Many larger-scale electromagnetic problems deal with the modeling of radar scattering from large metallic structures (targets like aircraft or vehicles) or the radiation of antennas in the presence of large scatterer platforms. Although a full-wave analysis of such open-boundary computational problems using the method of moments (MoM) is conceptually feasible, it may not be practical due to the enormous memory requirements for storage of the resulting moment matrices. To solve this class of problems, you may instead pursue asymptotic electromagnetic analysis methods.
Asymptotic methods are usually valid at high frequencies as k<sub>0</submath>k_0 R = 2pR2\pi R/?<sub\lambda_0 >0 > 1</submath>>> 1, where R is the distance between the source and observation points, k<sub>0 </sub>is the free-space propagation constant and ?λ<sub>0 </sub>is the free-space wavelength. Under such conditions, electromagnetic fields and waves start to behave more like optical fields and waves. Asymptotic methods are typically inspired by optical analysis. Two important examples of asymptotic methods are the Shoot-and-Bounce-Rays (SBR) method and Physical Optics (PO). The SBR method, which is featured in EM.Cube's [[Propagation Module]], is a ray tracing method based on Geometrical Optics (GO). An SBR analysis starts by shooting a number of ray tubes (or beams) off a source. It then traces all the rays as they propagate in the scene or bounce off the surface of obstructing scatterers. The uniform theory of diffraction (UTD) is used to model the diffraction of rays at the edges of the structure.
In the Physical Optics (PO) method, a scatterer surface is illuminated by an incident source, and it is modeled by equivalent electric and magnetic surface currents. This concept is based on the fundamental equivalence theorem of electromagnetics and the Huygens principle. The electric surface currents are denoted by '''J(r)''' and the magnetic surface currents are denoted by '''M(r)''', where '''r''' is the position vector. According to the Huygens principle, the equivalent electric and magnetic surface currents are derived from the tangential components of magnetic and electric fields on a given surface, respectively. This will be discussed in more detail in the next sections. In a classic PO analysis which involves only perfect electric conductors, only electric surface currents, related to the tangential magnetic fields, are considered.
[[File:PO2.png]]
where ?<submath>0\eta_0 = 120\pi \; \Omega</submath> = 120p O is the intrinsic impedance of the free space. Then, the electric and magnetic currents reduce to:
[[File:PO3.png]]
Two limiting cases of an impedance surface are perfect electric conductor (PEC) and perfect magnetic conductor (PMC) surface. For a PEC surface, Z = 0, a α = 1, and one can write:
[[File:PO4.png]]
while for a PMC surface, Z = 8, a α = -1, and one can write:
[[File:PO5.png]]
Another special case is a Huygens surface with equivalent electric and magnetic surface currents. In that case, Z = ?η<sub>0</sub>, a α = 0, and one can write:
[[File:PO10.png]]
[[File:PO6.png]]
where ''''''G<sub>EJ</sub>''', '''G<sub>EM</sub>''', '''G<sub>HJ</sub>''', '''G<sub>HM</sub>'''''' are the dyadic Green's functions of electric and magnetic fields due to electric and magnetic currents, respectively. In EM.Cube's PO Module, the background structure is the free space. Therefore, all these dyadic Green's functions reduce to the simple free-space Green's function of the form <math>\exp(-jk<sub>0</sub>rjk_0r)/(4pr4\pi r) </math> and the near fields reduce to:
[[File:PO7.png]]
where R = | '''r'''-'''r''''|, k<sub>0</sub> = 2p2π/?λ<sub>0</sub> and Z<sub>0</sub> = 1/Y<sub>0</sub> = ?η<sub>0</sub>.
When k<sub>0</sub>r >> 1, i.e. in the far-zone field of the scatterer, one can use the asymptotic form of the Green's functions and evaluate the radiation integrals using the stationary phase method to obtain far-field expressions for the electric and magnetic fields as follows:
[[File:PO16.png]]
where the summation over index ''j'' is carried out for all the elementary cells ?Δ<sub>j</sub> that make up the Huygens box. In EM.Cube Huygens surfaces are cubic and are discretized using a rectangular mesh. Therefore, ?Δ<sub>j</sub> represents any rectangular cell located on one of the six faces of Huygens box. Note that the calculated near-zone electric and magnetic fields act as incident fields for the scatterers in your PO Module project. The Huygens source data are normally generated in one of EM.Cube's full-wave computational modules like FDTD, Planar or MoM3D. Keep in mind that the fields scattered (or reradiated) by your physical structure do not affect the fields inside the Huygens source.
The far fields of the Huygens surface currents are calculated from the following relations:
# Verifying the mesh.
The objects of your physical structure are meshed based on a specified mesh density expressed in cells/?λ<sub>0</sub>. The default mesh density is 20 cells/?λ<sub>0</sub>. To view the PO mesh, click on the [[File:mesh_tool_tn.png]] button of the '''Simulate Toolbar''' or select '''Menu > Simulate > Discretization > Show Mesh''' or use the keyboard shortcut '''Ctrl+M'''. When the PO mesh is displayed in the project workspace, EM.Cube's mesh view mode is enabled. In this mode, you can perform view operations like rotate view, pan, zoom, etc. However, you cannot select or move or edit objects. While the mesh view is enabled, the '''Show Mesh''' [[File:mesh_tool.png]] button remains depressed. To get back to the normal view or select mode, click this button one more time, or deselect '''Menu > Simulate > Discretization > Show Mesh''' to remove its check mark or simply click the '''Esc Key''' of the keyboard.
"Show Mesh" generates a new mesh and displays it if there is none in the memory, or it simply displays an existing mesh in the memory. This is a useful feature because generating a PO mesh may take a long time depending on the complexity and size of objects. If you change the structure or alter the mesh settings, a new mesh is always generated. You can ignore the mesh in the memory and force EM.Cube to generate a mesh from the ground up by selecting '''Menu > Simulate > Discretization > Regenerate Mesh''' or by right clicking on the '''3-D Mesh''' item of the Navigation Tree and selecting '''Regenerate''' from the contextual menu.
* RCPz
The direction of incidence is defined through the ? θ and f φ angles of the unit propagation vector in the spherical coordinate system. The values of these angles are set in degrees in the boxes labeled '''Theta''' and '''Phi'''. The default values are ? θ = 180° and f φ = 0° representing a normally incident plane wave propagating along the -Z direction with a +X-polarized E-vector. In the TM<sub>z</sub> and TE<sub>z</sub> polarization cases, the magnetic and electric fields are parallel to the XY plane, respectively. The components of the unit propagation vector and normalized E- and H-field vectors are displayed in the dialog. In the more general case of custom linear polarization, besides the incidence angles, you have to enter the components of the unit electric '''Field Vector'''. However, two requirements must be satisfied: '''ê . ê''' = 1 and '''ê à k''' = 0 . This can be enforced using the '''Validate''' button at the bottom of the dialog. If these conditions are not met, an error message is generated. The left-hand (LCP) and right-hand (RCP) circular polarization cases are restricted to normal incidences only (?θ = 180°).
To define a plane wave source follow these steps:
Unlike the FDTD method, Physical Optics is an open-boundary technique. You do not need a far field box to perform near-to-far-field transformations. Nonetheless, you still need to define a far field observable if you want to plot radiation patterns. A far field can be defined by right clicking on the '''Far Fields''' item in the '''Observables''' section of the Navigation Tree and selecting '''Insert New Radiation Pattern...''' from the contextual menu. The Radiation Pattern dialog opens up. You can accept most of the default settings in this dialog. The Output Settings section allows you to change the '''Angle Increment''' in the degrees, which sets the resolution of far field calculations. The default value is 5 degrees. After closing the radiation pattern dialog, a far field entry immediately appears with its given name under the '''Far Fields''' item of the Navigation Tree.
After a PO simulation is finished, three radiation patterns plots are added to the far field node in the Navigation Tree. These are the far field component in ? θ direction, the far field component in f φ direction and the total far field defines as:
[[File:FDTD129.png]]
[[File:FDTD130.png]]
Three RCS quantities are computed: the ? θ and f φ components of the radar cross section as well as the total radar cross section, which are dented by sσ<sub>?θ</sub>, sσ<sub>fφ</sub>, and sσ<sub>tot</sub>. In addition, EM.Cube's PO Module calculates two types of RCS for each structure: '''Bi-Static RCS''' and '''Mono-Static RCS'''. In bi-static RCS, the structure is illuminated by a plane wave at incidence angles ?θ<sub>0</sub> and fφ<sub>0</sub>, and the RCS is measured and plotted at all ? θ and f φ angles. In mono-static RCS, the structure is illuminated by a plane wave at incidence angles ?θ<sub>0</sub> and fφ<sub>0</sub>, and the RCS is measured and plotted at the echo angles 180°-?θ<sub>0</sub>; and fφ<sub>0</sub>. It is clear that in the case of mono-static RCS, the PO simulation engine runs an internal angular sweep, whereby the values of the plane wave incidence angles ? θ and f φ are varied over the entire intervals [0°, 180°] and [0°, 360°], respectively, and the backscatter RCS is recorded.
To calculate RCS, first you have to define an RCS observable instead of a radiation pattern. Right click on the '''Far Fields''' item in the '''Observables''' section of the Navigation Tree and select '''Insert New RCS...''' to open the Radar Cross Section Dialog. Use the '''Label''' box to change the name of the far field or change the color of the far field box using the '''Color''' button. Select the type of RCS from the two radio buttons labeled '''Bi-Static RCS''' and '''Mono-Static RCS'''. The former is the default choice. The resolution of RCS calculation is specified by '''Angle Increment''' expressed in degrees. By default, the ? θ and f φ angles are incremented by 5 degrees. At the end of a PO simulation, besides calculating the RCS data over the entire (spherical) 3D space, a number of 2D RCS graphs are also generated. These are RCS cuts at certain planes, which include the three principal XY, YZ and ZX planes plus one additional constant f-cut. This latter cut is at f = 45° by default. You can assign another azimuth angle in degrees in the box labeled '''Non-Principal Phi Plane'''.
At the end of a PO simulation, the thee RCS plots sσ<sub>?θ</sub>, sσ<sub>fφ</sub>, and sσ<sub>tot</sub> are added under the far field section of the Navigation Tree. These plots are very similar to the three 3D radiation pattern plots. You can view them by clicking on their names in the navigation tree. The RCS values are expressed in m<sup>2</sup>. For visualization purposes, the 3D plots are normalized to the maximum RCS value, which is also displayed in the legend box. Keep in mind that computing the 3D mono-static RCS may take an enormous amount of computation time.
[[File:PO47.png]]