An estimate of the error caused by the plane-wave approximation in free-space dielectric measurement systems

In a free-space measurement system, the electrical constitutive parameters of a plane sheet of material are determined using incident, transmitted and reflected electromagnetic beams. The electrical parameters are almost always obtained from the measured transmission and reflection coefficients using a procedure that approximates each beam as a single plane wave whose wavefront is parallel to the surfaces of the sheet (the plane-wave approximation). A theoretical model is constructed to investigate, in a very general way, the limitations of the plane-wave approximation. The field of the incident beam is taken to be a Gaussian function at the center of the sheet and analytic expressions are obtained for the transmitted and reflected beams using the plane-wave spectrum technique. Power transmission and reflection coefficients are calculated and series expansions of these coefficients are used to determine under what conditions the plane-wave approximation applies. Graphical results are presented to show how the error associated with the plane-wave approximation depends on the various parameters for the problem: electrical size of the beam waist, electrical thickness of the sheet etc