Computational Electromagnetics and Fast Physical Optics Felipe Vico, Miguel Ferrando, Alejandro Valero, Jose Ignacio Herranz, Eva Antonino Instituto de Telecomunicaciones y Aplicaciones Multimedia (iTEAM) Universidad Politécnica de Valencia Building 8G, access D, Camino de Vera s/n 46022 Valencia (SPAIN) Corresponding author: [email protected] Abstract In mathematics and physics, the scattering theory is a framework for studying and understanding A new hybrid technique is used to compute the the scattering of waves and particles. Prosaically, RCS pattern of a spherical cap by means of PO wave scattering corresponds to the collision and approximation. The approach used here com- scattering of waves with some material object. bines analytical techniques with path deforma- More precisely, scattering consist of the study of tion techniques to compute accurately the full how solutions of partial differential equations, RCS diagram. The computation time is inde- propagating freely “in the distant past”, come to- pendent with frequency. This approach is faster gether and interact with one another or with a than standard numerical techniques (PO inte- boundary condition, and then propagate away gration) and more accurate than non-uniform “to the distant future”. In acoustics, the differen- asymptotic techniques. tial equation is the wave equation, and scatte- ring studies how its solutions, the sound waves, Keywords: Physical Optics, Radar Cross Section, scatter from solid objects or propagate through Highly oscillatory integrals, path deformation. non-uniform media (such as sound waves, in sea water, coming from a submarine). In our case (classical electrodynamics), the differential equa- 1. Introduction tions are Maxwell equations, and the scattering of light or radio waves is studied. In quantum Computational science and computational en- and particle physics, the equations are those of gineering have been developed from the very quantum electrodynamics QED, quantum chro- invention of the modern computer. Since the modynamics QCD and the Standard Model, the application of the Neumann’s machine to the solution of which correspond to fundamental solution of termonuclear and nuclear physics particles. In quantum chemistry, the solutions problems, the use of computers was extended to correspond to atoms and molecules, governed almost all engineering processes. by the Schrodinger equation. The Field of Electromagnetic Wave Theory pro- Computational techniques for scattering pro- duces a great number of challenging problems blems in classical electrodynamics can be divi- for Electrical and Electronic Engineers. Among ded into two different groups: them, scattering problems are particularly hard. New technologies on wireless communications, On one hand High Frequency Techniques. remote sensing, space communication systems, Notable among these are the geometrical earth observation satellites… require precise theory of diffraction (GTD) introduced by J. designs that usually overwhelm the accuracy B. Keller; the Physical Theory of Diffraction of the existing computational techniques. The (PTD) developed by P. Y. Ufimtsev; the Uni- international community on Antennas and Pro- form Asymptotic Theory (UAT) and the Uni- pagation has been developing for many years form Theory of Diffraction (UTD) formulated a lot of very interesting algorithms to solve fast by Lewis et al. and Kouyoumjian and Pathak; and efficiently a large amount of problems ari- and the Spectral Tory of Doffractopm (STD) sing in telecommunications: Antenna reflectors, introduced by Mittra and Others. RCS of planes … Nevertheless, in many cases the accuracy and the computation time are far form On the other hand Numerical Techniques, satisfactory. which are full wave methods. Notable among Waves · 2009 · year 1 / ISSN 1889-8297 155 these are the Method of Moments (MoM), Fi- with a reduced electrical size as the computatio- nite Element Method (FEM), Finite Difference nal complexity increase with frequency. On the Time Domain (FDTD). The first one is much other hand, High frequency methods are suita- more suitable for radiation problems while the ble for geometries with a detail which is much second one is suitable for guided problems. larger than the wavelength. Therefore we find The third is suitable for wideband problems. that high frequency and numerical techniques are complementary, in the sense that they are suitable for different problems. High Frequency Techniques Numerical Techniques Despites the great amount of problems that can Ray Methods: be handled by these techniques (numerical and Differential Equation high frequency), there are still some difficult si- Geométrical Optics (GO) tuations where no method can give a comple- Methods: tely satisfactory solution. Geometrical Theory of Diffraction Finite-Difference Methods (FDTD) Hybrid techniques try to combine numerical and (GTD) high frequency methods to both increase the Finite-Element Methods (FEM) Uniform Asymptotic Theories range of applicability of the numerical methods Stability and Accuracy and increase the accuracy of high frequency me- (UAT,UTD) thods. Nowadays, hybrid techniques are a hot Integral Equation Methods: spot in computational electromagnetics. Current Base Methods: Spatial Domain 1. Physical Optics Physical Optics (PO) Spectral Domain Physical Theory of Diffraction (PTD) Periodic MoM The Physical Optics (PO) method is a high fre- quency approximation that allows obtaining a Fast Methods (FMM, AIM) solution with a high accuracy and efficiency. The Incremental Methods: Physical optics method consist of approximate Modal Expansion Methods: the induced current on the scatterer (the solu- Boundary Waves Methods tion of the integral equation) by the, so called, PO Mode Matching Incremental Theory of Diffraction current, given by the following expression: Generalized Admitance Matrix (ITD) Complex Rays and Gaussian Beams: Now, the diffracted field can be obtained expli- citly by the integral of the PO current: Table 1. Computational Techniques The following table contains a summary of the main techniques: Methods from the first column are based on the The approximation of the induced current can expansion of the solution in terms of asymptotic be justified in different ways. The most typical series combined with the localization principle. way is using a reasoning based on fields. For these methods, the accuracy increases with frequency. As far as the computation time, the The next step to obtain an estimation of the first group (Ray Methods) is the fastest one with scattered field is to evaluate numerically the a CPU time not frequency dependent. The se- integral. For very large frequency, the integral cond group (current base methods) is not so fast obtained is a highly oscillatory integral. Due to but more robust since the solution is given by a the complexity of the integrand, the computa- highly oscillatory integral. tion time using a standard numerical quadrature rule is very large. Methods from the second column are based on the discretization of the Maxwell’s equations (the In some cases, the PO integral can be computed integral or the differential form) and the approxi- directly using brute force algorithm. In that case, mation of the solution by a finite dimensional the method is known as PO method. Neverthe- subspace. The accuracy is much higher for these less, in order to speed up the calculation of the- methods than for the high frequency methods. se integrals, a number of special methods have The weakpoint of these methods is the CPU time been developed. and the requirement in memory, especially in high frequencies. Next we show an example of fast method for computing the PO integral in the case of a semi Numerical methods are suitable for problems spherical cap surface: 156 ISSN 1889-8297 / Waves · 2009 · year 1 Figure 1. Spherical surface Figure 2. Highly oscillatory integrand (real part) 2. Approach Due to the highly oscillatory nature of the inte- Consider a PEC spherical cap as depicted in fig 1. grand (see fig. 2), the direct computation of the The parameterization is given in spherical coor- integral using standard quadrature rules is very dinates by the following formulas: inefficient. In order to compute efficiently the integral (6), a combined technique is used. First, the two di- mensional integral is reduced to a one dimensio-  (1) nal integral as follows: A rotation in the coordinate system is performed The incident field is given by the following for- around the x axis to change the ks vector into mula: the z unitary vector (see fig. 3). The integral (6) takes the following analytical expression:  (2) (7) We define the scattering amplitude pattern U for the polarization of interest via the copolarized component of the backscattered far field as:  (3) Using the PO approximation, the amplitude pat- tern can be computed by the following integral expression: (4) The vector ks is given by: (5) Using spherical coordinates, the integral (4) is given by: (6) Figure 3. Rotation of the coordinate system. Waves · 2009 · year 1 / ISSN 1889-8297 157 where the function θ’max is given by: The resulting integral is: (8) (12) The relation between φ and φ’ is given by: where: (9) The integral with θ’ can be computed analyti- cally: (13) the integral (12) can be written in a more suc- cinct way: (14) where: (10) (15) The surface highly oscillatory integral is changed into a line integral (see fig 4) again highly oscil- latory. Therefore a computation time improve- ment has been achieved.