Abstract:
A bandpass radome is described including inductive layers comprising periodic conductive grids. First and second capacitive patch layers may be disposed above, and third and fourth capacitive patch layers may be disposed below the inductive layer to realize a 2-pole bandpass radome. An additional inductive layer and a fifth and sixth capacitive patch layers may be added below the fourth capacitive layer to realize a 3-pole bandpass radome. Conductive posts may connect one of the uppermost patch layers to one of the lowermost patch layers without connecting to the intervening inductive conductive grids. The conductive posts may form a rodded medium to suppress transverse magnetic (TM) surface waves. The total thickness of the bandpass radome may be less than 1/30 of a free-space wavelength at the center of a passband frequency. More than one passband may be separated by a ratio of center frequencies exceeding 1.5.

Description:
This application claims the benefit of U.S. provisional application Ser. No. 60/830,515, filed on Jul. 13, 2006, and U.S. provisional application Ser. No. 60/860,510, filed on Nov. 20, 2006, each of which is incorporated herein by reference 
    
    
     TECHNICAL FIELD 
     This application relates to periodic metallo-dielectric structures. In particular, the metallo-dielectric structures may be used as frequency selective surfaces to filter electromagnetic waves. 
     BACKGROUND 
     Bandpass radomes constructed with Frequency Selective Surfaces (FSS) typically use resonant FSS elements that are approximately one half of a wavelength long in their largest dimension at the passband center frequency. Such half-wavelength elements typically exhibit multiple resonances such that, at normal incidence, a radome having a passband centered at f o  exhibits spurious resonances at 3f o , 5f o , 5f o , etc. At oblique incidence, spurious resonances may also occur near 2f o , 4f o , 6f o , etc. In addition, such resonant element radomes will typically support the propagation of undesired surface wave electromagnetic wave modes excited at edges of the structure or at other discontinuities. The surface waves can radiate energy to produce radiation pattern anomalies for an antenna system where the radome is used to cover the antenna. 
     Bandpass radomes may be used in antenna system applications where one desires to allow transmission of electromagnetic waves in one or more ranges of radio frequencies and to suppress the transmission of waves at other frequencies. Such bandpass radomes may have dielectric layers that are each approximately λ/4 (one-quarter of a free-space wavelength) in thickness. At high microwave frequencies, λ/4 is a relatively small dimension, but at UHF frequencies (300 MHz to 1 GHz) or even low microwave frequencies (1-3 GHz), λ/4 can be too large for some applications. Hence there exists a need for electrically-thin and physically thin bandpass radomes. Furthermore, thin bandpass radomes may have less mass than conventional bandpass radomes due to thinner dielectric layers. 
     SUMMARY 
     A frequency selective surface, which may be a frequency selective radome (FSR) is described, including a first and a second patch layer disposed in relatively close proximity to each other. The term “relatively close” will be understood by persons skilled in the art as being substantially less than a wavelength at a frequency within a transmission frequency window. Third and fourth FSS patch layers may be disposed in relatively close proximity to each other. A dielectric region may be disposed between the second and third patch layers the dielectric region containing a pair of parallel inductive grids. 
     The FSR may be a mechanically-balanced structure where the layers are symmetrical about a plane. The first and second patch layers as well as one of the inductive grid layers may be disposed above the plane of symmetry and the third and fourth FSS patch layer and a second inductive grid layer may be disposed below the plane of symmetry. 
     In an aspect, the FSR may include a first array of conducting posts that connect the first patch layer to the fourth patch layer, and may further include a second array of conducting posts that connect the second patch layer to the third patch layer. The conducting posts form a rodded medium and the spatial period and dimensions of the conductive posts may suppress TM (transverse magnetic) surface wave modes over a desired band of frequencies. 
     In another aspect, the patch layers use an array of rectangular patches. The term rectangular will be understood by a person of skill in the art to represent any structure having generally a regular shape and where the principal dimensions are roughly comparable, such as a square, circle, triangle, pentagon, or the like. For example, the rectangular patches may have rebated or mitered corners to provide clearance between the patches and conductive posts. In yet another aspect the patches may be formed with interdigitating portions. The conductive posts may be plated thru holes in a dielectric layer. 
     A dielectric layer of thickness t may separate the first and second patch layers. A first dielectric layer of thickness d 1  may separate the second patch layer from the upper inductive grid. A second dielectric layer of thickness d 2  may separate the upper and lower inductive grids, and third dielectric layer of thickness d 3  may separate the lower inductive grid from the third patch layer. A fourth dielectric layer of thickness t separates the third and fourth patch layers. The first and fourth dielectric layers may be comprised of a flexible laminate such as liquid crystal polymer (LCP), PET (Dupont Mylar™), or PTFE. Dielectric layers may b formed of any electrically suitable material, including air. 
     In yet another aspect, the conductive posts are disposed to pass through apertures in the inductive grids, and thus may not electrically connect to the inductive grids. Similarly conductive posts may pass through junctions of the inductive grids, the junctions having apertures formed therein so that the posts may not electrically connect to the grid. 
     In a further aspect, the inductive grids may have a period that is half of the period P of the patches, or a period that is equal to or greater than the period P of the patches. Inductive grids with periods of P or greater may have enhanced inductance and allow passbands that have lower center frequencies as compared to similar radomes with a grid period of P/2. 
     In still another aspect, the equivalent shunt capacitance of the capacitive patch layers, the equivalent shunt inductance of the grid layers, the separation distance between inductive grids, and the separation distance between capacitive FSS layers and inductive grid layers, may be selected to provide a plurality of distinct frequency passbands. 
     A lower passband center frequency may be adjusted independently of an upper passband through the design of the inductive grids. Alternatively, the upper passband center frequency may be adjusted independently of the lower passband center frequency by controlling the separation between inductive grids. 
     A 3-pole bandpass characteristic may be obtained by using 6, 8, or 10 metal layers. A 6-layer structure may include two exterior capacitive layers two interior capacitive layers, and two inductive layers. The exterior capacitive layers may include an inter-digital finger arrangement to increase the effective shunt capacitance. 
     In another aspect, the radome may have 8 metal layers that may result in a 3-pole bandpass filter characteristic. Six of the metal layers may be capacitive patches arranged in overlapping patterns to form three shunt capacitors. The remaining two metal layers contain may inductive grids to form two shunt inductors. 
     A 3-pole FSR radome may include 10 metal layers that result in a 3-pole bandpass filter characteristic. Six of the metal layers may be capacitive patches arranged in overlapping patterns to form three shunt capacitors. The remaining four metal layers may contain inductive grids to form four shunt inductors. 
     The eight or ten layer 3-pole FSR may include a first and second patch layer disposed in relatively proximity to each other, a third and fourth patch layer disposed in proximity to each other, and a fifth and six patch layer also disposed in proximity to each other. A first dielectric region may be disposed between the second and third patch layers, where this dielectric region contains a parallel inductive grid. A second dielectric region may be disposed between the fourth and fifth patch layers, where second dielectric region also contains a parallel inductive grid. 
     The eight or ten layer 3-pole FSR may include an array of conducting posts that may connect the first patch layer to the sixth patch layer, and may further include a second array of conductive posts that connect the second patch layer to the fifth patch layer. The conductive posts form a rodded medium and the spatial period and dimensions of the conductive posts may suppress TM (transverse magnetic) surface wave modes over a desired band of frequencies. This desired band of frequencies may include the passband. 
     The periodic distance P′ between conductive posts may exceed the period P between patches of the capacitive layers so as to broaden the TM mode surface wave stopband. 
     The conductive posts of the 3-pole FSR may be disposed to pass through apertures in the inductive grids and thus may electrically connect to the inductive grids. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  shows an edge view of a bandpass radome as an eight layer stackup; 
         FIG. 2  shows an edge view of the bandpass radome as a symmetric six layer stackup; 
         FIG. 3  shows (a) a plan view of the bandpass radome of  FIG. 2  where only the conductive posts and patch layers are shown; and, (b) a plan view of the bandpass radome of  FIG. 2  where only the conductive posts and inductive layers are shown; 
         FIGS. 4  ( a ) and (b) illustrate the transmission (S 21 ) and reflection (S 11 ) plots for the bandpass radome of  FIGS. 2 and 3 ; 
         FIG. 5  shows (a) a plan view of the bandpass radome of  FIG. 2  where only the conductive posts and patch layers are shown; and, (b) a plan view of another example of the bandpass radome of  FIG. 2  where only the conductive posts and inductive layers are shown; 
         FIGS. 6  ( a ) and ( b ) illustrate the transmission (S 21 ) and reflection (S 11 ) plots for the bandpass radome of  FIG. 5 ; 
         FIG. 7  shows (a) another example of the inductive layers of the bandpass radome of  FIG. 2  where the layers are comprised of aligned grids; and (b) another example of the inductive layers of the bandpass radome of  FIG. 2  where the layers are comprised of staggered grids; 
         FIG. 8  shows yet another example of the inductive layers of the bandpass radome of  FIG. 2 ; 
         FIG. 9  is an equivalent circuit model of the bandpass radome of  FIG. 1  for angles near normal incidence; 
         FIGS. 10  ( a ) and ( b ) are an equivalent circuit model of the bandpass radome of  FIG. 1  or  2  for a symmetrically fabricated radome, for angles near normal incidence. 
         FIG. 11  shows an edge view of a 3-pole bandpass radome as a ten layer stackup; 
         FIG. 12  is an approximate equivalent circuit model of the bandpass radome of  FIG. 11  for angles near normal incidence; 
         FIG. 13  shows an edge view of a 3-pole bandpass radome as an eight layer stackup; 
         FIG. 14  is an approximate equivalent circuit model of the bandpass radome of  FIG. 13  for angles near normal incidence; 
         FIG. 15  is a plan view showing of all the metal layers in an example of a 3-pole bandpass radome corresponding to  FIG. 13 ; 
         FIG. 16  illustrates transmission (S 21 ) and reflection (S 11 ) plots at normal incidence for the bandpass radome of  FIGS. 13 and 15 ; 
         FIG. 17  shows an edge view of a 3-pole bandpass radome as a six layer stackup; 
         FIG. 18  shows a plan view of a capacitive layer as an array of inter-digital capacitors; and 
         FIG. 19  shows an edge view of a 2-pole bandpass radome as a four layer stackup. 
     
    
    
     DETAILED DESCRIPTION 
     Reference will now be made in detail to several examples; however, it will be understood that claimed invention is not limited to such examples. In the following description, numerous specific details are set forth in the examples in order to provide a thorough understanding of the subject matter of the claims which, however, may be practiced without some or all of these specific details. In other instances, well known process operations or structures have not been described in detail in order not to unnecessarily obscure the description. 
     When describing a particular example, the example may include a particular feature, structure, or characteristic, but every example may not necessarily include the particular feature, structure or characteristic. This should not be taken as a suggestion or implication that the features, structure or characteristics of two or more examples should not or could not be combined, except when such a combination is explicitly excluded. When a particular feature, structure, or characteristic is described in connection with an example, a person skilled in the art may give effect to such feature, structure or characteristic in connection with other examples, whether or not explicitly described. 
     An example of an electrically-thin bandpass radome  100  is shown in  FIG. 1 . Bandpass radome or frequency selective radome (FSR)  100  is a multilayer structure which may be comprised of alternating conductive and dielectric layers. The layers having conductive components may be periodic in x and y directions and may be frequency selective surfaces (FSS) of either predominantly a capacitive type or predominantly an inductive type at frequencies within the electromagnetic bandpass. Conductive layers  102 ,  104 ,  106 ,  112 ,  114 , and  116  are two-dimensional arrays of isolated patches which may be capacitive. Layers  108  and  110  are inductive and comprised of a two-dimensional periodic conductive grid. In the following description, the terms “inductive layer” and “inductive grid” will be used to represent the same concept. The structure periods of the inductive layers may be less than, equal to, or greater than the periods of the capacitive layers. 
     The capacitive layers  102 ,  104 , and  106  are separated by dielectric layers  101  and  103  of thickness t. Capacitive layers  112 ,  114 , and  116  are separated by dielectric layers  111  and  113  of thickness t. Capacitive layer  106  and inductive layer  108  are separated by a dielectric layer  105  of thickness d 1 . Inductive layers  108  and  110  are separated from each other by a dielectric layer  107  of thickness d 2 . Inductive layer  110  and capacitive layer  112  are separated by a dielectric layer  109  of thickness d 3 . The thickness t may typically be substantially smaller than the thickness d 1  or d 3 . For example, the value of the thickness t may typically range from about 1/50 to about ⅕ of the thickness d 1 . In an example, the total radome thickness defined by 4t+d 1 +d 2 +d 3  plus the thickness of all eight metal layers, may be in the range of approximately λ/100 to approximately λ/30 at the radome passband center frequency. 
     Individual dielectric layers  101 ,  103 ,  105 ,  107 ,  109 ,  111 , and  113  may not be homogeneous dielectric regions. For example, each dielectric layer may be a core, a bonding layer such as a prepreg, or a combination of both types. 
     The bandpass radome  100  may also have arrays of conductive posts  128  and  130 . These posts may connect to selected patches of the capacitive layers, and may connect to a central portion of such patches. The array of conductive posts  128 , which may be periodic, may electrically connect to patches on layers  102 ,  106 ,  112 , and  116 . The periodic array of conductive posts  130  may electrically connect patches on layer  104  to patches on layer  114 . As shown in  FIG. 1 , the inductive grids  108  and  110  are disposed so as to avoid electrical contact with both arrays of conductive posts  128  and  130 . In this example, the inductive grids  108  and  110  are electrically isolated from the conductive posts. Alternatively, for example, the array of conductive posts  128  or  130  may be omitted. 
     There is no ground plane, such as is described in U.S. Pat. No. 6,476,771, “Electrically Thin Multi-Layer Bandpass Radome, issued to William E. McKinzie, III on Nov. 5, 2002, which is commonly assigned, and incorporated herein by reference. In the present examples, the inductive layer may not be directly connected to the conductive posts, and a slotted inductive grid may be used. Two inductive grids may be separated by a dielectric spacer layer  107 , and an upper frequency transmission pole may be adjusted independently of a lower frequency transmission pole by varying the thickness of the spacer layer. Moreover, the lower transmission pole may be adjusted independently of the upper transmission pole by adjusting the inductance of the grids: for example, by varying the size of the apertures in the inductive grids. 
     For simplicity of analysis and design, radomes may often be designed and optimized for a desired passband center frequency assuming a normal angle of incidence)(0°) of the electromagnetic wave on the surface of the radome. However, it may be desirable that the passband be stable in frequency even with changes in the angle of incidence away from the normal. The periodic conductive posts form an anisotropic rodded medium which may make the electrical length of the equivalent transmission lines associated with dielectric layers  105 ,  107 , and  109  fairly insensitive with respect to the angle of incidence. This may make the passband center frequency less sensitive to changes in angle of incidence. 
     In another aspect, the arrays of conductive posts  128  and  130  cut off parasitic TM surface-wave modes which may be excited at discontinuities such as edges of the radome surface. The arrays of conductive posts  128  and  130  may make the radome passband center frequency less sensitive to changes in angle of incidence. 
     The periodic array of conductive posts and patches within the radome forms an electromagnetic bandgap structure which may suppress TM mode surface waves along the radome structure. The TM mode has a normal (z-directed) component of electric field. A rodded medium with rods aligned in the z direction may cut off the dominant TEM mode (which has a z-directed electric field) from DC (direct current) to some cutoff upper frequency related to the rod diameter and spacing. TM modes in a surface waveguide (e.g., a bandpass radome structure) comprised of layers of rodded media may exhibit a negative effective dielectric constant for those layers. Such layers may be modeled as anisotropic effective media. 
     The term “effective media” will be understood by a person of skill in the art as being used to describe an equivalent homogeneous dielectric or magnetic media that is used in a numerical analysis or simulation to replace an inhomogeneous complex media, such as a periodic structure whose unit cell contains one or more dielectric regions and one or more metal regions. Dispersion equations for surface waves attached to this radome structure may be derived based on effective medium models. A surface wave analysis procedure is found in, “Design Methodology for Sievenpiper High-Impedance Surfaces: An Artificial Magnetic Conductor for Positive Gain Electrically Small Antennas,” Clavijo, Diaz and McKinzie, IEEE Trans. Antennas and Propagation, Vol. 51, No. 10, October 2003. When the spacing between conductive posts, and the radius of the conductive posts, is sufficiently small, TM mode waves may be cut off for the passband frequency range or frequency ranges. The conductive posts may be connected to the patches of the capacitive layers for surface-wave suppression. 
     The bandpass radome  100  may be fabricated, for example, as a multilayer printed circuit board (PCB). The materials selected may determine whether the PCB acts as a flexible or a rigid structure.  FIG. 1  illustrates an eight layer PCB where the number of layers refers to the number of conductive layers in the stackup. An even number of conductive layers, and a symmetrical arrangement of dielectric layers (in type and thickness) may be used to mitigate warping of the radome due to materials stresses. The arrays of conductive posts  128  and  130  may be, for example, plated vias. Although the conductive posts  130  in  FIG. 1  are shown as blind vias, the vias may be, for example, plated thru holes. The thru holes may be counter bored if the length of the vial is less that the total board thickness. 
     The purpose of the capacitive layers is to realize a desired value of effective capacitance, C fss , per unit square arising from the stored electrical energy between, for example, the patches of layers  102 ,  104 , and  106 . Energy is stored in the z-directed electric field between adjacent patches as in a parallel-plate capacitor. Energy is also stored in the fringing electric fields between adjacent edges of patches and may be termed edge capacitance. The parallel-plate capacitance may dominate the edge capacitance and, in some cases, the edge capacitance may be ignored in the design analysis. 
     The value of thickness t for dielectric layers  101 ,  103 ,  111 , and  113  may be selected to be as small as practical so as to maximize C fss  for a given patch size. Layers  112 ,  114 , and  116  on the other side of the radome may also used to realize a desired effective capacitance per unit square. The symbol t is used to represent the thickness of a dielectric layer, but this does not require that all such layers be of the same thickness t. 
     The bandpass radome may have a greater or lesser number of capacitive layers than are shown in  FIG. 1 . For example, the exterior layers  102  and  116  may be omitted to realize lower values of effective capacitance C fss  as shown in  FIG. 2 . In another example, some or all of the capacitive layers  102 ,  104 ,  114 , and  116  may be omitted. In this embodiment, C fss  is relatively small and dominated by edge capacitance. However, the edge capacitance may be increased by forming the patches of capacitive layers  106  and  112  into inter-digital fingers that couple to inter-digital fingers of adjacent coplanar patches. 
       FIG. 2  shows an example of a bandpass radome as a six-layer symmetric structure The dielectric layers  105  and  109  (not shown in  FIG. 2 , but disposed between layers  106  and  108  and  110  and  112 , respectively) may be equal in thickness, d 3 =d 1 , and of a similar or the same material composition. This example is symmetrical about a plane parallel to the layers and disposed equidistant from the outer layers. Where the term “plane of symmetry” is used, it will be understood by persons of skill in the art that only a local region need be planar. A radome surface may have a radius of curvature or other shaped profile so long as the variation of curvature parameters is consistent with the operating wavelength. Capacitive layers  102  and  116  are omitted in this example. 
       FIG. 2  is also a section view, section A-A, of  FIGS. 3(   a ) and  3 ( b ). The radial coordinate ρ corresponds to the radial direction from a via cut by the section line A-A. 
       FIG. 3(   a ) shows a plan view of the bandpass radome of  FIG. 2 , where only the arrays of conductive posts  128 ,  130  and capacitive layers  104 ,  106  are shown. Capacitive layers  114  and  112  may be the same as capacitive layers  104  and  106 , respectively, and are not shown. Layer  104  is comprised of a periodic array of patches of period P in the x and y directions. Layer  106  is also comprised of a periodic array of patches of period P in the x and y directions, but offset by P/2 in the x and y directions with respect to layer  104 . In this example, the patches of layers  104  and  106  are rebated at their corners so as to avoid contact with the conductive posts  128  and  130 , as well as associated via pads. The conductive posts may be fabricated, for example, as plated vias. The rebated corners on the patches shown in  FIG. 3(   a ) are mitered corners, but any shape may be used for rebating including a square cutout, a quarter circle, or the like. The effective capacitance of the surface may be estimated from: 
                     C   fss     ≅         ɛ   r     ⁢         ɛ   o     ⁡     (       P   /   2     -   g     )       2       t             (   1   )               
where g is the gap between patches, ∈ o  is the permittivity of free space, and ∈ r  is the relative dielectric constant of the dielectric layers  103  and  111 . The dielectric layers separate the lower capacitive layers ( 104  and  106 ) and the upper capacitive layers ( 112  and  114 ). The patches  104  and  106  shown in  FIG. 3(   a ) are essentially square, but the patches may take on any polygonal shape, even a circular shape, as long as sufficient effective capacitance is achieved.
 
       FIG. 3(   b ) shows a plan view of the bandpass radome of  FIG. 2 , where only the arrays of conductive posts  128 ,  130  and the inductive grid layers  108 ,  110  are shown. Both inductive grids  108  and  110  may be substantially the same shape and are aligned with each other so that only one grid is visible in the plan view of  FIG. 3(   b ). The inductive grids  108  and  110  are periodic in the x and y directions with period P′, which is the same period as the patches. The grid traces have width w. A lower bound on the value of inductance of each grid may be computed from: 
                     L   grid     ≥           μ   o     ⁢     P   ′         2   ⁢   π       ⁢     ln   ⁡     (     csc   ⁡     (       π   ⁢           ⁢   w       2   ⁢           ⁢     P   ′         )       )       ⁢       (       P   ′     -   w     )       P   ′                 (   2   )               
where μ o  is the permeability of free space.
 
     A more accurate grid inductance obtained by comparison of equivalent circuit models to full-wave electromagnetic simulations suggests that this formula for L grid  may underestimate the inductance by 50% to 70%. This may arise as equation (2) was derived for isolated grids in free space, and the grids  108  and  110  are both capacitively and inductively coupled to each other due to close proximity. 
     The arrays of conductive posts  128  and  130  may not electrically connect to either inductive layer  108  or  110 . The conductive posts  118  are located midway between the grids in the x and y directions. The posts  120  pass through the intersections of the grid “streets”, but are isolated from the inductive grid by antipads  221 , which are an absence of the conductive grid. The antipads  221  may be circular in shape as shown, square, or any convenient shape such that electrical isolation between the posts and grids is achieved. 
       FIGS. 4(   a ) and  4 ( b ) show an example of transmission (S 21 ) and reflection (S 11 ) plots at normal incidence for the bandpass radome of  FIGS. 2 and 3 . The S parameters were simulated using Microstripes™ version 7, a full-wave electromagnetic simulator marketed and licensed by Flomerics (Marlborough, Mass.). In this simulation, P=P′=8 mm, g=0.5 mm, d 1 =1.25 mm, d 2 =1 mm, t=0.05 mm, w=0.5 mm, and all vias are 0.5 mm diameter. The 0.5 mm wide “streets” of the grids intersect at conductive pads of 1.5 mm diameter. Concentric within each pad is a via of diameter 0.5 mm and an antipad of diameter 1 mm. The dielectric constant is ∈ r =2.9 for layers  103  and  111 , which may be realized, for example, using a flexible laminate of 2 mil Rogers RO3850 available from Rogers Corporation (Rogers, Conn.). This dielectric material is a liquid crystal polymer (LCP) material. Other choices of thin flexible laminates may also be used such as chemical compositions of PET (mylar) and PTFE (TEFLON). LCP and PTFE laminates may be suited for a radome application as they exhibit low loss tangents at RF frequencies, and are stable over varying temperature and humidity; however, other materials may be used, and new materials with suitable properties may be later developed. Dielectric layers  105 ,  107 , and  109  were simulated using Rogers RO4003 laminate which has a relative dielectric constant of ∈ r =3.38. Dielectric losses and conductor losses are both included, where copper is the conductor. 
     The S 21  plot shows two distinct passbands: one centered near 780 MHz, and another centered near 1430 MHz. The passbands are separated by a frequency ratio of about 1.8. This passband separation may be possible due to the relatively high inductance of the inductive layers  108  and  110 . Reducing the grid inductance of layers  108  and  11  may move the passbands toward each other. The broadband S 21  plot of  FIG. 4(   b ) shows that there are no spurious above-band responses in the simulation out to at least 14 GHz, which is at least a decade of frequency range above the highest passband frequency. The total thickness of the bandpass radome is only about 3.6 mm, ignoring the thickness of the thin metal layers. This thickness is approximately λ/58 at the center of the higher frequency passband, and approximately λ/107 at the center of the lower frequency passband. 
     The simulations show that certain design parameters may permit substantially independent control of the lower and upper passband center frequencies such that, for example:
         (a) the lower passband center frequency may be adjusted substantially independently of the upper passband center frequency by varying the effective inductance of the inductive layers  108  and  110 . For simplicity, the effective inductance may be equal for each layer. Increasing the effective inductance decreases the lower passband center frequency;   (b) the upper passband center frequency may be adjusted substantially independently of the lower passband center frequency by varying the distance d 2  between inductive grids  108  and  110 . A larger separation distance d 2  moves the upper passband lower in frequency; and   (c) both passband center frequencies may be increased or decreased in unison by either decreasing or increasing C fss , respectively.       

       FIG. 5(   b ) shows another example of the bandpass radome of  FIG. 2  where a lower value of inductance is used for the inductive layers  108  and  110 .  FIG. 5(   a ) repeats  FIG. 3(   a ).  FIG. 5(   b ) is a plan view of the inductive grids  108  and  110 , and the arrays of conductive posts,  128  and  130 . In this example, both inductive grids  108  and  110  may have the same shape and are aligned so that only one grid is visible in the plan view. The grids  108  and  110  are periodic in x and y directions with period P′=P/2 where P is the period of the patches in the x and y directions. The grid traces have a width w. The salient difference between this example and the example shown in  FIG. 3(   b ) is that the inductive grids of  FIG. 5(   b ) have a spatial period which is half of that of  FIG. 3(   b ). 
     Antipads  221  electrically isolate the conductive posts from both inductive grids  108  and  110  where the posts penetrate the “streets” of the grids. In an alternative, the grids may be offset by P/4 in both the x and y directions, so that the conductive posts may pass through the apertures of the inductive layers. 
       FIG. 6(   a ) shows the simulated transmission (S 21 ) and reflection (S 11 ) plots at normal incidence for an example of the bandpass radome of  FIGS. 2 and 5 . In this example, P=8 mm, P′=4 mm, g=0.5 mm, d 1 =1.524 mm, d 2 =1 mm, t=0.05 mm, w=1.25 mm, and all conductive posts are 0.5 mm diameter vias. As with the previous example simulation, dielectric layers  103  and  111  are modeled as 2 mil Rogers RO3850 LCP. Dielectric layers  105 ,  107 , and  109  are modeled as Rogers RO4003. 
     The previously distinct passbands have coalesced into one broader passband, centered near 1275 MHz, which may be a result of the reduction in grid inductance. The broadband S 21  plot of  FIG. 6(   b ) shows that there are no spurious responses in the simulated results out to at least 14 GHz, which is at least a decade in frequency range above the highest passband frequency. The total thickness of this embodiment of the bandpass radome is about 4.148 mm, ignoring the metal thickness, which is equivalent to approximately λ/56 thick at the center of the passband. 
     In another aspect,  FIG. 7(   a ) shows a plan view of an example of the inductive grid layers of  FIG. 2  where the inductive layers are comprised of aligned grids  108  and  110 . Only one grid is shown since, for simplicity, both grids are assumed to have the same width. The inductive grids may have a period of P equal to the patch period. The grid traces are positioned to run between arrays of conductive posts  118  and  120  so as to avoid electrical contact therebetween.  FIG. 7(   b ) shows a similar plan view where one of the inductive grids  110  has been offset in the x and y directions by one-half of the grid period, resulting in a staggered set of grid streets. The inductive grids  108  and  110  are routed between the arrays of conductive posts  128  and  130  so as to avoid electrical contact. 
       FIG. 8  shows yet another example of possible inductive grid designs, where the “streets” of the inductive grids  108  and  110  have been rotated by 45° with respect to the x and y coordinate axes. This orientation allows more space to run the grid streets between arrays of conductive posts  118  and  120 . As with the example of  FIG. 7(   b ), the grids are staggered horizontally, although this is not necessary. The period P′ of the grids  108  and  110  exceeds the period of patches P, which is the distance between collinear posts in the x or y direction. 
       FIGS. 1 and 2  show two inductive layers disposed near the center of the radome. This configuration permits an even number of layers for the entire stackup, but is not necessary. For example, either the inductive layer  108  or the inductive layer  110  may be omitted. The capacitive layers may be an odd number. 
     The performance of the bandpass radome  100  may be understood using equivalent circuit models instead of full-wave electromagnetic simulations.  FIG. 9  shows a multi-resonance equivalent circuit model  900  of the bandpass radome  100  for angles near normal incidence. In the circuit model  900 , the capacitive layers  102 ,  104 ,  106 ,  112 ,  114 , and  116  are modeled as equivalent circuits  902 ,  904 ,  906 ,  912 ,  914 , and  916  respectively. The topology of these equivalent circuits is a plurality of series RLC networks, connected in parallel. Each equivalent circuit is used to model the broadband behavior of a given capacitive layer. For each capacitive layer, the number of branches and the RLC values may be different. 
     In the circuit model  900 , the inductive layers  108  and  110  are modeled as equivalent circuits  908  and  910 . The topology of these equivalent circuits is a sequence of parallel RLC circuits connected in series. This series combination is connected in shunt across the equivalent TEM mode transmission line at the location of the inductive grid. In general, for each inductive layer, the number of parallel RLC circuits and the RLC values may be different. 
     In the circuit model  900 , the transmission lines  901 ,  903 ,  905 ,  907 ,  909 ,  911 , and  913  model a TEM mode traveling through dielectric layers  101 ,  103 ,  105 ,  107 ,  109 ,  111 , and  113  respectively. The modeled lengths of the transmission lines are the same as the thickness of each corresponding dielectric layer. The characteristic impedances of the transmission lines are modeled as √{square root over (μ o /(∈ o ∈ r ))} where ∈ r  is the relative dielectric constant of each dielectric layer. 
     The equivalent circuits of  902 ,  904 ,  906 ,  908 ,  910 ,  912 ,  914 , and  916  are each shown as a sequence of RLC resonators (either series or parallel resonators). These resonators are used to model the multiple resonances of the layers, where each RLC resonator models one resonance. In most cases, a layer is designed to be used in a frequency range where only one of these resonances may be expected to occur. In the radome examples described herein, the passbands are generally substantially lower in frequency than the resonant frequencies of the individual layers, and the multi-resonator equivalent circuit  900  may be simplified. 
       FIG. 10(   a ) shows one such simplified equivalent circuit model  1000  of the bandpass radome of  FIG. 1  or  2  for a symmetrically fabricated radome and angles near normal incidence. In the equivalent circuit  1000 , the shunt capacitance C fss  is a simplified equivalent circuit of  901 ,  902 ,  903 ,  904 , and  906 . This simplification may be appreciated as replacing the multiple series RLC networks of a given capacitive layer by one series network to model the dominant resonance. When operating far below this resonance, the one series inductance may be eliminated. Since the capacitive layers are typically copper (Cu) or some other highly conductive material, the series resistance in each layer may be eliminated from the model. As the period of the capacitive patches is less than a free space wavelength λ, grating lobe losses are absent. Since the transmission lines  901  and  903  are electrically short at the passband frequencies, on the order of λ/400 or less, the transmission line may be replaced with direct connections of zero length. This allows combination of parallel shunt capacitive networks into one net shunt capacitance of value C fss . These considerations may result in the reduction of equivalent circuits  911 ,  912 ,  913 ,  914 , and  916  into a shunt capacitance of C fss . 
     In the equivalent circuit  1000 , the shunt inductance L g  is the simplified equivalent circuit of networks  908  or  910 . One parallel RLC resonator may dominate and, far below the resonance thereof, the parallel capacitor may be eliminated in the model. The losses of the inductive layers may be negligible assuming good conductors, permitting the elimination of the parallel resistor in  908  and  910 . The remaining component is the parallel inductance denoted as L g  in  FIG. 10(   a ). 
     The equivalent circuit  1000  is sufficiently simplified to permit closed-form analysis of its transmission performance S 21 . Closed-form expressions are useful when one wishes to perform parametric studies of design variables or to optimize design parameters. The design may be refined or confirmed using full wave analysis. 
     The equivalent circuit  1000  may be analyzed by segmenting it into three cascaded subcircuits denoted as  1001 ,  1002 , and  1003 . The approach is to model each subcircuit with an ABCD matrix.  FIG. 10(   b ) shows an equivalent network representation  1010  for the symmetric radome where each of the three subcircuits  1001 ,  1002 , and  1003  have the ABCD matrices shown. Subcircuits  1001  and  1003  are the same, but the ports are reversed. That is, the ABCD parameters for subcircuits  1001  and  1003  contain the same elements but are rearranged. The ABCD parameters may be expressed as: 
                     A   1     =       cos   ⁡     (       β   1     ⁢     d   1       )       +         Z     o   ⁢           ⁢   1         ω   ⁢           ⁢     L   g         ⁢     sin   ⁡     (       β   1     ⁢     d   1       )                   (   3   )                 B   1   =jZ   o1  sin(β 1   d   3 )  (4) 
                     C   1     =         (       j   ⁢           ⁢   ω   ⁢           ⁢     C   fss       +     1     j   ⁢           ⁢   ω   ⁢           ⁢     L   g           )     ⁢     cos   ⁡     (       β   1     ⁢     d   1       )         +       j   ⁡     (       1     Z     o   ⁢           ⁢   1         +         C   fss       L   g       ⁢     Z     o   ⁢           ⁢   1           )       ⁢     sin   ⁡     (       β   1     ⁢     d   1       )                   (   5   )                 D   1 =cos(β 1   d   1 )−ω C   fss   Z   o1  sin(β 1   d   1 )  (6)   A   2 =cos(β 2   d   2 )  (7)   B   2   =jZ   o2  sin(β 2   d   2 )  (8) 
                     C   2     =       j     Z     o   ⁢           ⁢   2         ⁢     sin   ⁡     (       β   2     ⁢     d   2       )                 (   9   )                 D   2 =cos(β 2   d   2 )  (10) 
     Matrix multiplication of the ABCD parameters for each subcircuit, followed by the substitution of D 2 =A 2 , yields the ABCD parameters for the entire radome:
 
 A=A   1   A   2   D   1   +D   1   B   1   C   2   +A   1   B   2   C   1   +A   2   B   1   C   1   (11)
 
 B=A   1 ( A   1   B   2   +A   2   B   1 )+ B   1   2   C   2   +A   1   2   B   2   (12)
 
 C=B   2   C   1   2   +D   1   A   2 (2 C   1   +C   2 )  (13)
 
D=A.  (14)
 
Finally, the transmission response in dB for this symmetric radome may be expressed as
 
                     S   21     =       -   20     ⁢           ⁢   log   ⁢     {            1   2     ⁢     (       2   ⁢   A     +     B     Z   L       +       Z   L     ⁢   C       )            }     ⁢       (     d   ⁢           ⁢   β     )     .               (   15   )               
Z L  is the wave impedance of free space, 377Ω.
 
     The previous examples have illustrated the capacitive and inductive layers as isotropic patterns having equal equivalent circuits for electromagnetic waves polarized in both x and y directions. This may result in dual-polarized radomes with equal performance for both polarizations. However, anisotropic layers may be used such that the passbands may differ in center frequency as a function of polarization. 
     The previous examples have a passband performance which may be described as a 2-pole response, where two distinct frequencies are associated with peaks in the transmission response S 21 . Electrically thin bandpass radomes may also be configured, for example, for a 3-pole response characteristic. A 3-pole response radome may have a broader passband, typically about 10% to 16% bandwidth, and a larger filter shape factor, for better frequency selectivity. 
     An example of a 3-pole response bandpass radome  1100  is shown in  FIG. 11 . The features in the transverse (x and y) directions may be similar to the 2-pole examples, however, the stackup in the z direction may be more complex. Layers  104 ,  106 ,  112 ,  114 ,  120 , and  122  are capacitive layers comprised of two-dimensional arrays of isolated patches. Layers  108 ,  110 ,  116 , and  118  are inductive layers and may be comprised of two-dimensional periodic grids. The structure periods of the inductive layers may be less than, equal to, or greater than the periods of the capacitive layers. In addition, the periods of the capacitive layers may not be uniform. For instance, layers  104 ,  106 ,  120 , and  122  may have patch arrays with a smaller period than the patch arrays on the interior layers  112  and  114 . 
     In radome  1100 , capacitive layers  104  and  106  are separated by a dielectric layer  103  of thickness t 1 . Capacitive layers  112  and  114  are separated by a dielectric layer  111  of thickness t 2 . Capacitive layers  120  and  122  are separated by a dielectric layer  119  of thickness t 3 . Dielectric layers  105 ,  107 , and  109  space the two inductive layers  108  and  110  at pre-selected distances between capacitive layers  106  and  112 . Inductive layers  108  and  110  are spaced a distance d 2  apart, layers  106  and  108  are separated by a distance d 1 , and layers  110  and  112  are separated by a distance d 3 . Similarly, dielectric layers  113 ,  115 , and  117  space the two inductive layers  116  and  118  at pre-selected distances between capacitive layers  114  and  120 . Inductive layers  116  and  118  are spaced a distance d 5  apart, layers  114  and  116  are separated by a distance d 4 , and layers  118  and  120  are separated by a distance d 6 . The thicknesses t 1 , t 2 , and t 3  may typically range from about 1/50 to ⅕ of the dimensions of d 1  thru d 6 . In an example, the total radome thickness defined as t 1 +t 2 +t 3 +d 1 +d 2 +d 3 +d 4 +d 5 +d 6 , plus the thickness of all ten metal layers, may be in the range of approximately λ/150 to λ/10 at the radome passband center frequency, where λ, is the free-space wavelength. 
     Bandpass radome  1100  may also have arrays of conductive posts  128  and  130 , similar to conductive posts  128  and  130  of  FIGS. 1 through 8 . These posts may connect to capacitive layers, and they may connect to a central region of patches on capacitive layers. The posts  128  and  130  may or may not contact the conductive patches on capacitive layers  112  and  114 . As with the previous 2-pole response examples, the conductive posts  128  and  130  may be disposed so as to avoid electrical contact with conductors on inductive layers  108 ,  110 ,  116 , and  118 . The arrays of conductive posts  128  and  130 , which may be periodic, may electrically connect patches which reside on opposite sides of the radome. The periodic array of conductive posts  130  connects conductive patches on capacitive layer  104  to conductive patches on capacitive layer  122 . The periodic array of conductive posts  128  may connect conductive patches on capacitive layer  106  to conductive patches on capacitive layer  120 . The arrays of conductive posts may create a TM mode surface-wave stopband. One of the arrays of posts may be omitted to lower the stopband frequency range. The period of the array of posts  128  and  130  may exceed the period of the patches on corresponding capacitive layers resulting in some of the patches on these capacitive layers being isolated by not being connected to a post. 
     When connecting capacitive layers to arrays of conductive posts, the ordering of the exterior capacitive layers may not be significant. For example, in the 3-pole FSR of  FIG. 11 , layer  104  may be connected to layer  120 , and layer  106  may be connected to layer  122 . For the 2-pole FSR of  FIG. 2 , layer  104  may be connected to layer  112 , and layer  106  may be connected to layer  114 . A rodded medium (array of posts) may terminate on whatever capacitive layers combine to form the exterior shunt capacitance. 
     Individual dielectric layers  103 ,  105 ,  107 ,  109 ,  111 ,  113 ,  115 ,  117 , and  119  in radome  1100  need not be homogeneous dielectric regions. A layer, for example, may be a core, prepreg, a bonding layer, or a combination thereof. Dielectric layers may be isotropic or anisotropic, as with honeycomb materials. 
     The 3-pole radome  1100  may be fabricated as a multi-layer printed circuit board. The 10 metal layer structure of  FIG. 11  may be fabricated as a mechanically-balanced structure where t 1 =t 3 , d 1 =d 6 , d 2 =d 5 , d 3 =d 4 , and a plane of symmetry would exist midway between capacitive layers  112  and  114 . The conductive posts may be, for example, plated vias. 
     A simplified equivalent circuit  1200  for radome  1100  is shown in  FIG. 12 . This equivalent circuit may be appropriate for relatively low frequencies such as in the passband frequency range, and for angles of incidence near normal. Capacitive layers  104  and  106  are modeled as a shunt capacitor C fss1 . Capacitive layers  112  and  114  are modeled as a shunt capacitor C fss2 . Capacitive layers  120  and  122  are modeled as a shunt capacitor C fss3 . Inductive layers  108 ,  110 ,  116 , and  118  are modeled as shunt inductances L g1 , L g2 , L g3 , and L g4  respectively. Transmission lines  1205 ,  1207 ,  1209 ,  1213 ,  1215 , and  1217  model plane waves traveling through dielectric layers  105 ,  107 ,  109 ,  113 ,  115 , and  117 , respectively. These transmission lines are modeled as having the same physical length as the corresponding dielectric regions. Characteristic impedances are modeled in the same manner as discussed for the 2-pole radome example. 
     Higher order bandpass filters may be realized, for example, by adding alternating inductive and capacitive layers to the stackup of lower-order bandpass filters. 
     One of the three poles of radome  1100  may be widely separated in frequency from the other two poles to produce a dual-band radome. However, if a single transmission band is desired, then a simpler 3-pole bandpass radome, shown as radome  1300  in  FIG. 13  may be employed. The dielectric regions  107  and  115 , and the inductive layers  110  and  118  have been omitted. The result is an eight-layer radome that may be thinner, lighter, and less expensive to manufacture than radome  1100 . A simplified equivalent circuit  1400  for radome  1300  is shown in  FIG. 14 . Inductive layers  108  and  116  are modeled as shunt inductances L g1  and L g3  respectively. The other circuit element definitions are the same as in  FIG. 12 . 
     An example of radome  1300  is shown in plan views for the individual capacitive and inductive layers in  FIG. 15 . Each view shows a square unit cell of dimensions P×P. This results in a mechanically-balanced structure where capacitive layers  104  and  122  are the same, capacitive layers  106  and  120  are the same, and inductive layers  108  and  116  are the same. In this example P=8 mm, g 1 =1.65 mm, g 2 =0.5 mm, P′=4 mm, a=2.2 mm, the arrays of posts  128  and  130  are modeled as 0.5 mm square, and the posts are isolated from the inductive grids by 1.5 mm square antipads. The patches on layer  114  have rebated corners defined by square antipads of size ml=2 mm. Dielectric layers  103 ,  111 , and  119  have a dielectric constant of 2.9 and a thickness of t 1 =t 2 =t 3 =0.05 mm which is approximately 2 mils. Dielectric layers  105 ,  109 ,  113 , and  117  have a dielectric constant of 3.38 and a thickness of d 1 =d 3 =d 4 =d 5 =1.7 mm which is approximately 67 mils. 
     The simulated S parameter performance is shown in  FIG. 16 . Transmission and reflection S-parameters are calculated using a full-wave 3D EM simulator (Microstripes 7.1) for normal incidence. The passband is centered near about 1790 MHz, and the −10 dB return loss bandwidth is about 190 MHz or 10.6%. The passband bandwidth may be increased with a trade-off of greater passband ripple. Radome  1300  has a total thickness of approximately 6.95 mm (274 mils), ignoring the thickness of the eight metal layers. This corresponds to a normalized thickness of about λ/24 at the center of the passband, resulting in an electrically-thin radome. The small size of the unit cell places the computed spurious responses above 20 GHz. An equivalent circuit simulation was optimized to determine the effective values of the shunt inductors and capacitors, which were determined to be: C fss1 =C fss3 =3.32 pF/sq., C fss2 =6.62 pF/sq., and L g1 =L g3 =0.282 nH/sq. 
     The closely spaced overlapping patch layers shown in  FIGS. 1 ,  2 ,  11  and  13  may be used to form a equivalent shunt capacitance. If the patch period is sufficiently large, and the passband center frequency sufficiently high, then the edge capacitance available between patches in a capacitance layer may be sufficient to achieve the desired value of capacitance. In this case, 2 or 3 closely spaced patch layers may be replaced by a single patch. Edge capacitance may be enhanced by designing patches with inter-digital fingers as taught by Rogers, McKinzie, and Mendolia in “AMCs Comprised of an Interdigital Capacitor FSS Layer Enable Lower Cost Applications,” 2003  IEEE Antennas and Propagation International Symposium , Columbus, Ohio, Jun. 22-27, 2003, Vol. 2, pp. 411-414. Examples shown in the reference include inter-digital FSS structures of capacitance value 1.4 pF/sq. and 4.7 pF/sq., where the corresponding periods between centers of adjacent patches are 315 mils (8 mm) and 700 mils (17.8 mm) respectively. 
       FIG. 17  shows a profile view of an example of a 3-pole response bandpass radome  1700  where the capacitive layers  104  and  122  have sufficient shunt capacitance to realize desired values of C fss1  and C fss3 . In this example, the capacitive layers  106  and  120  have been omitted, and the dielectric layers  103  and  119  have been omitted. The elimination of patch layers  106  and  120  results in eliminating a need for the array of conductive posts  128 . The remaining layers and posts are the same as in  FIG. 13 . Layers  104  and  122  may be etched as an array of patches having inter-digital fingers as shown in  FIG. 18  to result in capacitances C fss1  and C fss2 . Patches  1806  and fingers  1802  mesh on layer  104  with slots in adjacent patches to increase the edge capacitance. Posts  130  connect to a central region of the patches. A similar pattern of inter-digital capacitors may be used in capacitive layer  122 . Using inter-digital capacitors may reduce the 3-pole bandpass radome to a 6 layer design, however above-band spurious responses may have to be considered. 
     Inter-digital capacitors may be also used to reduce the number of metal layers in a 2-pole bandpass radome.  FIG. 19  shows a profile view of radome  1900  where the capacitive layers  106  and  112  may be the inter-digital design of  FIG. 18 . Layers  108  and  110  are inductive grids. An array of conductive posts  128  connects the patches of layers  106  to  112 , for TM surface wave suppression. Antipads  221  isolate the array of posts  128  from the inductive grids on the inductive layers  108  and  110 . 
     Thus, an array of posts may be used to electrically connect patches on opposite (exterior) sides of a bandpass radome. The posts are electrically isolated from the grids on intervening inductive layers. The posts cooperate with the capacitive layers to result in a TM mode surface wave stopband that may be designed to coincide with a desired passband. The passband may then be free of undesired coupling to TM surface-wave modes that may be excited at discontinuities such as radome edges and corners. 
     Although the foregoing has been a description and illustration of specific examples of embodiments of the invention, various modifications and changes can be made by persons skilled in the art without departing from the scope and spirit of the invention. For example, the dielectric materials used to separate the conductive FSS layers can have different dielectric or mechanical properties. For instance, a dielectric layer may be inhomogeneous or anisotropic. The dielectric layers may not be “solid” but might be a honeycomb structure or substantially open structure to save weight. The inductive layers may contain patterns more elaborate than simple square grids, such as meandering lines. Furthermore the apertures in the inductive grids may not be essentially rectangular, but may take on more complex shapes such as circular, elliptical, or a general polygon. Some of the patches of the capacitive layers may be left floating as opposed to being connected to conductive posts. Accordingly, the invention is defined by, and limited only by, the following claims.