We give an explicit construction of a fundamental solution for an arbitrary non-degenerate partial differential equation with smooth coefficients.

Ultra-short and high-peak-power laser pulses have important industrial and scientific applications. While direct laser amplification can lead to peak powers of several million watts, higher values than these cannot be achieved without causing damage to the amplifier material. Chirped pulse amplification technique is thus invented to break this barrier. By temporally stretching pulses before entering amplifier, the pulse peak power is significantly reduced and thus becomes safe to be passed through the amplifier. After amplification, a compressor is used to recover the pulse width, and high-power ultra-short laser pulses are produced. Chirped pulse amplification technology increases the pulse energy by transferring the damaging effects of high-peak power laser pulses from the vulnerable amplifier to a relatively robust compressor system. The compressor is therefore a crucial device for producing high peak powers. However, there are some major drawbacks associated with it. First, compressors in high-energy laser system are usually over 1 cubic meter in size. For many applications, this large and cumbersome size is a limiting factor. Second, compressors are sensitive to outside disturbances; a little misalignment can lead to failure of pulse compression process. Third, gratings with large uniformly ruled area are difficult to fabricate, which impose a limit on achievable peak powers and pulse durations of laser pulses through the use of conventional compressors. In this project, we present a grating-based monolithic optical compressor that offers a way around some of the major problems of existing compressors. By integrating the key optical components, one can make a robust and monolithic compressor that requires no alignment. In the new scheme, folding the optical path with reflective coatings allows one to design a compressor of significantly reduced size by minimizing both the longitudinal and transverse dimensions of the device. The configuration and operation mechanism of this novel compressor are described. A method for calculating the volume of the compressor is investigated. This is validated by computing the size of a specific monolithic compressor. Simulation results obtained through finite-difference time-domain method are presented, proving that the new compressor provides a compact, portable, and robust means for temporally compressing long duration pulses.

Devido às transformações que o ensino atravessa, o surgimento de novas tecnologias digitais utilizadas no ensino e diante da falta de motivação por parte dos alunos no aprendizado da matemática e da física, essa pesquisa foi concebida com o objetivo de verificar, as potencialidades do uso do software Geogebra no ensino interdisciplinar, através de atividades simuladas no Geogebra para resolução de problemas em óptica geométrica. / Due to new technologies introduced in teaching and a lack of motivation among students to learn mathematics and physics, inspired this research to verify the potential of Geogebra software in interdisciplinary learning through activities simulated in Geogebra to solve problems on/about geometrical optics.

Devido às transformações que o ensino atravessa, o surgimento de novas tecnologias digitais utilizadas no ensino e diante da falta de motivação por parte dos alunos no aprendizado da matemática e da física, essa pesquisa foi concebida com o objetivo de verificar, as potencialidades do uso do software Geogebra no ensino interdisciplinar, através de atividades simuladas no Geogebra para resolução de problemas em óptica geométrica. / Due to new technologies introduced in teaching and a lack of motivation among students to learn mathematics and physics, inspired this research to verify the potential of Geogebra software in interdisciplinary learning through activities simulated in Geogebra to solve problems on/about geometrical optics.

We introduce two models to design near field reflectors in R^3 that solve an inverse problem in radiometry, taking into account the inverse square law of irradiance. The problem leads to a Monge-Ampere type inequality. The surfaces in the first model are strictly convex and require to be far from the source to avoid obstruction. In the second model, the reflectors are neither convex nor concave and do not block the rays even if they are close to the source. / Mathematics

In 2014, Astala, Päivärinta, Reyes, and Siltanen conducted numerical experiments reconstructing a piecewise continuous conductivity. The algorithm of the shortcut method is based on the reconstruction algorithm due to Nachman, which assumes a priori that the conductivity is Hölder continuous. In this dissertation, we prove that, in the presence of infinite-precision data, this shortcut procedure accurately recovers the scattering transform of an essentially bounded conductivity, provided it is constant in a neighborhood of the boundary. In this setting, Nachman’s integral equations have a meaning and are still uniquely solvable. To regularize the reconstruction, Astala et al. employ a high frequency cutoff of the scattering transform. We show that such scattering transforms correspond to Beltrami coefficients that are not compactly supported, but exhibit certain decay at infinity. For this class of Beltrami coefficients, we establish that the complex geometric optics solutions to the Beltrami equation exist and exhibit the same subexponential decay as described in the 2006 work of Astala and Päivärinta. This is a first step toward extending the inverse scattering map of Astala and Päivärinta to non-compactly supported conductivities.

In this thesis, we analyze the simplest optical imaging system: a ball lens. The traditional method of using a geometric optics analysis on an optical system only gives the roughest qualitative solution due to the lack of consideration of wave properties. Therefore, for accurate quantitative results, we need to analyze said system with a complete wave theory approach. The reason that we chose a ball lens as the focus of this research is due to its spherical symmetry properties which allows us to rigorously investigate it with different analytic methods. We will apply geometric optics, Fourier optics, scalar wave optics, and electromagnetic optics methods to compute the point spread functions (PSF) of a ball lens under the assumption that the point source is isotropic. We will follow up by predicting the spot sizes that correspond to each mean. First, with geometric optics (GO), we apply the analytic ray tracing method to correlate the origins of light rays passing through the ball lens to their respective positions on the receptive end. We can then evaluate the energy distribution function by gathering the density of rays on image plane. Second, in the theory of Fourier optics (FO), to obtain the analytic formula of the point spread function, the integral kernel can be approximated as the Fresnel integral kernel by means of paraxial approximation. Compared to GO, the results from FO are superior due to the inclusion of wave characteristics. Furthermore, we consider scalar wave optics by directly solving the inhomogeneous Helmholtz equation which the scalar light field should satisfy. However, the light field is not assigned to an exact physical meaning in the theory of scalar wave optics, so we reasonably require boundary conditions where the light field function and its first derivative are continuous everywhere on the surface of ball lens. Finally, in the theory of electromagnetic optics (EMO), we consider the polarization of the point source, and the two kinds of Hertz vectors and , both of which satisfy inhomogeneous Helmholtz equation, and are derived from Maxwell¡¦s equations in spherical structures. In contrast with the scalar wave optics, the two Hertz vectors are defined concretely thus allowing us to assign exact boundary conditions on the interface. Then the fields corresponding to and are averaged as the final point spread function.

Modeling the single-scattering properties of nonspherical particles in the atmo¬sphere (in particular, ice crystals and dust aerosols) has important applications to climate and remote sensing studies. The first part of the dissertation (Chapters II¬V) reports a combination of exact numerical methods, including the finite-difference time-domain (FDTD), the discrete-dipole-approximation (DDA), and the T-matrix methods, and an approximate method-the physical-geometric optics hybrid (PGOH) method-in the computation of the optical properties of the non-spherical particles in a complete range of size parameters. The major advancements are made on the modeling capabilities of the PGOH method, and the knowledge of the electromag¬netic tunneling effect – a semi-classical scattering effect. This research is important to obtain reliable optical properties of nonspherical particles in a complete range of size parameters with satisfactory accuracy and computational efficiency. The second part (Chapters VI-VII) of the dissertation is to investigate the de¬pendence of the optical properties of ice crystals and mineral dust aerosols in the atmosphere on the spectrum, the particle size and the morphology based on compu¬tational models. Ice crystals in the atmosphere can be classified to be simple regular faceted particles (such as hexagon columns, plates, etc.) and imperfect ice crystals. Modeling of the scattering by regular ice crystals is straightforward, as their morphologies can be easily defined. For imperfect ice crystals, the morphology is quite diverse, which complicates the modeling process. We present an effective approach of using irregular faceted particle to characterize the imperfectness of ice crystals. As an example of application, less-than-unity backscattering color ratio of cirrus clouds is demonstrated and explained theoretically, which provides guidance in the calibra¬tion algorithm for 1.064-µm channel on the Calipso lidar. Dust aerosols have no particular morphology. To develop an approach to modeling the optical properties of realistic dust particles, the principle of using simple shapes (triaxial ellipsoids and nonsymmetric hexahedra) to represent irregular dust particles is explored. Simulated results have been compared with those measured in laboratory for several realistic aerosol samples. Agreement between simulated results and measurement suggests the potential applicability of the two aforementioned aerosol models. We also show the potential impact of the present study to passive and active atmospheric remote sensing and future research works.

Submitted by Aelson Maciera (aelsoncm@terra.com.br) on 2017-08-16T20:13:36Z No. of bitstreams: 1 DissDES.pdf: 2143823 bytes, checksum: ac10374208d2c147182c161c8983ba29 (MD5) / Approved for entry into archive by Ronildo Prado (bco.producao.intelectual@gmail.com) on 2018-01-30T16:41:34Z (GMT) No. of bitstreams: 1 DissDES.pdf: 2143823 bytes, checksum: ac10374208d2c147182c161c8983ba29 (MD5) / Approved for entry into archive by Ronildo Prado (bco.producao.intelectual@gmail.com) on 2018-01-30T16:41:43Z (GMT) No. of bitstreams: 1 DissDES.pdf: 2143823 bytes, checksum: ac10374208d2c147182c161c8983ba29 (MD5) / Made available in DSpace on 2018-01-30T16:46:48Z (GMT). No. of bitstreams: 1 DissDES.pdf: 2143823 bytes, checksum: ac10374208d2c147182c161c8983ba29 (MD5) Previous issue date: 2017-03-22 / Não recebi financiamento / Facing the students’ lack of motivation during Physics classes and the presence of new digital technologies that can aid and improve the teaching approach nowadays, this present work consists of proposing a methodology capable of promoting Geometric Optics learning, specifically for the Reflection theme, through the Geogebra software as a pedagogical resource. Based on the idea that computer simulations allow the development of several skills, involve concentration on the theoretical concepts and help to contextualize theory and practice, the official proposals (PCNs and Curriculum of the State of São Paulo) which aim to guide the teacher, indicating the necessary abilities that students must acquire when graduating high school were studied and investigated. A test was applied after the activities and we could verify a satisfactory result of the methodology. / Diante de um quadro de desmotivação por parte dos alunos durante as aulas de física e a presença de novas tecnologias digitais que podem auxiliar no ensino, o presente trabalho consiste em propor uma metodologia capaz de promover uma aprendizagem em Óptica Geométrica, especificamente o tema Reflexão, através do software Geogebra, como recurso pedagógico. Partindo da ideia que simulações computacionais possibilitam o desenvolvimento de várias habilidades, envolvem a concentração nos conceitos teóricos e auxiliam na contextualização entre teoria e prática, investigou-se as propostas dos documentos oficiais (PCNs e Currículo do Estado de São Paulo) que objetivam auxiliar o professor, indicando as competências e habilidades necessárias para o aluno adquirir ao ser formado na educação básica. Após as atividades, um questionário foi aplicado e a partir dele verificamos uma eficiência satisfatória da metodologia.

Submitted by André Luigi Amaral Di Salvo (andre.amaral@ifsuldeminas.edu.br) on 2018-08-16T20:49:37Z No. of bitstreams: 1 salvo_alad_dr_rcla.pdf: 9802752 bytes, checksum: 172b8278f611240bbd74d06ce473f0b6 (MD5) / Rejected by Ana Paula Santulo Custódio de Medeiros null (asantulo@rc.unesp.br), reason: Solicitamos que realize uma nova submissão seguindo as orientações abaixo: - Página de rosto: falta nome do orientador Agradecemos a compreensão on 2018-08-17T13:22:06Z (GMT) / Submitted by André Luigi Amaral Di Salvo (andre.amaral@ifsuldeminas.edu.br) on 2018-08-17T23:42:28Z No. of bitstreams: 1 di_salvo_lb_dr_rcla.pdf: 9807717 bytes, checksum: b8e283551078a579f75583dfbf863241 (MD5) / Approved for entry into archive by Adriana Aparecida Puerta null (dripuerta@rc.unesp.br) on 2018-08-20T13:10:35Z (GMT) No. of bitstreams: 1 disalvo_ala_dr_rcla.pdf: 9290721 bytes, checksum: cac26883aa35bc0bcad10429f06c8dc0 (MD5) / Made available in DSpace on 2018-08-20T13:10:36Z (GMT). No. of bitstreams: 1 disalvo_ala_dr_rcla.pdf: 9290721 bytes, checksum: cac26883aa35bc0bcad10429f06c8dc0 (MD5) Previous issue date: 2018-07-03 / A educação no Século XXI passa por um grande desafio: de um lado, alunos que nasceram e cresceram imersos em um mundo de tecnologias digitais; do outro, professores que, em sua maioria, foram formados por sistemas nos quais essas tecnologias não estavam presentes, nem foram posteriormente integradas à sua formação. Tal contraponto tem provocado uma série de descontentamentos em ambos os lados. Para tentar minimizar esse problema, este trabalho apresenta um novo tipo de ferramenta digital: um jogo educacional que utiliza os preceitos de mundos virtuais, laboratórios de simulação e jogos de entretenimento, para explorar os conteúdos de óptica geométrica. A ferramenta foi concebida, construída e testada por uma equipe multidisciplinar composta por professores, pedagogos e alunos, usando-se as premissas do Design Participativo. Para testar sua eficiência, tanto no ensino regular quanto em ambientes de ensino a distância, quatro experimentos distintos foram conduzidos nas instalações do Instituto Federal de Educação, Ciência e Tecnologia do Sul de Minas Gerais – Campus Inconfidentes, contando com a participação de 190 alunos pertencentes aos quatro cursos técnicos ofertados pelo campus – Agropecuária, Alimentos, Agrimensura e Informática. Os alunos foram divididos em dois grupos: controle (sem acesso ao software) e experimental (com acesso ao software). Todos foram avaliados em dois momentos distintos: pré e pós-teste. Os dados coletados durante o período de intervenção foram submetidos a análises estatísticas descritivas e inferenciais. Os resultados indicam que os alunos que tiveram acesso ao software apresentaram ganhos significativos de média, fato este não observado no grupo que não teve acesso. Além dessa análise, o software foi submetido à escala EGameFlow, que avaliou a presença de uma série de características necessárias a um jogo digital, dentre as quais imersão, concentração e motivação. Os resultados dessa análise mostraram-se também favoráveis à utilização desse tipo de ferramenta como recurso didático, com a maioria dos entrevistados indicando que ele auxilia o processo ensino/aprendizagem. / Education in the 21st century is facing a huge challenge: on one hand are the students who were born to and are growing up immersed in a technological world; on the other are teachers formed by systems in which technology was not present, nor was it integrated to their formation later on. Such counterpoint has been the cause of much discontentment on both sides. As an attempt to minimize the problem, this work presents a new type on digital tool: an educational game that utilizes the precepts of virtual worlds, simulation laboratories and leisure games, to explore contents of Geometric Optics, in order to enhance the teaching-learning process. This tool was conceived, constructed and tested by a multi-disciplinary team consisting of teachers, pedagogues, and students, under the Participative Design approach. Four experiments were carried out in order to test its efficiency, both in traditional and distance learning environments, using the facilities at the Instituto Federal de Educação, Ciência e Tecnologia do Sul de Minas Gerais – Campus Inconfidentes. 190 students from all of the four technical courses offered in the campus – Farming, Food Processing, Land Mapping e Informatics - agreed to attend the experiment. They were separated into two groups: a control group (without access to the software), and an experimental group (with full access to the software). All the students were evaluated in two distinct moments: prior to test and post-test. Data collected during intervention were submitted to statistical descriptive inferential analyses, and the results indicate that the students who had full access to the software averaged significantly higher, as compared to the other group. Besides the tests for efficiency in the teaching-learning process, the software was also evaluated using the EGameFlow scale as for the required features for a digital game, among which are immersion, concentration and motivation. The results were also favorable in respect to the utilization of this type of tool as a didactic resource, with the majority of those surveyed pointing out that it does enhance the teaching-learning process.