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Figure from:Dispersion Characteristics of a Metamaterial-Based Parallel-Plate Ridge Gap Waveguide Realized by Bed of Nails

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Dispersion Characteristics of a Metamaterial-Based Parallel-Plate Ridge Gap Waveguide Realized by Bed of Nails
Dispersion Characteristics of a Metamaterial-Based Parallel-Plate Ridge Gap Waveguide Realized by Bed of Nails
Fig. 1. Geometry of the ridge gap waveguide embedded in a bed of nails present formulation, we will not describe the field inside the bed of nails itself, but instead represent the latter by a spatially dis- persive anisotropic homogeneous medium characterized by an equivalent homogeneous reflection coefficient according to [4]. The assumed field expressions in the region above the ridge and the two side regions above the surrounding pin surfaces are used to establish point matching continuity at the interfaces between the regions, and thereby to obtain a dispersion equation.
Fig. 1. Geometry of the ridge gap waveguide embedded in a bed of nails present formulation, we will not describe the field inside the bed of nails itself, but instead represent the latter by a spatially dis- persive anisotropic homogeneous medium characterized by an equivalent homogeneous reflection coefficient according to [4]. The assumed field expressions in the region above the ridge and the two side regions above the surrounding pin surfaces are used to establish point matching continuity at the interfaces between the regions, and thereby to obtain a dispersion equation.
Fig. 2. Bed of nails basic geometry. (left) 3D-view; (right) top-view. As a fist step we analyze the phenomenology associated with the bed of nails shown in Fig. 2. The nails are constituted by small metallic cylinders of height d, with radius b, and spacing a in both x and y directions. For completeness, we assume the nails to be embedded in a host medium with permittivity €;,; in practical application however, it is better to have nails immersed in free space. In [4] the same geometry has been studied by assuming a plane wave with wave vector k; incident on the tex- tured surface. Following [4], the bed of nails is regarded as a spatial and frequency dispersive medium whose permittivity is characterized by the tensor
Fig. 2. Bed of nails basic geometry. (left) 3D-view; (right) top-view. As a fist step we analyze the phenomenology associated with the bed of nails shown in Fig. 2. The nails are constituted by small metallic cylinders of height d, with radius b, and spacing a in both x and y directions. For completeness, we assume the nails to be embedded in a host medium with permittivity €;,; in practical application however, it is better to have nails immersed in free space. In [4] the same geometry has been studied by assuming a plane wave with wave vector k; incident on the tex- tured surface. Following [4], the bed of nails is regarded as a spatial and frequency dispersive medium whose permittivity is characterized by the tensor
has been done in [17], but one of the parallel-plate faces was there constituted by an anisotropic hard/soft surface along the direction of propagation, o btained as a periodic structure made up of corrugations. Here, the fakir’s bed homogenized surface is comp etely 3D, and then the modal field configuration sup- ported is more complicated. The wave bouncing in the gap be- tween the two faces can sti ficients solution of the eigenvalue vanishing of the tangentia [evkuy + electric [TTMTE (ky)eJhu¥] electric field y=h background in the = 0. bed of nails, TE mode solution is that 1 be described by the reflection coef- Tr? (k,) and [7"(k,), where now the k, value is the problem obtained by imposing the at the upper wall, i.e. In absence of the di- associated with the resonance between the upper and lower PEC walls, ic. ky = 1/(h + da). For TM modes, the equation to be solved can be conveniently rewritten by employing (4), which leads to The above equation can be easily interpreted as a circuit series resonance equation of the kind
has been done in [17], but one of the parallel-plate faces was there constituted by an anisotropic hard/soft surface along the direction of propagation, o btained as a periodic structure made up of corrugations. Here, the fakir’s bed homogenized surface is comp etely 3D, and then the modal field configuration sup- ported is more complicated. The wave bouncing in the gap be- tween the two faces can sti ficients solution of the eigenvalue vanishing of the tangentia [evkuy + electric [TTMTE (ky)eJhu¥] electric field y=h background in the = 0. bed of nails, TE mode solution is that 1 be described by the reflection coef- Tr? (k,) and [7"(k,), where now the k, value is the problem obtained by imposing the at the upper wall, i.e. In absence of the di- associated with the resonance between the upper and lower PEC walls, ic. ky = 1/(h + da). For TM modes, the equation to be solved can be conveniently rewritten by employing (4), which leads to The above equation can be easily interpreted as a circuit series resonance equation of the kind
Fig. 3. Reflection phenomenology supported by the bed of nails surface covered by a PEC plate. The guiding phenomenon on the bed of nails surface is strongly modified by the presence of a metallic plate. They form together a parallel plate waveguide where one face is PEC and the other is reactive through the homogenized surface de- scribed in Section II (see Fig. 3). For simplicity, hereinafter we consider free-space embedded nails (c€;, = 1). The same
Fig. 3. Reflection phenomenology supported by the bed of nails surface covered by a PEC plate. The guiding phenomenon on the bed of nails surface is strongly modified by the presence of a metallic plate. They form together a parallel plate waveguide where one face is PEC and the other is reactive through the homogenized surface de- scribed in Section II (see Fig. 3). For simplicity, hereinafter we consider free-space embedded nails (c€;, = 1). The same
Fig. 4. Equivalent resonant circuit associated with the dispersion equation of the bed of nails surface covered by a top metal plate at height h.
Fig. 4. Equivalent resonant circuit associated with the dispersion equation of the bed of nails surface covered by a top metal plate at height h.
(fta=r/4); f(d+-n=r/2))» Where ky is real and greater than k, that implies attenuation along any direction along the surface, and the second region ( f(a4n=/2); f(a=d/2)) Where the propagation is admitted along the sur face. The first range is the stop band of the structure, whose upper bound is the cut-off frequency of the first TE mode. The uppe to the lower cut-off freq stop band, a dispersion d where k, = ,/k? — kz, r cut-off of the stop band, corresponds uency. In order to better identify the iagram f versus k, is shown in Fig. 6 with ky = ky for TM solution (dotted line), and ky = ky for T E solution (dashed line). The light line Fig. 6. Frequency versus real part of kz for TM (black dots) and TE (dashed line) solutions of the pertinent eigenvalues problems. The light line is also shown (solid line). These results have been obtained for the same geometri- cal parameters used for Fig. 5. line), and ky = hey for TE solution (dashed line). The light line
(fta=r/4); f(d+-n=r/2))» Where ky is real and greater than k, that implies attenuation along any direction along the surface, and the second region ( f(a4n=/2); f(a=d/2)) Where the propagation is admitted along the sur face. The first range is the stop band of the structure, whose upper bound is the cut-off frequency of the first TE mode. The uppe to the lower cut-off freq stop band, a dispersion d where k, = ,/k? — kz, r cut-off of the stop band, corresponds uency. In order to better identify the iagram f versus k, is shown in Fig. 6 with ky = ky for TM solution (dotted line), and ky = ky for T E solution (dashed line). The light line Fig. 6. Frequency versus real part of kz for TM (black dots) and TE (dashed line) solutions of the pertinent eigenvalues problems. The light line is also shown (solid line). These results have been obtained for the same geometri- cal parameters used for Fig. 5. line), and ky = hey for TE solution (dashed line). The light line
1e reference system is centered in the middle of the width z ‘the ridge. In equations (11), Arajg is an unknown coefficient: hile k, is the eigenvalue solution of equation (8). Thus, th spersion relation for this mode is k* = (—j@,)? + kj + kz here the attenuation constant a, and the propagation constan Under the approximations used above, we can now write the ex- pressions for the fields. In particular, the TMy evanescent mode fields in the region above the bed of nails are,
1e reference system is centered in the middle of the width z ‘the ridge. In equations (11), Arajg is an unknown coefficient: hile k, is the eigenvalue solution of equation (8). Thus, th spersion relation for this mode is k* = (—j@,)? + kj + kz here the attenuation constant a, and the propagation constan Under the approximations used above, we can now write the ex- pressions for the fields. In particular, the TMy evanescent mode fields in the region above the bed of nails are,
Fig. 7. Normalized amplitude of the vertical (along y) electric field for differenti frequencies calculated through CST Microwave Studio k, are also unknowns. A second x-evanescent TEy mode is as- sumed to be excited in the bed of nails region, whose expression is give by
Fig. 7. Normalized amplitude of the vertical (along y) electric field for differenti frequencies calculated through CST Microwave Studio k, are also unknowns. A second x-evanescent TEy mode is as- sumed to be excited in the bed of nails region, whose expression is give by
hat, added to the ones in (16), allow us to find the dispersion >quation To find the remaining three equations, we enforce a razor blade continuity o walls x = a f the three field components across the separation tw/2. The matched field components Ey, Hz are namely all t hose belonging to the quasi-TEM mode. The non- matched components E,,, HL, and H,, (only present in the bed of nails regions) are very weak, since the boundary conditions at the top co three matchi ver impose vanishing of them. Thus, we obtain the ng field equations
hat, added to the ones in (16), allow us to find the dispersion >quation To find the remaining three equations, we enforce a razor blade continuity o walls x = a f the three field components across the separation tw/2. The matched field components Ey, Hz are namely all t hose belonging to the quasi-TEM mode. The non- matched components E,,, HL, and H,, (only present in the bed of nails regions) are very weak, since the boundary conditions at the top co three matchi ver impose vanishing of them. Thus, we obtain the ng field equations
Fig. 8. Dispersion diagram for the ridge waveguide (dotted line). The geometry is the same as that reported in caption of Fig. 6, but including an insertec a 5mm wide ridge. The dispersion diagram is compared with the one ob: tained through a CST Microwave Studio simulation (diamond line) of th reference structure inserted above the graph. The light line is also shown. The solution is found numerically and plotted as the disper- sion diagram in Fig. 8, where the ridge width is w = 5mm. Again, the Matlab FSOLVE routine is employed [18]. The ge- ometry is the same as that used in the previous results (see cap- tion of Fig. 6). The curve has been successfully compared with the dispersion diagram obtained by a CST Microwave Stu- dio simulation (diamond line). The CST reference structure is shown above the graph in Fig. 8. Our approximated solution tends to fail when the frequency is larger than the upper parallel- plate cut-off frequency, because we have only included one fun- damental mode for each region, and this is not sufficient. Higher order modes could be accounted for at higher frequencies.
Fig. 8. Dispersion diagram for the ridge waveguide (dotted line). The geometry is the same as that reported in caption of Fig. 6, but including an insertec a 5mm wide ridge. The dispersion diagram is compared with the one ob: tained through a CST Microwave Studio simulation (diamond line) of th reference structure inserted above the graph. The light line is also shown. The solution is found numerically and plotted as the disper- sion diagram in Fig. 8, where the ridge width is w = 5mm. Again, the Matlab FSOLVE routine is employed [18]. The ge- ometry is the same as that used in the previous results (see cap- tion of Fig. 6). The curve has been successfully compared with the dispersion diagram obtained by a CST Microwave Stu- dio simulation (diamond line). The CST reference structure is shown above the graph in Fig. 8. Our approximated solution tends to fail when the frequency is larger than the upper parallel- plate cut-off frequency, because we have only included one fun- damental mode for each region, and this is not sufficient. Higher order modes could be accounted for at higher frequencies.
Fig. 9. Dispersion diagram obtained by CST Microwave Studio including all modes due to enclosure resonances. The diamond marked lines are the same as in Fig. 8. The light line is also shown.
Fig. 9. Dispersion diagram obtained by CST Microwave Studio including all modes due to enclosure resonances. The diamond marked lines are the same as in Fig. 8. The light line is also shown.
Fig. 11. Normalized amplitude (in dB) of E,, (solid line) and H (dashed line) field distributions at 13G Hz, computed along the middle of the gap region (y=d+h/2) Fig. 10. Normalized amplitude of the vertical (along y) electric field for differ- ent frequencies calculated by the expressions in (11), (13) and (15)
Fig. 11. Normalized amplitude (in dB) of E,, (solid line) and H (dashed line) field distributions at 13G Hz, computed along the middle of the gap region (y=d+h/2) Fig. 10. Normalized amplitude of the vertical (along y) electric field for differ- ent frequencies calculated by the expressions in (11), (13) and (15)
Fig. 12. Transverse attenuation constant in x-direction plotted as dB/Xo versus frequency compared with the one obtained through a CST simulation where the field decaying is captured by a couple of 6-spaced probes located as shown in the inset. a “a x Ae = \/k2+ki —k? anda, = \/k? +k, — k? are plotted in Fig. 12 in the stop band of the parallel-plate region, corre- sponding to the working bandwidth of the ridge gap waveguide. While a, is strongly varying with frequency and has a value greater than 100dB/Xo almost over the entire bandwidth, On is almost a constant in the same bandwidth. Indeed, since k, ~ k. Ay ky = 1/(h + d), and the resulting value is smaller over the bandwidth. This implies that H, component has a weaket decay, dictated by the attenuation factor @,. Secondly, regard- ing the CST H,, component, we see oscillations in the nails re- gion which are due to the actual periodicity of the pins, that we neglect in the surface impedance model. It is evident from Fig. 11 that the CST EH, component does not show the same ripples. Indeed, E,, has a stronger decay along x than H,, and this prevents E,, from being influenced by the periodicity of the pins. The results are given in dB/Xg, where Ao is calculated at
Fig. 12. Transverse attenuation constant in x-direction plotted as dB/Xo versus frequency compared with the one obtained through a CST simulation where the field decaying is captured by a couple of 6-spaced probes located as shown in the inset. a “a x Ae = \/k2+ki —k? anda, = \/k? +k, — k? are plotted in Fig. 12 in the stop band of the parallel-plate region, corre- sponding to the working bandwidth of the ridge gap waveguide. While a, is strongly varying with frequency and has a value greater than 100dB/Xo almost over the entire bandwidth, On is almost a constant in the same bandwidth. Indeed, since k, ~ k. Ay ky = 1/(h + d), and the resulting value is smaller over the bandwidth. This implies that H, component has a weaket decay, dictated by the attenuation factor @,. Secondly, regard- ing the CST H,, component, we see oscillations in the nails re- gion which are due to the actual periodicity of the pins, that we neglect in the surface impedance model. It is evident from Fig. 11 that the CST EH, component does not show the same ripples. Indeed, E,, has a stronger decay along x than H,, and this prevents E,, from being influenced by the periodicity of the pins. The results are given in dB/Xg, where Ao is calculated at