SIEVENPIPER et al. : HIGH-IMPEDANCE ELECTROMAGNETIC SURFACES 2065 M Fig 9. Square geometry studied using the finite-element model Fig. 10. Surface nd structure of the high-impedance surface, model. The radiation broadening of the te hase than out-of-phase. It represents the maximum usable above the light Ii error width of a flush-mounted antenna on a resonant surface of this type The relative bandwidth Aw/w is proportional to VL/C thus, if the capacitance is increased, the bandwidth suffers Surface Since the thickness is related to the inductance. the more the Under esonance frequency is reduced for a given thickness, the more the bandwidth is diminished Microwave Absorber V. FINITE-ELEMENT MODEL In the effective surface impedance model described above Coax he properties of the surface are summarized into a single Probe parameter, namely the surface impedance. Such a model cor- rectly predicts the reflection properties of the high-impedance surface and some features of the surface-wave bands. how- ver, it does not predict an actual bandgap. Neverthele we have found experimentally that the surface-wave bandgap edges occur where the reflection phase is equal to +T/2, thus this generally corresponds to the width of the surface-wave bandgap. Within this region, surface currents radiate It is necessary to obtain more accurate results using a finite- (a)TM surface-wave measurement using vertical monopole probe element model, in which the detailed geometry of the surface he probes couple to the vertical electric field of TM surface waves. nt using horizontal monopole probe antennas. texture is included explicitly. In the finite-element model, the couple to the horizontal electric field of TE surface waves metal and dielectric regions of one unit cell are discretized on a grid. The electric field at all points on the grid can be reduced to an eigenvalue equation, which may be solved numerically the graph by a dotted line. These qualitatively Bloch boundary conditions are used, in which the fields at with the effective medium model. The finite-element method one edge of the cell are related to the fields at the opposite also predicts higher frequency bands that are seen in the frequencies for a particular wave vector, and the procedure mode ements, but do not appear in the effective medium edge by the wave vector. The calculation yields the allowed is repeated for each wave vector to produce the dispersion According to the finite-element model, the TM band does diagram. The structure analyzed by the finite-element method not reach the TE band edge, but stops below it, forming a was a two-layer high-impedance surface with square geometry, bandgap. The Te band slopes upward upon crossing the light shown in Fig 9. The lattice constant was 2.4 mm, the spacing line. Thus, the finite-element model predicts a surface-wave between the plates was 0.15 mm, and the width of the vias bandgap that spans from the edge of the TM band,to was 0.36 mm. The volume below the square plates was filled point where the TE band crosses the light line. The resonance with e=2.2. and the total thickness was 1.6 mm frequency is centered in the forbidden bandgap The results of the finite-element calculation are shown in In both the TM and tE bands, the k=0 state represents 10. The TM band follows the light line up to a certain a continuous sheet of current. The lowest TM mode, at zero frequency, where it suddenly becomes very flat. The TE band frequency, is simply a sheet of constant current-a dc mode begins at a higher frequency, and continues upward with a The highest TM mode, at the brillouin zone edge, is a standing slope of less than the vacuum speed of light, which is indicated wave in which each row of metal protrusions has oppositeSIEVENPIPER et al.: HIGH-IMPEDANCE ELECTROMAGNETIC SURFACES 2065 Fig. 9. Square geometry studied using the finite-element model. in-phase than out-of-phase. It represents the maximum usable bandwidth of a flush-mounted antenna on a resonant surface of this type. The relative bandwidth is proportional to , thus, if the capacitance is increased, the bandwidth suffers. Since the thickness is related to the inductance, the more the resonance frequency is reduced for a given thickness, the more the bandwidth is diminished. V. FINITE-ELEMENT MODEL In the effective surface impedance model described above, the properties of the surface are summarized into a single parameter, namely the surface impedance. Such a model cor￾rectly predicts the reflection properties of the high-impedance surface, and some features of the surface-wave bands. How￾ever, it does not predict an actual bandgap. Nevertheless, we have found experimentally that the surface-wave bandgap edges occur where the reflection phase is equal to , thus, this generally corresponds to the width of the surface-wave bandgap. Within this region, surface currents radiate. It is necessary to obtain more accurate results using a finite￾element model, in which the detailed geometry of the surface texture is included explicitly. In the finite-element model, the metal and dielectric regions of one unit cell are discretized on a grid. The electric field at all points on the grid can be reduced to an eigenvalue equation, which may be solved numerically. Bloch boundary conditions are used, in which the fields at one edge of the cell are related to the fields at the opposite edge by the wave vector. The calculation yields the allowed frequencies for a particular wave vector, and the procedure is repeated for each wave vector to produce the dispersion diagram. The structure analyzed by the finite-element method was a two-layer high-impedance surface with square geometry, shown in Fig. 9. The lattice constant was 2.4 mm, the spacing between the plates was 0.15 mm, and the width of the vias was 0.36 mm. The volume below the square plates was filled with , and the total thickness was 1.6 mm. The results of the finite-element calculation are shown in Fig. 10. The TM band follows the light line up to a certain frequency, where it suddenly becomes very flat. The TE band begins at a higher frequency, and continues upward with a slope of less than the vacuum speed of light, which is indicated Fig. 10. Surface-wave band structure of the high-impedance surface, calcu￾lated using a finite-element model. The radiation broadening of the TE modes above the light line is indicated by error bars. (a) (b) Fig. 11. (a) TM surface-wave measurement using vertical monopole probe antennas. The probes couple to the vertical electric field of TM surface waves. (b) TE surface-wave measurement using horizontal monopole probe antennas. The probes couple to the horizontal electric field of TE surface waves. on the graph by a dotted line. These results agree qualitatively with the effective medium model. The finite-element method also predicts higher frequency bands that are seen in the measurements, but do not appear in the effective medium model. According to the finite-element model, the TM band does not reach the TE band edge, but stops below it, forming a bandgap. The TE band slopes upward upon crossing the light line. Thus, the finite-element model predicts a surface-wave bandgap that spans from the edge of the TM band, to the point where the TE band crosses the light line. The resonance frequency is centered in the forbidden bandgap. In both the TM and TE bands, the state represents a continuous sheet of current. The lowest TM mode, at zero frequency, is simply a sheet of constant current—a dc mode. The highest TM mode, at the Brillouin zone edge, is a standing wave in which each row of metal protrusions has opposite