an integral that is subsequently evaluated asymptotically, for high frequencies, in a uniform manner. The resulting solution is then expressed in the desired UTD ray form. This solution is uniform in the sense that it has the important property that it remains continuous across the transition region on either side of the surface shadow boundary. Outside the shadow boundary transition region it recovers the purely ray optical incident and reflected ray fields on the deep lit side of the shadow boundary and to the modal surface diffracted ray fields on the deep shadow side. The scattered field is seen to have a cross-polarized component due to the coupling between the TEz and TMz waves (where z is the cylinder axis) resulting from the IBC. Such cross-polarization vanishes for normal incidence on the cylinder, and also in the deep lit region for oblique incidence where it properly reduces to the GO or ray optical solution. This UTD solution is shown to be very accurate by a numerical comparison with an exact reference solution. Then, an effective IBC is developed for the EM scattered field on a coated PEC circular cylinder illuminated by an obliquely incident plane wave. Two surface impedances are derived in a direct relation with the TM and TE surface and creeping wave modes excited on a coated cylinder. The TM and TE surface impedances are coupled at oblique incidence, and depend on the geometry of the problem and the wave numbers. Nevertheless, a constant surface impedance is found, although with a different value when the observation point lays in the lit or in the shadow region. Then, a UTD solution for the scattering of an obliquely incident plane wave on an electrically large smooth convex coated PEC cylinder is introduced, via a generalization of the canonical circular cylinder problem. The asymptotic solution is uniform because it remains continuous across the transition region, in the vicinity of the shadow boundary, and it recovers the ray optical solution in the deep lit region and the creeping wave formulation within the deep shadow region. When a coating is present a cross-polar field term is excited, which vanishes at normal incidence and in the deep lit region. The limitations of the effective surface impedance formulas are discussed, and the UTD solution is compared with some reference solutions where a very good agreement is met. And in third place, an effective surface impedance approach is introduced for determining surface fields on an electrically large coated metallic circular cylinder. Differences in analysis of rigorouslytreated coated metallic cylinders and cylinders with an IBC are discussed. While for the impedance cylinder case a single constant or uniform surface impedance is considered, for the coated metallic cylinder case two surface impedances are derived. These are associated with the TM and TE creeping wave modes excited on a cylinder and depend on observation and source positions and orientations. With this in mind, a UTD based method with IBC is derived for the surface fields by taking into account the surface impedance variation. The asymptotic expansion is performed, via the Watson transformation, over the appropriate series representation of the Green’s functions, thus avoiding higher-order derivatives of Fock-type integrals, and yielding a fast and an accurate solution. Numerical examples reveal a very good accuracy for large cylinders when the separation between the observation and the source point is large. Thus, this solution could be efficiently applied in mutual coupling analysis, along with the method of moments (MoM), of large conformal microstrip array antennas. The proposed UTD methods for scattered and surface EM field analysis on a coated PEC cylinder with an effective IBC are considered the first steps toward the generalization of a UTD solution for large arbitrarily convex smooth metallic surfaces covered by a material coating and illuminated by an arbitrary EM source.

Performance Comparison Between Serrated Edge and Rolled Edge Reflectors Inside CATR Facilities

CATR facilities are attractive antenna measurement facilities. Main reasons which contribute to this fact lie on its inherent reduced volume, on-the-fly measurements and the extension of both to a wide range of frequencies. However, these features rely on the assumption that the field collimation scheme is able to generate a plane wave distribution (quiet zone) where the AUT is to be placed and operated in RX mode. Unfortunately, electromagnetic theory states that this field distribution is not possible to be generated by a finite size scatterer operated as the collimator of a nonzero wavelength time-harmonic propagating field. This is the background of this paper, where two well-known electromagnetic field collimators will be discussed: the serrated edge reflector and the blended rolled edge reflector. To reach this purpose, electromagnetic hybrid analysis techniques developed at Technical University of Madrid will be applied.

Phase transitions in number theory: from the birthday problem to Sidon sets

In this work, we show how number theoretical problems can be fruitfully approached with the tools of statistical physics. We focus on g-Sidon sets, which describe sequences of integers whose pairwise sums are different, and propose a random decision problem which addresses the probability of a random set of k integers to be g-Sidon. First, we provide numerical evidence showing that there is a crossover between satisfiable and unsatisfiable phases which converts to an abrupt phase transition in a properly defined thermodynamic limit. Initially assuming independence, we then develop a mean-field theory for the g-Sidon decision problem. We further improve the mean-field theory, which is only qualitatively correct, by incorporating deviations from independence, yielding results in good quantitative agreement with the numerics for both finite systems and in the thermodynamic limit. Connections between the generalized birthday problem in probability theory, the number theory of Sidon sets and the properties of q-Potts models in condensed matter physics are briefly discussed

Modelling relativistic solitary wave interactions in over-dense plasmas: a perturbed nonlinear Schröndinger equation framework

We investigate the dynamics of localized solutions of the relativistic cold-fluid plasma model in the small but finite amplitude limit, for slightly overcritical plasma density. Adopting a multiple scale analysis, we derive a perturbed nonlinear Schrödinger equation that describes the evolution of the envelope of circularly polarized electromagnetic field. Retaining terms up to fifth order in the small perturbation parameter, we derive a self-consistent framework for the description of the plasma response in the presence of localized electromagnetic field. The formalism is applied to standing electromagnetic soliton interactions and the results are validated by simulations of the full cold-fluid model. To lowest order, a cubic nonlinear Schrödinger equation with a focusing nonlinearity is recovered. Classical quasiparticle theory is used to obtain analytical estimates for the collision time and minimum distance of approach between solitons. For larger soliton amplitudes the inclusion of the fifth-order terms is essential for a qualitatively correct description of soliton interactions. The defocusing quintic nonlinearity leads to inelastic soliton collisions, while bound states of solitons do not persist under perturbations in the initial phase or amplitude

Calculation Methodology and Fabrication Procedures for Particle Accelerator Strip-Line Kickers: Application to the CTF3 Combiner Ring Extraction Kicker and TL2 Tail Clippers

A particle accelerator is any device that, using electromagnetic fields, is able to communicate energy to charged particles (typically electrons or ionized atoms), accelerating and/or energizing them up to the required level for its purpose. The applications of particle accelerators are countless, beginning in a common TV CRT, passing through medical X-ray devices, and ending in large ion colliders utilized to find the smallest details of the matter. Among the other engineering applications, the ion implantation devices to obtain better semiconductors and materials of amazing properties are included. Materials supporting irradiation for future nuclear fusion plants are also benefited from particle accelerators. There are many devices in a particle accelerator required for its correct operation. The most important are the particle sources, the guiding, focalizing and correcting magnets, the radiofrequency accelerating cavities, the fast deflection devices, the beam diagnostic mechanisms and the particle detectors. Most of the fast particle deflection devices have been built historically by using copper coils and ferrite cores which could effectuate a relatively fast magnetic deflection, but needed large voltages and currents to counteract the high coil inductance in a response in the microseconds range. Various beam stability considerations and the new range of energies and sizes of present time accelerators and their rings require new devices featuring an improved wakefield behaviour and faster response (in the nanoseconds range). This can only be achieved by an electromagnetic deflection device based on a transmission line. The electromagnetic deflection device (strip-line kicker) produces a transverse displacement on the particle beam travelling close to the speed of light, in order to extract the particles to another experiment or to inject them into a different accelerator. The deflection is carried out by the means of two short, opposite phase pulses. The diversion of the particles is exerted by the integrated Lorentz force of the electromagnetic field travelling along the kicker. This Thesis deals with a detailed calculation, manufacturing and test methodology for strip-line kicker devices. The methodology is then applied to two real cases which are fully designed, built, tested and finally installed in the CTF3 accelerator facility at CERN (Geneva). Analytical and numerical calculations, both in 2D and 3D, are detailed starting from the basic specifications in order to obtain a conceptual design. Time domain and frequency domain calculations are developed in the process using different FDM and FEM codes. The following concepts among others are analyzed: scattering parameters, resonating high order modes, the wakefields, etc. Several contributions are presented in the calculation process dealing specifically with strip-line kicker devices fed by electromagnetic pulses. Materials and components typically used for the fabrication of these devices are analyzed in the manufacturing section. Mechanical supports and connexions of electrodes are also detailed, presenting some interesting contributions on these concepts. The electromagnetic and vacuum tests are then analyzed. These tests are required to ensure that the manufactured devices fulfil the specifications. Finally, and only from the analytical point of view, the strip-line kickers are studied together with a pulsed power supply based on solid state power switches (MOSFETs). The solid state technology applied to pulsed power supplies is introduced and several circuit topologies are modelled and simulated to obtain fast and good flat-top pulses.

Raman spectroscopy in Group IV nanowires and nanowire axial heterostructures

The control of the SiGe NW composition is fundamental for the fabrication of high quality heterostructures. Raman spectroscopy has been used to analyse the composition of SiGe alloys. We present a study of the Raman spectrum of SiGe nanowires and SiGe/Si heterostructures. The inhomogeneity of the Ge composition deduced from the Raman spectrum is explained by the existence of a Ge-rich outer shell and by the interaction of the NW with the electromagnetic field associated with the laser beam.

Comparativa de las medidas de emisiones electromagnéticas en las normativas europea y americana para teléfonos duales

El objetivo del presente Proyecto Fin de Grado es la realización de una comparativa de los resultados obtenidos en las medidas de SAR (tasa de absorción específica) y potencia utilizando la normativa europea CENELEC (basada en la del IEC) y la americana FCC (basada en la del IEEE) para distintos dispositivos móviles duales. En primer lugar se ha realizado el estudio de sus características de potencia y de SAR, viendo la variación que hay con respecto a las distintas normativas y rangos de frecuencias. En algunos casos, aunque funcionan sin problema alguno en las diferentes bandas, la diferencia de niveles de emisiones obtenidos en las diferentes bandas puede ser notable. Como se ha comentado al comienzo, se han utilizado diferentes dispositivos duales y se realizaron en ellos las medidas pertinentes utilizando las diferentes normativas y el procedimiento indicado en cada una de ellas. Para observar las diferencias concretas que pudieran existir al utilizar las diferentes normativas, las medidas se han realizado en todas las bandas de funcionamiento del dispositivo y para cada una de ellas se han aplicado las diferentes normativas. Con esto se pretende evaluar si pudiera existir alguna discrepancia en los dispositivos que tienen el certificado de conformidad para una norma concreta cuando se utiliza en otra banda que no es exactamente la banda de utilización de esa norma. En resumen, se quería comprobar que un dispositivo que está certificado con la norma de una región, por ejemplo Europa (900 y 1800 MHz), también cumple si se utiliza en otra región, por ejemplo América (850 y 1900 MHz). La realización práctica del presente PFG se ha hecho en las instalaciones del Laboratorio de Radiofrecuencia de la Secretaría de Estado de Telecomunicación y para la Sociedad de la Información ubicado en El Casar (Guadalajara). En concreto se utilizó el robot articulado movible DASY4 y el software para su control DASY 52.8, disponible en el “banco de medida de Campos Electromagnéticos (Sala de SAR)”. La duración estimada del presente PFG ha sido aproximadamente de cuatro meses. Para llevar a cabo el presente PFG, se dispuso de todo tipo de material y recursos puestos a disposición por el Ministerio de Industria, Turismo y Comercio en el propio laboratorio, así como de los distintos terminales móviles duales con los que se realizaron las medidas pertinentes. Como bibliografía básica se han utilizado las diferentes normas indicadas anteriormente, es decir la norma europea CENELEC (basada en la del IEC) y la americana FCC (basada en la del IEEE), así como manuales de los equipos implicados en el bando de medida de SAR: analizadores de redes, robot articulado y software de control, así como el resto de dispositivos utilizados en las medidas. ABSTRACT. The goal of this Final Degree Project is to perform a comparison of the results obtained in SAR measurements (specific absorption rate) and power using the European standards CENELEC (based on IEC Regulation) and the American FCC (based on IEEE Regulation) to different mobile dual devices. If first place it was made the study of its power and SAR features, seeing that there is shift with respect to the different standards and frequency ranges. In some cases, although they work without any problem in different bands, the difference in levels of emissions obtained in the different bands can be significant. As mentioned at the beginning, different dual devices were used and relevant measurements were taken from them using the different standards and the procedure in each one of them. To see the specific differences that may exist when using the different standards, the measurements were made in all bands of the device operation and to each one it has been applied in the different standards. This attempted to assess whether there could be some discrepancy in the devices that have the certificate of compliance to a specify standard when used in another band that is not exactly the used band of this standard. To sum up, it was required to verify that a device which is certified to the standard of a region, for example (900 and 1800 MHz), also verifies if it is used in another region, for example America (850 and 1900 MHz). The practical realization of this Final Degree Project was made in the facilities of the Radio Frequency Laboratory of the Ministry of State for Telecommunications and the Information Society located in El Casar (Guadalajara). Specifically, the movable articulated robot DASY4 was used and the control software DASY 52.8, available in the “Measure Electromagnetic Field testbench (SAR room)”. The duration of this Final Degree Project has benn about four months. To carry out the present project, all kinds of materials and resources were provided by the Ministry of Industry, Tourism and Trade in its own laboratory, as well as the different mobile dual terminals with which relevant measurements were made. As basic references the different standards indicated above has been used, that is to say the European standard CENELEC (based on IEC standard) and the American FCC (based on IEEE standard), as well as the equipment manuals involved in the SAR measure testbench: network analyzers, articulated robot and control software, as well as the rest of the devices used in the measurements.

Sobre la solución del problema de la radiación electromagnética

En la presente Tesis Doctoral se propone una posible solución al problema de la radiación electromagnética. Este es un problema científico estudiado desde hace mucho tiempo y que hasta el presente no se ha podido resolver de manera exacta. Aquí se redefine nuevamente el problema y se plantea el mismo partiendo un modelo de radiador formado por dos placas planas circulares paralelas, que por sus reducidas dimensiones eléctricas, se considerará para el estudio como un dipolo simétrico corto. Se obtienen las soluciones de las ecuaciones de Maxwell para la radiación electromagnética emitida por dicho sistema y se propone lo que sería el principio de formación de las ondas electromagnéticas. Con ayuda de este se ofrece una posible interpretación de las soluciones obtenidas. Luego a dicho sistema se le realizan correcciones en sus dimensiones a fin de ajustar las soluciones obtenidas al caso real de un dipolo simétrico de media onda. Además se analiza de una manera sencilla la naturaleza material del campo electromagnético y su estructura, lo que hace posible proponer una manera original de comprender las ecuaciones de Maxwell. Finalmente se propone un modelo sencillo mediante el cual se puede demostrar la Ley de Planck para la distribución espectral de la energía electromagnética radiada por los cuerpos calientes, pero desde las soluciones de las ecuaciones de Maxwell para la radiación de las antenas. Con esto podría ser aceptado que la solución del problema de la radiación electromagnética es única tanto para las antenas como para los cuerpos calientes en la naturaleza. ABSTRACT In this PhD thesis a possible solution to the problem of electromagnetic radiation is proposed. This is a scientific problem which has been studied for a long time and to this day has still not exactly been solved. The problem is redefined here again and it is stated using a model formed by two flat parallel circular plates as the radiator system, which due to its reduced electrical dimensions will be considered for the study as a short symmetrical dipole. The solutions of the Maxwell equations for emitted electromagnetic radiation are obtained by said system, and that which could be the principle formation of electromagnetic waves is proposed. Using this principle, a possible interpretation of the solutions obtained is presented. Then the system will perform corrections in size to adjust the solutions obtained to the real case of a symmetrical half-wave dipole. The material nature of the electromagnetic field and its structure is also analyzed in a simple way, making it possible to present an original way to understand the Maxwell equations. Finally, a simple model which can be used to prove Planck's law of spectral distribution of the electromagnetic energy radiated by hot bodies is proposed, but parting from the solutions of the Maxwell equations for the antennas radiation. With this it could be accepted that the solution to the problem of electromagnetic radiation is unique to both antennas and hot bodies on nature.