Hybrid method of moments (MoM) physical optics (PO) computational technique in Antenna Toolbox™ allows you to model antennas near large scatterers such as parabolic reflectors. The antenna element is modeled using MoM while the effect of electrically large structures is considered using PO.

The familiar Rao Wilton Glisson (RWG) basis functions on triangles are based on [2].

In the image, for two arbitrary triangular patches
*tr _{n}^{+}*
and

where is the vector drawn from the free vertex of the
triangle *tr _{n}^{+}*
to the observation point ; is the vector drawn from the observation point to
the free vertex of the triangle

From [1], along with
the standard definition, this method requires two unit normal vectors and two-unit vectors also shown in the figure. Vector is the plane of triangle
*tr _{n}^{+}*;
both vectors are perpendicular to the edge

are also shown in the figure. This technique assumes that the normal vectors are properly (angle between adjacent must be less than 180 degrees) and uniquely defined. Specific vector orientation (e.g. outer or inner normal vectors) does not matter. We then form two cross product vectors ,

and establish that both such unit vectors directed along the edge are identical,

Only vector is eventually needed.

The surface current density, , on the entire metal surface is expanded into
*N* RWG basis functions. However, a part of such basis
functions belongs to the MoM region (or "exact region") while another part will
belong to the PO region (or "approximate region"). These basis functions (or
regions) can overlap and be arbitrarily distributed in space (not necessarily be
contiguous). The method assumes that *N _{MoM}*
basis functions from the MoM region up front in the list and

If there is no PO region, you can solve the entire problem using the MoM with single square MoM system matrix , which may be subdivided into 4 matrices as shown.

The figure shows the matrix interpretation of the hybrid MoM-PO solution and its comparison with plain MoM solution. The method assumes the antenna feeds gives the vector, that describes the excitation, which belongs to the MoM region only.

The hybrid solution keeps submatrices and . In other words, the method solves the standard system of the linear equations for the MoM region where radiation from the PO region via is considered.

The hybrid solution ignores the submatrices, entirely. Here, the currents in the PO region do
not interact with each other. They are found via the radiated magnetic field, , from the MoM region, using PO approximation [1].
A new matrix describes this operation, , and negative identity matrix,
*E*, which replaces .

The suitable PO approximation has the form [1]

where δ accounts for the shadowing effects. If the observation point lies in the shadow region, δ must be zero. Otherwise it equals ±1 depending on the direction of incidence with respect to the orientation normal vector . Using second Eq.(4) yields:

Reference [1] outlines an
elegant way to express unknowns
*I _{n}^{PO}*
explicitly, using an interesting variation of the collocation method. First, we
consider a collocation point that tends to the edge center of a certain basis function and is in its plus triangle. We then multiply Eq.
(7) by vector . Since the normal component of the basis function
under interest at the edge is one and all other basis functions sharing the same
triangle have no normal component at this edge, the result becomes

Repeat the same operation with the minus triangle and obtain

Add both Eqs. (8a) and (8b) together, divide the result by two, and transform the triple vector product to obtain

Therefore, according to Eqs. (2) and (3),

To complete the derivation, the H-field radiated by the MoM region is always written in the form

where are given by individual basis function contributions. In the simplest case, every such contribution is the dipole radiation [3]. Substitution of Eq. (11) into Eq. (10) yields

According to the second figure, the coupled system of equations has the form

The direct solution method results in the substitution of the expression for the PO current into the first equation,

[1] U. Jakobus and F. M.
Landstorfer, “Improved PO-MM Formulation for Scattering from Three-Dimensional
Perfectly Conducting Bodies of Arbitrary Shape,” *IEEE Trans.
Antennas and Propagation*, vol. AP-43, no. 2, pp. 162-169, Feb.
1995.

[2] S. M. Rao, D. R. Wilton,
and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,”
*IEEE Trans. Antennas and Propagation*, vol.
AP-30, no. 3, pp. 409-418, May 1982.

[3] S. Makarov, *Antenna and EM Modeling in MATLAB*, Wiley, New York,
2002.