2060 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL, 47, NO Il, NOVEMBER 1999 They are arranged in a two-dimensional visualized as mushrooms or thumbtacks ng from the surface. The structure is easily fabricated board technology. The protrusions are formed as metal patches on the top surface of the board, connected to the solid lower conducting surface by metal plated vias If the protrusions are small compared to the operating wave- length, their electromagnetic properties can be described using Fig. 2. Surface wave bound to a dielectric interface, decaying exponentially lumped-circuit elements--capacitors and inductors. They be- away from the surface ive as a network of parallel resonant LC circuits, which act as a two-dimensional electric filter to block the flow of surface impedance is very high, the tangential magnetic field solved to obtain the following results / .equations can be currents along the sheet. In the frequency range where the or the waves described above. maxwells is small, even with a large electric field along the surface. Such a structure is sometimes described as a"magnetic conductor V1+E Due to this unusual boundary condition, the high-impedance surface can function as a unique new type of ground plane for low-profile antennas. The image currents are in-phase, rather adjacent to the surface, while still radiating efficiently, F o. than out-of-phase, allowing radiating elements to lie direct 1+eC (5) xample, a dipole lying flat against a high-impedance ground If e is positive then a and y are imaginary, and the waves plane is not shorted out as it would be on an ordinary metal do not decay with distance from the surface; they are simply sheet. Furthermore,in a forbidden frequency band, the high- plane waves propagating through the dielectric interface. Thus, impedance ground plane does not support propagating surface TM surface waves do not exist on nonconductive dielectric waves,thus, the radiation pattern is typically smooth, and free materials. On the other hand. if E is less than -1. or if it rom the effects of multipath interference along the ground is imaginary, the solution describes a wave that is bound to the surface. These TM surface waves can occur on metals or other materials with nonpositive dielectric constants. The IL. SURFACE WAVES solution for te surface waves can also be obtained from the foregoing analysis by the principle of duality [1]. If the electric dissimilar materials, such as metal and free space. They are the solution above can be applied to the le case uted for E, Surface waves can occur on the interface between two and magnetic fields are exchanged, and u is substituted for e bound to the interface, and decay exponentially into the sur- rounding materials. At radio frequencies, the fields associated B. Metal Surfaces with these waves can extend thousands of wavelengths into the surrounding space, and they are often best described as surface The effective. relative dielectric constant of a metal can be currents. They can be modeled from the viewpoint of an expressed in the following form [71 effective dielectric constant, or an effective surface impedance A. Dielectric Interfaces o is the conductivity, which is given by the following equation To derive the properties of surface waves on a dielectric interface [3],[7], begin with a surface in the Y Z plane, as X>0is filled with 1+ while the region X <0 is filled with dielectric E. Assume T is the mean electron collision time, q is the electron a wave that is bound to the surface, decaying in the +X- charge, and m and n are the effective mass and the density, direction with decay constant a, and in the -X-direction with respectively, of the conduction electrons decay constant y. The wave propagates in the Z-direction with For frequencies much lower than 1/ T, including the mi- propagation constant k. For a TM polarized surface wave, crowave spectrum, the conductivity is primarily real, and much Ey=0. The electric field in the upper half-space has the greater than unity, thus, the dielectric constant is a large following form imaginary number. Inserting (6) into(3)leads to a simple dispersion relation for surface waves at radio frequencies En=(iELr + 2El)eJu (1)follows In the lower half-space, the electric field has a similar form as follows: Thus, surface waves propagate at nearly the speed of light in vacuum, and they travel for many wavelengths along the metal E2=(E2x+2E2 ahaz+a (2) surface with little attenuation. By inserting()into(4),we can2060 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 47, NO. 11, NOVEMBER 1999 They are arranged in a two-dimensional lattice, and can be visualized as mushrooms or thumbtacks protruding from the surface. The structure is easily fabricated using printed-circuit￾board technology. The protrusions are formed as metal patches on the top surface of the board, connected to the solid lower conducting surface by metal plated vias. If the protrusions are small compared to the operating wave￾length, their electromagnetic properties can be described using lumped-circuit elements—capacitors and inductors. They be￾have as a network of parallel resonant circuits, which act as a two-dimensional electric filter to block the flow of currents along the sheet. In the frequency range where the surface impedance is very high, the tangential magnetic field is small, even with a large electric field along the surface. Such a structure is sometimes described as a “magnetic conductor.” Due to this unusual boundary condition, the high-impedance surface can function as a unique new type of ground plane for low-profile antennas. The image currents are in-phase, rather than out-of-phase, allowing radiating elements to lie directly adjacent to the surface, while still radiating efficiently. For example, a dipole lying flat against a high-impedance ground plane is not shorted out as it would be on an ordinary metal sheet. Furthermore, in a forbidden frequency band, the high￾impedance ground plane does not support propagating surface waves, thus, the radiation pattern is typically smooth, and free from the effects of multipath interference along the ground plane. II. SURFACE WAVES Surface waves can occur on the interface between two dissimilar materials, such as metal and free space. They are bound to the interface, and decay exponentially into the sur￾rounding materials. At radio frequencies, the fields associated with these waves can extend thousands of wavelengths into the surrounding space, and they are often best described as surface currents. They can be modeled from the viewpoint of an effective dielectric constant, or an effective surface impedance. A. Dielectric Interfaces To derive the properties of surface waves on a dielectric interface [3], [7], begin with a surface in the plane, as shown in Fig. 2. The region is filled with vacuum, while the region is filled with dielectric . Assume a wave that is bound to the surface, decaying in the - direction with decay constant , and in the -direction with decay constant . The wave propagates in the -direction with propagation constant . For a TM polarized surface wave, . The electric field in the upper half-space has the following form: (1) In the lower half-space, the electric field has a similar form as follows: (2) Fig. 2. Surface wave bound to a dielectric interface, decaying exponentially away from the surface. For the waves described above, Maxwell’s equations can be solved to obtain the following results [7]: (3) (4) (5) If is positive, then and are imaginary, and the waves do not decay with distance from the surface; they are simply plane waves propagating through the dielectric interface. Thus, TM surface waves do not exist on nonconductive dielectric materials. On the other hand, if is less than 1, or if it is imaginary, the solution describes a wave that is bound to the surface. These TM surface waves can occur on metals, or other materials with nonpositive dielectric constants. The solution for TE surface waves can also be obtained from the foregoing analysis by the principle of duality [1]. If the electric and magnetic fields are exchanged, and is substituted for , the solution above can be applied to the TE case. B. Metal Surfaces The effective, relative dielectric constant of a metal can be expressed in the following form [7]: (6) is the conductivity, which is given by the following equation: (7) is the mean electron collision time, is the electron charge, and and are the effective mass and the density, respectively, of the conduction electrons. For frequencies much lower than , including the mi￾crowave spectrum, the conductivity is primarily real, and much greater than unity, thus, the dielectric constant is a large imaginary number. Inserting (6) into (3) leads to a simple dispersion relation for surface waves at radio frequencies as follows: (8) Thus, surface waves propagate at nearly the speed of light in vacuum, and they travel for many wavelengths along the metal surface with little attenuation. By inserting (6) into (4), we can