Detection of scalar gravitational waves with a hollow spherical antenna

We study the response and cross sections for the absorption of GW energy generated in a Jordan-Brans-Dicke theory by a resonant mass detector shaped as a hollow sphere. As a source of the GW we take a binary system in the Newtonian approximation. For masses of the stars of the order of the solar mass, the emitted GW sweeps a range of frequencies which include the first resonant mode of the detector.

The asymmetrically excited spherical antenna /

"Contract No. AF33(616)-310 RDO No. R-112-110 SR-6f2"

The Schenberg spherical gravitational wave detector: the first commissioning runs

Here we present a status report of the first spherical antenna project equipped with a set of parametric transducers for gravitational detection. The Mario Schenberg, as it is called, started its commissioning phase at the Physics Institute of the University of Sao Paulo, in September 2006, under the full support of FAPESP. We have been testing the three preliminary parametric transducer systems in order to prepare the detector for the next cryogenic run, when it will be calibrated. We are also developing sapphire oscillators that will replace the current ones thereby providing better performance. We also plan to install eight transducers in the near future, six of which are of the two-mode type and arranged according to the truncated icosahedron configuration. The other two, which will be placed close to the sphere equator, will be mechanically non-resonant. In doing so, we want to verify that if the Schenberg antenna can become a wideband gravitational wave detector through the use of an ultra-high sensitivity non-resonant transducer constructed using the recent achievements of nanotechnology.

The Schenberg data acquisition and analysis: results from its first commissioning run

The Mario Schenberg gravitational wave detector has started its commissioning phase at the Physics Institute of the University of Sao Paulo. We have collected almost 200 h of data from the instrument in order to check out its behavior and performance. We have also been developing a data acquisition system for it under a VXI System. Such a system is composed of an analog-to-digital converter and a GPS receiver for time synchronization. We have been building the software that controls and sets up the data acquisition. Here we present an overview of the Mario Schenberg detector and its data acquisition system, some results from the first commissioning run and solutions for some problems we have identified.

Reduction of truncation errors in planar, cylindrical and partial spherical near-field antenna measurements

A method to reduce truncation errors in near-field antenna measurements is presented. The method is based on the Gerchberg-Papoulis iterative algorithm used to extrapolate band-limited functions and it is able to extend the valid region of the calculated far-field pattern up to the whole forward hemisphere. The extension of the valid region is achieved by the iterative application of a transformation between two different domains. After each transformation, a filtering process that is based on known information at each domain is applied. The first domain is the spectral domain in which the plane wave spectrum (PWS) is reliable only within a known region. The second domain is the field distribution over the antenna under test (AUT) plane in which the desired field is assumed to be concentrated on the antenna aperture. The method can be applied to any scanning geometry, but in this paper, only the planar, cylindrical, and partial spherical near-field measurements are considered. Several simulation and measurement examples are presented to verify the effectiveness of the method.

Input impedance of a spherical ferrite antenna with a latitudinal current /

"Contract No. AF33(616)-310 RDO No. R-112-110 SR-6f2"

What can we learn about gravitational wave physics with an elastic spherial antenna?

A general formalism is set up to analyze the response of an arbitrary solid elastic body to an arbitrary metric gravitational wave (GW) perturbation, which fully displays the details of the interaction antenna wave. The formalism is applied to the spherical detector, whose sensitivity parameters are thereby scrutinized. A multimode transfer function is defined to study the amplitude sensitivity, and absorption cross sections are calculated for a general metric theory of GW physics. Their scaling properties are shown to be independent of the underlying theory, with interesting consequences for future detector design. The GW incidence direction deconvolution problem is also discussed, always within the context of a general metric theory of the gravitational field.

Contribution to the Analysis, Design and Assessment of Compact Antenna Test Ranges At Millimeter Wavelengths - Contribución al Análisis, Diseño y Caracterización de Sistemas de Medida de Antenas en Rango Compacto para Frecuencias Milimétricas

This Doctoral Thesis entitled Contribution to the analysis, design and assessment of compact antenna test ranges at millimeter wavelengths aims to deepen the knowledge of a particular antenna measurement system: the compact range, operating in the frequency bands of millimeter wavelengths. The thesis has been developed at Radiation Group (GR), an antenna laboratory which belongs to the Signals, Systems and Radiocommunications department (SSR), from Technical University of Madrid (UPM). The Radiation Group owns an extensive experience on antenna measurements, running at present four facilities which operate in different configurations: Gregorian compact antenna test range, spherical near field, planar near field and semianechoic arch system. The research work performed in line with this thesis contributes the knowledge of the first measurement configuration at higher frequencies, beyond the microwaves region where Radiation Group features customer-level performance. To reach this high level purpose, a set of scientific tasks were sequentially carried out. Those are succinctly described in the subsequent paragraphs. A first step dealed with the State of Art review. The study of scientific literature dealed with the analysis of measurement practices in compact antenna test ranges in addition with the particularities of millimeter wavelength technologies. Joint study of both fields of knowledge converged, when this measurement facilities are of interest, in a series of technological challenges which become serious bottlenecks at different stages: analysis, design and assessment. Thirdly after the overview study, focus was set on Electromagnetic analysis algorithms. These formulations allow to approach certain electromagnetic features of interest, such as field distribution phase or stray signal analysis of particular structures when they interact with electromagnetic waves sources. Properly operated, a CATR facility features electromagnetic waves collimation optics which are large, in terms of wavelengths. Accordingly, the electromagnetic analysis tasks introduce an extense number of mathematic unknowns which grow with frequency, following different polynomic order laws depending on the used algorithmia. In particular, the optics configuration which was of our interest consisted on the reflection type serrated edge collimator. The analysis of these devices requires a flexible handling of almost arbitrary scattering geometries, becoming this flexibility the nucleus of the algorithmia’s ability to perform the subsequent design tasks. This thesis’ contribution to this field of knowledge consisted on reaching a formulation which was powerful at the same time when dealing with various analysis geometries and computationally speaking. Two algorithmia were developed. While based on the same principle of hybridization, they reached different order Physics performance at the cost of the computational efficiency. Inter-comparison of their CATR design capabilities was performed, reaching both qualitative as well as quantitative conclusions on their scope. In third place, interest was shifted from analysis - design tasks towards range assessment. Millimetre wavelengths imply strict mechanical tolerances and fine setup adjustment. In addition, the large number of unknowns issue already faced in the analysis stage appears as well in the on chamber field probing stage. Natural decrease of dynamic range available by semiconductor millimeter waves sources requires in addition larger integration times at each probing point. These peculiarities increase exponentially the difficulty of performing assessment processes in CATR facilities beyond microwaves. The bottleneck becomes so tight that it compromises the range characterization beyond a certain limit frequency which typically lies on the lowest segment of millimeter wavelength frequencies. However the value of range assessment moves, on the contrary, towards the highest segment. This thesis contributes this technological scenario developing quiet zone probing techniques which achieves substantial data reduction ratii. Collaterally, it increases the robustness of the results to noise, which is a virtual rise of the setup’s available dynamic range. In fourth place, the environmental sensitivity of millimeter wavelengths issue was approached. It is well known the drifts of electromagnetic experiments due to the dependance of the re sults with respect to the surrounding environment. This feature relegates many industrial practices of microwave frequencies to the experimental stage, at millimeter wavelengths. In particular, evolution of the atmosphere within acceptable conditioning bounds redounds in drift phenomena which completely mask the experimental results. The contribution of this thesis on this aspect consists on modeling electrically the indoor atmosphere existing in a CATR, as a function of environmental variables which affect the range’s performance. A simple model was developed, being able to handle high level phenomena, such as feed - probe phase drift as a function of low level magnitudes easy to be sampled: relative humidity and temperature. With this model, environmental compensation can be performed and chamber conditioning is automatically extended towards higher frequencies. Therefore, the purpose of this thesis is to go further into the knowledge of millimetre wavelengths involving compact antenna test ranges. This knowledge is dosified through the sequential stages of a CATR conception, form early low level electromagnetic analysis towards the assessment of an operative facility, stages for each one of which nowadays bottleneck phenomena exist and seriously compromise the antenna measurement practices at millimeter wavelengths.

Contribution to the quality improvement of the antenna measurement using post-processing techniques = Contribución a la mejora de la calidad de medida de antena usando técnicas de post-procesado

One important task in the design of an antenna is to carry out an analysis to find out the characteristics of the antenna that best fulfills the specifications fixed by the application. After that, a prototype is manufactured and the next stage in design process is to check if the radiation pattern differs from the designed one. Besides the radiation pattern, other radiation parameters like directivity, gain, impedance, beamwidth, efficiency, polarization, etc. must be also evaluated. For this purpose, accurate antenna measurement techniques are needed in order to know exactly the actual electromagnetic behavior of the antenna under test. Due to this fact, most of the measurements are performed in anechoic chambers, which are closed areas, normally shielded, covered by electromagnetic absorbing material, that simulate free space propagation conditions, due to the absorption of the radiation absorbing material. Moreover, these facilities can be employed independently of the weather conditions and allow measurements free from interferences. Despite all the advantages of the anechoic chambers, the results obtained both from far-field measurements and near-field measurements are inevitably affected by errors. Thus, the main objective of this Thesis is to propose algorithms to improve the quality of the results obtained in antenna measurements by using post-processing techniques and without requiring additional measurements. First, a deep revision work of the state of the art has been made in order to give a general vision of the possibilities to characterize or to reduce the effects of errors in antenna measurements. Later, new methods to reduce the unwanted effects of four of the most commons errors in antenna measurements are described and theoretical and numerically validated. The basis of all them is the same, to perform a transformation from the measurement surface to another domain where there is enough information to easily remove the contribution of the errors. The four errors analyzed are noise, reflections, truncation errors and leakage and the tools used to suppress them are mainly source reconstruction techniques, spatial and modal filtering and iterative algorithms to extrapolate functions. Therefore, the main idea of all the methods is to modify the classical near-field-to-far-field transformations by including additional steps with which errors can be greatly suppressed. Moreover, the proposed methods are not computationally complex and, because they are applied in post-processing, additional measurements are not required. The noise is the most widely studied error in this Thesis, proposing a total of three alternatives to filter out an important noise contribution before obtaining the far-field pattern. The first one is based on a modal filtering. The second alternative uses a source reconstruction technique to obtain the extreme near-field where it is possible to apply a spatial filtering. The last one is to back-propagate the measured field to a surface with the same geometry than the measurement surface but closer to the AUT and then to apply also a spatial filtering. All the alternatives are analyzed in the three most common near-field systems, including comprehensive noise statistical analyses in order to deduce the signal-to-noise ratio improvement achieved in each case. The method to suppress reflections in antenna measurements is also based on a source reconstruction technique and the main idea is to reconstruct the field over a surface larger than the antenna aperture in order to be able to identify and later suppress the virtual sources related to the reflective waves. The truncation error presents in the results obtained from planar, cylindrical and partial spherical near-field measurements is the third error analyzed in this Thesis. The method to reduce this error is based on an iterative algorithm to extrapolate the reliable region of the far-field pattern from the knowledge of the field distribution on the AUT plane. The proper termination point of this iterative algorithm as well as other critical aspects of the method are also studied. The last part of this work is dedicated to the detection and suppression of the two most common leakage sources in antenna measurements. A first method tries to estimate the leakage bias constant added by the receiver’s quadrature detector to every near-field data and then suppress its effect on the far-field pattern. The second method can be divided into two parts; the first one to find the position of the faulty component that radiates or receives unwanted radiation, making easier its identification within the measurement environment and its later substitution; and the second part of this method is able to computationally remove the leakage effect without requiring the substitution of the faulty component. Resumen Una tarea importante en el diseño de una antena es llevar a cabo un análisis para averiguar las características de la antena que mejor cumple las especificaciones fijadas por la aplicación. Después de esto, se fabrica un prototipo de la antena y el siguiente paso en el proceso de diseño es comprobar si el patrón de radiación difiere del diseñado. Además del patrón de radiación, otros parámetros de radiación como la directividad, la ganancia, impedancia, ancho de haz, eficiencia, polarización, etc. deben ser también evaluados. Para lograr este propósito, se necesitan técnicas de medida de antenas muy precisas con el fin de saber exactamente el comportamiento electromagnético real de la antena bajo prueba. Debido a esto, la mayoría de las medidas se realizan en cámaras anecoicas, que son áreas cerradas, normalmente revestidas, cubiertas con material absorbente electromagnético. Además, estas instalaciones se pueden emplear independientemente de las condiciones climatológicas y permiten realizar medidas libres de interferencias. A pesar de todas las ventajas de las cámaras anecoicas, los resultados obtenidos tanto en medidas en campo lejano como en medidas en campo próximo están inevitablemente afectados por errores. Así, el principal objetivo de esta Tesis es proponer algoritmos para mejorar la calidad de los resultados obtenidos en medida de antenas mediante el uso de técnicas de post-procesado. Primeramente, se ha realizado un profundo trabajo de revisión del estado del arte con el fin de dar una visión general de las posibilidades para caracterizar o reducir los efectos de errores en medida de antenas. Después, se han descrito y validado tanto teórica como numéricamente nuevos métodos para reducir el efecto indeseado de cuatro de los errores más comunes en medida de antenas. La base de todos ellos es la misma, realizar una transformación de la superficie de medida a otro dominio donde hay suficiente información para eliminar fácilmente la contribución de los errores. Los cuatro errores analizados son ruido, reflexiones, errores de truncamiento y leakage y las herramientas usadas para suprimirlos son principalmente técnicas de reconstrucción de fuentes, filtrado espacial y modal y algoritmos iterativos para extrapolar funciones. Por lo tanto, la principal idea de todos los métodos es modificar las transformaciones clásicas de campo cercano a campo lejano incluyendo pasos adicionales con los que los errores pueden ser enormemente suprimidos. Además, los métodos propuestos no son computacionalmente complejos y dado que se aplican en post-procesado, no se necesitan medidas adicionales. El ruido es el error más ampliamente estudiado en esta Tesis, proponiéndose un total de tres alternativas para filtrar una importante contribución de ruido antes de obtener el patrón de campo lejano. La primera está basada en un filtrado modal. La segunda alternativa usa una técnica de reconstrucción de fuentes para obtener el campo sobre el plano de la antena donde es posible aplicar un filtrado espacial. La última es propagar el campo medido a una superficie con la misma geometría que la superficie de medida pero más próxima a la antena y luego aplicar también un filtrado espacial. Todas las alternativas han sido analizadas en los sistemas de campo próximos más comunes, incluyendo detallados análisis estadísticos del ruido con el fin de deducir la mejora de la relación señal a ruido lograda en cada caso. El método para suprimir reflexiones en medida de antenas está también basado en una técnica de reconstrucción de fuentes y la principal idea es reconstruir el campo sobre una superficie mayor que la apertura de la antena con el fin de ser capaces de identificar y después suprimir fuentes virtuales relacionadas con las ondas reflejadas. El error de truncamiento que aparece en los resultados obtenidos a partir de medidas en un plano, cilindro o en la porción de una esfera es el tercer error analizado en esta Tesis. El método para reducir este error está basado en un algoritmo iterativo para extrapolar la región fiable del patrón de campo lejano a partir de información de la distribución del campo sobre el plano de la antena. Además, se ha estudiado el punto apropiado de terminación de este algoritmo iterativo así como otros aspectos críticos del método. La última parte de este trabajo está dedicado a la detección y supresión de dos de las fuentes de leakage más comunes en medida de antenas. El primer método intenta realizar una estimación de la constante de fuga del leakage añadido por el detector en cuadratura del receptor a todos los datos en campo próximo y después suprimir su efecto en el patrón de campo lejano. El segundo método se puede dividir en dos partes; la primera de ellas para encontrar la posición de elementos defectuosos que radian o reciben radiación indeseada, haciendo más fácil su identificación dentro del entorno de medida y su posterior substitución. La segunda parte del método es capaz de eliminar computacionalmente el efector del leakage sin necesidad de la substitución del elemento defectuoso.

Analysis and suppression of reflections in far-field antenna measurement ranges

This paper presents the analysis of the reflections in two kind of spherical far field ranges: one if the classical acquisition where the AUT is rotated and the second one corresponds to the systems where the AUT is fixed and the antenna probe is rotated. In large far field systems this is not possible, but this can be used to the measurement of small antennas, for instance, with the SATIMO StarGate system. In both cases, it is assumed that only one frequency is acquired and the results should be improved cut by cut, in order not to lose the advantages or far field measurements. Finally, some practical results are studied using measurements of one antenna in the outdoor far field facility of LIT INPE in Brazil.

New methods for the synthesis of radiation patterns of coupled antenna arrays = Nuevos métodos de síntesis de diagramas de radiación de agrupaciones de antenas acopladas

El principal objetivo de esta tesis es el desarrollo de métodos de síntesis de diagramas de radiación de agrupaciones de antenas, en donde se realiza una caracterización electromagnética rigurosa de los elementos radiantes y de los acoplos mutuos existentes. Esta caracterización no se realiza habitualmente en la gran mayoría de métodos de síntesis encontrados en la literatura, debido fundamentalmente a dos razones. Por un lado, se considera que el diagrama de radiación de un array de antenas se puede aproximar con el factor de array que únicamente tiene en cuenta la posición de los elementos y las excitaciones aplicadas a los mismos. Sin embargo, como se mostrará en esta tesis, en múltiples ocasiones un riguroso análisis de los elementos radiantes y del acoplo mutuo entre ellos es importante ya que los resultados obtenidos pueden ser notablemente diferentes. Por otro lado, no es sencillo combinar un método de análisis electromagnético con un proceso de síntesis de diagramas de radiación. Los métodos de análisis de agrupaciones de antenas suelen ser costosos computacionalmente, ya que son estructuras grandes en términos de longitudes de onda. Generalmente, un diseño de un problema electromagnético suele comprender varios análisis de la estructura, dependiendo de las variaciones de las características, lo que hace este proceso muy costoso. Dos métodos se utilizan en esta tesis para el análisis de los arrays acoplados. Ambos están basados en el método de los elementos finitos, la descomposición de dominio y el análisis modal para analizar la estructura radiante y han sido desarrollados en el grupo de investigación donde se engloba esta tesis. El primero de ellos es una técnica de análisis de arrays finitos basado en la aproximación de array infinito. Su uso es indicado para arrays planos de grandes dimensiones con elementos equiespaciados. El segundo caracteriza el array y el acoplo mutuo entre elementos a partir de una expansión en modos esféricos del campo radiado por cada uno de los elementos. Este método calcula los acoplos entre los diferentes elementos del array usando las propiedades de traslación y rotación de los modos esféricos. Es capaz de analizar agrupaciones de elementos distribuidos de forma arbitraria. Ambas técnicas utilizan una formulación matricial que caracteriza de forma rigurosa el campo radiado por el array. Esto las hace muy apropiadas para su posterior uso en una herramienta de diseño, como los métodos de síntesis desarrollados en esta tesis. Los resultados obtenidos por estas técnicas de síntesis, que incluyen métodos rigurosos de análisis, son consecuentemente más precisos. La síntesis de arrays consiste en modificar uno o varios parámetros de las agrupaciones de antenas buscando unas determinadas especificaciones de las características de radiación. Los parámetros utilizados como variables de optimización pueden ser varios. Los más utilizados son las excitaciones aplicadas a los elementos, pero también es posible modificar otros parámetros de diseño como son las posiciones de los elementos o las rotaciones de estos. Los objetivos de las síntesis pueden ser dirigir el haz o haces en una determinada dirección o conformar el haz con formas arbitrarias. Además, es posible minimizar el nivel de los lóbulos secundarios o del rizado en las regiones deseadas, imponer nulos que evitan posibles interferencias o reducir el nivel de la componente contrapolar. El método para el análisis de arrays finitos basado en la aproximación de array infinito considera un array finito como un array infinito con un número finito de elementos excitados. Los elementos no excitados están físicamente presentes y pueden presentar tres diferentes terminaciones, corto-circuito, circuito abierto y adaptados. Cada una de estas terminaciones simulará mejor el entorno real en el que el array se encuentre. Este método de análisis se integra en la tesis con dos métodos diferentes de síntesis de diagramas de radiación. En el primero de ellos se presenta un método basado en programación lineal en donde es posible dirigir el haz o haces, en la dirección deseada, además de ejercer un control sobre los lóbulos secundarios o imponer nulos. Este método es muy eficiente y obtiene soluciones óptimas. El mismo método de análisis es también aplicado a un método de conformación de haz, en donde un problema originalmente no convexo (y de difícil solución) es transformado en un problema convexo imponiendo restricciones de simetría, resolviendo de este modo eficientemente un problema complejo. Con este método es posible diseñar diagramas de radiación con haces de forma arbitraria, ejerciendo un control en el rizado del lóbulo principal, así como en el nivel de los lóbulos secundarios. El método de análisis de arrays basado en la expansión en modos esféricos se integra en la tesis con tres técnicas de síntesis de diagramas de radiación. Se propone inicialmente una síntesis de conformación del haz basado en el método de la recuperación de fase resuelta de forma iterativa mediante métodos convexos, en donde relajando las restricciones del problema original se consiguen unas soluciones cercanas a las óptimas de manera eficiente. Dos métodos de síntesis se han propuesto, donde las variables de optimización son las posiciones y las rotaciones de los elementos respectivamente. Se define una función de coste basada en la intensidad de radiación, la cual es minimizada de forma iterativa con el método del gradiente. Ambos métodos reducen el nivel de los lóbulos secundarios minimizando una función de coste. El gradiente de la función de coste es obtenido en términos de la variable de optimización en cada método. Esta función de coste está formada por la expresión rigurosa de la intensidad de radiación y por una función de peso definida por el usuario para imponer prioridades sobre las diferentes regiones de radiación, si así se desea. Por último, se presenta un método en el cual, mediante técnicas de programación entera, se buscan las fases discretas que generan un diagrama de radiación lo más cercano posible al deseado. Con este método se obtienen diseños que minimizan el coste de fabricación. En cada uno de las diferentes técnicas propuestas en la tesis, se presentan resultados con elementos reales que muestran las capacidades y posibilidades que los métodos ofrecen. Se comparan los resultados con otros métodos disponibles en la literatura. Se muestra la importancia de tener en cuenta los diagramas de los elementos reales y los acoplos mutuos en el proceso de síntesis y se comparan los resultados obtenidos con herramientas de software comerciales. ABSTRACT The main objective of this thesis is the development of optimization methods for the radiation pattern synthesis of array antennas in which a rigorous electromagnetic characterization of the radiators and the mutual coupling between them is performed. The electromagnetic characterization is usually overlooked in most of the available synthesis methods in the literature, this is mainly due to two reasons. On the one hand, it is argued that the radiation pattern of an array is mainly influenced by the array factor and that the mutual coupling plays a minor role. As it is shown in this thesis, the mutual coupling and the rigorous characterization of the array antenna influences significantly in the array performance and its computation leads to differences in the results obtained. On the other hand, it is difficult to introduce an analysis procedure into a synthesis technique. The analysis of array antennas is generally expensive computationally as the structure to analyze is large in terms of wavelengths. A synthesis method requires to carry out a large number of analysis, this makes the synthesis problem very expensive computationally or intractable in some cases. Two methods have been used in this thesis for the analysis of coupled antenna arrays, both of them have been developed in the research group in which this thesis is involved. They are based on the finite element method (FEM), the domain decomposition and the modal analysis. The first one obtains a finite array characterization with the results obtained from the infinite array approach. It is specially indicated for the analysis of large arrays with equispaced elements. The second one characterizes the array elements and the mutual coupling between them with a spherical wave expansion of the radiated field by each element. The mutual coupling is computed using the properties of translation and rotation of spherical waves. This method is able to analyze arrays with elements placed on an arbitrary distribution. Both techniques provide a matrix formulation that makes them very suitable for being integrated in synthesis techniques, the results obtained from these synthesis methods will be very accurate. The array synthesis stands for the modification of one or several array parameters looking for some desired specifications of the radiation pattern. The array parameters used as optimization variables are usually the excitation weights applied to the array elements, but some other array characteristics can be used as well, such as the array elements positions or rotations. The desired specifications may be to steer the beam towards any specific direction or to generate shaped beams with arbitrary geometry. Further characteristics can be handled as well, such as minimize the side lobe level in some other radiating regions, to minimize the ripple of the shaped beam, to take control over the cross-polar component or to impose nulls on the radiation pattern to avoid possible interferences from specific directions. The analysis method based on the infinite array approach considers an infinite array with a finite number of excited elements. The infinite non-excited elements are physically present and may have three different terminations, short-circuit, open circuit and match terminated. Each of this terminations is a better simulation for the real environment of the array. This method is used in this thesis for the development of two synthesis methods. In the first one, a multi-objective radiation pattern synthesis is presented, in which it is possible to steer the beam or beams in desired directions, minimizing the side lobe level and with the possibility of imposing nulls in the radiation pattern. This method is very efficient and obtains optimal solutions as it is based on convex programming. The same analysis method is used in a shaped beam technique in which an originally non-convex problem is transformed into a convex one applying symmetry restrictions, thus solving a complex problem in an efficient way. This method allows the synthesis of shaped beam radiation patterns controlling the ripple in the mainlobe and the side lobe level. The analysis method based on the spherical wave expansion is applied for different synthesis techniques of the radiation pattern of coupled arrays. A shaped beam synthesis is presented, in which a convex formulation is proposed based on the phase retrieval method. In this technique, an originally non-convex problem is solved using a relaxation and solving a convex problems iteratively. Two methods are proposed based on the gradient method. A cost function is defined involving the radiation intensity of the coupled array and a weighting function that provides more degrees of freedom to the designer. The gradient of the cost function is computed with respect to the positions in one of them and the rotations of the elements in the second one. The elements are moved or rotated iteratively following the results of the gradient. A highly non-convex problem is solved very efficiently, obtaining very good results that are dependent on the starting point. Finally, an optimization method is presented where discrete digital phases are synthesized providing a radiation pattern as close as possible to the desired one. The problem is solved using linear integer programming procedures obtaining array designs that greatly reduce the fabrication costs. Results are provided for every method showing the capabilities that the above mentioned methods offer. The results obtained are compared with available methods in the literature. The importance of introducing a rigorous analysis into the synthesis method is emphasized and the results obtained are compared with a commercial software, showing good agreement.

Melting and freezing of spherical bismuth nanoparticles confined in a homogeneous sodium borate glass

The melting temperature and the crystallization temperature of Bi nanoclusters confined in a sodium borate glass were experimentally determined as functions of the cluster radius. The results indicate that, on cooling, liquid Bi nanodroplets exhibit a strong undercooling effect for a wide range of radii. The difference between the melting temperature and the freezing temperature decreases for decreasing radius and vanishes for Bi nanoparticles with a critical radius R = 1.9 nm. The magnitude of the variation in density across the melting and freezing transitions for Bi nanoparticles with R = 2 nm is 40% smaller than for bulk Bi. These experimental results support a basic core-shell model for the structure of Bi nanocrystals consisting of a central crystalline volume surrounded by a structurally disordered shell. The volume fraction of the crystalline core decreases for decreasing nanoparticle radius and vanishes for R = 1.9 nm. Thus, on cooling, the liquid nanodroplets with R < 1.9 nm preserve, across the liquid-to-solid transformation, their homogeneous and disordered structure without crystalline core.

Dynamical Casimir effect for a massless scalar field between two concentric spherical shells with mixed boundary conditions

In this work we analyze the dynamical Casimir effect for a massless scalar field confined between two concentric spherical shells considering mixed boundary conditions. We thus generalize a previous result in literature [Phys. Rev. A 78, 032521 (2008)], where the same problem is approached for the field constrained to the Dirichlet-Dirichlet boundary conditions. A general expression for the average number of particle creation is deduced considering an arbitrary law of radial motion of the spherical shells. This expression is then applied to harmonic oscillations of the shells, and the number of particle production is analyzed and compared with the results previously obtained under Dirichlet-Dirichlet boundary conditions.