NASA-CR-195140 Report KO: TRC-EM-CAB-9402 HIGH-FREQUENCY TECHNIQUES FOR RCS PREDICTION OF PLATE GEOMETRIES AND A PHYSICAL OPTICS/EQUIVALENT CURRENTS MODEL FOR THE RCS OF TRIHEDRAL CORNER REFLECTORS Semiannual Progress Report PART I Constantine A. Balanis and Lesley A. Polka August 1, 1993 - January 31, 1994 PART I1 Constantine A. Balanis and Anastasis C. Polycarpou August 1, 1993 - January 31, 1994 Telecommunications Research Center College of Engineering and Applied Science Arizona State University Tempe, AZ 85287-7206 Grant No. NAG-1-562 National Aeronautics and Space Administration Langley Research Center Hampton, VA 23665 Report No: TRC-EM-CAB-9402 HIGH-FREQUENCY TECHNIQUES FOR RCS PREDICTION OF PLATE GEOMETRIES AND A PHYSICAL OPTICS/EQUIVALENT CURRENTS MODEL FOR THE RCS OF TRIHEDRAL CORNER REFLECTORS Semiannual Progress Report PART I Constantine A. Balanis and Lesley A. Polka August 1, 1993 - January 31, 1994 PART I1 Constantine A. Balanis and Anastasis C. Polycarpou August 1, 1993 - January 31, 1994 Telecommunications Research Center College of Engineering and Applied Science Arizona State University Tempe, AZ 85287-7206 Grant No. NAG-1-562 National Aeronautics and Space Administration Langley Research Center Hampton, VA 23665 Abstract Part I of this report includes formulations for scattering from the coated plate and the coated dihedral corner reflector. A coated plate model based upon the Uniform Theory of Diffraction (UTD) for impedance wedges was presented in the last report. In order to resolve inaccura- cies and discontinuities in the predicted patterns using the UTD-based model, an improved model that uses more accurate diffraction coeffi- cients is presented in this report. A Physical Optics (PO) model for the coated dihedral corner reflector is presented as an intermediary step in developing a high-frequency model for this structure. The PO model is based upon the reflection coefficients for a metal-backed lossy material. Preliminary PO results for the dihedral corner reflector suggest that, in addition to being much faster computationally, this model may be more accurate than existing moment method (MM) models. Part I1 of this report presents an improved Physical Optics (PO) / Equivalent Currents model for modeling the Radar Cross Section (RCS) of both square and triangular, perfectly conducting, trihedral corner reflectors. The new model uses the PO approximation at each reflection for the first- and second-order reflection terms. For the third- order reflection terms, a Geometrical Optics (GO) approximation is used for the first reflection; and PO approximations are used for the remaining reflections. The previously reported model used GO for all reflections except the terminating reflection. Using PO for most of the reflections results in a computationally slower model because many in- tegrations must be performed numerically, but the advantage is that the predicted RCS using the new model is much more accurate. Com- parisons between the two PO models, Finite-Difference Time-Domain (FDTD) and experimental data are presented for validation of the new model. 1 I. HIGH-FREQUENCY TECHNIQUES FOR RCS PREDICTION OF PLATE GEOMETRIES A. INTRODUCTION Recent reports [l, 2, 3, 4, 5, 6, 7, 81 have dealt with the use of the Uniform Theory of Diffraction (UTD) for impedance wedges [9, 101 to model the principal-plane radar cross section (RCS) of a coated conducting plate. The initial goal was to apply the knowledge gained from modeling this simple structure to more compli- cated geometries, specifically the coated dihedral corner reflector. As the modeling process has evolved, however, the importance of modeling the plate has grown as a problem in and of itself, independent from the dihedral corner reflector problem. Specifically, the UTD plate model presented in the previous report [l]y ielded fairly good results near and at normal incidence; however, angles closer to the transi- tion regions, near grazing incidence, presented problems. The problems involved inaccuracies and discontinuities in the predicted patterns in the regions near the transition from the coated side to the uncoated side of the plate. The resolution of these problems requires the use of more accurate diffraction coefficients in these regions. Modifications to the model presented in the last report are discussed in this report. Because diffraction terms are the predominant contributors to the overall RCS of the coated plate, the formulation of an accurate high-frequency model for this geometry requires careful analysis of diffraction mechanisms and methods of modeling various diffraction mechanisms for coated structures. The information garnered from this analysis is useful for applications to other coated structures for which diffractions are the predominate contribution to the overall RCS; however, this analysis is not very useful for the analysis of the coated dihedral corner reflector because the main scattering mechanisms for this structure are single and double 2 Observation Figure 1: Impedance wedge geometry. reflections. In order to resolve inaccuracies in a previously reported UTD model for the coated dihedral corner reflector [ll],a Physical Optics (PO) model for this structure is presented in this report. Results are compared to Moment Method (MM) data. B. THEORY AND RESULTS - 1. Coated Plate UTD Analysis The UTD model for coated plate scattering presented in the previous report [l] used the UTD diffraction coefficients formulated by Tiberio, et al. , and Griesser and Balanis [9, 101 for an impedance wedge, shown in Fig. 1. The coated plate, shown in Fig. 2, was modeled as the joining of two half planes with a coating of finite thickness on the upper wedge faces. The coating was incorporated into the model using an equivalent impedance approximated by the impedance of a comparable short-circuited transmission-line. Results presented in the last report demonstrated that the UTD model formulated in this manner is very accurate for 3 Incident Y t + w + Figure 2: Geometry for principal-plane scattering from a strip/plate with a finite- thickness coating backed by a perfect conductor. most scattering angles. The necessity of incorporating higher-order diffraction and surface-wave terms was also demonstrated. Terms accounting for multiple diffractions between the edges improve the model; however, the way in which these terms were incorporated into the model presented in the last report leads to inaccuracies and discontinuities in the region of transition from the coated to the uncoated side of the plate. Specifically, the diffraction coefficient for the impedance wedge goes to zero when the source is on the face of the wedge and for the reciprocal case of an observation point on the wedge face. In order to use this coefficient for higher-order diffractions, the coefficient was calculated for a point slightly off the wedge face. This worked remarkably well for most scattering angles; however, improvement in the grazing regions is desired. To more effectively account for higher-order diffractions, a more precise coef- ficient that does not go to zero on the face of the wedge must be used. Tiberio, et al., formulated the necessary diffraction coefficient in [9]. The general form of the resulting diffracted field is given in Eq. (16) of [9]. This can be greatly simplified 4 for the case of a half plane [la] so that the resulting expression that will be included in a new version of the UTD model for the coated plate is: X $2 (2- 4 + $0) $2 (7- eo) $2 (a + 4 + 02) $2 ($ - 82) $2 (F - 4 - $0) $2 (7t eo) $2 (; + 4 - 02) $2 (F + e2) The usual definitions for the variables apply; Le., 00,2 are he Brewster angles for the designated faces, fl is the angle of incidence, 4 is the angle of observation, p is the distance between diffraction points, F[z]i s the Fresnel transition function extended to complex arguments as explained in [la], and f2(t) is the expression: + sin t - 2&sin(t/2) 2t T f2(t) = - (2) 8T cos t In general, fn(t)i s an infinite integral [9]. Fortunately, this integral reduces to closed form for the case of a half plane (n = 2) and is given above as fZ(t). This updated version of the UTD model will be coded in the next reporting period and numerical results obtained and compared to measured data. Another modification that will be explored is the use of the diffraction coefficients reported in [13, 141. These coefficients are for cylindrical-wave incidence and should model interactions between edges more accurately than the previously used coefficients, which are theoretically only for plane-wave incidence on a wedge. 5 L b L 0b servation \ source Figure 3: Dihedral corner reflector geometry. 2. Coated Dihedral Corner Reflector The coated dihedral corner reflector, shown in Fig. 3, is an important structure to analyze because it supports most of the basic scattering mechanisms. Specifically, mechanisms which must be included in a high-frequency RCS model are first-order diffractions for both exterior and interior wedges; single, double, and triple reflec- tions; and reflection-diffraction terms. The most logical approach to formulating a high-frequency model for this geometry is to combine UTD, to account for diffrac- tions, and Geometrical Optics (GO), to account for reflections. This model was formulated and reported on by Griesser, et al., in [Ill; however, since the appear- ance of this paper, inaccuracies due to the reflection terms have been discovered [7]. In order to isolate the source of the inaccuracy, a PO model for the reflector 6 is examined in this report. Although the PO model cannot account for diffraction mechanisms, it is a good model to use to study the reflector because reflection terms dominate the scattering pattern of this geometry. A brief summary of the model is given in the next section, followed by a results section, which includes MM data for comparison. In addition to being a computationally intensive model, the MM model is also highly inaccurate at some points. These difficulties with the MM model further emphasize the need for an accurate high-frequency model for the coated dihedral corner reflector. PO Analysis: The PO model for the coated reflector is based upon the PO model for the perfectly conducting reflector, reported upon in [15]. Obtaining re- sults for the coated reflector simply involves multiplying the fields for the perfectly conducting geometry by the appropriate reflection coefficients. The reflection co- efficient for a coated, flat plane backed by a perfect conductor is used as the fundamental reflection coefficient. The short-circuited transmission-line approxi- mation is used to account for the coating impedance. The expressions for the basic reflection coefficients are, thus, given by: Soft Polarization Hard Polarization r]eq\lw (4) \ , + cos 8 PCCC where The angle of incidence with respect to the surface normal is 8; pc and cc are the relative permeability and permittivity, respectively, of the coating material; and t is the coating thickness in free-space wavelengths. 7 Coefficients for multiple reflections are formed as a product of this basic co- efficient. To obtain the appropriate reflection terms for each plate of the reflector, the incident angles measured from the normal to each plate must be known at each reflection. The reflection coefficients are a product of the basic coefficient from above evaluated at the appropriate angles. The following table summarizes the reflection coefficients for single, double, and triple reflections from both plates of the reflector. Referring to Fig. 3, the left-hand plate is Plate I and the right- hand plate is Plate 11. The angle of incidence with respect to the given coordinate system is 4; and a is half of the total interior angle between the plates of the reflector. Angular Range Angle to the Normal Reflection Coefficent Plate I - Single Reflection 1 Plate I - Double Reflect ion I I Reflection Reflection I I I I Reflection 8