Simulation of Ultrafast Optical Transitions using Genetic Algorithm

The purpose of this paper is to investigate the possibility of application of the genetic algorithm to quantum control of electronic transitions between energy bands in solids. In particular, the hole transitions between valence bands induced by ultrashort (femtosecond duration) electric eld pulse will be considered. Examples are presented to illustrate the effciency of the algorithm in this case

Theoretical analysis of electronic band structure of 2-to-3-nm Si nanocrystals

We introduce a general method which allows reconstruction of electronic band structure of nanocrystals from ordinary real-space electronic structure calculations. A comprehensive study of band structure of a realistic nanocrystal is given including full geometric and electronic relaxation with the surface passivating groups. In particular, we combine this method with large scale density functional theory calculations to obtain insight into the luminescence properties of silicon nanocrystals of up to 3 nm in size depending on the surface passivation and geometric distortion. We conclude that the band structure concept is applicable to silicon nanocrystals with diameter larger than $\approx$ 2 nm with certain limitations. We also show how perturbations due to polarized surface groups or geometric distortion can lead to considerable moderation of momentum space selection rules

Square root of a multivector in 3D Clifford algebras

The problem of square root of multivector (MV) in real 3D (n = 3) Clifford algebras Cl3;0, Cl2;1, Cl1;2 and Cl0;3 is considered. It is shown that the square root of general 3D MV can be extracted in radicals. Also, the article presents basis-free roots of MV grades such as scalars, vectors, bivectors, pseudoscalars and their combinations, which may be useful in applied Clifford algebras. It is shown that in mentioned Clifford algebras, there appear isolated square roots and continuum of roots on hypersurfaces (infinitely many roots). Possible numerical methods to extract square root from the MV are discussed too. As an illustration, the Riccati equation formulated in terms of Clifford algebra is solved.&nbsp

Exponential and logarithm of multivector in low-dimensional (n = p + q < 3) Clifford algebras

The aim of the paper is to give a uniform picture of complex, hyperbolic, and quaternion algebras from a perspective of the applied Clifford geometric algebra. Closed form expressions for a multivector exponential and logarithm are presented in real geometric algebras Clp;q when n = p + q = 1 (complex and hyperbolic numbers) and n = 2 (Hamilton, split, and conectorine quaternions). Starting from Cl0;1 and Cl1;0 algebras wherein square of a basis vector is either –1 or +1, we have generalized exponential and logarithm formulas to 2D quaternionic algebras Cl0;2, Cl1;1, and Cl2;0. The sectors in the multivector coefficient space, where 2D logarithm exists are found. They are related with a square root of the multivector

Spatial homogeneity of optically switched semiconductor photonic crystals and of bulk semiconductors

This paper discusses free carrier generation by pulsed laser fields as a mechanism to switch the optical properties of semiconductor photonic crystals and bulk semiconductors on an ultrafast time scale. Requirements are set for the switching magnitude, the time-scale, the induced absorption as well as the spatial homogeneity, in particular for silicon at lambda= 1550 nm. Using a nonlinear absorption model, we calculate carrier depth profiles and define a homogeneity length l_hom. Homogeneity length contours are visualized in a plane spanned by the linear and two-photon absorption coefficients. Such a generalized homogeneity plot allows us to find optimum switching conditions at pump frequencies near v/c= 5000 cm^{-1} (lambda= 2000 nm). We discuss the effect of scattering in photonic crystals on the homogeneity. We experimentally demonstrate a 10% refractive index switch in bulk silicon within 230 fs with a lateral homogeneity of more than 30 micrometers. Our results are relevant for switching of modulators in absence of photonic crystals

Exponentials of general multivector in 3D Clifford algebras

Closed form expressions to calculate the exponential of a general multivector (MV) in Clifford geometric algebras (GAs) Clp;q are presented for n = p + q = 3. The obtained exponential formulas were applied to find exact GA trigonometric and hyperbolic functions of MV argument. We have verified that the presented exact formulas are in accord with series expansion of MV hyperbolic and trigonometric functions. The exponentials may be applied to solve GA differential equations, in signal and image processing, automatic control and robotics