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RADIO SCIENCE, VOL. 42, RS6S23, doi:10.1029/2007RS003655, 2007

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Incremental theory of diffraction for complex point source illumination A. Polemi,1 G. Carluccio,2 M. Albani,2 A. Toccafondi,2 and S. Maci2 Received 2 March 2007; revised 3 August 2007; accepted 30 August 2007; published 11 December 2007.

[1] The complex point source (CPS) is a solution of the Helmholtz equation obtained by

analytical continuation of the free-space Green’s function for complex position of the point source. The CPS representation of radiated fields can be used within a ray code to predict the interaction between an antenna and its actual environment, when standard diffraction formulations are extended to the CPS illumination. In the past, ray-based diffraction theories such as the geometrical theory of diffraction and its uniform version (UTD) were extended to complex point source fields, leaving, however, open some problematic issues concerning the ‘‘complex ray tracing’’. In this paper, the generalization of the incremental theory of diffraction (ITD) to CPS is formulated. The total field scattered by the object is given in terms of line integration along edge discontinuities of ITD diffraction coefficients plus the discontinuous geometrical optics (GO). An incremental form of the discontinuous GO is also proposed to overcome GO ‘‘complex ray tracing’’ difficulties. The final formulation is very simple and leads to accurate results that are successfully validated by comparison against a method of moment solution. Citation: Polemi, A., G. Carluccio, M. Albani, A. Toccafondi, and S. Maci (2007), Incremental theory of diffraction for complex point source illumination, Radio Sci., 42, RS6S23, doi:10.1029/2007RS003655.

1. Introduction [2] As it is well known, ray diffraction theories, such as the geometrical theory of diffraction (GTD) [Keller, 1962] and its uniform version (UTD) [Kouyoumjian and Pathak, 1974], give a description of the diffracted field in terms of rays emanating from diffraction points located on the scatterers. The diffraction point locations are calculated using ray tracing based on the minimum path Fermat principle. A field contribution is associated with each ray by resorting to a canonical problem that fits the local geometry at the diffraction point. Conversely, edge wave theories such as the incremental length diffraction coefficients (ILDC) [Mitzner, 1974; Michaeli, 1984; Knott, 1985; Shore and Yaghjian, 1988], the physical theory of diffraction (PTD) [Ufimtsev, 1991] and incremental theory of diffraction (ITD) [Tiberio et al., 2004], build up the diffracted field by integral superposition of incremental field contributions emanating from every point of the edge. At each incremental point, the contri1 Department of Information Engineering, University of Modena and Reggio Emilia, Modena, Italy. 2 Department of Information Engineering, University of Sienna, Sienna, Italy.

Copyright 2007 by the American Geophysical Union. 0048-6604/07/2007RS003655$11.00

bution is again calculated by invoking the high-frequency localization principle, and by resorting to a canonical problem that locally fits the actual geometry. The superposition of incremental contributions is performed via a numerical integration; this integration substitutes the ray tracing operation and overcomes the GTD/UTD impairment associated with curved edge caustic. When performing the analytical continuation of the UTD for complex point source (CPS) [Green et al., 1979; Suedan and Jull, 1989, 1991; Heyman and Ianconescu, 1995], the diffracted field description requires a more general ‘‘complex ray tracing’’; this consists of finding complex diffraction points whose determination and geometrical interpretation are no longer straightforward. This feature may be a strong drawback when implementing raytracing algorithms, where the estimate of diffraction points positions must be performed at first. Conversely, this problem does not appear using incremental theories. ILDC, PTD and ITD contributions are readily calculated by analytical continuation of the source coordinates into complex space. The main feature of the CPS is that its field satisfies the Helmholtz wave equation everywhere in space [Deschamps, 1971; Shin and Felsen, 1974; Felsen, 1976; Heyman and Felsen, 2001]. An appropriate CPS expansion can match more complicated wavefield in localized region of space; furthermore its paraxial approximation leads to a Gaussian beam field, whose

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observation point r and the source point r0, k is the wavenumber and an ejwt time harmonic dependence is intended and suppressed. Consider now a real vector b  (xb, yb, zb)  (b sin b b cos fb, b sin b b sin fb, b cos bb) and define the analytical continuation ~r0 of the source position into complex space through ~r0 ¼ r0  jb  ð~x0 ; ~y0 ; ~z0 Þ; ~x0 ¼ r0 sin b0 cos f0  jb sin b b cos fb ; ~y0 ¼ r0 sin b0 sin f0  jb sin bb sin fb ; 0

0

ð2Þ

0

~z ¼ r cos b  jb cos b b :

Figure 1. Reference systems and coordinates of CPS ^ and disk of singularity C. effectiveness in treating electromagnetic scattering problem has been widely demonstrated in the literature [Maciel and Felsen, 1989; Chou and Pathak, 1997; Chou et al., 2001; Chou and Pathak, 2004]. In this paper, we formulate the generalization of the ITD to CPS. In particular, section 2 describes the model and the properties of the vectorial CPS used in the formulation. Section 3 briefly summarizes the ITD formulation for (real) point source. In section 4 the ITD generalization to complex source is formulated. In order to gain physical insight, the canonical problem of an infinite straight wedge is considered in section 5 by reconstructing the canonical response using ITD integration. There it is shown that the present formulation leads to a different definition of the shadow boundaries of the geometrical optics (GO) field, which agrees with Albani et al. [2005a, 2005b]. Section 6 presents the derivation of the GO in an incremental form to simplify the implementation and shows the accuracy of the present formulation by a comparison with method of moments results.

2. Scalar and Vectorial Complex Point Source [3] Let us consider both a cartesian (x, y, z) and spherical (r, b, f) coordinate system in IR3 and let r0  (x0, y0, z0)  (r0 sin b 0 cos f0, r0 sin b0 sin f0, r0 cos b0) be the position in the real space of a spherical point source, as depicted in Figure 1. Let P be the observation point defined by r  (x, y, z)  (r sin b cos f, r sin b sin f, r cos b). The scalar free-space Green’s function G(r, r0) is defined by Gðr; r0 Þ ¼

From now on, a tilde on a variable will denote its analytical continuation in complex space. If we replace in (1) the complex source location vector (2), we obtain the analytical continuation of the scalar free-space Green’s function (still solution of the scalar Helmholtz wave equation) ~

Gðr; ~r0 Þ ¼

ð1Þ

where R = jr  r0j is the distance between a generic

ð3Þ

~ = jr  ~r0j = [(x  ~x0)2 + (y  ~y0)2 + (z  ~z0)2]1/2. where R ~ is The analytical continuation of the complex distance R a multivalued function. To satisfy the Sommerfeld radiation condition, the branch cut is defined so that ~  0. This choice implies singularities located on Re {R} ^ lying on the plane perpendicular to b, a circular disk C, that is centered at r0 (Figure 1) with radius b = jbj. At the ~ vanishes at its circumference of C^ the complex distance R branch point and the CPS presents a field singularity. Conversely, in the inner part of the disk the branch cut ~ results in a field discontinuity between introduced by R the two faces that may correspond to an equivalent sheet of currents radiating on the disk aperture. The CPS exhibits an increasing directivity for increasing b. In the half-space defined by (r  r0) b > 0 and for observation points in the neighborhood of the CPS axis, the paraxial approximation of the CPS provides a scalar rotationally ^ = b/b, symmetric Gaussian beam (GB) directed along b with waist centered in r0 and width determined by b [Heyman and Felsen, 2001]. Note that the GB is an approximate paraxial solution of the wave equation, whereas the CPS is an exact solution of the wave equation. Let us consider now the electric field radiated in the far zone by a small electric dipole p located at r0; i.e.,
 jkR ^ R ^ p e ; Ei ðrÞ ¼ jkz R 4pR

jkR

e ; 4pR

ejk R ; ~ 4pR

ð4Þ

^ = (r  r0)/jr  r0j. The electric field of a where R vectorial CPS is asymptotically obtained as a modulation of the CPS (3) by the dipole real vectorial factor in (4);

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^b ^ 0 þ Dh ðn; f; f0 Þf ^f ^0 Dðn; f; f0 Þ ¼ De ðn; f; f0 Þb

ð7Þ

is the incremental diffraction dyad, composed by soft (e) and hard (h) incremental diffraction coefficients, De;h ðn; f; f0 Þ ¼ ½d ðn; f  f0 Þ þ d ðn; f þ f0 Þ  ½d ðn; f þ f0 Þ þ d ðn; f  f0 Þ;

ð8Þ

in which the upper/lower sign applies to the soft (e)/hard (h) case, respectively, and Figure 2. Perfectly conducting wedge illuminated by a complex point source. Geometry of the local canonical problem and associated edge fixed reference system.

  sin pF 1
 n  ; d ðn; FÞ ¼ 2n cos nn  cos pF n

ð9Þ

    0  b b n ¼ j log tan = tan : 2 2

ð10Þ

in which

i.e.,  jk R~ ^ R ^ p e E ðrÞ ¼ jkz R : ~ 4pR ~i




ð5Þ

In the formulation presented next, only the Green’s function is continued into complex space, while the vectorial factor is kept real. This approximation is asymptotically good for moderately narrow CPS’ beam.

3. ITD Formulation for Real Source [4] Let us consider an arbitrarily shaped edge discontinuity in a perfectly conducting object, which is large in terms of the wavelength. According to a well established locality principle at high-frequency, a locally tangent wedge canonical problem is defined, at each point Ql along the curved edge l, which has an infinite uniform cylindrical configuration with an np exterior wedge angle. A local rectangular coordinate system is defined, with its z axis along the tangent to the edge at Ql, and x axis tangent to the upper face of the canonical wedge. The spherical coordinates (r, b, f) and (r0, b 0, f0) define the position of the observation point and of the source point, respectively, in this local reference system (see Figure 2 with b = 0). The following dyadic representation for the incremental diffracted field is obtained [Tiberio et al., 2004] dEd ðrÞ ¼ Dðn; f; f0 Þ Ei ðrl Þ

ejkr dl; 2pr

where rl identifies the incremental point Ql, and

ð6Þ

The total diffracted electric field is obtained by distributing and integrating the incremental contributions along the actual edge discontinuity l; i.e., Z ejkr d dl: ð11Þ E ð rÞ ¼ Dðn; f; f0 Þ Ei ðrl Þ 2pr l From (10), it is apparent that n vanishes at b = b0, i.e., on the Keller diffraction cone. Therefore incremental field singularities occur at b = b 0, (1  2nN)p = ±f ± f0, with N = 0, 1; i.e., at GO shadow boundaries (SBs). Such singularities in the integrand of (11) result in a discontinuity of the diffracted field that precisely compensate the relevant GO discontinuities at the SBs, thus providing a smooth, continuous total field.

4. ITD Formulation for Complex Source [5] Let us now consider a CPS as in (5) to illuminate the actual wedge (Figure 2). The procedure for defining the ITD field contributions for locally tangent planar wedges, when illuminated by a complex field, essentially consists of an analytical continuation of the dyadic diffraction coefficients defined in (7) – (10). Thus the CPS incremental diffracted field (CPS ITD) becomes

 jkr ~ d ð rÞ ¼ D ~ n~; f; f~0 E ~ i ð rl Þ e dl; ð12Þ dE 2pr ~ 0) has the same formal expression as in ~ n , f, f where D(~

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~ 0 obtained by analytical (7) – (9), with arguments n~ and f 0 continuation of n and f , and defined by "   !# ~0 b b = tan n~ ¼ j log tan ; 2 2 0 0 ~ 0 ¼ ~y ~ 0 ¼ ~x ; sin f ; cos f ~0 ~0 ~r0 sin b ~r0 sin b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~r0 ¼ ~x02 þ ~y02 þ ~z02 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~x02 þ ~y02 ~z0 0 0 ~ ~ cos b ¼  0 ; sin b ¼ : ~r ~r0

ð13Þ

In (13) pffi the branch of the square root is taken with 0 and the logarithmic function is defined by log(s) = logjsj + j arg(s) with arg(s) 2 [p, p].

5. Construction of the Canonical Wedge Field Response Via CPS ITD [6] In order to provide physical insight in the diffraction mechanism, it is useful to show how the canonical field scattered by an infinite wedge is reconstructed by integrating the incremental CPS ITD contributions in (12), adiabatically distributed on a straight edge, i.e., Z þ1

 jkrðz00 Þ ~ d ð rÞ ¼ ~ n~; f; f~0 E ~ i ðz00^zÞ e dz00 ð14Þ E D 00 Þ 2pr ð z 1 where z00 is the spatial integration variable running along the edge, and 0

~ 0) in (9). It exhibits two poles zpi, i = 1, 2 in the d(~ n, f  f 00 z complex plane defined by  zpi ¼ z0 r2 þ z~ r02 þ r~ r0 ð z þ ~z0 Þ cos n~p  ð1Þi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~02 þ jz  ~z0 j2

jr~ r0 sin n~p r2 þ 2r~ r0 cos n~p þ r
1 ~02 ; ði ¼ 1; 2Þ;

r2 þ 2r~ r0 cos n~p þ r ð17Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~0 = ~x02 þ ~y02 and n~p = where r = x2 þ y2 and r 0 ~ 0) in (7) also ~ ). The term d(~ n, f + f p  (f  f ~ 0). exhibits pairs of poles, found at n~ = n~p = p  (f + f These poles are associated with GO reflected field from the upper face of the wedge. Similarly, when y0 < 0, pair ~ 0) and n~ = n~p = p + of poles found at n~ = n~p = p + (f  f 0 ~ (f + f )  2np are relevant to the incident and reflected field respectively, from the np face of the wedge. The residues associated with these poles are equal to one half of the GO incident or reflected field. A complex saddle point zs, satisfying the equation @ 0 00 ½~r ðz Þ þ rðz00 Þz00 ¼zs ¼ 0; @z00

zs ¼

00

ð19Þ

~0 r ; ~r0 ðz00 Þ r sin bðz00 Þ ¼ 00 ; r ðz Þ

~ 0 ðz00 Þ ¼ sin b ð15Þ

In (14) and (15), qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~r0 ðz00 Þ ¼ ~x02 þ y~02 þ ðz00  ~z0 Þ2 ; 0

~r0 z þ r~z0 : ~0 rþr

If we define

jk~r ðz Þ

0

ð18Þ

is found at

00

~ i ðz00^zÞ ¼ jkz ½^r0 ðz00 Þ  ^r0 ðz00 Þ  p e : E 4p~r0 ðz00 Þ

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ð20Þ

equation (19) may be rewritten implicitly as ~ 0 ðzs Þ; b ðzs Þ ¼ b

ð21Þ

0

ðx ; y ; z  z Þ ^r0 ðz00 Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; x02 þ y02 þ ðz00  z0 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rðz00 Þ ¼ x2 þ y2 þ ð z  z00 Þ2 ;

ð16Þ

ð x; y; z  z00 Þ ^rðz00 Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; x2 þ y2 þ ð z  z00 Þ2

and (~x0, ~y0, ~z0) are defined in (2). 5.1. Analysis of the z00 Complex Plane [7] It is important to investigate pole and branch cut topology in the complex plane, which is the analytical continuation of the spatial integration variable z00 defined along the edge (Figure 2). Let us consider first the term

which is the analytical continuation of the condition of diffracted rays belonging to the Keller cone. The complex value zs in (19) is therefore the continuation of the diffraction point in complex space and the condition (21) defines the ‘‘complex ray tracing’’ as analytical continuation of the Fermat principle in complex space. The steepest descent path (SDP) associated with the saddle point zs is also depicted in Figure 3. The uniform asymptotic evaluation of the integral in (14) after contour deformation onto the SDP leads to the CPS extension of UTD. This uniform asymptotic is not reported here because it is out of the scope of the present paper. We just emphasize a basic difference with respect to the real source case. Indeed, for the conventional real point source case, the poles are located on the complex plane and the saddle point on the

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complex z00 plane, thus implying a bifurcation of the line of singularities in the real space. By considering the ~ 0), associated with the compencontribution d(~ n, f  f sation of the incident CPS field, it is found that the incident shadow boundary lines of singularities (ISB LSs) start from the incremental point z00 in parallel with ^i  (sin b i cos fi, sin b i sin fi, cos the two directions u bi), (i = 1, 2), where h i i ~0 bi ¼ 2 arctan jtjeð1Þ =mff g ;  0 ~ þ ð1Þi arg½t : fi ¼ p þ > > > 1 > ~ r ðrÞ f4  f  f3 ~ i ð rÞ þ E > E > > < 2 ~ GO ðrÞ ¼ E ~ i ð rÞ ð24Þ E f3 < f < f1 > > > 1 > > ~ i ð rÞ > E f1  f  f2 > > 2 > : 0 f2 < f < 2p Finally, the total field is represented by sum of the ITD diffracted field and the generalized GO field in (24), i.e., Figure 5. Branching of the RSB surfaces from the edge of a half plane illuminated by a CPS for r0 = 2l, b 0 = 90, 0 = 60 e b = 2l, bb = 90, b = 220. (a) 3D view. (b) 2D view.

^i(z00), (i = 1, 2), defines a pair of directions of singularity u ISB surfaces, denoted by Si, (i = 1, 2), (Figure 4). These two surfaces intersect into a curved line which belongs to ^ of the CPS. If the a plane parallel to the beam direction b beam direction exactly points to the edge at z00, the surfaces intersection becomes a straight line defined by ^ When the incremental ^2 = b. ^1 = u the unit vector u contributions are integrated along the edge, the ISB surfaces become discontinuity surfaces of the diffracted field, to compensate for the GO discontinuity. Analogous considerations hold for the reflection shadow boundary, obtained by analyzing the denominator of the contribu~ 0); the reflection shadow boundary lines tion d(~ n, f + f

~ ð rÞ ¼ E ~ GO ðrÞ þ E ~ d ðrÞ: E

ð25Þ

5.4. Numerical Results [10] As a first example, the field predicted by (25) is compared with those obtained by a UTD-based ray method obtained as a generalization to a vector problem of the formulation by Suedan and Jull [1991], since it is valid for skewed beam direction. An alternative UTD formulation can be obtained by asymptotic evaluation of the integral in (14). This evaluation is however outside the scope of the present paper and will be the subject of a future work. The example of a half plane illuminated by a vectorial CPS is considered first. The source is directed ^  ^z, located at r0 = 8l, f0 = 60°, b 0 = 50° and along p with b = l, fb = 230°, b b = 50°. The observation is performed at a distance r = 2.5l on a cone with b = 50°. In order to apply the UTD ray method it is necessary to locate a real diffraction point on the structure (zs real); ~ 0 real. this is obtained in this example by maintaining b Figure 6 shows the b component of the electric diffracted

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Figure 6. Normalized Eb component of the total electric field. Comparison with UTD solution. Each curve is normalized to the maximum value of the ITD field.

field as a function of the f angle of observation for ITD (solid line) and UTD (dash-dotted line) calculation. Each curve is normalized to the maximum value of the ITD calculation. It is worth noting the very good agreement between the two curves. [11] In the second example, a more general configuration is considered in which the half plane is illuminated by a vectorial CPS located as in the previous example but with a beam pointing with b = l, fb = 220°, b b = 60°. Observation is performed on the xy plane at a distance of r = 6.5l. Since the diffraction point is now complex, the UTD formulation proposed in literature cannot be applied; thus we present the ITD field in its individual contributions. Figure 7a shows the compensation mechanism of the GO field in (24) and the ITD diffracted field. Each curve is normalized to the maximum value of the total field. When the observation point crosses the RSB and ISB surfaces, the diffracted ITD field (dashdotted line) exhibits a discontinuity that compensates for the GO field (dashed line) discontinuity in order to restore a continuous total field (solid line). The magnification of Figure 7b clarifies the ISB compensation mechanisms. In particular, it is shown that GO incident field (dashed line) first reduces to one half just before f = 240° and then vanish just above, as illustrated in the previous paragraph. As a consequence, the diffracted field obtained by ITD calculation (dash-dotted line),

exhibits two discontinuities that proper compensate the GO field discontinuities.

6. Application to Planar Contoured Scatterers [12] The ITD formulation presented here can be applied to arbitrarily contoured planar object. The major difficulty in this case is the definition of the generalized ISB and RSB defined in section 5. For arbitrary contours, the individuation of these surfaces can be numerically inefficient. To improve the formulation, it is useful to define an incremental version of the generalized GO field as in [Albani et al., 2005a, 2005b]. 6.1. Incremental Form of the GO Field (IGO) [13] Let us consider a planar perfectly conductor scatterer with rim G (Figure 8) and tangent unit vector ^l. Let be S and P the source and the observation points, respectively, and Ql the incremental point. Let r0 = jQl  Sj and r = jP  Qlj, while ^r0 and ^r will denote the unit vector relevant to r0 and r, respectively. Consider now the electric far field radiated by a small electric dipole, with momentum p. The infinite plane containing the scatterer divides the space into two regions, A and B (see Figure 8). Region A includes the source p and region B its image p*. According to

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boundary regions, and the total field can be finally expressed as ~ ð rÞ ¼ E ~ i ð rÞ þ E ~ r ð rÞ þ E ~ d ðrÞ: E IGO IGO

ð28Þ

Another advantage in using the IGO field representation is that, the total field can be reconstruct by an integration of a unique integrand which is the summation of ITD and IGO incremental contributions, and this integrand is also regular throughout the space. The main consequence is to increase the numerical efficiency of the procedure, also when applying standard integration techniques. 6.2. Numerical Results

Figure 8. Incremental geometrical optics (IGO) reference geometry for direct field. Albani et al. [2005a, 2005b], the GO incident and reflected fields are obtained as
 jk R~ i ~ i ðrÞ ¼ jkz R ^i  R ^i  p e E IGO ~i 4pR Z ^ i  ^r ^l 1c R



2 dl; ^ G 2pr 1  ^ ~ r R

ð26Þ

i


 jk R~ r ~ r ðrÞ ¼ jkz R ^r  R ^ r  p e E IGO ~r 4pR Z ^ ~ r  ^r ^l R c



2 dl; 2pr ^ G ~r 1  ^r R

ð27Þ

~ i identify the real and the complex vector where Ri, R between the complex source and the observation point, ~ r identify the real and the complex vector and Rr, R between the complex source image and the observation point. In (26) – (27) c is a unit step function which equals 0 in Region A or 1 in Region B. The GO field is seen as a superposition of incremental contributions associated with each infinitesimal element of the rim. The main evidence is that the GO regions of existence are automatically reconstructed by the integration process. In particular for CPS, the IGO formulation prevents to know a priori the reflection and incidence shadow

[14] As a first example, the scattering from a perfectly conducting circular disk is considered. The geometrical configuration is shown in the inset of Figure 9. A horizontal y directed dipole is placed at a distance of h = 8l from a disk with radius a = 2.5l. The vector associated with the axis of the beam is defined by b = l, fb = 0°, Jb = 160°, so the beam points toward the disk. The observation is made at a distance r = 10l on the yz plane. In Figure 9 the J component of the electric field is shown obtained with IGO ITD calculation. Each curve is normalized to the maximum value of the total field. As explained in section 5.2, this particular configuration emphasize the branching of both the reflection and the incidence shadow boundaries. Again, the CPS ITD calculation provides the correct compensation of the CPS IGO branched discontinuities. The circular disk configuration has also been tested to compare the present formulation with a method of moment (MoM) technique. MoM calculations are obtained using an electromagnetic dipole as illuminating source, with proper far field pattern function. This latter was calculated in the far field for a CPS dipole, sampled over the solid angle sphere, and finally imported into MoM code as an arbitrary radiation pattern. Again, a horizontal y directed dipole is placed at a distance of h = 8l from the disk with radius a = 2.5l. The beam pointing vector is defined by b = l, fb = 90°, Jb = 160°. The observation is made at a distance r = 10l on the yz plane. Figure 10 shows the comparison between IGO ITD (solid line) and MoM (dash-dotted line) calculation of the total electric field J component. Each curve is normalized to the maximum value of the IGO ITD field. A good agreement between MoM and CPS incremental technique is found except for the region close to grazing, where double diffraction coefficients need to be introduced to compensate for the discontinuity of incremental diffraction

Figure 7. (a) Normalized Eb component of the electric field with both GO and ITD curves. (b) GO ITD compensation mechanism which is emphasized around the ISB. Each curve is normalized to the maximum value of the total field. 9 of 13

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Figure 9. Normalized E# component of the electric field. IGO ITD compensation mechanism. Each curve is normalized to the maximum value of the total field. coefficient in (6), as expected. This is particularly evident for J = 90°, while for J = 90° the discrepancies between the two curves are masked by the GO field. Two final examples relevant to a square plate of perfectly conducting material illuminated by a CPS are considered. Also in these cases, in order to verify the effectiveness of the proposed formulation, comparisons with MoM calculation have been made. The geometrical configuration relevant to the first example is shown in the inset of Figure 11. The plate has a side of l = 4l and the CPS dipole source is placed at h = 8l from the plate. The beam axis points toward the center of the plate, with b = l. The observation is made at a distance r = 10l on the yz plane. Figure 11 shows the comparison between IGO ITD (solid line) and MoM (dash-dotted line) calculations for the total electric field. Each curve is normalized to the maximum value of the IGO ITD field. Throughout the

scan plane, they are found in good agreement except for observation near grazing, where again double diffraction contributions need to be introduced to improve the accuracy of the prediction. Finally in the example of Figure 12, the beam axis is tilted toward a side of the plate, with b = l, fb = 90° and Jb = 160°. The observation is made at a distance r = 10l on the plane defined by the relations f = 50° and 180° < J < 180°. Also in this last example, the curves obtained by IGO ITD formulation has been found in good agreement with that obtained by MoM simulations.

7. Conclusions [15] In this paper, the extension of the incremental theory of diffraction to complex points source was presented. This ITD extension has been motivated by

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Figure 10. Normalized E# component of the electric field. Comparison between IGO ITD and MoM solutions. Both curves are normalized to the IGO ITD field. the need of an accurate and efficient way to describe the overall diffraction of fields generated by antennas in complex environment, where the radiation pattern can be synthesized by Gaussian beams in the paraxial region, or, more generally, by complex point sources (whose scalar Green’s function is solution of Helmholtz wave equation everywhere). The formulation is obtained by a proper analytical continuation in complex space of the dyadic

diffraction coefficients obtained originally in real space. It is found that diffraction coefficients experience branched singularities across reflection and incidence shadow boundaries. When integrated along the edge, these singularities properly compensate for the discontinuities of the GO field. In order to improve the accuracy of numerical integration, a complex source incremental GO was proposed. The CPS IGO has the same disconti-

Figure 11. Normalized E# component of the electric field. Comparison between IGO ITD and MoM solutions. Both curves are normalized to the IGO ITD field. 11 of 13

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Figure 12. Normalized E# component of the electric field. Comparison between IGO ITD and MoM solutions. Both curves are normalized to the IGO ITD field. nuities of the CPS ITD and when added to this latter, provides a regular contribution of the total field, which can be distributed and integrated very efficiently. Furthermore, the ITD representation of the diffraction process seems to allow a quite straightforward interpretation of the physical mechanism involved in this extension, thus ITD is a good candidate to be widely employed inside codes for the description of the scattering and diffraction from complex antenna systems. The formulation was tested with both an extension to complex source of the standard UTD formulation and a MoM technique. Results associated with a perfectly conducting disk and a square plate showed the effectiveness of this incremental formulation for describing radiation and scattering phenomena in presence of an illuminating complex source. [16] Acknowledgment. The work was supported by the Antenna Center of Excellence (ACE), EU contract 026957.

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M. Albani, G. Carluccio, S. Maci, and A. Toccafondi, Department of Information Engineering, University of Sienna, Via Roma 56, Sienna, 53100, Italy. ([email protected]; [email protected]; [email protected]; albertot@ dii.unisi.it) A. Polemi, Department of Information Engineering, University of Modena and Reggio Emilia, Via Vignolese 905, Modena, 41100, Italy. ([email protected])

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