Abstract:
A steerable artificial impedance surface antenna steerable in phi and theta angles including a dielectric substrate, a plurality of metallic strips on a first surface of the dielectric substrate, the metallic strips spaced apart across a length of the dielectric substrate and each metallic strip extending along a width of the dielectric substrate, and surface wave feeds spaced apart along the width of the dielectric substrate near an edge of the dielectric substrate, wherein the dielectric substrate is substantially in an X-Y plane defined by an X axis and a Y axis, wherein the phi angle is an angle in the X-Y plane relative to the X axis, and wherein the theta angle is an angle relative to a Z axis orthogonal to the X-Y plane.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application is related to the disclosure of U.S. patent application Ser. No. 12/939,040 filed Nov. 3, 2010, and U.S. patent application Ser. No. 13/242,102 filed Sep. 23, 2011, the disclosures of which are hereby incorporated herein by reference. 
     
    
     TECHNICAL FIELD 
       [0002]    This disclosure relates to artificial impedance surface antennas (AISAs). 
       BACKGROUND 
       [0003]    An antenna whose primary gain lobe can be electronically steered in two dimensions is desirable in many applications. In the prior art the two dimensional steering is most commonly provided by phased array antennas. Phased array antennas have complex electronics and are quite costly. 
         [0004]    In the prior art, various electronically steered artificial impedance surface antennas (AISAs) have been described that have one dimensional electronic steering, including the AISAs described in U.S. Pat. Nos. 7,245,269, 7,071,888, and U.S. Pat. No. 7,253,780 to Sievenpiper. These antennas are useful for some applications, but are not suitable for all applications that need two dimensional steering. In some applications mechanical steering can be used to provide steering of a 1D electronically steered antenna in a second dimension. However, there are many applications where mechanical steering is very undesirable. The antennas described by Sievenpiper also require vias for providing voltage control to varactors. 
         [0005]    A two dimensionally electronically steered AISA has been described in U.S. Pat. No. 8,436,785, issued on May 7, 2013, to Lai and Colburn. The antenna described by Lai and Colburn is relatively costly and is electronically complex, because to steer in two dimensions a complex network of voltage control to a two dimensional array of impedance elements is required so that an arbitrary impedance pattern can be created to produce beam steering in any direction. 
         [0006]    Artificial impedance surface antennas (AISAS) are realized by launching a surface wave across an artificial impedance surface (AIS), whose impedance is spatially modulated across the AIS according a function that matches the phase fronts between the surface wave on the AIS and the desired far-field radiation pattern. 
         [0007]    In previous references, listed below, references [1]-[6] describe artificial impedance surface antennas (AISA) formed from modulated artificial impedance surfaces (AIS). Patel [1] demonstrated a scalar AISA using an end-fire, flare-fed one-dimensional, spatially-modulated AIS consisting of a linear array of metallic strips on a grounded dielectric. Sievenpiper, Colburn and Fong [2]-[4] have demonstrated scalar and tensor AISAs on both flat and curved surfaces using waveguide- or dipole-fed, two-dimensional, spatially-modulated AIS consisting of a grounded dielectric topped with a grid of metallic patches. Gregoire [5]-[6] has examined the dependence of AISA operation on its design properties. 
         [0008]    Referring to  FIG. 1 , the basic principle of AISA operation is to use the grid momentum of the modulated AIS to match the wave vectors of an excited surface-wave front to a desired plane wave. In the one-dimensional case, this can be expressed as 
         [0000]        k   sw   =k   o  sin θ o   −k   p   (1)
 
         [0009]    where k o  is the radiation&#39;s free-space wavenumber at the design frequency, θ o  is the angle of the desired radiation with respect to the AIS normal, k p =2π/p is the AIS grid momentum where p is the AIS modulation period, and k sw =n o k o  is the surface wave&#39;s wavenumber, where n o  is the surface wave&#39;s refractive index averaged over the AIS modulation. The SW impedance is typically chosen to have a pattern that modulates the SW impedance sinusoidally along the SWG according to 
         [0000]        Z ( x )= X+M  cos(2π x/p )  (2)
 
         [0010]    where p is the period of the modulation, X is the mean impedance, and M is the modulation amplitude. X, M and p are chosen such that the angle of the radiation θ in the x-z plane w.r.t the z axis is determined by 
         [0000]      θ=sin −1 ( n   0 −λ 0   /p )  (3)
 
         [0011]    where n 0  is the mean SW index, and λ 0  is the free-space wavelength of radiation. n 0  is related to Z(x) by 
         [0000]    
       
         
           
             
               
                 
                   
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                               1 
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                            
                           
                               
                           
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                           X 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
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         [0012]    The AISA impedance modulation of Eqn. (2) can be generalized for an AISA of any shape as 
         [0000]        Z ({right arrow over ( r )})= X+M  cos( k   o   n   o   r−{right arrow over (k)}   o   □{right arrow over (r)} )  (5)
 
         [0013]    where {right arrow over (k)} o  is the desired radiation wave vector, {right arrow over (r)} is the three-dimensional position vector of the AIS, and r is the distance along the AIS from the surface-wave source to {right arrow over (r)} along a geodesic on the AIS surface. This expression can be used to determine the index modulation for an AISA of any geometry, flat, cylindrical, spherical, or any arbitrary shape. In some cases, determining the value of r is geometrically complex. 
         [0014]    For a flat AISA, it is simply r=√{square root over (x 2 +y 2 )}. 
         [0015]    For a flat AISA designed to radiate into the wave vector at {right arrow over (k)}=k o (sin θ o {circumflex over (x)}+cos θ o {circumflex over (z)}), with the surface-wave source located at x=y=0, the modulation function is 
         [0000]        Z ( x,y )= X+M  cos( k   o ( n   o   r−x  sin θ o ))  (6)
 
         [0016]    The cos function in Eqn. (2) can be replaced with any periodic function and the AISA will still operate as designed, but the details of the side lobes, bandwidth and beam squint will be affected. 
         [0017]    The AIS can be realized as a grid of metallic patches on a grounded dielectric. The desired index modulation is produced by varying the size of the patches according to a function that correlates the patch size to the surface wave index. The correlation between index and patch size can be determined using simulations, calculation and/or measurement techniques. For example, Colburn [3] and Fong [4] use a combination of HFSS unit-cell eigenvalue simulations and near field measurements of test boards to determine their correlation function. Fast approximate methods presented by Luukkonen [7] can also be used to calculate the correlation. However, empirical correction factors are often applied to these methods. In many regimes, these methods agree very well with HFSS eigenvalue simulations and near-field measurements. They break down when the patch size is large compared to the substrate thickness, or when the surface-wave phase shift per unit cell approaches 180°. 
         [0018]    In the prior art electronically-steerable AIS antennas described in [8] and [9], the AIS is a grid of metallic patches on a dielectric substrate. The surface-wave impedance is locally controlled at each position on the AIS by applying a variable voltage to voltage-variable varactors connected between each of the patches. It is well known that an AIS&#39;s SW impedance can be tuned with capacitive loads inserted between impedance elements [8], [9]. Each patch is electrically connected to neighboring patches on all four sides with voltage-variable varactor capacitor. The voltage is applied to the varactors though electrical vias connected to each impedance element patch. Half of the patches are electrically connected to the groundplane with vias that run from the center of each patch down through the dielectric substrate. The rest of the patches are electrically connected to voltage sources that run through the substrates, and through holes in the ground plane to the voltage sources. 
         [0019]    Computer control allows any desired impedance pattern to be applied to the AIS within the limits of the varactor tunability and the AIS SW property limitations. One of the limitations of this method is that the vias can severely reduce the operation bandwidth of the AIS because the vias also impart an inductance to the AIS that shifts the SW bandgap to lower frequency. As the varactors are tuned to higher capacitance, the AIS inductance is increased and this further reduces the SW bandgap frequency. The net result of the SW bandgap is that it does not allow the AIS to be used above the bandgap frequency. It also limits the range of SW impedance that the AIS can be tuned to. 
       REFERENCES 
       [0000]    
       
         1. Patel, A. M.; Grbic, A., “A Printed Leaky-Wave Antenna Based on a Sinusoidally-Modulated Reactance Surface,” Antennas and Propagation, IEEE Transactions on, vol. 59, no. 6, pp. 2087, 2096, June 2011 
         2. D. Sievenpiper et al, “Holographic AISs for conformal antennas”, 29th Antennas Applications Symposium, 2005 
         3. D. Sievenpiper, J. Colburn, B. Fong, J. Ottusch and J. Visher., 2005 IEEE Antennas and Prop. Symp. Digest, vol. 1B, pp. 256-259, 2005. 
         4. B. Fong et al, “Scalar and Tensor Holographic Artificial Impedance Surfaces,” IEEE TAP., 58, 2010 
         5. D. J. Gregoire and J. S. Colburn,  Artificial impedance surface antennas , Proc. Antennas Appl. Symposium 2011, pp. 460-475 
         6. D. J. Gregoire and J. S. Colburn,  Artificial impedance surface antenna design and simulation , Proc. Antennas Appl. Symposium 2010, pp. 288-303 
         7. O. Luukkonen et al, “Simple and accurate analytical model of planar grids and high-impedance surfaces comprising metal strips or patches”, IEEE Trans. Antennas Prop., vol. 56, 1624, 2008 
         8. Colburn, J. S.; Lai, A.; Sievenpiper, D. F.; Bekaryan, A.; Fong, B. H.; Ottusch, J. J.; Tulythan, P.; “Adaptive artificial impedance surface conformal antennas,” Antennas and Propagation Society International Symposium, 2009. APSURSI &#39;09. IEEE, vol., no., pp. 1-4, 1-5 Jun. 2009 
         9. Sievenpiper, D.; Schaffner, J.; Lee, J. J.; Livingston, S.; “A steerable leaky-wave antenna using a tunable impedance ground plane,” Antennas and Wireless Propagation Letters, IEEE, vol. 1, no. 1, pp. 179-182, 2002. 
       
     
         [0029]    What is needed is an electronically steered artificial impedance surface antenna (AISA) that can be steered in two dimensions, while being lower cost. The embodiments of the present disclosure answer these and other needs. 
       SUMMARY 
       [0030]    In a first embodiment disclosed herein, a steerable artificial impedance surface antenna steerable in phi and theta angles comprises dielectric substrate, a plurality of metallic strips on a first surface of the dielectric substrate, the metallic strips spaced apart across a length of the dielectric substrate and each metallic strip extending along a width of the dielectric substrate, and surface wave feeds spaced apart along the width of the dielectric substrate near an edge of the dielectric substrate, wherein the dielectric substrate is substantially in an X-Y plane defined by an X axis and a Y axis, wherein the phi angle is an angle in the X-Y plane relative to the X axis, and wherein the theta angle is an angle relative to a Z axis orthogonal to the X-Y plane. 
         [0031]    In another embodiment disclosed herein, a steerable artificial impedance surface antenna steerable in phi and theta angles comprises a dielectric substrate, a plurality of metallic strips on a first surface of the dielectric substrate, the metallic strips spaced apart across a length of the dielectric substrate, the metallic strips having equally spaced centers, the metallic strips varying in width with a period of p, and each metallic strip extending along a width of the dielectric substrate, and surface wave feeds spaced apart along a width of the dielectric substrate near an edge of the dielectric substrate, wherein the dielectric substrate is substantially in an X-Y plane defined by an X axis and a Y axis, wherein the phi angle is an angle in the X-Y plane relative to the X axis, and wherein the theta angle is an angle relative to a Z axis orthogonal to the X-Y plane. 
         [0032]    These and other features and advantages will become further apparent from the detailed description and accompanying figures that follow. In the figures and description, numerals indicate the various features, like numerals referring to like features throughout both the drawings and the description. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0033]      FIG. 1  shows surface waves propagating outward from a source interact with the modulated impedance to produce radiation in a narrow beam in accordance with the prior art; 
           [0034]      FIG. 2A  shows an electronically steered artificial impedance surface antenna (AISA), and  FIG. 2B  shows a side elevation view of an AISA in accordance with the present disclosure; 
           [0035]      FIG. 3  is a diagram of a spherical coordinate system showing the angles and the transformations to Cartesian coordinates in accordance with the prior art; 
           [0036]      FIG. 4  shows another electronically steered artificial impedance surface antenna (AISA) in accordance with the present disclosure; 
           [0037]      FIG. 5  shows yet another electronically steered artificial impedance surface antenna (AISA) in accordance with the present disclosure; 
           [0038]      FIG. 6  shows another side elevation view of an AISA in accordance with the present disclosure; and 
           [0039]      FIG. 7  shows yet another side elevation view of an AISA in accordance with the present disclosure. 
       
    
    
     DETAILED DESCRIPTION 
       [0040]    In the following description, numerous specific details are set forth to clearly describe various specific embodiments disclosed herein. One skilled in the art, however, will understand that the presently claimed invention may be practiced without all of the specific details discussed below. In other instances, well known features have not been described so as not to obscure the invention. 
         [0041]      FIG. 2A  shows an electronically steered artificial impedance surface antenna (AISA) in accordance with the present disclosure that is relatively low cost and capable of steering in both theta (θ) and phi (φ) directions.  FIG. 3  is a diagram of a spherical coordinate system showing the theta (θ) and phi (φ) angles. In  FIG. 3  the phi (φ) angle is the angle in the x-y plane, and the theta (θ) angle is the angle from the z axis. Because the primary gain lobe of the electronically steered artificial impedance surface antenna (AISA) in accordance with the present disclosure is capable of steering in both theta (θ) and phi (φ) directions, those skilled in the art refer to it as a 2D electronically steered artificial impedance surface antenna (AISA). 
         [0042]    The electronically steered artificial impedance surface antenna (AISA) of  FIG. 2A  includes a tunable artificial impedance surface antenna (AISA)  101 , a voltage control network  102 , and a one-dimensional 1D radio frequency (RF) feed network  103 . When the tunable artificial impedance surface antenna (AISA)  101  is in the X-Y plane of  FIG. 3 , the steering of the primary gain lobe of the electronically steered artificial impedance surface antenna (AISA) is controlled in the phi (φ) direction by changing the relative phase difference between the RF surface wave feeds  108  of the 1D RF feed network  103 . The theta steering is controlled by varying or modulating the surface wave impedance of the tunable artificial impedance surface antenna (AISA)  101 . 
         [0043]    The artificial impedance surface antenna (AISA)  101  in the embodiment of  FIG. 2A  includes a dielectric substrate  106 , a periodic array of metallic strips  107  on one surface of the dielectric substrate  106 , varactors  109  electrically connected between the metallic strips  107 , and a 1D array of RF surface wave feeds  108 . The impedance of the AISA  101  may be varied or modulated by controlling voltages to the metallic strips  107  on the tunable artificial impedance surface antenna (AISA)  101 . The voltages on the metallic strips  107  change the capacitance of varactors  109  between the metallic strips  107 , which changes the impedance of the AISA  101 , thereby steering the primary gain lobe in the theta direction. 
         [0044]    The voltage control network  102  applies direct current (DC) voltages to the metallic strips  107  on the AISA structure. Control bus  105  provides control for the voltage control network  102 . The control bus  105  may be from a microprocessor, central processing unit, or any computer or processor. 
         [0045]    Control bus  104  provides control for the 1D RF feed network  103 . The control bus  104  may be from a microprocessor, central processing unit, or any computer or processor. 
         [0046]      FIG. 2B  shows a side elevation view of  FIG. 2A . As shown varactors  109  are between the metallic strips  107 , which are on the surface of the dielectric substrate  106 . The dielectric substrate  106  may or may not have a ground plane  119  on a surface opposite to the surface upon which the metallic strips  107  are located. As further described below, in one embodiment shown in  FIG. 6 , varactors are not between the metallic strips  107 . In another embodiment, shown in  FIG. 7 , and further described below, varactors are again not used; however, the dielectric substrate  106  may further include a material  404  with tunable electrical properties, such as a liquid crystal. When a voltage is applied to the impedance elements, such as the metallic strips  107 , which may be formed, deposited, printed, or pasted onto the dielectric substrate  106 , the properties of the dielectric substrate  106 , or the material  404  with tunable electrical properties may change. In particular the dielectric constant may change, thereby changing the impedance between the metallic strips  107 , and thereby steering a beam in the theta direction. 
         [0047]    A varactor is a type of diode whose capacitance varies as a function of the voltage applied across its terminals, which makes it useful for tuning applications. When varactors  109  are used between the metallic strips  107 , as shown in  FIG. 2A , by controlling the voltage applied to the varactors  109  via the metallic strips  107 , the capacitances of the varactors  109  vary, which in turn varies or modulates the capacitive coupling and the impedance between the metallic strips  107  to steer a beam in the theta direction. 
         [0048]    The polarities of the varactors  109  are aligned such that all the varactor connections to any one of the metallic strips  107  are connected with the same polarity. One terminal on a varactor may be referred to as an anode, and the other terminal as a cathode. Thus, some of the metallic strips  107  are only connected to anodes of varactors  109 , and other metallic strips  107  are only connected to cathodes of varactors  109 . Further, as shown in  FIG. 2A , adjacent metallic strips  107  on the AISA  101  alternate in being connected to anodes or cathodes of varactors  109 . 
         [0049]    The spacing of the metallic strips  107  in one dimension of the AISA, which may, for example, be the X axis of  FIG. 3 , may be a fraction of the RF surface wave (SW) wavelength of the RF waves that propagate across the AISA from the RF surface wave feeds  108 . In a preferred embodiment, the spacing of the metallic strips  107  may be at most ⅕ of the RF surface wave (SW) wavelength of the RF waves. Typically the fraction may be only about 1/10 of the RF surface wave (SW) wavelength of the RF waves. 
         [0050]    The spacing between varactors  109  connected to the metallic strips  107  in a second dimension of the AISA, which is generally orthogonal to the first dimension of the AISA and which may be the Y axis of  FIG. 3 , is typically about the same as the spacing between metallic strips. 
         [0051]    The RF SW feeds  108  may be a phased array corporate feed structure, or may be conformal surface wave feeds, which are integrated into the AISA, such as by using micro-strips. Conformal surface wave feeds that may be used include those described in U.S. patent application Ser. No. 13/242,102 filed Sep. 23, 2011, or those described in “Directional Coupler for Transverse-Electric Surface Waves”, published in IP.com Prior Art Database Disclosure No. IPCOM000183639D, May 29, 2009, which are incorporated herein by reference as though set forth in full. 
         [0052]    The spacing between the RF SW feeds  108  in the second dimension of the AISA or the y dimension of  FIG. 3 , may be based on rules of thumb for phased array antennas that dictate they be no farther apart than ½ of the free-space wavelength for the highest frequency signal to be transmitted or received. 
         [0053]    The thickness of the dielectric substrate  106  is determined by its permittivity and the frequency of radiation to be transmitted or received. The higher the permittivity, the thinner the substrate can be. 
         [0054]    The capacitance values of the varactors  109  are determined by the range necessary for the desired AISA impedance modulations to obtain the various angles of radiation. 
         [0055]    An AISA operating at about 10 GHz may use for the dielectric substrate  106 , a 50-mil thick Rogers Corp 3010 circuit board material with a relative permittivity equal to 11.2. The metallic strips  107  may be spaced 2 millimeters (mm) to 3 mm apart on the dielectric substrate  106 . The RF surface wave feeds  108  may be spaced 1.5 centimeters (cm) apart and the varactors  109  may be spaced 2 mm to 3 mm apart. The varactors  109  vary in capacitance from 0.2 to 2.0 pico farads (pF). Designs for different radiation frequencies or designs using different substrates will vary accordingly. 
         [0056]    To transmit or receive an RF signal, transmit/receive module  110  is connected to the feed network  103 . The feed network  103  can be of any type that is known to those skilled in the state of the art of phased array antennas. For the sake of illustration, the feed network  103  shown in  FIG. 2A  includes a series of RF transmission lines  111  connected to the transmit/receive module  110 , power dividers  112 , and phase shifters  113 . The phase shifters  113  are controlled by voltage control lines  118  from a digital to analog converter (DAC)  114  that receives digital control signals  104  to control the steering in the phi (φ) direction. 
         [0057]    The antenna main lobe is steered in the phi direction by using the feed network  103  to impose a phase shift between each of the RF SW feeds  108 . If the RF SW feeds  108  are spaced uniformly, then the phase shift between adjacent RF SW feeds  108  is constant. The relation between the phi (φ) steering angle, and the phase shift may be calculated using standard phased array methods, according to equation, 
         [0000]      φ=sin −1 (λΔψ/2π d )  (7)
 
         [0000]    where λ is the radiation wavelength, d is the spacing between SW feeds  108 , and Δψ is the phase shift between SW feeds  108 . The RF SW feeds  108  may also be spaced non-uniformly, and the phase shifts adjusted accordingly. 
         [0058]    The antenna lobe is steered in the theta (θ) direction by applying voltages to the varactors  109  between the metallic strips  107  such that AISA  101  has surface-wave impedance Z sw , that is modulated or varied periodically with the distance (x) away from the SW feeds  108 , according to equation, 
         [0000]        Z   sw   =X+M  cos(2π x/p )  (8)
 
         [0000]    where X and M are the mean impedance and the amplitude of its modulation respectively, and p is the modulation period. The variation of the surface-wave impedance Z sw  may be modulated sinusoidally. The theta steering angle θ, is related to the impedance modulation by the equation, 
         [0000]      θ=sin −1 ( n   sw   −λ/p )  (9)
 
         [0000]    where λ is the wavelength of the radiation, and 
         [0000]        n   sw =√{square root over (( X/ 377) 2 +1)}  (10)
 
         [0000]    is the mean surface-wave index. 
         [0059]    The beam is steered in the theta direction by tuning the varactor voltages such that X, M, and p result in the desired theta θ. The dependence of the surface wave (SW) impedance on the varactor capacitance is calculated using transcendental equations resulting from the transverse resonance method or by using full-wave numerical simulations. 
         [0060]    In the embodiment of  FIG. 2A , voltages are applied to the varactors  109  by grounding alternate metallic strips  107  to ground  120  and applying tunable voltages via voltage control lines  116  to the rest of the strips  107 . The voltage applied to each voltage control line  116  is a function of the desired theta (θ), and may be different for each voltage control line  116 . The voltages may be applied from a digital-to-analog converter (DAC)  117  that receives digital controls  105  from a controller for steering in the theta direction. The controller may be a microprocessor, central processing unit (CPU) or any computer, processor or controller. 
         [0061]    An advantage of grounding half of the metallic strips  107  is that only half as many voltage control lines  116  are required as there are metallic strips  107 . A disadvantage is that the spatial resolution of the voltage control and hence the impedance modulation is limited to twice the spacing between metallic strips. 
         [0062]      FIG. 4  shows another electronically steered artificial impedance surface antenna (AISA) in accordance with the present disclosure that is essentially the same as the embodiment described with reference to  FIG. 2A , except in the embodiment of  FIG. 4 , a voltage is applied to each of the metallic strips  207  by voltage control lines  216 . Twice as many control voltages are required compared to the embodiment of  FIG. 2A , however, the spatial resolution of the impedance modulation is doubled. The voltage applied to each voltage control line  216  is a function of the desired theta (θ) angle, and may be different for each voltage control line  216 . The voltages are applied from a digital-to-analog converter (DAC)  217  that receives digital controls  205  from an outside source, which may be a microprocessor, central processing unit (CPU) or any computer or processor, for steering in the theta direction. 
         [0063]    The antenna main lobe is steered in the phi direction by using the feed network  203  to impose a phase shift between each of the RF SW feeds  208  in the same manner as described with reference to  FIG. 1 . 
         [0064]      FIG. 5  illustrates a preferred embodiment where the theta θ angle control DACs  117  and  217  of  FIGS. 2A and 4  are replaced by a single control voltage from a variable voltage source  350 . As the voltage of variable voltage source  350  is varied, the AISA radiation angle varies between a minimum and maximum theta angle that is determined by the details of the AISA design. The voltage is applied though voltage control lines  352  and  354  to the metallic strips  340  on the surface of the AISA. Voltage control line  354  may be a ground with the voltage control line  352  being a variable voltage. Across the x dimension, the metallic strips  340  are alternately tied to voltage control line  352  or to voltage control line  354 . 
         [0065]    One or more varactors diodes  309  may be in each gap between adjacent metallic strips  340  and electrically connected to the metallic strips in the same manner as shown in  FIG. 2A . 
         [0066]    The metallic strips may have centers that are equally spaced in the x dimension, with the widths of the metallic strips  340  periodically varying with a period p  346 . The number of metallic strips in a period  346  can be any number, although 10 to 20 is reasonable for most designs. The width variation is designed to produce surface-wave impedance with a periodic modulation in the X-direction with period p  346 , for example, the sinusoidal variation of equation (8) above. 
         [0067]    The surface-wave impedance at each point on the AISA is determined by the width of the metallic strips and the voltage applied to the varactors  309 . The relation between the surface-wave impedance and these parameters is well understood and documented in the references [1]-[9]. 
         [0068]    The capacitance of the diode varactors  309  varies with the applied voltage. When the voltage is 0 volts, the diode capacitance is at its maximum value of C max . The capacitance decreases as the voltage is increased until it reaches a minimum value of C min . As the diode capacitance is varied, the impedance modulation parameters, X and M in Eqn. (8) vary also from minimum values X min  and M min  to maximum values of X max  and M max . Likewise, the mean surface-wave index of Eqn. (10) varies from n min =√{square root over ((X min /377) 2 +1)} to n max =√{square root over ((X max /377) 2 +1)}. 
         [0069]    Then from Eqn. (9), the range that the AISA&#39;s radiation angle can be scanned varies from a minimum of 
         [0000]      θ min =sin −1 ( n   min   −λ/p )  (11)
 
         [0000]    to a maximum of 
         [0000]      θ max =sin −1 ( n   max   −λ/p )  (12)
 
         [0000]    with variation of a single control voltage. 
         [0070]    In another embodiment shown in the elevation view of  FIG. 6 , the substrate  401 , which may be used for dielectric substrates  106 ,  206  or  306 , is a material whose electrical permittivity is varied with application of an electric field. As described above, no varactors  109 ,  209  or  309  are used in this embodiment. When a voltage is applied to metallic strips  402  on the AISA, an electric field is produced between adjacent strips and also between the strips and the substrate ground plane  403 . The electric field changes the permittivity of the substrate material, which results in a change in the capacitance between adjacent metallic strips  402 . As in the other embodiments, the capacitance between adjacent metallic strips  402  determines the surface-wave impedance. 
         [0071]    In a variation on this, shown in the elevation view of  FIG. 7 , a voltage differential may be applied to adjacent metallic  402  strips, which creates an electric field between the metallic strips  402  and produces a permittivity change in a variable material  404  between the metallic strips  402 . The variable material  404  may be any electrically variable material, such as liquid crystal material or barium strontium titanate (BST). It may be necessary, especially in the case of using liquid crystals, to embed the variable material  404  in pockets within an inert substrate  405 , as shown in  FIG. 7 . 
         [0072]    The antenna main lobe is steered in the phi direction by using the feed network  303  to impose a phase shift between each of the RF SW feeds  308  in the same manner as described with reference to  FIG. 2A . 
         [0073]    Having now described the invention in accordance with the requirements of the patent statutes, those skilled in this art will understand how to make changes and modifications to the present invention to meet their specific requirements or conditions. Such changes and modifications may be made without departing from the scope and spirit of the invention as disclosed herein. 
         [0074]    The foregoing Detailed Description of exemplary and preferred embodiments is presented for purposes of illustration and disclosure in accordance with the requirements of the law. It is not intended to be exhaustive nor to limit the invention to the precise form(s) described, but only to enable others skilled in the art to understand how the invention may be suited for a particular use or implementation. The possibility of modifications and variations will be apparent to practitioners skilled in the art. No limitation is intended by the description of exemplary embodiments which may have included tolerances, feature dimensions, specific operating conditions, engineering specifications, or the like, and which may vary between implementations or with changes to the state of the art, and no limitation should be implied therefrom. Applicant has made this disclosure with respect to the current state of the art, but also contemplates advancements and that adaptations in the future may take into consideration of those advancements, namely in accordance with the then current state of the art. It is intended that the scope of the invention be defined by the Claims as written and equivalents as applicable. Reference to a claim element in the singular is not intended to mean “one and only one” unless explicitly so stated. Moreover, no element, component, nor method or process step in this disclosure is intended to be dedicated to the public regardless of whether the element, component, or step is explicitly recited in the Claims. No claim element herein is to be construed under the provisions of 35 U.S.C. Sec. 112, sixth paragraph, unless the element is expressly recited using the phrase “means for . . . ” and no method or process step herein is to be construed under those provisions unless the step, or steps, are expressly recited using the phrase “comprising the step(s) of . . . . ”