SIEVENPIPER et al. : HIGH-IMPEDANCE ELECTROMAGNETIC SURFACES 2063 Below resonance, TM surface waves are supported. At low frequencies, they lie very near the light line, and the fields 25 extend many wavelengths beyond the surface, as they do a flat metal surface. Near the resonant frequency, the surface waves are tightly bound to the sheet, and have a very low roup velocity, as seen by the fact that the dispersion curve esonance frequency is bent over, away from the light line. In the effective surface impedance limit, there is no Brillouin zone boundary, and the TM dispersion curve approaches the resonance frequency asymptotically. Thus, this approximation fails to predict the Above the resonance frequency, the surface is capacitive, and TE waves are supported. The lower end of the dispersion curve is close to the light line, and the waves are weakly bound to the surface, extending far into the surrounding space. As the Wave Vector [1/cm] frequency is increased, the curve bends away from the light Fig. 7. Dispersion diagram of the high-impedance surface, calculated using line, and the waves are more tightly bound to the surface the effective surface impedance The slope of the dispersion curve indicates that the waves feel an effective index of refraction that is greater than unity This is because a significant portion of the electric field is of lumped parameters to describe electromagnetic structures concentrated in the capacitors. The effective dielectric constant is valid as long as the wavelength is much longer than the size of the individual features. This short wave vector range of a material is enhanced if it is permeated with capacitor-like is also the regime of effective medium theory. The effective structures The te waves that lie to the left of the light line exist as surface impedance model can predict the reflection properties leaky waves that are damped by radiation. Radiation occurs from surfaces with a real impedance, thus, the leaky modes to the bandgap itself, which by definition must extend to large the left-hand side of the light line occur at the resonance fre- wave vector quency. The radiation from these leaky tE modes is modeled as a resistor in parallel with the high-impedance surface, which A. Surface Waves blurs the resonance frequency. Thus, the leaky waves actually We can determine the dispersion relation for surface waves radiate within a finite bandwidth, as shown in Fig.7.The in the context of the effective surface impedance model by damping resistance is the impedance of free space, projected inserting(1)into Maxwells equations. The wave vector k is onto the surface according to the angle of radiation. Small related to the spatial decay constant a and the frequency, w wave vectors represent radiation perpendicular to the surface by the following expression: while wave vectors near the light line represent radiation at k2=pe02+a2 grazing angles. For a tE polarized plane wave, the magnetic (18) field, H, projected on the surface at angle 0 with respect to For TM waves we can combine(18)with(14)to find the normal is H(0)=Hoods(0), while the electric field is just E(0= Eo. The impedance of free space, as seen by the a function of w, n=VHo/Eo is the impedance of free space and c=1/V/HOE0 expression surface for radiation at an angle, is given by the following is the speed of light in vacuum 2x(O)=B(6 1) (19) Thus. the radiation resistance is 377 3 for small wave We can find a similar expression for TE waves by combining vectors and normal radiation, but the damping resistance (18) with(15)as follows approaches infinity for wave vectors near the light line. Infinite resistance in a parallel resonant circuit corresponds to no damping, so the radiative band is reduced to zero width ATE= (20) for grazing angles near the light line. The high-impedance radiative region is shown as a shaded area, representing the By inserting(16)into(19)and(20), we can plot the disper- blurring of the leaky waves by radiation damping. In place of sion diagram for surface waves. in the context of the effective a bandgap, the effective surface impedance model predicts a surface impedance model. Depending on geometry, typical frequency band characterized by radiation damping values for the shee and sheet ing 0 layer structure are about 0.05 pF2, and 2 nH2, respectively. The B. Reflection Phase complete dispersion diagram, calculated using the effective The surface impedance determines the boundary condition medium model, is shown in Fig. 7 at the surface for the standing wave formed by incident andSIEVENPIPER et al.: HIGH-IMPEDANCE ELECTROMAGNETIC SURFACES 2063 Fig. 7. Dispersion diagram of the high-impedance surface, calculated using the effective surface impedance model. of lumped parameters to describe electromagnetic structures is valid as long as the wavelength is much longer than the size of the individual features. This short wave vector range is also the regime of effective medium theory. The effective surface impedance model can predict the reflection properties and some features of the surface-wave band structure, but not the bandgap itself, which by definition must extend to large wave vectors. A. Surface Waves We can determine the dispersion relation for surface waves in the context of the effective surface impedance model by inserting (1) into Maxwell’s equations. The wave vector is related to the spatial decay constant and the frequency, by the following expression: (18) For TM waves we can combine (18) with (14) to find the following expression for as a function of , in which is the impedance of free space and is the speed of light in vacuum: (19) We can find a similar expression for TE waves by combining (18) with (15) as follows: (20) By inserting (16) into (19) and (20), we can plot the disper￾sion diagram for surface waves, in the context of the effective surface impedance model. Depending on geometry, typical values for the sheet capacitance and sheet inductance of a two￾layer structure are about 0.05 pF , and 2 nH , respectively. The complete dispersion diagram, calculated using the effective medium model, is shown in Fig. 7. Below resonance, TM surface waves are supported. At low frequencies, they lie very near the light line, and the fields extend many wavelengths beyond the surface, as they do on a flat metal surface. Near the resonant frequency, the surface waves are tightly bound to the sheet, and have a very low group velocity, as seen by the fact that the dispersion curve is bent over, away from the light line. In the effective surface impedance limit, there is no Brillouin zone boundary, and the TM dispersion curve approaches the resonance frequency asymptotically. Thus, this approximation fails to predict the bandgap. Above the resonance frequency, the surface is capacitive, and TE waves are supported. The lower end of the dispersion curve is close to the light line, and the waves are weakly bound to the surface, extending far into the surrounding space. As the frequency is increased, the curve bends away from the light line, and the waves are more tightly bound to the surface. The slope of the dispersion curve indicates that the waves feel an effective index of refraction that is greater than unity. This is because a significant portion of the electric field is concentrated in the capacitors. The effective dielectric constant of a material is enhanced if it is permeated with capacitor-like structures. The TE waves that lie to the left of the light line exist as leaky waves that are damped by radiation. Radiation occurs from surfaces with a real impedance, thus, the leaky modes to the left-hand side of the light line occur at the resonance fre￾quency. The radiation from these leaky TE modes is modeled as a resistor in parallel with the high-impedance surface, which blurs the resonance frequency. Thus, the leaky waves actually radiate within a finite bandwidth, as shown in Fig. 7. The damping resistance is the impedance of free space, projected onto the surface according to the angle of radiation. Small wave vectors represent radiation perpendicular to the surface, while wave vectors near the light line represent radiation at grazing angles. For a TE polarized plane wave, the magnetic field, , projected on the surface at angle with respect to normal is , while the electric field is just . The impedance of free space, as seen by the surface for radiation at an angle, is given by the following expression: (21) Thus, the radiation resistance is 377 for small wave vectors and normal radiation, but the damping resistance approaches infinity for wave vectors near the light line. Infinite resistance in a parallel resonant circuit corresponds to no damping, so the radiative band is reduced to zero width for grazing angles near the light line. The high-impedance radiative region is shown as a shaded area, representing the blurring of the leaky waves by radiation damping. In place of a bandgap, the effective surface impedance model predicts a frequency band characterized by radiation damping. B. Reflection Phase The surface impedance determines the boundary condition at the surface for the standing wave formed by incident and