Smale's Conjecture on Mean Values of Polynomials and Electrostatics

2000 Mathematics Subject Classification: Primary 30C10, 30C15, 31B35.A challenging conjecture of Stephen Smale on geometry of polynomials is under discussion. We consider an interpretation which turns out to be an interesting problem on equilibrium of an electrostatic field that obeys the law of the logarithmic potential. This interplay allows us to study the quantities that appear in Smale’s conjecture for polynomials whose zeros belong to certain specific regions. A conjecture concerning the electrostatic equilibrium related to polynomials with zeros in a ring domain is formulated and discussed.Research supported by the Brazilian foudations CNPq under Grant 304830/2006-2 and FAPESP under Grant 03/01874-2

Photonic Crystals with Alternate Arrays of Rods and Holes in a Low Dielectric-Index Material

This thesis theoretically deals with the propagation of electromagnetic waves (light beams) in periodically modulated dielectric material structures based on Maxwell’s equations. We are interested in novel light propagation characteristics in these man-made dielectric material structures for practical applications, especially on optical communications and computations. Since the wavelength range of light is on the same order of magnitude as the modulation periods of dielectric materials, an analogy of the light propagation in dielectric-constant modulated structures with the electron transport in solid-state crystals is used throughout my thesis by using a term “photonic crystals (PhCs)” referring to these dielectric structures. I started my work on two-dimensional (2D) PhCs. A new type of PhCs is proposed which consists of alternate arrays of rods and holes (AARH), embedded in a low dielectric-constant material such as ultravioletcurable polymer. By modeling them as 2D PhCs, it is discovered that this type of PhCs exhibits overlapped photonic band gaps (PBGs) for both transverse electric (TE) and transverse magnetic (TM) polarized light beams. This discovery is important for many practical applications related to light manipulation. It opens the door to more effective optical computing elements, as well as 1 better wave guides, LEDs and micro-lasers. It is also found that new AARH PhCs possess many interesting near-band-edge properties such as left-hand-material characteristics manifested by perfect reflection, negative refraction, and superlensing. I then extended my work to three dimensional (3D) counterparts of the discovered PhCs, which are called photonic crystal slabs (PCSs). I found that the overlapped TE and TM PBGs persist in these PCSs although they are restricted to a partial k-space. By manipulating certain structure parameters such as the thickness of PCS and its cladding, it is possible to achieve overlapping incomplete PBGs exactly in the frequency range predicted by 2D simulations. Hence, one can use fast and cheap 2D simulation instead of slow and expensive 3D and still engineer complex 3D photonic structures. The AARH PCSs also exhibit negative refraction and near zero effective refractive index. These effects allow for a strong control over the light propagation in PhCs. Additionally, the edge and surface modes of proposed PCSs are observed that effectively enlarge slabs’ PBGs