Patent Publication Number: US-6670932-B1

Title: Multi-resonant, high-impedance surfaces containing loaded-loop frequency selective surfaces

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation in part of application Ser. No. 09/678,128, filed Oct. 4, 2000 and commonly assigned with the present application. 
    
    
     BACKGROUND 
     The present invention relates generally to high-impedance surfaces. More particularly, the present invention relates to a multi-resonant, high-impedance electromagnetic surface. 
     A high impedance surface is a lossless, reactive surface whose equivalent surface impedance,            Z   s     =       E   tan       H   tan         ,                   
     approximates an open circuit and which inhibits the flow of equivalent tangential electric surface current, thereby approximating a zero tangential magnetic field, H tan ≈0. E tan  and H tan  are the electric and magnetic fields, respectively, tangential to the surface. High impedance surfaces have been used in various antenna applications. These applications range from corrugated horns which are specially designed to offer equal E and H plane half power beamwidths to traveling wave antennas in planar or cylindrical form. However, in these applications, the corrugations or troughs are made of metal where the depth of the corrugations is one quarter of a free space wavelength, λ/4, where λ is the wavelength at the frequency of interest. At high microwave frequencies, λ/4 is a small dimension, but at ultra-high frequencies (UHF, 300 MHz to 1 GHz), or even at low microwave frequencies (1-3 GHz), λ/4 can be quite large. For antenna applications in these frequency ranges, an electrically-thin (λ/100 to λ/50 thick) and physically thin high impedance surface is desired. 
     One example of a thin high-impedance surface is disclosed in D. Sievenpiper, “High-impedance electromagnetic surfaces,” Ph.D. dissertation, UCLA electrical engineering department, filed January 1999, and in PCT Patent Application number PCT/US99/06884. This high impedance surface  100  is shown in FIG.  1 . The high-impedance surface  100  includes a lower permittivity spacer layer  104  and a capacitive frequency selective surface (FSS)  102  formed on a metal backplane  106 . Metal vias  108  extend through the spacer layer  104 , and connect the metal backplane to the metal patches of the FSS layer. The thickness h of the high impedance surface  100  is much less than λ/4 at resonance, and typically on the order of λ/50, as indicated in FIG.  1 . 
     The FSS  102  of the prior art high impedance surface  100  is a periodic array of metal patches  110  which are edge coupled to form an effective sheet capacitance. This is referred to as a capacitive frequency selective surface (FSS). Each metal patch  110  defines a unit cell which extends through the thickness of the high impedance surface  100 . Each patch  110  is connected to the metal backplane  106 , which forms a ground plane, by means of a metal via  108 , which can be plated through holes. The periodic array of metal vias  108  has been known in the prior art as a rodded media, so these vias are sometimes referred to as rods or posts. The spacer layer  104  through which the vias  108  pass is a relatively low permittivity dielectric typical of many printed circuit board substrates. The spacer layer  104  is the region occupied by the vias  108  and the low permittivity dielectric. The spacer layer is typically 10 to 100 times thicker than the FSS layer  102 . Also, the dimensions of a unit cell in the prior art high-impedance surface are much smaller than λ at the fundamental resonance. The period is typically between λ/40 and λ/12. 
     A frequency selective surface is a two-dimensional array of periodically arranged elements which may be etched on, or embedded within, one or multiple layers of dielectric laminates. Such elements may be either conductive dipoles, patches, loops, or even slots. As a thin periodic structure, it is often referred to as a periodic surface. 
     Frequency selective surfaces have historically found applications in out-of-band radar cross section reduction for antennas on military airborne and naval platforms. Frequency selective surfaces are also used as dichroic subreflectors in dual-band Cassegrain reflector antenna systems. In this application, the subreflector is transparent at frequency band f 1  and opaque or reflective at frequency band f 2 . This allows one to place the feed horn for band f 1  at the focal point for the main reflector, and another feed horn operating at f 2  at the Cassegrain focal point. One can achieve a significant weight and volume savings over using two conventional reflector antennas, which is critical for space-based platforms. 
     The prior art high-impedance surface  100  provides many advantages. The surface is constructed with relatively inexpensive printed circuit technology and can be made much lighter than a corrugated metal waveguide, which is typically machined from a block of aluminum. In printed circuit form, the prior art high-impedance surface can be 10 to 100 times less expensive for the same frequency of operation. Furthermore, the prior art surface offers a high surface impedance for both x and y components of tangential electric field, which is not possible with a corrugated waveguide. Corrugated waveguides offer a high surface impedance for one polarization of electric field only. According to the coordinate convention used herein, a surface lies in the xy plane and the z-axis is normal or perpendicular to the surface. Further, the prior art high-impedance surface provides a substantial advantage in its height reduction over a corrugated metal waveguide, and may be less than one-tenth the thickness of an air-filled corrugated metal waveguide. 
     A high-impedance surface is important because it offers a boundary condition which permits wire antennas conducting electric currents to be well matched and to radiate efficiently when the wires are placed in very close proximity to this surface (e.g., less than λ/100 away). The opposite is true if the same wire antenna is placed very close to a metal or perfect electric conductor (PEC) surface. The wire antenna/PEC surface combination will not radiate efficiently due to a very severe impedance mismatch. The radiation pattern from the antenna on a high-impedance surface is confined to the upper half space, and the performance is unaffected even if the high-impedance surface is placed on top of another metal surface. Accordingly, an electrically-thin, efficient antenna is very appealing for countless wireless devices and skin-embedded antenna applications. 
     FIG. 2 illustrates electrical properties of the prior art high-impedance surface. FIG.  2 ( a ) illustrates a plane wave normally incident upon the prior art high-impedance surface  100 . Let the reflection coefficient referenced to the surface be denoted by Γ. The physical structure shown in FIG.  2 ( a ) has an equivalent transverse electromagnetic mode transmission line shown in FIG.  2 ( b ). The capacitive FSS  102  (FIG. 1) is modeled as a shunt capacitance C and the spacer layer  104  is modeled as a transmission line of length h which is terminated in a short circuit corresponding to the backplane  106 . FIG.  2 ( c ) shows a Smith chart in which the short is transformed into the stub impedance Z stub  just below the FSS layer  102 . The admittance of this stub line is added to the capacitive susceptance to create a high impedance Z in  at the outer surface. Note that the Z in  locus on the Smith Chart in FIG.  2 ( c ) will always be found on the unit circle since our model is ideal and lossless. So Γ has an amplitude of unity. 
     The reflection coefficient Γ has a phase angle θ which sweeps from 180° at DC, through 0° at the center of the high impedance band, and rotates into negative angles at higher frequencies where it becomes asymptotic to −180°. This is illustrated in FIG.  2 ( d ). Resonance is defined as that frequency corresponding to 0° reflection phase. Herein, the reflection phase bandwidth is defined as that bandwidth between the frequencies corresponding to the +90° and −90° phases. This reflection phase bandwidth also corresponds to the range of frequencies where the magnitude of the surface reactance exceeds the impedance of free space: |X|≧η 0 =377 ohms. 
     A perfect magnetic conductor (PMC) is a mathematical boundary condition whereby the tangential magnetic field on this boundary is forced to be zero. It is the electromagnetic dual to a perfect electric conductor (PEC) upon which the tangential electric field is defined to be zero. A PMC can be used as a mathematical tool to create simpler but equivalent electromagnetic problems for slot antenna analysis. PMCs do not exist except as mathematical artifacts. However, the prior art high-impedance surface is a good approximation to a PMC over a limited band of frequencies defined by the +/−90° reflection phase bandwidth. So in recognition of its limited frequency bandwidth, the prior art high-impedance surface is referred to herein as an example of an artificial magnetic conductor, or AMC. 
     The prior art high-impedance surface offers reflection phase resonances at a fundamental frequency, plus higher frequencies approximated by the condition where the electrical thickness of the spacer layer, βh, in the high-impedance surface  100  is nπ, where n is an integer. These higher frequency resonances are harmonically related and hence uncontrollable. If the prior art AMC is to be used in a dual-band antenna application where the center frequencies are separated by a frequency range of, say 1.5:1, we would be forced to make a very thick AMC. Assuming a non-magnetic spacer layer (μ D =1), the thickness h must be h=λ/14 to achieve at least a 50% fractional frequency bandwidth where both center frequencies would be contained in the reflection phase bandwidth. Alternatively, magnetic materials could be used to load the spacer layer, but this is a topic of ongoing research and nontrivial expense. Accordingly, there is a need for a class of AMCs which exhibit multiple reflection phase resonances, or multi-band performance, that are not harmonically related, but at frequencies which may be prescribed. 
     BRIEF SUMMARY 
     By way of introduction only, in a first aspect, an artificial magnetic conductor includes a frequency selective surface having a frequency dependent permeability μ 1z  in a direction normal to the frequency dependent surface, a conductive ground plane, and a rodded medium disposed between the frequency selective surface and the conductive ground plane. 
     In another aspect, an artificial magnetic conductor includes a conductive ground plane and a spacer layer disposed on the ground plane. One or more arrays of coplanar loops are resonant at two or more frequency bands, each loop having a similar shape and similar size. The one or more arrays of coplanar loops produce a frequency dependent normal permeability μ z . 
     In another aspect, a disclosed electrical apparatus includes a conductive ground plane and a dielectric layer perforated by conductive rods in electrical contact with the conductive ground plane. The electrical apparatus further includes a frequency selective surface (FSS) disposed on the dielectric layer. The FSS includes a first layer of capacitively coupled loops resonant at a first frequency, a dielectric spacer layer and a second layer of capacitively coupled loops resonant at a second frequency. The frequency selective surface has a frequency dependent permeability in a direction substantially normal to the frequency selectively surface. 
     The foregoing summary has been provided only by way of introduction. Nothing in this section should be taken as a limitation on the following claims, which define the scope of the invention. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a perspective view of a prior art high impedance surface; 
     FIG. 2 illustrates a reflection phase model for the prior art high impedance surface; 
     FIG. 3 is a diagram illustrating surface wave properties of an artificial magnetic conductor; 
     FIG. 4 illustrates electromagnetic fields of a TE mode surface wave propagating in the x direction in the artificial magnetic conductor of FIG. 3; 
     FIG. 5 illustrates electromagnetic fields of a TM mode surface wave propagating in the x direction in the artificial magnetic conductor of FIG. 3; 
     FIG. 6 illustrates top and cross sectional views of a prior art high impedance surface; 
     FIG. 7 presents a new effective media model for the prior art high-impedance surface of FIG. 6; 
     FIG. 8 illustrates a first embodiment of an artificial magnetic conductor; 
     FIG. 9 illustrates a second, multiple layer embodiment of an artificial magnetic conductor; 
     FIG. 10 is a cross sectional view of the artificial magnetic conductor of FIG. 9; 
     FIG. 11 illustrates a first physical embodiment of a loop for an artificial magnetic molecule; 
     FIG. 12 illustrates a multiple layer artificial magnetic conductor using the loop of FIG.  11 ( d ); 
     FIG. 13 shows y-polarized electromagnetic simulation results for the normal-incidence reflection phase of the artificial magnetic conductor illustrated in FIG. 12; 
     FIG. 14 shows y-polarized electromagnetic simulation results for the normal-incidence reflection phase of the artificial magnetic conductor very similar to that illustrated in FIG. 12, except the gaps in the loops are now shorted together; 
     FIG. 15 shows the TEM mode equivalent circuits for the top layer, or FSS layer, of a two layer artificial magnetic conductor of FIG. 8; 
     FIG. 16 illustrates the effective relative permittivity for a specific case of a multi-resonant FSS, and the corresponding reflection phase; for an AMC which uses this FSS as its upper layer. 
     FIG. 17 shows an alternative embodiment for a frequency selective surface implemented with square loops; 
     FIG. 18 shows measured reflection phase data for an x polarized electric field normally incident on the AMC of FIG. 17; 
     FIG. 19 shows measured reflection phase data for a y polarized electrical field normally incident on the AMC of FIG. 17; 
     FIG. 20 shows additional alternative embodiments for a frequency selective surface implemented with square loops; 
     FIG. 21 shows additional alternative embodiments for a frequency selective surface implemented with square loops; 
     FIG. 22 shows measured reflection phase data for an x polarized electric field normally incident on the AMC of FIG. 21; 
     FIG. 23 shows measured reflection phase data for ay polarized electrical field normally incident on the AMC of FIG. 21; 
     FIG. 24 illustrates another embodiment of a capacitive frequency selective surface structure consisting of a layer of loops closely spaced to a layer of patches; 
     FIG. 25 illustrates an alternative embodiment of a capacitive frequency selective surface structure using hexagonal loops; 
     FIG. 26 illustrates an alternative embodiment of a capacitive frequency selective surface structure using hexagonal loops; 
     FIG. 27 illustrates an alternative embodiment of a capacitive frequency selective surface structure using hexagonal loops; 
     FIG. 28 illustrates an effective media model for an artificial magnetic conductor; 
     FIG. 29 illustrates a prior art high impedance surface; 
     FIG. 30 illustrates Lorentz and Debye frequency responses for the capacitance of an FSS used in a multi-resonant AMC; 
     FIG. 31 illustrates an artificial magnetic conductor including a multiple layer frequency selective surface; and 
     FIG. 32 illustrates a top view of the multiple-layer frequency selective surface of FIG.  31 . 
    
    
     DETAILED DESCRIPTION OF THE PRESENTLY PREFERRED EMBODIMENTS 
     A planar, electrically-thin, anisotropic material is designed to be a high-impedance surface to electromagnetic waves. It is a two-layer, periodic, magnetodielectric structure where each layer is engineered to have a specific tensor permittivity and permeability behavior with frequency. This structure has the properties of an artificial magnetic conductor over a limited frequency band or bands, whereby, near its resonant frequency, the reflection amplitude is near unity and the reflection phase at the surface lies between +/−90 degrees. This engineered material also offers suppression of transverse electric (TE) and transverse magnetic (TM) mode surface waves over a band of frequencies near where it operates as a high impedance surface. The high impedance surface provides substantial improvements and advantages. Advantages include a description of how to optimize the material&#39;s effective media constituent parameters to offer multiple bands of high surface impedance. Advantages further include the introduction of various embodiments of conducting loop structures into the engineered material to exhibit multiple reflection-phase resonant frequencies. Advantages still further include a creation of a high-impedance surface exhibiting multiple reflection-phase resonant frequencies without resorting to additional magnetodielectric layers. 
     This high-impedance surface has numerous antenna applications where surface wave suppression is desired, and where physically thin, readily attachable antennas are desired. This includes internal antennas in radiotelephones and in precision GPS antennas where mitigation of multipath signals near the horizon is desired. 
     An artificial magnetic conductor (AMC) offers a band of high surface impedance to plane waves, and a surface wave bandgap over which bound, guided transverse electric (TE) and transverse magnetic (TM) modes cannot propagate. TE and TM modes are surface waves moving transverse or across the surface of the AMC, in parallel with the plane of the AMC. The dominant TM mode is cut off and the dominant TE mode is leaky in this bandgap. The bandgap is a band of frequencies over which the TE and TM modes will not propagate as bound modes. 
     FIG. 3 illustrates surface wave properties of an AMC  300  in proximity to an antenna or radiator  304 . FIG.  3 ( a ) is an ω-β diagram for the lowest order TM and TE surface wave modes which propagate on the AMC  300 . Knowledge of the bandgap over which bound TE and TM waves cannot propagate is very critical for antenna applications of an AMC because it is the radiation from the unbound or leaky TE mode, excited by the wire antenna  304  and the inability to couple into the TM mode that makes bent-wire monopoles, such as the antenna  304  on the AMC  300 , a practical antenna element. The leaky TE mode occurs at frequencies only within the bandgap. 
     FIG.  3 ( b ) is a cross sectional view of the AMC  300  showing TE waves radiating from the AMC  300  as leaky waves. Leakage is illustrated by the exponentially increasing spacing between the arrows illustrating radiation from the surface as the waves radiate power away from the AMC  300  near the antenna  304 . Leakage of the surface wave dramatically reduces the diffracted energy from the edges of the AMC surface in antenna applications. The radiation pattern from small AMC ground planes can therefore be substantially confined to one hemisphere, the hemisphere above the front or top surface of the AMC  300 . The front or top surface is the surface proximate the antenna  304 . The hemisphere below or behind the AMC  300 , below the rear or bottom surface of the AMC  300 , is essentially shielded from radiation. The rear or bottom surface of the AMC  300  is the surface away from the antenna  304 . 
     FIG. 4 illustrates a TE surface wave mode on the artificial magnetic conductor  300  of FIG.  3 . Similarly, FIG. 5 illustrates a TM surface wave mode on the AMC  300  of FIG.  3 . The coordinate axes in FIGS. 4 and 5, and as used herein, place the surface of the AMC  300  in the xy plane. The z axis is normal to the surface. The TE mode of FIG. 4 propagates in the x direction along with loops of an associated magnetic field H. The amplitude of the x component of magnetic field H both above the surface and within the surface is shown by the graph in FIG.  4 . FIG. 5 shows the TM mode propagating in the x direction, along with loops of an associated electric field E. The relative amplitude of the x component of the electric field E is shown in the graph in FIG.  5 . 
     The performance and operation of the AMC  300  will be described in terms of an effective media model. An effective media model allows transformation all of the fine, detailed, physical structure of an AMC&#39;s unit cell into that of equivalent media defined only by the permittivity and permeability parameters. These parameters allow use of analytic methods to parametrically study wave propagation on AMCs. Such analytic models lead to physical insights as to how and why AMCs work, and insights on how to improve them. They allow one to study an AMC in general terms, and then consider each physical embodiment as a specific case of this general model. However, it is to be noted that such models represent only approximations of device and material performance and are not necessarily precise calculations of that performance. 
     First, the effective media model for the prior art high-impedance surface is presented. Consider a prior art high-impedance surface  100  comprised of a square lattice of square patches  110  as illustrated in FIG.  6 . Each patch  110  has a metal via  108  connecting it to the backplane  106 . The via  108  passes through a spacer layer  102 , whose isotropic host media parameters are ε D  and μ D . 
     FIG. 7 presents a new effective media model for substantially characterizing the prior art high-impedance surface of FIG.  6 . Elements of the permittivity tensor are given in FIG.  7 . The parameter α is a ratio of areas, specifically the area of the cross section of the via  108 , πd 2 /4, to the area of a unit cell, a 2 =A. Each unit cell has an area A and includes one patch  110 , measuring b×b in size, plus the space g in the x and y directions to an adjacent patch  110 , for a pitch or period of a, and with a thickness equal to the thickness of the high impedance surface  100 , or h+δ in FIG.  6 . Note that α is typically a small number much less than unity, and usually below 1%. 
     In the cross sectional view of FIG.  6 ( b ), the high impedance surface  100  includes a first or upper region  602  and a second or lower region  604 . The lower region  604 , denoted here as region  2 , is referred to as a rodded media. Transverse electric and magnetic fields in this region  604  are only minimally influenced by the presence of the vias or rods  108 . The effective transverse permittivity, ε 2x  and permeability, μ 2x , are calculated as minor perturbations from the media parameters of the host dielectric. This is because the electric polarisability of a circular cylinder, πd 2 /2, is quite small for the thin metal rods whose diameter is small relative to the period a. Also note that effective transverse permittivity, ε 2x , and permeability, μ 2x , are constant with frequency. However, the normal, or z-directed, permittivity is highly dispersive or frequency dependent. A transverse electromagnetic (TEM) wave with a z-directed electric field traveling in a lateral direction (x or y), in an infinite rodded medium, will see the rodded media  102  as a high pass filter. The TEM wave will experience a cutoff frequency, f c , below which ε 2x  is negative, and above this cutoff frequency, ε 2z  is positive and asymptotically approaches the host permittivity ε D . This cutoff frequency is essentially given by          f   c     =     1     2                 π            ɛ   D          ɛ   0          μ   D          μ   0            A     4                 π            [       ln        (     1   α     )       +   α   -   1     ]                               
     The reflection phase resonant frequency of the prior art high-impedance surface  100  is found well below the cutoff frequency of the rodded media  102 , where ε 2z  is quite negative. 
     The upper region  602 , denoted as region  1 , is a capacitive FSS. The transverse permittivity, ε 1x  or ε 1y , is increased by the presence of the edge coupled metal patches  110  so that ε 1x =ε 1y &gt;&gt;1, typically between 10 and 100 for a single layer frequency selective surface such as the high-impedance surface  100 . The effective sheet capacitance, C=ε 0 ε 1x t, is uniquely defined by the geometry of each patch  110 , but ε 1x  in the effective media model is somewhat arbitrary since t is chosen arbitrarily. The variable t is not necessarily the thickness of the patches, which is denoted as δ. However, t should be much less than the spacer layer  604  height h. 
     The tensor elements for the upper layer  602  of the prior art high-impedance surface  100  are constant values which do not change with frequency. That is, they are non-dispersive. Furthermore, for the upper layer  602 , the z component of the permeability is inversely related to the transverse permittivity by μ 1z =2/ε 1x . Once the sheet capacitance is defined, μ 1z  is fixed. 
     It is useful to introduce the concept of an artificial magnetic molecule. An artificial magnetic molecule (AMM) is an electrically small conductive loop which typically lies in one plane. Both the loop circumference and the loop diameter are much less than one free-space wavelength at the useful frequency of operation. The loops can be circular, square, hexagonal, or any polygonal shape, as only the loop area will affect the magnetic dipole moment. Typically, the loops are loaded with series capacitors to force them to resonate at frequencies well below their natural resonant frequency 
     A three dimensional, regular array or lattice of AMMs is an artificial material whose permeability can exhibit a Lorentz resonance, assuming no intentional losses are added. At a Lorentz resonant frequency, the permeability of the artificial material approaches infinity. Depending on where the loop resonance is engineered, the array of molecules can behave as a bulk paramagnetic material (μ r &gt;1) or as a diamagnetic material (μ r &lt;1) in the direction normal to the loops. AMMs may be used to depress the normal permeability of the FSS layer, region  1 , in AMCs. This in turn has a direct impact on the TE mode cutoff frequencies, and hence the surface wave bandgaps. 
     The prior art high impedance surface has a fundamental, or lowest, resonant frequency near f 0 =1(2π{square root over (μ D μ 0 hC)}), where the spacer layer is electrically thin, (βh&lt;&lt;1 where ={square root over (μ D μ 0 ε D ε 0 )}). Higher order resonances are also found, but at much higher frequencies where βh≈bπ and n=1,2,3, . . . The n=1 higher order resonance is typically 5 to 50 times higher than the fundamental resonance. Thus, a prior art high impedance surface designed to operate at low microwave frequencies (1-3 GHz) will typically exhibit its next reflection phase resonance in millimeter wave bands (above 30 GHz). 
     There is a need for an AMC which provides a second band or even multiple bands of high surface impedance whose resonant frequencies are all relatively closely spaced, within a ratio of about 2:1 or 3:1. This is needed, for example, for multi-band antenna applications. Furthermore, there is a need for an AMC with sufficient engineering degrees of freedom to allow the second and higher reflection phase resonances to be engineered or designated arbitrarily. Multiple reflection phase resonances are possible if more than two layers (4, 6, 8, etc.) are used in the fabrication of an AMC. However, this adds cost, weight, and thickness relative to the single resonant frequency design. Thus there is a need for a means of achieving multiple resonances from a more economical two-layer design. In addition, there is a need for a means of assuring the existence of a bandgap for bound, guided, TE and TM mode surface waves for all of the high-impedance bands, and within the +/−90° reflection phase bandwidths. 
     FIG. 8 illustrates an artificial magnetic conductor (AMC)  800 . The AMC  800  includes an array  802  that is in one embodiment a coplanar array of resonant loops or artificial magnetic molecules  804  which are strongly capacitively coupled to each other, forming a capacitive frequency selective surface (FSS). The resonant loops  804  in the illustrated embodiment are uniformly spaced and at a height h above a solid conductive ground plane  806 . An array of electrically short, conductive posts or vias  808  are attached to the ground plane  806  only and have a length h. Each loop  804  includes a lumped capacitive load  810 . The one or more layers of artificial magnetic molecules (AMMs) or resonant loops of the artificial magnetic conductor  800  create a frequency dependent permeability in the z direction, normal to the surface of the AMC  800 . 
     An AMC  800  with a single layer of artificial magnetic molecules  804  is shown in FIG.  8 . In this embodiment, each loop and capacitor load are substantially identical so that all loops have substantially the same resonant frequency. In alternative embodiments, loops having different characteristics may be used. In physical realizations, due to manufacturing tolerances and other causes, individual loops and their associated resonant frequencies will not necessarily be identical. 
     An AMC  900  with multiple layers of artificial magnetic molecules  804  is shown in FIG.  9 . FIG. 10 is a cross sectional view of the artificial magnetic conductor  900  of FIG.  9 . The AMC  900  includes a first layer  902  of loops  804  resonant at a first frequency f 1 . The AMC  900  includes a second layer  904  of loops  804  resonant at a second frequency f 2 . Each loop  804  of the first layer  902  of loops includes a lumped capacitive load C 1    908 . Each loop  804  of the second layer  904  of loops includes a lumped capacitive load C 2    906 . The lumped capacitances may be the same but need not be. In combination, the first layer  902  of loops  804  and the second layer  906  of loops  904  form a frequency selective surface (FSS) layer  910  disposed on a spacer layer  912 . In practical application, the low frequency limit of the transverse effective relative permittivity, ε 1x  and ε 1y , for the multiple layer AMC  900  lies between 100 and 2000. Accordingly, strong capacitive coupling is present between loops  902  and  904 . A practical way to achieve this coupling is to print two layers of loops on opposite sides of an FSS dielectric layer as shown in FIG.  10 . Other realizations may be chosen as well. 
     FIG. 11 illustrates a first physical embodiment of a loop  1100  for use in an artificial magnetic conductor such as the AMC  800  of FIG.  8 . Conducting loops such as loop  1100  which form the artificial magnetic molecules can be implemented in a variety of shapes such as square, rectangular, circular, triangular, hexagonal, etc. In the embodiment of FIG. 11, the loop  1100  is square in shape. Notches  1102  can be designed in the loops to increase the self inductance, which lowers the resonant frequency of the AMMs. Notches  1102  and gaps  1104  can also be introduced to engineer the performance of the loop  1100  to a particular desired response. For example, the bands or resonance frequencies may be chosen by selecting a particular shape for the loop  1100 . In general, a gap  1104  cuts all the way through a side of the loop  1100  from the center of the loop  1100  to the periphery. In contrast, a notch cuts through only a portion of a side between the center and periphery of the loop  1100 . FIG. 11 illustrates a selection of potential square loop designs. 
     FIG. 12 illustrates a portion of a two layer artificial magnetic conductor whose FSS layer uses a square loop of FIG.  11 ( d ). Wide loops with relatively large surface area promote capacitive coupling between loops of adjacent layers when used in a two-layer overlapping AMC, as illustrated in FIG.  12 . An overlap region  1202  at the gap  1104  provides the series capacitive coupling required for loop resonance. 
     In one preferred embodiment, loops of the type illustrated in FIGS. 11 and 12 are formed on surfaces of dielectric materials using conventional printed circuit board (PCB) manufacturing techniques. For example, a metallic layer is deposited on a surface of the PCB and subsequently patterned by chemical etching or other technique. Such processes provide precise control of sizes, spacing and uniformity of printed features. 
     FIG.  13  and FIG. 14 show simulation results for the normal-incidence reflection phase of the AMC illustrated in FIG.  12 . In both simulations, the incident electric field is y-polarized. In the simulation illustrated in FIG. 13, P=10.4 mm, h=6 mm, t=0.2 mm, s=7.2 mm, w=1.6 mm, g 2 =0.4 mm, ε r1 =ε r2 =3.38. FIG. 13 shows a fundamental resonance near 1.685 GHz, and a second resonance near 2.8 GHz. In FIG. 14, when the gap in the loops is eliminated so that the loops are shorted and g 2 =0 in FIG. 12, then only one resonance is obtained. The reason that the AMC  800  with gaps  1104  has a second resonance is that the effective transverse permittivity of the frequency selective surface has become frequency dependent. A simple capacitive model is no longer adequate. 
     FIG. 15 shows equivalent circuits for portions of the artificial magnetic conductor  800  of FIG.  8 . FIG.  15 ( a ) illustrates the second Foster canonical form for the input admittance of a one-port circuit, which is a general analytic model for the effective transverse permittivity of complex frequency selective surface (FSS) structures. FIG.  15 ( b ) gives an example of a specific equivalent circuit model for an FSS whereby two material or intrinsic resonances are assumed. FIG.  15 ( c ) shows the TEM mode equivalent circuit for plane waves normally incident on a two layer AMC, such as AMC  900  of FIG.  9 . As noted above, the models developed herein are useful for characterizing, understanding, designing and engineering devices such as the AMCs described and illustrated herein. These models represent approximations of actual device behavior. 
     Complex loop FSS structures, such as that shown in FIG. 12, have a dispersive, or frequency dependent, effective transverse permittivity which can be properly modeled using a more complex circuit model. Furthermore, analytic circuit models for dispersive dielectric media can be extended in applicability to model the transverse permittivity of complex FSS structures. The second Foster canonical circuit for one-port networks, shown in FIG.  15 ( a ), is a general case which should cover all electrically-thin FSS structures. Each branch manifests an intrinsic resonance of the FSS. For an FSS made from low loss materials, R n  is expected to be very low, hence resonances are expected to be Lorentzian. 
     The effective sheet capacitance for the loop FSS shown in FIG. 12 has a Lorentz resonance somewhere between 1.685 GHz and 2.8 GHz. In fact, if the transverse permittivity of this FSS is modeled using only a three-branch admittance circuit, as shown in FIG.  15 ( b ), the ε 1y  curve  1602  shown in the upper graph of FIG. 16 is obtained. Two FSS material resonances are evident near 2.25 GHz and 3.2 GHz. The ε 1y  curve  1604  is the transverse relative permittivity required to achieve resonance for the AMC, a zero degree reflection phase. This curve  1604  is simply found by equating the capacitive reactance of the FSS, X c =1/(ωC)=1/(ωε 1y ε 0 t), to the inductive reactance of the spacer layer, X L =ωL=ωμ 2x μ 0 h, and solving for transverse relative permittivity: ε 1y =1/(ω 2 μ 2x μ 0 ε 0 ht). Intersections of the curve  1602  and the curve  1604  define the frequencies for reflection phase resonance. The reflection phase curve shown in the lower graph of FIG. 16 was computed using the transmission line model shown in FIG.  15 ( c ) in which the admittance of the FSS is placed in parallel with the shorted transmission line of length h representing the spacer layer and backplane. This circuit model predicts a dual resonance near 1.2 GHz and 2.75 GHz, which are substantially the frequencies of intersection in the ε 1y  plot. Thus the multiple resonant branches in the analytic circuit model for the FSS transverse permittivity can be used to explain the existence of multiple AMC phase resonances. Any realizable FSS structure can be modeled accurately using a sufficient number of shunt branches. 
     There are many additional square loop designs which may be implemented in FSS structures to yield a large transverse effective permittivity. More examples are shown in FIGS. 17,  20  and  21  where loops of substantially identical size and similar shape are printed on opposite sides of a single dielectric layer FSS. Reflection phase results for x and y polarized electric fields applied to an AMC of the design shown in FIG. 17 are shown in FIGS. 18 and 19. In this design, P=400 mils, g 1 =30 mils, g 2 =20 mils, r=40 mils, w=30 mils, t=8 mils, and h=60 mils. ε r =3.38 in both FSS and spacer layers since this printed AMC is fabricated using Rogers R04003 substrate material. In the center of each loop, a via is fabricated using a 20 mil diameter plated through hole. 
     FIG. 18 shows measured reflection phase data for an x polarized electric field normally incident on the AMC of FIG.  17 . Resonant frequencies are observed near 1.6 GHz and 3.45 GHz. Similarly, FIG. 19 shows measured reflection phase data for a y polarized electric field normally incident on the AMC of FIG.  17 . Resonant frequencies are observed near 1.4 GHz and 2.65 GHz. 
     In FIGS. 18 and 19, a dual resonant performance is clearly seen in the phase data. For the specific case fabricated, each polarization sees different resonant frequencies. However, it is believed that the design has sufficient degrees of freedom to make the resonance frequencies polarization independent. 
     FIG. 21 shows an additional alternative embodiment for a frequency selective surface implemented with square loops. The illustrated loop design of FIG. 21 has overlapping square loops  2100  on each layer  902 ,  904  with deep notches  2102  cut from the center  2104  toward each corner. Gaps  2106 ,  2108  are found at the 4:30 position on the upper layer and at the 7:30 position on the lower layer respectively. This design was also fabricated, using h=60 mils and t=8 mils of Rogers R04003 (ε r =3.38) as the spacer layer and FSS layer thickness respectively. AMC reflection phase for the x and y directed E field polarization is shown in FIGS. 22 and 23 respectively. Again, dual resonant frequencies are clearly seen. 
     An alternative type of dispersive capacitive FSS structure can be created where loops  2402  are printed on the one side and notched patches  2404  are printed on the other side of a single dielectric layer FSS. An example is shown in FIG.  24 . 
     In addition to the square loops illustrated in FIGS. 17,  20 ,  21  and  24 , hexagonal loops can be printed in a variety of shapes that include notches which increase the loop self inductance. These notches may vary in number and position, and they are not necessarily the same size in a given loop. Furthermore, loops printed on opposite sides of a dielectric layer can have different sizes and features. There are a tremendous number of independent variables which uniquely define a multilayer loop FSS structure. 
     Six possibilities of hexagonal loop FSS designs are illustrated in FIGS. 25,  26  and  27 . In each of FIGS. 25,  26  and  27 , a first layer  902  of loops is capacitively coupled with a second layer of loops  904 . The hexagonal loops presented here are intended to be regular hexagons. Distorted hexagons could be imagined in this application, but their advantage is unknown at this time. 
     FIG. 28 illustrates an effective media model for a high impedance surface  2800 . The general effective media model of FIG. 28 is applicable to high impedance surfaces such as the prior art high impedance surface  100  of FIG.  1  and the artificial magnetic conductor (AMC)  800  of FIG.  8 . The AMC  800  includes two distinct electrically-thin layers, a frequency selective surface (FSS)  802  and a spacer layer  804 . Each layer  802 ,  804  is a periodic structure with a unit cell repeated periodically in both the x and y directions. The periods of each layer  802 ,  804  are not necessarily equal or even related by an integer ratio, although they may be in some embodiments. The period of each layer is much smaller than a free space wavelength λ at the frequency of analysis (λ/10 or smaller). Under these circumstances, effective media models may be substituted for the detailed fine structure within each unit cell. As noted, the effective media model does not necessarily characterize precisely the performance or attributes of a surface such as the AMC  800  of FIG. 8 but merely models the performance for engineering and analysis. Changes may be made to aspects of the effective media model without altering the overall effectiveness of the model or the benefits obtained therefrom. 
     As will be described, the high impedance surface  2800  for the AMC  800  of FIG. 8 is characterized by an effective media model which includes an upper layer and a lower layer, each layer having a unique tensor permittivity and tensor permeability. Each layer&#39;s tensor permittivity and each layer&#39;s tensor permeability have non-zero elements on the main tensor diagonal only, with the x and y tensor directions being in-plane with each respective layer and the z tensor direction being normal to each layer. The result for the AMC  800  is an AMC resonant at multiple resonance frequencies. 
     In the two-layer effective media model of FIG. 28, each layer  2802 ,  2804  is a bi-anisotropic media, meaning both permeability μ and permittivity ε are tensors. Further, each layer  2802 ,  2804  is uniaxial meaning two of the three main diagonal components are equal, and off-diagonal elements are zero, in both μ and ε. So each layer  2802 ,  2804  may be considered a bi-uniaxial media. The subscripts t and n denote the transverse (x and y directions) and normal (z direction) components. 
     Each of the two layers  2802 ,  2804  in the bi-uniaxial effective media model for the high impedance surface  2800  has four material parameters: the transverse and normal permittivity, and the transverse and normal permeability. Given two layers  2802 ,  2804 , there are a total of eight material parameters required to uniquely define this model. However, any given type of electromagnetic wave will see only a limited subset of these eight parameters. For instance, uniform plane waves at normal incidence, which are a transverse electromagnetic (TEM) mode, are affected by only the transverse components of permittivity and permeability. This means that the normal incidence reflection phase plots, which reveal AMC resonance and high-impedance bandwidth, are a function of only ε 1t , ε 2t , μ 1t , and μ 2t  (and heights h and t). This is summarized in Table 1 below. 
     
       
         
           
               
               
               
             
               
                 TABLE 1 
               
               
                   
               
               
                 Wave Type 
                 Electric Field Sees 
                 Magnetic Field Sees 
               
               
                   
               
             
            
               
                 TEM, normal incidence 
                 ε 1t , ε 2t   
                 μ 1t , μ 2t   
               
               
                 TE to x 
                 ε t1 , ε 2t   
                 μ 1t , μ 2t , μ 1n , μ 2n   
               
               
                 TM to x 
                 ε 1t , ε 2t , ε 1n , ε 2n   
                 μ 1t , μ 2t   
               
               
                   
               
            
           
         
       
     
     A transverse electric (TE) surface wave propagating on the high impedance surface  2800  has a field structure shown in FIG.  4 . By definition, the electric field (E field) is transverse to the direction of wave propagation, the +x direction. It is also parallel to the surface. So the electric field sees only transverse permittivities. However, the magnetic field (H field) lines form loops in the xz plane which encircle the E field lines. So the H field sees both transverse and normal permeabilities. 
     The transverse magnetic (TM) surface wave has a field structure shown in FIG.  5 . Note that, for TM waves, the role of the E and H fields is reversed relative to the TE surface waves. For TM modes, the H field is transverse to the direction of propagation, and the E field lines (in the xz plane) encircle the H field. So the TM mode electric field sees both transverse and normal permittivities. 
     The following conclusions may be drawn from the general effective media model of FIG.  28 . First, ε 1n  and ε 2n  are fundamental parameters which permit independent control of the TM modes, and hence the dominant TM mode cutoff frequency. Second, μ 1n  and μ 2n  are fundamental parameters which permit independent control of the TE modes, and hence the dominant TE mode cutoff frequency. 
     One way to distinguish between prior art high impedance surface  100  of FIG.  1  and an AMC such as AMC  800  (FIG. 8) or AMC  900  (FIG. 9, FIG. 10) is by examining the differences in the elements of the            μ   i     _                   and                     ɛ   i     _                     
     tensors. FIG. 29 shows a prior art high impedance surface  100  whose frequency selective surface  102  is a coplanar layer of square conductive patches of size b×b, separated by a gap of dimension g. In the high impedance surface  100 , ε D  is the relative permittivity of the background or host dielectric media in the spacer layer  104 , μ D  is the relative permeability of this background media in the spacer layer  104 , and α is the ratio of cross sectional area of each rod or post to the area A of the unit cell in the rodded media or spacer layer  104 . The relative permittivity          ɛ   avg     =       1   +     ɛ   D       2                     
     is the average of the relative dielectric constants of air and the background media in the spacer layer  104 . C denotes the fixed FSS sheet capacitance. 
     The permittivity tensor for both the high-impedance surface  100  and the AMCs  800 ,  900  is uniaxial, or ε ix =ε iy =ε it ≠ε iz =ε in ; i=1, 2 with the same being true for the permeability tensor. The high impedance surface  100  has a square lattice of both rods and square patches, each having the same period. Therefore, unit cell area A=(g+b) 2 . Also, α=(πd 2 /4)/A, where d is the diameter of the rods or posts. The dimensions of the rods or posts are very small relative to the wavelength at the resonance frequencies. The rods or posts may be realized by any suitable physical embodiment, such as plated-through holes or vias in a conventional printed circuit board or by wires inserted through a foam. Any technique for creating a forest of vertical conductors (i.e., parallel to the z axis), each conductor being electrically coupled with the ground plane, may be used. The conductors or rods may be circular in cross section or may be flat strips of any cross section whose dimensions are small with respect to the wavelength λ in the host medium or dielectric of the spacer layer. In this context, small dimensions for the rods are generally in the range of λ/1000 to λ/25. 
     In some embodiments, the AMC  800  has transverse permittivity in the y tensor direction substantially equal to the transverse permittivity in the x tensor direction. This yields an isotropic high impedance surface in which the impedance along the y axis is substantially equal to the impedance along the x axis. In alternative embodiments, the transverse permittivity in the y tensor direction does not equal the transverse permittivity in the x tensor direction to produce an anisotropic high impedance surface, meaning the impedances along the two in-plane axes are not equal. Examples of the latter are shown in FIGS. 17 and 21. 
     Effective media models for substantially modelling both the high impedance surface  100  and an AMC  800 ,  900  are listed in Table 2. Two of the tensor elements are distinctly different in the AMC  800 ,  900  relative to the prior art high-impedance surface  100 . These are the transverse permittivity ε 1x , ε 1y  and the normal permeability  1z , both of the upper layer or frequency selective surface. The model for the lower layer or spacer layer is the same in both the high impedance surface  100  and the AMC  800 ,  900 . 
     
       
         
           
               
               
               
             
               
                   
                 TABLE 2 
               
               
                   
                   
               
               
                   
                 High impedance surface 100 
                 AMC 800, 900 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                 FSS  Layer  (upper  layer) 
                 
                   
                     
                       
                         
                           ɛ 
                           
                             1 
                              
                             x 
                           
                         
                         = 
                         
                           
                             ɛ 
                             
                               1 
                                
                               y 
                             
                           
                           = 
                           
                             C 
                             
                               
                                 ɛ 
                                 o 
                               
                                
                               t 
                             
                           
                         
                       
                     
                     
                     
                         
                     
                   
                 
                 
                   
                     
                       
                         
                           ɛ 
                           
                             1 
                              
                             x 
                           
                         
                         = 
                         
                           
                             ɛ 
                             
                               1 
                                
                               y 
                             
                           
                           = 
                           
                             
                               Y 
                                
                               
                                 ( 
                                 ω 
                                 ) 
                               
                             
                             
                               j 
                                
                               
                                   
                               
                                
                               
                                 ωɛ 
                                 o 
                               
                                
                               t 
                             
                           
                         
                       
                     
                     
                     
                         
                     
                   
                 
               
               
                   
                 
                   
                     
                       
                         
                           ɛ 
                           
                             1 
                              
                             z 
                           
                         
                         = 
                         1 
                       
                     
                     
                     
                         
                     
                   
                 
                 
                   
                     
                       
                         
                           ɛ 
                           
                             1 
                              
                             z 
                           
                         
                         = 
                         1 
                       
                     
                     
                     
                         
                     
                   
                 
               
               
                   
                 μ 1x  = μ 1y  = 1 
                 μ 1x  = μ 1y  = 1 
               
               
                   
                 
                   
                     
                       
                         
                           μ 
                           
                             1 
                              
                             z 
                           
                         
                         = 
                         
                           
                             2 
                              
                             
                               ɛ 
                               avg 
                             
                           
                           
                             ɛ 
                             
                               1 
                                
                               x 
                             
                           
                         
                       
                     
                     
                     
                         
                     
                   
                 
                 
                   
                     
                       
                         
                           μ 
                           
                             1 
                              
                             z 
                           
                         
                         = 
                         
                           
                             Z 
                              
                             
                               ( 
                               ω 
                               ) 
                             
                           
                           
                             j 
                              
                             
                                 
                             
                              
                             
                               ωμ 
                               o 
                             
                              
                             t 
                           
                         
                       
                     
                     
                     
                         
                     
                   
                 
               
               
                 Spacer  layer  (lower  layer) 
                 
                   
                     
                       
                         
                           ɛ 
                           
                             2 
                              
                             x 
                           
                         
                         = 
                         
                           
                             ɛ 
                             
                               2 
                                
                               y 
                             
                           
                           = 
                           
                             
                               ɛ 
                               D 
                             
                              
                             
                               ( 
                               
                                 
                                   1 
                                   + 
                                   α 
                                 
                                 
                                   1 
                                   - 
                                   α 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                     
                         
                     
                   
                 
                 
                   
                     
                       
                         
                           ɛ 
                           
                             2 
                              
                             x 
                           
                         
                         = 
                         
                           
                             ɛ 
                             
                               2 
                                
                               y 
                             
                           
                           = 
                           
                             
                               ɛ 
                               D 
                             
                              
                             
                               ( 
                               
                                 
                                   1 
                                   + 
                                   α 
                                 
                                 
                                   1 
                                   - 
                                   α 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                     
                         
                     
                   
                 
               
               
                   
                 
                   
                     
                       
                         
                           ɛ 
                           
                             2 
                              
                             z 
                           
                         
                         = 
                         
                           
                             ɛ 
                             D 
                           
                           - 
                           
                             1 
                             
                               
                                 ω 
                                 2 
                               
                                
                               
                                 ɛ 
                                 o 
                               
                                
                               
                                 μ 
                                 o 
                               
                                
                               
                                 μ 
                                 D 
                               
                                
                               
                                 
                                   A 
                                   
                                     4 
                                      
                                     π 
                                   
                                 
                                  
                                 
                                   [ 
                                   
                                     
                                       ln 
                                        
                                       
                                         ( 
                                         
                                           1 
                                           α 
                                         
                                         ) 
                                       
                                     
                                     + 
                                     α 
                                     - 
                                     1 
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                     
                     
                     
                         
                     
                   
                 
                 Same as High  impedance  surface 100 
               
               
                   
                 
                   
                     
                       
                         
                           μ 
                           
                             2 
                              
                             x 
                           
                         
                         = 
                         
                           
                             μ 
                             
                               2 
                                
                               y 
                             
                           
                           = 
                           
                             
                               
                                 ɛ 
                                 D 
                               
                               
                                 ɛ 
                                 
                                   2 
                                    
                                   x 
                                 
                               
                             
                              
                             
                               μ 
                               D 
                             
                           
                         
                       
                     
                     
                     
                         
                     
                   
                 
                 
                   
                     
                       
                         
                           μ 
                           
                             2 
                              
                             x 
                           
                         
                         = 
                         
                           
                             μ 
                             
                               2 
                                
                               y 
                             
                           
                           = 
                           
                             
                               
                                 ɛ 
                                 D 
                               
                               
                                 ɛ 
                                 
                                   2 
                                    
                                   x 
                                 
                               
                             
                              
                             
                               μ 
                               D 
                             
                           
                         
                       
                     
                     
                     
                         
                     
                   
                 
               
               
                   
                 μ 2z  = (1 − α)μ D   
                 μ 2z  = (1 − α)μ D   
               
               
                   
               
            
           
         
       
     
     In Table 2, Y(ω) is an admittance function written in the second Foster canonical form for a one port circuit:          Y        (   ω   )       =       j                 ω                   C   ∞       +     1     j                 ω                   L   o         +       ∑     n   =   1     N                     1       R   n     +     j                 ω                   L   n       +     1     j                 ω                   C   n                                 
     This admittance function Y(ω) is related to the sheet capacitance (C=ε 1t ε o t) of the FSS  802  of the AMC 800, 900 by the relation Y=jωC. The high impedance surface  100  has an FSS capacitance which is frequency independent. However, the AMC  800 ,  900  has an FSS  802  whose capacitance contains inductive elements in such a way that the sheet capacitance undergoes one or more Lorentz resonances at prescribed frequencies. Such resonances are accomplished by integrating into the FSS  802  the physical features of resonant loop structures, also referred to as artificial magnetic molecules. As the frequency of operation is increased, the capacitance of the FSS  802  will undergo a series of abrupt changes in total capacitance. 
     FIG. 30 illustrates sheet capacitance for the frequency selective surface  802  of the AMC  800  of FIG.  8  and the AMC  900  of FIG.  9 . FIG.  30 ( a ) shows that the capacitance of the FSS  802  is frequency dependent. FIG.  30 ( b ) shows a Debye response obtained from a lossy FSS where R n  is significant. In FIG. 30, two FSS resonances (ω=1/{square root over (L n C n )}, N=2) are defined. The drop in capacitance across each resonant frequency is equal to C n , the capacitance in each shunt branch of Y(ω). Although the regions of rapidly changing capacitance around a Lorentz resonance may be used to advantage in narrowband antenna requirements, some embodiments may make use of the more slowly varying regions, or plateaus, between resonances. This FSS capacitance is used to tune the inductance of the spacer layer  804 , which is a constant, to achieve a resonance in the reflection coefficient phase for the AMC  800 ,  900 . This multi-valued FSS capacitance as a function of frequency is the mechanism by which multiple bands of high surface impedance are achieved for the AMC  800 ,  900 . 
     In contrast, the two-layer high impedance surface  100  will offer reflection phase resonances at a fundamental frequency, plus higher frequencies near where the electrical thickness of the bottom layer is nπ and n is an integer. These higher frequency resonances are approximately harmonically related, and hence uncontrollable. 
     A second difference in the tensor effective media properties for the high impedance surface  100  and AMC  800  is in the normal permeability component μ 1n . The high impedance surface  100  has a constant μ 1n , whereas the AMC  800 ,  900  is designed to have a frequency dependent μ 1n . The impedance function Z(ω) can be written in the first Foster canonical form for a one-port circuit.          Z        (   ω   )       =       j                 ω                   L   ∞       +     1     j                 ω                   C   o         +       ∑     n   =   1     N                     1       G   n     +     j                 ω                   C   n       +     1     j                 ω                   L   n                                 
     This impedance function is sufficient to accurately describe the normal permeability of the FSS  802  in an AMC  800 ,  900  regardless of the number and orientation of uniquely resonant artificial magnetic molecules. 
     The prior art high-impedance surface  100 , whose FSS  102  is composed of metal patches, has a lower bound for μ 1n . This lower bound is inversely related to the transverse permittivity according to the approximate relation μ 1n ≈2/ε 1t . Regardless of the FSS sheet capacitance, μ 1n  is anchored at this value for the prior art high-impedance surface  100 . However, a normal permeability which is lower than μ 1n =2/ε 1t  is needed to cut off the guided bound TE mode in all of the high-impedance bands of a multi-band AMC such as AMC  800  and AMC  900 . 
     The overlapping loops used in the FSS  802  of the AMC  800 ,  900  allow independent control of the normal permeability. Normal permeabilities may be chosen so that surface wave suppression occurs over some and possibly all of the +/−90° reflection phase bandwidths in a multi-band AMC such as AMC  800  and AMC  900 . The illustrated embodiment uses arrays of overlapping loops as the FSS layer  802 , or in conjunction with a capacitive FSS layer, tuned individually or in multiplicity with a capacitance. This capacitance may be the self capacitance of the loops, the capacitance offered by adjacent layers, or the capacitance of external capacitors attached to the FSS. such as chip capacitors. The loops and capacitance are tuned so as to obtain a series of Lorentz resonances across the desired bands of operation. Just as in the case of the resonant FSS transverse permittivity, the resonances of the artificial magnetic molecules affords the designer a series of staircase steps of progressively dropping normal permeability. Again, the region of rapidly changing normal permeability around the resonances may be used to advantage in narrowband operations. However, the illustrated embodiment uses plateaus of extended depressed normal permeability to suppress the onset of guided bound TE surface waves within the desired bands of high-impedance operation. 
     In summary, the purpose of the resonance in the effective transverse permittivities ε 1t  is to provide multiple bands of high surface impedance. The purpose of the resonances in the normal permeabilityμ 1n  is to depress its value so as to prevent the onset of TE modes inside the desired bands of high impedance operation. 
     In some applications, an artificial magnetic conductor having more than two layers of loops separated by more than a single dielectric layer may provide important performance advantages. FIG. 31 illustrates an artificial magnetic conductor  3100  including a multiple layer frequency selective surface (FSS)  3102 . The AMC  3100  further includes a conductive ground plane  3104  and a rodded media forming a spacer layer  3106  disposed between the FSS  3102  and the conductive ground plane  3104 . The FSS  3102  has a frequency dependent permeability μ 1z  in a direction normal to the frequency dependent surface  3102 . Exemplary dimensions and coordinate axes are shown in FIG.  31 . 
     The FSS  3102  includes three arrays of substantially coplanar artificial magnetic molecules. The artificial magnetic molecules are preferably implemented as overlapping capacitively coupled loops. In the embodiment of FIG. 31, the FSS  3102  includes a first array  3112 , a second array  3114  and a third array  3116  of artificial magnetic molecules. A first dielectric layer  3118  separates the first array  3112  of artificial magnetic molecules from the second array  3114  of artificial magnetic molecules. 
     The arrays  3112 ,  3114 ,  3116  are shown as being coplanar in respective planes. This arrangement is particularly well suited to manufacture using conventional printed circuit board (PCB) manufacturing techniques of depositing a metallic layer on a PCB surface and etching with a chemical or other process. In other embodiments, other manufacturing techniques, some of which will produce arrays of artificial magnetic molecules which are not substantially coplanar, may be substituted. 
     Also, the AMC  3100  includes three layers  3112 ,  3114 ,  3116  of loops separated by two dielectric layers  3118 ,  3120 . In other embodiments, other combinations of layers of loops and dielectric layers may be used. In general, a FSS in accordance with the disclosed embodiments will include n layers of loops and n−1 dielectric layers isolating the layers of loops. 
     The spacer layer  3106  includes metallic rods  3108  periodically positioned in a dielectric material. Preferably, each loop of each array of loops  3112 ,  3114 ,  3116  is associated with a rod  3108  of the spacer layer  3106 . Any suitable manufacturing method, for example, as described above, may be used to manufacture the rodded media of the spacer layer  3108 . 
     FIG. 32 illustrates a top view of the multiple-layer frequency selective surface  3102  of FIG.  31 . FIG. 32 shows the first array  3112 , the second array  3114  and the third array  3116  of the frequency selective surface  3102 . A portion only of each array is visible to illustrate the layering of the respective arrays. 
     In FIG. 32, each of the arrays  3112 ,  3114 ,  3116  includes substantially identical hexagonal loops periodically spaced on the FSS  3102 . Each loop is notched to tailor the self-inductance of the loop and includes a gap to tailor the resonant frequency of the loop. The embodiment of FIGS. 31 and 32 is illustrative only. In other embodiments, different size and shape loops may be used along with different numbers of layers or arrays. 
     From the foregoing, it can be seen that the present embodiments provide a variety of high-impedance surfaces or artificial magnetic conductors which exhibit multiple reflection phase resonances, or multi-band performance. The resonant frequencies for high surface impedance are not harmonically related, but occur at frequencies which may be designed or engineered. This is accomplished by designing the tensor permittivity of the upper layer to have a behavior with frequency which exhibits one or more Lorentzian resonances. 
     While a particular embodiment of the present invention has been shown and described, modifications may be made. Other methods of making or using anisotropic materials with negative axial permittivity and depressed axial permeability, for the purpose of constructing multiband surface wave suppressing AMCs, such as by using artificial dielectric and magnetic materials, are extensions of the embodiments described herein. Any such method can be used to advantage by a person ordinarily skilled in the art by following the description herein for the interrelationship between the Lorentz material resonances and the positions of the desired operating bands. Accordingly, it is therefore intended in the appended claims to cover such changes and modifications which follow in the true spirit and scope of the invention.