## Asymptotic-Expansion MethodsThe techniques described in the previous sections are exact methods in that the error in the numerical solution only comes from the discretization. The numerical solution approaches the exact solution as the discretization is refined. However, as the number of unknowns grows, the demand for computer memory and calculation time also grows. This prohibits these methods from being applied to high frequency problems where the size of the object is much larger than the wavelength. The methods described in this section are based on asymptotic high-frequency expansions of Maxwell's equations. They are high frequency methods that are only accurate when the dimensions of the objects being analyzing are large compared to the wavelength of the field. The asymptotic techniques introduced in the following sections include physical optics, geometrical optics, geometrical theory of diffraction, and uniform theory of diffraction. ## Physical OpticsThe (1) where H represent the incident and reflected magnetic field
components evaluated on the surface.
is the unit vector normal to the surface.
If the surface can be approximated as an infinite plane surface, then by image theory,^{r}(2) and Equation (1) reduces to (3) The electric and magnetic field radiated by the surface current on the illuminated side of the reflector can be determined by [3], (4) (5) where . Equation (3) is exact only when the surface is infinitely large. The accuracy of the approximation depends on the transverse dimensions of the reflecting surface, the radius of curvature, location of edges, and the angle of the incident field. Generally, PO works well for large, smooth surfaces with low curvature. The implicit assumption for the physical optics approximation is that the incident field is treated as a locally planar wave. Also, it assumes that the reflector surface is perfectly conducting. It has been found that PO provides an accurate prediction of far-field patterns of reflected antennas in the main beam region and out to several side lobes [5]. The major disadvantage of PO is that the integration over the surface of the reflector may be quite complicated and time consuming when the feed is placed off-axis or the feed pattern is asymmetric [197]. Moreover, the radiation integral has to be evaluated each time the observation point is changed. Fast and efficient evaluation of the radiation integral was proposed using a fast series approach [5], incorporating a multilevel fast multipole method [6], or decomposing the scatterer into subdomains [7]. Initially applied in the frequency domain, PO has also been extended into the time domain [8]. ## Geometrical Optics (Ray Optics)
The relationship between GO and PO was demonstrated in [10]. It was shown that the PO integral can be represented as a summation of many Fourier transforms, such that the first few terms resemble the GO representation. Using the "extinction theorem" [3], the fields predicted by the integration of PO surface currents were shown to agree with the geometrical optics aperture fields on the aperture plane to within the local plane wave approximation. It was concluded that the accuracy obtained by the two methods is comparable. ## Geometrical Theory of DiffractionThe approximations in both physical optics and geometrical optics are based on the following assumptions [3]: - The current density is zero on the shadow side of the reflector
- The discontinuity of the current density over the rim of the reflector is neglected
- Direct radiation from the feed and aperture blockage by the feed are neglected.
Both PO and GO ignore the edge diffractions which are highly dependent on the whether the edges of the reflector are flared, sharp, absorber lined or serrated. Thus, they cannot accurately predict the far fields beyond the first few side lobes. For predicting the patterns more accurately in all regions, geometrical diffraction techniques are required. As an extension of GO, the Two major advantages of GTD over other high frequency asymptotic techniques are that it provides insight into the radiation and scattering mechanisms from the various parts of the structure, and it can yield more accurate results. The method has attracted increasing attention; especially for applications to reflector antennas [13-18]. Unfortunately, GTD fails in the transition region adjacent to the shadow boundary, at caustics (points through which all the rays of a wave pass), or in close proximity to the surface of the scatterer. In these zones, the field cannot be treated as a plane wave. Thus, ray techniques become invalid. To deal with this problem, a number of alternative approaches have been proposed: uniform solutions [20-21], methods for dealing with caustic curves [22-24], physical theory of diffraction (PTD) [25], and the spectral theory of diffraction (STD) [26-27]. A comprehensive introduction to these methods can be found in [19]. ## Uniform Theory of DiffractionThe ## References[1] W. V. T. Rusch and P. D. Potter, [2] C. A. Balanis, [3] A. D. Yaghjian, "Equivalence of surface current and aperture field integrations
for reflector antennas," [4] J. F. Kauffman, W. F. Croswell, and L. J. Jowers, "Analysis of the radiation
patterns of reflector antennas," [5] Y. Rahmat-Samii and V. Galindo, "Shaped reflector antenna analysis using the Jacobi-Bessel
series," [6] T. F. Eibert, "Modeling and design of offset parabolic reflector antennas using
physical optics and multilevel fast multipole method accelerated method of
moments," [7] A. Boag, "A fast physical optics (FPO) algorithm for high frequency scattering,"
[8] E. Y. Sun and W. V. T. Rusch, "Time-domain physical-optics," [9] R. Mittra, Y. Rahmat-Samii, V. Galindo-Israel, R. Norman, "An efficient technique
for the computation of vector secondary patterns of offset reflectors," [10] Y. Rahmat-Samii. "A comparison between go/aperture-field and physical optics
methods for offset reflectors,"
[11] G. A. Deschamps, "Ray techniques in electromagnetics,"
[12] J. B. Keller, "Geometrical theory of diffraction," [13] D. L. Hutchins, "Asymptotic series describing the diffraction of a plane wave by a two-dimensional wedge of arbitrary angle," Ph.D. Dissertation, The Ohio State University, Dept. of Electrical Engineering, 1967. [14] C. A. Balanis and L. Peters, Jr., "Analysis of aperture radiation from an axially
slotted circular conducting cylinder using GTD," [15] G. L. James and V. Kerdemelidis,
"Reflector antenna radiation pattern analysis by
equivalent edge currents," [16] C. A. Mentzer and L. Peter, Jr.,
"A GTD analysis of far-out sidelobes of Cassegrain
antennas," [17] P .H. Pathak and R. G. Kouyoumjian,
"An analysis of the radiation from aperture on
curved surfaces by the geometrical theory of diffraction,"
[18] A. W. Rudge, "Offset-parabolic-reflector antennas: A review,"
[19] D. P. Bouche, F. A. Molinet and R. Mittra,
"Asymptotic and hybrid techniques for
electromagnetic scattering," [20] R. C. Kouyoumjian and P. H. Pathak,
"A geometrical theory of diffraction for an edge
in a perfectly conducting surface,"
[21] S. W. Lee and G. A. Deschamps,
"A uniform asymptotic theory of electromagnetic
diffraction by a curved wedge,"
[22] R. W. Ziolkowski and G. A. Deschamps, "Asymptotic evaluation of high frequency fields
near a caustic: An introduction to Maslov's method,"
[23] Y. Kravtsov and Y. Orlov,
"Caustics, catastrophes and wave fields,"
[24] H. Ikuno and L. B. Felsen,
"Complex ray interpretation of reflection from concave-convex
surfaces," IEEE [25] P. Y. Ufimtsev,
"Elementary edge waves and the physical theory of diffraction,"
[26] R. Mittra, Y Rahmat-Samii, and W. L. Ko, "Spectral theory of diffraction,"
[27] Y. Rahmat-Samii and R. Mittra, "A spectral domain interpretation of high-frequency
diffraction phenomena," [28] R. G. Kouyoumjian and P. H. Pathak, "A uniform geometrical theory of diffraction for
an edge in a perfectly conducting surface," |