Patent Publication Number: US-2017373404-A1

Title: Equatorially and near-equatorially radiating arc-shaped polarization current antennas and related methods

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims priority under 35 U.S.C. §119 to U.S. Provisional Patent Application Ser. No. 62/355,478, filed Jun. 28, 2016, and to U.S. Provisional Patent Application Ser. No. 62/399,716, filed Sep. 26, 2016, the entire content of both of which are incorporated herein by reference in their entireties. 
    
    
     BACKGROUND 
     Antennas that include a dielectric radiator that is excited using a series of polarization devices are known in the art. Such antennas are referred to herein as “polarization current antennas.” An example of such a polarization current antenna is disclosed in European Patent No. 1112578 titled “Apparatus for Generating Focused Electromagnetic Radiation,” filed on Sep. 6, 1999. Each polarization device may comprise, for example, a pair of electrodes that are positioned on opposite sides of a ring-shaped dielectric radiator. The dielectric radiator may be a continuous dielectric element, and the electrode pairs may be positioned side-by-side on inner and outer sides thereof. Each pair of electrodes and the portion of the dielectric radiator therebetween forms a “polarization element” of the polarization current antenna. 
     The above-described polarization current antenna may operate as follows. When a voltage is applied across one of the electrode pairs, an electric field is generated across the portion of the dielectric radiator therebetween. The electric field generates a displacement current within the dielectric radiator. This displacement current may be referred to as a “volume polarization current” because the current is generated by polarizing the portion of the dielectric material that is between the electrode pair throughout its volume. The generated volume polarization current emits electromagnetic radiation. A volume polarization current distribution pattern may be generated in the dielectric radiator by applying different voltages across multiple of the electrode pairs. Moreover, this volume polarization current distribution pattern may be caused to propagate within the dielectric radiator by appropriate sequencing of the energization of the electrode pairs. One example of a moving volume polarization current distribution pattern is a polarization current wave such as, for example, a sinusoidal polarization current wave that propagates through the dielectric radiator. This polarization current wave can be made to propagate through the dielectric radiator in a direction orthogonal to a vector extending between the electrodes of an electrode pair. Polarization current antennas that have dielectric radiators that are driven by individual amplifiers are known in the art. See U.S. Pat. No. 8,125,385, titled “Apparatus and Methods for Phase Fronts Based on Superluminal Polarization Current,” filed Aug. 13, 2008, which is incorporated herein by reference. Polarization current antennas that are driven by passive feed networks are also known in the art. See International Patent Publication No. WO/2014/100008, which is also incorporated herein by reference. Polarization current antennas differ from conventional antennas in that their emission of electromagnetic radiation arises from a polarization current rather than a conduction or convection electric current. 
     Polarization current antennas that generate polarization current waves that move faster than the speed of light in a vacuum have been experimentally realized. One example of such a polarization current antenna that has already been constructed and tested functions by generating a rotating polarization current wave in a dielectric radiator that is implemented as a ring-shaped block of dielectric material. By phase-controlled excitation of the voltages that are applied to electrodes that surround the dielectric radiator, a volume polarization current can be generated that has a moving distribution pattern (i.e., a polarization current wave that travels along the dielectric radiator) that changes faster than the speed of light and exhibits centripetal acceleration. See, e.g., U.S. Patent Publication No. 2006/0192504 (“the &#39;504 publication”); see also, U.S. patent application Ser. No. 13/368,200, titled “Superluminal Antenna” filed on Feb. 7, 2012, the disclosures of each of which are incorporated herein by reference. It should be noted that while the polarization current wave travels faster than the speed of light, the movements of the underlying charged particles that create the polarization current wave are subluminal. 
       FIG. 1  is a perspective view of the polarization current antenna  1  that is disclosed in the &#39;504 publication. As shown in  FIG. 1 , the polarization current antenna  1  includes a ring-shaped dielectric radiator  2  that has a plurality of inner electrodes  4  that are disposed on an inner surface of the ring-shaped dielectric radiator  2  and a plurality of outer electrodes  6  that are disposed on an outer surface of the ring-shaped dielectric radiator  2 . The ring-shaped dielectric radiator  2  circles an axis of rotation z. The polarization current antenna  1  of  FIG. 1  produces tightly-focused packets of electromagnetic radiation that are fundamentally different from the emissions of conventional antennas. 
     Polarization current antennas that generate polarization current waves that move faster than the speed of light can make contributions at multiple “retarded times” to a signal received instantaneously at a location remote from the polarization current antenna. The location where the electromagnetic radiation is received may be referred to herein as an “observation point,” and each “retarded time” refers to the earlier time at which a specific portion of the electromagnetic radiation that is received at the observation point at the observation time was generated by the volume polarization current. The contributions to the electromagnetic radiation made by the volume elements of the polarization current that approach the observation point, along the radiation direction, with the speed of light and zero acceleration at the retarded time, may coalesce and give rise to a focusing of the received waves in the time domain. In other words, waves of electromagnetic radiation that were generated by a volume element of the polarization current at different points in time can arrive at the same time at the observation point. The interval of time during which a particular set of electromagnetic waves is received at the observation point is considerably shorter than the interval of time during which the same set of electromagnetic waves is emitted by the polarization current antenna. As a result, part of the electromagnetic radiation emitted by the polarization current antenna possesses an intensity that decays non-spherically with a distance d from the antenna as 1/d α  with 1&lt;α&lt;2 rather than as the conventional inverse square law, 1/d 2 . This does not contravene the physical law of conservation of energy. The constructively interfering waves from the particular set of volume elements of the polarization current that are responsible for the non-spherically decaying signal at a given observation point constitute a radiation beam for which the time-averaged value of the temporal rate of change of energy density is always negative. For this non-spherically decaying radiation, the flux of energy into a closed region (e.g., into the volume bounded by two large spheres centered on the source) is smaller than the flux of energy out of it because the amount of energy contained within the region decreases with time. (The area subtended by the beam increases as d 2 , so that the flux of energy increases with distance as d 2−α  across all cross sections of the beam.) In that it consists of caustics and so is constantly dispersed and reconstructed out of other electromagnetic waves, the beam in question has temporal characteristics radically different from those of a conventional beam of electromagnetic radiation. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         FIG. 1  is a schematic perspective view of a known polarization current antenna. 
         FIGS. 2 and 3  are schematic side views of a device that includes a dielectric radiator, a ground plane and a single upper electrode that illustrate how a volume polarization current can be induced in a dielectric radiator. 
         FIGS. 4 and 5  are schematic side views of a polarization current antenna that includes a dielectric radiator, a ground plane and a plurality of upper electrodes that illustrate how a volume polarization current distribution pattern can be generated and made to move within the dielectric radiator. 
         FIG. 6  is a graph illustrating the application of discretized sinusoidally varying voltages to the polarization elements of a polarization current antenna at four equally-spaced consecutive time intervals. 
         FIG. 7  is a schematic perspective view of a polarization current antenna according to embodiments of the present invention that has a closed arc or “ring” shape. 
         FIG. 7A  is a schematic perspective view of the dielectric radiator included in the polarization current antenna of  FIG. 7 . 
         FIG. 8  is a schematic perspective view of a polarization current antenna according to further embodiments of the present invention that has an arc shape. 
         FIG. 8A  is a schematic perspective view of the dielectric radiator included in the polarization current antenna of  FIG. 8 . 
         FIG. 9  is a schematic diagram that illustrates the azimuthal beamwidth of the polarization current antenna of  FIG. 8 . 
         FIG. 10  is a graph of the directive gain for a polarization current antenna according to embodiments of the present invention as a function of the polar coordinate of observation points that are at a fixed distance from the antenna. 
         FIG. 10A  is a graph comparing the radiation distribution pattern of  FIG. 10  to the radiation distribution pattern of a stationary source. 
         FIG. 11  is a graph that illustrates the location of the dielectric radiator of a polarization current antenna according to embodiments of the present invention and how the projection of the cusp associated with a selected observation point onto a meridional plane may pass through the dielectric radiator dividing it into two regions with differing radiative properties. 
         FIG. 12  is a graph that translates the data included in the graphs of  FIG. 10  and  FIG. 13  into a polar coordinate system. 
         FIG. 13  is graph of the directive gain for a polarization current antenna according to embodiments of the present invention as a function of the polar coordinate of observation points that are at six different distances from the antenna. 
         FIG. 14  is a schematic diagram illustrating the different components of the radiation emitted by a polarization current antenna according to embodiments of the present invention in one direction. 
         FIG. 15  is logarithmic plot of the radial component of the normalized Poynting vector versus the normalized distance along the generating line of a cone inside the solid angle where the radiation decays non-spherically. 
         FIG. 16  is graph of the exponent α of the distance dependence    P   −α  of the radial component of the Poynting vector over the range    P =10 to    P =10 6  as a function of the polar angle θ P  for values of θ P  in the range 56.4&lt;θ P ≦90°. 
     
    
    
     DETAILED DESCRIPTION 
     Pursuant to embodiments of the present invention, arc-shaped (including ring-shaped) polarization current antennas are provided that emit electromagnetic radiation along or near an equator of the arc. These polarization current antennas may comprise an arc-shaped dielectric radiator and a plurality of polarization devices that together form a plurality of polarization elements. Each polarization device may comprise, for example, a pair of electrodes. The electrodes may, for example, be disposed on top and bottom surfaces of the arc-shaped dielectric radiator to facilitate equatorial (or near-equatorial) emission of electromagnetic radiation by the polarization current antenna. The radius of the arc-shaped dielectric radiator may define a circle that lies in a horizontal plane. The “equator” of the arc lies in this horizontal plane. A vertical axis of rotation z (see  FIG. 1 ) may be defined at the center of the circle defined by the radius. The arc-shaped dielectric radiator extends around this axis of rotation z. 
     The peak emission of the electromagnetic radiation emitted by the polarization elements of the polarization current antenna may be directed at an angle from the horizontal plane that is referred to herein as an “elevation angle.” The phase difference between the oscillations of the elements of the arc-shaped dielectric radiator and various other parameters of the polarization current antenna may be selected based on a center frequency of a signal that is to be transmitted by the polarization current antenna to achieve peak emission at a desired elevation angle. In some embodiments, the desired elevation angle may be an elevation angle of between −10° and 10° with respect to the equatorial plane. In some embodiments, the desired elevation angle may be an elevation angle of substantially zero with respect to the equatorial plane. In some embodiments, the desired elevation angle may be an elevation angle of between −5° and 5° with respect to the equatorial plane. In some embodiments, polarization current antennas having arc-shaped dielectric radiators are provided that are configured so that the polarization current waves generated therein travel at a speed that is less than c within a first portion of the arc-shaped dielectric radiator and at a speed that is greater than or equal to c within a second portion of the arc-shaped dielectric radiator, where c is the speed of light in vacuum. The first portion may be an inner portion of the arc-shaped dielectric radiator and the second portion may be an outer portion of the arc-shaped dielectric radiator. In some embodiments, the polarization current antennas may be configured so that the polarization current wave travels at the speed of between c and 1.02*c within a portion of the dielectric radiator. In each of the above cases this configuration may be designed to result in equatorial or near-equatorial emission. In some embodiments, the polarization current antennas may be configured so that the polarization current wave travels at the speed of light within at least a portion of the dielectric radiator in order to, for example, cause the polarization current antenna to emit radiation equatorially. As shown herein, enhanced emission may be obtained with equatorial and/or near equatorial emission. A height of the arc-shaped dielectric radiator (i.e., a distance that the arc extends in a direction parallel to the axis of rotation z) may be selected to provide a desired elevation beamwidth in some embodiments. 
     The polarization current antennas according to embodiments of the present invention may include a plurality of polarization elements that together form a volume polarization current distribution radiator. Each polarization element may comprise a pair of electrodes (or other polarization device) and an associated segment of a dielectric radiator. In some embodiments, a single continuous dielectric radiator may be used, respective segments of which comprise parts of the individual polarization elements. In other embodiments, the dielectric radiator may comprise a plurality of discrete dielectric elements that together form the dielectric radiator (e.g., each polarization element may have its own discrete dielectric element and these dielectric elements may together form the dielectric radiator). The polarization elements may be arranged in an arc having a radius r about the axis of rotation z. The polarization elements may be oriented such that the dielectric radiator faces outwardly, away from the axis of rotation z, and the electrodes may be placed on top and bottom surfaces of the dielectric radiator. The dielectric radiator has a finite polarization region that is created by selectively applying a voltage to one or more electrodes. In some embodiments, the electrodes are excited such that a polarization current wave propagates along the dielectric radiator at about the speed of light. The polarization current wave propagates from polarization element to polarization element about the axis of rotation z. 
     Thus, pursuant to embodiments of the present invention, polarization current antennas having arc-shaped dielectric radiators are provided which emit a beam of electromagnetic radiation from an outer surface of the arc-shaped dielectric radiator. As the electrodes or other polarization devices may be disposed adjacent the top and bottom surfaces of the arc-shaped dielectric radiator, the electrodes may not block or otherwise interfere with the beam of electromagnetic radiation that is emitted from the outer surface of the arc-shaped dielectric radiator. In some embodiments, these polarization current antennas may be configured so that a polarization current wave is generated in the arc-shaped dielectric radiator that travels at the speed of light within a portion (e.g., the center) of the dielectric radiator. In some embodiments, the polarization current wave may travel subluminally in other (e.g., inner) portions of the dielectric radiator and may travel superluminally in still other (e.g., outer) portions of the dielectric radiator. 
     Before describing various embodiments of the present invention in greater detail, the configuration and operation of polarization current antennas will first be described in more detail. 
     In a conventional phased array antenna, a plurality of dipole, patch or other radiating elements are used to transmit and receive radio frequency (RF) signals. In these conventional antennas, each radiating element may be considered a point source of electromagnetic radiation. The radiating elements may be separated by a distance that is proportional to the wavelength of an RF signal that is emitted by the radiating elements. The electromagnetic radiation is generated by surface currents, such as surface currents generated on the dipole or patch radiating elements. 
     In contrast to such point-source electromagnetic radiation sources, the polarization current antennas according to embodiments of the present invention produce a continuous, moving source of electromagnetic radiation that is distributed over a volume. In some embodiments, this source may be a polarization current wave that flows through a dielectric radiator. 
     The production and propagation of electromagnetic radiation in the polarization current antennas according to embodiments of the present invention is described by the following two of Maxwell equations: 
       ∇× E=∂B/∂t   (1)
 
       ∇× H=J   free +∈ 0   ∂E/∂t+∂P/∂t   (2)
 
     In Equations (1) and (2), H is the magnetic field strength, B is the magnetic induction, P is polarization, and E is the electric field, and all terms are in SI units. The (coupled) terms in B, E and H of Equations (1) and (2) describe the propagation of electromagnetic radiation. The generation of electromagnetic radiation is encompassed by the source terms J free  (the current density of free charges) and ∂P/∂t (the polarization current density). An oscillating J free  is the basis of conventional radio transmission. The charged particles that make up J free  have finite rest mass, and therefore cannot move with a speed that exceeds the speed of light in vacuo. Practical polarization current antennas employ a volume polarization current to generate electromagnetic radiation, which is represented by the volume polarization current density ∂P/∂t. 
     The principles of such polarization current antennas will now be described with reference to  FIGS. 2-5 .  FIGS. 2 and 3  schematically illustrate a device  10  that includes a dielectric radiator  12 . An electrode  14  is provided on one side of the dielectric radiator  12  and a ground plane  16  is provided on the other (opposite) side of the dielectric radiator  12 . The dielectric radiator  12  is an electrical insulator that may be polarized by applying an electric field thereto. When the electric field is applied to the dielectric radiator  12 , electric charges in the portion of the dielectric radiator  12  effected by the electrical field shift from their average equilibrium positions causing polarization in this portion of the dielectric radiator  12 . When the dielectric radiator  12  is polarized, positive charges are displaced in the same direction as that of the electric field and negative charges shift in the opposite direction away from the electric field. 
     In the example of  FIG. 2 , no electric field is applied across the dielectric radiator  12  via the electrode  14  and ground plane  16 , so the charges in the dielectric radiator  12  are shown as being randomly distributed to indicate that they are in their average equilibrium positions. In the example of  FIG. 3 , a voltage has been applied to the electrode  14  to generate an electric field across the dielectric radiator  12 . As shown in  FIG. 3 , in response to this voltage, the positive and negative charges shift slightly from their average equilibrium positions (see  FIG. 2 ) to move in opposite directions with the negative charges shifting towards the applied voltage and the positive charges shifting away from the applied voltage. A finite polarization P has therefore been induced in the dielectric radiator  12 . A changing state of polarization P corresponds to charge movement, and so is equivalent to current. Thus, changes to the state of the polarization P of the dielectric radiator  12 —such as the change shown between  FIGS. 2 and 3 —may generate electromagnetic radiation. 
       FIGS. 4 and 5  illustrate a portion of a polarization current antenna  100 . The polarization current antenna  100  is similar to the device  10  of  FIGS. 2 and 3 , and includes a dielectric radiator  112  and a ground plane  116  that may be identical to the dielectric radiator  12  and the ground plane  16 , respectively, of the device  10  of  FIGS. 2-3 . The polarization current antenna  100  of  FIGS. 4 and 5  differs from the device  10  of  FIGS. 2-3  in that the common electrode  14  of the device  10  of  FIGS. 2-3  has been replaced with a plurality of smaller, individual electrodes labeled  114 - 1  through  114 - 11  (which are collectively referred to herein as the electrodes  114 ) that are arranged in a side-by-side relationship. Each electrode  114 , in conjunction with a portion of the dielectric radiator  112  and a portion of the ground plane  116 , forms a polarization element  118  of the polarization current antenna  100 . One such polarization element  118  is shown in the dashed box in  FIG. 4 . The polarization current antenna  100  has a total of twenty polarization elements  118 , but only the first eleven polarization elements  118  are shown to simplify the drawings. 
     As a plurality of separate electrodes  114  are provided in the polarization current antenna  100  of  FIGS. 4-5 , a spatially-varying electric field may be applied across the dielectric radiator  112  by simultaneously applying different voltages to different ones of the electrodes  114 . Moreover, the distribution pattern of the electric field can be made to move by, for example, applying voltages in sequence to the electrodes  114 . In particular, if the distribution pattern of the spatially-varying electric field is made to move, then the polarized region moves with it; thereby producing a traveling “wave” of P that moves along the dielectric radiator  112  (and also, by virtue of the time dependence imposed by movement, a traveling wave of ∂P/∂t). As noted above, this traveling “wave” of P may be referred to herein as a “polarization current wave.” This polarization current wave generates electromagnetic radiation as it moves along the dielectric radiator  112 . While the description that follows will primarily focus on polarization current waves that move through a dielectric radiator, it will be appreciated that volume polarization current distribution patterns other than polarization current waves may be made to move through the dielectric radiator. Embodiments of the present invention encompass such moving but non-wave-like volume polarization current distribution patterns. 
       FIGS. 4 and 5  illustrate how a polarization current wave may be created and made to move through the dielectric radiator  112 . In particular,  FIG. 4  illustrates the position of a polarized region of the dielectric radiator  112  at time t 1 . As shown in  FIG. 4 , at time t 1  electrodes  114 - 1  through  114 - 3  and  114 - 8  through  114 - 11  are not energized (shown by the “0” above the individual electrodes  114 ), while a voltage is applied to electrodes  114 - 4  through  114 - 7  (shown by the “+” above the individual electrodes  114 ). In this state, an electric field exists between electrodes  114 - 4  through  114 - 7  and the ground plane  116 , and therefore a polarized region also exists in the dielectric radiator  112  adjacent to electrodes  114 - 4  through  114 - 7 . The state of the antenna  100  at time t 2  is illustrated in  FIG. 5 . At time t 2 , the voltage is removed from electrode  114 - 4  and a voltage is applied to electrode  114 - 8 . The electric field, and therefore the polarized region, has moved one electrode  114  to the right. In other words,  FIGS. 4-5  illustrate the movement of a polarization current wave from the region of dielectric radiator  112  underneath electrodes  114 - 4  through  114 - 7  (see  FIG. 4 ) to the region of dielectric radiator  112  underneath electrodes  114 - 5  through  114 - 8  (see  FIG. 5 ). Note that this polarization current wave can move arbitrarily fast (Including faster than the speed of light in vacuo) because the polarization current wave is generated by movement of charges in a first direction (i.e., the vertical direction in  FIGS. 4-5 ) while the polarization current wave moves in a second direction that is orthogonal to the first direction (i.e., the horizontal direction in  FIGS. 4-5  as the polarization current wave moves along the dielectric radiator  112 ). Thus, the individual charges do not themselves move faster than the speed of light, while the polarization current wave may be made to move faster than the speed of light. As a simple example, this phenomenon is akin to a “wave” that is created by fans standing up and sitting down in a stadium during an athletic event. The speed at which the wave moves through the stadium is a function of a number of factors, only one of which is the speed at which the individual spectators stand up and sit down, and hence the speed of the wave can be made to be faster than the speed at which the individuals creating the wave move. 
     The polarization current antenna  100  may be used, for example, to transmit an information signal. Typically, radio frequency communications involves modulating an information signal onto a carrier signal, where the carrier signal is typically a sinusoidal signal having a frequency in a desired frequency band of operation. By way of example, the various different cellular communications networks have fixed frequency bands of operation in which the signals that are transmitted between base stations and mobile terminals are transmitted at frequencies within the specified frequency band. One way to use the polarization current antenna  100  to transmit an information signal is to modulate the information signal onto a sinusoidal waveform that oscillates at a desired radio frequency (“RF”) such as, for example, 2.5 GHz, and to use this modulated RF signal to excite the electrodes of the polarization current antenna  100 . This can be accomplished using, for example, a passive corporate feed network in some embodiments. The corporate feed network is used to divide the modulated RF signal into a plurality of sub-components with differing phases. The number of sub-components may be equal to the number of polarization elements  118  included in the polarization current antenna  100 , so that a sub-component of the modulated RF signal is applied to, for example, each electrode  114 . In some embodiments, the magnitude of each sub-component of the RF signal may be proportional to that of the modulated RF signal to be transmitted. 
     With this approach, at any given point in time, a sub-component of the modulated RF signal is applied to each of the polarization elements  118 . At a first point in time t 1 , the applied modulated RF signal will have a given amplitude. However, the sub-components of the modulated RF signal that are applied to different polarization elements  118  have different phase offsets, and hence their magnitude will vary as the modulated RF signal varies with time. At a subsequent point in time t 2 , the magnitude of the modulated RF signal at any given polarization element  118  will have changed in a known manner based on the frequency of the signal and the time difference t 2 −t 1 . This is shown graphically in  FIG. 6 . 
     In particular,  FIG. 6  illustrates the voltages V j  that may be applied to the upper electrodes  114  of the polarization current antenna  100 . The lower electrode  116  may be connected to a constant reference voltage such as a ground voltage. The four separate curves in  FIG. 6  illustrate the voltages V j  applied to the upper electrodes  114  of the twenty polarization elements  118  (note that  FIGS. 4 and 5  only illustrate the first eleven polarization elements  118  to simplify the drawings) at four equally-spaced consecutive times (t 1 &lt;t 2 &lt;t 3 &lt;t 4 ). The polarization elements  118  are identified in  FIG. 6  according to the azimuthal coordinate φ j  (j=1, 2, 3, . . . , 20) of the center of each polarization element  118 . Thus, in  FIG. 6 , the horizontal axis corresponds to the position of each of the twenty polarization elements  118  along the dielectric radiator  112  (which extends along a y-axis, as shown in  FIGS. 4-5 ) and the vertical axis shows the voltages V j  that are applied to the twenty polarization elements  118 . The four curves show the respective voltages applied to the twenty polarization elements  118  at the four different points in time t 1  through t 4 . As can be seen in  FIG. 6 , over time a sinusoidally varying excitation signal is applied to the polarization elements  118 . 
     In  FIG. 6 , V j ∝ cos [ω(t−jΔt)] where Δt is the time difference between the instants at which the oscillatory voltages applied to adjacent polarization elements attain their maximum amplitude. Accordingly, the constant phase difference ωΔt between the oscillations of adjacent polarization elements  118  results in a sinusoidal polarization current wave that propagates to the right through the dielectric radiator  112  with the speed Δl/Δt, where Δl is the distance between the centers of adjacent polarization elements  118 . While sinusoidal curves are illustrated in the example of  FIG. 6 , it will be appreciated that the embodiments of the present invention discussed herein are not limited to sinusoidal curves. In particular, other waveforms may be employed to achieve any desired polarization current wave. Additionally, while a polarization current wave is one type of volume polarization current distribution pattern that may be made to propagate through the dielectric radiator  112 , it will be appreciated that embodiments of the present invention are not limited to volume polarization current distribution patterns that are waves that are made to propagate through the dielectric radiator  112 . 
     It will also be appreciated that polarization devices other than a series of upper electrodes  114  and a ground plane  116  may be used to apply an electric field across a portion of the dielectric radiator  112 . For example, in other embodiments, the ground plane  116  may be replaced with a plurality of individual lower electrodes which may or may not be connected to ground. Note that herein the term “electrode” is used broadly to encompass the ground plane  116  as well as upper and lower electrodes. In still other embodiments, structures other than electrodes may be used to polarize the dielectric radiator  112 . The polarization devices are preferably sized such that a plurality of polarization devices may be located closely adjacent to each other so that, when excited in sequence, the polarization devices apply a stepped approximation of a continuous electric field distribution to the dielectric radiator  112  as shown in the example of  FIG. 6  above. 
     Various embodiments of the present invention will now be discussed in greater detail with respect to  FIGS. 7-9 . 
     A first arc-shaped equatorially radiating polarization current antenna  200  according to embodiments of the present invention is illustrated in  FIG. 7 . The polarization current antenna  200  has a dielectric radiator  212  that extends a full 360 degrees to form a ring. It should be noted that herein a “ring-shaped” or “circular” structure is considered to be an “arc-shaped” structure where the arc is a closed arc that extends for a full 360 degrees. 
     Referring to  FIG. 7 , the polarization current antenna  200  includes a dielectric radiator  212 , a plurality of upper electrodes  214  and a plurality of lower electrodes  216 . The dielectric radiator  212  is arranged in an arc about a vertical axis of rotation z, and extends fully around the axis of rotation z in the example of  FIG. 7 .  FIG. 7A  is a schematic perspective view of the dielectric radiator  212  included in the antenna  200  of  FIG. 7 . As shown in  FIG. 7A , the dielectric radiator  212  has an outer surface  220 , an inner surface  222 , a top surface  224 , and a bottom surface  226 . The outer surface  220  may comprise the front surface of the polarization current antenna  200  and may be the surface of the polarization current antenna  200  through which electromagnetic radiation is emitted. The upper electrodes  214  may be on the top surface  224  of dielectric radiator  212  and the lower electrodes  216  may be on the bottom surface  226  of dielectric radiator  212 . The upper and lower electrodes  214 ,  216  may be vertically aligned so as to be arranged in pairs. Each pair of an upper electrode  214  and a lower electrode  216  and the portion of the dielectric radiator  212  disposed therebetween forms a respective polarization element  218 . The electrodes  214 ,  216  may be replaced with other polarization devices in other embodiments. 
     The dielectric radiator  212  in the example of  FIG. 7  comprises a continuous dielectric block that is formed in the shape of a ring. As shown in  FIGS. 7 and 7A , the dielectric radiator  212  has a mean radius r 0 , a thickness Δr, and a height Δz. Each upper electrode  214  has the same angular width around the arc, and the upper electrodes  214  are spaced apart by uniform amounts. In particular, the center of each upper electrode  214  is spaced apart from the centers of adjacent upper electrodes  214  by a constant (arc-length) distance Δl. Likewise, each lower electrode  216  has the same angular width around the arc, and the lower electrodes  216  are spaced apart by uniform amounts so that the center of each lower electrode  216  is spaced apart from the centers of adjacent lower electrodes  216  by the constant distance Δl. While the dielectric radiator  212  is depicted as a continuous block in  FIG. 7 , it will be appreciated that a plurality of discrete dielectric radiators  212  may be used instead in other embodiments, which may or may not touch one another. As the polarization current wave that is generated when the dielectric radiator  212  is polarized in sequence moves from a first polarization element  218  to a second polarization element  218 , it travels in an annular strip with radii r 0 +/−½Δr about the axis of rotation z (note that the mean radius r 0  is measured to the center of the dielectric radiator  212 ). The direction in which the polarization current antenna  200  emits electromagnetic radiation is controlled by the velocity of the polarization current wave, as will be described in further detail below. 
     In the ring-shaped polarization current antenna  200  of  FIG. 7 , the polarization current wave within the dielectric radiator  212  rotates with the angular frequency: 
     
       
         
           
             
               
                 
                   ω 
                   = 
                   
                     
                       2 
                        
                       π 
                        
                       
                           
                       
                        
                       v 
                     
                     m 
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     where ν is the frequency of oscillations of the applied voltages (e.g., 2.5 GHz) and m is the length around the ring-shaped dielectric radiator  212  of the polarization current antenna  200  in terms of the number of wavelengths L p  of the polarization current wave. Referring again to  FIG. 6 , the length around the ring-shaped dielectric radiator  212  in terms of the number m of wavelengths L p  of the polarization current wave refers to how many wavelengths L p  the polarization current wave passes through in passing through the complete arc of the dielectric radiator  212  once. In the example of  FIG. 6 , the wavelength L p  of the polarization current wave corresponds to twenty polarization elements, and hence m is equal to one. If, for example, the polarization current antenna associated with the graph of  FIG. 6  instead included sixty polarization elements, then m would be equal to three. Improved performance may be possible in some cases if m is an integer, although embodiments of the present invention are not limited to polarization current antennas for which m is an integer value. 
     The speed of the polarization current wave (which acts as the source of the electromagnetic radiation emitted by the polarization current antenna  200 ) has the value u=rω at a radius r within the ring-shaped dielectric radiator  212 . The non-spherically decaying electromagnetic radiation (i.e., the electromagnetic radiation that does not decay with distance d from the source according to the inverse square law 1/d 2 ) that is generated by this polarization current wave at the radius r within the dielectric radiator  212  is emitted at the polar angles: 
       θ P =arcsin( c/u ) and θ P =π−arcsin( c/u )  (4)
 
     above and below the equatorial plane θ P =π/2, where θ P  denotes the angle between the axis of rotation z and the direction at which the electromagnetic radiation is emitted and c is the speed of light in vacuum. The emitted waves constructively interfere to form cusps along the above two values of θ P  because the volume elements of the distribution pattern of the polarization current that move with the superluminal speed u approach a far-field observer located at these values of θ P  with the speed of light and zero acceleration at the retarded time. 
     Pursuant to embodiments of the present invention, the electric field may be applied to the arc-shaped dielectric radiator  212  in such a way that u equals c within the radial thickness Δr of the arc-shaped dielectric radiator  212 . In some embodiments, the electric field is applied to the dielectric radiator  212  such that u equals c at the center of the dielectric radiator  212 . This results in the emission of electromagnetic radiation into the plane of rotation θ P =π/2. Note that when u equals c at the center of the dielectric radiator  212 , then the polarization current wave will move subluminally at the inner radius of the arc-shaped dielectric radiator  212  (i.e., at a speed that is less than the speed of light) while the polarization current wave will move superluminally at the outer radius of the arc-shaped dielectric radiator  212  (i.e., at a speed that is faster than the speed of light). 
     There is a strong beam of electromagnetic radiation whose angular width in a direction that is normal to the plane of rotation is given by: 
       Δθ P =arctan(Δ z/R   P )  (5)
 
     where Δz is the thickness of the arc-shaped dielectric radiator  212  along the direction parallel to the axis of rotation (i.e., perpendicular to the circle defined by the radius of the arc-shaped dielectric radiator  212 ) and R P  is the distance of the observation point P from the center of the arc-shaped dielectric radiator  212 . The intensity of a portion of the emitted electromagnetic radiation diminishes as 1/R P   α  with 1&lt;α&lt;2 as the distance R P  from the antenna  200  increases. The direction of emission of the electromagnetic radiation within the equatorial plane θ P =712 is everywhere tangent to the arc-shaped dielectric radiator  212 . The radiation will emit in a full 360 degree circle in the example of  FIG. 7  as the dielectric radiator  212  extends through a full 360 degrees. 
     The polarization current antenna  200  may be designed so that the electric field that is applied to the arc-shaped dielectric radiator  212  will generate a polarization current wave that has a velocity u that equals c within the radial thickness Δr of the dielectric radiator  212  for a given frequency ν of an input signal by selecting the mean radius r 0  of the dielectric radiator, the number of polarization elements N and the time difference Δt between the instants at which the input signals are applied to adjacent polarization elements  218  attain their maximum amplitudes. In other words, an antenna designer may select the following four parameters for the polarization current antenna having a circular dielectric radiator:
         N is the total number of polarization elements included in the antenna;   r 0  is the mean radius of the circular dielectric radiator;   Δt=the time difference between the instants at which the input signals applied to adjacent polarization elements attain maximum amplitude; and   ν=the frequency of the input signal (i.e., the signal to be transmitted) that is applied to the polarization elements.       

     Based on the above parameters, the following parameters of the polarization current antenna  200  may be determined:
         Δl=the center-to-center distance between adjacent polarization elements=(2π)r 0 /N;   ΔΦ=360*ν*Δt=phase difference between the oscillations of adjacent polarization elements in degrees;   m=the number of wavelengths of the polarization current wave that fit around the circumference of the circular dielectric radiator=ΔΦ*N/360; and   (1/c)*(Δl/Δt)=(360*ν*Δl)/(c*ΔΦ)=propagation speed of the polarization current wave in units of the speed of light in vacuo (c).       

     Thus, the values of N, r 0  and Δt may be selected for a given ν in order to design the polarization current antenna so that it will generate a polarization current wave that has a desired propagation speed through the circular dielectric radiator  212  such as, for example, a propagation speed equal to the speed of light in vacuo (c). 
     A second arc-shaped polarization current antenna  300  according to embodiments of the present invention is illustrated in  FIG. 8 . The polarization current antenna  300  includes a dielectric radiator  312  that extends in an arc of φ=120 degrees, or more generally, of 0&lt;φ&lt;360 degrees. Thus, the polarization current antenna  300  differs from the polarization current antenna  200  of  FIGS. 7-7A  in that the polarization current antenna  300  does not extend through a full circle. The polarization current antenna  300  further includes a plurality of upper electrodes  314  and a plurality of lower electrodes  316 . The dielectric radiator  312  is arranged in an arc about a vertical axis of rotation z, but in this case extends around only about 120 degrees of the axis of rotation z.  FIG. 8A  is a schematic perspective view of the dielectric radiator  312  included in the antenna  300  of  FIG. 8 . As shown in  FIG. 8A , the dielectric radiator  312  has an outer surface  320 , an inner surface  322 , a top surface  324 , a bottom surface  326  and a pair of end surfaces  328 . Electromagnetic radiation is emitted through the outer surface  320 . The upper and lower electrodes  314 ,  316  may be vertically aligned so as to be arranged in pairs to form respective polarization elements  318 . The electrodes  314 ,  316  may be replaced with other polarization devices in other embodiments. The dielectric radiator  312  comprises a continuous dielectric block again having a mean radius r 0 , a thickness Δr, and a height Δz. Each upper electrode  314  and each lower electrode  316  have the same angular width and are spaced apart by uniform amounts. The centers of each upper and lower electrode  314 ,  316  are spaced apart from the centers of respective adjacent upper and lower electrodes  314 ,  316  by a constant distance Δl. As the polarization current wave moves from polarization element  318  to polarization element  318  it travels in an annular strip with radii r 0 +/−½Δr about the axis of rotation z. As with the polarization current antenna  200  of  FIG. 7  discussed above, the direction in which electromagnetic radiation is emitted from the polarization current antenna  300  is controlled by the velocity of the polarization current wave. 
     In the arc-shaped polarization current antenna  300  of  FIG. 8 , the polarization current wave rotates within the dielectric radiator  312  with the angular frequency shown by Equation (3) above and the speed of the polarization current wave has the value u=rω at a radius r within the arc-shaped dielectric radiator  312 . The non-spherically decaying electromagnetic radiation generated by the portion of the polarization current wave that rotates through the dielectric radiator  312  at the radius r is emitted at the polar angles θ P  given by Equation (4) above. 
     As with the polarization current antenna of  FIG. 7  discussed above, the electric field may be applied to the arc-shaped dielectric radiator  312  in such a way that u equals c within the radial thickness Δr of the arc-shaped dielectric radiator  312  so that the electromagnetic radiation is emitted into the plane of rotation θ P =π/2. The beamwidth of the stronger portion of the non-spherically decaying electromagnetic radiation that propagates into the plane of rotation is given by Equation (5) above. The direction of emission of the electromagnetic radiation within the equatorial plane θ P =π/2 is everywhere tangent to the arc-shaped dielectric radiator  312 . Referring to  FIG. 9 , it can be seen that the azimuthal beamwidth Δφ P  of the emitted electromagnetic radiation has the same value as the angle subtended by the arc-shaped dielectric radiator  312 . Thus, in the present case, the azimuth beamwidth will be about 120 degrees since the arc-shaped dielectric radiator  312  extends through an arc of about 120 degrees. 
     Pursuant to some embodiments of the present invention, the polarization current antennas of  FIGS. 7-8  may have the following characteristics:
         The polarization current antennas  200 ,  300  may be designed so that the velocity of the polarization current wave is equal to the speed of light in vacuum within at least a portion of the dielectric radiator  212 ,  312  (for example, the polarization current wave may travel at a speed that is less than or equal to the speed of light in vacuum along an inner radius of the dielectric radiator and may travel at a speed that is greater than the speed of light along an outer radius of the dielectric radiator);   The polarization current antennas  200 ,  300  may include at least five polarization elements per wavelength of the polarization current wave; and   The arc defined by the arc-shaped dielectric radiator  212 ,  312  extends over a distance of at least 2 wavelengths of the polarization current wave (i.e., m≧2), and, in some embodiments, the distance that the arc extends may be substantially equal to an integral multiple of such wavelengths.       

     TABLE 1 below sets forth various example embodiments of arc-shaped polarization current antennas according to embodiments of the present invention that may be similar or identical to the polarization current antenna  300  of  FIG. 8 . 
     
       
         
           
               
               
               
               
               
               
               
               
               
             
               
                 TABLE 1 
               
               
                   
               
               
                   
                   
                   
                   
                   
                   
                 Inner 
                 Mean 
                 Mean Speed 
               
               
                 Frequency 
                 ΔΦ 
                 Δl 
                   
                 Angle of Arc 
                   
                 Radius 
                 Radius 
                 (in units 
               
               
                 (GHz) 
                 (deg) 
                 (cm) 
                 m 
                 φ (deg) 
                 N 
                 (cm) 
                 r 0  (cm) 
                 of c) 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 2.5 
                 27.7 
                 1.015 
                 10 
                 360 
                 130 
                 19 
                 21 
                 1.1 
               
               
                 2.5 
                 13.85 
                 0.507 
                 5 
                 360 
                 130 
                 10.4 
                 11.5 
                 1.1 
               
               
                 1.25 
                 13.85 
                 1.015 
                 5 
                 360 
                 130 
                 19 
                 21 
                 1.1 
               
               
                 5 
                 27.7 
                 0.507 
                 10 
                 360 
                 130 
                 10.4 
                 11.5 
                 1.1 
               
               
                 2 
                 22.5 
                 1 
                 5 
                 360 
                 80 
                 11.9 
                 12.7 
                 1.066 
               
               
                 1 
                 11.25 
                 1 
                 5 
                 120 
                 160 
                 71.6 
                 76.4 
                 1.066 
               
               
                 2 
                 11.25 
                 0.5 
                 5 
                 120 
                 160 
                 35.8 
                 38.2 
                 1.066 
               
               
                 1.75 
                 20 
                 1 
                 10 
                 120 
                 180 
                 81.8 
                 85.9 
                 1.05 
               
               
                 2.5 
                 30 
                 1 
                 20 
                 120 
                 240 
                 113 
                 114.6 
                 1 
               
               
                   
               
            
           
         
       
     
     The polarization current antennas according to some embodiments of the present invention may have the electrodes disposed adjacent the top and bottom surfaces of the arc-shaped dielectric radiator. The top and bottom electrodes may lie in first and second parallel planes. The first plane defined by the upper electrodes (e.g., electrodes  314 ) and the second plane defined by the lower electrodes (e.g., electrodes  316 ) are not only parallel to each other, but also are parallel to a third plane that is parallel to the direction of propagation of the polarization current wave. In contrast, some known circular polarization current antennas position the electrodes on the inner and outer surfaces of the ring-shaped dielectric radiator. In these known polarization current antennas, the direction of a vector perpendicular to the exposed face of the dielectric radiator is parallel to the axis of rotation of the displacement current. 
     Referring again to  FIG. 8 , the polarization current antenna  300  only extends through an angle φ of about 120 degrees. The polarization elements  318  thereof are excited in sequence starting, for example, with the first polarization element on a first end  328 - 1  of the polarization current antenna  300 . Each polarization element  318  is excited in turn with a constant time delay interval. Eventually the last polarization element  318  on the second end  328 - 2  of the polarization current antenna  300  will be reached. When this occurs, the first polarization element  318  is then excited at the constant delay interval in turn as if it were at the next polarization element  318  in sequence. 
     The polarization current antennas according to embodiments of the present invention may include a feed network that is used to energize the polarization devices of the polarization elements progressively with a constant time delay interval (i.e., the time period between when a first polarization element is energized and a second, adjacent polarization element is energized is constant across all polarization elements). When such a feed network is used, the angular speed of the polarization current wave will be constant. However, even though the speed of the polarization current wave is constant, by virtue of the geometry of the antenna, when such a feed network is applied to a curved or circular array of polarization elements, the rotating volume elements of the polarization current wave are centripetally accelerated. The polarization elements  318  may be continuously excited in this fashion. 
     Accordingly, pursuant to embodiments of the present invention, polarization current antennas having arc-shaped dielectric radiators are provided that may be designed to emit electromagnetic radiation equatorially or near equatorially. In some embodiments, the polarization current antennas may be designed according to the following parameters:
         φ=the angular extent of the arc-shaped dielectric radiator in degrees;   Δl=the center-to-center distance between adjacent polarization elements;   Δt=the time difference between the instants at which the input signal applied to adjacent polarization elements attains maximum amplitude;   ν=the frequency of the input signal (i.e., the signal to be transmitted) that is applied to the polarization elements;   ΔD=360*ν*Δt=phase difference between the oscillations of adjacent polarization elements in degrees;   (1/c)*(Δl/Δt)=(360*ν*Δl)/(c*ΔΦ)=propagation speed of the polarization current wave in units of the speed of light in vacuo (c);   m=the number of wavelengths of the polarization current wave that fit around the circumference of the arc-shaped dielectric radiator;   N=360*m/ΔΦ=the total number of polarization elements in the antenna; and   r 0 =360*N*Δl/2π*φ=the mean radius of the arc-shaped dielectric radiator.       

     In some embodiments, the polarization current antennas may be configured so that the polarization current wave travels at the speed of between c and 1.02*c within a portion of the dielectric radiator, where c is the speed of light in vacuo (c). This may result in equatorial or near-equatorial emission. In some embodiments, the polarization current antennas may be configured so that the polarization current wave travels at the speed of light within at least a portion of the dielectric radiator in order to, for example, cause the polarization current antenna to emit radiation equatorially. 
     Enhanced performance may be achieved when the polarization current antennas described herein are configured for equatorial emission or near equatorial emission. This is because an additional mechanism of focusing comes into play if there are volume elements of the distribution pattern of the polarization current whose speeds u are close to the speed of light c. As u approaches the value c, the two polar angles appearing in Equation (4) both approach the value 90 degrees, i.e., both approach the equatorial plane. As a result, an observer whose z coordinate is small enough to match the z coordinates of the source elements that approach the observer with the speed of light and zero acceleration receives waves that are further focused by the coalescence of the two arms of the cusps described in Equation (4). A higher degree of focusing of the received waves in turn implies an enhanced intensity for the resulting radiation. 
     A computational program such as Mathematica may be used to solve Maxwell&#39;s equations to determine the radiation field that is generated by an arc-shaped polarization current antenna according to embodiments of the present invention. Maxwell&#39;s equations were solved to determine the radiation field emitted by a ring-shaped polarization current antenna. 
     In performing the above-described computational analysis, it was assumed that the polarization current antenna had the general design of the polarization current antenna  200  that is described above with reference to  FIGS. 7 and 7A . Accordingly, the description below will use the reference numerals shown in  FIGS. 7 and 7A  to describe this polarization current antenna  200 . It will be appreciated that minor variations may exist between the antenna  200  pictured in  FIGS. 7 and 7A  and the exact antenna design used in the computation analysis such as, for example, the number of upper and lower electrodes  214 ,  216 . 
     The polarization current antenna  200  that was modelled in the computational analysis had a ring-shaped dielectric radiator  212  with the following parameters: 
     Average radius (r 0 )=21 cm; 
     Radial width (Δr)=3.8 cm; 
     Height (Δz)=3.8 cm. 
     The (arc-length) distance (Δl) between the centers of adjacent upper electrodes  214  was assumed to be Δl=1.015 cm. The circumference of the above-described antenna  200  is 2πr=131.945 cm. Since each upper electrode  214  extends for a distance of 1.015 cm, the antenna  200  has 130 electrode pairs. 
     The polarization current flows in the dielectric radiator  212  in a direction that is parallel to the axis of rotation z and was assumed to have an oscillation frequency of ν=2.5 GHz. The density of the polarization current was assumed to be 2.5 amps/m 2 . The resultant polarization current wave that would be generated in the dielectric radiator  212  of antenna  200  has a sinusoidal shape, and this polarization current wave travels through ten wavelengths when travelling once around the full circumference of the dielectric radiator  212  (i.e., m=10). The polarization current wave rotates with an angular frequency of ω=1.57×10 9  radians/second. The above-described physical parameters for the dielectric radiator  212 , the electrodes  214  and the oscillation frequency ν for the polarization current were selected so that the speed u=rω of the polarization current wave is equal to the speed of light in a vacuum (c) at the inner radius of the dielectric radiator  212  and the speed of the polarization current wave is 1.2*c at the outer radius of the dielectric radiator  212 . These speeds u may be experimentally realized in the above-described polarization current antenna  200  by setting the phase difference ΔΦ between the oscillations of the voltages on adjacent pairs of electrodes  214 ,  216  equal to 27.7°. 
     The time-averaged value of the component of the Poynting vector along the radiation direction was solved for the polarization current antenna  200  having the above-described parameters. The time-averaged value of the component of the Poynting vector along the radiation direction represents the power emitted by the polarization current antenna  200  that propagates across a unit area normal to the radiation direction at a given observation point P.  FIG. 10  is a graph of the time-averaged value of the component of the Poynting vector along the radiation direction divided by the average value of the power that propagates across a sphere of radius R P =10 c/ω per unit solid angle as a function of the polar coordinate θ P  of the observation point P (i.e., the location where the power of the emitted radiation is measured), where R P  is the spherical polar coordinate of the observation point P. The parameter c/ω corresponds to the radius of the light cylinder for the polarization current antenna  200 . A light cylinder refers to the cylinder on which the linear speed rω of a rotational motion equals the speed of light c, and the radius thereof may be a convenient unit for expressing the distance to selected observation points P when evaluating the performance of a polarization current antenna. For the polarization current antenna  200  having the above-described parameters, the light cylinder has a radius of c/ω=19.1 cm. Thus,  FIG. 10  shows the directive gain of the antenna  200  at various observation points P that are each at a distance of 1.91 meters from the antenna  200 , as a function of the polar angle θ P  of the observation point P. Note that a logarithmic unit of measurement is used along the vertical axis in  FIG. 10  so that changes in the plotted quantity (i.e., the time-averaged value of the component of the normalized Poynting vector along the radiation direction) are shown in decibels. 
     In  FIG. 10 , the curve formed by the data points (labelled curve  300 ) represents the received power per unit area of the electromagnetic radiation emitted by the polarization current antenna  200 . As should be clear from the discussion above, curve  300  includes received power that is generated by both (1) source elements of antenna  200  (i.e., portions of the polarization current wave) that have a velocity component along the direction the electromagnetic radiation travels to the observation point P that is less than the speed of light in vacuo (c) and by (2) source elements of antenna  200  that have a velocity component along the direction the electromagnetic radiation travels to the observation point P that is greater than or equal to the speed of light in vacuo (c). Note that  FIG. 10  shows only half of the radiation distribution, since the radiation pattern is symmetric with respect to the equatorial plane θ P =90°. It therefore will be appreciated that, for polar angles of 90&lt;θ P ≦180°, the radiation distribution will be the mirror image with respect to the equatorial plane θ P =90° of the radiation distribution shown in  FIG. 10 . 
     The electromagnetic radiation emitted by the polarization current antenna  200  that is generated by source elements that have a velocity component along the direction the electromagnetic radiation travels to the observation point P that is less than the speed of light in vacuo (c) represents the power per unit area of the conventional electromagnetic radiation emitted by polarization current antenna  200  (i.e., radiation that decays with distance d from the source according to the inverse square law, 1/d 2 ). As shown in  FIG. 10 , a sharp increase in the power per unit area of the emitted electromagnetic radiation occurs at the polar angle of θ P =56.4°. As is explained in more detail below, this sharp increase occurs because at polar angles of 56.4°≦θ P ≦123.6° there are source elements that have a velocity component along the direction the electromagnetic radiation travels to the observation point P that is greater than or equal to the speed of light in vacuo (c). The electromagnetic radiation emitted by such source elements is referred to herein as “non-spherically decaying electromagnetic radiation” as it has different properties from conventional radiation including the fact that it does not decay according to the inverse square law as does conventional electromagnetic radiation. 
     As shown in  FIG. 10 , the non-spherically decaying electromagnetic radiation is only emitted at polar angles of 123.6°≧θ P ≧56.4° given the particular design (described above) of the polarization current antenna  200  and the oscillation frequency of the polarization current. This is consistent with Equation (4) above, which shows that the non-spherically decaying electromagnetic radiation from each volume element of the antenna  200  is emitted at the polar angles: 
       θ P =arcsin( c/u ) and θ P =180°−arcsin( c/u )
 
     above and below the equatorial plane θ P =π/2, where c is the speed of light in vacuo and u is the speed of the volume element in question of the polarization current wave in units of the speed of light in vacuo. Here, the maximum speed u of the polarization current wave occurs at the outer radius of the dielectric radiator  212 , where the speed of the polarization current wave is u max =1.2*c. The minimum speed of the polarization current wave, which occurs at the inner radius of the dielectric radiator  212 , is u min =c. Thus, filling these speeds u into Equation (4) it can be seen that the non-spherically decaying electromagnetic radiation is emitted between the polar angles of 56.4°≦θ P ≦123.6°. 
     As the above discussion makes clear, the angular elevation beamwidth of polarization antenna  200  will be a function of the speed of the polarization current wave generated in the arc-shaped dielectric radiator, where angular elevation beamwidth refers to the range of polar angles into which the non-spherically decaying electromagnetic radiation is emitted. In the example above, the non-spherically decaying electromagnetic radiation is emitted into polar angles in the range of 56.4°≦θ P ≦123.6°, which corresponds to an angular elevation beamwidth of 67.2°. So long as the polarization current wave has a speed equal to the speed of light in vacuo at some point within the dielectric radiator  212 , then the angular elevation beamwidth of the of non-spherically decaying electromagnetic radiation emitted by the polarization current antenna  200  will be equal to 180°−2*arcsin(c/u max ), where c is the speed of light in vacuo and u max  is the speed of the polarization current wave at the outer radius of the dielectric radiator  212 . 
     Based on the relationship between the speed of the polarization current wave and the angular elevation beamwidth, a method of operating the above-described polarization current antennas according to embodiments of the present invention is to generate a polarization current wave in the arc-shaped dielectric radiator thereof that has a pre-selected speed at the outer radius of the arc-shaped dielectric radiator that is selected so that the beam of non-spherically decaying electromagnetic radiation that is generated by the polarization current wave has a pre-selected angular elevation beamwidth. In example embodiments for which the speed u min  of the polarization current wave at the inner radius of the arc-shaped dielectric radiator is smaller or equal to the speed of light in vacuo, the pre-selected speed u max  of the polarization current wave at the outer radius of the arc-shaped dielectric radiator may be between the speed of light in vacuo and 1.2 times the speed of light in vacuo, which results in an angular elevation beamwidth of 67.2° or less. In another example embodiment, the pre-selected speed u max  of the polarization current wave at the outer radius of the arc-shaped dielectric radiator may be between the speed of light in vacuo and 1.02 times the speed of light in vacuo, which results in an angular elevation beamwidth of 22.8° or less. Any appropriate speed may be selected to achieve a desired angular elevation beamwidth. 
     Reference is now made to  FIG. 10A , which is a graph comparing the radiation distribution pattern of  FIG. 10  (curve  300 ) to the radiation distribution pattern of a stationary source (curve  350 ). In particular, curve  350  shows the distribution of the time-averaged radial component of normalized Poynting vector, as a function of the polar coordinate θ P  of observation points P that are at a distance    P =10, for emission from a polarization current that is identical to the polarization current used to generate  FIG. 10 , except that the polarization current used to generate curve  350  has a stationary distribution pattern (i.e., it does not rotate around the dielectric radiator). In generating curve  350  it was assumed that the polarization current had the same sinusoidal distribution pattern, the same current density and the same oscillation frequency as the polarization current used to generate curve  300 , and that the polarization current was generated in a dielectric radiator having the same dimensions as the dielectric radiator  212  discussed above. Moreover, the normalization factor used in  FIG. 10A  is the same as that in  FIG. 10 : namely the average value of the power arising from the rotating source that propagates across a sphere of radius    P =10 per unit solid angle. The emission from the stationary source decays spherically as predicted by the inverse square law. 
     As can also be seen from  FIG. 10A , the power of the non-spherically decaying electromagnetic radiation (curve  300 ) is more than 18 dB greater than the power of the conventional electromagnetic radiation (curve  350 ) across the full range of polar angles θ P  where the non-spherically decaying electromagnetic radiation is emitted. Thus, even at very small distances (here the magnitude of the electromagnetic radiation is measured at an observation point P that is less than 2θ meters from the polarization current antenna  200 ), the magnitude of the non-spherically decaying electromagnetic radiation exceeds the magnitude of the conventional radiation by more than a factor of sixty at all polar angles at which the non-spherically decaying electromagnetic radiation is emitted. Moreover, as can also be seen from  FIG. 10A , the degree to which the non-spherically decaying electromagnetic radiation exceeds the conventional electromagnetic radiation increases dramatically at polar angles θ P  that are close to 90°. In fact, in the example, of  FIG. 10A , the non-spherically decaying electromagnetic radiation (curve  300 ) exceeds the conventional electromagnetic radiation (curve  350 ) by a factor of more than 200 for an observation point P at the polar angle of θ P =90°. 
     The rapid change in the intensity of the total electromagnetic radiation that occurs for observation points P at polar angles of θ P ≧56.4° reflects the penetration of the cusps associated with these observation points P into the source distribution across its outer boundary, where the outer boundary is the outermost radius r U  of the dielectric radiator  212 . The cusp is the locus of source elements at which (1) the component of the velocity of the polarization current wave along the direction of the electromagnetic radiation (i.e., along the line from the source element to the selected observation point P) equals the speed of light in vacuo and (2) the component of acceleration of the polarization current wave along the direction of the electromagnetic radiation equals zero. In other words, when the observation point P is located at a polar angle of θ P =56.4 degrees, there is a volume element of the ring in the polarization current wave travelling through the dielectric radiator  212  at the speed of u max =1.2*c (i.e., the portion of the polarization current wave at the outer radius r U  of the dielectric radiator  212 ) that will have a velocity component in the direction of the observation point P that is equal to the speed of light in vacuo and an acceleration vector that is perpendicular to the direction of the line from its retarded position to the observation point P. Consequently, that particular source element on the portion of the polarization current wave that is travelling through the outer radius r U  of dielectric radiator  212  will emit non-spherically decaying electromagnetic radiation in the direction of the observation point P. As the polar angle θ P  between the axis of rotation z of antenna  200  and the observation point P increases beyond 56.4°, portions of the polarization current wave in the dielectric radiator  212  that lie to the right of the cusp in  FIG. 11  will have a velocity component in the direction of the observation point P that is greater than or equal to the speed of light in vacuo, and hence these additional portions of the dielectric radiator will emit non-spherically decaying electromagnetic radiation in the direction of the selected observation point P. This effect is illustrated graphically in  FIG. 11 . 
     In particular,  FIG. 11  is a graph that illustrates the location of a cross-section of the dielectric radiator  212  (labelled “Source” in  FIG. 11 ) of the polarization current antenna  200  by a meridional plane. The dielectric radiator  212  has an inner radius r L  and an outer radius r U . As described above, the polarization current wave travels within the dielectric radiator  212  with a fixed angular velocity ω [its linear velocity rω varies linearly between r L ω) and r U ω (c and 1.2*c in the selected example) depending upon the radial location of the polarization current wave within the dielectric radiator  212 ]. The relative velocity of the polarization current wave with respect to a selected observation point P will vary based upon the polar angle θ P  between the selected observation point P and the polarization current antenna  200 . The curve labelled  400  in  FIG. 11  is the projection of the cusp associated with a selected observation point P onto the meridional plane of  FIG. 11 . In the graph of  FIG. 11 ,  =rω/c and {circumflex over (z)}=zω/c are cylindrical polar coordinates based on the axis of rotation in units of c/ω, and thus the coordinates in  FIG. 11  represent both speed (in units of c) and position (in units of c/ω). Whether the cusp  400  will intersect the dielectric radiator  212  (the source distribution), as shown in  FIG. 11 , or alternatively lies to the left or the right of the dielectric radiator  212 , is dictated by the polar coordinate θ P  of the selected observation point P. In  FIG. 11 , the selected observation point was placed close to the source (dielectric radiator  212 ), namely at    P ={circumflex over (z)} P =3 so that the point of intersection of the cusp  400  with the light cylinder ( =1) falls within the figure. Note that the {circumflex over (z)} coordinate of the point of intersection of the cusp with the light cylinder (shown by the green dashed vertical line in  FIG. 11 ) has the same value as the {circumflex over (z)} coordinate {circumflex over (z)} P  of the observation point P. 
     The portion of the polarization current wave that at the observation time occupies the portion of the dielectric radiator  212  that lies to the right of the cusp  400  in  FIG. 11  will have a velocity along the radiation direction to the observation point P that exceeds the speed of light in vacuo, thus generating non-spherically decaying electromagnetic radiation. The portion of the polarization current wave that at the observation time occupies the portion of the dielectric radiator  212  that lies to the left of the cusp  400  in  FIG. 11  will have a velocity along the radiation direction to the observation point P that is less than the speed of light in vacuo, and hence will only generate conventional electromagnetic radiation. The portion of the polarization current wave that travels through the portion of the dielectric radiator  212  that lies along the cusp  400  in  FIG. 11  will have a velocity along the radiation direction to the observation point P that equals the speed of light in vacuo, with zero acceleration, and hence will emit waves of electromagnetic radiation that interfere constructively at the observation point P. 
     Referring again to  FIG. 10 , the antenna only emits conventional radiation in the direction of observation points P that lie at polar angles θ P  of less than 56.4°, resulting in the relatively lower values for the radial component of the Poynting vector for observation points P in these locations. However, for observation points P that lie at polar angles θ P ≧56.4°, the cusps  400  are such that they penetrate the dielectric radiator  212  in the manner shown in  FIG. 11 , and hence a rapid increase in the intensity of the received electromagnetic radiation at the observation point P is observed, as shown in  FIG. 10 . As it is only those source elements that lie along (or very close to) the cusp  400  that exhibit the constructive interference, penetration of the cusp  400  farther into the dielectric radiator  212  only provides a limited additional increase in the radiation intensity, as is also shown in  FIG. 10 . 
     It should be noted that  FIG. 11  does not represent the above-described polarization current antenna  200  that was modelled to generate the results shown in  FIG. 10 , as the polarization current antenna  200  used in the modelling was designed to generate a polarization current wave having a speed equal to the speed of light in vacuo at the inner radius r L  of the dielectric radiator  212  thereof. The graph of  FIG. 11  represents a more general case where the speed of the polarization current wave exceeds the speed of light in vacuo throughout the entire cross-section of the dielectric radiator. 
     The integral of the radial component of the time-averaged Poynting vector over a sphere having a radius of    P =10 for the emission of curve  300  in  FIG. 10A  is about 9.9 Watts. This integral represents the total power emitted by the polarization current antenna  200  that propagates across a sphere of radius    P =10 when operated in the manner described above to generate the emission shown in  FIG. 10 . In contrast, the integral of the radial component of the time-averaged Poynting vector over the same sphere for the emission of curve  350  in  FIG. 10A , which represents the corresponding total power emitted by a polarization current antenna that is identical to the polarization current antenna  200  except that it is operated to have a stationary source, is 0.11 Watts. This shows that a superluminal source is a more efficient radiator by a factor of 90 (i.e., more than 19 dB) than its stationary counterpart. Moreover,  FIGS. 10 and 13  shows that the polarization current antenna when operated in the manner described above has a directive gain that exceeds 8 dBi at a distance of 1.91 meters (   P =10) and 13 dBi at a distance of 1.91*10 5  meters (   P =10 6 ). It will be appreciated that since the electromagnetic radiation emitted by the polarization current antennas according to embodiments of the present invention does not decay spherically with distance, the gain of these antennas will be a function of distance. 
     Curve a of  FIG. 12  is another representation of the data regarding the intensity of the electromagnetic radiation emitted by the polarization current antenna  200  as a function of the polar coordinate of the observation point P that was obtained through the above-described modelling. In particular, curve a of  FIG. 12  illustrates the same data shown in curve  300  of  FIG. 10 , but  FIG. 12  plots this data in a polar coordinate system. In  FIG. 12  the data has been normalized by adding 30 dB to all data points so that the radial coordinates of the points in the graph of  FIG. 12  have positive values across a sufficiently wide range of angles. The three dimensional radiation pattern for the polarization current antenna  200  at    P =10 may be obtained from curve a of  FIG. 12  by adding the reflection of curve a shown in  FIG. 12  across the equatorial plane (the horizontal axis) and then rotating the plotted data about the z-axis. Curves c and f of  FIG. 12  show the counterparts of curve a of this figure at  P=10 3  and    P =10 6  respectively, i.e., plot the data shown in curves c and f of  FIG. 13  in a polar coordinate system.  FIG. 12  plots the radiation pattern for the polarization current antenna  200  in a more traditional form. 
       FIG. 15  is logarithmic plot of the radial component of the normalized Poynting vector versus the normalized distance along the generating line of a cone inside the solid angle where the radiation decays non-spherically. In the example of  FIG. 15 , the plotted generating line is for the cone corresponding to θ P =60.9°. The points shown in  FIG. 15  are extracted from  FIG. 12 . The best fit to these points has a slowly varying slope of −1.54. Thus,  FIG. 15  illustrates that the decay rate of the radial component of the Poynting vector along the direction θ P =60.9° is given by    P   −1.54  (instead of    P   −2 ) over the range    P =10 to    P =10 6 . This decay rate itself slowly changes with distance. The plot of  FIG. 15  is based on the polarization current antenna  200  that has the parameters listed above that were used in the computational modelling analysis. 
       FIG. 16  is graph of the exponent α of the distance dependence    P   −α  of the radial component Poynting vector over the range    P =10 to    P =10 6  as a function of the polar angle θ P  for values of θ P  in the range 56.4&lt;θ P ≦90°. The plot of  FIG. 16  was obtained by applying the procedure discussed above with reference to  FIG. 15  to the entire set of computed points shown in  FIG. 12 . The plot of  FIG. 16  is also based on the polarization current antenna  200  that has the parameters listed above that were used in the computational modelling analysis. 
     The conventional radiation that is emitted by the polarization current antenna  200  has an angular distribution that is independent of the distance of the observation point P from the polarization current antenna  200 . In other words, while the power density of the conventional radiation emitted by polarization current antenna  200  decays with distance d from the source according to the inverse square law, 1/d 2 , the angular distribution of this conventional radiation remains constant. In contrast, the non-spherically decaying electromagnetic radiation, which is emitted into the region 56.4°≦θ P ≦90° (focusing solely on observation points above the equatorial plane) has a dependence on θ P  that varies with the distance of the observation point P from the polarization current antenna  200 . This is illustrated graphically in  FIG. 13 . 
     In particular,  FIG. 13  is a graph showing the radial component of the time-averaged Poynting vector divided by the average power that propagates across a sphere of radius    P =10 per unit solid angle as a function of the polar coordinate θ P  of the observation point P for observation points P at six different distances from the polarization current antenna  200 . In  FIG. 13 , the results are only shown for observations points P at polar angles of 56.4°≦θ P ≦90° degrees in order to provide better resolution in the figure. The six curves  500 ,  510 ,  520 ,  530 ,  540 ,  550  shown in  FIG. 13  correspond to observation points P at distances of 10, 100, 1,000, 10,000, 100,000 and 1,000,000 light cylinder radii from the polarization current antenna  200 , respectively. This corresponds to physical distances of 1.91 meters (curve  500 ), 19.1 meters (curve  510 ), 191 meters (curve  520 ), 1.91 km (curve  530 ), 19.1 km (curve  540 ) and 191 km (curve  550 ). In  FIG. 13 , curves  510 ,  520 ,  530 ,  540 ,  550  have been vertically shifted upwardly by the respective amounts of 20 dB, 40 dB, 60 dB, 80 dB and 100 dB with respect to curve  500 . These vertical shifts to curves  510 ,  520 ,  530 ,  540 ,  550  represent the amount that the radiation intensity would have changed for conventional radiation based on the different distances of the observation points P (i.e., the decrease, in decibels, of the value of the plotted quantity if its decay had obeyed the inverse square law). 
     Since the results shown in  FIG. 13  have been normalized with respect to distance according to the inverse square law, if conventional radiation were plotted in  FIG. 13  for observation points P at the five different distances listed above, the five curves would all fall on top of each other due to the above-described normalization. The separation between the curves in  FIG. 13  is a measure of the degree to which the emission by the polarization current antenna  200  decays more slowly with distance than predicted by the inverse square law. Moreover, the improvement in terms of the decrease in the rate of decay of the intensity of the electromagnetic radiation from that predicted by the inverse square law persists with increasing distance (i.e., curve  510  falls above curve  500 , curve  520  falls above curve  510 , etc.). Thus, for example, for an observation point P that is 10 light cylinder radii from the antenna  200  and located at polar angle of θ P =56.4°, the non-spherically decaying electromagnetic radiation exceeds the conventional spherically-decaying electromagnetic radiation arising from an identical stationary source by about 13 dB (see  FIG. 10A ). When the observation point P is at a distance of 1,000,000 light cylinder radii from the antenna  200  and again at a polar angle of θ P =56.4°, the combination of  FIGS. 10 and 13  illustrate that the non-spherically decaying electromagnetic radiation exceeds the conventional electromagnetic radiation by about 33 dB. 
       FIG. 13  further illustrates that the rate of decay of the emitted electromagnetic radiation with distance depends on the polar coordinate θ P  of the observation point P. This characteristic of the polarization current antenna  200  also differs from that of conventional antennas. 
     More specifically,  FIG. 13  shows that the rate of decay of the emitted electromagnetic radiation with distance increases as the observation point P is moved to polar coordinates that are closer to the equatorial plane (i.e., toward θ P =90 degrees). For example, for observation points P at polar angles of θ P =56.4°, the difference between the normalized value of the radial component of the Poynting vector for an observation point P at a distance of 10 light cylinder radii (see curve  500 ) and the normalized value of the radial component of the Poynting vector for an observation point P at a distance of 1,000,000 light cylinder radii (see curve  550 ) is about 20 dB. In contrast, for observation points P at polar angles of θ P =90°, the difference between the normalized value of the radial component of the Poynting vector for an observation point P at a distance of 10 light cylinder radii (see curve  500 ) and the normalized value of the radial component of the Poynting vector for an observation point P at a distance of 1,000,000 light cylinder radii (see curve  550 ) is about 5 dB. In other words,  FIG. 13  shows that the curves  500 ,  510 ,  520 ,  530 ,  540 ,  550  move closer together as θ P  increases toward 90°. The closer the curves  500 ,  510 ,  520 ,  530 ,  540 ,  550  are together, the closer the rate of decay of the radiation is to the rate of decay predicted by the inverse square law. 
     As can also be seen in  FIG. 13 , for each of the six distances to the observation point P (i.e., for curves  500 ,  510 ,  520 ,  530 ,  540 ,  550 ), the radial component of the Poynting vector is largest at θ P =90°. Moreover, a sharp increase in the Poynting vector occurs as the polar angle θ P  of the observation point approaches 90°. This sharp increase is observed because an additional mechanism of focusing occurs when the coordinate {circumflex over (z)} P  of the observation point P falls within the {circumflex over (z)}-extent (−{circumflex over (z)} 0 ≦{circumflex over (z)}≦{circumflex over (z)} 0 ) of the source distribution. In other words, this additional focusing mechanism occurs for observation points P having a {circumflex over (z)}-coordinate that is within the range of the {circumflex over (z)}-coordinates of the dielectric radiator  212 . This focused beam of electromagnetic radiation propagates into a solid angle encompassing the equatorial plane (θ P =90°), where the polar width of this solid angle is: 
       π/2−arcsin( {circumflex over (z)}   0 /   P )≦θ P ≦π/2+arcsin( {circumflex over (z)}   0 /   P )  (6)
 
     As    P  is the distance to the observation point P, in units of c/ω, Equation (6) shows that the width of the focused beam of electromagnetic radiation decreases linearly with distance (i.e., at a rate of 1/   P ) in the far zone. Since the area subtended by the narrowing solid angle into which the above-described focused beam of electromagnetic radiation propagates decreases linearly with distance (i.e., at a rate of 1/   P ), while the rate of decay of the magnitude of the Poynting vector with distance for emission into the equatorial plane (θ P =90°) is close to 1/   P   2  (see  FIG. 13  where the data points at θ P =90° are close together), it can be seen that the integral of the Poynting vector over any area subtending the narrowing solid angle into which this focused beam of electromagnetic radiation encompassing the equatorial plane propagates decreases with distance. This means that the above-described increase with distance in the flux of energy across surfaces subtending the fixed solid angle within which the Poynting vector decays more slowly than predicted by the inverse square law (i.e., decays at a rate &lt;1/   P   2 ), which in this example is the angle 56.4°≦θ P ≦123.6°, is partly compensated by the above-described corresponding decrease with distance in the flux of energy across surfaces subtending the narrowing solid angle π/2−arcsin ({circumflex over (z)} 0 /   P )≦θ P ≦n/2+arcsin({circumflex over (z)} 0 /   P ), 0≦θ P ≦2π, into which the stronger beam of equatorial electromagnetic radiation propagates. In other words, the general increase in the flux of energy with distance that occurs across the full fixed angle into which the non-spherically decaying electromagnetic radiation propagates (56.4°≦θ P ≦123.6° in the present example) is partly offset by a decrease in the flux of energy with distance that occurs because the beamwidth of the focused beam of electromagnetic radiation into the equatorial plane narrows. There is also another way in which the non-spherically decaying radiation meets the requirements of the conservation of energy. In the case of a conventional radiation, the flux of energy into any closed region (e.g., into the volume bounded by two spheres centered on the source) equals the flux of energy out of that region at any given time. In the present case, on the other hand, the time-averaged rate of change of the energy density of the non-spherically decaying radiation contained within a closed region of space is always negative, so that the flux of energy out of that region can be greater than the flux of energy into it. 
       FIG. 14  is a schematic diagram illustrating the different components of the electromagnetic radiation emitted by the polarization current antenna  200 . To simplify the figure, the radiation pattern is only shown in one direction. It will be appreciated that the electromagnetic radiation will be emitted throughout a full 360°. The actual radiation pattern of the antenna  200  may be obtained by rotating the three radiation patterns  600 ,  610 ,  620  shown in  FIG. 14  about the axis of rotation z of the polarization current antenna  200 . 
     As shown in  FIG. 14 , conventional radiation will be omitted over a broad range of polar angles θ P . The radiation pattern or “antenna beam” formed by this conventional radiation is illustrated in  FIG. 14  by the antenna beam  600 . Depending upon the design of the antenna, the electrodes  214 ,  216  of antenna  200  will tend to block the radiation for some range of polar angles near θ P =0° and near θ P =180°, and hence the antenna beam  600  is not shown as encompassing these polar angles. It will be appreciated that a small amount of conventional radiation will be emitted at these polar angles, and that a small amount of conventional radiation (also not shown in antenna beam  600 ) will also be emitted into the interior of the dielectric radiator  212  of antenna  200 . 
     As is further shown in  FIG. 14 , non-spherically decaying electromagnetic radiation will be emitted over a smaller range of polar angles θ P . For the example polarization current antenna  200  having the design described above, this smaller range of polar angles is 56.4°≦θ P ≦123.6°. This non-spherically decaying electromagnetic radiation is represented in  FIG. 14  by the antenna beam  610 . The magnitude of the non-spherically decaying electromagnetic radiation per unit area may exceed the magnitude of the conventional radiation by an order of magnitude or more. Finally, the non-spherically decaying electromagnetic radiation includes a very focused beam of radiation which is represented in  FIG. 14  by antenna beam  620 . 
     Thus, as shown in  FIG. 14 , pursuant to embodiments of the present invention polarization current antennas are provided that include a dielectric radiator that extends along an arc, where a radius of the arc defines an equatorial plane. These polarization current antennas are configured to emit a first beam of focused electromagnetic radiation into the equatorial plane, the first beam having an angular elevation beamwidth that decreases with increasing distance from the polarization current antenna (i.e., beam  620  in  FIG. 14 ). These polarization current antennas also are configured to emit a second beam of electromagnetic radiation that decays non-spherically with increasing distance from the polarization current antenna (i.e., beam  610  in  FIG. 14 ). 
     In the above-described polarization current antennas, the angular elevation beamwidth of the second beam (beam  610 ) exceeds the angular elevation beamwidth of the first beam (beam  620 ). The physical elevation beamwidth of the first beam, which refers to the elevation beamwidth of the first beam as measured in unit length along a direction perpendicular to the equatorial plane, may be substantially fixed as a function of distance from the polarization current antenna in some embodiments. The physical elevation beamwidth of the first beam may be substantially equal to a height of the dielectric radiator in the direction perpendicular to the equatorial plane. 
     The angular elevation beamwidth of the second beam may be based on a speed of a portion of a polarization current wave that travels through the dielectric radiator at the outer radius of the dielectric radiator during operation of the polarization current antenna. In particular, the angular elevation beamwidth of the second beam may be equal to: 
       Elevation Beamwidth=180°−2*arcsin( c/u   max ),  (7)
 
     where c is the speed of light in vacuo and u max  is the speed of the polarization current wave at the outer radius of the dielectric radiator. 
     The polarization current antenna emits a third beam of electromagnetic radiation that decays spherically with increasing distance from the polarization current antenna. An angular elevation beamwidth of the third beam may be greater than the angular elevation beamwidth of the second beam. 
     As noted above, the calculations of the Poynting vector that are provided in  FIGS. 10, 12 and 13  are based on a polarization current antenna that has a sinusoidally shaped polarization current wave that travels 10 wavelengths in circling the dielectric radiator  212  once (i.e., m=10). As can be seen in  FIG. 13 , the gain in the magnitude of the Poynting vector owing to its non-spherical decay over the range in distances to the observation point P from 100 light cylinder radii to 1,000,000 light cylinder radii is about 20 dB for a polar angle of θ P =56.4°. This can be seen in  FIG. 13  from the fact that the radial component of the normalized Poynting vector for an observation point P 1  at the polar angle of θ P =56.4° that is at a distance of 100 light cylinder radii is about −11 dB, while the magnitude of the Poynting vector for an observation point P 2  at the same polar angle and a distance of 1,000,000 light cylinder radii is about 9 dB. However, this gain is larger—i.e., the rate of decay of the magnitude of the Poynting vector is slower—the larger the value of the parameter m, which is the number of wavelengths of the polarization current wave that fit around the circumference of the dielectric radiator  212 . Since the parameter m is a function of the antenna design, it is possible to increase the gain by designing the antenna to have a larger value of m. 
     Since the rate of decay of the Poynting vector decreases with increasing values of m, it follows that higher values of m may be desirable for communications over large distances and, in particular, for point-to-point communications over large distances. In other words, by increasing the value of m one can make the rate of decay of the non-spherically decaying electromagnetic radiation be closer to 1/d, and hence higher antenna gain may be achieved at these large distances than would conventionally be possible. Based on this, a method of designing the above-described polarization current antennas according to embodiments of the present invention is to select the number of polarization elements, the frequency of the input signal and the time difference between the instants at which the input signal is applied to adjacent polarization elements attains maximum value so that the polarization current antenna will generate a polarization current wave that will have a pre-selected number of wavelengths that fit around the circumference of the arc-shaped dielectric radiator, where the pre-selected number of wavelengths is selected based at least in part on a distance to an antenna that is to receive signals transmitted by the polarization current antenna. In some embodiments, the pre-selected number of wavelengths may be at least ten wavelengths. In other embodiments, the pre-selected number of wavelengths may be greater than fifteen wavelengths. In still other embodiments, the pre-selected number of wavelengths may be greater than twenty wavelengths. In yet other embodiments, the pre-selected number of wavelengths may be greater than twenty-five wavelengths. 
     As described above, the polarization current antenna  200  has, among other things, the following properties that are different than the properties of conventional, non-polarization current antennas:
         1. The flux of energy decays more slowly with distance than predicted by the inverse square law;   2. The rate of decay of the flux of energy with distance increases as the observation point moves closer to the equatorial plane; and   3. The general increase in the flux of energy with increasing distance that occurs across the full fixed angle into which the non-spherically decaying electromagnetic radiation propagates is partly offset by a decrease in the flux of energy with increasing distance that occurs as the beamwidth of the focused beam of electromagnetic radiation into the equatorial plane narrows.   4. The time-averaged rate of change of the energy density of the non-spherically decaying radiation contained within a closed region of space is always negative, so that the flux of energy out of that region can be greater than the flux of energy into it.       

     The above-described properties of the equatorially emitting polarization current antenna have a number of interesting implications for purposes of antenna design. For instance,  FIG. 10  shows that the strongest emission occurs in the direction of observation points P that are in or near the equatorial plane. However, as the distance to the observation point P increases, the elevation beamwidth of the focused equatorial beam narrows linearly with distance. Thus elevation beamwidth requirements may limit the distances where this component of the radiation from the polarization current antenna is suitable for certain applications. 
     As discussed above, the angular elevation beamwidth for the non-spherically decaying electromagnetic radiation emitted by polarization current antennas according to embodiments of the present invention may be controlled by designing the antenna to generate a polarization current wave that has a speed at the outer radius of the dielectric radiator that provides a desired angular elevation beamwidth. The physical elevation beamwidth of the focused beam of electromagnetic radiation that is emitted into the equatorial plane that has an angular elevation beamwidth that decreases with increasing distance from the polarization current antenna (i.e., beam  620  in  FIG. 14  above) may be controlled by selection of the height Δz of the dielectric radiator. Moreover, the angular azimuth beamwidth of the non-spherically decaying electromagnetic radiation emitted by polarization current antenna will be equal to the angular arc length φ of the arc-shaped dielectric radiator. 
     Accordingly, the polarization current antennas according to embodiments of the present invention may be designed to have desired azimuth and elevation beamwidths by (1) selecting an angular arc length of the arc-shaped dielectric radiator to provide a pre-selected angular azimuth beamwidth for a beam of electromagnetic radiation that is emitted by the polarization current antenna that is non-spherically decaying with distance from the polarization current antenna and (2) selecting properties of the polarization current in the arc-shaped dielectric radiator to provide a pre-selected elevation beamwidth for the beam of electromagnetic radiation that is emitted by the polarization current antenna that is non-spherically decaying with distance from the polarization current antenna. 
     In some cases, the goal may be to select a desired elevation beamwidth for the focused beam of electromagnetic radiation  620  that is discussed above with reference to  FIG. 14 . In such embodiments, the pre-selected elevation beamwidth may be a pre-selected physical elevation beamwidth, and the property of the arc-shaped dielectric radiator that is selected to set the physical elevation beamwidth is the height Δz of the arc-shaped dielectric radiator. 
     In other cases, the pre-selected elevation beamwidth may be a pre-selected angular elevation beamwidth. In such cases, properties of the polarization elements and properties of signals supplied to the polarization elements may also be selected so as to provide the pre-selected angular elevation beamwidth for the beam of electromagnetic radiation that is emitted by the polarization current antenna that is non-spherically decaying with distance from the polarization current antenna. The properties of the arc-shaped dielectric radiator that are selected may comprise a radius of the arc-shaped dielectric radiator. The properties of the polarization elements that are selected may comprise a number of polarization devices and a distance between adjacent polarization elements. The properties of the signals supplied to the polarization elements that are selected may comprise a frequency of the signals and a phase difference between the oscillations of adjacent polarization elements. 
     The polarization current antennas according to embodiments of the present invention have unique properties that may be particularly well-suited for certain applications. For example, conventional antennas typically emit a main beam of electromagnetic radiation in a given direction along with a plurality of less intense beams of electromagnetic radiation that are emitted in directions on either side of the main beam. These less intense beams of electromagnetic radiation are typically referred to as “sidelobes.” Sidelobes are undesirable in many applications where a goal is to provide coverage to an area using the main beams of multiple different antennas, where each main beam covers a “sector” of the coverage area, as the sidelobes of an antenna beam that covers a first sector may fall within one or more adjacent sectors. When this occurs, the sidelobes may appear as interference to the main beams in the adjacent sectors. This is a common issue, for example, in cellular communications systems. A common technique to mitigate this issue is to transmit on different frequencies in adjacent sectors in order to avoid the interference problem. 
     The electromagnetic radiation or “beam” patterns of the polarization current antennas according to embodiments of the present invention do not have sidelobes in the traditional sense, although they may be designed to emit three distinct types of radiation as discussed above with reference to  FIG. 14 , namely a first beam  600  of conventional (i.e., spherically decaying) electromagnetic radiation, a second beam  610  of non-spherically decaying electromagnetic radiation, and a third very narrow beam  620  of highly focused non-spherically decaying electromagnetic radiation that subtends a smaller angle with increasing distance. By adjusting the properties of the polarization current waves that flow in the dielectric radiators of the polarization current antennas according to embodiments of the present invention (which can be done by adjusting physical properties of the polarization current antenna and/or the signal fed thereto in the manner described above), the shapes and other properties of the three above-described beams  600 ,  610 ,  620  of electromagnetic radiation may be adjusted. Thus, the polarization current antennas according to embodiments of the present invention may be designed to have reduced “sidelobes” when used in applications where sidelobes are problematic. 
     For example, as discussed above, the polar angles subtended by the second beam  610  of non-spherically decaying electromagnetic radiation shown in  FIG. 14  may be readily changed by adjusting the speed of the polarization current wave within the dielectric radiator of the polarization current antenna. Moreover, as shown in  FIGS. 10 and 13 , the second beam  610  may have a relatively constant magnitude across most polar angles into which the second beam  610  is emitted, as the increase in intensity that occurs at polar angles of about 90° is primarily due to the third beam  620  of highly focused non-conventional radiation, at least for the range of distances shown by the curves included in  FIG. 13  (i.e., distances of 10 to 1,000,000 light cylinder radii). For example, it can be seen that between the polar angles of θ P =60° and θ P =80° the change in intensity of the non-conventional (i.e., non-spherically decaying) electromagnetic radiation (i.e., the combination of antenna beams  610  and  620 ) is only about 5 dB in the example of  FIG. 10 , and  FIG. 13  shows that the intensity varies even less with increasing distance. Moreover, while the conventional electromagnetic radiation represented by beam  600  of  FIG. 14  may be considered to be akin to a sidelobe,  FIG. 10  shows that the intensity of beam  600  is significantly lower than the intensity of the non-conventional radiation (i.e., the combination of beams  610  and  620 ), and that the intensity of beam  600  drops off very rapidly. For example, the maximum value of beam  600  in the example of  FIG. 10  is almost 13 dB less than the minimum value of the combination of beams  610  and  620 , and the magnitude of beam  600  falls off by nearly an additional 10 dB over the next 10° of elevation beamwidth (i.e., the magnitude of the “sidelobes” of the polarization current antenna fall off sharply). 
     Moreover, it is possible to design the polarization current antennas to control the ratio of the intensity of the non-conventional (beams  610  and  620  of  FIG. 14 ) and conventional (beam  600  of  FIG. 14 ) electromagnetic radiation. For example, by increasing the value of the parameter m, this ratio may be increased. As described above, other parameters may also be adjusted that impact the ratio of the intensity of the non-conventional and conventional electromagnetic radiation. Thus, the polarization current antennas according to embodiments of the present invention have the unusual property that the ratio of the magnitude of the main lobe of electromagnetic radiation to the magnitude of the sidelobes may be adjusted. This is a highly desirable property, as in most applications the sidelobes at a minimum represent “lost” radiation that does not produce any benefit and reduces the amount of radiation within the main lobe, and in many cases the sidelobes may also represent an interfering signal for an antenna covering an adjacent sector that acts to degrade the performance of this adjacent antenna. 
     As discussed above, the parameter m is a function of (1) the number of polarization elements, (2) the frequency of an input signal to the polarization current antenna and (3) a time difference between the instants at which the input signal is applied to adjacent polarization elements attains maximum value. Thus, one or more of these parameters may be selected so that a portion of the spherically decaying electromagnetic radiation that is emitted by the polarization current antenna that is emitted outside a range of polar angles where the non-spherically decaying electromagnetic radiation is emitted has a maximum intensity that is at least a pre-selected level lower than a maximum intensity of the non-spherically decaying electromagnetic radiation at a first distance from the polarization current antenna. In other words, with reference to  FIG. 10 , parameters of the polarization current antenna may be selected to ensure that the step function in the Poynting vector that occurs at =56.4° is at least a pre-selected minimum value such as, for example, 10 dB, 12 dB, 15 dB, 18 dB, 20 dB, 25 dB, 30 dB, 35 dB or 40 dB in various embodiments. As described above, this may be done to ensure that the sidelobe levels of the antenna are at suitably low levels. The first distance may be, for example, a distance to a second antenna that is configured to receive signals transmitted by the polarization current antenna. 
     The above-discussed properties of the polarization current antennas according to embodiments of the present invention may be used in designing the properties of the polarization current antennas for certain applications. For example, in designing the parameters of a polarization current antenna for point-to-multipoint applications, the antenna may be designed to generate a polarization current wave that travels in the dielectric radiator of the antenna at a speed such that the antenna will emit non-spherically decaying electromagnetic radiation over a range of polar angles that corresponds to a desired elevation beamwidth of the antenna. As shown in Equation (6) above, this may be accomplished by designing the antenna so that the speed of the polarization current wave at the outer radius of the dielectric radiator generates a beam  610  of non-spherically decaying electromagnetic radiation that has a desired elevation beamwidth. As discussed above, the speed of the polarization current wave is a function of (1) various parameters of the arc-shaped dielectric radiator (e.g., radius, thickness, etc.), (2) various parameters of the polarization devices (e.g., distance between adjacent polarization devices, the total number of polarization devices, etc.) and (3) the feed network (e.g., the time difference between the instants at which the input signals applied to adjacent polarization devices attain maximum amplitude). Thus, these parameters may be designed so that the polarization current antenna generates a polarization current wave having a speed at the outer radius of the dielectric radiator that results in the polarization current antenna emitting non-spherically decaying electromagnetic radiation over a desired elevation beamwidth. Likewise, the angle φ of the arc defined by the arc-shaped dielectric radiator may be chosen to select an azimuth beamwidth for the polarization current antenna. Additionally, various parameters of the polarization current antenna such as the parameter m may be selected so that a maximum intensity of a portion of a beam of conventional radiation that is emitted outside the range of polar angles at which the non-spherically decaying electromagnetic radiation is emitted is below a pre-selected level. As discussed above, this portion of the conventional spherically decaying electromagnetic radiation corresponds to the “sidelobe” of the polarization current antenna. Thus, the polarization current antennas according to embodiments of the present invention may be readily designed to have sidelobes that are at or below pre-selected levels with respect to, for example, the maximum intensity value of the main beam at a pre-selected distance (or a range of distances). 
     Another property of the polarization current antennas according to embodiments of the present invention is that the design of the antenna may be adjusted to increase the directive gain of the antenna at a given distance. In fact, the polarization current antennas according to embodiments of the present invention may achieve directive gain values that are comparable to very large parabolic dish reflector antennas while having an antenna size that is a small fraction of the size of such a parabolic dish reflector antenna. In particular, by varying one or more parameters of the arc-shaped dielectric radiator, the polarization devices and/or the feed network, the directive gain of the antenna in a direction of peak emission may be changed in known, predictable ways. Thus, it is possible to design the antennas to have at least a pre-selected directive gain value at a pre-selected distance. As noted above, one parameter that may have a significant impact on the directive gain is the parameter m, which is the number of wavelengths of the polarization current wave that fit within one rotation of the dielectric radiator. By changing the value of the parameter m, the directive gain of the antenna, at a given distance, may be changed. The parameter m is a function of the number of polarization elements, the frequency of the input signal and the time difference between the instants at which the input signal is applied to adjacent polarization elements attains maximum value, as is discussed above. 
     Yet another property of the polarization current antennas according to embodiments of the present invention is that they generate a very focused beam of non-spherically decaying electromagnetic radiation corresponding to the beam  620  in  FIG. 14 . This beam of radiation has an angular beamwidth (in the embodiments discussed above, an angular elevation beamwidth) that narrows with increasing distance from the polarization current antenna. Thus, pursuant to further embodiments of the present invention, polarization current antennas are provided that include (1) a dielectric radiator and (2) a plurality of polarization devices that are configured to generate an electric field within the dielectric radiator, where the polarization current antenna is configured to generate a beam of electromagnetic radiation that has an angular beamwidth that narrows with increasing distance from the dielectric radiator. This narrow beam of focused radiation may have a very high intensity in the near field. 
     While example embodiments of the present invention have been described above, it will be appreciated that many modifications may be made to these example embodiments without departing from the scope of the present invention. For example, while the polarization current antennas that are discussed above have arc-shaped dielectric radiators that have a constant radius, it will be appreciated that embodiments of the present invention are not limited thereto. In particular, in other embodiments, the radius of the arc may vary along the length of the arc to provide a curved dielectric radiator having a non-constant radius. 
     As another example, the dielectric radiators that are discussed above are in the form of an arc-shaped strip. While such a strip is a convenient shape for the dielectric radiator, it will be appreciated that other shapes may also be used to support a travelling volume polarization current distribution pattern. Thus, it is contemplated that electrodes or other polarization devices may be on, embedded in or otherwise coupled to dielectric radiators having shapes other than arc-shaped strips. For example, s-shaped dielectric radiators could be used in some embodiments. Many other shapes are possible. As yet another example, electrodes (including ground planes) are used as examples of polarization devices that may be used to polarize the dielectric radiator. It will be appreciated, however, that any suitable polarization devices may be used in further embodiments of the present invention. 
     While the present invention has been described above primarily with reference to the accompanying drawings, it will be appreciated that the invention is not limited to the illustrated embodiments; rather, these embodiments are intended to fully and completely disclose the invention to those skilled in this art. In the drawings, like numbers refer to like elements throughout. Thicknesses and dimensions of some components may be exaggerated for clarity. 
     Spatially relative terms, such as “under”, “below”, “lower”, “over”, “upper”, “top”, “bottom” and the like, may be used herein for ease of description to describe one element or feature&#39;s relationship to another element(s) or feature(s) as illustrated in the figures. It will be understood that the spatially relative terms are intended to encompass different orientations of the device in use or operation in addition to the orientation depicted in the figures. For example, if the device in the figures is turned over, elements described as “under” or “beneath” other elements or features would then be oriented “over” the other elements or features. Thus, the exemplary term “under” can encompass both an orientation of over and under. The device may be otherwise oriented (rotated 90 degrees or at other orientations) and the spatially relative descriptors used herein interpreted accordingly. As one specific example, various features of the communications jacks of the present invention are described as being, for example, adjacent a top surface of a dielectric radiator. It will be appreciated that if elements are adjacent a bottom surface of a dielectric radiator, they will be located adjacent the top surface if the device is rotated 180 degrees. Thus, the term “top surface” can refer to either the top surface or the bottom surface as the difference is a mere matter of orientation. 
     Well-known functions or constructions may not be described in detail for brevity and/or clarity. As used herein the expression “and/or” includes any and all combinations of one or more of the associated listed items. 
     The terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention. As used herein, the singular forms “a”, “an” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the terms “comprises”, “comprising”, “includes” and/or “including” when used in this specification, specify the presence of stated features, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, operations, elements, components, and/or groups thereof. 
     Herein, the terms “on,” “attached,” “connected,” “contacting,” “mounted” and the like can mean either direct or indirect attachment or contact between elements, unless stated otherwise. 
     Although exemplary embodiments of this invention have been described, those skilled in the art will readily appreciate that many modifications are possible in the exemplary embodiments without materially departing from the novel teachings and advantages of this invention. Accordingly, all such modifications are intended to be included within the scope of this invention as defined in the claims. The invention is defined by the following claims, with equivalents of the claims to be included therein.