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

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BACCARELLI et al.: MODAL PROPERTIES OF SURFACE AND LEAKY WAVES PROPAGATING 39 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
40 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 1, JANUARY 2005 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
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