In this paper the Laguerre-Gaussian (LG) series representation of the twodimensional fractional Fourier transform is derived from conventional ordinary Fourier transform in polar coordinates. The kernel of this series representation is constituted by Laguerre-Gaussian functions, from which the series representation of a fractional Hankel transform can be easily derived. The connection between the gradient-index medium and the LG series representation is illustrated as an example of its applications.

We present a careful study of the energetics of vacancy and substitutional impurities in aluminum in both the bulk and small cluster environments. The calculations are done within the framework of the local-densityfunctional formalism and are based on the pseudopotential method with plane-wave expansion and periodic boundary conditions. Both the ionic and electronic degrees of freedom are fully relaxed. The electronic structure problem is treated with a preconditioned conjugate-gradient method that applies equally well to insulators and metals, and is suitable for parallel computing. We have considered up to 216 atoms in the supercell, and we show that reliable results can be obtained with 108-atom cells with proper k-point sampling. Vacancy-formation energy, heats of solution of the impurities and the relaxations near the defects are in good agreement with available experimental data. The energetics of substitution in small clusters was found to be rather different from bulk.

We present a local density functional calculation of the effect of an external electric field on the work function change of Pd and Ag adsorption on an Al(001) surface. The adsorption of K has also been considered for comparison. We found that the work functions for all the systems increased linearly when the strength of the external electric field was increased. Since the polarized dlectrons at the interstitial regions between the adsorbate and substrate for Pd/Al(001) and Ag/Al(001) react to the external electric field differently, the subtle differences between Pd/Al(001 and Ag/Al(001) bondings has been characterized through the comparison of the slopes of the work function change versus electric field.

We demonstrate that the propagation of a light wave in an elliptic gradient-index medium can be regarded as a joint fractional Fourier transform. The distinctive features of the Wigner distributon function and the recurrence of the original function are analyzed qualitatively. q1998 Elsevier Science B.V. All rights reserved.

The trends of the conduction band minima with the alloy compositions for the pseudomorphic Si1-x-yGexCy alloys grown on (110) Si and Ge substrates are investigated theoretically with the use of the ab initio pseudopotential method and the virtualcrystal approximation. It is found that the minimum energy gaps are all indirect and not monotonically dependent on the compositions in some cases. In the case of the tensile and compressive strains, the minimum energy gap decreases and increases, respectively, as the C fraction increases and the Ge fraction is constant. Our result is in good agreement with other theoretical results for the Si12xGex system.