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

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BACCARELLI et al.: MODAL PROPERTIES OF SURFACE AND LEAKY WAVES PROPAGATING 41 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
42 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 1, JANUARY 2005 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.
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Modal Properties of Surface and Leaky Waves ... - La Sapienza

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