An electromagnetic formulation for treating optical reflections from graded-material surfaces

Abstract

The reflection of a monochromatic plane wave falling obliquely upon the surface of an arbitrary, flat, depth-dependent material is investigated theoretically. The complex reflection coefficient for either principal (s or p) polarization of the field is shown to satisfy a non-linear differential equation of the Ricatti type. An alternate formulation based on the wave immittance (i.e., jmpedance or a dmittance) functions is also presented. The immittance functions are shown to satisfy Ricatti differential equations of their own. The reflection coefficient formulation and the wave immittance formulations are related via a bilinear algebraic transformation. Singularities appearing in the reflection coefficient formulation may be suppressed in the immittance formulation, and viceversa. The advantage of either formulation is that the reflection coefficients for an arbitrary, depth-dependent medium can be obtained directly without having to solve Maxwell's equations for the internal field configurations.

Metrological consequences of the hard optical boundary assumption

“…For example, with the appropriate modification of eqs. (4-5) and (4)(5)(6) for k(z) and T(z) [4] the effects of oxide and contaminant films on optical reference surfaces can also be explored.…”

“…The electron and ion densities at a vacuum-metal interface are graphed in figure 4 from data provided in the original paper by Lang and Kohn [5]. The electron density profile presented in this figure corresponds to the metal sodium which has a Wigner-Seitz radius [6] is calculated as the difference between the phase shift determined for the Lang-Kohn model and the phase shift determined for a step transition (i.e., hard boundary) corresponding the same interior electron density.…”

“…A rigorous theoretical proof is given in reference [4]. N is the electron density, and m is the mass of an electron.…”

“…(3)(4)(5)(6) and (3)(4)(5)(6)(7) for the transmitted fields and eq . for the permittivity function, become R exp…”

“…The phase shift in the electric field upon its reflection at the surface In a companion paper [4] a differential formulation was described for the s-polarized plane wave reflection coefficient R(0) at the surface to an arbitrary vacuum-material transition. For present purposes the medium is a jellium metal having a specified, depth-dependent plasma The depth z g , at which eq .…”