2064 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL, 47, NO Il, NOVEMBER 1999 Although the surface exhibits high impedance, it is not actually devoid of current. (If there were no current, electro- magnetic waves would be transmitted right through the ground plane. )However, the resonant structure provides a phase shift, thus. the image currents in the surface reinforce the currents in the antenna, instead of canceling them To the left-hand side of the light line in Fig. 7, we can determine the frequency range over which the radiation effi- ciency is high by using a circuit model, in which the antenna is modeled as a current source. The textured surface is modeled as an LC circuit in parallel with the antenna, and the radiation into free space is modeled as a resistor with a value of po/Eo/cos(0)= 377 S/cos(0). The amount of power dissipated in the resistor is a measure of the radiation efficiency of the antenna The maximum power dissipated in istor occurs at Fig 8. Reflection phase of the high-impedance surtace, calculated using the the LC resonance frequency of the ffective surface the surface reactance crosses through frequencies, or at very high frequencies, the current is shunted reflected waves. If the surface has low impedance, such as through the inductor or the capacitor, and the power flowing in the case of a good conductor, the ratio of electric field to to the resistor is reduced. It can be shown that the frequencies magnetic field is small. The electric field has a node at the where the radiation drops to half of its maximum value occur surface, and the magnetic field has an antinode. Conversely, when the magnitude of the surface impedance is equal to the for a high impedance surface, the electric field has an antinode impedance of free space. For normal radiation, we have the at the surface, while the magnetic field has a node. Another following equation term for such a surface is an artificial "magnetic conductor Recent work involving grounded frequency selective surfaces has also been shown to mimic a magnetic conductor [37] 2LC=7 However, these structures do not possess a complete surface- This can be solved for w to yield the following equation wave bandgap, since they lack the vertical conducting vias, which interact with the vertical electric field of tm surface (23) waves Typical parameters for a two-layer ground plane are 2 nH2 For typical geometries, L is usually on the order of I nH, of inductance, and 0.05 pF2 of capacitance. For these values, and C is in the range of 0.05-10 pF. With these values, the the reflection phase is plotted in Fig 8. At very low frequen- terms involving 1/rC are much smaller than the 1/LC cies, the reflection phase is T, and the structure behaves like terms, so we will eliminate them. This approximation yields an ordinary flat metal surface. The reflection phase slopes the following expression for the edges of the operating band downward, and eventually crosses through zero at the reso- nance frequency. Above the resonance frequency, the phase u=Wb11士 to -T. The phase falls within T/2 and -T/2 when the ude of the surface impedance exceeds the impedance The resonance frequency is wb=1/VLC, and Zo=VL/C rather than out-of-phase, and antenna elements may lie directly is the characteristic impedance of the LC circuit. With the adjacent to the surface without being shorted out parameters for L, C, and n given above, Zo is usually significantly smaller than n. Thus, the square root can be expanded in the following approximation (1±1z An antenna lying parallel to the textured surface will see the impedance of free space on one side, and the impedance The two frequencies designated by the t signs delimit the of the ground plane on the other side. Where the textured range over which an antenna would radiate efficiently or surface has low impedance, far from the resonance frequency, such a surface. The total bandwidth is roughly equal to the antenna current is mirrored by an opposing current in the characteristic impedance of the surface divided by the the surface. Since the antenna is shorted out by the nearby impedance of free space conductor, the radiation efficiency is very low. Within the forbidden bandgap near resonance, the textured surface has much higher impedance than free space, so the antenna is not shorted out. In this range of frequencies, the radiation This is also the bandwidth over which the reflection coefficient ency is high falls between +r/2 and -T/2, and image currents are more
2064 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 47, NO. 11, NOVEMBER 1999 Fig. 8. Reflection phase of the high-impedance surface, calculated using the effective surface impedance model. reflected waves. If the surface has low impedance, such as in the case of a good conductor, the ratio of electric field to magnetic field is small. The electric field has a node at the surface, and the magnetic field has an antinode. Conversely, for a high impedance surface, the electric field has an antinode at the surface, while the magnetic field has a node. Another term for such a surface is an artificial “magnetic conductor.” Recent work involving grounded frequency selective surfaces has also been shown to mimic a magnetic conductor [37]. However, these structures do not possess a complete surfacewave bandgap, since they lack the vertical conducting vias, which interact with the vertical electric field of TM surface waves. Typical parameters for a two-layer ground plane are 2 nH of inductance, and 0.05 pF of capacitance. For these values, the reflection phase is plotted in Fig. 8. At very low frequencies, the reflection phase is , and the structure behaves like an ordinary flat metal surface. The reflection phase slopes downward, and eventually crosses through zero at the resonance frequency. Above the resonance frequency, the phase returns to . The phase falls within and when the magnitude of the surface impedance exceeds the impedance of free space. Within this range, image currents are in-phase, rather than out-of-phase, and antenna elements may lie directly adjacent to the surface without being shorted out. C. Radiation Bandwidth An antenna lying parallel to the textured surface will see the impedance of free space on one side, and the impedance of the ground plane on the other side. Where the textured surface has low impedance, far from the resonance frequency, the antenna current is mirrored by an opposing current in the surface. Since the antenna is shorted out by the nearby conductor, the radiation efficiency is very low. Within the forbidden bandgap near resonance, the textured surface has much higher impedance than free space, so the antenna is not shorted out. In this range of frequencies, the radiation efficiency is high. Although the surface exhibits high impedance, it is not actually devoid of current. (If there were no current, electromagnetic waves would be transmitted right through the ground plane.) However, the resonant structure provides a phase shift, thus, the image currents in the surface reinforce the currents in the antenna, instead of canceling them. To the left-hand side of the light line in Fig. 7, we can determine the frequency range over which the radiation effi- ciency is high by using a circuit model, in which the antenna is modeled as a current source. The textured surface is modeled as an circuit in parallel with the antenna, and the radiation into free space is modeled as a resistor with a value of . The amount of power dissipated in the resistor is a measure of the radiation efficiency of the antenna. The maximum power dissipated in the resistor occurs at the resonance frequency of the ground plane, where the surface reactance crosses through infinity. At very low frequencies, or at very high frequencies, the current is shunted through the inductor or the capacitor, and the power flowing to the resistor is reduced. It can be shown that the frequencies where the radiation drops to half of its maximum value occur when the magnitude of the surface impedance is equal to the impedance of free space. For normal radiation, we have the following equation: (22) This can be solved for to yield the following equation: (23) For typical geometries, is usually on the order of 1 nH, and is in the range of 0.05–10 pF. With these values, the terms involving are much smaller than the terms, so we will eliminate them. This approximation yields the following expression for the edges of the operating band: (24) The resonance frequency is , and is the characteristic impedance of the circuit. With the parameters for , and given above, is usually significantly smaller than . Thus, the square root can be expanded in the following approximation: (25) The two frequencies designated by the signs delimit the range over which an antenna would radiate efficiently on such a surface. The total bandwidth is roughly equal to the characteristic impedance of the surface divided by the impedance of free space (26) This is also the bandwidth over which the reflection coefficient falls between and , and image currents are more
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 opposite
SIEVENPIPER 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 correctly predicts the reflection properties of the high-impedance surface, and some features of the surface-wave bands. However, 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 finiteelement 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, calculated 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