SIEVENPIPER et al. : HIGH-IMPEDANCE ELECTROMAGNETIC SURFACES find the following expression for a, the decay constant of the in which the surface impedance is given by the following fields into the surrounding space: expression Zs(TM)=2c Conversely, TE waves can occur on a capacitive surface, with For good conductors at microwave frequencies, the urface the following impe ance waves extend a great distance into the surrounding space For example, on a copper surface, the electromagnetic fields Zs(TE)==oke (15) associated with a 10-GHz surface wave extend about 70m 2300 wavelengths into free space. Hence, at microwave The surface impedance of the textured metal surface described frequencies they are often described simply as surface currents, in this paper is characterized by an equivalent parallel resonant rather than surface waves. These surface currents are nothing LC circuit. At low frequencies it is inductive, and supports more than the normal alternating currents that can occur on TM waves. At high frequencies it is capacitive, and supports any conductor TE waves. Near the LC resonance frequency, the surface We can also determine ?, the surface-wave penetration impedance is very high. In this region, waves are not bound to depth into the metal. By inserting(6)into(5), we obtain the the surface; instead, they radiate readily into the surrounding following T≈(1+j) 00_(1+) (10) IIL. TEXTURED SURFACE Ght The concept of suppressing surface waves on metals is not Thus, we have derived the skin depth d from the surface-wave new. It has been done before using several geometries, such penetration depth [2]. The surface currents penetrate only a as a metal sheet covered with small bumps [81, [9], or a very small distance into the metal. For example, at 10 GHz, corrugated metal slab [11H-[19]. The novelty of this study the skin depth of copper is less than 1 um is the application of an array of lumped-circuit elements to From the skin depth, we can derive the surface impedance produce a thin two-dimensional structure that must generally of a fiat metal sheet [3].Using(10), we can express the current be described by band structure concepts, even though the in terms of the skin depth, assuming Eo is the electric field thickness and preiodicity are both much smaller than the at the surface operating wavelength. J(x)=aE2(x)=0Ec-m(1+)/ (ID)A. Bumpy Surfaces The magnetic field at the surface is found by integrating along Surface waves can be eliminated from a metal surface over a path surrounding the thin surface layer of current, extending a finite frequency band by applying a periodic texture, such far into the metal beyond the skin depth as follows as a lattice of small bumps. As surface waves scatter from the rows of bumps, the resulting interference prevents them J(r)da (12) from propagating, producing a two-dimensional electromag- netic bandgap. Such a structure has been studied at optical frequencies by Barnes et al.[8] and Kitson et al.[9] using a Thus, the surface impedance of a flat sheet of metal is derived triangular lattice of bumps, patterned on a silver film as follows When the wavelength is much longer than the period of two- dimensional lattice, the surface waves barely notice the small (13)bumps. At shorter wavelengths. the surface waves feel the ffects of the surface texture. When one-half wavelength fits The surface impedance has equal positive real and positive between the rows of bumps, this corresponds to the Brillouin imaginary parts, so the small surface resistance is accompanied zone boundary [10] of the two-dimensional lattice. At this by an equal amount of surface inductance. For example, the wavelength, a standing wave on the surface can have two surface impedance of copper at 10 GHz is 0.03(1+j) possible positions: with the wave crests centered on the bumps or with the nulls centered on the bumps, as shown in Fig 3 C. High-Impedance Surfaces These two modes have slightly different frequencies, separated By applying a texture to the metal surface, we can alter its by a small bandgap, within which surface waves cannot surface impedance, and thereby change its surface-wave prop- an extension of this"bumpy surface" in which the bandgap erties. The behavior of surface waves on a general impedance urface is derived in several electromagnetics textbooks [2] has been lowered in frequency by capacitive loading 3]. The derivation proceeds by assuming a wave that decays exponentially away from a boundary, with decay constant a. B. Corrugated Surfaces The boundary is characterized by its surface impedance. It The high-impedance surface can also be understood by can be shown that TM waves occur on an inductive surface. examining a similar structure. the corrugated surface. whichSIEVENPIPER et al.: HIGH-IMPEDANCE ELECTROMAGNETIC SURFACES 2061 find the following expression for , the decay constant of the fields into the surrounding space: (9) For good conductors at microwave frequencies, the surface waves extend a great distance into the surrounding space. For example, on a copper surface, the electromagnetic fields associated with a 10-GHz surface wave extend about 70 m, or 2300 wavelengths into free space. Hence, at microwave frequencies they are often described simply as surface currents, rather than surface waves. These surface currents are nothing more than the normal alternating currents that can occur on any conductor. We can also determine , the surface-wave penetration depth into the metal. By inserting (6) into (5), we obtain the following: (10) Thus, we have derived the skin depth from the surface-wave penetration depth [2]. The surface currents penetrate only a very small distance into the metal. For example, at 10 GHz, the skin depth of copper is less than 1 m. From the skin depth, we can derive the surface impedance of a flat metal sheet [3]. Using (10), we can express the current in terms of the skin depth, assuming is the electric field at the surface (11) The magnetic field at the surface is found by integrating along a path surrounding the thin surface layer of current, extending far into the metal beyond the skin depth as follows: (12) Thus, the surface impedance of a flat sheet of metal is derived as follows: (13) The surface impedance has equal positive real and positive imaginary parts, so the small surface resistance is accompanied by an equal amount of surface inductance. For example, the surface impedance of copper at 10 GHz is . C. High-Impedance Surfaces By applying a texture to the metal surface, we can alter its surface impedance, and thereby change its surface-wave prop￾erties. The behavior of surface waves on a general impedance surface is derived in several electromagnetics textbooks [2], [3]. The derivation proceeds by assuming a wave that decays exponentially away from a boundary, with decay constant . The boundary is characterized by its surface impedance. It can be shown that TM waves occur on an inductive surface, in which the surface impedance is given by the following expression: (14) Conversely, TE waves can occur on a capacitive surface, with the following impedance: (15) The surface impedance of the textured metal surface described in this paper is characterized by an equivalent parallel resonant circuit. At low frequencies it is inductive, and supports TM waves. At high frequencies it is capacitive, and supports TE waves. Near the resonance frequency, the surface impedance is very high. In this region, waves are not bound to the surface; instead, they radiate readily into the surrounding space. III. TEXTURED SURFACES The concept of suppressing surface waves on metals is not new. It has been done before using several geometries, such as a metal sheet covered with small bumps [8], [9], or a corrugated metal slab [11]–[19]. The novelty of this study is the application of an array of lumped-circuit elements to produce a thin two-dimensional structure that must generally be described by band structure concepts, even though the thickness and preiodicity are both much smaller than the operating wavelength. A. Bumpy Surfaces Surface waves can be eliminated from a metal surface over a finite frequency band by applying a periodic texture, such as a lattice of small bumps. As surface waves scatter from the rows of bumps, the resulting interference prevents them from propagating, producing a two-dimensional electromag￾netic bandgap. Such a structure has been studied at optical frequencies by Barnes et al. [8] and Kitson et al. [9] using a triangular lattice of bumps, patterned on a silver film. When the wavelength is much longer than the period of two￾dimensional lattice, the surface waves barely notice the small bumps. At shorter wavelengths, the surface waves feel the effects of the surface texture. When one-half wavelength fits between the rows of bumps, this corresponds to the Brillouin zone boundary [10] of the two-dimensional lattice. At this wavelength, a standing wave on the surface can have two possible positions: with the wave crests centered on the bumps, or with the nulls centered on the bumps, as shown in Fig. 3. These two modes have slightly different frequencies, separated by a small bandgap, within which surface waves cannot propagate. Our high-impedance surface can be considered as an extension of this “bumpy surface,” in which the bandgap has been lowered in frequency by capacitive loading. B. Corrugated Surfaces The high-impedance surface can also be understood by examining a similar structure, the corrugated surface, which