Open access peer-reviewed chapter

# High Frequency Techniques: the Physical Optics Approximation and the Modified Equivalent Current Approximation (MECA)

By Javier Gutiérrez-Meana, José Á. Martínez-Lorenzo and Fernando Las-Heras

Submitted: October 20th 2010Reviewed: March 10th 2011Published: July 5th 2011

DOI: 10.5772/17307

## 1. Introduction

In most of the electromagnetic problems, the number of unknowns to evaluate the scattered fields grows whenever the size of the antenna, device or scenario increases or the working frequency becomes higher. In this context, the rigorous full-wave methods –e.g. Method of Moments (MoM), Fast Multipole Method (FMM) (Engheta et al., 1992), Finite-Difference Time-Domain (FDTD) (Taflove & Umashankar, 1987) or Finite-Difference Frequency-Domain (FDFD) (Rappaport & McCartin, 1991), Finite Element Method (FEM) (Kempel et al., 1998) – can not tackle the analysis of such problems beyond an upper limit determined by the computational requirements in terms of time and memory. High frequency techniques consist in the asymptotic evaluation of the Maxwell’s equations. As a consequence, they provide good accuracy when dealing with electrically large geometries meanwhile the computational needs diminish with respect to the aforementioned methods.

Within the high frequency techniques, the Geometrical Optics (GO) and the Physical Optics (PO) approximation are the most extended methods due to the successful results obtained in various fields such as Radar Cross Section (RCS), design of reflector antennas or radioelectric coverage calculation. Since the Physical Optics approximation is detailed in the following section, the Geometrical Optics is briefly summarised.

The main interest in the GO lies in the fact that incident, reflected and transmitted electromagnetic waves are studied based on the conservation of the energy flux along a ray tube between a source and an observation point. Therefore, the Geometrical Optics is usually referred to as Ray Optics. The GO comprises two different methodologies (Rossi & Gabillet, 2002): Ray Tracing (Glassner, 1989) – the starting point is the receiver or observation point and a path to the source is sought analysing the reflections on walls, buildings, mountains – and Ray Launching – multiple rays are launched from the source, so they are independently followed until an observation point or the receiver is reached. One of the common applications of the GO is the evaluation of radio electric coverage or the channel characterization in urban scenarios.

Both the GO and the PO techniques require of an additional method to compute the contribution due to the diffraction phenomenon. The GO can be complemented by means of the Geometrical Theory of Diffraction (GTD) (Keller, 1962) or the Uniform Theory of Diffraction (UTD) (Pathak & Kouyoumjian, 1974). On the other hand, the Physical Theory of Diffraction (PTD) (Ufimtsev, 1962) is applied in joint with the PO formulation.

This chapter focuses on the PO approximation and especially on its extension to dielectric and lossy materials, namely the Modified Equivalent Current Approximation (MECA) method. In section 2, the PO approximation is presented and the formulation is obtained from the equivalence principle. Then the MECA method is introduced, so the equivalent current densities are obtained at the end of section 3.1.3. In order to complete the expressions, the reflection coefficients are then calculated and the determination of the electromagnetic field levels at the observation points based on an analytical solution of the radiation integral is accomplished in section 3.2. A couple of validation examples are studied before continuing to describe the algorithms for solving the visibility problem and, specifically, the Pyramid method in section 4. Afterwards, some of the applications of MECA are listed in “Application examples” where an example of radio electric coverage evaluation is also shown. “Conclusion” and “References” closes the chapter.

## 2. The Physical Optics (PO) approximation

The Physical Optics (PO) approximation (Harrington, 2001; Balanis, 1989) is a well-known high frequency technique based on the determination of the equivalent current densities induced on the surface of an illuminated perfect electric conductor (PEC) plane. The final expressions can be achieved through the equivalence principle as it will be shown in section 2.1, so once these equivalent or PO current densities are obtained, both electric and magnetic field levels can be calculated from the corresponding radiation integrals.

In order to be in compliance with the constraints introduced by the PO approximation, some aspects have to be evaluated prior to the selection of this technique to tackle any electromagnetic problem:

• The geometry must be made of electrically large obstacles with a smooth variation of their surfaces.

• Given a radiating source, a distinction between lit and non-lit regions must be possible to perform.

The latter implies the need of a complementary algorithm to identify line of sight (LOS) directions from a specific point of view. Some methods will be commented in section 4.

Henceforth, for the sake of simplicity, a time-harmonic variation,ejωt, of the electromagnetic fields and current densities is assumed. Likewise, the spatial dependence of those variables is not explicitly written.

### 2.1. PO formulation

The expressions for the electric and magnetic equivalent current densities, JPOand MPOrespectively, can be derived from the physical equivalent (Balanis, 1989) in Fig. 1(left), where a scheme consisting of two different media is presented: a PEC on the left and a medium with permittivity ε1, permeability μ1 and conductivity σ1 on the right. Consequently, the boundary Sbetween the first and the second media is a PEC plane with the outward normal unit vector n^pointing to the right.

The incident electric and magnetic fields due to external sources and in absence of any obstacle are E1incandH1inc. On the other hand, the total fields inside the PEC are null (EPECtot=HPECtot=0), while in the second medium E1totand H1totare calculated by adding those incident fields to the reflected ones denoted by E1refandH1ref. Therefore the electric and magnetic induced current densities, JandM, at the boundary S can be obtained from the tangential components of the total fields as:

J=n^×(H1totHPECtot)|S=n^×H1tot|S=n^×(H1inc+H1ref)|SM=n^×(E1totEPECtot)|S=n^×E1tot|S=n^×(E1inc+E1ref)|S=0E1

Fig. 1 (centre) shows the equivalent problem for the non-PEC medium. In order to keep the same boundary conditions, this medium has been extended replacing the PEC part and the total fields are now E1incand H1incat the left side of S and E1refand H1refat its right side.

Considering the characterization of the boundary Sas a PEC plane in Fig. 1(right) and taking into account the expressions in Eq. (1), the Physical Optics approximation states that H1incand H1refat the boundary Sare in phase and also have the same amplitude. Thus, JPOand MPOcan be expressed as:

JPO=n^×(H1inc+H1ref)|S2n^×H1inc|SMPO=n^×(E1inc+E1ref)|S=0E2

At this point, an additional consideration to complete the PO formulation is included: when dealing with finite geometries, the PO current density is null in the regions not illuminated by the source:

JPO={2n^×H1inc|S0litregionnonlitregionE3

This is the reason why the distinction between shadowed and illuminated parts of the scenario is one of the aforementioned constraints to correctly apply the PO approximation.

## 3. The Modified Equivalent Current Approximation (MECA) method: a PO extension for penetrable and non-metallic objects

Even though several electromagnetic problems fulfil the restrictions by the PO approximation in terms of electric size or radius of curvature, the characterization of the obstacles as perfect electric conductors reduces the scope of application of this high frequency technique.

Previous works (Rengarajan & Gillespie, 1988; Hodges & Rahmat-Samii, 1993, Sáez de Adana et al., 2004) agree that the extension of the PO approximation for penetrable and non-metallic objects have to account for the reflection coefficients Rin order to apply the boundary conditions at the interface between the two media –one of the media uses to be the air. In this context, it is convenient to accomplish a decomposition of the incident fields into their transversal electric (TE) and transversal magnetic (TM) components to independently insert RTEand RTMin the evaluation of the equivalent current densities:

Einc=ETEinc+ETMinc=ETEince^TE+ETMince^TME4

where e^TEand e^TMare the unit vectors in the direction of the TE and TM components of the incident electric field respectively. Likewise, the incident magnetic field can be written in an analogous way.

The Modified Equivalent Current Approximation (MECA) method (Meana et al., 2010) is a new high frequency technique based on the evaluation of a set of equivalent currents to calculate electromagnetic field levels at any observation point. The most important features of this method are:

• MECA deals with electrically large scenarios consisting of dielectric and lossy surfaces due to both electric and magnetic equivalent currents, JMECAandMMECA, are taken into account.

• MECA reduces to the PO expressions when considering PEC obstacles. Therefore, it can be seen as an extension and an improvement of the classic Physical Optics approximation.

• Reflection coefficients are calculated in a rigorous manner.

• A constant amplitude and linear phase current density distribution on the surface of the facets is adopted to analytically solve the radiation integral.

### 3.1. MECA formulation

Let us suppose a skew plane wave impinging on the interface Swith an angle of incidence denoted by θinc and a unit incident vectork^inc. The two media are now characterised by their respective permittivities (ε1,ε2), permeabilities (μ1,μ2) and conductivities (σ1,σ2). The analysis of the TE and TM components will be presented separately.

#### 3.1.1. TE component

For the incident wave, the coordinate system [k^inc,e^TM,e^TE]where

e^TE=k^inc×n^|k^inc×n^|e^TM=e^TE×k^incE5

has been chosen (see Fig. 2). Substituting k^incby the unit reflection vectork^ref, the coordinate system [s^TM,k^ref,s^TE=e^TE]is defined. Then, the incident electric and magnetic fields are rewritten in terms of these unit vectors:

ETEinc=ETEince^TEHTEinc=HTEince^TM=1η1ETEinc(k^inc×e^TE)E6

where the intrinsic impedance of the first medium η1isηi=jωμijωεi+σi,i=1,2.

From the Snell law, the angle of reflection θref is equal to θinc. Therefore, k^refcan be expressed as:

k^ref=k^inc2n^(k^incn^)E7

Inserting the TE reflection coefficient RTE, the electric ETErefand magnetic HTErefreflected fields are:

ETEref=RTEETErefe^TEHTEref==1η1RTEETEincs^TM=1η1RTEETEinc(k^ref×e^TE)E8

Considering that

ETE,TMtot=ETE,TMinc+ETE,TMrefHTE,TMtot=HTE,TMinc+HTE,TMrefE9

and introducing Eq. (8) in Eq. (9), the total fields for the TE component are:

ETEtot=ETEinc(1+RTE)e^TEHTEtot=1η1ETEinc[(k^inc×e^TE)+RTE(k^ref×e^TE)]E10

#### 3.1.2. TM component

In Fig. 3 the same directions of incidence, reflection and transmission as well as the boundary Sare again depicted. In addition, the TM components of the fields are plotted. In accordance with the reasoning in the previous section, the expressions for the total fields can be obtained, leading to:

ETMtot=ETMinc[e^TM+RTM(k^ref×e^TE)]HTMtot=1η1ETMinc(1RTM)e^TEE11

#### 3.1.3. MECA equivalent currents

Adding the contributions of the TE and TM components in Eqs. (10) and (11), the total electric and magnetic fields can be written as a function of the incident electric field, the reflection coefficients, the intrinsic impedance of the first medium, the propagation vectors and the unit vectore^TE:

Etot=Einc+RTEETEince^TE+RTMETMinc(k^ref×e^TE)Htot=1η1k^inc×Einc+1η1[RTEETEinc(k^ref×e^TE)RTMETMince^TE]E12

Going back to the boundary conditions, the MECA equivalent current densities are reckoned from the total fields at Sin Eq. (12), being the magnetic ones no longer zero:

JMECA=n^×Htot|S==1η1{ETEinccosθinc(1RTE)e^TE+ETMinc(1RTM)(n^×e^TE)}|SMMECA=n^×Etot|S={ETEinc(1+RTE)(e^TE×n^)+ETMinccosθinc(1+RTM)e^TE}|SE13

As it can be seen and remarked in the previous expressions, the amplitude and phase of the current densities depend on the polarization and also on the angle of incidence θinc of the incident wave.

#### 3.1.4. TE and TM reflection coefficients

In order to complete the evaluation of the MECA current densities, RTEand RTMhave to be determined before moving to the calculation of the electromagnetic fields at the observation points. For this task, the boundary condition for the continuity of the tangential components of the electric and magnetic fields at the boundary Sis utilised.

The general dispersion relation applied to the second isotropic medium allows the evaluation of the transmission propagation vector k^tra(its real part is depicted in Fig. 2 and Fig. 3) due to

kk=ω2μi(εijσiω)E14

where k=[k1k2k3]denotes any propagation vector, the subscript irefers to the first (i=1) or second medium (i=2) andω=2πf, with fbeing the working frequency. Ifσi0, the right hand side of Eq. (14) is a complex number and the analysis is better performed introducing the real and imaginary parts of the vectork, denoted by {k}and {k}respectively. Particularising for the reflection and transmission propagation vectors:

{|{kref}|2|{kref}|2=ω2μ1ε12{kref}{kref}=0|{ktra}|2|{ktra}|2=ω2μ2ε22{ktra}{ktra}=ωμ2σ2E15

The continuity condition at Sfor the first component of the propagation vector means that

{k1inc}={k1ref}={k1tra}=2πλ1sinθincE16

where λ1is the wavelength in the first medium and

{k1inc}={k1ref}={k1tra}=0E17

Inserting Eqs. (16) and (17) in Eq. (15), the real and imaginary parts of the third component of the transmission propagation vector are obtained as:

{k3tra}=12[{ktra2}(2πλ1)2sin2θinc+{ktra2}(2πλ1)2sin2θinc+{ktra2}]12{k3tra}={ktra2}2{k3tra}E18

Once k3tra={k3tra}+j{k3tra}is determined, RTEis (Staelin et al., 1993)

RTE=μ22πλ1cosθincμ1k3traμ22πλ1cosθinc+μ1k3traE19

and the corresponding RTMis

RTM=(ε2jσ2ω)2πλ1cosθincε1k3tra(ε2jσ2ω)2πλ1cosθinc+ε1k3traE20

In case of working with non-metallic obstacles, the reflection coefficients become the well-known and simpler Snell’s expressions where the calculation of RTEand RTMis straightforward.

When the second medium is a PEC (σ2), the absolute value of the third component of ktratends to infinite and consequently RTE=1as well asRTM=1. Substituting these values in Eq. (12), the total magnetic field equals 2Hincat the boundary S. Therefore, JMECAand MMECAin Eq. (13) reduces to JPOand MPOin Eq. (2).

### 3.2. Electromagnetic field levels

Once JMECAand MMECAcan be calculated with the inclusion of the reflection coefficients in the previous section, the evaluation of the scattered fields due to this set of equivalent current densities on the surface of an obstacle requires solving a radiation integral. For arbitrary distributions and geometries, an analytical solution can not be reached. As a consequence, two assumptions are imposed (Arias et al. 2000; Lorenzo et al., 2005):

• The incident wave is a plane wave.

• The obstacle consists of flat triangular facets.

With the objective of distinguishing between the reflected fields at the boundary and the scattered fields at the observation points, the latter are denoted by EsandHs. Note that these do not take into account the contribution of the diffraction phenomenon.

#### 3.2.1. Incident plane wave approximation

The position vector r'of a point on the surface of a flat triangular facet can be written as:

r'=ri+r''E21

where riis the position vector of the barycentre of the i-th facet and r''is the vector from the barycentre to the source point (see Fig. 4).

The current densities are particularised for the i-th facet, JiMECAandMiMECA, having constant amplitudes,Ji0MECA=JiMECA|r=riand Mi0MECA=MiMECA|r=riand a linear phase distribution which varies with the unit direction of the Poynting vector as:

p^i={Eiinc×(Hiinc)*}|{Eiinc×(Hiinc)*}|E22

where Eiincand Hiincare the incident electric and magnetic fields at rirespectively, and * denotes the complex conjugate value of the complex vector in brackets. Therefore, in a medium where the working wavelength is λ1,

JiMECA=Ji0MECAexp{j2πλ1p^ir''}MiMECA=Mi0MECAexp{j2πλ1p^ir''}E23

The reason to introduce the linear phase variation is that this technique allows employing larger facets than when assuming a constant phase distribution.

#### 3.2.2. Application to flat triangular facets

In order to analytically evaluate the radiation integral, the observation points rare supposed to be in the far field of the radiating facet – but not necessarily in the far field of the whole scenario. Most of times this constraint is not an additional restriction because applications, such us radar cross section, directly impose huge distances between the object under test and the coordinates where the fields are determined.

Mathematically, the magnitude of the vector between the source point r'and the observation point ris approximated (Balanis, 1989) by the magnitude of the latter|R|=|rr'||r|r. The phase is taken asR=rr'r-r^r', wherer^=r|r|.

The formulation to compute the scattered fields can be derived from the Maxwell’s equations, e.g. using the magnetic vector potential Aand the electric potential vector Fseparately and combining both solutions. The resulting expressions are written in terms of the Green’s function in unbounded media Gand its gradientG. The far field approximation allows expressing these functions as:

G=14πexp{j2πλ1|R|}|R|14πexp{j2πλ1r}rexp{j2πλ1r^r'}G=j2πλ1GR^(1+1j2πλ1R)j2πλ1Gr^E24

Consequently, the scattered fields are:

Esj2λ1exp{j2πλ1r}rS[r^×MMECA(r^×η1JMECA×r^)]exp{j2πλ1r^r'}s'Hsj2λ1exp{j2πλ1r}rS[r^×JMECA(r^×r^×1η1MMECA)]exp{j2πλ1r^r'}s'E25

and they can be simplified inserting the auxiliary vectors Eaand Ha

Esj2λ1exp{j2πλ1r}r[Eaη1Ha×r^]Hsj2λ1exp{j2πλ1r}r[Ha1η1r^×Ea]E26

with

Ea=Sr^×MMECAexp{j2πλ1r^r'}s'Ha=Sr^×JMECAexp{j2πλ1r^r'}s'E27

Considering that the expressions of Eaand Haare quite similar, the following reasoning is only performed for the magnetic fields, so the contribution due to the electric current density JMECAon the i-th facet at the observation point rkis given by:

Hika=exp{j2πλ1r^ikri}Si(r^ik×JiMECA)exp{j2πλ1r^ikr''}si'E28

where Sirefers to the surface of the i-th facet and r^ikis the unit vector from the barycentre of that facet to the observation point k. Substituting JiMECAby its expression in Eq. (23), where the constant amplitude and linear phase variation distributions were inserted and extracting the terms from the integrand, Eq. (28) is rewritten as:

Hika=exp{j2πλ1r^ikri}(r^ik×Ji0MECA)I(r^ik)I(r^ik)=Siexp{j2πλ1(r^p^i)r''}si'E29

In case of dealing with flat triangular facets, r''can be expressed as a function of the vectors vmnin Fig. 4:

r''=v01+fv12+gv13E30

withvmn=PnPm, being Plthe position vector of the vertices l=1,2,3of the facet andP0, the position vector of the barycentre. fand gare real coefficients. Taking into account this transformation, the integral in Eq. (29) is converted into:

I(r^)=2Aexp{-ja+b3}0101fexp{j(af+bg)}gfE31

where Ais the area of the facet and the values of aand bare defined as:

a=2πλ1v12(r^p^i)b=2πλ1v13(r^p^i)E32

The solution to the Eq. (31) is obtained by parts, but in order to overcome some singularities, five different cases are detailed (Arias et al. 2000) in Table 1.

 a b I(r^) Any Any 2A exp{-ja+b3}[aexp{jb}−bexp{ja}+b−a(a-b)ab] 0 Any 2A exp{-jb3}[1+jb−exp{jb}b2] Any 0 2A exp{-ja3}[1+ja−exp{ja}a2] 0 0 A Any a 2A exp{-j2a3}[exp{ja}(1−ja)−1a2]

### Table 1.

Solution to the integral in Eq. (31). Different cases are presented to overcome the singularities in the original expression.

The formulation of the Modified Equivalent Current Approximation method is completed once all the intermediate results that have been described in this section are combined to evaluate the scattered fields at a specific observation point because of one radiating facet. Until this point, the steps to follow are summarised in:

• Decompose the incident electric field into its TE and TM components.

• Obtain the reflection coefficients in Eqs. (19) and (20).

• Calculate the MECA electric and magnetic current densities in Eq. (13).

• Particularise the expressions for the barycentre of every radiating facet.

• Compute the integral I(r^ik)with the help of Table 1.

• Evaluate the auxiliary fields Eaand Haintroducing the expressions of the MECA current densities and the result of the radiation integral.

• Calculate the scattered fields Esand Hs

Afterwards, the superposition theorem is employed to add the contribution due to all the radiating facets in the geometry for the observation point k:

Hks=iHiksE33

If an evaluation of the line of sight between the radiating sources and the obstacle were necessary because of its shape, this task would be faced at the beginning of the whole.

### 3.3. Validation schemes

In this section, two canonical examples for different constitutive parameters and geometries are presented: one assures the good behaviour in high frequency and the other consists in the analysis of electrically large surfaces. In the first scenario, the validation is accomplished in terms of the Radar Cross Section (RCS) which is calculated as:

RCS=4πlimrr2|E|2|Einc|2E34

Since the field levels due to the diffraction contribution are not significant in the selected examples, this effect can be neglected in the computation of the RCS. Consequently, the term in the numerator Eonly takes into account the electric field levels owing to reflections on the surface.

The methods to contrast the results provided by MECA are an analytical solution taken from the references and the full-wave technique the Method of Moments (MoM) (Medgyesi-Mitschang et al., 1994).

#### 3.3.1. High frequency behaviour

As a high frequency technique, MECA has to show an accurate behaviour when the dimensions of the obstacle are large in comparison with the working wavelength. A good example to test this is a sphere, whose monostatic RCS can be obtained theoretically for both PEC (Balanis, 1989) and non-metallic (Van-Bladel, 2007) characterisations. A frequency sweep is performed for a sphere of radius awith relative permittivity and permeability ofεr=2.2, μr=1.1respectively and conductivity ofσ=7ωε0, beingε0=8.8541012F/m, for the lossy case. The region in Fig. 5 for 2πλr>10corresponds to the high frequency zone where a great coincidence in the results can be observed.

#### 3.3.2. Electrically large surfaces

In this setup, a resonant horizontal dipole is placed at a fixed position whose height is of h=1.5metres above an electrically large flat surface as depicted in Fig. 6. The position of the observation point Pobsis determined by two correlated variables: the horizontal distance from the source toPobs, denoted by dand the height ofPobs, denoted byH. The angle υ is defined with the purpose of representing the results.

For the working frequency of 1800 MHz and the values of Hranging from 1 to 35 metres, the magnitude of the total electric field|Etot|, including the direct illumination from the source, is evaluated. In Fig. 6 MECA results are compared with those by MoM (Medgyesi-Mitschang, 1994) for the following constitutive parameters:εr=3.5, μr=1andσ=0.5S/m. On the left, |Etot|is plotted as a function of υ, while on the right the variable is the logarithm of the distance din wavelengths, dλ, so the small deviation –less than 2dB in the worst case- can be appreciated. Consequently, a high degree of overlapping in the curves of MoM and MECA clearly demonstrates the accuracy of the high frequency technique.

## 4. Fast visibility algorithm for solving the visibility problem

Because MECA (also PO) predicts null equivalent current densities in shadowed regions and non-null equivalent current densities in those regions with line of sight from the source point, a distinction between non-illuminated and illuminated surfaces is mandatory prior to the evaluation of electromagnetic levels at the observation points. This distinction is known as the visibility problem and it is widely found in computer graphics (Dewey, 1988; Foley, 1992; Bittner & Wonka, 2003), e.g. virtual reality or games, but also in the context of electromagnetic problems such as radio electric coverage evaluation or tracking applications in radar.

Within the classic algorithms, the Z-buffer and the Painter’s algorithm can be cited. Assuming the screen in the XY plane, both are based on the identification of the z-coordinate of the elements in the geometry to finally project the nearest parts of the objects. Another method is the Binary Space Partitioning (BSP) (Fuch et al. 1980; Gordon & Chen, 1991) which recursively divides the space into front and back semi-spaces, creating a binary tree at the same time. Afterwards, and in accordance with the target direction, the tree is walked and a priority list of the facets can be built.

Most of the techniques to solve the visibility problem are thought to project an image onto a display unit, so the algorithm itself has not better resolution than the pixel size. Therefore, an error is introduced, although almost imperceptible to the human eye. In order to overcome it, solutions where no approximations are made can be implemented. For example, in the Trimming method (Meana et al., 2009) when a piece of surface is partially occluded by other one, only the region in shadow is trimmed and removed from the original geometry. At the end, the remaining surface is the exact part that can be seen from a specific point of view. This type of algorithms is very time consuming but its precision counteracts this disadvantage.

To speed up the computation of the existing techniques, some modifications in the phase of preprocessing can be inserted. One option to carry out this is to split the scenario into parallelepipedic or conic macrodomains. Then the facets are classified in one or other macrodomain depending on their position. As a consequence, a preliminary discrimination is accomplished by blocks, so the facets are discarded faster. In this category, the Angular Z-buffer (AZB) (Cátedra et al., 1998) or the Space Volumetric Partitioning (SVP) (Cátedra & Arriaga, 1999) is included.

Additional acceleration can be achieved when the same algorithms are developed in Graphic Processing Units (GPUs) instead of Central Process Units (CPUs) (Ricks & Kuhlen, 2010). Fortunately, this is not restricted to visibility algorithms but it is suitable for all kind of algorithms consisting of loops, matrix operations, etc. As a consequence, generic GPUs are being extensively utilised to run the codes with smaller costs in comparison with CPU parallelisation. Even more, thanks to the use of libraries, e.g. DirectX or OpenGL (Shreiner, 2004) or the interaction between Matlab® and the graphic cards by means of the free plug-in by Nvidia® or Jacket by AccelerEyes®, the implementation of the routines turns into less complex programming.

In the following section the Pyramid method (Meana et al., 2009) is described due to its suitability to deal with flat triangular facets. Briefly, a right pyramid is built so that its vertex coincides with the source point. The i-th wall is the plane containing the source point and the i-th edge of the facet under test. Thus, a point behind this facet and inside the walls of the pyramid is occluded. This is a fast operation that is accomplished by substituting the coordinates of the observation point in the equation of the plane for all the walls.

### 4.1. Pyramid method

The scenario consists of Nfacets which have been sorted by their distance from the source point sto their barycentres, so the closest facet, denoted byF1, is always seen. Additionally, the origin of the coordinate system has been moved to the coordinates ofs.

In order to know whether a facet Fjoccludes a generic point in the geometryP, the plane in which Fjis contained is defined:

n^j([xyz]P0j)=0E35

where n^jis the unit outward normal vector and P0jis the barycentre ofFj. Similarly, the three planes containing one of the edges of the Fjand sare written as:

V2j×V1j|V2j×V1j|P=0,V3j×V2j|V3j×V2j|P=0,V1j×V3j|V1j×V3j|P=0E36

with Vijbeing the position vector of the i-th vertex (i=1,2,3) ofFj. These planes constitute the walls of the pyramid whose base is the facet under test. There are two conditions to conclude that Pcan not be seen from the source point:

1. P
2. P
i=13sign{|Vi+1j×Vij|Vi+1j×Vij|P|}=3E38

where V4j=V1jand the vertices are supposed to be in clockwise or counterclockwise order.

The algorithm determines the occlusion of points by a triangular facet as depicted in Fig. 8, but the Pyramid method can be applied to determine the existence of line of sight among facets. The process to perform this is based on simplifying the representation of the triangles choosing some important points like the barycentre, vertices, inner points and executing the algorithm for them.

Even though the expressions have been particularised for the case of triangular facets, the implementation can be accomplished for any polygon by including additional walls in the pyramid and substituting the upper limit of the summatory in Eq. (38).

A quite similar algorithm to the Pyramid method is the Cone method (Meana et al., 2009), but a right cone is defined instead of a right pyramid. Its radius is a mean value of the distances from the barycentre of the facet under test to each of its vertices. Analogically, any point behind that facet and inside the cone is occluded. This is a faster algorithm because only one operation comparing the cone angle and the angle between the observation point and the axis of the cone is carried out. On the other hand, the exactness diminishes.

## 5. Application examples

The scope of application of the Modified Equivalent Current Approximation comprises all the problems which have been analysed by means of the Physical Optics traditionally, but also the extension to dielectric and lossy materials. In the following paragraphs some of these fields where MECA could be employed are summarised.

The RCS computation is one of the most cited topics in the literature: from canonical geometries (e.g. spheres, flat plates and dihedrals) (Griesser & Balanis, 1987; Ross, 1966) for contrasting and validating results to electrically large random surfaces and complex targets (ships or airplanes) (Adana et al., 2000; Uluisik et al., 2008) with a remarkable decrease in the computational time in comparison to full wave techniques. The consideration of absorbing materials allows studying the reduction in the radar signature of aircrafts or missiles.

In order to deal with more realistic scenarios in the RCS computation, the analysis of open-ended cavities including reflections and resonances (Burkholder & Lundin, 2005) is also a line of investigation MECA can face. Likewise, MECA has proven a good accuracy when dealing with electrically large rough surfaces (Meana et al., 2010) that can model the sea surface or the orography of a rural environment. Therefore, the evaluation of the electromagnetic levels for RCS or other applications can consider not only the target itself but also its surroundings.

Another field of application is the design of reflector antennas (Boag & Letrou, 2003; Lorenzo et al., 2005) with or without dielectric radomes because of their electric size. They are usually fed by an array of antennas whose radiation pattern is determined to compute the equivalent current densities and to calculate the fields at distances much larger than the wavelength afterwards.

On the other hand, the Physical Optics approximation and its extensions can also provide some reference values to validate new algorithms or the results by some imaging or shape reconstruction techniques (Saeedfar & Barkeshli, 2006). Due to the fact that the illuminating sources are known a priori, the simulation of a set of constitutive parameters and the shape of the objects can be tackled in a fast manner.

Although the high frequency techniques have lost relevance in the recent years due to the Fast Multipole Method (FMM) –developed by Rohklin in the field of acoustic dispersion (Rohklin, 1985; Rohklin, 1990) and then extended to electromagnetic dispersion (Engheta et al., 1992) – in joint with multilevel schemes and other acceleration methodologies, the advances in the terahertz band will become notable again in the foreseeable future. This means similar applications but working at higher frequencies where the wavelength is smaller than 1 mm and, as a consequence, most of the obstacles and objects is electrically large. Some papers have already proven the validity of the asymptotic approximations at these frequencies.

In addition to the previously enumerated fields of application of the MECA method and in spite of the typical employment of Ray Tracing and Launching in joint with empirical models to deal with the radioelectric coverage evaluation (Papkelis et al., 2007), this problem must be added to that list. The reason is that it fulfils all the requisites detailed through this chapter. In the next section an application example will be shown.

Network planning and optimization in rural and urban environments can be studied for different radiocommunication systems (Global System for Mobile Communications–GSM –, Universal Mobile Telecommunications System–UMTS – or radio and television broadcasting). This example focuses on the radioelectric coverage evaluation of the General Packet Radio Service–GPRS – at the working frequency of 1800 MHz, where the scenario consists of a rural terrain of 17 km long by 10 km wide with a uniform soil characterization, εr=3, μr=1,σ=0.001S/m. Some relevant information about the scenario is that is mainly compound of mountains and valleys where there are not big villages. Thus, regions with no coverage are expected there while, on the other hand, a town on the right and a freeway at the bottom of the map suggest a better coverage in these areas.

The radio electric stations are located at their real emplacements (see Fig. 10) simulating their electromagnetic behaviour based on the available data, such as configuration parameters (e.g. power, gain), provided by the telecommunications company. As a consequence, the radiation patterns were synthesised assuming a dipole array with a different number of elements for every different antenna. Once the illumination from the sources is explicitly determined, the MECA equivalent current densities are computed and the electromagnetic field levels are calculated.

The power density at the observation points situated at 1.5 metres above the terrain is represented in Fig. 10. A threshold has been obtained by taking into account the sensibility of the receivers, so that the minimum in the colorbar has been fixed in a lower value. This means that any region in dark blue can not initiate or maintain a communication with the base station. In order to improve the coverage in specific zones, new base stations could be added and a fast evaluation can be accomplished by computing only the direct illumination due to those stations.

## 6. Conclusion

In this chapter “High frequency techniques: the Physical Optics approximation and the Modified Equivalent Current Approximation (MECA)” the whole process to compute electromagnetic field levels based on the high frequency technique Modified Equivalent Current Approximation has been presented. MECA is an extension of the Physical Optics formulation for penetrable and non-metallic objects based on the equivalence principle, so not only electric but also magnetic equivalent current densities are taken into account for dielectric and lossy materials.

One of the most relevant features of MECA is that the reflection coefficients are calculated from the general dispersion relation for the transversal electric and transversal magnetic components independently. Therefore any conductivity and relative permittivity and permeability are allowed. Then, they are inserted in the corresponding TE/TM analysis which was shown in Fig. 2 and Fig. 3, to be finally combined in order to obtain the MECA equivalent current densities. With the objective of determining the electromagnetic fields at the observation points analytically, the incident wave is supposed to be a plane wave which impinges on the surface and generates a current density distribution with constant amplitude and linear phase variation. Assuming a flat triangular facet, the radiation integral can be solved by parts.

The good behaviour was proven in the validation examples, where the results from the frequency sweep in the high frequency region agreed the theoretical values. Likewise, an excellent overlapping was obtained for different angles of incidence when dealing with a non-PEC electrically large surface.

Because one of the constraints to employ PO and MECA is the determination of line of sight between the source and the observation points, some algorithms to solve the visibility problem were described. The classic methods are complemented by acceleration techniques and then, they are translated into the GPU programming languages. The Pyramid method was explained as an example of fast algorithm which was specifically developed for evaluating the occlusion by flat facets. Undoubtedly, this can be employed in joint with the MECA formulation, but the Pyramid method can also be helpful in other disciplines of engineering.

Throughout the section “Application examples”, the way MECA becomes a powerful and efficient method to tackle different scattering problems for electrically large scenarios was satisfactorily demonstrated by means of the example consisting in the evaluation of the radio electric coverage in a rural environment. In addition to this, other fields of application were suggested from the RCS computation to imaging techniques, covering a wide range of electromagnetic problems.

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Javier Gutiérrez-Meana, José Á. Martínez-Lorenzo and Fernando Las-Heras (July 5th 2011). High Frequency Techniques: the Physical Optics Approximation and the Modified Equivalent Current Approximation (MECA), Electromagnetic Waves Propagation in Complex Matter, Ahmed Kishk, IntechOpen, DOI: 10.5772/17307. Available from:

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