David Bermudez

Light-emitting diode spherical packages: an equation for the light transmission efficiency

Virtually all light-emitting diodes (LEDs) are encapsulated with a transparent epoxy or silicone gel. Here we analyze the optical efficiency of spherical encapsulants. We develop a quasi-radiometric equation for the light transmission efficiency, which incorporates some ideas of Monte Carlo ray tracing into the context of radiometry. The approach includes the extended source nature of the LED chip and the chip radiance distribution. The equation is an explicit function of the size and the refractive index of the package, and also of several chip parameters such as shape, size, radiance, and location inside the package. To illustrate the use of this equation, we analyze several packaging configurations of practical interest, for example, a hemispherical dome with multiple chips, a flat encapsulation as a special case of the spherical package, and approximate calculations of an encapsulant with a photonic crystal LED or a photonic quasi-crystal LED. These calculations are compared with Monte Carlo ray tracing, giving almost identical values.

Factorization method and new potentials from the inverted oscillator

Abstract In this article we will apply the first- and second-order supersymmetric quantum mechanics to obtain new exactly-solvable real potentials departing from the inverted oscillator potential. This system has some special properties; in particular, only very specific second-order transformations produce non-singular real potentials. It will be shown that these transformations turn out to be the so-called complex ones. Moreover, we will study the factorization method applied to the inverted oscillator and the algebraic structure of the new Hamiltonians.

Supersymmetric quantum mechanics and Painlevé equations

In these lecture notes we shall study first the supersymmetric quantum mechanics (SUSY QM), specially when applied to the harmonic and radial oscillators. In addition, we will define the polynomial Heisenberg algebras (PHA), and we will study the general systems ruled by them: for zero and first order we obtain the harmonic and radial oscillators, respectively; for second and third order the potential is determined by solutions to Painleve IV (PIV) and Painleve V (PV) equations. Taking advantage of this connection, later on we will find solutions to PIV and PV equations expressed in terms of confluent hypergeometric functions. Furthermore, we will classify them into several solution hierarchies, according to the specific special functions they are connected with.

Non-hermitian Hamiltonians and the Painlevé IV equation with real parameters

Abstract In this Letter we will use higher-order supersymmetric quantum mechanics to obtain several families of complex solutions g ( x ; a , b ) of the Painleve IV equation with real parameters a , b . We shall also study the algebraic structure, the eigenfunctions and the energy spectra of the corresponding non-hermitian Hamiltonians.

Wronskian differential formula for confluent supersymmetric quantum mechanics

Abstract A Wronskian differential formula, useful for applying the confluent second-order SUSY transformations to arbitrary potentials, will be obtained. This expression involves a parametric derivative with respect to the factorization energy which, in many cases, is simpler for calculations than the previously found integral equation. This alternative mechanism will be applied to the free particle and the single-gap Lame potential.

Simple function for intensity distribution from LEDs

Since its beginnings, light-emitting diode (LED) has progressed toward greater performance. Today, LEDs are everywhere, in many shapes, and with a wide range of radiation patterns. We propose a general analytic representation for the angular intensity distribution of the light emitted from an LED. The radiation pattern equation is determined by adding a Gaussian or a power cosine expression for contributions from the emitting surfaces (chip, chip arrays, or for some cases a phosphor surface), and the light redirected by the reflecting cup and the encapsulating lens. Mathematically, the pattern is described as a sum of Gaussian or of cosine-power functions. The resulting equation is widely applicable for any kind of LED of practical interest. We successfully model the radiation patterns from several manufacturer datasheets.

Hawking spectrum for a fiber-optical analog of the event horizon

Hawking radiation has been regarded as a more general phenomenon than in gravitational physics, in particular in laboratory analogs of the event horizon. Here we consider the fiber-optical analog of the event horizon, where intense light pulses in fibers establish horizons for probe light. Then, we calculate the Hawking spectrum in an experimentally realizable system. We found that the Hawking radiation is peaked around group-velocity horizons in which the speed of the pulse matches the group velocity of the probe light. The radiation nearly vanishes at the phase horizon where the speed of the pulse matches the phase velocity of light.

Solution Hierarchies for the Painlevé IV Equation

We will obtain real and complex solutions of the Painleve IV equation through supersymmetric quantum mechanics. The real solutions will be classified into several hierarchies, and a similar procedure will be followed for the complex solutions.

Complex solutions to the Painlevé IV equation through supersymmetric quantum mechanics

In this work, supersymmetric quantum mechanics will be used to obtain complex solutions to the Painleve IV equation with real parameters. We will also focus on the properties of the associated Hamiltonians, i.e. the algebraic structure, the eigenfunctions and the energy spectra

Linearized coherent states for Hamiltonian systems with two equidistant ladder spectra

A simple way to construct exactly solvable Hamiltonians whose spectra contain two equidistant ladders, one finite and another infinite, appears when applying supersymmetric quantum mechanics to the harmonic oscillator. Some of those supersymmetric partners have third order differential ladder operators, although the order of the transformation is higher than one. In this work the linearized coherent states for these specific Hamiltonians are studied. To each SUSY partner Hamiltonian corresponds two families of linearized coherent states: one inside the subspace associated with the isospectral part of the spectrum and another one in the finite subspace generated by the states inserted through the SUSY technique.