Several numerical integration methods are compared in order to search out the most effective method for the Kramers-Kronig transformation, using the analytical formula of the Kramers-Kronig transformation of a Lorentzian function as a reference. The methods to be compared involve the use of (1) Maclaurin's formula, (2) trapezium formula, (3) Simpson's formula, and (4) successive double Fourier transform methods. It is found that Maclaurin's formula, in which no special approximation is necessary for the pole part of the integration, gives the most accurate results, and also that its computation time is short. Successive Fourier transform is less accurate than the other methods, but it takes the least time when used without zero-filling. These results have important relevance for programs used to obtain optical constant spectra and to analyze spectral data.

Comparison Among Several Numerical Integration Methods for Kramers-Kronig Transformation

The CIRCLE ATR accessory has been used to measure the optical and dielectric constants of organic liquids and water. The method, based on Fresnel's equations, is described in detail, and the agreement between the results obtained and literature values is shown to be adequate for chemical use. The utility of optical and dielectric constants for the calculation of traditional infrared intensities in liquids and of dipole moment derivatives is outlined.

Infrared Intensities of Liquids I: Determination of Infrared Optical and Dielectric Constants by FT-IR Using the CIRCLE ATR Cell:

The basic optical properties of solids [1, 2], despite their great diversity, are rigorously limited by nature. These limitations take the form of sum rules and dispersion relations that reflect the physical laws governing the dynamics of matter and its interaction with light. Many of these limitations were enumerated in the first half of this century, but in the early 1970s, the discovery of a large number of new sum rules [3-15] deepened our understanding of these constraints and their application. Thus, it is now known that in addition to the celebrated f sum rule for absorption processes, there are companion sum rules for dispersive processes. For example, the real part of the complex refractive index N(ω) = n(ω) + ik(ω) satisfies [3, 4] ∫ 0 ∞ [ n ( ω ) − 1 ] d ω = 0 (1). That is, the refractive index averaged over all frequencies must be unity. Moreover, it may be shown that this restriction arises in part from the inertial property of matter. Related rules—with a similar physical interpretation—have been shown to apply to the complex dielectric function and its inverse.

Dispersion Theory, Sum Rules, and Their Application to the Analysis of Optical Data

A method is presented to deal with the numerical evaluation of Kramers–Kronig transforms (the Hilbert transforms of even and odd functions on the positive real axis). The general Hilbert transform is also treated. The functions involved must be continuous on the integration interval with suitable asymptotic behavior for large values of the argument and must have an appropriate functional form in the vicinity of the singularity of the integrand of the transform. The approach is based on a specialized Gaussian quadrature technique that uses the weight function log x-1. This choice allows the region in the vicinity of the singularity to be swept into the quadrature weights and abscissa values. Application to the Lorentzian and Gaussian line profiles is discussed.

Efficient numerical approach to the evaluation of Kramers–Kronig transforms

A general method based upon reflectance and transmittance measurements in the infrared region has been developed for the determination of the optical constants n(v) and k(v) of thin films deposited on any substrate (transparent or not). The corresponding computer program, written in FORTRAN 77, involves three main parts: (1) a matrix formalism to compute reflection and transmission coefficients of multilayered systems; (2) an iterative Newton-Raphson method to estimate the optical constants by comparison of the calculated and experimental values; and (3) a fast Kramers-Kronig transform to improve the accuracy of calculating the refractive index. The first part of this program can be used independently to simulate reflection and transmission spectra of any multilayered system using various experimental conditions. Two practical examples are given for illustration. Simulation of reflection spectra at grazing incidence for thin films deposited on a metal surface and determination of the optical constants for thin CaF2 layers deposited on a silicon substrate are presented.

Thin-Film Optical Constants Determined from Infrared Reflectance and Transmittance Measurements

A scheme based on conjugate Fourier series is proposed for the determination of the dissipative mode of an optical constant from the corresponding dispersive mode, and vice versa. The connection between the conjugate Fourier series and Fourier integral method is discussed. The advantages of the method in relation to the Kramers-Kronig approach are outlined.

Federick W. King

Papers

Analysis of optical data by the conjugate Fourier-series approach