Abstract:
A thin electromagnetic phase shifting element, named phase and amplitude shifting surface (PASS), is disclosed. The PASS is capable of independently altering both the phase and the amplitude distribution of the electromagnetic fields propagating through the structure. The element comprises a few patterned metallic layers separated by dielectric layers. The patterns of the metallic layers are tuned to locally alter the phase and/or the amplitude of an incoming electromagnetic wave to a prescribed set of desired values for the outgoing electromagnetic wave. The PASS can be applied to design components such as gratings, lenses, holograms, and various types of antennas in the microwave, millimeter wave and sub-millimeter wave.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     The present invention claims priority from U.S. Provisional Patent Application No. 61/230,180, filed Jul. 31, 2009, and Canadian Patent Application No. 2,674,785, filed Aug. 4, 2009, which are incorporated herein by reference. 
    
    
     TECHNICAL FIELD 
     The present invention relates to devices for altering a wavefront of an electromagnetic wave, and in particular to phase or amplitude shifting elements for redirecting, focusing, or collimating an electromagnetic wave. 
     BACKGROUND OF THE INVENTION 
     Electromagnetic waves are widely used in areas ranging from communication and radiolocation to TV broadcasting, imaging, medical treatment, and food processing. Electromagnetic waves in the microwave, millimeter-wave, and in sub-millimeter wave ranges are particularly useful for the purposes of communication due to relatively high carrier frequency and associated ease of beaming the wave and comparatively large information carrying capacity. Electromagnetic waves in the sub-millimeter wavelength region, or so-called terahertz radiation, are presently used in non-invasive through-imaging applications. 
     Devices for collimating or focusing electromagnetic waves are important elements of many types of antenna devices. Quite often, the term “antenna” is used in the prior art to denote the collimating or focusing elements themselves. Antennas vary widely in their construction. One of the most traditional and well-known antennas is a reflective antenna, such as paraboloid reflector antennas for a satellite TV reception. Transmission antennas, such as hyperboloid dielectric lenses, are also known, although they are not as widely used due to their larger mass as compared to reflective antennas. Furthermore, holographic principles have been applied at microwave frequencies for designing low-profile scattering surfaces for high-gain reflective and transmissive antenna applications. A review of microwave holography can be found in an article by W. E. Kock entitled “Microwave Holography”,  Microwaves , vol. 7, no. 11, pp. 46-54, November 1968, which is incorporated herein by reference. 
     Reflective antennas can be constructed in form of concave reflectors, Fresnel zone plates, or in form of so called “artificial impedance” reflectors. Fresnel zone plates and artificial impedance reflectors are thinner than concave reflectors, however they generally suffer from a lower aperture efficiency, as well as somewhat limited bandwidth. A considerable effort has been devoted to developing a so-called “reflectarray” technology, where the curved reflector surfaces are replaced by thin flat panels of microstrip patches. For example, in U.S. Pat. No. 4,684,952 by Munson et al., which is incorporated herein by reference, a microstrip reflectarray for satellite communications is disclosed. The major disadvantage of reflectarrays is their limited bandwidth. Furthermore, due to the difficulty in reproducing the required interference patterns at microwave frequencies, holographic antennas, artificial impedance antennas, and reflectarrays generally suffer from lower aperture efficiencies than conventional reflectors or dielectric lenses. 
     Each technology has its strengths and weaknesses, and the requirements of the application will usually dictate the type of antenna to be selected. Conventional paraboloid reflectors and dielectric lenses in general have a higher radiation efficiency, but they require a larger volume than planar arrays. Planar arrays are attractive for their low profile and capability for electronic beam scanning, but these advantages come at the expense of complex feed network design and reduced radiation efficiency. For fixed-beam applications, conventional reflectors and lenses usually have superior electrical performance and would probably always have been selected if it were not for the larger volumes they occupy. 
     Regardless of technology used, transmissive antennas offer certain advantages over reflective ones, such as the elimination of aperture blockage by the feed antenna and reduced sensitivity to manufacturing tolerances, both of which are important for higher frequency designs. However, less work has been carried out on reducing the volume of transmissive antennas. In most cases, transmissive antennas are lenses formed out of dielectric material with a plano-hyperbolic cross-section. These lenses are relatively thick, especially for designs with small focal length-to-diameter (F/D) ratios. The lenses can be zoned to reduce the overall thickness, but the zoning results in reduced bandwidth and aperture efficiency of the lens. 
     Referring to  FIG. 1A , a prior-art dielectric lens  100  is shown. The dielectric lens  100  is shaped to transform a planar wavefront  101  into a spherical wavefront  102  or vice versa. The dielectric lens  100  introduces a phase delay at its center  103  larger than a phase delay at its edge  104 , whereby the spherical wavefront  102  is formed. Factors affecting the performance of the dielectric lens  100  include the impedance mismatch at the boundaries and the transmission loss in the material of the dielectric lens  100 . As noted above, the dielectric lens  100  has a comparatively good performance, although it is too bulky and heavy for many applications. 
     Referring now to  FIG. 1B , a prior-art Fresnel zone plate  105  is shown. The Fresnel zone plate  105  operates by blocking portions of the incoming wavefront  101  by metal structures  106  to obtain the spherical wavefront  102  due to diffraction. The Fresnel zone plate  105 , although being much thinner and lighter than the dielectric lens  100 , has a focusing efficiency of about 6 dB worse than a dielectric lens  100 , because approximately half of the incoming electromagnetic energy is blocked by the metal structures  106  of the Fresnel zone plate  105 . 
     An effort has been undertaken in the prior art to provide a transmissive antenna that would combine the compactness of the Fresnel zone plate  105  with the performance of the dielectric lens  100 . Two approaches have been tried in the prior art. One approach is to use so called “artificial dielectric” as a material for the lens. The artificial dielectric is a composite material consisting of a dielectric host containing an array of metal inclusions, thus modifying an effective dielectric constant of the composite material. By spatially varying the density of the inclusions to make the effective refractive index of the lens higher at the lens center than at its edges, the desired focusing property of the lens can be achieved without having to make the lens as thick as the traditional dielectric lens  100 . 
     Volume holographic elements can also be created using the artificial dielectric approach. For example, in U.S. Pat. No. 6,987,591 by Shaker et al., incorporated herein by reference, an artificial dielectric-type volume hologram is disclosed. Disadvantageously, the artificial dielectric approach, although reducing the antenna thickness, still results in a relatively thick, heavy, and expensive antenna device, because many layers of metal inclusions, typically 80 or more, are required for a satisfactory performance to be obtained. 
     Another prior-art approach to reduce thickness of a transmissive-type antenna is to use an array of microstrip patches. Turning to  FIG. 1C , a microstrip device  107  is schematically shown in a side view. The microstrip device  107  includes an array of resonant patch receivers  108 A,  108 B,  108 C,  108 D, and  108 E coupled to an array of microstrip delay lines  109 A,  109 B,  109 C,  109 D, and  109 E coupled to an array of resonant patch transmitters  110 A,  110 B,  110 C,  110 D and  110 E, respectively. The delay time of the microstrip delay lines  109 A to  109 E varies, increasing in going from the top microstrip delay line  109 A to the center microstrip delay line  109 C, and decreasing in going from the center microstrip delay line  109 C to the bottom microstrip delay line  109 E. The microstrip delay times are selected so that the planar wavefront  101  is transformed into the spherical wavefront  102  or vice-versa. One drawback of the microstrip device designs is that the resonant patch receivers  108 A to  108 E have to be spatially separated from corresponding resonant patch transmitters  110 A to  110 E by a gap  111  containing the microstrip delay lines  109 A to  109 E, which increases the mechanical complexity and thickness of the microstrip device  107 . Furthermore, the separation between the elements on the same layer ( 108 A to  108 E;  110 A to  110 E) has to be about one wavelength, which results in a significant quantization error. Another drawback is a reduced bandwidth due to the resonant character of the patch receivers  108 A to  108 E and the patch transmitters  110 A to  110 E. 
     Yet another prior-art approach to create a low-profile transmissive antenna is to use so-called transmitarrays. Transmitarrays use a small number of thin dielectric layers to emulate a lens behaviour. A prototype transmitarray consisting of four dielectric sheets upon which thin cross dipoles were printed was demonstrated by M. R. Chaharmir et al. in an article entitled “Cylindrical Multilayer Transmitarray Antennas,”  International URSI Commission B Electromagnetic Theory Symposium , EMTS-2007, Ottawa, Canada, July 2007, incorporated herein by reference. One drawback with current transmitarray designs is the requirement for an air gap between dielectric layers of one tenth of a wavelength or more, to maximize radiation efficiency. This increases the mechanical complexity of the device and does not allow for achieving an optimum thickness reduction. Nevertheless, a transmitarray is usually much thinner than the shaped dielectric lens  100 . 
     Finally, it is important to mention research carried out on the use of holographic techniques for designing low-profile antennas and lenses at microwave frequencies, as disclosed in an article by K. Iizuka et al. entitled “Volume-Type Holographic Antenna,”  IEEE Transactions on Antennas and Propagation , vol. 23, no. 6, pp. 807-810, November 1975; in an article by K. Lévis et al. entitled “Ka-band Dipole Holographic Antennas,”  IEE Proceedings on Microwaves, Antennas and Propagation , vol. 148, no. 2, pp. 129-132, April 2001, and in an article by M. Elsherbiny et al. entitled “Holographic Antenna Concept, Analysis, and Parameters,”  IEEE Transactions on Antennas and Propagation , Vol. 52, No. 3, pp. 830-839, March 2004, all of which are incorporated herein by reference. Disadvantageously, due to the difficulty in recording the phase pattern at microwave frequencies, these antennas were all of the amplitude type and consequently suffered from low aperture efficiencies, similar to Fresnel zone plate lenses disclosed by A. Petosa et al. in an article entitled “Comparison of an Elementary Hologram and Fresnel Zone Plate,”  The Radio Science Bulletin , no. 324, pp. 29-36, March 2008, which is incorporated herein by reference. 
     An ideal electromagnetic lens device would work in transmission, have minimal reflection and transmission losses, operate over a wide bandwidth, have a thin flat profile, be lightweight and inexpensive to manufacture. The prior art is lacking a transmission antenna device that would have a relatively high efficiency, while being inexpensive, lightweight, and thin. 
     The present invention provides a transmissive phase element that is electrically thin, inexpensive, and lightweight, while being capable of introducing a predetermined arbitrary phase shift pattern into an electromagnetic wave for focusing, collimating, redirecting, or splitting the electromagnetic wave in almost arbitrary manner. This versatile performance is achieved without introducing an excessive loss in the path of the electromagnetic wave. Furthermore, a phase element of the present invention can also introduce a predetermined arbitrary amplitude shift pattern in addition to the phase shift pattern. The amplitude shifting property can be used, for example, for electromagnetic beam shaping and pattern synthesis. 
     SUMMARY OF THE INVENTION 
     A phase element of the present invention generates a pattern of phase shifts using an approach similar to newspaper printing, where the grey tones are obtained from different size of small black dots in a given, predetermined array. Instead of the ink on paper used in the newspaper printing process, metallic patches are etched on a dielectric substrate layer. A single layer of metallic patches does not produce a significant phase shift range and is always associated with a considerable amplitude shift pattern dependent on the phase shift pattern. Accordingly, adding more thin dielectric layers has been initially expected to result in a corresponding increase of the transmission loss, which was undesired. Quite unexpectedly, however, adding more layers under certain conditions resulted in a significant decoupling of achievable phase and amplitude shift patterns. The decoupling allowed one to obtain a very low amplitude shift, or transmission loss, of the phase element. This allowed the inventors to produce low-loss, thin, and lightweight phase elements using a low-cost, mature and efficient process of metal etching on a dielectric substrate. 
     In accordance with the invention there is provided a phase element for introducing a predetermined phase shift pattern into an electromagnetic wave propagating therethrough, the phase element comprising stack of alternating conductive and dielectric layers each having a thickness, 
     wherein the conductive layers are patterned throughout the thickness thereof, the patterned conductive layers having a spatially varying feature, so as to obtain the predetermined phase shift pattern, 
     wherein the thickness of each of the dielectric layers are smaller than one tenth of a wavelength of the electromagnetic wave, and 
     wherein the total number of the layers in the stack, including the conductive and the dielectric layers, is more than two but less than nine. 
     The spatially varying feature can be a conductive strip of varying width, or a rectangle of varying size, or some other conductive shape having a spatially varying dimension, orientation, or position relative to other shapes. 
     In accordance with another aspect of the invention there is further provided a phase element for introducing a predetermined phase shift pattern into an electromagnetic wave propagating therethrough, the phase element comprising stack of alternating conductive and dielectric layers each having a thickness, the stack including first and second neighboring conductive layers, 
     wherein the first and the second conductive layers are patterned throughout the thickness thereof, so as to form a plurality of conductive shapes capacitively coupled to their respective neighboring shapes disposed in the same conductive layer, thereby forming two-dimensional patterns of first and second capacitances, respectively, and 
     wherein the conductive shapes of the first conductive layer are capacitively and inductively coupled to their respective neighboring conductive shapes disposed in the second conductive layer of the stack, 
     whereby the conductive shapes of the first and the second conductive layers form a two-dimensional pattern of transmission lines going through the stack, wherein each transmission line comprises a succession of a first capacitance of the two-dimensional pattern of first capacitances, capacitively and inductively coupled to a second capacitance of the two-dimensional pattern of second capacitances, 
     wherein the first and the second capacitances are selected so as to introduce the predetermined phase shift pattern into the electromagnetic wave propagating through the phase element. 
     The phase element of the invention can be used in a low-profile antenna or in an antenna that is hidden from view. 
     In accordance with the invention there is further provided a method of manufacture of the phase element, comprising: 
     (a) selecting a material and a thickness for each of the layers of the stack; 
     (b) selecting the number of the conductive layers in the stack; 
     (c) performing an electromagnetic simulation of the stack to obtain a dependence of a phase shift value on the spatially varying feature; and 
     (d) patterning the conductive layers to obtain the predetermined phase shift pattern, based on the dependence obtained in step (c). 
     In accordance with the invention there is further provided a method of manufacture of the phase element, comprising: 
     (a) selecting a material and a thickness for each of the layers of the stack; 
     (b) selecting the total number of the conductive layers in the stack; 
     (c) performing an electromagnetic simulation of the conductive shapes of the first and the second conductive layers, so as to obtain a dependence of a phase shift magnitude on dimensions and a relative position of the conductive shapes; 
     (d) based on the dependence obtained in step (c), determining the dimensions and the relative position of the conductive shapes of the first and the second conductive layers, required to obtain the pre-determined phase shift pattern; and 
     (e) patterning the first and the second conductive layers to obtain the predetermined phase shift pattern, based on the dimensions and the relative position of the conductive shapes, obtained in step (d). 
     The patterning of the conducting layers of the phase element is preferably performed by etching through the conductive layers. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Exemplary embodiments will now be described in conjunction with the drawings in which: 
         FIGS. 1A ,  1 B, and  1 C are schematic side views of prior-art lens, Fresnel zone plate, and microstrip patch antenna devices, respectively; 
         FIG. 2A  is a cross-sectional side view of a phase element of the present invention; 
         FIG. 2B  is a plan view of the upper surface of the phase element of  FIG. 2A ; 
         FIG. 2C  is a magnified view of a portion of the upper surface of  FIG. 2B , disposed between vertical dashed lines A and A′ therein; 
         FIG. 3A  is a cross-sectional side view of  FIG. 2A  having superimposed thereupon an equivalent electrical circuit; 
         FIG. 3B  is a schematic of the equivalent electrical circuit of  FIG. 3A ; 
         FIG. 3C  is a schematic of a single transmission line of the circuit of  FIG. 3B ; 
         FIG. 3D  is a graph showing a simulated best-case amplitude vs. phase performance of the phase element of  FIGS. 2A to 2C ; 
         FIG. 4  is a schematic side view of a microwave diffraction grating of the invention; 
         FIG. 5  is a photograph of a prototype of the diffraction grating of  FIG. 4 ; 
         FIG. 6  is an angular dependence of electromagnetic power transmitted through the prototype diffraction grating of  FIG. 5 ; 
         FIG. 7  is a schematic side view of a microwave cylindrical lens of the invention; 
         FIG. 8  is a photograph of a prototype of the cylindrical lens of  FIG. 7 ; 
         FIG. 9  is an angular dependence of gain of the prototype cylindrical lens of  FIG. 8 ; 
         FIGS. 10A and 10B  are side and front views, respectively, of a lens of the present invention for operating in a single polarization; 
         FIGS. 11A and 11B  are photographs of two prototypes of the lens of  FIGS. 10A and 10B ; 
         FIG. 12  is an angular dependence of gain of the prototype lens of  FIG. 11A ; 
         FIG. 13  is a frequency dependence of gain of the prototype lens of  FIG. 11A ; 
         FIG. 14  is a photograph of a prototype of a polarization-independent lens of the present invention; 
         FIG. 15  is an angular dependence of gain of the prototype lens of  FIG. 14 ; 
         FIG. 16  is a dependence of gain of the lens of  FIG. 15  on roll angle of the lens; and 
         FIG. 17  is a block diagram of steps for manufacturing a phase element of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     While the present teachings are described in conjunction with various embodiments and examples, it is not intended that the present teachings be limited to such embodiments. On the contrary, the present teachings encompass various alternatives, modifications and equivalents, as will be appreciated by those of skill in the art. 
     Throughout the specification, the phase element of the invention is called a “phase and amplitude shifting surface” (PASS) or a “phase shifting surface” (PSS). Herein, the word “surface” is used to refer to an “electrically thin” element, that is an element whose lateral dimensions are much smaller than a wavelength of the electromagnetic wave propagating through the element. The PSS is an element that alters phase distribution of the electromagnetic wave, while introducing a negligible transmission loss. 
     Referring to  FIGS. 2A to 2C , a PSS, or a phase element  200  ( FIG. 2A ) of the invention is shown in side and plan views. In  FIG. 2A , a side cross-sectional view of the phase element  200  of the present invention is shown. The phase element  200  has a stack  210  of alternating conductive layers  201 ,  203 , and  205  and dielectric layers  202  and  204  having a dielectric constant ∈ r . The symbol ∈ r  denotes dielectric constant in  FIGS. 2A ,  2 B,  2 C, and  10 A. The conductive layers  201 ,  203 , and  205  are patterned, for example etched, on individual surfaces of the dielectric layers  202  and  204 , forming conductive features  201 A,  203 A, and  205 A, respectively, on individual surfaces of the dielectric layers  202  and  204 . The conductive features  201 A,  203 A, and  205 A are strips having spatially varying width. The width of the strips  201 A and  205 A is denoted as a 1 , and the width of the strip  203 A is denoted as a 2 . The gap between neighboring strips  201 A, or between neighboring strips  205 A, is denoted as g 1 . The gap between neighboring strips  203 A is denoted as g 2 . By spatially varying a 1  and/or g 1  and/or a 2  and/or g 2 , the phase element  200  can be made to introduce a predetermined phase shift pattern into an incoming or incident electromagnetic wave  212 . 
     Referring to  FIG. 2B  with further reference to  FIG. 2A , a frontal view of the conductive layer  201  is presented. The strips  201 A are represented by dashed areas. The strips  201 A ( FIGS. 2A and 2B ),  203 A ( FIG. 2A ), and  205 A ( FIG. 2A ) extend perpendicular to  FIG. 2A  and along an X-axis  206  shown as a horizontal arrow in  FIG. 2B . The width a 1  of the strips  201 A ( FIGS. 2A and 2B ) and  205 A ( FIG. 2A ), shown as dashed areas in  FIG. 2B , varies along the X-axis  206 , as shown in solid lines outlining the dashed areas. The width a 2  ( FIG. 2A ) of the strips  203 A can also vary along the X-axis  206 , or it can stay constant, depending on a particular phase shift pattern required. The strips  203 A and  205 A are not shown in  FIG. 2B , for clarity of the picture. The widths a 1  and a 2  of the conductive strips  201 A,  203 A, and  205 A and the thicknesses of the dielectric layers  202  and  204  are smaller than one half of a wavelength of the incident electromagnetic wave  212 , as shown in  FIG. 2A . Although having only two conductive layers  201  and  203  separated by one dielectric layer  202  are sufficient to construct a phase element of the present invention, a better performance is achieved with three conductive layers  201 ,  203 , and  205 , and two dielectric layers  202  and  204  therebetween ( FIG. 2A ). 
     In operation, as shown in  FIG. 2A , the incident electromagnetic wave  212  polarized along a Y-axis  207 , as indicated by the electric field vector E parallel to the Y-axis  207  ( FIGS. 2A ,  2 B, and  2 C), impinges on the front conductive layer  201  of the stack  210  inducing electric currents in the conductive strips  201 A,  203 A, and  205 A. The conductive strips  201 A,  203 A, and  205 A are electrically coupled to each other, the magnitude of electric coupling between the neighboring conductive strips  201 A being dependent on the value of the gap g 1  at a constant cell height s ( FIGS. 2A ,  2 B, and  2 C). In the stack  210 , the conductive strips  205 A have the same shape as the conductive strips  201 A, although in general it is not required, i.e. the conductive layers  201  and  205  can have different dimensions of the corresponding features, for example, different strip widths. 
     The electric coupling and associated gap g 1  variation along the X axis  206  ( FIG. 2B ) and the Y axis  207  are selected so as to cause a predetermined X, Y pattern of phase shift of a transmitted electromagnetic wave  214  as compared to the incoming electromagnetic wave  212 . Intuitively, one could expect that a continuous phase variation would require a continuous, smooth variation of the gap g 1  shown at  216  ( FIG. 2B ; “Continuous Phase Profile”). In practice, however, the continuous gap variation  216  can be replaced with a “digitized” gap variation  218  ( FIG. 2B ; “Discretized Phase Profile”), wherein the gap width g 1  stays constant across a single “zone” A-A′ having a width w, as shown in  FIG. 2B . The phase shift quantization error becomes negligible when the width w is sufficiently small, for example smaller than one half of the wavelength of the electromagnetic wave  212 , and preferably smaller than one tenth of the wavelength. Referring to  FIG. 2C , a single “zone” A-A′ of the conductive strip  201 A having the width w along the X-axis  206  and a height s along the Y-axis  207  is shown. Across the zone A-A′, the strip width a 1  and the gap width g 1  do not change. 
     In general, a reflected electromagnetic wave  213  is also formed as shown in  FIG. 2A . Its magnitude can be minimized upon proper impedance matching of the stack  210  to that of the environment (typically free space), thus improving the transmission loss performance of the phase element  200 . The impedance matching can be achieved upon a proper selection of the widths a 1  of the conductive strips  201 A and/or the gaps g 1  between the conductive strips  201 A, the widths a 2  of the conductive strips  203 A and/or the gaps g 2  between neighboring strips  203 A, the thickness h ( FIG. 2A ), and the dielectric constant ∈ r  ( FIGS. 2B ,  2 C). Particular examples of phase elements (PSS and PASS) construction, including thicknesses of layers, dielectric constants, feature shapes and dimensions, as well as resulting achievable magnitude of phase shift and associated transmission loss, will be given below. 
     Electromagnetic simulations of single unit cells have been performed to determine the resulting amplitude and phase in transmission as a function of the strip widths a 1  and a 2  and the corresponding gaps g 1  and g 2 . The electromagnetic finite-difference time-domain (FDTD) simulations were performed under an assumption of infinite periodicity along the X-axis  206  and the Y-axis  207  and a normal incidence of a plane electromagnetic wave. Other simulation methods can also be used to generate the results, including a finite element method (FEM) and a method of moments (MoM). 
     Turning now to  FIG. 3A , an equivalent electrical circuit of the stack  210  of  FIG. 2A  is shown including dielectric layers  202 ,  204 . The circuit has capacitances C 1  between neighboring strips  201 A, and also between the neighboring strips  205 A which, as was noted above, repeat the shape of the strips  201 A; capacitances C 2  between the strips  203 A; as well as capacitances C 3  and inductances L 1  between the respective neighboring conductive shapes disposed in neighboring conductive layers of the stack  210 . Thus, the conductive layers  201 ,  203 , and  205  form a two-dimensional pattern of transmission lines  300  going through the stack  210 . Referring to  FIG. 3B , the equivalent electrical circuit is redrawn for convenience, so that the transmission lines  300  can be better seen. Each transmission line  300  has a succession of the capacitances C 1  capacitively and inductively coupled through the capacitance C 3  and the inductance L 1 , respectively, to the capacitance C 2  and to capacitance C 1 . The strip widths/gaps and the resulting capacitances are selected so as to introduce the predetermined phase shift pattern into the electromagnetic wave  212  ( FIGS. 2A ,  2 B) propagating through the phase element to form the transmitted electromagnetic wave  214  ( FIGS. 3A ,  3 B). The electromagnetic wave  212  is polarized orthogonally to the strips  201 A,  203 A, and  205 A, as indicated with an arrow  311  ( FIG. 3A ). 
     Referring to  FIG. 3C , the transmission line  300  is redrawn for convenience. Amplitude and phase performance of the transmission line  300  can be calculated from the capacitances C 1 , C 2 , and C 3  and the inductance L 1  using analytical methods well known to those of skill in the art. Since the capacitances C 1 , C 2 , and C 3  and the inductance L 1  are determined by geometrical dimensions of the conductive strips  201 A,  203 A, and  205 A, as well as thickness and dielectric constant ∈ r  of the dielectric layers  202  and  204 , one can establish a relationship between these parameters and the produced phase shift magnitude, for example in form of a database. From this database, the shape of the conductive strips  201 A,  203 A, and  205 A and the gaps therebetween, required to obtain the pre-determined phase shift pattern, can be determined. 
     Turning now to  FIG. 3D , results of FDTD simulations of the phase element  200  are presented. The FDTD simulations were performed using Empire XCcel™ simulation software commercially available from IMST GmbH, Dusseldorf, Germany. The simulations were conducted over a frequency band including the frequency of 30 GHz, at cell width s (see  FIG. 2B ) of 3 mm, dielectric constant ∈ r  of 2.2 and the total thickness h of 1 mm. The results are shown in form of a graph of a best-case transmission loss in dB units (“Amplitude (dB)”) vs. achievable phase performance at 30 GHz in degrees (“Normalized Phase (Degrees)”) assuming infinite periodicity of the strips  201 A,  203 A, and  205 A and normal angle of incidence of the incident electromagnetic wave  212 . The phase shift introduced by areas having with no strips, that is, areas having only the dielectric layers  202  and  204 , is taken to be zero. Operating points close to the top-end of the vertical axis correspond to the case of 0 dB loss, or 100% transmission. Operating points away from this axis correspond to a reduced transmission coefficient, which translates into a lower efficiency of the phase element  200 .  FIG. 3D  shows the best transmission cases for different phase values. Inspection of  FIG. 3D  allows one to determine the transmission amplitude variation obtainable for a given range of transmission phase values. In  FIG. 3D , letters A, B, C, D, and E denote ranges of phase variation obtainable at different values of maximum allowable transmission loss. These ranges and values are summarized in the following Table 1. 
     
       
         
               
               
               
             
               
               
               
             
           
               
                 TABLE 1 
               
               
                   
               
               
                 Region 
                 Transmission Loss, dB 
                 Phase Shift Range, degrees 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 A 
                 0 
                 100 
               
               
                 B 
                 0.2 
                 120 
               
               
                 C 
                 0.4 
                 220 
               
               
                 D 
                 1.15 
                 295 
               
               
                 E 
                 2.2 
                 305 
               
               
                   
               
             
          
         
       
     
     The results of simulation presented in  FIG. 3D  and Table 1 prove that the phase element  200 , having total of 5 layers in the stack  210 , including three conductive layers  201 ,  203 , and  205 , can provide a wide range of achievable phase shift at moderate transmission loss penalties. By way of example, with three patterned conductive layers  301 ,  303 , and  305 , the practically achievable phase shift range is from 0 degrees to 300 degrees of phase and the transmission loss is less than 2.5 dB. 
     With four different patterned conductive layers, the achievable phase shift range can vary from zero degrees to slightly beyond 360 degrees of phase and the transmission loss is less than 2 dB, as the following Table 2 indicates. 
     
       
         
               
               
               
               
             
               
               
               
               
             
           
               
                 TABLE 2 
               
               
                   
               
               
                 Transmission 
                 Minimum Phase 
                 Maximum Phase 
                 Phase Shift Range, 
               
               
                 Loss, dB 
                 Shift, degrees 
                 Shift, degrees 
                 degrees 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 0 
                 −50 
                 −300 
                 250 
               
               
                 0.2 
                 −40 
                 −325 
                 285 
               
               
                 0.7 
                 0 
                 −335 
                 335 
               
               
                 1.7 
                 0 
                 −360 
                 360 
               
               
                   
               
             
          
         
       
     
     The relative permittivity ∈ r  of the dielectric layers  202  and  204  is preferably selected to be low, for example between 2 and 3; a value of 2.2 was used by the inventors for prototyping. The thickness h is to be kept relatively small, typically 1-1.5 mm at 30 GHz (Ka band), which corresponds to 0.1-0.15 of the free-space wavelength of the incoming electromagnetic wave  212 . In one embodiment, the thickness h is less than one third of the wavelength. The thickness of individual dielectric layers  202  and  204  is preferably less than one tenth of the wavelength. This combination of the dielectric constant ∈ r  and thickness h in the given range allows for achieving a large phase shift range with relatively high amplitude transmission. If high electromagnetic transparency is required, the phase element  200  can be designed to minimize the reflection and maximize the transmission of the incident wave  212 . 
     The phase element  200  is simple to fabricate using conventional etching processes resulting a thin, low-cost and lightweight antenna. When used as a lens, the phase element  200  offers similar performance to a dielectric piano-hyperbolic lens antenna over a reasonable bandwidth. When optimized for other applications, including amplitude control, the phase element  200  allows independent phase and amplitude shifting. The inventors discovered that, due to the inductive and capacitive coupling between the conductive layers  201 ,  203 , and  205 , a large phase shift, of the order of 300 degrees of phase, can be achieved; furthermore, quite remarkably, this large phase shift can be achieved at a low transmission loss of less than 2.5 dB. Furthermore, with four conductive layers, the phase shift range of 360 degrees is achievable at a transmission loss of below 2 dB. 
     Even with two electrically coupled conductive layers, the transmission loss of a PSS or a PASS element can be lessened, the phase shifting range of 120 degrees still being achievable. The electrical coupling between the neighboring conductive layers is characterized by the interlayer capacitance C 3 . For the reduction of the transmission loss, it is preferable that C 3  be equal to or greater than 20% of C 1  or C 2 , whichever is less. This is only possible when the thickness of the dielectric layer is small, typically less than a tenth of a free-space wavelength. In general, for multi-layer phase elements of the invention, it is preferable that the interlayer capacitance is equal to or greater than 20% of the capacitance between adjacent spatially varying features of the same patterned conductive layer. 
     Conventional photolithographic process has a limited achievable smallest gap size, thereby limiting a range of the capacitances C 1  and C 2  that are achievable in practice. If the unit cell height s (defined in  FIG. 2B ) is too small, the capacitances C 1  and C 2  will be very low and no practical phase shift range could be achieved. However, for the same gap size, C 1  and C 2  can be increased if the unit cell size is increased. A smaller unit cell size allows for a smaller quantization error, but it may lead to a smaller phase shift range, depending on the achievable gap size. Consequently, it may be preferred to increase the unit cell size, despite an increase of the quantization error. In the prototypes designed at 30 GHz, the unit cell size is 3 mm, or about one third of the wavelength. 
     To verify the performance of a phase element of the present invention, a number of prototypes of PSS and PASS elements were constructed and tested. One of the simplest phase elements is a phase diffraction grating. Referring to  FIG. 4 , a schematic side view of a phase diffraction grating  400  is shown. The phase diffraction grating  400  has parallel lines  410  introducing a periodic non-zero phase delay P into a wavefront of an incoming reference beam  402  having the electric field vector E directed as shown. The grating lines  410  are evenly spaced apart with a period Λ and extend in a direction perpendicular to the XZ plane (that is, in Y-direction going in and out of plane of  FIG. 4 ). A phase delay introduced by the grating lines  410  is constant in going along the grating lines  410  (that is, along the Y-direction). The “reference”beam  402  striking the grating  400  is diffracted on the hologram grating lines  410 , splitting into a “desired” beam  404  and a “mirror” beam  406 , which “mirrors” the desired beam  404 . A fraction of the reference beam  402  exits the diffraction grating  400  as an undiffracted, or zero-order beam  408 . This terminology comes from holography, wherein a reference beam is made to interfere with a desired beam to record a hologram, which upon subsequent illumination with the reference beam recreates the desired beam through the phenomenon of diffraction. The angle α of the desired beam depends on the wavelength of the reference beam  402  and the grating period Λ, as is well known to those of skill in the art. 
     Turning now to  FIG. 5 , a photograph of a prototype  500  of the diffraction grating  400  is shown in a plan view. The prototype  500  has the general structure of the phase element  200  of  FIGS. 2A to 2C , consisting of three conductive layers and two dielectric layers therebetween. The difference between the prototype  500  and the phase element  200  is that the conductive features  501 A have a fixed width, to have a fixed phase delay along grating lines  510 . The array of the conductive features  501 A and the array of the conductive features underlying the features  501 A together form the grating lines  510 . The grating lines  510  are spaced apart along the X-axis and run parallel to an Y axis  520 . The grating lines  510  correspond to the grating lines  410  of the diffraction grating  400 . A ruler  505  is shown in  FIG. 5  next to the diffraction grating  500  to show the scale of the diffraction grating  500 . The ruler  505  shows length in centimeters. 
     The phase delay introduced by the grating lines  410  depends on a gap  515  between neighboring conductive features  501 A, as well as a gap between the conductive features underlying the features  501 A. The gaps between the conductive features in the prototype grating  500  and the thicknesses and the dielectric constants of the dielectric layers were selected so as to minimize the transmission loss. The dielectric constant ∈ r  of the dielectric layers  502  was 2.2, and the total thickness h was about 1 mm. 
     The following Table 3 shows the ideal and the simulated transmitted phase shift values, as well as associated strip width parameters a 1  and a 2 . 
     
       
         
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
             
           
               
                 TABLE 3 
               
               
                   
               
               
                   
                 Ideal 
                 Ideal 
                 Simulated 
                 Simulated 
                   
                   
               
               
                   
                 Transmitted 
                 Transmitted 
                 Transmitted 
                 Transmitted 
               
               
                   
                 Phase, 
                 Amplitude, 
                 Phase, 
                 Amplitude, 
               
               
                 Region 
                 degrees 
                 dB 
                 degrees 
                 dB 
                 a 1   
                 a 2   
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 no 
                 0 
                 0 
                 0 
                 −0.437 
                 0 
                 0 
               
               
                 strips 
               
               
                 strips 
                 −180 
                 0 
                 −178.2 
                 −0.073 
                 2.55 
                 2.85 
               
               
                 510 
               
               
                   
               
             
          
         
       
     
     Referring now to  FIG. 6  with further reference to  FIG. 4  and  FIG. 5 , results of testing of the prototype  500  are presented. In  FIG. 6 , transmitted power at the frequency of 30 GHz (“Normalized Amplitude of Transmitted Power (dB)”) is plotted in dB units as a function of receiver angle in degrees (“Angle (degrees)”) for three types of diffraction gratings: the prototype  500  (shown in “+” signs), a dielectric phase grating (shown in circles), and an amplitude grating (shown in “X” signs). The diffracted beam  404  of  FIG. 4  is denoted in  FIG. 6  as “n=−1”. The undiffracted beam  408  of  FIG. 4  is denoted in  FIG. 6  as “n=0”. The incoming electric field E is shown in  FIG. 5  as parallel to the grating lines  510  and the Y-axis  520 . The transmitted power is normalized by the power of the reference beam  402  of  FIG. 4 . It is seen that the prototype  500  of  FIG. 5  outperforms the dielectric grating at the angle α≈47 degrees of the first-order diffracted “desired beam”  404  of  FIG. 4 . In fact, the fraction of power of the first-order diffracted beam  404  approaches a maximum attainable power to within 2% (39% for the prototype  500  compared to 41% for the maximum attainable power), which proves that a phase element of the invention is indeed capable of achieving very high efficiency and low transmission loss. 
     A cylindrical phase correcting Fresnel zone plate lens antenna, hereafter called a “cylindrical lens”, has also been built according to the invention. Referring now to  FIG. 7 , a schematic side view of a “cylindrical lens”  700  is shown. The “cylindrical lens”  700  is in fact a phase element (a PSS) of the invention, introducing into an incoming electromagnetic wave a pattern of phase shifts that is similar to one introduced by a dielectric lens of a cylindrical shape having a 90 degree step phase correcting pattern. Herein, it is called a “cylindrical lens”  700  for brevity. The cylindrical lens  700  has parallel bars  721 ,  722 ,  723 , and  724  introducing phase delays P 1  to P 4 , respectively, into a wavefront of an incoming reference beam  702  at a frequency of 30 GHz, emitted by a feed horn  701  disposed one focal length F away from the cylindrical lens  700  and polarized as shown by the electric field vector E. Similarly to the diffraction grating  400 , the bars  721  to  724  run parallel to the Y-axis, that is, in and out of the plane of  FIG. 7 . The effect of the cylindrical lens  700  is to collimate the beam  702  in the XZ plane. The phase delays P 1  to P 4  are equal to −270 degrees; −180 degrees; −90 degrees; and 0 degrees, respectively. The widths of the bars  721  to  724  are determined using Fresnel “zoning rule” for flat surfaces in geometric approximation given by 
     
       
         
           
             
               
                 
                   
                     
                       r 
                       i 
                     
                     = 
                     
                       
                         
                           
                             2 
                             ⁢ 
                             iF 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               λ 
                               0 
                             
                           
                           P 
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 i 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   λ 
                                   0 
                                 
                               
                               P 
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                   , 
                   
                     i 
                     = 
                     1 
                   
                   , 
                   2 
                   , 
                   … 
                   ⁢ 
                   
                       
                   
                   , 
                   N 
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     wherein r i  is the size of the i-th Fresnel zone along the axis r, F is the focal length, λ 0  is the free-space wavelength, P is the number of corrections, and N is the total number of zones. In the cylindrical lens  700 , P=4 and F=76.2 mm. The following Table 4 summarizes the ideal and the simulated transmitted phase shift values, as well as associated width parameters a 1  and a 2  of the bars  721  to  724 . The cell height s (see  FIG. 2B ) is 3.00 mm. 
     
       
         
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
             
           
               
                 TABLE 4 
               
               
                   
               
               
                   
                 Ideal 
                 Ideal 
                 Simulated 
                 Simulated 
                   
                   
               
               
                   
                 Transmitted 
                 Transmitted 
                 Transmitted 
                 Transmitted 
               
               
                 Bar 
                 Phase Delay, 
                 Amplitude, 
                 Phase Delay, 
                 Amplitude, 
               
               
                 numeral 
                 degrees 
                 dB 
                 degrees 
                 dB 
                 a 1   
                 a 2   
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 724 
                 0 
                 0 
                 0 
                 −0.437 
                 0 
                 0 
               
               
                 723 
                 −90 
                 0 
                 −89.3 
                 0 
                 2.32 
                 2.00 
               
               
                 722 
                 −180 
                 0 
                 −175.6 
                 −0.049 
                 2.54 
                 2.85 
               
               
                 721 
                 −270 
                 0 
                 −264.8 
                 −0.098 
                 2.83 
                 2.83 
               
               
                   
               
             
          
         
       
     
     Turning to  FIG. 8 , a photograph of a prototype  800  of the cylindrical lens  700  is shown in a perspective view. The prototype  800  has the general structure of the phase element  200  of  FIGS. 2A to 2C , consisting of three patterned conductive layers and two dielectric layers therebetween. Parallel bars  821 ,  822 ,  823 , and  824  of the prototype  800  correspond to the parallel bars  721  to  724  of the cylindrical lens  700  of  FIG. 7 , and the nominal widths a 1  and a 2  of the conductive strips of the parallel bars  821 ,  822 ,  823 , and  824  are the same as the widths a 1  and a 2  of the conductive strips of the parallel bars  721  to  724  of  FIG. 7 , shown in the two rightmost columns in Table 4 above. As is evident from  FIGS. 2B and 2C , the gap width g 1  is calculated as the difference between the cell width s and the parameter a 1 ; therefore, the smaller is the a 1  parameter, the larger is the gap g 1 . An insert  810  shows the structure of the parallel bars  821 ,  822 ,  823 , and  824  in more detail, the bars  821  to  823  differ by a size of respective gap  831 ,  832 , and  833  between conductive (copper) features, the gap  831  being the smallest and the gap  833  being the largest, in accordance with the corresponding values a 1  of Table 4 for the bars  731  to  733 . 
     The far-field gain patterns of the cylindrical lens  800  were measured in an anechoic chamber. A traditional cylindrical dielectric Fresnel zone plate lens of the same exact size in the XY plane but with a thickness of 15 mm and made of Plexiglas™, or poly(methyl methacrylate) (PMMA) was used for comparison. Referring now to  FIG. 9 , a far-field radiation pattern with the cylindrical lens  800  in the beam path is presented, wherein the far-field gain of the cylindrical lens  800  is plotted in dBi units as a function of the measurement angle in degrees. The beam frequency was 29.5 GHz, and the beam polarization is shown by the electric vector E in  FIG. 8 . The “+” signs ( FIG. 9 ) denote measurement points taken with the prototype cylindrical lens  800 ; the circles denote the measurements performed with the dielectric lens; and the “×” signs denote a reference measurement of the radiation emitted by the feed horn  701  (FIG. D, with the lenses removed from the beam path. One can see from  FIG. 9  that the transmission loss and beam width performance of the prototype lens  800  matches closely that of the dielectric lens, the beam width being slightly larger than that of the dielectric lens, and the sidelobe performance of the prototype cylindrical lens  800  being better than that of the dielectric lens. 
     Flat lenses with 90 degree, 45 degree and continuous phase correction according to the present invention were fabricated and tested. Referring to  FIGS. 10A and 10B , side and a plan views of a 90 degree phase-correcting flat lens  1000  ( FIG. 10B ) are shown, respectively. Referring to  FIGS. 11A and 11B  with further reference to  FIGS. 10A and 10B , photographs of prototypes  1100 A ( FIG. 11A) and 1100B  ( FIG. 11B ) of 90-degree and continuous phase-correcting lenses are presented, respectively. The prototype lenses  1100 A and  1100 B have a thickness t ( FIG. 10A ) of about 1.0 mm, which corresponds to 0.1 of a free-space wavelength at the frequency of 30 GHz, a diameter D ( FIG. 10A ) of 152.4 mm, and a focal distance F of 76.2 mm, which corresponds to the F/D ratio of 0.5. 
     Turning now to  FIGS. 12 and 13 , measured angular and frequency dependencies of gain obtained using the prototype lens  1100 A are shown, respectively. In  FIG. 12 , the gain of the prototype lens  1100 A is plotted in dBi units as a function of the measurement angle in degrees. In  FIG. 13 , the gain of the prototype lens  1100 A is plotted in dBi units as a function of frequency in GHz. In  FIGS. 12 and 13 , the performance of the prototype lens  1100 A is compared to the performance of a dielectric plano-hyperbolic lens, a 90 degree phase-correcting Fresnel zone plate, and a Fresnel zone plate. The squares (“ ”) correspond to the dielectric plano-hyperbolic lens; the crosses (“+”) correspond to the phase-correcting Fresnel zone plate; the circles correspond to the Fresnel zone plate antenna; and the cross signs (“×”) correspond to the prototype lens  1100 A. One can see by comparing the far-field gain profiles obtained with these lenses that the prototype lens  1100 A shows impressive results. Its boresight gain is only 0.3 dB less than the gain of the dielectric plano-hyperbolic lens at 30 GHz, with a weight reduction of almost ten times, a thickness reduction close to 40 times, and improved cross-polarization performance. The prototype lens  1100 A outperforms the 90 degree phase-correcting Fresnel zone plate in almost every aspect with significant practical advantages; its boresight gain is almost 1 dB higher at 30 GHz, its weight is more than 5 times less and its thickness is reduced by a factor of almost 8. Therefore, the results presented in  FIGS. 12 and 13  prove that a phase element of the present invention is a very viable and practical alternative for many applications. 
     The phase elements  200 ,  400 ,  500 ,  700 ,  800 ,  1000 ,  1100 A, and  1100 B are designed to operate in a single polarization perpendicular to the conductive stripes  201 A,  203 A, and  205 A, as shown by the direction of the electric field vector E in  FIGS. 2A-2C ,  4 ,  5 ,  7 ,  8 , and  10 B. This provides an advantage of combining the phase element and a polarizer in a single element. 
     A phase element of the present invention can also be constructed to operate with an unpolarized or randomly polarized electromagnetic wave. Referring to  FIG. 14 , a photograph of a polarization-insensitive lens prototype  1400  is shown. The conductive layers of the lens  1400  are patterned with not stripes but squares, thus achieving polarization insensitivity. The lens  1400  has three metal layers etched on two thin dielectric sheets which are then bonded using a bonding film and pressed together. The substrates have a dielectric constant of 2.2 and thickness of approximately 0.05 free-space wavelength each. Thus the total thickness is about 0.1 free-space wavelength, which leads to a very thin, flat, light-weight and low-cost lens configuration. The unit cell size for the square elements is 3×3 mm. The diameter D of the lens is 152.4 mm and the focal distance F is 76.2 mm, yielding the F/D ratio of 0.5. 
     To verify the performance of the lens  1400 , far-field radiation patterns of electromagnetic radiation at 30 GHz collimated with the lens  1400  were measured. Referring to  FIGS. 15 and 16 , the gain of the lens  1400  is plotted in dBi units as a function of the measurement angle in degrees in  FIG. 15  and the lens “roll” angle in degrees in  FIG. 16 , for different polarization configurations. In  FIG. 15 , squares correspond to the H-plane co-polarized (“co-pol”) measurements; “×” signs correspond to the H-plane cross-polarized (“X-pol”) measurements; “+” signs correspond to the E-plane co-polarized (“co-pol”) measurements; and circles correspond to the E-plane cross-polarized “X-pol”) measurements. In  FIG. 16 , squares correspond to realized gain in dBi, and “×” signs correspond to maximum cross-polarization level in dBi. In  FIG. 16 , the roll angle values are illustrated by four images of the lens at the top of the graph. 
     The boresight gain measured is 29.9 dBi and the maximum cross-polarization level is −8 dBi at 30 GHz at a lens rotation, or “roll angle”, of 0 degrees. A maximum realized gain of 30 dBi occurs at 29.9 GHz; by accounting for the return loss of 17.8 dB at that frequency, the corresponding aperture efficiency is calculated to be 44.6%. The realized gain results are marginally higher than that of the strip-based lens  1100 A, whereas the cross-polarization performance is slightly degraded. 
     Referring again to  FIG. 16 , a measured dependence of gain and cross-polarization on the roll angle is presented. Small sketches on top of the plot are added to help visualize the rotation angle.  FIG. 16  shows a marginal gain degradation of about 0.25 dB when the lens is rotated from 0 degrees to 45 degrees. The cross-polarization was found to increase by 13.5 dB as the lens  1400  was rotated from 0 degrees to 45 degrees, reaching a maximum value of 5.5 dBi. The cross-polar sidelobe level is still within an acceptable range for the worst-case at 45 degrees, with a value of −24 dB. 
     The inventors have determined that three conductive layers are sufficient in most cases to build a PSS, or a phase shifting element. If an independent phase and amplitude shifting (PASS) is required, then the number of conductive layers is preferably 4 or more. The electric coupling between neighboring layers facilitates decoupling of achievable amplitude and phase shift patterns. 
     The phase patterns achievable using a phase element of the present invention can be used to split the incoming electromagnetic beam into two or more beams, reshape/apodize/redirect the beam and so on. In general, any beam transformation achievable with a holographic element is also achievable with an element of the present invention which, in this respect, functions as a holographic element. Flat (low-profile) antennas and antennas hidden from view can be constructed using a phase element of the present invention. 
     The following general steps (a) to (d) can be followed to manufacture a phase element of the present invention: 
     (a) selecting a material and a thickness for each of the layers of the stack of alternating conductive and dielectric layers; 
     (b) selecting the number of the conductive layers in the stack; 
     (c) performing an electromagnetic simulation of the stack to obtain a dependence of a phase shift value on the spatially varying feature; and 
     (d) patterning the conductive layers to obtain the predetermined phase shift pattern, based on the dependence obtained in step (c). 
     Referring now to  FIG. 17 , a detailed breakdown of steps (a) to (d) for manufacturing the phase elements  200 ,  400 ,  500 ,  700 ,  800 ,  1000 ,  1100 A,  1100 B, or  1400  is presented. 
     At a step  1701 , the desired amplitude and phase (denoted as A and Φ) profile of the phase element is determined. This can be done using any readily available standard analytical electromagnetic or optical technique used for a lens or a grating design. 
     At a step  1702 , the substrate dielectric constant and thickness to be used in the phase element are selected. In practice, these values are selected based on commonly available microwave substrate materials. Typically, the dielectric constant of between 2 and 3 is selected, but higher values can be used as well. The substrate thickness depends on the wavelength of the electromagnetic beam. A value of 0.05 of the wavelength is typical. 
     At a step  1703 , the number of conducting (typically copper) layers is selected. The number of layers will depend on the required phase and or amplitude shift ranges. If only a phase shifting is required, then a minimum of two conductive layers are needed. Three layers are usually required to realize the full range of phase values with minimum transmission loss. If both phase and amplitude variations are required, then a minimum of three conductive layers is needed, but four conductive layers are preferable to achieve a much broader range of phase and amplitude variation. 
     At a step  1704 , an appropriate unit cell size is selected. For example, the cell height s is selected at this step. The unit cell is used in the subsequent analysis of the phase element. The phase element is analyzed by placing the various cell elements in an infinite periodic two dimensional array. Typical unit cell dimensions are on the order of a half-wavelength or less, to avoid high quantization errors. 
     At a step  1705 , full-wave electromagnetic simulations of the unit cell are run with proper electromagnetic boundaries to emulate an infinite periodic structure. The simulations of conducting features of dimensions and shapes are performed. 
     At a step  1706 , a database mapping the dimensions of the various sets of conducting shapes to the resulting amplitude and phase variations is generated. 
     At a step  1707 , the surface of the phase element is subdivided into unit cells of dimensions corresponding to the simulation cases in the steps  1704  and  1705 . For each subdivision or unit cell, the amplitude and phase profile (usually at cell center) is determined based on the amplitude and phase profile pre-determined in the step  1701 . In other words, the pre-determined amplitude and phase profile is broken into subdivisions corresponding to the unit cell size. 
     At a step  1708 , the conducting shape dimensions are matched to the corresponding amplitude and phase requirements using the database generated in step  1706 , for each conductive layer. 
     At a step  1709 , a layout of each conductive layer of the phase element is generated using a computer-aided design (CAD) tool or any other two-dimensional layout tool. The layout is generated based on the results obtained in the step  1708 . When using a photolithographic process such as wet chemical etching, layouts of masks can be generated for each conductive layer to be patterned. 
     The foregoing description of one or more embodiments of the invention has been presented for the purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise form disclosed. Many modifications and variations are possible in light of the above teaching. It is intended that the scope of the invention be limited not by this detailed description, but rather by the claims appended hereto.