Increase of refractive index induced by absorbing centres

An analytical expression of the complex permittivity is derived for absorbing centres featuring inhomogeneous absorption-line broadening. Such an expression gives the dispersion law of the real part of the permittivity when the imaginary part has a Gaussian lineshape. Our mathematical approach starts from an overlap integral of Lorentzian-type dielectric susceptibilities weighted by a Gaussian probability distribution of the resonance absorption energies. The analytical solution found is consistent with the Kramers–Kronig relation. We demonstrate that, like in the case of homogeneous absorption-line broadening, the refractive index increases at photon energies lower than the resonance absorption energy also for inhomogeneous absorption-line broadening; if the absorbing centres emit Stokes-shifted radiation, such an increase can be exploited for passive and active waveguiding applications. An example is reported regarding active waveguides based on colour centres in a lithium fluoride crystal.     

Optical reflectance and transmittance of a multilayer coating affected by refractive-index inhomogeneity,interface roughness,and thickness wedge

A theoretical model to evaluate normal-incidence specular, direct, hemispherical, and diffuse reflectance and transmittance of a single layer affected by refractive-index inhomogeneity along the growth axis, roughness, and thickness wedge was introduced in recent papers. In the present work, the model is extended to the case of a multilayer optical coating. To that purpose, suitable transfer matrices that account for layer-index inhomogeneity and random deviations of the interfaces from smooth planes are introduced, and statistical averages are applied to the resulting amplitude and intensity coefficients. Eventually, thickness wedge is included as a deterministic average on the intensity coefficients.     

Commutative hypercomplex numbers and functions of hypercomplex variable: a matrix study

Systems of hypercomplex numbers, which had been studied and developed at the end of the 19^th century, are nowadays quite unknown to the scientific community. It is believed that study of their applications ended just before one of the fundamental discoveries of the 20^th century, Einstein’s equivalence between space and time. Owing to this equivalence, not-defined quadratic forms have got concrete physical meaning and have been recently recognized to be in strong relationship with a system of bidimensional hypercomplex numbers. These numbers (called hyperbolic) can be considered as the most suitable mathematic language for describing the bidimensional space-time, in spite of some unfamiliar algebraic properties common to all the commutative hypercomplex systems with more than two dimensions: they are decomposable systems and there are non-zero numbers whose product is zero. With respect to the famous Hamilton quaternions, one can introduce the differential calculus for the hyperbolic numbers and for all the commutative hypercomplex systems; moreover, one can even introduce functions of hypercomplex variable. The aim of this work is to study the systems of commutative hypercomplex numbers and the functions of hypercomplex variable by describing them in terms of a familiar mathematical tool, i.e. matrix algebra.     

Effects of F2 Color Center Spatial Distribution on the Emission Properties of Optical Microcavities Based on LiF Films

Highly stable F₂ color centers are very efficiently produced in lithium fluoride (LiF) by electron beam irradiation at room temperature. We have fabricated optical microcavities in which the active medium is a low-energy electron beam irradiated LiF film, whose optical thickness is comparable with the peak wavelength (～668nm) of the F₂ broad photoluminescence band. By selecting the proper electron beam energy, one can control the F₂ color center depth distribution. This distribution influences the photoemission angular distribution of the microcavity, whose resonance properties are determined by the coupling of the depth profile of the defects with the pump electromagnetic field and microcavity modes.     

The parabolic analytic functions and the derivative of real functions

Among the bidimensional hypercomplex-number systems defined as the parabolic (dual) numbers are introduced with the rule α = 0. As well as the functions of a complex variable, the analytic functions of a parabolic variable can be introduced as analytic continuation of the real functions of a real variable. These functions hold the property that the “imaginary” part is linked to the derivative of the “real” part. In this paper we will show how this property allows one to demonstrate in an algebraic way some rules of the differential calculus for the real functions of a real variable.     

Cooperative optical effects in volumes embedded in layered media

A cluster of point sources can generate optical radiation in a manner substantially different from what characterizes the emission of a single point source. Such differences are mainly caused by the cooperation of the sources and are even more remarkable under particular electromagnetic boundary conditions. Furthermore, the geometry of the problem cannot be ignored as it makes an important contribution to the contrast between single source and collective behaviour. This paper tries to explore the subject in view of its applications to coherent scattering that is typical of non‐linear Raman processes. To this end, the classical theories of electromagnetic radiation from a point source (treated as a randomly oriented Hertzian dipole) and of partial coherence are joined into a unified formalism to evaluate light emission from a volume seen as a collection of point sources embedded in a layered medium. In certain reasonable circumstances and beyond the undeniable complexity of the problem, the formalism leads to the relevant advantage of analytical results for the power radiated outside the medium. In other more general cases, fast Fourier transforms can be used in principle. The possibility and convenience of using the theory to model micro‐CARS imaging are illustrated and discussed. Copyright © 2010 John Wiley & Sons, Ltd.