2062 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL, 47, NO Il, NOVEMBER 1999 Fig. 5. Origin of the capacitance and inductance in the high-impedance surfac netal sheet has a narrow surface-wave bandgap (a of the bandgap, in which the electric field wraps Layer 2 field extends across the bumps 6. ency for a given thickness by using capacitive loading, but it also suffers a reduction in bandwidth high impedance over a predetermined frequency band. Peri- odic two- or three-dimensional dielectric [2024], metallic [251[28], or metallodielectric [29H-33] structures that prevent 入 the propagation of electromagnetic waves are known as pho. tonic crystals [34-36 The high-impedance surface can be considered as a kind of two-dimensional photonic crystal that prevents the propagation of radio-frequency surface currents Fig. 4. Corrugated metal slab has high impedance at the top surface if the within the bandgap corrugations are one quarter-wavelength deer As the structure illustrated in Fig. 5 interacts with electro- magnetic waves, currents are induced in the top metal plates [1][12]. Numerous authors have also contributed general to build up on the ends of the plates, which can be described treatments of corrugated surfaces [13H-[15], and specific stud- as a capacitance. As the charges slosh back and forth, they ies of important structures [161-[19]. A corrugated surface flow around a long path through the vias and bottom plate a metal slab, into which a series of vertical slots have been an inductance Associated with these currents is a magnetic field and. thu cut, as depicted in Fig. 4. The slots are narrow, so that many We assign to the surface a sheet impedance equal to the of them fit within one wavelength across the slab. Each slot impedance of a parallel resonant circuit, consisting of the sheet can be regarded as a parallel-plate transmission line, running capacitance and the sheet inductance lown into the slab. and shorted at the bottom. If the slots are one quarter-wavelength deep, then the short circuit at the 2 bottom end is transformed by the length of the slots into an 1-∞2LC open-circuit at the top end. Thus, the impedance at the top The surface is inductive at low frequencies, and capacitive surface is very high at high frequencies. The impedance is very high near the If there are many slots per wavelength, the structure can resonance frequency wb be assigned an effective surface impedance equal to the impedance of the slots. The behavior of the corrugations is reduced to a single parameter-the boundary condition at the top surface. If the depth of the slots is greater than one We associate the high impedance with a forbidden frequency quarter-wavelength, the surface impedance is capacitive, and bandgap. In the two-layer geometry shown in Fig. 1,the TM surface waves are forbidden. Furthermore, a plane wave capacitors are formed by the fringing electric fields between polarized with the electric field perpendicular to the ridges adjacent metal patches, and the inductance is fixed by the will appear to be reflected with no phase reversal since the thickness of the structure. A three-layer design shown in Fig. 6 effective reflection plane is actually at the bottom of the slots, achieves a lower resonance frequency for a given thickness one quarter-wavelength away by using capacitive loading. In this geometry, parallel-plate formed by the top two C. High-Impedance Surfaces The high-impedance surface described here is an abstraction IV EFFECTIVE SURFACE IMPEDANCE MODEL of the corrugated surface, in which the corrugations have been Some of the properties of the high-impedance surface can folded up into lumped-circuit elements, and distributed in a be explained using an effective surface impedance model two-dimensional lattice. The surface impedance is modeled The surface is assigned an impedance equal to that of a as a parallel resonant circuit, which can be tuned to exhibit parallel resonant LC circuit, derived by geometry. The use2062 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 47, NO. 11, NOVEMBER 1999 (a) (b) Fig. 3. Bumpy metal sheet has a narrow surface-wave bandgap. (a) Mode at the upper edge of the bandgap, in which the electric field wraps around the bumps. (b) Mode at the lower edge of the bandgap, in which the electric field extends across the bumps. Fig. 4. Corrugated metal slab has high impedance at the top surface if the corrugations are one quarter-wavelength deep. is discussed in various textbooks [2], [3], and review articles [11], [12]. Numerous authors have also contributed general treatments of corrugated surfaces [13]–[15], and specific stud￾ies of important structures [16]–[19]. A corrugated surface is a metal slab, into which a series of vertical slots have been cut, as depicted in Fig. 4. The slots are narrow, so that many of them fit within one wavelength across the slab. Each slot can be regarded as a parallel-plate transmission line, running down into the slab, and shorted at the bottom. If the slots are one quarter-wavelength deep, then the short circuit at the bottom end is transformed by the length of the slots into an open-circuit at the top end. Thus, the impedance at the top surface is very high. If there are many slots per wavelength, the structure can be assigned an effective surface impedance equal to the impedance of the slots. The behavior of the corrugations is reduced to a single parameter—the boundary condition at the top surface. If the depth of the slots is greater than one quarter-wavelength, the surface impedance is capacitive, and TM surface waves are forbidden. Furthermore, a plane wave polarized with the electric field perpendicular to the ridges will appear to be reflected with no phase reversal since the effective reflection plane is actually at the bottom of the slots, one quarter-wavelength away. C. High-Impedance Surfaces The high-impedance surface described here is an abstraction of the corrugated surface, in which the corrugations have been folded up into lumped-circuit elements, and distributed in a two-dimensional lattice. The surface impedance is modeled as a parallel resonant circuit, which can be tuned to exhibit Fig. 5. Origin of the capacitance and inductance in the high-impedance surface. Fig. 6. Three-layer high-impedance surface achieves a lower operating fre￾quency for a given thickness by using capacitive loading, but it also suffers a reduction in bandwidth. high impedance over a predetermined frequency band. Peri￾odic two- or three-dimensional dielectric [20]–[24], metallic [25]–[28], or metallodielectric [29]–[33] structures that prevent the propagation of electromagnetic waves are known as pho￾tonic crystals [34]–[36]. The high-impedance surface can be considered as a kind of two-dimensional photonic crystal that prevents the propagation of radio-frequency surface currents within the bandgap. As the structure illustrated in Fig. 5 interacts with electro￾magnetic waves, currents are induced in the top metal plates. A voltage applied parallel to the top surface causes charges to build up on the ends of the plates, which can be described as a capacitance. As the charges slosh back and forth, they flow around a long path through the vias and bottom plate. Associated with these currents is a magnetic field and, thus, an inductance. We assign to the surface a sheet impedance equal to the impedance of a parallel resonant circuit, consisting of the sheet capacitance and the sheet inductance (16) The surface is inductive at low frequencies, and capacitive at high frequencies. The impedance is very high near the resonance frequency (17) We associate the high impedance with a forbidden frequency bandgap. In the two-layer geometry shown in Fig. 1, the capacitors are formed by the fringing electric fields between adjacent metal patches, and the inductance is fixed by the thickness of the structure. A three-layer design shown in Fig. 6 achieves a lower resonance frequency for a given thickness by using capacitive loading. In this geometry, parallel-plate capacitors are formed by the top two overlapping layers. IV. EFFECTIVE SURFACE IMPEDANCE MODEL Some of the properties of the high-impedance surface can be explained using an effective surface impedance model. The surface is assigned an impedance equal to that of a parallel resonant circuit, derived by geometry. The use