# ANALYSIS METHODS AND EVALUATION Geometrical and Physical Optics Formulations

In both the geometrical optics (GO) and physical optics (PO) formulations, the ultimate goal is to determine equivalent currents, which can then be integrated to obtain the far-field patterns, a process well described in the literature on aper­ture antennas (20). We focus our attention on the assump­tions and approximations inherent in each of these formula­tions, as well as on their intrinsic differences.

The GO technique yields the aperture fields, assuming equal angles of incidence and reflection. The far-field patterns can then be calculated using a Fourier transformation di­rectly, which is equivalent to obtaining equivalent currents and then integrating, as described later in this subsection. With the use of image theory, it is necessary to know only the electric field distribution over the reflector projected aper­ture, Er, which is computed from the incident electric field E{ (i. e., the feed radiation), with (20)

Er = 2(n • E;)n — E; (31)

where A is the unit vector normal to the surface; see Eq. (5). Equation (31) assumes that at the point of reflection the re­flector is planar and perfectly conducting. In addition, the in­cident wave from the feed antenna is treated locally as a

plane wave. These same assumptions are also used by the PO

technique to determine the surface currents, Js, over the re­flector as follows:

Js = 2n x H; (32)

where H is the incident magnetic field from the feed antenna and can be computed from Eq. (14), recalling that in the far field H = (r X E)/^ (where ^ is the free-space characteristic impedance). The PO approximation assumes that currents ex­ist only over the side of the reflector directly illuminated by the feed antenna.

Table 6. GBT Dual Offset Reflector Configuration and Computed Performance Values

Main Reflector Configuration

Shape: Offset paraboloid Projected diameter D: 100 m Parent reference diameter Dp: 208 m Focal length F: 60 m Offset of reflector center, H: 54 m Angle fi, 5.58°

Subreflector Configuration

Shape: Offset ellipsoid Projected height DS: 7.55 m Parameter c of ellipse: 5.9855 m Parameter fS of ellipse: 5.3542 m Eccentricity e: 0.5278

Feed Configuration (On Focus; GRASP Calculation)

Polarization: Linear (xf)

Pattern shape: Gaussian, Eqs. (14) and (15)

Gain Gf: 21.31 dBi 10 dB beamwidth: 30°

Angle a: 17.91°

Angle y: 12.33°

System Performance (GRASP Calculation)

Gain G: 82.83 dBi

Cross-polarization (XPOL) level: —43.01 dB Sidelobe level (SLL): —22.56 dB Aperture efficiency eap: 77.76%

The far-field pattern can then be determined by summing the individual contributions of each current point over the surface, taking into account the different amplitudes and phases due to the excitation and spatial location. Antenna theory shows that a unit point source of current radiates a spherical wave, which is normally referred to as the free — space Green’s function (e—jkr/4wr); see Ref. 20 for further de­tails. In the limit as the current distribution becomes continu­ous, such as the one given by Eq. (32), the weighted sum of spherical waves becomes an integral, yielding the radiated patterns.

Note that the integration process for obtaining the pat­terns is the same as the one employed by the GO technique, given that once the aperture distribution is determined from Eq. (31), equivalent currents can then be obtained and inte­grated over the reflector aperture. This process is equivalent to computing the Fourier transform of the aperture distribu­tion given in Eq. (31). One difference between GO and PO is that PO currents are determined over the reflector curved surface and the GO equivalent currents over the planar pro­jected aperture, with the latter already in a format more ap­propriate for integration through a Fourier transform. How­ever, the use of a Jacobian transformation (3,4) maps the PO currents over the reflector curved surface to the planar aper­ture, yielding the possibility of also using Fourier transforma­tions for performing the integration. Analytical integration is only possible for symmetrical reflectors (4,8), and numerical techniques are normally required to evaluate offset reflectors, as discussed in the next subsection.

The PO formulation is generally considered more accurate than GO to evaluate offset reflectors, especially if XPOL as­sessment is a main concern. However, pattern accuracy as determined from both techniques degrades beyond the main beam and near-in sidelobes. The pattern in the far-out region is dominated by diffraction effects, especially scattering from the reflector and/or subreflector edges. This is taken into ac­count by augmenting GO with the geometrical theory of diffraction (GTD) or augmenting PO with edge currents through the physical theory of diffraction (PTD); see Refs. 20 and 30 for details. However, the near-in pattern region is, most of the time, the region of interest when analyzing high — gain antennas such as the reflector antennas considered here.

Updated: 02.05.2014 — 10:01