In this book we do not discuss how to develop numerical techniques, but it is important to understand how to apply methods. Whether you develop your own codes or use commercial codes, certain rules should be applied. Consider Eq. (1).
 (1)
The normal to the surface points in the direction of the incident wave: outward. If the normal pointed inward, the sign of induced electric current density would change. Most codes have made the assumption that the normal points outward, but some codes may check on the direction of the normal relative to the incident wave and make the necessary sign change. We must keep track of the direction of the normal, and it may be necessary to rotate the normal depending on the expected direction of the incident wave. If an object can have radiation from both sides, it may be necessary to use two objects in the analysis.
Many codes store each object as a separate entity in a disk file. In some cases we need to store an object multiple times. Take, for example, a Cassegrain dual reflector. The feed antenna illuminates the subreflector and induces currents on it. These currents radiate and excite currents on the main reflector. When the main reflector–induced currents radiate, the subreflector intercepts or blocks part of the fields. We account for this blockage by using a second subreflector object on which the code calculates a new set of induced currents by using the main reflector currents as the source. We could add these currents to the existing disk file object or merely keep the second object. We want to keep the second object separate so that we can calculate additional currents induced on the main reflector using these currents as sources. These currents will be reduced from the initial set, but they are an important contribution to the fields radiated behind the reflector. This example illustrates iterative PO. When objects face each other significantly, iterative PO is necessary to calculate correct patterns. The method converges rapidly in most cases.
Figure 1 illustrates the geometry of a corner reflector. A half-wavelength-long dipole is placed between two metal plates usually bent to form a 90◦ angle.
FIGURE 1 Corner reflector with a dipole located between two flat plates.
We can use other angular orientations between the plates, but this is the usual design. The figure does not show the feed line to the dipole, which usually starts at the juncture of the two plates and runs up to the dipole. Although the figure shows the plates as solid, many implementations use metal rods to reduce weight and wind loading.
The analysis starts with assumed currents on the dipole. We divide the plates analytically into small rectangular patches, which can be small (≈ λ/8 to λ/4) on a side since it takes only a few to cover the plates. You should repeat the analysis with different-sized current patches to determine if the analysis has converged. In a similar manner, we break down the current on the dipole into short linear segments, each with constant amplitude. By using a near-field version of Eq. (2), we calculate the magnetic field incident on each patch on the plates. This field induces electric currents on the plates calculated from Eq. (1). Remember that we combine radiation from the source dipole with that radiated from the induced currents to reduce the radiation behind the antenna. The currents were induced to satisfy the boundary condition of the plate, but only with both radiations present. Figure 2-a illustrates this process of inducing currents. Figure 3 shows the antenna pattern calculated using these currents. The E-plane pattern drawn as a solid line produces a null at 90◦ because the dipole pattern has this null. The plates cause the narrowing of the beam in the H-plane. The plates reduced the back radiation to −22 dB relative to the forward radiation, called the front-to-back ratio (F/B). The gain has increased from the 2.1 dB expected from a dipole to 9.3 dB. An equivalent geometric optics analysis uses two images in the plates, as shown in Figure 2-b, for the analysis.
 (2)
FIGURE 2 Cross-sectional view of a corner reflector: (a) magnetic field radiated from a dipole induces currents on plates; (b) plate currents replaced with image dipoles.
FIGURE 3 Pattern calculated from a combination of dipole and plate currents in a corner reflector with 1 × 0.9λ plates without induced current iteration.
If you look at Figure 1 or 2, you should notice that the two plates face each other. Currents on one plate will radiate toward the other plate and induce another set of currents on it. We could ignore these induced currents if the radiation was insignificant, but to produce correct patterns we must include them. The solution to this problem calls for an iterative technique where we calculate the radiation from the currents on the first plate and induce incremental currents on the second plate. These incremental currents produce further radiation that induces additional currents on the other plate. The method converges rapidly. Figure 4 gives the antenna pattern after the iterations have been completed and we include radiation from all currents. The actual F/B ratio of the antenna is 29 dB, and the additional currents increased the gain by 0.7 dB to 10 dB. Adding the two plates in the original analysis increased the gain by 7.2 dB, whereas the iterative technique had a much smaller effect. Figure 5 illustrates the iterative technique and shows that the equivalent geometric optics analysis adds a third image to represent the reflection between the plates. Remember when you mount the antenna in an application, the structure will change the realized pattern, but the high F/B ratio reduces this effect. The mounting structure used when measuring the antenna changes the pattern as well, which limits our knowledge of the real pattern.
FIGURE 4 Pattern of corner reflector with 1 × 0.9λ plates with induced current iteration equivalent to multiple reflectors between the plates.
FIGURE 5 (a) Wall currents on plates radiate magnetic fields that induce additional currents on facing plates; (b) added induced currents equivalent to additional image dipole.
Physical optics can determine the impedance effects of the limited images in the ground planes, such as the corner reflector. The local nature of impedance effects allows the use of images to calculate the mutual impedance effects of ground planes. We use impedance calculations not only to determine the bounds of ground-plane effects on input impedance, but to calculate the total power radiated by the antenna. The images (excited currents on ground planes) radiate but do not receive input power. A ground plane at least λ/2 on a side located about λ/4 away from the antenna produces nearly the same impedance effects as an infinite ground plane, but the ground plane alters the radiation pattern greatly because it restricts possible radiation directions.
It has commonly been thought that physical optics could compute the field only in the main beam pattern direction of a paraboloidal reflector. The method can determine this pattern region accurately by using only a few patches, each one being many wavelengths on a side. As the processing power of computers increases, the patch size can be shrunk until PO can calculate the pattern in every direction, including behind the reflector. It is important to remember to include the feed pattern behind the reflector even though its radiation is obviously blocked by the main reflector. Physical optics uses induced currents to cancel the fields inside objects when the incident fields and the radiation from the induced currents are added. We can calculate the pattern behind a reflector using UTD (GTD), the uniform (geometric) theory of diffraction. This geometric optics-based method blocks the radiation from the feed and uses diffractions from the rim edge to calculate the pattern behind the reflector. A comparison of UTD and physical optics calculations of the pattern behind shows that the two methods match.
The dashed curve of Figure 6 plots the results of the PO analysis of a 20λ-diameter centrally fed paraboloidal reflector. The feed antenna radiation tapers to −12 dB at the reflector rim. Figure 6 shows the feed power spillover peaking at angles off the boresight near 100◦. PO analysis computes the currents on a patch by assuming that it is embedded in an infinite plate. The reflector rim violates this assumption and we need extra terms to calculate the pattern behind the reflector accurately. Adding PTD (the physical theory of diffraction) to PO improves the match between the two methods behind the reflector as shown by the solid curve on Figure 6. PTD handles caustic regions of PO in a manner similar to the equivalent current method based on diffraction coefficients of UTD with geometric optics for shadow and reflection boundaries. For this example, the additional PTD currents add with the same phase because of the symmetry of the reflector geometry and produce the maximum effect. The PTD currents on the rim of an offset reflector will not add and produce a peak effect behind the reflector but will produce a more defuse effect. We only need PTD over a limited pattern angular range to reduce error, and the cost of implementing the fix may exceed the necessity of knowing the pattern in these regions. Similarly, UTD needs the addition of edge currents for accurate calculation of the radiation near 180◦, behind the reflector. Although any model for the feed pattern can be used with PO, results matching UTD exactly occur only over all regions of the back radiation when the feed satisfies Maxwell’s equations in the near and far fields. One such feed is the Gaussian beam approximation. Again, like PTD fixes, the small errors when using other feed antenna approximations occur only at limited pattern regions that may be unimportant.
FIGURE 6 Physical optics analysis of a 20λ-aperture-diameter paraboloidal reflector (dashed curve) compared to analysis that includes PTD (solid curve). |
