International Scholarly Research Notices

Volume 2015 (2015), Article ID 657254, 9 pages

http://dx.doi.org/10.1155/2015/657254

## Effects of Curved Wavefronts on Conductor-Backed Reflection-Only Free-Space Material Characterization Techniques

^{1}Loyola University Maryland, Baltimore, MD 21210, USA^{2}Michigan State University, East Lansing, MI 48824, USA

Received 8 March 2015; Accepted 6 July 2015

Academic Editor: Vincent Ji

Copyright © 2015 Raenita A. Fenner and Edward J. Rothwell. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A true plane wave is often not physically realizable in a laboratory environment. Therefore, wavefront curvature introduces a form of systematic error into Free-space material characterization methods. Free-space material characterization is important to the determination of the electric permittivity and magnetic permeability of conductor-backed and in situ materials. This paper performs an error analysis of the impact on wavefront curvature on a Free-space method called the *two-thickness* method. This paper compares the extracted electric and magnetic permeability computed with a plane wave versus a line source for a low-loss dielectric and magnetic radar absorbing material. These steps are conducted for TE and TM plane waves and electric and magnetic line sources.

#### 1. Introduction

Error analysis is the study and evaluation of uncertainty in measurement [1]. The determination of the amount of propagated error is an essential step for any experimentally determined quantity. Most error analysis is performed to establish the impact of* random error* on an experimentally determined quantity. Random error is inherently present in all measurements and is due to unpredictable fluctuations in reading of measurement apparatus. Error analysis can also be performed to establish the impact of* systematic error*. Systematic error is due to reproducible inaccuracies in the measurement procedure.

Material characterization is the process of determining the relative magnetic permeability, , and electric permittivity, , of a material, where and . Accurate knowledge of and is necessary for many applications which include dielectric resonator antennas [2] and the design of RFID tag antennas [3]. There are many different material characterization methods. One category of material characterization methods is free-space methods. Free-space methods utilize reflection and transmission data from plane wave illumination to determine and [4]. However, for many applications transmission data cannot be obtained; an example application is for a conductor-backed material sample. When transmission data cannot be obtained, reflection-only free-space methods are required.

A plane wave is defined as a constant-frequency wave whose wavefronts (surfaces of constant phase) are infinite parallel planes of constant amplitude normal to the phase velocity vector [5]. Since in practice a true plane wave cannot be produced, the assumption of plane wave incidence used in free-space methods introduces a source of systematic error to these methods.

The purpose of this paper is to investigate the effect of the plane wave assumption on reflection-only free-space methods. Work has been done in [6] to quantify the impact of the plane wave assumption for free-space methods which use both reflection and transmission, but work has not been done to address the impact of wave curvature on conductor-backed reflection-only free methods. In this paper, the* two-thickness method* is used as a sample test method. To examine the extent of this form of systematic error, the reflection coefficient is calculated due to a nonplanar incident-field wavefront and used in the extraction scheme in the exact manner as the reflection coefficient due to a plane wave. The extracted and computed using the nonplanar incident-field wavefront reflection coefficients and plane wave reflection coefficients are then compared.

The reflection coefficient due to a 2-dimensional curved wavefront is determined from examination of the canonical problem of a line source above a material slab. Past similar investigations include examination of a two-dimensional problem of a conducting cylinder with a uniform material coating in [7] and examination of whether the late-time component of the field reflected by a planar slab under line-source illumination can be characterized by a natural mode series [8]. In this work, the scattered electric field produced by both electric and magnetic line sources above a conductor-backed material under test (MUT) is computed via the Sommerfeld integral approach and used to determine a reflection coefficient for the case of a nonplanar incident-field wavefront by maintaining Snell’s law of reflection. By varying the height of the line source and the MUT thickness, the effect of wavefront curvature on the accuracy of the extracted material properties can be explored for several types of materials.

#### 2. Reflection Coefficients due to Electric and Magnetic Line Sources

The analysis necessary for this paper requires derivation of the reflection coefficient due to electric and magnetic line sources. The reflection coefficients for both electric and magnetic line sources are required in order to emulate both perpendicular and parallel polarization of a plane wave. To derive the reflection coefficients, the scattered electric field due the line sources is initially computed.

For simplicity, the derivation steps are listed in detail in Section 2.1 for the electric line source only. The derivation steps for the magnetic line source are similar and are determined by duality; the derivation for the magnetic line source is summarized in Section 2.2.

##### 2.1. Electric Line Source

The scattered electric field produced by an electric line source above a conductor-backed MUT is computed via the Sommerfeld integral approach. Figure 1 depicts the arrangement of the electric line source and MUT used for calculating the scattered electric field.