Optical constants at complex energies: local deconvolution

Juan I. Larruquert and Luis V. Rodríguez de Marcos

Abstract

Causality implies that the optical constants of any material continue in the upper complex plane of photon energies or wavelengths as an analytic function. This is the basis for Kramers-Kronig dispersion relations to obtain e1 from e2, or n from k. However, there have not been attempts to explore this continuation. This research focuses on such continuation and on applications thereof. An interesting property has been found: optical constants progressively smoothen when entering the upper complex plane. The continuation to complex energies is found to result in an average of the optical constants with a Lorentzian weight function.

This optical-constant smoothening originated in a shift to the upper complex plane is naturally produced in optical constants that have been obtained by means of an optical instrument with a Lorentzian slit function. This smoothening results in reduced resolution through convolution with the slit function. A procedure that takes advantage of optical constants at complex energies is developed for optical-constant deconvolution. Deconvolution is performed locally, i.e., with no integration, and it consists in shifting the energy of the optical constants by an imaginary amount by means of a Taylor series expansion. The first correction term involves the derivative of the other optical constant. Even though deconvolution of optical constants measured with a Gaussian slit cannot be directly performed with the present method, an approach based on powers of the Lorentz function is also proposed. This procedure could be implemented as an analysis tool of a spectrophotometer or an ellipsometer; this tool would enable one to measure optical constants with a modest resolution and to improve it by post—processing them with the present scheme.