Kramers-Kronig Analysis of Relative Reflectance Spectra Measured at an Oblique Angle

Abstract

Relative specular reflectance R is defined as R = R(s)/R(w), where R(s) and R(w) are absolute reflectances of a sample material s and a material w for which the index of refraction n(w) and the extinction coefficient k(w) are known quantities. An algorithm was developed for computing n(s)and k(s) from the sample's R spectrum measured for radiant flux polarized perpendicular to the plane of incidence and reflected at oblique angle theta. Kramers-Kronig analysis of the R spectrum provides DeltaPhi the difference between phase shifts for electromagnetic waves reflected at the surfaces of materials s and w. Real and imaginary parts of a Fresnel equation for relative reflectivity provide equations for computing n(s) and k(s) when theta, DeltaPhi n(w), and k(w) are known quantities. Optical constants for aqueous solutions containing NaCl were computed in this manner; distilled water was the reflectance standard.

The optical constants of several atmospheric aerosol species: Ammonium sulfate, aluminum oxide, and sodium chloride

“…The relation permits both n and K to be found from a numerical integration of the measured reflectivity. The technique has been extended by Hale et al [ 1973] to include measurements of relative reflectance and reflectance at nonnormal incidence. Indeed, the Kramers-Kronig technique is quite general, and for example, a relation similar to (4b) can be written between n and K so that if K is known, n can be found.…”

Optical Properties of Natural Minerals and Other Materials in the 350-50,000 CM-1 Spectral Region.

“…It was found that the relative reflectance spectra for the X and Z directions contained a hump which centered for both at approximately 2600 cm 4 . The anomalous feature in the X spectrum was quite distinct and extended from roughly 2300 cm 1 to 2900 cm I .…”

“…Hale and Querry (1973) previously published values of the complex refractive index of water in the 200 mm to 200 pm wavelength range. Their investigation was based on a KK analysis of a k spectrum for water compiled from about 80 papers in the scientific literature.…”

“…Tam and Patel 7 gave values for a (v) for the interval 1.44xi04 to 2.24xi0 4 cm -(446 to 694 r wavelength) in a paper published in 1979. The technique used was pulsed dye-laser cptoacousic spectrosccpy.…”

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

“…In the case of reflection in a medium other than vacuum [81][82][83][84] or at nonnormal incidence [81,[84][85][86], the dispersion relations must be modified to account for more complex terms in the phase shift [46]. Dispersion relations [89] and sum rules [90] for the real and imaginary parts of the ellipsometric function have been developed. While this more complex situation has led to discussions of the validity of dispersion analysis [87,88], sum rules have not been investigated for these more general cases except in the case of ellipsometry.…”