Flat Optics for Surface Waves - IEEE Xplore

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 64, NO. 1, JANUARY 2016

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Flat Optics for Surface Waves Enrica Martini, Senior Member, IEEE, Mario Mencagli, Jr., David González-Ovejero, Member, IEEE, and Stefano Maci, Fellow, IEEE Abstract—The name flat optics (FO) has been introduced in a recent paper by Capasso’s group for denoting light-wave manipulations through a general type of penetrable or impenetrable metasurfaces (MTSs). There, the attention was focused on plane waves, whereas here we treat surface waves (SWs) excited on impenetrable impedance surfaces. Space variability of the boundary conditions imposes a deformation of the SW wavefront, which addresses the local wavector along not-rectilinear paths. The ray paths are subjected to an eikonal equation analogous to the one for geometrical optics (GO) rays in graded index materials. The basic relations among ray paths, ray velocity, and transport of energy for both isotropic and anisotropic boundary conditions are presented for the first time. This leads to an elegant formulation which allows for closed form analysis of flat operational devices (lenses or beam formers), giving a new guise to old concepts. It is shown that when an appropriate transformation is found, the ray paths can be conveniently controlled without the use of ray tracing, thus simplifying the problem and leading to a flat version of transformation optics, which is framed here in the general FO theory. Index Terms—Geometrical optics, metasurfaces, ray tracing, surface waves, transformation optics.

I. I NTRODUCTION

M

ETASURFACES (MTSs) [1], [2] are thin metamaterials constituted by a dense periodic texture of subwavelength elements printed on a dielectric slab. These MTSs support the propagation of surface waves (SWs). By averaging the tangential fields of the SW, the MTS can be macroscopically described through impedance boundary conditions (IBCs). Two types of MTS can be distinguished in this concern, depending on the presence or not of a ground plane below the dielectric slab. When the ground plane is present (impenetrable MTSs), the IBCs impose a relationship between the average tangential electric and magnetic fields at the interface with the free space [1], [2]. In absence of the ground plane (penetrable MTSs), IBCs impose an impedance-type relationship between the tangential electric field and the discontinuity of the tangential magnetic field across the MTS [3], [6]. The impenetrable MTS

Manuscript received March 06, 2015; revised October 18, 2015; accepted November 06, 2015. Date of publication November 12, 2015; date of current version December 31, 2015. The work of David González-Ovejero has been supported by a Marie Curie International Outgoing Fellowship within the 7th European Community Framework Programme. E. Martini is with the Department of Information Engineering and Mathematics, University of Siena, 53100 Siena, Italy, and also with the Wave Up S.r.l., 50126 Firenze, Italy (e-mail: [email protected]). M. Mencagli, Jr. and S. Maci are with the Department of Information Engineering and Mathematics, University of Siena, 53100 Siena, Italy (e-mail: [email protected]; [email protected]; [email protected]). D. González-Ovejero was with the Department of Information Engineering and Mathematics, University of Siena, 53100 Siena, Italy. He is now with the Astronomy Department, California Institute of Technology, Pasadena, CA 91125 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2015.2500259

is also known as tensor impedance surface [4], [5], [7] and it is the one we are referring to in this paper. When the shape of the constituent elements is regular enough, the impedance is a scalar and the effect of the IBC is isotropic with respect to the propagation direction of the SW. When the shape contains asymmetric features, such as slots, grooves, or cuts [7]–[9], the impedance is a tensor, thus introducing a characterization similar to anisotropy in volumetric media. This anisotropy can be exploited to extend the design capability in practical MTS-based devices [7]–[11]. Changing the dimensions of the elements arranged on a regular lattice leads to a modulation of the IBC that can be locally treated by assuming each element as immersed in a periodic environment. The spatial variability of the IBC imposes a deformation of the SW wavefront, which addresses the local wavevector along not-rectilinear paths. A simple example is illustrated in Fig. 1. It shows an MTS consisting of a regular lattice of printed square patches modulated in size along x, with larger sizes for increasing x. Assume that this MTS is excited at y = 0 with an SW beam, represented in Fig. 1 by parallel equi-length wavevectors kt . As the wave progresses along y, the increasing value of reactance along x imposes a decrease of the local phase velocity, thus producing a bending of the wavefront toward higher levels of impedance, consistently with the dispersion equation determined by the local reactance value. This phenomenon is very similar to the one described in geometrical optics (GO) for graded index materials, and leads to a poorly explored branch of wave theory, which some scientists have referred to as flat optics (FO). To our knowledge, the name FO has been introduced in a recent paper by Capasso’s group [12]; in the latter, however, more emphasis is given to space waves manipulations through MTSs, while here we use this terminology with reference to SW manipulation. Several works have been published on SW propagation on modulated MTSs [1], [10], [11], [13]–[24], some of them addressing these phenomena in the framework of transformation optics (TO) [10], [15]–[21] or leaky-wave antennas [7], [8], [22]–[24]. However, a rigorous treatment of all aspects relevant to SW FO such as ray tracing, transport of energy, and ray velocity have not been treated in literature to our knowledge, and are rigorously derived here for both cases of isotropic and anisotropic IBCs. In the isotropic case, the formulation can be seen as an adaptation to MTSs of the GO for evanescent waves introduced by Felsen in [25]. The formulation presented here opens new design possibilities for a large number of microwave devices based on SWs. In particular, whenever the objective of the design can be described through a mathematical transformation, FO can be treated trough flat TO. This extension of the more popular volumetric

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Fig. 1. Example of curved-wavefront SW supported by a modulated IBC obtained by changing the patch dimensions.

TO [26] has been introduced in previous papers [10], [19]–[21], and it is here framed in the general FO theory. It is shown here that flat TO constitutes a valid alternative to the solution of the differential equation for ray path. In [10], TO for SWs is introduced by using a Poynting vector, looking similar to what we present in Section III; however, we introduce here the transport of energy formula that allows for treating analytically the global distribution of power density on the surface, which is untreated in the literature. The paper is organized as follows. Section II presents FO for isotropic IBC. Section III extends the theory to anisotropic IBC. Section IV outlines the basic principles of flat transformation optics. Section V discusses the retrieval of the reactance from a full-wave analysis. Section VI presents numerical results. Conclusions are drawn in Section VII. II. FO FOR I SOTROPIC M ETASURFACES Let us assume that impenetrable, scalar, continuous, lossless IBCs are imposed on a planar surface on the plane z = 0 of a Cartesian reference system (x,y,z) with unit vectors x ˆ, y ˆ, ˆ z z × Ht ]z=0+ . Et |z=0+ = jX(ρ) · [ˆ

(1)

Free-space is assumed in the half-space z > 0. In (1), Et and Ht are the transverse electric and magnetic fields. The IBC in (1) is characterized by a lossless, reactive, continuous impedance Z (ρ) = jX (ρ) where the reactance X (ρ) is a scalar function of the observation point ρ = xˆ x + yˆ y on the plane z = 0. We assume that the functional variability with ρ is smooth. In the following, we will assume that the boundary conditions (1) are weakly space dispersive. Although not strictly necessary, we assume that a vertical elementary electric dipole is placed on the surface at the origin of the reference system. This vertical dipole launches an SW on the MTS. Since in practical cases the impedance is created by subwavelength patches printed on a grounded slab, X is inductive, and therefore, the SW is transverse magnetic (TM) with respect to z. Our goal is to find an asymptotic structure of the SW fields “far enough” from the dipole source that in practice means at a distance of at least one free-space wavelength from the dipole A. Eikonal Equation At a distance larger than a free-space wavelength λ = 2π/k from the exciting dipole, we assume that a z-directed magnetic

potential is of the form Az (ρ, z) = I (ρ, z) exp (−jkΨ (ρ, z)) where I (ρ, z) and Ψ (ρ, z) are complex functions of the space variable ρ assumed weakly dependent on k, as normally assumed in GO (see, e.g., weak variabil[27], pag. 111). This ity should be quantify as ∂Ψ/∂k > 1 and kz < 1, the quantities f /k and g/k 2 are negligible in (4), and the wave equation in (3) is satisfied when (7) −∇t s · ∇t s + ξ 2 + 1 = 0. The electromagnetic field can be derived from the magnetic potential in (2) as ∂2 ∂Az z (8) E = −jkζ ∇t 2 + 1 + 2 2 Az ˆ k ∂z k ∂z H = ∇t Az × zˆ

(9)

where ζ is the free-space impedance. Under the asymptotic assumptions ks >> 1 and kz < 1, one has

z Az E ∼ −jkζ j∇t sξ + 1 + ξ 2 ˆ (10) H ∼ −jk∇t s × ˆ zAz .

(11)

MARTINI et al.: FO FOR SURFACE WAVES

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Applying the boundary conditions in (1) implies z × Ht ]z=0+ = jX[ˆ z × Ht ]z=0+ Et |z=0+ = jξζ[ˆ which leads to X/ζ = ξ and then, from (7), to 2 . |∇t s| = 1 + (X (ρ) /ζ) = neq (ρ) .

(12)

(13)

The quantity neq > 1 plays the same role of the refractive index in GO. Therefore, (13) can be referred to as “eikonal equation”, in analogy with GO [27]. We note that since neq > 1, the SW satisfies the local nonradiative condition assumed at the beginning. Many results valid for GO apply also to the description of the SW supported by modulated IBCs. To derive these results, it is convenient to introduce the unit vector normal to the wavefront ˆt = ∇t s/ |∇t s| = ∇t s/neq k

(14)

which denotes the local ray propagation direction. The ray path ˆt can be defined as the trajectory described by the unit vector k orthogonal at each point ρ0 to the wavefront s = s (ρ0 ). Using (13) allows for writing (10) and (11) as

ˆt neq − jkζneq 2ˆ z exp (−jks − zkX/ζ) (15) E ∼ I0 kX k ˆt × ˆ z exp (−jks − zkX/ζ) . H ∼ −jkI0 neq k

(16)

Once the eikonal equation is solved, (15) and (16) are entirely written in terms of the local impedance, except for the smoothly varying function I0 . This latter will be defined in Section IID by imposing the conservation of energy along the ray path; alternatively, it can be derived by equating the terms of the subdominant asymptotic orders. In contrast with GO rays, which are locally transverse electromagnetic (TEM) waves, here the electric field has a component along the direction of ˆt , due to the TM nature of the SW. propagation k B. Ray Tracing and Phase Velocity Denoting with ρ the position vector of a point P and with d the elementary length of a curvilinear ray path at P, one has ˆt · dρ = d and thus the difference between the phases at ρ k and ρ0 is obtained as ρ ρ ρ ˆ kt neq ·dρ = s(ρ) − s(ρ0 ) = ∇t s(ρ ) · dρ = neq d. ρ0

ρ0

ρ0

(17) The quantity in (17) is equal to the “optical path” between the same points, which is the length of the ray-trajectory weighted by the local equivalent refractive index (Fig. 2). If ρ0 is the position of the point source and the zero phase is in there, (17) is the phase to be set in (2). The differential version of (17) is ds/d = neq ; applying ∇t to the latter, along with (14), leads to d ˆ kt neq = ∇t neq (18) d which is the trajectory equation for the ray path. Equation (18), used for ray-tracing, can be reduced to an ordinary differential equation that can be solved by the Runge–Kutta method [28].

Fig. 2. Curved ray paths for a SW launched by a dipole source on an isotropic MTS with modulated reactance X(ρ). The zone around the dipole is assumed uniform enough to launch radial rays, which eventually bend toward the area with a higher level of reactance.

It is natural to define the phase velocity of the ray as ˆt d/dt = k ˆt c/neq υ=k

(19)

where dt is the time needed for the wavefront to travel the disˆt . Like in GO, this velocity is equal to the tance d along k speed of light in free space, weighted by the inverse of the local equivalent refractive index. Due to the relationship in (13), (19) implies a lower phase velocity for a higher reactance. Hence, the portion of a wavefront on a region characterized by a higher reactance decelerates and the ray trajectories always tend to bend toward increasing impedance regions (see Fig. 1). C. Poynting Vector The real part of the Poynting vector (time-average power density per unit surface) is obtained by using (15) and (16) as 1 2 2 3 ˆ X 1 ∗ (20) S = Re [E × H ] ∼ k I0 ζneq kt exp −2zk 2 2 ζ ˆt . We will see in the next which is always directed along k section that this property does not hold for anisotropic surface impedances. We define as W the time-average energy density per unit volume on the IBC plane, obtained by summing up the electric and magnetic contributions; i.e., 1 1 2 2 |E| ε0 + |H| μ0 = W. 4 4

(21)

Using (15) and (16) in (21), it is straightforward to find W υ = S.

(22)

The latter is a well-known GO relationship, which interprets the Poynting vector as transporting energy along the curvilinear path with velocity υ. D. Transport of Energy Equation The EM field solution in (15) and (16) is not completely identified by the impedance value, since the slowly varying amplitude function I0 is not defined by the first-order asymptotic. The space dependence of this function is found by imposing conservation of energy in an elementary strip of infinitesimal width along each ray (see dashed area in Fig. 2). We denote this

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elementary strip as “ray-strip,” in analogy with the GO “raytube.” Conservation of energy along the ray-strip is satisfied by imposing ∇t · S = 0; this indeed ensures that the energy flowing on two elementary portions of wavefronts intercepted by the ray-strip at ρ1 and ρ2 is conserved (see Fig. 2), since the lat2 eral flux vanishes. Rewriting (20) as S = k 2 ζ(I0 neq ) ∇ t s/2, 2

the condition ∇t · S = 0 becomes ∇t · (I0 neq ) ∇t s = 0, namely 2

2I0 neq ∇t (I0 neq ) · ∇t s + (I0 neq ) ∇t 2 s = 0.

(23)

The latter can be also rewritten as 1 ∇t (ln (I0 neq )) · ∇t s = − ∇t 2 s. 2

(25)

Integrating (25) along the ray path from ρ0 to ρ, yields 1 ρ ∇2t s I0 (ρ0 ) neq (ρ0 ) exp − I0 (ρ) = d (26) neq (ρ) 2 ρ0 neq The previous equation is the FO version of the transport equation [27]. The amplitude diverges in those points where ∇2t s is singular; this happens where a ray bundle converges. There, the FO description in (26) loses its validity. For uniform impedance (neq = n0 = constant), the ray paths are along the radial direction, and therefore ∇2t s = n0 ρ−1 , where ρ is the distance from the source. Therefore, the integral in the exponent in (26) becomes ln(ρ0 /ρ) thus leading to I0 (ρ) = √ √ I0 (ρ0 ) ρ0 / ρ, that is the expected relationship for cylindrical spreading. Note that, being I0 (ρ) inversely proportional to neq (ρ), the Poynting vector in (20) is linearly proportional to neq (ρ). III. FO FOR A NISOTROPIC MTS Anisotropic MTSs are characterized by the IBC z × Ht )z=0+ . Et |z=0+ = j X (ρ) · (ˆ

The dominant SW mode supported by the anisotropic BC is not a purely TM mode, since the anisotropy of the BC implies a not-orthogonality of the tangential components of electric and magnetic fields. The mode is, therefore, hybridized with a TEquote and the analysis cannot leave a part the electric potential Fz . We assume that both the magnetic Az and the electric Fz potentials have the following functional dependence Az (ρ) ∼ I0 (ρ) exp (−jks (ρ)) exp (−zkξ (ρ))

(29)

Fz (ρ) ∼ V0 (ρ) exp (−jks (ρ)) exp (−zkξ (ρ)) .

(30)

The two potentials are related to the field by ∂ ∂2 ˆ + ∇t 2 Az − ∇t Fz × z ˆ E = −jkζ 1 + 2 2 Az z k ∂z k ∂z (31) 2 ∂ −jk ∂ ˆ ˆ + ∇t 2 Fz + ∇t Az × z H= 1 + 2 2 Fz z ζ k ∂z k ∂z (32) and they asymptotically satisfy the wave equation. By following the same steps as in Section II, we still arrive at the same conclusion as in (7) (33) −∇t s · ∇t s + ξ 2 + 1 = 0. To find the relationship between ξ and X, (33) is used in (31) and (32); after neglecting the higher order asymptotic contribution, one obtains

z Az + jk∇t s × ˆ E ∼ −jkζ j∇t sξ + 1 + ξ 2 ˆ zFz (34)

z Fz . H ∼ −jk∇t s × ˆ zAz − j (k/ζ) j∇t sξ + 1 + ξ 2 ˆ (35) Assuming

(27)

In absence of losses, jX satisfies the anti-Hermitian con † dition jX = − jX [29], where superscript † denotes a transpose-conjugate operation. In the cases, where the printed element exhibits two orthogonal axes of symmetry, X is real and symmetric. As such, it admits real principal values (eigenvalues) and orthogonal principal vectors (eigenvectors), thus allowing for the following representation e1ˆ e1 X1 + ˆ e2ˆ e2 X2 X=ˆ

A. Eikonal Equation

(24)

For an arbitrary function Φ, one has ∇t Φ · ∇t s = ∇t Φ · ˆt neq = neq ∂Φ/∂, where is the curvilinear length along the k ray path. Therefore, one may rewrite (24) as ∂ 1 (ln (I0 neq )) = − ∇t 2 s. ∂ 2neq

SW direction of propagation. In the following, we will make this assumption.

ˆt ; k ˆ⊥ ˆt ∇t s/neq = k z×k t =ˆ

where neq = ξ 2 + 1, (34) and (35) become 2 ˆt Az − j k ˆ⊥ E ∼ ζξ k zAz (37) t Fz kneq − jkζneq ˆ 2 ˆ⊥ ˆ H ∼ jk zFz . (38) t Az + (ξ/ζ) kt Fz kneq − j (k/ζ) neq ˆ From (37) and (38), one has

(28)

where ˆ en are the unit eigenvectors and Xn the eigenvalues. In practical MTSs, we may assume that, at least in a certain frequency range, the representation in (28) does not depend on the

(36)

Et = −jχ · ˆ z × Ht

(39)

. ⊥ ˆt ξ + k ˆ ˆt k ˆ⊥ k χ = ζ −k 1/ξ . t t

(40)

with


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Fig. 3. Representation of the eikonal equation for anisotropic media in the ∇t s plane.

Evaluating (39) for z = 0 and subtracting it from (27), yields X + χ · (ˆ z × Ht ) =0 (41) z=0+

which is satisfied for non-zero field only if det X + χ = 0.

(42)

The above equation can be solved as (ξ − ξ− ) (ξ + ξ+ ) = 0

(43)

where

⎤ ⎡ 2 2 2 ee 1 − X1 X2 /ζ (1 − X1 X2 /ζ ) X ξ∓ = ⎣ ∓ + + hh ⎦ 2X hh /ζ 2X hh /ζ X (44) ˆt (TM–TM) and k ˆ⊥ ˆt k ˆ⊥ and X ee and X hh are the k t kt (TE– TE) components of X. These are expressed in terms of the eigenvalues as X ee = X1 cos2 ψ + X2 sin2 ψ X

hh

2

2

= X1 sin ψ + X2 cos ψ

difference due to the intrinsic dependence of the equivalent ˆt ; this leads to a difficulty in tracing rays refractive index on k (we address this issue in Section III-C). In practical realizations where the impedance is obtained by printing patches over a grounded dielectric substrate, the impedance may be spatially dispersive, and both the eigenen of X may depend on values Xn and the eigenvectors ˆ the local direction of propagation. In the following, we disregard this case, assuming that the diagonal form of the tensor impedance is weakly dependent on the propagation direction. In such a case, the eigenvectors’ direction essentially depends on the geometry of the printed elements. If the printed elements exhibit two orthogonal symmetry axes, ˆ en are aligned with those axes. We note, however, that also under this assumption ˆt . neq does depend on k If the angle ψ is varied at a fixed frequency, the tip of ∇t s describes a curve in accordance with (48). This curve approximately represents an ellipse with principal axes ori 2

ented along the eigenvectors ˆ en and semi-axes namely

cos2 ψ

2

neq ≈

1 + (X1 /ζ)

2

+

1 + (Xn /ζ) , −1

sin2 ψ 1 + (X2 /ζ)

2

.

(49)

The error function f Xζ1 , Xζ2 = maxX1 ,X2 |Δneq /neq | expresses the maximum relative error between (48) and (49) under the assumption that the principal values of the impedance tensor are independent of kt . In [19], it is found that 2 2 X1 X2 Xζ1 − Xζ2 − X2 − X1 f Xζ1 , Xζ2 = (50) ζ ζ X1 X2 4 ζ + X1 + ζ X2 from which it can be verify that in a range of normalized eigenvalues from 12 to 2 the maximum error is 2.5%.

(45) (46)

ˆt where ψ is the angle between the direction of propagation k ˆt · ˆ ˆt · ˆ and the eigenvector ˆ e1 (cos ψ = k e1 , sin ψ = k e2 ). It can be seen that, up to a certain frequency, the only solution of (43) is ξ = ξ− . Under this assumption, we have ˆt (47) |∇t s| = neq ρ, k ˆt = (Xeq /ζ)2 + 1 neq ρ, k (48) where we have defined Xeq = ζξ− . Equation (48) is the generalized eikonal equation for anisotropic surface impedance. ˆt ; k ˆ⊥ ˆt with neq z×k Accordingly, we write ∇t s/neq = k t =ˆ defined by (48). In the case X1 = X2 = X, one has ξ− = X/ζ, and the eikonal equation reduces to (13). A representation of the ˆt plane is given in Fig. 3. eikonal equation in the ∇t s = neq k Although (47) is formally equal to (13) there is a significant

B. Poynting Vector The generalized eikonal equation (48) allows us to find a relationship between Az (ρ) and Fz (ρ), and therefore, between the amplitude functions I0 (ρ) and V0 (ρ). To this end, we use (39) in (41) to have

ˆt I0 + (ξ/ζ) k ˆ⊥ X + χ · jk (51) t V0 = 0. The reactance X is represented in the TM-TE basis as ⊥ hh eh ˆ ˆ⊥ ⊥ˆ ˆt X ee + k ˆ ˆt k ˆ⊥ ˆ k (52) k k k X=k X + X + k t t t t t t where X ee , X hh are given in (45) and (46) and X eh = sin ψ cos ψ(X1 − X2 ). Inserting (52) into (51) leads to V0 =

jζX eh I0 . (ξ− X hh + ζ)

(53)

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The above relationship allows for writing the fields in terms of the magnetic potential Az only ζX eh 2 ˆt + k ˆ⊥ E∼ ζξ− k − jkζn ˆ z Az kn eq t eq ξ− X hh + ζ (54) ˆ⊥ ˆ H∼ j k t + kt

ξ− X eh ξ− X hh + ζ

kneq +

kX eh n2eq ˆ z Az ξ− X hh + ζ (55)

where Az ∼ I0 exp (−jks) exp(−zkξ− ). The time-average Poynting vector can be derived from the previous expression as Xeq 1 2 3 2 ˆ S ∼ k ζneq I0 σ kr exp −2zk (56) 2 ζ where Xeq = ζξ− with ξ− defined in (44), and ˆr = k ˆt cos γ + k ˆ⊥ k t sin γ 2 X eh 1 1+ cos γ = σ ξ− X hh + ζ sin γ = −2 σ = 1+

ξ− X eh 1 σ (ξ− X hh + ζ) 2

X eh (ξ− X hh + ζ)

2

+

2ξ− X eh (ξ− X hh + ζ)

(57) (58) (59) 2 . (60)

ˆr is perpendicular to the pseudoellipse It can be seen that k representing the solution of the eikonal equation at a given frequency (see Fig. 3). C. Ray Tracing and Ray Velocity We note that the Poynting vector is aligned with the unit vecˆt due to the anisotropy of the ˆr , which is not parallel to k tor k IBC. We define a “ray velocity” as c ˆ k υr = (61) ˆr · k ˆt r neq k which differs from the phase velocity in (19), because of the ˆt at the denominator. This terminology is the ˆr · k product k same as that used in GO for anisotropic media (see, e.g., [27], p. 667). It is straightforward to demonstrate that W υr = S

(62)

where W is the electromagnetic energy for unit volume at the surface defined as in (21), but using the expressions (54) and (55) for the fields. This substantiates the definition in (61) and ˆr . The identifies the ray-tracing with the tracing of vector k eikonal equation (47) is formally equivalent to (13), thus, (17) and (18) are still valid d ˆ ˆt ˆt . kr neq ρ, k = ∇t neq ρ, k (63) d

Fig. 4. Ray tracing for an anisotropic surface.

ˆt , (63) cannot be simDue to the dependence of neq on k ply reduced to an ordinary differential equation like in the isotropic case. Furthermore, we are now interested in tracing ˆr , which is written as a function of k ˆt through (57). Therefore, k the ray tracing can be done by the algorithm described in [30]. The graphical tracing is helped by depicting eikonal ellipses along discrete sets of points on the surface. Fig. 4 illustrates an example. A group of rays launched by a point source within a narrow angular region are graphically traced through the ˆr are initially aligned, ˆt and k Poynting vector. The two vectors k assuming that a region around the source is almost isotropic. Afterward, the region becomes anisotropic and the eikonal circles become gradually elliptical. The Poynting vector starts to deviate from the phase gradient maintaining its direction ˆr form the angle ˆt and k orthogonal to the eikonal ellipse; i.e., k γ. In the case in Fig. 4, the wavefront remains almost flat, while the rays proceed along curved trajectories. The rotation of the eikonal ellipses (depicted in figure) gives the information on the local rotation of the Poynting vector, while the amplitude follows the generalization of the transport of energy equation described next.

D. Transport of Energy Equation Equations (54) and (55) completed by (47) give the expression of the field as a function of the reactance. However, the space-dependent amplitude I0 is still unknown. As done in Section II-D, we define this function by imposing conservation of energy along a ray-strip. To this end, it is observed that the Poynting vector is formally equal to the isotropic case, except ˆt is replaced by neq σ k ˆr . Therefore, the transport that neq k equation in (24) is modified as ⎡

ρ ∇t · neq σ k ˆr 1

I0 (ρ0 ) neq (ρ0 ) exp ⎣− I0 (ρ) = neq (ρ) 2

ρ0

neq σ

⎤ dr ⎦ (64)

where the integration is performed along the ray-trajectory r . The above expression allows for having the complete FO definition of the SW field when used in (54) and (55).


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IV. F LAT T RANSFORMATION O PTICS Although the design of flat lenses or other flat devices can be carried out by using ray-tracing, the same design can be nicely simplified using a flat version of transformation optics [31], [32]. In its volumetric version [26], TO establishes a rigorous analytical rule for obtaining a control on wave propagation within inhomogeneous anisotropic metamaterials. This control is achieved by properly changing the equivalent macroscopic constitutive tensors of the metamaterial on the basis of the differential parameters of a spatial coordinate transformation. The most famous use of TO is concerned with invisibility cloaks [33], [34]. The basic principle of TO for SWs is given in [19] and applied to special cases in [20], [21]. While the details of flat TO are given in the above references, here we focus the attention on the interpretation of rays within the mathematical transformation. To this end, let us consider two half-spaces: a “virtual” one, identified by a primed posiˆ + y y ˆ + z ˆ z = ρ + z ˆ z , z > 0 and a tion vector r = x x “real” one, whose points are identified by an unprimed vector r = xˆ x + yˆ y + zˆ z = ρ + zˆ z, z > 0. The two vectors are expressed in Cartesian coordinates and unit vectors of their respective spaces. Both half-spaces z > 0 and z > 0 are filled by free space, but with two different boundary conditions at z = 0 and z = 0. We assume that the virtual space possesses boundary conditions described by a scalar reactive impenetrable uniform reactance X and refraction index neq related each other by the eikonal relation (13). In the real space, instead, we define at z = 0 two complementary surfaces S ∗ and S, separated by a continuous line boundary ∂S, whose summation covers the entire plane (see Fig. 5). In S ∗ , we assume the same IBC described by the surface reactance X. Next, coordinate transformations are defined, which map the real space into the virtual space and viceversa, leaving unchanged the z-coordinate. In particular, the transformation γ maps univocally the virtual space into the real space and its inverse transformation Γ maps the real space into the virtual space. We also impose that the transformations are identities in the region of space S ∗ , where S ∗ also includes a circular region close to the source (see Fig. 5) γ (ρ ) : {x , y } → {x(x , y ), y(x , y )} in S ρ= ρ in S ∗ (65) Γ (ρ) : {x, y} → {x (x, y), y (x, y)} in S ρ = (66) ρ in S ∗ . Since the coordinate transformation γ is the identity in S ∗ , the primed space is coincident with the real space there. The Jacobian MΓ of the transformation Γ and its inverse MΓ−1 = Mγ can be expressed in covariant and contravariant bases as MΓ = x ˆ gx + y ˆ gy = γx x ˆ + γy y ˆ Mγ = gx x ˆ + gy y ˆ = x ˆγ x + y ˆγ y

S

kˆ t '

yy'

S*

x'

γ

(a)

Γ kˆ r

kˆ t

y

S*

x (b)

Fig. 5. Geometry for MTS transformation and representation of the local isofrequency dispersion curves in: (a) virtual space (circle), and (b) real space (ellipse).

phase centered at the source. At a certain distance from the source, the asymptotic phase is of type exp −jkneq ρ , where ρ = x2 + y 2 and neq satisfies the eikonal equa 2 tion neq = 1 + (X /ζ) . The SW in the transformed space is obtained the transformation to the phase fac by applying tor exp −jkneq ρ = exp −jkneq Γ(ρ) = exp (−jks(ρ)). . ˆt is the local wavevector of the SW Hence, k∇t s (ρ) = kneq k in the transformed space. Therefore, one has ˆt = neq MT · k ˆt neq k Γ

(68)

ˆt = ρˆ = [x x ˆ + y y ˆ] /ρ . where k Premultiplying both sides of (68) by MΓ−1 = Mγ and equalizing the amplitude of the result, leads to the following eikonal equation

2 ˆt · α · k ˆt n2eq = (n eq ) / k (69) where

(67)

where gx = ∂ρ/∂x , gx = ∇t x , γx = ∂ρ /∂x, γ x = ∇t x, with analogous definitions for gy , gy , γy , and γ y . Consider a TM–SW launched by a dipole source in the virtual space. Since the reactive impedance jX is uniform in space, the surface wave is a cylindrical wave,

2

2

ˆx ˆ(γ x ) + y ˆy ˆ(γ y ) + (ˆ xy ˆ+y ˆx ˆ) γ x · γ y . α = Mγ · MTγ = x (70) The eikonal equation (69) represents ellipses with axes aligned with the eigenvectors of α. By denoting these eigenvecei (ρ) , (i = 1, 2), one can write α in the diagonal tors as ˆ ei = ˆ

162


form α = ˆ e1ˆ e1 σ1 + ˆ e2ˆ e2 σ2 , where σi are the eigenvalues of α. This allows for rewriting (69) in the form

−1 2 n2eq = (neq ) cos2 ψσ1 + sin2 ψσ2 (71) ˆt . In the polar plane (neq , ψ), (71) reprewhere cos ψ = ˆ e1 · k √ sents an ellipse with semi-axes neq / σi (see Fig. 5). When compared with (49), (71) can be interpreted as the eikonal equation associated with the following anisotropic IBC (neq )2 (neq )2 −1 +ˆ e2ˆ −1 . e1ˆ e1 ζ e2 ζ X (ρ) = ˆ σ1 (ρ) σ2 (ρ) (72) Therefore, in order to implement the desired transformation, one should use a variable impedance locally satisfying (72), which is fully defined by the transformation. We underlined that the representation is not exact, since it is based on the approximation (49) for the eikonal equation. Note that, although we have considered here a point source, the above procedure is general and valid for any type of wavefront in the virtual space [35]. A. Rays and Wavefronts The transformation of phase through Γ (ρ) identifies the ray paths as mapping of the radial lines emerging from the point source. The radial line associated with ρˆ in a certain direction ϕ0 (which traces a ray from the source) is parameterized as x = cos ϕ0 , y = sin ϕ0 with ranging from 0 to infinity. Mapping this straight-line with γ onto the real plane leads to the parametric curve x = x( cos ϕ0 , sin ϕ0 ), y = y( cos ϕϕ0 , sin ϕϕ0 ), which represents the curvilinear ray path starting from the dipole in direction ϕ0 [see Fig. 5(b)]. Similarly, the wave-front at ρ0 , which is defined in the virtual plane by x = ρ0 cos t, y = ρ0 sin t, with t ∈ (0, 2π) is transformed in the curved wavefront with the parametric form x = x(ρ0 cos t, ρ0 sin t), y = y(ρ0 cos t, ρ0 sin t), where t ranges from 0 to 2π. This allows for avoiding the sometimes complex issue of ray tracing. B. Conformal and Quasi-Conformal Mappings Particular cases of transformations are conformal mappings. These mappings satisfy the condition that angles of intersection between two lines are maintained after mapping. If we assume that the transformations γ and Γ are conformal, they satisfy the Cauchy–Riemann (CR) conditions. For the transformation γ (ρ ), these conditions are ∂x/∂x = ∂y/∂y , ∂x/∂y = −∂y/∂x which imply γ x · γ y = 0, |γ x | = |γ y | 2 2 2 and ∇t x = ∇t y . Therefore, α = ∇t x I where I is the identity dyad, and from (71) one has neq = neq / ∇t x, namely, the impedance is isotropic, and therefore, realizable through geometrically regular printed elements. If the mapping is quasi-conformal, one can have a good approximation by using neq = neq / |∇t x × ∇t y|. We note that in this case the Poynting vector is orthogonal to the wavefronts.

V. R ETRIEVAL OF MTS R EACTANCE F ROM P RINTED E LEMENTS The present formulation assumes a realistic substance when one possesses practical tools for retrieving the nonuniform surface reactance tensor from a numerical model of subwavelength patches printed on a substrate. This retrieval normally starts from the assumption of local periodicity, namely from the assumption that each element is locally surrounded by equal, periodically distributed elements. The local periodicity assumption allows one to reduce the analysis to a single unit cell with periodic boundary conditions. The use of a periodic method of moments (MoM) based on the grounded slab array Green’s function is particularly useful. In principle, the analysis should be repeated for any couple of parameters (ω, kt ) and for various geometries for constructing a database; however, it is possible to significantly reduce the computational burden by resorting to some kind of physics-based parametric modeling of the equivalent impedance. Among them, the pole-zero matching method [38] approximates the eigenvalues of the tensor impedance by a pole-zero Foster-type expansion. An approximation of the previous method is possible when the ground plane is accessible to only one higher order Floquet mode. The entries of the MTS reactance can be in such a case expressed in a TE-TM basis; this directly leads to an equivalent analytical form of capacitance and inductance as a function of the Fourier spectra of the low-frequency irrotational and solenoidal components of the currents [9]. In a recent paper, a closed form solution for the equivalent reactance has been derived as a function of a single parameter [39]. The approximation is given for isotropic elements, or for direction of incidence along the principal axes of the reactance tensor. The extension to anisotropic elements can be derived as described in [35]. Other retrieval methods for impedance are found in [10].

VI. N UMERICAL R ESULTS We will consider here examples in which the equivalent refraction index and the ray paths are known in analytical form. We start from isotropic surface, treated first by the present theory and next by a realistic model that uses subwavelength printed patches, the latter analysed by a fast MoM. We consider first a Maxwell’s fish-eye (MFE) lens [27] and a Luneburg lens (LL) [13], [14]. These lenses are defined by the following equivalent refraction index neq (ρ) =

2

2n0 /(1 + (ρ/a) ) (MFE) 2 1/2

n0 [2 − (ρ/a) ]

(LL)

(73)

for 0 < ρ < a and neq = n0 for ρ > a, where a is the lens radius. The geometrical and analytical details concerning the ray paths are given in Appendix B, for both MFE and LL. The z-component of the electric field for an impedance reactance surface X = ζ 1 − n2eq illuminated by a point source placed at x = −a, y = 0 is given by (15), (17), and (26) as


163

Fig. 6. Maps of the real part of the z-component of the electric field for the MFE lens at 7.5 GHz, with radius a = 125mm (3λ), observed at a distance of 5 mm (0.125 λ) from the interface and calculated by (72) with cos(γ) tapering and an ideal reactance X (top picture), and compared with full-wave results of a modulated MTS obtained with circular patches and slotted circular patches in the insets (from [21]). The circumference delimits the printed region. Excitation is provided by an x-directed dipole placed on the left hand side.

Ez (ρ) = Ez0 neq exp

ρ ρ0

−zk 1 − n2eq

−jkneq −

∇2t s 2neq

d (74)

where Ez0 is a constant, ρ0 ≡ (x= −a, y=0) is the position of the source point, and the integral is performed along the ray path. Caustic points are avoided just stopping the integration close to them. The other components of the fields can be derived from (15) and (16). For both MFE and LL, the ray path and ∇2t s can be evaluated in analytical form as shown in Appendix. The MFE lens is a circular lens that focuses the rays coming from a point source placed on the lens rim on the diametrically opposite point [27]. Numerical results are presented in Fig. 6 for a single x-directed dipole. In the analytical calculations, in order to simulate the single dipole in FO framework, the rays emerging from the source point have been angularly weighted by cos γ [Fig. 6(a)] where γ is the angle formed by the rays launched by the source with the x axis (see Appendix). A small region around the focusing point is excluded from the FO calculation, since the FO model is not valid there. The results obtained by using (74) [Fig. 6(a)] are compared with numerical results obtained through full-wave simulation of an MTS lens implemented by using printed patches [Fig. 6(b)]. The details of this latter design are given in [21]. The full-wave analysis has been carried out with the MoM-based ADF [36] Fig. 7 shows a comparison between FO and full-wave analysis for an LL. The details of the geometry are given in the caption. We note that, while in the LL simple circular patches are used, the MFE lens requires the use of circular patches with a hole inside to increase the dynamic range of reactance.

Fig. 7. Maps of the real part of the z-component of the electric field for the Luneburg lens at 7.5 GHz, with radius a = 125 mm (3λ), observed at a distance of 5 mm (0.125 λ) from the interface and calculated by (72) with cos(γ) tapering and an ideal reactance X (top picture), and compared with full-wave analysis of an MTS realized through circular patches in the inset. The circumference delimits the printed region. Excitation is provided by an x-directed dipole placed on the left hand side.

Consider now results relevant to a point source placed on top of an anisotropic uniform MTS with principal axes forming an angle δ = 41◦ with the x-axis and ratio of the eigenvalues equal to X1 /X2 = 2.22. The effect of the anisotropy can be also seen as a linear coordinate transformation in Cartesian coordinates x = x ; y = − tan θ0 x + y , where ρ ≡ (x , y ) is the virtual plane and ρ ≡ (x, y) is the real plane. The angle θ0 of the transformation depends on δ, X1 /X2 and on neq . For neq = 1.26, one has θ0 = 15◦ . The effect of the linear Cartesian transformation on the cylindrical coordinates is shown in Fig. 8. The circles of the virtual plane are transformed into ellipses with major axis rotated by an angle δ = 41◦ with respect to the x-axis. The radial lines are transformed in radial straight lines. This means, in our interpretation, that the circular wavefronts become elliptical wavefronts with Poynting vectors still along radial lines. In the same figure, the eikonal ellipses, resulting from (71), are plotted in a lattice of intersection points among rays and wavefronts. Since the anisotropy is uniform, all the eikonal ellipses are oriented in the same direction δ. In the same ˆt and the normal to the picture, the normal to the wavefront k eikonal ellipses are also depicted. It is visible that these latter ˆr ≡ vectors are coincident with the radial unit vector, namely k ρ, ˆ as expected. Fig. 9 presents the real part of the vertical SW electric field excited by a horizontal elementary dipole. The result obtained with FO [Fig. 9(a)] is compared with the one obtained from a full-wave MoM analysis. The latter is relevant to printed elements on a substrate with εr = 14 [Fig. 9(b). The printed elements are subwavelength elliptical patches all oriented along ψ = 41◦ (see inset and caption for the exact dimensions). The full wave results have been obtained by horizontal (along x)

164


Fig. 8. Virtual isotropic plane (x < 0) and plane transformed by the Cartesian linear transformation x = x ; y = − tan (15◦ ) x + y (x > 0). The circles in the virtual plane (extended ideally for x > 0) are transformed into ellipses. The eikonal ellipses (inclined of an angle δ = 41◦ from the x-axis) are ˆr and the normal to the depicted, as well as the Poynting vector direction k ˆt . wavefront k

Fig. 10. Geometry like in Fig. 9, with excitation given by seven halfwavelength spaced dipoles. The amplitude excitations of the dipoles are weighted by a cosine functions with nulls at half-wavelengths from the final dipoles.

Fig. 9. Real part of the vertical electric field excited by a horizontal electric dipole at the MTS interface at f = 9 GHz. Upper figure, analytical FO results with a constant anisotropic impedance (X1 = 0.47ζ, X2 = 1.04 ζ, ˆ e1 = cos δˆ x + sin δˆ y, ˆ e2 = − sin δˆ x + cos δˆ y; δ = 41◦ ). Lower figure: method of moments results relevant to an MTS obtained by printing the central zone with elliptical metallic dipoles shown in the inset.

printed dipoles. The analytical FO has been modified by weighting the rays emerging from the source point by cos γ, as for the results in Figs. 6 and 7. The wavefront of the SW is elliptical as predicted in Fig. 8. The operational frequency is 9 GHz; the anisotropic-element

region is located in a strip of width 30 cm; outside this strip, the substrate is unprinted; the height h = 1.5 mm is such as to realize an equivalent refractive index neq = 1.26 as the one for the virtual plane. In both cases, the field is observed at h = 5 mm, introducing in FO the field attenuation factor exp (−hkXeq /ζ). The agreement is found excellent; the very small interference found in MoM is due to the second interface with the bare slab, not present in the FO model. Fig. 10 shows the effect of an alignment of 7 x-directed dipoles separated by a half-wavelength and weighted by a cosine function. The global effect—well replicated by our theoretical results [Fig. 10(b)]—is to orient the beam in direction θ◦ = 15◦ , while preserving the local wavefront flat. Actually, the angular deviation of the beam is due to the Huygens-like envelope of the elliptical wavefronts of each dipole contribution. This example is equivalent to the simulation of an SW beam shifter [10], [11], [20]. Using dual orientation of the printed ellipses for y positive and negative leads to a beam splitter [21]. VII. C ONCLUSION A general methodology has been presented for analysing flat devices realized using MTSs. In these devices, the wavefront of the SW is controlled by designing boundary conditions. A general FO theory describing ray paths, ray velocity, Poynting vector, and transport of energy is derived for both isotropic and anisotropic IBCs. In particular, it is shown, for both isotropic


and anisotropic case, that the ray paths are governed by eikonal equations analogous to the one for GO rays. The link with flat transformation optics is also illustrated, on the bases of the previous results obtained by the authors. Whenever the space modulated impedance is known, the approximation of the FO field is obtained by the following steps: 1) determination of the phase path and equi-phase contours from differential form of the eikonal equation (18) or its anisotropic equivalent in (63); 2) determination of the phase by integrating along the ray path; 3) determination of the distribution of energy over the device through the transport of energy equation (26) for isotropic case or (64) for the anisotropic case. When the equivalent isotropic or anisotropic impedance is found through transformation optics, steps 1) and 2) can be avoided, since the trajectory of rays for both isotropic and anisotropic cases can be defined from the analytical transformation. Concerning step 3), we emphasize that the present formulation allows for solving an issue which is always skipped in the literature, namely estimating the amplitude field distribution over the surface for anisotropic nonuniform, impenetrable impedance tensor. The theory presented here is useful to construct a large number of planar microwave and optical devices. In this concern, one can make reference to the recent papers [12], [15], [17], [32]. Finally, we mention that we have not treated here the problem of transformation from surface to leaky wave when the surface is rapidly modulated. This issue is important in metasurface antenna design and it is the subject of a future article. A PPENDIX MFE AND LL D IFFERENTIAL G EOMETRY The ray paths for an MFE lens and an LL, are given by F (x, y, b) = 0 where 2 2 x + (|y| + b) − a2 − b2 MFE F (x, y, b) = (75) 2 2 2 b ρ − a − 2x |y| b + 2y 2 LL where a is the radius and ρ = x2 + y 2 . The parameter b is related to the angle γ between the ray and the x-axis at the source point (see Fig. 11) as tan−1 (a/b) MFE (76) γ= LL. tan−1 b For the MFE lens, the ray path are portion of circles centred at (x, y) = (0, −b); for the LL, they are portions of ellipses centred at the origin of the reference system and rotated of an angle γ/2. The unit tangent to the ray path is given by ˆt (b) = (∇t F × ˆ k z) / |∇t F | where b is kept constant in makˆt as a function of x and ing the gradient. The expression of k y is obtained by expressing b as a function of (x,y) through ˆt (b) (this is found with F (x, y, b) = 0, and substituting it in k the following choice for b = (a2 − ρ2 )/2 |y| for the branches: MFE lens and b = |y| −x + 2 (a2 − y 2 ) + x2 / a2 − ρ2

165

Fig. 11. Geometrical construction of the ray paths (red lines) of (a) Maxwell’s fish-eye and (b) Luneburg lens.

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Enrica Martini (S’98–M’02–SM’13) was born in Spilimbergo (PN), Italy, in 1973. She received the Laurea degree (cum laude) in telecommunication engineering and the Ph.D. degree in informatics and telecommunications from the University of Florence, Florence, Italy, in 1998 and 2002, and the Ph.D. degree in electronics from the University of NiceSophia Antipolis, Nice, in 2002. She was with the University of Florence under a one-year research grant from the Alenia Aerospazio Company, Rome, Italy, until 1999. In 2002, she was appointed as a Research Associate with the University of Siena, Siena, Italy. In 2005, she received the Hans Christian Ørsted Postdoctoral Fellowship from the Technical University of Denmark, Lyngby, Denmark, and she joined the Electromagnetic Systems Section, Ørsted DTU Department until 2007. Since 2007, she has been a Postdoctoral Fellow with the University of Siena. In 2012, she cofounded the start-up Wave Up Srl, Florence, Italy. Her research interests include metamaterial characterization, metasurfaces, electromagnetic scattering, antenna measurements, finite element methods, and tropospheric propagation.

Mario Mencagli Jr. was born in Sinalunga (SI), Italy, in 1983. He received the B.Sc. and M.Sc. degrees (cum laude) in telecommunications engineering from the University of Siena, Siena, Italy, where he is currently working toward the industrial Ph.D. degree financed by Thales Research and Technology. His research interests include periodic structures, metasurfaces, light matter interaction, and transformation optics.

David González-Ovejero (S’01–M’13) was born in Gandía, Spain, in 1982. He received the telecommunication engineering degree from the Universidad Politécnica de Valencia, Valencia, Spain, in 2005, and the Ph.D. degree in electrical engineering from the Université catholique de Louvain, Louvain-la-Neuve, Belgium, in 2012. From January 2006 to September 2007, he was as a Research Assistant with the Universidad Politécnica de Valencia, Valencia, Spain. In October 2007, he joined the Université catholique de Louvain, Leuven, Belgium, where he was a Research Assistant until October 2012. From November 2012 to October 2014, he was a Research Associate with the University of Siena, Siena, Italy. Since November 2014, he has been a Marie Curie Postdoctoral Fellow with the Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA. His research interests include the fields of computational electromagnetics, the analysis and design of antenna arrays and metasurfaces.

Stefano Maci (M’92–SM’99–F’04) received the Laurea degree (cum laude) in electronics engineering from the University of Florence, Florence, Italy, in 1987. He is currently a Full Professor of antennas with the University of Siena, Siena, Italy, and a Director of the Ph.D. School of Information Engineering and Science, which presently includes about 60 Ph.D. students. His research interests include high-frequency and beam representation methods, computational electromagnetics, large phased arrays, planar antennas, reflector antennas and feeds, metamaterials and metasurfaces. His research activity is documented in 10 book chapters, 120 papers published in international journals (among which 80 on IEEE journals) and about 300 papers in proceedings of international conferences. His h-index is 28, with more than 3000 citations (source Google Scholar). Prof. Maci has been a member of the Technical Advisory Board of 11 international conferences and a member of the Review Board of six international journals, since 2000. He has organized 23 special sessions in international conferences and held 10 short courses about metamaterials, antennas, and computational electromagnetics in IEEE Antennas and Propagation Society (AP-S) Symposia. He has been responsible for five projects funded by the European Union (EU). From 2004 to 2007 he was WP leader of the Antenna Center of Excellence (ACE, FP6) and in 2007010 he was International Coordinator of a 24-institution consortium of a Marie Curie Action (FP6). He was the Founder of the European School of Antennas (ESoA), a post-graduate school that presently comprises 30 courses on antennas, propagation, electromagnetic theory, and computational electromagnetics, conducted by 150 teachers coming from 15 different countries. He has also been a member of the AdCom of the IEEE Antennas and Propagation Society (IEEE AP-S), an Associate Editor of the IEEE T RANSACTIONS ON A NTENNAS AND P ROPOGATION, the Chair of the Award Committee of IEEE AP-S, and a member of the Board of Directors of the European Association on Antennas and Propagation (EurAAP). He is presently the Director of ESoA, a member of the Technical Advisory Board of the URSI Commission B, a member of the Governing Board of the European Science Foundation (ESF) project NewFocus, a member of the Italian Committee for Professor Promotion, Distinguished Lecturer of the IEEE Antennas and Propagation Society, and a member of the Antennas and Propagation Executive Board of the Institution of Engineering and Technology (IET, U.K.). He has been recipient of several awards, among which the EurAAP Carrier Award 2014, and other Awards for Best Papers in conferences and journals.