Construções geométricas como recurso pedagógico no ensino médio

This work aims to provide high school students an development in his mathematical and geometrical knowledge, through the use of Geometric Constructions as a teaching resource in Mathematics classes. First a literature search to understand how it emerged and evolved the field of geometry was carried out and the Geometric Constructions. The ways in which the teaching of geometry happened in our country, also were studied some theories related to learning and in particular the Van Hiele theory which deals with the geometric learning also through the literature search were diagnosed. Two forms of the Geometric Constructions approach are analyzed in class: through the design of hand tools - ruler and compass - and through the computational tool - geometric software - being that we chose to approach using the ruler and compass instruments. It is proposed a workshop with nine Geometric Construction activities which was applied with a group of 3rd year of high school, the Escola de Educac¸ ˜ao B´asica Professor Anacleto Damiani in the city of Abelardo Luz, Santa Catarina. Each workshop activity includes the following topics: Activity Goals, Activity Sheet, Steps of Construction Activity Background and activity of the solution. After application of the workshop, the data were analyzed through content analysis according to three categories: Drawing Instruments, angles and their implications and Parallel and its Implications. Was observed that most of the students managed to achieve the research objectives, and had an development in their mathematical and geometrical knowledge, which can be perceived through the analysis of questionnaires administered to students, audio recordings, observations made during the workshop and especially through the improvement of the students in the development of the proposed activities.

Acousto-optic modulation of optical reflection properties in fiber Bragg gratings and the application in fiber lasers

This thesis aims to investigate the interaction of acoustic waves and fiber Bragg gratings (FBGs) in standard and suspended-core fibers (SCFs), to evaluate the influence of the fiber, grating and modulator design on the increase of the modulation efficiency, bandwidth and frequency. Initially, the frequency response and the resonant acoustic modes of a low frequency acousto-optic modulator (f < 1.2 MHz) are numerically investigated by using the finite element method. Later, the interaction of longitudinal acoustic waves and FBGs in SCFs is also numerically investigated. The fiber geometric parameters are varied and the strain and grating properties are simulated by means of the finite element method and the transfer matrix method. The study indicates that the air holes composing the SCF cause a significant reduction of the amount of silica in the fiber cross section increasing acousto-optic interaction in the core. Experimental modulation of the reflectivity of FBGs inscribed in two distinct SCFs indicates evidences of this increased interaction. Besides, a method to acoustically induce a dynamic phase-shift in a chirped FBG employing an optimized design of modulator is shown. Afterwards, a combination of this modulator and a FBG inscribed in a three air holes SCF is applied to mode-lock an ytterbium doped fiber laser. To improve the modulator design for future applications, two other distinct devices are investigated to increase the acousto-optic interaction, bandwidth and frequency (f > 10 MHz). A high reflectivity modulation has been achieved for a modulator based on a tapered fiber. Moreover, an increased modulated bandwidth (320 pm) has been obtained for a modulator based on interaction of a radial long period grating (RLPG) and a FBG inscribed in a standard fiber. In summary, the results show a considerable reduction of the grating/fiber length and the modulator size, indicating possibilities for compact and faster acousto-optic fiber devices. Additionally, the increased interaction efficiency, modulated bandwidth and frequency can be useful to shorten the pulse width of future all-fiber mode-locked fiber lasers, as well, to other photonic devices which require the control of the light in optical fibers by electrically tunable acoustic waves.