Modal Properties of Surface and Leaky Waves ... - La Sapienza

36 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 1, JANUARY 2005 

Modal Properties of Surface and Leaky Waves 

Propagating at Arbitrary Angles Along a Metal 

Strip Grating on a Grounded Slab 

Paolo Baccarelli, Member, IEEE, Paolo Burghignoli, Member, IEEE, Fabrizio Frezza, Senior Member, IEEE, 

Alessandro Galli, Member, IEEE, Paolo Lampariello, Fellow, IEEE, Giampiero Lovat, Student Member, IEEE, and 

Simone Paulotto, Student Member, IEEE 

Abstract—In this paper, a full-wave analysis is presented of the 

dispersion properties of modes supported by a grounded dielectric 

slab periodically loaded with metal strips, which represents a 

canonical configuration employed in planar microwave antennas 

and arrays and in the realization of artificially hard and soft surfaces. 

Propagation of surface and leaky modes at arbitrary angles 

is considered here, without any restrictive assumption on the 

values of the involved physical and geometrical parameters. Spectral 

properties of modes are studied, by deriving generalized conditions 

for establishing the proper or improper nature of the spatial 

harmonics in the Floquet representation of the fields. The proposed 

approach, based on a full-wave moment-method discretization of 

the relevant electric-field integral equation in the spectral domain, 

is validated through comparisons with the available data in the literature. 

Novel results are presented which illustrate the continuous 

evolution of modes as a function of the propagation angle along the 

grating, both in surface and leaky propagation regimes. 

Index Terms—Artificially hard and soft surfaces, leaky waves, 

metal-strip grating, spectral-domain approach, surface waves, unit 

cell. 

I. INTRODUCTION AND BACKGROUND 

AGROUNDED dielectric slab periodically loaded with 

metal strips is a canonical structure employed in a variety 

of microwave components and antennas. Significant examples 

are leaky-wave antennas based on the excitation of a TE leaky 

mode propagating orthogonally to the strips [1], [2], strip-element 

phased arrays [3], polarizers for two-dimensional (2-D) 

arrays with low cross-polarization [4], [5], and artificially 

hard and soft surfaces [6]–[8]. Scattering of plane waves from 

metallic structures covered with a dielectric layer periodically 

loaded by metal strips has also been studied, with particular 

reference to wedge configurations [8]–[10]. 

A basic problem in the analysis of structures involving the 

metal-strip grating on a grounded dielectric slab (MSG-GDS) 

is the determination of the dispersion properties of its modes. 

Many results are available on the dispersion properties of 

modes propagating orthogonally to the strips: in this case, 

the 2-D nature of the problem allows us to decouple the TE 

and TM polarizations; moreover, effective multimode network 

Manuscript received January 14, 2004; revised September 17, 2004. 

The authors are with the Department of Electronic Engineering, “La 

Sapienza” University of Rome, Rome 00184, Italy (e-mail: lampariello@mail.die.uniroma1.it). 

Digital Object Identifier 10.1109/TAP.2004.840529 

0018-926X/$20.00 © 2005 IEEE 

representations exist which enable a rapid and accurate determination 

of the TE and TM modal properties via a transverse 

resonance technique (TRT), in both guided and leaky propagation 

regimes [11]–[13]. The study of modes propagating 

parallel to the strips is considerably more involved, because 

of their hybrid nature: a TRT may be employed also in this 

case, but the relevant transverse network is accurate only for 

values of the strip width large with respect to the spatial period 

[14]; however, valid results can be obtained by a full-wave 

moment-method approach in the unit cell with perfect electric 

or magnetic conductors as lateral walls (see, e.g., [15], [16], 

where linear arrays of microstrip leaky-wave antennas based 

on the excitation of the leaky mode are considered). 

Very few results are available on propagation of modes 

at arbitrary angles with respect to the metal strips. To our 

knowledge, the only existing study of this generalized problem 

has been presented by Bellamine and Kuester in [17]. In their 

paper, the authors model the periodically-loaded air-dielectric 

interface by means of an equivalent homogeneous anisotropic 

boundary condition, under the assumption that both the spatial 

period and the substrate thickness are small with respect to the 

free-space wavelength. In this way, a closed-form dispersion 

equation can be obtained, whose validity is however restricted 

by the above-mentioned conditions only to sufficiently low 

frequencies. Some results on surface waves propagating at 

arbitrary angles have also been presented in [4] and in [18]. 

In [4] a full-wave analysis is performed, but surface waves 

are only briefly addressed in connection with scan-blindness 

effects, while in [18] asymptotic boundary conditions valid for 

spatial periods small with respect to wavelength are adopted. 

In this paper, we aim at studying modal properties of waves 

guided by the MSG-GDS propagating at arbitrary angles with 

respect to the strips without any restrictive assumption on the 

values of the involved geometrical and physical parameters. 

Both real and complex modes are considered. For complex 

modes, it is assumed that at least one spatial harmonic in the 

Floquet representation of the field propagates as a uniform 

plane wave in the grating plane; this allows us to formulate an 

eigenvalue problem in terms of only one complex propagation 

constant (as it was done, e.g., in [19] for a different planar 

structure). Since the structure is transversely open, each spatial 

harmonic may be either proper or improper, i.e., attenuating 

or growing at infinity in the air region above the grating plane, 

respectively; generalized conditions are derived here which

BACCARELLI et al.: MODAL PROPERTIES OF SURFACE AND LEAKY WAVES PROPAGATING 37 

allow us to determine when each spatial harmonic changes its 

proper or improper character as one of the involved physical 

parameters (e.g., frequency or propagation angle) is varied, 

thus changing the spectral character of the considered mode. 

Numerical results are provided, based on a full-wave moment-method 

discretization of the relevant electric-field integral 

equation (EFIE) in the unit cell in the presence of an arbitrary 

phase shift between adjacent cells. The adopted numerical 

approach, originally developed in [16] for the analysis of linear 

phased arrays of microstrip leaky-wave antennas, has been 

adapted here to solve the homogeneous eigenvalue problem 

of determining surface and leaky waves propagating along an 

arbitrary oblique direction in the grating plane. The accuracy 

of the proposed approach is validated by comparison with the 

dispersion data available in the literature. Novel interesting 

results are presented which illustrate the continuous evolution 

of modes from propagation orthogonal to propagation parallel 

to the strips, as a function of the angle, in different frequency 

ranges and propagation regimes. 

The paper is organized as follows. In Section II the Floquet 

representation of the sought modal fields is described and 

the main features of the adopted moment-method approach 

are summarized. In Section III spectral properties of modes 

are studied. In Section IV numerical results are presented for 

different structures. Finally, in Section V conclusions are drawn 

on the various results of this study. 

II. ANALYSIS 

The MSG-GDS considered here is shown in Fig. 1(a) with the 

relevant coordinate system; it consists of an infinite array of perfectly-conducting 

metal strips of width and negligible thickness, 

periodically arranged along the direction with spatial 

period ,infinitely long in the direction, placed on a grounded 

lossless dielectric slab of relative permittivity , relative permeability 

, and thickness . Modal fields of the MSG-GDS 

will be considered in the following, with a time dependence 

assumed and suppressed throughout. 

A. Homogeneous Modes Propagating at an Arbitrary Angle 

The MSG-GDS is uniform along the direction and periodic 

along the direction. Therefore, the sought modal fields 

have an exponential dependence on , with propagation constant 

, and they admit a Floquet representation along 

involving an infinite number of spatial harmonics with propagation 

constants ; 

the electric field may be expressed as [20] 

(and similarly for the magnetic field ), where 

is the wave vector in the plane of the th spatial harmonic 

and is the position vector in the same 

plane. 

(1) 

Fig. 1. Reference MSG-GDS: (a) Edge view, with geometrical and physical 

parameters and (b) 3-D view, with the phase vector and attenuation vector 

of the aHspatial harmonic of a mode propagating along the plane in 

the direction of the unit vector at an angle 0 with respect to the positive 

axis. 

By letting , it can be observed that each spatial 

harmonic propagates in the plane with a different phase 

vector but with the same attenuation vector . We aim here 

at studying homogeneous modes of the MSG-GDS, defined as 

modes with at least one spatial harmonic (the , without 

loss of generality) propagating as an homogeneous plane wave 

at an angle in the plane [19]. This corresponds to assume that 

where is a complex scalar propagation constant and 

is the real unit vector whose direction makes an angle with 

respect to the positive axis [see Fig. 1(b)]. Therefore, from (2) 

and (3) we obtain 

The assumption of a homogeneous mode is certainly satisfied 

for propagation at (i.e., across the strips) and 

(i.e., along the strips), and also when , i.e., for 

real modes; moreover, it allows us to formulate an eigenvalue 

problem in terms of only one complex unknown (i.e., the propagation 

constant ), as described in the next subsection. 

(2) 

(3) 

(4)


Fig. 2. Phase vectors of the spatial harmonics aH, 61 of a physical proper real mode in the plane of the phase constants ( , ). The circles (shaded areas) 

are the visible spaces for the spatial harmonics. 

B. Spectral-Domain Approach in the Unit Cell 

The periodicity of the structure along allows us to restrict 

the analysis to only one spatial period (unit cell), e.g., the one 

limited by the planes and [20]. The modal 

current on the strip in the unit cell can be written as 

The unknown can be represented as a linear combination 

of a finite number of basis functions , , 

whose coefficients may be found through a Galerkin momentmethod 

discretization of the relevant electric-field integral equation 

(EFIE) [16]. This leads to a linear algebraic system whose 

coefficient-matrix elements are given by infinite series 

In (6) the tilde indicates a Fourier transform with respect to 

and is the spectral dyadic Green’s function of the 

background GDS, which is known in a simple closed form [21]. 

Once a propagation direction has been fixed, the matrix elements 

are seen to depend on the propagation constant 

through , , according to 

(4). The eigenvalues are then obtained as zeroes of the determinant 

of the coefficient matrix and the modal currents 

are the corresponding eigenvectors. 

We have adopted here standard weighted Chebishev polynomials 

as basis functions for the longitudinal and transverse components 

of the unknown current [20]; for details on the implementation, 

including numerical acceleration of the series in 

(6), we refer to [16]. 

III. SPECTRAL PROPERTIES OF HOMOGENEOUS MODES 

A basic point of the modal analysis is related to the behavior 

of the electromagnetic field of each spatial harmonic in the air 

region above the grating plane. In such a region, the 

field of the th spatial harmonic depends on the transverse 

coordinate as 

given by 

, where the propagation constant is 

(5) 

(6) 

(7) 

The presence of a square root in (7) makes the transverse 

propagation constant a two-valued function of the modal 

propagation constant . The determination of with 

corresponds to fields exponentially 

decaying at infinity in the direction (proper determination), 

while the determination with corresponds 

to fields exponentially growing at infinity (improper 

determination). The proper or improper nature of each spatial 

harmonic in the infinite series in (6) determines the spectral 

character of the modal solution [22]. 

The simplest case is that of a mode with a real propagation 

constant . From (7) we then find that is either 

purely real or purely imaginary. The mode is physical only if 

it is attenuated in the air region at infinity along , i.e., is 

purely imaginary with for all . This is possible if the 

argument of the square-root function in (7) is negative for all , 

i.e., is outside the visible space of the th spatial harmonic, 

defined by the circle in the , plane 

for all (see Fig. 2), and the proper determination is chosen for 

all the spatial harmonics. 

As is well known from the theory of open periodic structures, 

by varying a physical parameter (e.g., the frequency or, in 

the present case, the propagation angle ), a proper real mode 

may evolve into a complex mode (e.g., in stopband or in radiative 

ranges), and one or more spatial harmonics may become 

improper [22]. In the latter case, the mode changes its spectral 

character, and this may happen only if the solution crosses the 

branch point or branch cut associated to one (or more) of the 

infinite square-root functions in (7) in the plane. 

In the following analysis, conditions for branch-point and 

branch-cut crossing will be derived. They will be expressed in 

geometrical terms with reference to the evolution of the phase 

vector , by varying one physical parameter, in the plane of 

the phase constants ( , ). 

A. Conditions for Branch-Point and Branch-Cut Crossings 

The branch cut (BC) in the complex plane which separates 

the proper and improper Riemann sheets for the squareroot 

function in (7) is defined by the conditions , 

(8)


as 

. By letting , these can be written 

(9a) 

(9b) 

Branch-point (BP) crossing occurs when the equal sign holds in 

(9b). 

In order to satisfy (9a), we see that two cases are possible, 

namely 

or 

Let us consider them separately. 

1) Case 1: When (10) holds, from (7) we have 

(10) 

(11) 

(12) 

Therefore, by considering the definition of visible space given 

in (8), it can easily be seen that (9b) is satisfied when the curve 

described by the phase vector in the ( , ) plane lies in 

the visible space of the th spatial harmonic and the mode has 

a zero attenuation constant. In particular, BP crossing occurs 

at the edge of such a visible space. In the special case 

(propagation across the strips) the well-known condition for BP 

crossing is recovered [22]. 

2) Case 2: When (11) holds, by substituting it in (9b) a 

second-order inequality in is obtained 

(13) 

The right-hand side of (13) is negative inside (and positive outside) 

the angular sector delimited by the half-lines tangent to the 

th visible-space edge through the origin of the ( , ) plane 

(see Fig. 3), so that (13) is certainly satisfied inside this sector for 

any value of . On the other hand, (11) corresponds to a circle 

in the ( , ) plane centered at with radius 

(see Fig. 4). Therefore, in this case the BC is crossed when the 

curve described by the phase vector in the ( , ) plane 

crosses the circle of (11): i) inside the th visible space with 

any attenuation constant [see Fig. 4(a)], or ii) outside the th 

visible space for any value of the attenuation constant such that 

(14) 

[see Fig. 4(b)]. In the latter case, BP crossing occurs when the 

equal sign holds in (14). 

It can be observed that (11) is equivalent to the condition 

of orthogonality between the phase vectors and (see 

Fig. 4). In fact 

(15) 

Fig. 3. Illustration of Case 2 of BC or BP crossing, for the a 0I spatial 

harmonic: angular range (shaded area) where the right-hand side of (13) is 

negative. 

Fig. 4. Illustration of Case 2 of BC or BP crossing, for the a 0I spatial 

harmonic: orthogonality of the phase vectors and when lies on 

the circle of (11) (dashed line). (a) Case of inside the a 0I visible space; 

(b) case of outside the a 0I visible space. 

In the special case (propagation across the strips) the 

well-known condition for branch-cut crossing is recovered, 

which corresponds to the th spatial harmonic radiating at 

broadside [22]. 

IV. NUMERICAL RESULTS 

In this section, numerical results on modal properties of 

different structures will be presented. In Section IV-A, comparisons 

are shown with dispersion diagrams available in the 

literature, obtained with different techniques, in order to validate 

the accuracy of the proposed full-wave moment-method


approach. In Section IV-B, results on two specific structures 

are presented in different frequency ranges, which show various 

peculiar features of modal propagation at arbitrary angles. 

Illustrative results on the novel spectral features discussed in 

Section III are given in Section IV-C. 

A. Comparisons With Independent Results 

In Fig. 5, the normalized phase constant 

a function of the normalized propagation angle 

is reported as 

for a real proper mode of a structure with , , 

, and , corresponding to Fig. 2 of [17] 

(case , of that paper). The mode is classified 

as a ‘grating mode’ [4], [17], [18] since its phase constant 

tends to infinity as tends to 0 . The agreement with 

the reference results in Fig. 2 of [17] is excellent. In Fig. 6, the 

normalized phase constant and the normalized attenuation 

constant are reported as a function of the normalized 

propagation angle for a complex improper 

(leaky) mode (with the spatial harmonic improper) of a 

structure with , , , and , 

corresponding to Fig. 4 of [17]. Again, the agreement is excellent 

in the whole angular range. 

As a further test on the accuracy of our full-wave approach, 

comparisons have been performed with results obtained through 

a TRT approach for propagation at . In Fig. 7, a Brillouin 

diagram ( vs. ) is shown for the spatial 

harmonic of the fundamental mode propagating on 

a MSG-GDS with , , , 

, corresponding to Fig. 4 of [23]. The corresponding 

unperturbed modes of the parallel-plate waveguide (PPW) 

and grounded dielectric slab (GDS) are also reported. Again the 

comparison shows exact agreement both in the proper range, 

between and , and in the improper 

range (with the 

. 

spatial harmonic improper), below 

B. Modal Features of Propagation at Different Angles 

In order to illustrate the continuous mode evolution from 

propagation across the strips to propagation along the 

strips , we consider here a MSG-GDS with , 

, 

chosen as in [24]). 

, and (parameters 

In the case , as mentioned in Section I, modes are either 

TE or TM and may be regarded as the TE and TM modes of 

a GDS perturbed by the presence of the metal-strip grating or, 

alternatively, as the TE and TM modes of a PPW perturbed by 

the presence of slots in the upper plate. In Fig. 8(a), the Brillouin 

diagrams of the fundamental (perturbed) and modes 

are reported in the first Brillouin zone . The diagram 

is symmetric with respect to the vertical axis ; 

thick lines correspond to the direct spatial harmonic, 

which contributes to field representation for positive values of 

the coordinate; thin lines correspond to the reverse 

spatial harmonic, which contributes to field representation for 

negative . The mode has cutoff at zero frequency, while 

the mode has cutoff at , i.e., between 

Fig. 5. Full-wave normalized phase constant a as a function of the 

normalized propagation angle @WH 0 0AaIVH for a real proper mode of 

‘grating’ type of a structure with 4 aPXP, aHXP, a aHXWW, and 

a aIXH, corresponding to Fig. 2 of [17] (case —a aHXHI, a aIof 

that paper). 

Fig. 6. Full-wave normalized phase constant a (solid line) and 

normalized attenuation constant a (dashed line) as a function of the 

normalized propagation angle @WH 0 0AaIVH for a complex improper (leaky) 

mode (with the aHspatial harmonic improper) of a structure with 4 aPXP, 

aHXU, a aHXW, and a aIXH, corresponding to Fig. 4 of [17]. 

(GDS cutoff) and 

(PPW cutoff). Real proper branches are represented with 

solid lines; complex proper branches are represented with short 

dashed lines: they occur in the stopband regions (shaded areas), 

where the modes are bound and attenuated, and outside the 

bound-mode triangle, where the modes are radiative due to the 

spatial harmonic radiating in the backward quadrant 

[22]. 

When the propagation angle is , the structure may 

be regarded as an infinite linear array of microstrip lines in the 

absence of a phase shift between elements. In this case modes 

are hybrid; the planes , , that divide each 

metal strip, are symmetry planes which may be either PMC or 

PEC; modes are classified accordingly as even or odd modes. 

It is to be noted that symmetry requires that also the planes 

, , located in the middle of adjacent 

strips, are PMC for even modes while they are PEC for odd 

modes, i.e., all the symmetry planes are of the same kind. The 

latter property directly affects the nature of leaky modes; in fact, 

the air region of the unit cell above the MSG-GDS, bounded 

by such PMC or PEC walls, can be regarded as a parallel-plate


Fig. 7. Full-wave Brillouin diagram ( a% vs. a%) for the aHspatial 

harmonic of the fundamental TE mode propagating on a structure with 4 a 

PH, a HXIR , a HXQQV , a a HXT, corresponding to Fig. 4 of 

[23]. Legend: proper solutions (solid line); improper solutions (dashed line). The 

edges of the first bound-mode triangle are represented by light dotted lines. The 

unperturbed „i modes of the parallel-plate waveguide (PPW) and grounded 

dielectric slab (GDS) are represented with gray dotted lines. 

waveguide, which supports a TEM wave with propagation constant 

equal to and with the magnetic or the electric field orthogonal 

to the PMC or PEC planes, respectively. A leaky mode 

may radiate into such a TEM wave propagating at an angle in 

the plane, so that modal cutoff for all the higher-order modes 

occurs when . 

In Fig. 8(b) the dispersion diagrams of the first two even 

modes and of the first odd mode are reported in a frequency 

range from 0 to 18 GHz. The mode labeled corresponds 

to the fundamental mode of the isolated microstrip, perturbed 

by the presence of the other array elements. It is always 

real proper and has cutoff at zero frequency. The mode labeled 

is the perturbed first higher-order mode of the isolated 

microstrip line [16]. It is proper real above its cutoff frequency 

and its phase constant is close to the phase 

constant of the unperturbed mode of the GDS above its 

cutoff frequency ; in fact, the TE modes of the 

GDS are perturbed very little by narrow strips placed along their 

direction of propagation, since their electric field is transverse 

to the strips. Below cutoff, the mode has a real improper 

continuation (with the spatial harmonic improper); such 

real improper solution originates, at lower frequencies, a complex 

improper solution. The mode labeled is an additional 

even mode with cutoff at zero frequency, where its phase constant 

is . It exists due to the presence of PMC lateral 

walls in the unit cell, thus it does not have a counterpart in the 

isolated microstrip line. 

In the next three figures, the evolution of the modes reported 

in Figs. 8(a) and (b) is shown as a function of the propagation 

angle in the range [0 ,90 ] at different frequencies. Results 

will be presented in the plane of the normalized phase constants 

( , ); by virtue of the symmetries of the structure, 

results for propagation angles outside the range [0 ,90 ] may 

be obtained by specular reflection of the reported curves with 

respect to the horizontal and vertical axes, which correspond to 

propagation at and at , respectively. 

Fig. 8. (a) Brillouin diagram ( a% vs. a%) for the fundamental TM and 

TE modes propagating across the strips @0 aH A; direct modes (thick lines), 

reverse modes (thin lines); the shaded areas represent their stopband regions. 

The edges of the first bound-mode triangle are represented by light dotted lines. 

On the right vertical axis also frequencies are reported in GHz. (b) Dispersion 

diagram ( a and a vs. frequency ) for the first two even and the 

first odd modes propagating along the strips @0 aWH A. Physical parameters: 

4 a PH, a HXIR , a HXQQV , a a HXP. Legend: normalized 

phase constant a or a%: proper real (solid line); improper real (dotted 

line); complex proper (short dashed line); complex improper (long dashed line). 

Normalized attenuation constant a : dashed line. 

The case is considered first. From Fig. 8(a) 

it may be seen that at this frequency the mode is real 

proper while the mode is below cutoff; on the other 

hand, from Fig. 8(b) it may be seen that at this frequency the 

modes and are above cutoff, while the mode 

is below cutoff in a real improper range. In Fig. 9 the 

evolution of these modes by varying the propagation angle 

from to may be observed in the range 

[0 ,90]. In particular, the mode becomes a ’grating 

mode,’ whose phase constant tends to infinity as tends 

to 0 , describing a slowly-varying periodic line almost parallel 

to the horizontal axis. On the other hand, the mode 

remains proper real with a bounded phase constant for 

all propagation angles, describing an approximately-elliptical 

curve outside the visible space, and finally becomes 

the fundamental mode at [see Fig. 8(a)]. This 

means that the surface wave related to this proper real mode 

may propagate at all angles on this structure. Finally, two real 

improper branches originate from the vertical axis


Fig. 9. Modal evolution in the ( a , a ) plane by varying the 

propagation angle 0 in the range [0 ,90] for a structure as in Fig. 8 at 

a IH qr. Legend: proper real solutions (black solid line); improper 

real solutions (dotted line); complex improper solution (dashed line). The 

visible-space edge for the a H spatial harmonic is represented by a gray 

solid line. 

Fig. 10. Same as in Fig. 9, at a IQ qr. 

in Fig. 9: the lower one is the continuation of the mode 

[see Fig. 8(b)], while the upper one is superimposed to the 

mode at . By decreasing , the two real improper 

branches merge to form a complex improper solution, which 

ends at on a higher-order TM leaky mode [not shown 

in Fig. 8(a)]; such a leaky mode is physical, since the curve is 

located inside the visible space of the spatial harmonic, 

however its attenuation constant (not shown here) is very high 

so that it is expected to give a negligible contribution to the 

representation of fields excited by finite sources. 

We consider now a higher frequency ; from 

Fig. 8 it may be seen that again at the only real proper 

mode is the mode, while at the mode is now 

above cutoff. In Fig. 10 the relevant modal evolution is reported. 

The fundamental mode again becomes a grating mode; 

however, the mode now is proper real and it remains proper 

real with a bounded propagation constant for all , becoming at 

the mode. On the other hand, the mode remains 

proper real only down to about , then its curve 

Fig. 11. Same as in Fig. 9, at a IV qr. 

becomes tangent to the visible-space edge and the 

spatial harmonic becomes improper. The mode thus becomes 

real improper, and it merges with the other real improper solution 

(superimposed to the mode at ) giving rise 

to a complex improper solution, which ends on a higher-order 

complex TM mode at ; such a leaky mode is nonphysical, 

since the curve is located outside the visible space of the 

spatial harmonic. It is interesting to observe that, by comparing 

the behaviors of the and modes in Figs. 9 and 10, it 

seems that they have exchanged their roles in relation with the 

mode propagating at . 

By further increasing the frequency to , from 

Fig. 8(a) it may be seen that the mode is above cutoff and 

proper real; moreover, the mode is now above its stopband. 

These features significantly change the picture of modal evolution 

as a function of the propagation angle, as shown in Fig. 11. 

In this case, the evolution of both direct and reverse modes at 

is shown in the range ;in 

this range, the real proper solutions (solid lines) are symmetric 

with respect to the vertical axis . It may be 

seen that the direct mode at remains proper real 

at all , becoming at the mode. Instead, both 

the direct and the reverse modes remain proper real up 

to about , then they merge to form a complex proper 

solution (dashed line) upto . The latter solution represents 

a mode with a very high attenuation constant (not shown 

here), which can be considered to be in a stopband angular region. 

For the sake of completeness, the reverse mode is also 

shown, which is proper real up to , then becomes complex 

proper, with an evolution similar to that already described 

for the mode. It is interesting to observe that, at this frequency, 

both the mode and the mode at 

become grating modes by varying the propagation angle. 

As a general comment on modal evolution as a function of 

the propagation angle, it may be observed that at all the considered 

frequencies in Figs. 9–11, at least one real proper solution 

exists at all angles with a bounded phase constant , which 

corresponds to a bound mode propagating along the MSG-GDS 

in all the directions. Moreover, the fundamental mode at 

always evolves into a grating mode.


Fig. 12. (a) Modal evolution in the ( a , a ) plane for an improper 

complex (leaky) mode with the aHspatial harmonic improper (dashed line) 

on a structure as in Fig. 7 at a PIXW qr. (b) Normalized phase constant 

a (long dashed line) and normalized attenuation constant a (dashed 

line) as a function of the propagation angle 0. 

As concerns leaky modes, in Figs. 9 and 10 complex improper 

(leaky) solutions have been shown, which exist inside an angular 

neighborhood of . The existence of physical leaky 

modes propagating at all angles (as the mode reported in Fig. 6) 

has been investigated; it has been found that these may exist and 

that reasonably-low values of their attenuation constant may be 

obtained. As an example, in Fig. 12 the case of a leaky mode on 

a structure as in Fig. 7 is reported. This mode is the continuation 

of the perturbed improper leaky mode propagating at 

[23]; at it is the continuation of a higher order 

even leaky mode. In Fig. 12(a) the relevant curve in the ( , 

) plane may be seen to entirely lie inside the visible 

space, so that the mode is physical at all propagation angles. In 

Fig. 12(b) the corresponding behavior of the normalized phase 

and attenuation constants is shown; the values of the attenuation 

constant are such that directive conical radiation is to be 

expected due to the excitation of this mode by finite sources. 

C. Spectral Evolution as a Function of the Propagation Angle 

In order to illustrate some of the novel spectral features 

discussed in Section III, the same structure as in Fig. 8 is 

considered again here at higher frequencies. In particular, in 

Fig. 13 a dispersion diagram is reported for a mode propagating 

Fig. 13. (a) Normalized phase constant a and normalized attenuation 

constant a as a function of frequency for a complex (leaky) mode 

propagating at 0 aIH on a structure as in Fig. 8. (b) The same, in a narrower 

frequency range, with the BC line of the a 0I spatial harmonic (gray 

dashed-dotted line). Legend: normalized phase constant: proper solution (solid 

line); improper solution (dotted line). Normalized attenuation constant: dashed 

line. 

at , which is the continuation of the mode 

propagating at [see Fig. 8(a)]. In Fig. 13(a), the 

normalized phase constant (proper: solid line; improper: dotted 

line) and the normalized attenuation constant (dashed line) 

are reported in a wide frequency range from 25 to 30 GHz. 

At the mode is proper real; at , 

the mode becomes proper complex and radiative (leaky), due 

to the spatial harmonic being fast. The attenuation 

constant in this range first increases, then decreases to zero, 

and finally raises steeply (at about ). The 

fundamental point to be stressed here is illustrated in the 

enlarged plot reported in Fig. 13(b), where the change of 

the spatial harmonic from proper to improper is 

shown. In this plot, the quantity is also reported 

(gray dashed-dotted line), which represents the BC line of 

the spatial harmonic: according to (11), when the 

phase-constant curve crosses this line, the spatial 

harmonic becomes improper. It is important to note that this 

happens before the sign of has changed and also before 

the attenuation constant has become zero; according to the 

discussion in Section III-A, this happens when the phase vectors


Fig. 14. (a) Phase vector of the aHspatial harmonic in the ( a , a ) 

plane for a mode as in Fig. 13 at a PQXUQ qr. (b) Normalized phase 

constant a and normalized attenuation constant a as a function of 

the propagation angle 0. Legend: visible-space edge of the a 0I spatial 

harmonic (thick gray line); BC line of the a 0I spatial harmonic (thin gray 

line); other curves: as in Fig. 13. 

and are orthogonal. Instead, when propagation at 

is considered, the change of the spectral character of 

the th spatial harmonic occurs exactly when and 

the attenuation constant is equal to zero [22]. 

It is also interesting to consider the spectral evolution of 

modes by varying the propagation angle. In Fig. 14 the same 

mode as in Fig. 13 is considered, at the fixed frequency 

. In Fig. 14(a) the modal curve in the ( , 

) plane is reported, together with the visible-space 

edge (thick gray line) and the circle of (11) (thin 

gray line); as pointed out in Section III-A, by crossing this 

circle inside the visible space, the branch cut is crossed 

and the spatial harmonic changes its spectral character. 

In Fig. 14(b), the relevant normalized phase and attenuation 

constants are plotted as a function of the propagation angle 

.At the mode is improper complex; by increasing 

, the curve in Fig. 14(a) crosses the gray circle (at ) 

and the mode becomes proper complex (see Fig. 13(b), where 

). By further increasing , the curve again crosses the 

circle (at ) and the mode becomes improper complex. 

Finally, the curve touches the visible-space edge [see 

Fig. 14(a)] with a zero attenuation constant (at ), the 

mode becomes proper real and exits the visible space in a 

bound regime. 

V. CONCLUSION 

In this paper, an investigation on the fundamental properties 

of modes which propagate at arbitrary angles along a 

MSG-GDS has been presented, without any restrictive assumption 

on the physical or geometrical parameters of the structure. 

Homogeneous modes, i.e., modes with at least one spatial harmonic 

propagating as a homogeneous plane wave in the grating 

plane have been considered, by determining their propagation 

constant and modal current configuration through a full-wave, 

spectral-domain moment-method approach in the unit cell. Particular 

attention has been devoted to the issue of the spectral 

character of the considered modes, which is related to the proper 

or improper nature of the spatial harmonics: novel conditions 

which allow to continuously track the relevant modal solutions 

on the different Riemann sheets have been presented, generalizing 

those already known for propagation across or along the 

metal strips. 

Numerical results have been provided for different structures 

and frequency ranges in order to assess the accuracy of the 

proposed approach, by comparison with the available results 

in the literature and with different full-wave techniques, and to 

illustrate the main features of surface and leaky modes as the 

propagation angle in the grating plane is varied. In particular, 

it has been found that both surface modes and leaky modes 

that propagate at all angles may exist, and the investigation of 

high-frequency regions has elucidated novel spectral features 

of the considered modes. 

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Paolo Baccarelli (S’96–M’01) received the Laurea 

degree in electronic engineering and the Ph.D. degree 

in applied electromagnetics from “La Sapienza” 

University of Rome, Rome, Italy, in 1996 and 2000, 

respectively. 

In 1996, he joined the Department of Electronic 

Engineering, “La Sapienza” University of Rome, 

where he has been a Contract Researcher since 2000. 

From April 1999 to October 1999, he was at the 

University of Houston, Houston, TX, as a Visiting 

Scholar. His research interests concern analysis and 

design of planar leaky-wave antennas, numerical methods, periodic structures, 

and propagation and radiation in metamaterials and anisotropic media. 

Paolo Burghignoli (S’97–M’01) was born in Rome, 

Italy, on February 18, 1973. He received the Laurea 

degree (cum laude) in electronic engineering and the 

Ph.D. degree in applied electromagnetics from “La 

Sapienza” University of Rome, in 1997 and 2001, respectively. 

In 1997, he joined the Electronic Engineering 

Department of “La Sapienza” University of Rome, 

where he is currently a Contract Researcher. From 

January 2004 to July 2004, he was a Visiting 

Research Assistant Professor at the University of 

Houston, Houston, Texas. His scientific interests include analysis and design 

of planar leaky-wave antennas, numerical methods for the analysis of passive 

guiding and radiating microwave structures, periodic structures, and propagation 

and radiation in metamaterials. 

Dr. Burghignoli received the Graduate Fellowship Award from IEEE Microwave 

Theory and Techniques Society, in 2003. 

Fabrizio Frezza (S’87–M’90–SM’95) was born in 

Rome, Italy, on October 31, 1960. He received the 

Laurea degree (cum laude) in electronic engineering 

and the Doctorate degree in applied electromagnetics 

from “La Sapienza” University of Rome, Rome, Italy, 

in 1986 and 1991, respecitvely. 



where he was a Researcher from 1990 to 1998, a 

temporary Professor of Electromagnetics from 1994 

to 1998, and an Associate Professor of Electromagnetics 

since 1998. His main research activity concerns guiding structures, 

antennas and resonators for microwaves and millimeter waves, numerical 

methods, scattering, optical propagation, plasma heating, and anisotropic 

media. 

Dr. Frezza is a Member of Sigma Xi, the European Microwave Association 

(EuMA), the Electrical and Electronic Italian Association (AEIT), and the 

Italian Society of Aeronautics and Astronautics (AIDAA). 

Alessandro Galli (S’91–M’96) received the Laurea 

degree in electronic engineering and the Ph.D. degree 

in applied electromagnetics from “La Sapienza” University 

of Rome, Rome, Italy, in 1990 and 1994, respectively. 



where he became an Assistant Professor in 2000 and 

an Associate Professor of “Electromagnetic Fields” 

for Telecommunications Engineering in 2002. His 

scientific interests mainly involve electromagnetic 

theory and applications, particularly regarding analysis and design of passive 

devices and antennas (dielectric and anisotropic waveguides and resonators, 

leaky-wave antennas, etc.) for microwaves and millimeter waves. He is also 

active in bioelectromagnetics (modeling of interaction mechanisms with living 

matter, health safety problems for low-frequency applications and mobile 

communications, etc.). 

Dr. Galli received the Quality Presentation Recognition Award by the IEEE 

Microwave Theory and Techniques Society in 1994 and 1995.


Paolo Lampariello (M’73–SM’82–F’96) was born 

in Rome, Italy, on May 17, 1944. He received the 

“Laurea” degree (cum laude) in electronic engineering 

from the University of Rome, in 1971. 

In 1971, he joined the Institute of Electronics, 

University of Rome, where he was an Assistant 

Professor of Electromagnetic Fields. Since 1976, he 

has been engaged in educational activities involving 

electromagnetic field theory and in 1986 he was 

made Professor of Electromagnetic Fields. From 

November 1988 to October 1994, he served as Head 

of the Department of Electronic Engineering of “La Sapienza” University of 

Rome, where, since November 1993, he has been the President of the Council 

for Electronic Engineering Curriculum, and since September 1995, he has 

been President of the Center Interdepartmental for Scientific Computing. 

From September 1980 to August 1981, he was a NATO Postdoctoral Research 

Fellow at the Polytechnic Institute of New York, Brooklyn. He has been 

engaged in research in a wide variety of topics in the microwave field, including 

electromagnetic and elastic wave propagation in anisotropic media, thermal 

effects of electromagnetic waves, network representations of microwave 

structures, guided-wave theory with stress on surface waves and leaky waves, 

traveling-wave antennas, phased arrays, and, more recently, guiding and 

radiating structures for the millimeter and near-millimeter wave ranges. 

Prof. Lampariello is Member of the Associazione Elettrotecnica de Elettronica 

Italiana (AEI). He is a past Chairman of the Central and South Italy 

Section of the IEEE and a past Chairman of the Microwave Theory and 

Techniques/Antennas Propagation Societies Joint Chapter of the same Section. 

Presently he is the President of the Specialist Group “Electromagnetism” of 

the AEI. 

Giampiero Lovat (S’02) was born in Rome, Italy, on 

May 31, 1975. He received the Laurea degree (cum 

laude) in electronic engineering from “La Sapienza” 

University of Rome, in 2001, where he is working 

toward the Ph.D. degree in applied electromagnetics 

in the Electronic Engineering Department. 

From January 2004 to July 2004, he was a Visiting 

Scholar at the University of Houston, Houston, TX. 

His scientific interests include theoretical and numerical 

studies on leakage phenomena in planar structures 

and guidance and radiation in metamaterials and 

general periodic structures. 

Simone Paulotto (S’97) received the Laurea degree 

(cum laude and honorable mention) in electronic engineering 

from “La Sapienza” University of Rome, 

Rome, Italy, in 2002, where he is currently working 

toward the Ph.D. degree in applied electromagnetics. 


Department of “La Sapienza” University of Rome. 

His scientific interests focus on electromagnetic 

propagation/radiation in planar structures and scattering 

theory.

Modal Properties of Surface and Leaky Waves ... - La Sapienza

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