WO2017040830A1

WO2017040830A1 - Tunable metamaterial systems and methods

Info

Publication number: WO2017040830A1
Application number: PCT/US2016/049965
Authority: WO
Inventors: Eric Black; Brian Mark Deutsch; Alexander Remley Katko; Melroy Machado; Jay Howard MCCANDLESS; Yaroslav A. Urzhumov
Original assignee: Elwha Llc
Priority date: 2015-09-04
Filing date: 2016-09-01
Publication date: 2017-03-09

Abstract

The present disclosure provides system and methods for optimizing the tuning of impedance elements associate with sub-wavelength antenna elements to attain target radiation and/or field patterns. Both static and variable (tunable) antenna systems may be manufactured. Static embodiments may be entirely passive in some embodiments. A scattering matrix (S-Matrix) of field amplitudes for each of a plurality of modeled lumped ports, N_a, may be determined that includes a plurality of lumped antenna ports, Na, with impedance values corresponding to the impedance values of associated impedance elements and at least one modeled external port, Ne, located external to the antenna system at a specified radius vector. Impedance values may be identified through an optimization process, and the impedance elements may be tuned (dynamically or statically) to attain a specific target radiation pattern.

Description

TUNABLE METAMATERIAL SYSTEMS AND METHODS

[0001] All subject matter of the Priority Application(s) is incorporated herein by reference to the extent such subject matter is not inconsistent herewith.

TECHNICAL FIELD

[0002] This disclosure relates to tunable metamaterial devices and the optimization of variable impedance elements associated therewith to attain target radiation and/or field patterns.

BRIEF DESCRIPTION OF THE DRAWINGS

[0003] FIG. 1 is a flow chart of one embodiment of a method for radiation patterning by optimizing variable impedance values associated with an S-Matrix that includes at least one lumped port external to an antenna system.

[0004] FIG. 2 illustrates an antenna system comprising an array of sub- wavelength antenna elements, according to one simplified embodiment.

[0005] FIG. 3A illustrates a close-up view of a section of an array of sub- wavelength antenna elements with associated variable impedance elements, according to one simplified embodiment.

[0006] FIG. 3B illustrates a view of a conceptual model of a single sub- wavelength antenna element with an associated impedance element, according to one simplified embodiment.

[0007] FIG. 4A illustrates an array of sub-wavelength antenna elements and associated variable impedance elements modeled as lumped ports, N_a, in an S-Matrix with a single external port, N_e, located physically external to the antenna system, according to one simplified embodiment.

[0008] FIG. 4B illustrates a radiation pattern formed to maximize a field amplitude of an S-Matrix element associated with an external port, N_e, located physically external to the antenna system by adjusting the impedance values associated with each of the lumped ports, N_a, defined by the sub-wavelength antenna elements and associated impedance elements, according to one embodiment. [0009] FIG. 4C illustrates a radiation pattern formed to maximize a field amplitude of S-Matrix elements associated with two external ports, N_e, located physically external to the antenna system and by minimizing the field amplitude of three other external ports N_e, according to one embodiment.

[0010] FIG. 5A illustrates an antenna system comprising an array of sub- wavelength antenna elements and associated variable impedance elements with two intended targets for radiation patterning.

[0011] FIG. 5B illustrates one embodiment showing the modeling of the antenna system in an S-Matrix of field amplitudes of a plurality of ports, N, including lumped antenna ports, N_a, and two lumped external ports, N_e.

[0012] FIG. 5C graphically illustrates the results of adjusting one or more variable impedance control inputs to modify one or more impedance values of one or more of the variable impedance elements to attain a desired radiation pattern, according to one embodiment.

[0013] FIG. 6 illustrates an edge-fed metamaterial surface antenna for wireless power transfer transmitting a convergent beam to an electrically small antenna, according to one embodiment.

[0014] FIG. 7 illustrates a center-fed metamaterial surface antenna for wireless power transfer transmitting a convergent beam to an electrically small antenna, according to another embodiment.

[0015] FIG. 8A illustrates a metamaterial surface antenna wireless power transmission system with utilizing two center-fed metamaterial surface antennas (one as a transmitter and one as a receiver) with a transfer efficiency approximating 100% at one-half of the Fraunhofer distance.

[0016] FIG. 8B illustrates a zoomed in view of the transmitter of the metamaterial surface antenna wireless power transmission system.

[0017] FIG. 9A illustrates a relatively inefficient metamaterial surface antenna with two sub-wavelength antenna elements per wavelength attaining a resolution similar to a conventional phases array.

[0018] FIG. 9B illustrates a more efficient metamaterial surface antenna with four sub-wavelength antenna elements per wavelength attaining an improved resolution for a more efficient wireless power transfer. [0019] FIG. 9C illustrates an even more efficient metamaterial surface antenna with eight sub-wavelength antenna elements per wavelength attaining even greater resolution for improved wireless power transfer.

[0020] FIG. 10A illustrates two relatively low-density metamaterial surface antennas used as a transmitter and a receiver in a wireless power transmission system.

[0021] FIG. 10B illustrates a graph of power transmittance for a range of frequencies using the relatively low-density metamaterial surface antenna wireless power transmission system, according to one embodiment.

[0022] FIG. 1 1 A illustrates two relatively high-density metamaterial surface antennas being used as transmitter and receiver in a wireless power transmission system.

[0023] FIG. 1 1 B illustrates a graph of improved power transmittance for a range of frequencies using the relatively high-density metamaterial surface antenna wireless power transmission system, according to another embodiment.

[0024] FIG. 12 illustrates a steerable metamaterial surface antenna wireless power transmission system, according to one embodiment.

[0025] FIG. 13A illustrates a graph of a metamaterial surface antenna wireless power transmission system with one metamaterial surface antenna acting as a transmitter and two metamaterial surface antennas acting as receivers, according to one embodiment.

[0026] FIG. 13B illustrates a graph of power transmittance relative to frequency for the dual receiver wireless power transmission system of FIG. 13A, according to one embodiment.

[0027] FIG. 14 illustrates an examples of unmanned aerial vehicle being wirelessly powered using a metamaterial surface antenna, according to one example embodiment.

[0028] FIG. 15 is a flow chart of a wireless power transfer method utilizing at least one metamaterial surface antenna, according to various embodiments.

[0029] FIG. 16 is a flow chart of a method of manufacturing a static wireless power transmitter, according to one embodiment. [0030] FIG. 17A illustrates a view of a conceptual model of a single sub- wavelength antenna element with an associated variable impedance element, according to one simplified embodiment.

[0031] FIG. 17B illustrates a view of a conceptual model of a single sub- wavelength antenna element with an associated static or fixed impedance element, according to one simplified embodiment.

[0032] FIG. 18 is a flow chart of a method of manufacturing a variable or tunable wireless power transmitter, according to one embodiment. DETAILED DESCRIPTION

[0033] Various embodiments, systems, apparata, and methods are described herein that relate to radiation and electromagnetic field patterning for communication, wireless power transfer, and other uses. Tunable metamaterial devices may be used to solve various electromagnetic field-based issues. By tuning individual elements of a densely packed metamaterial array, a wide variety of customizable radiation patterns may be attained. In many instances of this disclosure, metamaterial elements are used as example embodiments of sub- wavelength antenna elements. It is, however, appreciated that any of a wide variety of sub-wavelength antenna elements may be utilized that may or may not be classified as metamaterials.

[0034] Optimizing the tuning of the individual sub-wavelength antenna elements or groups of elements to attain a target radiation pattern may be done in a wide variety of manners. Many of these approaches, however, result in one or a small number of potential tuning solutions, without giving any assurance that any of these solutions represent the best solution (global optimum) and/or without providing any indication of how close to the global optimum the solution might be. Exhaustive computations using traditional methods may be too computationally intensive and/or infeasible for real-time tuning and for switching.

[0035] The complexity of the optimization problem may increase rapidly with the complexity of the device. In many embodiments, the complexity increases exponentially with the number of tunable or selectable elements. Thus, standard optimization approaches for tuning elements of an array of sub-wavelength antenna elements may require cost functions to be evaluated a large number of times. The number of tunable elements of the antenna system may be expressed as the degrees of freedom (DoF) of an antenna device. The DoF may be based on the number of antenna elements, associated tunable elements, and/or other tunable or adjustable components associated with an antenna system. As the DoF increases, the complexity is likely to increase exponentially, leading to optimization problems for which global or even quasi-global solutions are prohibitively computationally expensive for even moderate device complexity.

[0036] The present systems and methods provide optimization solutions for arrays of antenna elements and associated tunable (i.e., variable) lumped impedance elements in which the optimization solutions are rational multivariate functions. Accordingly, globally optimal solutions may be found by solving optimization problems that scale linearly with the DoF. The optimization approach can be simplified by making the cost function dependent on one matrix-value input (such as an impedance matrix, Z-Matrix) that can be calculated by performing no more than N linear system simulations. In the present application, N is an integer corresponding to the number of variable (e.g., tunable) impedance elements associated with an antenna system.

[0037] The cost function, although still nonlinear, may have a specific rational form that permits exhaustive enumeration of all local extrema. A global maximum (or minimum) can be selected from the local extrema. For rational function, the extrema are found by solving multivariate polynomial equations. Root enumeration and/or numerical calculations of the multivariate polynomial equations may allow for specialized treatment.

[0038] Tunable metamaterials, including two-dimensional metasurface devices, may comprise an array of unit cells. Each unit cell may be modeled as a sub- wavelength antenna element associated with one or more variable impedance elements. Each variable impedance element may be associated with one or more sub-wavelength antenna elements. Each impedance element or group of impedance elements may be variably controlled based on one or more impedance control inputs. The tuning may be a one-time static tuning that is performed during the manufacturing of the antenna device, or the tuning may be a dynamic process that occurs during operation by modifying one or more control inputs. [0039] As an example of static tunability, a metamaterial device may be manufactured using a 3D printer and the tuning may comprise selecting a material or combination of materials that results in a specific electromagnetic or electrical property for each of the impedance elements. By uniquely selecting the material or combination of materials for each of the unit cells, a metamaterial antenna device may be statically tuned to a specific radiation pattern. Alternatively, each unit cell may be modeled to include a lumped impedance element with (at least) one input and (at least) one output. The input(s) may be dynamically manipulated during operation to dynamically tune the antenna device in real-time to allow for a wide range of selectable target radiation patterns.

[0040] As previously described, the system may be modeled to include lumped impedance elements that can be passive, active, or variably passive-active. At a given frequency, each impedance element may be fully described by the complex value of its impedance "z." A positive integer N may be used to describe the number of tunable or variable lumped impedance elements in an antenna system. A diagonal square matrix of size N may have diagonal elements z_n representative of the nth elements of the antenna system. Alternatively, an N-dimensional complex vector, {z_n}, can be used to represent the n-valued list of impedance values.

[0041] Each variable impedance element may be modeled as a port (e.g., a lumped port and/or a wave port). A plurality of lumped ports, N, may include a plurality of lumped antenna ports, N_a, with impedance values corresponding to the impedance values of each of the variable impedance elements, and at least one lumped external port, N_e, that may or may not have a variable impedance or any impedance at all. That is, the z value of the modeled lumped external port, N_e, may be zero and represent an idealized shorted port. Alternatively, the z value of the modeled lumped external port, N_e, may be infinity and represent an idealized open port. In many embodiments, the z value of the external port, N_e, may be a complex value with a magnitude between zero and infinity.

[0042] Regardless of the impedance values of each of the lumped ports, N, including the lumped antenna ports, N_a, and the at least one lumped external port, Ne, each of the lumped ports (or in some embodiments wave ports) may have its own self-impedance and the network of ports may be described by an N x N impedance matrix (Z-Matrix) or by the equivalent inverse admittance matrix (Y- Matrix) where Y=Z^~1. Additionally, the network of ports can be modeled as an S- parameter matrix or scattering matrix (S-Matrix). The Z-Matrix and its inverse the Y-Matrix are independent from the specific z values of the ports because the matrix elements are defined as Znm=V_n/lm, where V_n and l_m are the voltage at port n and the current at port m, measured with all other ports open. That is, assuming port currents lk=0 for all k not equal to m or n. Similarly, for the admittance matrix, Ynm=lm/Vn, measured with all other ports open. Again, that is assuming port currents lk=0 for all k not equal to m or n.

[0043] The S-Matrix is expressible through the Z or Y matrices and the values of the lumped impedance elements as follows:

[0044] s = (fizfi - (jyzj + 1)^"1 = (l - ¾ ?)(i + ¾ ?f ¹

[0045] In the equation above, the "1 " represents a unit matrix of size N. The S- Matrix models the port-to-port transmission of off -diagonal elements of the N-port antenna system. In a lossless system, the S-Matrix is necessarily unitary. If elements s_n are the singular values of the S-Matrix, which are the same as the magnitudes of the eigenvalues, it can be stated that in a lossless system, all s_n=1 . In general, if s_max is the largest singular value, then for a passive lossy system it can be stated that s_n≤ s_max≤1 .

[0046] In an active system, these bounds still hold, however Smax can now exceed unity, representing an overall power gain for at least one propagation path. The Z and Y matrices are diagonalized in the same basis represented by a unitary matrix

U (t/t = tr¹ ), such that Z = U Z_dU, Y = U Y_dU, where the subscript d indicates a diagonal matrix, the elements of which are complex-valued eigenvalues of the corresponding matrix.

[0047] Generally speaking, unless Vz is proportional to a unit matrix (i.e., all lumped element impedances are equal), the S-Matrix will not be diagonal in the U- basis. In the U-basis, the general form of the S-Matrix is S = t/^†(l - ζΥ_άζ)(1 + ζΥ_αζ)^~1ϋ, where a new non-diagonal matrix ζ = t/Vzt/^† is used such that Vz = υ^ζυ, and Y_d is diagonal, though not generally commutative with ζ.

[0048] The S-Matrix of the system can be numerically evaluated with any desired accuracy by solving exactly N linear system problems (e.g., Znm=V_n/lm or Ynm= lm/Vn and the associated open port conditions described above). Such problems may be solved with Finite Element Methods (FEM) or finite-difference time-domain (FDTD) based solvers for linear electromagnetic systems. Examples of commercially available solvers include ANSYS HFSS, COMSOL, and CST. These numerical simulations incorporate various fine effects of the near-field and far-field interactions between various parts of the system, regardless of complexity.

[0049] The Z-Matrix and/or the Y-Matrix can be evaluated based on a knowledge of the S-matrix and the impedance values. With many FEM solvers, it is also possible to directly evaluate the Z-Matrix or the Y-Matrix, by solving N² linear problems. This approach, however, is N times less efficient than calculating the S-Matrix with a fixed set of port impedance values (known as reference impedance values), and transforming it to Z and/or Y.

[0050] In various embodiments, an antenna system may include a plurality of sub-wavelength antenna elements. The sub-wavelength antenna elements may each have a maximum dimension that is less than half of a wavelength of the smallest frequency within an operating frequency range. One or more of the sub-wavelength antenna elements may comprise a resonating element. In various embodiments, some or all of the sub-wavelength antenna elements may comprise metamaterials. In other embodiments, an array of the sub-wavelength antenna elements (e.g., resonating elements) may be collectively considered a metamaterial.

[0051 ] In various embodiments in which at least some of the sub-wavelength antenna elements comprise resonating elements, the resonating elements may have quality factors exceeding 10, 100, or even 1000 depending on the application. Some resonating elements may include superconductors to achieve the specific quality factors.

[0052] The resonating elements may have at least one electric dipole resonance at a frequency within or proximate the operational frequency band. The resonating elements have at least one magnetic dipole resonance at a frequency within or proximate the operational frequency band. In various embodiments, the resonance frequencies of the resonating elements are controlled by the impedance values of the lumped impedance elements. [0053] The sub-wavelength antenna elements may have inter-element spacings that are substantially less than a free-space wavelength corresponding to an operating frequency or frequency range. For example, the inter-element spacings may be less than one-half or one-quarter of the free-space operating wavelength. The inter-element spacings, or antenna element density, may be significant based on the intended use. For example, it is understood that a Gaussian beam is the tightest-waist beam possible given the laws of diffraction. The minimum waste diameter of a Gaussian beam is roughly one wavelength.

[0054] Many of the descriptions and illustrates provided herein are described relative to two-dimensional modeling (mathematically and/or graphically) that can be extrapolated for a three-dimensional model or system by one of skill in the art. Given the similarities between two-dimensional models and illustrates and three- dimensional models, the disclosure utilizes terms like "diameter" and "radius" that are applicable to the three-dimensional embodiments to describe attributes of the two-dimensional embodiments. Moreover, a discussion of antenna element density above may be understood with reference to the illustrates and two- dimensional descriptions as a linear density. However, it is appreciated that the same description can be understood in the context of a three-dimensional array of antenna elements as relating to a three-dimensional array of antenna elements.

[0055] In various embodiments, a waist equation for Gaussian beams may be used determine the spot size (or two-dimensional equivalents) for various far-field propagation distances. The waist equation is given by: w(z) =

where z_R = nw^/A is the Rayleigh range. The Rayleigh range can be understood as approximately the same as the Fraunhofer/far-field distance for an aperture equal to the minimum waist size, w₀.

[0056] For metamaterial surface antennas utilizing an array of sub-wavelength antenna elements, diffraction still limits the minimum possible beam waist to about half-wavelength. To achieve Gaussian beams with the tightest possible waists, the spatial Fourier spectrum of the radiated mode must contain transverse wavenumbers close to maximum free space number with an accurate spectrum sampling. If the sub-wavelength antenna element spacing is approximately two per wavelength, then the waist of the minimum waist size of a generated beam will be relatively large. Such an embodiment with approximately two sub-wavelength antenna per wavelength would correspond well to a conventional phases array. Such an embodiment might be suitable for some far-field applications, but is less useable for radiative near field applications.

[0057] In contrast, by increasing the element density to four, eight, or even more sub-wavelength antenna elements per wavelength, the tighter Gaussian beams can be formed with a greater power density concentrated within a minimum possible waste size.

[0058] The antenna system may be configured to operate in a wide variety of operating frequency ranges, including, but not limited to, microwave frequencies. The presently described systems and methods may be adapted for use with other frequency bands, including those designated as very low frequency, low frequency, medium frequency, high frequency, very high frequency, ultra-high frequency, superhigh frequency, and extremely high frequency or millimeter waves.

[0059] In various embodiments, for efficient wireless power transfer it may be desirable to utilize frequencies that propagate well within an expected usage environment. Molecular absorption and small-particle scattering increase rapidly with frequency. However, the smaller wavelengths (higher frequencies) allow for decreased aperture sizes of the array of sub-wavelength antenna elements. The tradeoff between keeping the aperture size small and having reasonably low attenuation may be decided on a case-by-case basis based on an expected usage environment and application sizing requirements. In some embodiments, the W-band may be particularly useful due to reduced atmospheric absorption between approximately 70 and 1 10 GHz. For example, frequencies between 93 and 96 GHz might be used in one particular embodiment.

[0060] Various embodiments, usage scenarios, applications, and system variables may make it desirable to operate a communication and/or wireless power transfer system in any of a wide variety of frequency ranges, including but not limited to the following known ranges: 30-300 kHz, 0.3-3 MHz, 3-30 MHz, ISM band (e.g., centered at 13.56 MHz), 6.78 MHz, and ISM band centered at 40.68 MHz. Additional possible operating frequency ranges include, but again are not limited to: between 300 MHz and 300 GHz, a decimeter wave frequency, the 2.45 GHz ISM band, a centimeter wave frequency, the 5.8 GHz ISM band, the 24.125 GHz ISM band, the 61 .25 GHz ISM band, the 94-96 GHz atmospheric transparency window, and/or a millimeter wave frequency.

[0061] In some embodiments, each of the sub-wavelength antenna elements is associated with at least one lumped impedance element. A common transmission line (TL) may be coupled to the sub-wavelength antenna elements via the lumped impedance elements. Alternative waveguides may be used instead of or in addition to TLs. Each lumped impedance element may have a variable impedance value that may be at least partially based on the connected sub-wavelength antenna element(s) and/or a connected TL or other waveguide(s). A waveguide or TL may be modeled as another port in the S-Matrix in some embodiments, such as in Heretic-like architectures with variable couplers.

[0062] In other embodiments, an array or other plurality of sub-wavelength antenna elements may be configured to scatter received electromagnetic fields transmitted by an independent and/or separate antenna transmitter and/or to an independent and/or separate antenna receiver. Thus, for many of the various embodiments described herein, it is appreciated that the array or other plurality of sub-wavelength antenna elements may be used to directly transmit and/or receive electromagnetic radiation or alternatively scatter (in a controlled manner) electromagnetic radiation to or from another antenna. Thus, the electromagnetic fields received by the plurality of sub-wavelength antenna elements may be generated by a source external to the wireless power transmitter or via a transmission line (e.g., a waveguide).

[0063] In embodiments in which the sub-wavelength antenna elements receive electromagnetic fields from a transmission line, the sub-wavelength antenna elements may receive the electromagnetic fields from one or locations on the transmission line to which the wireless power transmitter is evanescently coupled. The sub-wavelength antenna elements may receive electromagnetic fields generated by a slot in a transmission line, an inductive element coupled to a transmission line, a capacitive element coupled to a transmission, and/or other combination of discretely coupled or integrated devices.

[0064] Examples of transmissions lines include, but are not limited to, rectangular waveguides, parallel-plate waveguides, microstrip lines, and coaxial lines. Electromagnetic fields may also be received for scattering by the plurality of sub-wavelength antenna elements from one or more of a short dipole antenna, a resonant dipole antenna, a have-wavelength dipole antenna, a loop antenna, a patch antenna, a horn antenna, and a dish antenna.

[0065] The impedance of each of the lumped impedance elements may be variably adjusted through one or more impedance control inputs. The number of sub-wavelength antenna elements, associated impedance elements, and the number of impedance control inputs may be a 1 : 1 :1 ratio or an X:Y:Z, where X, Y, and Z are integers that may or may not be equal. For instance, in one embodiment there may be a 1 :1 mapping of impedance elements to sub- wavelength antenna elements while there is only one-tenth the number of impedance control inputs.

[0066] In various embodiments, the modeled lumped external port, N_e, may or may not be associated with a variable impedance element. In some embodiments, the lumped external port, N_e, is modeled as an external port with an infinitesimal volume located at a particular radius-vector relative to the antenna device. The lumped external port, N_e, may be in the far-field of the antenna device, the radiative near-field of the antenna device, or the reactive near-field of the antenna device.

[0067] In some embodiments, the lumped external port, N_e, may comprise a virtual port, an external region of space assumed to be a void, a region of space assumed to be filled with a dielectric material, and/or a location in space assumed to be filled with a conductive, radiative, reactive, and/or reflective material. In at least some embodiments, the lumped external port, N_e, comprises a receiving antenna.

[0068] The lumped external port, N_e, may also be modeled as a virtual external port, comprises a field probe, as measured by a non-perturbing measurement. In other embodiments, the virtual external port may represent a numerical field probe, as calculated using a numerical simulation. The field probe may correspond to a field amplitude measured by anon-perturbing measurement or a numerical field probe. The numerical field probe may correspond to an electromagnetic field amplitude calculated using a numerical simulation.

[0069] As previously described, in some embodiments, a unique lumped impedance element may be associated with each sub-wavelength antenna element. In other embodiments, a plurality of sub-wavelength antenna elements may be grouped together and associated with a single, variable, lumped impedance element. Conversely, a plurality of lumped impedance elements may be associated with a single sub-wavelength antenna element. In such an embodiment, the impedance of each of the plurality of lumped impedance elements may be controlled individually, or only some of them may be variable. In any of the above embodiments, X impedance control inputs may be varied to control the impedance of Y lumped impedance elements, where X and Y are integers that may or may not be equal.

[0070] As a specific example, 1 ,000 unique impedance control inputs may be provided for each of 1 ,000 unique lumped impedance elements. In such an embodiment, each of the impedance control inputs may be varied to control the impedance of each of the lumped impedance elements. As an alternative example, 1 ,000 unique lumped impedance elements may be controlled to be variably addressed by a binary control system with 10 inputs.

[0071] In some embodiments, one or more of the impedance control inputs may utilize the application of a direct current (DC) voltage to variably control the impedance of the lumped impedance element based on the magnitude of the applied DC voltage. In other embodiments, an impedance control input may utilize one or more of an electrical current input, a radiofrequency electromagnetic wave input an optical radiation input, a thermal radiation input, a terahertz radiation input, an acoustic wave input, a phonon wave input, a mechanical pressure input, a mechanical contact input, a thermal conduction input, an electromagnetic input, an electrical impedance control input, and a mechanical switch input. In various embodiments, the lumped impedance elements may be modeled as two-port structures with an input and an output.

[0072] The lumped impedance elements may comprise one or more of a resistor, a capacitor, an inductor, a varactor, a diode, a MEMS capacitor, a BST capacitor, a tunable ferroelectric capacitor, a tunable MEMS inductor, a pin diode, an adjustable resistor, an HEMT transistor, and/or another type of transistor. Any of a wide variety of alternative circuit components (whether in discrete or integrated form) may be part of a lumped impedance element. A lumped impedance element may comprise a liquid-crystal-filled tunable capacitor, a liquid- crystal-filled tunable resonator, a split ring resonator, an electric inductor-capacitor (ELC or electric-LC) resonator, a complementary split ring resonator, a tunable ferroelectric capacitor, a tunable MEMS inductor, and/or a complementary ELC resonator.

[0073] In some embodiments, at least one of the lumped impedance elements may comprise a memory element, such as a volatile or non-volatile memory element. A volatile memory element may include a transistor, a capacitor, and/or a flip-flop circuit. Volatile memory elements may be addressable using, for example, row and column addressing schemes.

[0074] Similarly, at least one of the lumped impedance elements may comprise a non-volatile memory element. The non-volatile memory element may include a floating-gate transistor, a non-volatile ferroelectric RAM element, and/or a nonvolatile magnetoresistive RAM element. Non-volatile memory elements may be addressable using, for example, row and column addressing schemes.

[0075] One or more of the lumped impedance elements may be connected to and collocated with a tunable microelectronic circuit. In various embodiments, some of the lumped impedance elements are connected to and collocated with tunable microelectronic circuits addressable using a row and column addressing scheme. In some embodiments, one or more of the lumped impedance elements are connected to and collocated with a tunable microelectronic circuit located at the intersection of a word line and a bit line, and connected to both the word line and the bit line.

[0076] One or more hardware, software, and/or firmware solutions may be employed to perform operations for radiation patterning by controlling the impedance values of the lumped impedance elements via the one or more impedance control inputs. For instance, a computer-readable medium (e.g., a non-transitory computer-readable medium) may have instructions that are executable by a processor to form a specific radiation pattern. The executed operations or method steps may include determining a scattering matrix (S-Matrix) of field amplitudes for each of a plurality of lumped ports, N.

[0077] The lumped ports, N, may include a plurality of lumped antenna ports, N_a, with impedance values corresponding to the impedance values of the plurality of physical impedance elements. In at least some embodiments, the modeled lumped ports, N, include at least one external port, N_e, that is located physically external to the antenna system. In some embodiments, the lumped ports, N, also include a TL or other waveguide as another lumped port for the calculation of the S-Matrix.

[0078] The S-Matrix is expressible in terms of an impedance matrix, Z-Matrix, with impedance values, z_n, of each of the plurality of lumped ports, N. Thus, by modifying one or more of the impedance values, z_n, associated with one or more of the plurality of lumped ports, N, a desired S-Matrix of field amplitudes can be attained. The operations or method steps may include identifying a target radiation pattern of the antenna system defined in terms of target field amplitudes in the S-Matrix for the at least one lumped external port, N_e.

[0079] An optimized port impedance vector {z_n} of impedance values z_n for each of the lumped antenna ports, N_a, may be calculated that results in S-Matrix elements for the one or more lumped external ports, N_e, that approximates the target field amplitude for a given operating frequency. Once an optimized {z_n} is identified that will result in the desired field amplitude values for the S-Matrix elements of the one or more lumped external ports, N_e, the variable impedance control inputs may be adjusted as necessary to attain the optimized {z_n}.

[0080] As an example, a target field amplitude in the S-Matrix for a lumped external port, N_e, may correspond to a null in the field amplitude of the target radiation pattern. Alternatively, the target field amplitude in the S-Matrix for a lumped external port, N_e, may be maximized.

[0081] Any number of lumped external ports, N_e, may be used as part of the S- Matrix calculation. Using a plurality of lumped external ports, N_e, may allow for the definition of a radiation pattern having a plurality of side lobes, main lobes, and/or nulls. Thus, the S-Matrix may be calculated with a plurality of lumped external ports located external to the antenna device. The target field amplitudes in the S-Matrix for each of the lumped external ports may correspond to a target radiation pattern for the antenna device for a specific frequency range.

[0082] In various embodiments, at least one of the plurality of lumped antenna ports, N_a, is strongly mutually coupled to at least one other lumped antenna port, N_a. In some embodiments, at least one of the lumped external ports, N_e, is mutually coupled to one or more of the lumped antenna ports, N_a. Strongly mutually coupled devices may be those in which an off-diagonal Z-Matrix element Zij, is greater in magnitude than one-tenth of the max(|Zii|, |¾|).

[0083] Determining an optimized {z_n} may include calculating an optimized Z- Matrix using one or more of a variety of mathematical optimization techniques. For example, the optimized {z_n} may be determined using a global optimization method involving a stochastic optimization method, a genetic optimization algorithm, a Monte-Carlo optimization method, a gradient-assisted optimization method, a simulated annealing optimization algorithm, a particle swarm optimization algorithm, a pattern search optimization method, a Multistart algorithm, and/or a global search optimization algorithm. Determining the optimized {z_n} may be at least partially based on one or more initial guesses.

[0084] Depending on the optimization algorithm used, the optimized values may be local optimizations based on initial guesses and may not in fact be true global optimizations. In other embodiments, sufficient optimization calculations are performed to ensure that a true globally optimized value is identified. In some embodiments, a returned optimization value or set of values may be associated with a confidence level or confidence value that the returned optimization value or set of values corresponds to global extrema as opposed to local extrema.

[0085] For gradient-assisted optimization, a gradient may be calculated analytically using an equation relating an S-parameter of the S-Matrix to the Z- Matrix and the optimized {z_n}. In some embodiments, a Hessian matrix calculation may be utilized that is calculated analytically using the equation relating the S- parameter to the Z-Matrix and the optimized {z_n}. A quasi-Newton method may also be employed in some embodiments. In the context of optimization, the Hessian matrix may be considered a matrix of second derivatives of the scalar optimization goal function with respect to the optimization variable vector.

[0086] In some embodiments, the global optimization method may include exhaustively or almost exhaustively determining all local extrema by solving a multivariate polynomial equation and selecting a global extrema from the determined local extrema. Alternative gradient-based methods may be used, such as conjugate gradient (CG) methods and steepest descent methods, etc. In the context of optimization, a gradient may be a vector of derivatives of the scalar optimization goal function with respect to the vector of optimization variables. [0087] Exhaustively determining all local extrema may be performed by splitting the domain based on expected roots and then splitting it into smaller domains to calculate a single root or splitting the domain until a domain with a single root is found. Determining the optimized {z_n} may include solving the optimization problem in which a simple case may include a clumped function scalar function with one output and N inputs. The N inputs could be complex z_n values and the optimized Z-Matrix may be calculated based on an optimization of complex impedance values of the z_n vectors.

[0088] The optimized {z_n} may be calculated by finding an optimized Z-Matrix based on an optimization of complex impedance values z_n. The optimized {z_n} may be calculated by finding an optimized Z-Matrix based on an optimization of roots of complex values of the impedance values z_n. The optimized {z_n} may be calculated by finding an optimized Z-Matrix based on an optimization of reactances associated with the impedance values of the impedance values z_n. The optimized {z_n} may be calculated by finding an optimized Z-Matrix based on an optimization of resistivities associated with the impedance values of the impedance values z_n.

[0089] In some embodiments, the optimization may be constrained to allow only positive or inductive values of reactances, or only negative or capacitive values of reactances. In other embodiments, the optimization of resistivities may be constrained to only allow for positive or passive values of resistivities.

[0090] The optimized {z_n} may be calculated by finding an optimized Z-Matrix based on an optimization of the impedance control inputs associated with the lumped impedance elements of each of the sub-wavelength antenna elements. The optimized {z_n} may be calculated by optimizing a nonlinear function. The nonlinear function may relate impedance values for each of the lumped antenna ports, N_a, as modeled in the S-Matrix and the associated impedance control inputs. In some embodiments, the nonlinear function may be fitted to a lower- order polynomial for optimization.

[0091] Mapping the Z-Matrix values to the S-Matrix values may comprise a nonlinear mapping. In some instances, the mapping may be expressible as a single- or multivariate polynomial. The polynomial may be of a relatively low order (e.g., 1 -5). The S-Matrix may comprise N values and the Z-Matrix may comprise M values, where N and M are both integers and equal to one another, such that there is a 1 : 1 mapping of S-Matrix values and Z-Matrix values. Any of a wide variety of mappings are possible. For example, the S-Matrix may comprise N values and the Z-Matrix may comprise M values, where N squared is equal to M. Alternatively, there may be a 2:1 or 3: 1 mapping or a 1 :3 or 2: 1 mapping.

[0092] The physical location of the at least one lumped external port, N_e, may be associated with a single-path or multipath propagation channel that is electromagnetically reflective and/or refractive. The multipath propagation channel may be in the near-field. In a radiative near-field, the multipath propagation pattern may be in the reactive near-field.

[0093] As previously described, the field amplitudes in the S-Matrix may be used to define a target radiation pattern. In some embodiments, the target radiation pattern of the antenna device may be defined in terms of a target field amplitude for a single linear field polarization. The target radiation pattern may be defined in terms of a plurality of field amplitudes for a plurality of lumped external ports, N_e. The target radiation pattern may be defined in terms of a target field amplitude for at least two linear polarizations.

[0094] The target field amplitudes for one or more lumped external ports, N_e, may be selected to decrease far-field sidelobes of the antenna device, decrease a power level of one or more sidelobes of the antenna device, change a direction of a strongest sidelobe of the antenna device, increase a uniformity of a radiation profile in the near-field, and/or minimize a peak value of field amplitudes in the near-field. The system may utilize a minimax approximation algorithm to minimize a peak value of field amplitudes in the near-field.

[0095] Determining the optimized {z_n} of impedance values for each of the lumped antenna ports, N_a, may include determining an optimized set of control values for the plurality of impedance control inputs that results in a field amplitude for the at least one lumped external port, N_e, in the S-Matrix that approximates the target field amplitude for a given operating frequency or frequency range.

[0096] In conformity with the antenna systems and associated methods described above, a plurality of lumped antenna ports, N_a, with impedance values corresponding to the impedance values of each of the plurality of lumped impedance elements may be considered jointly with one or more external ports, N_e, whose purpose is to account for the field intensity at a particular location exterior to the antenna system. The external port, N_e, may represent an actual receive antenna, in which case a known input impedance of that port may be assigned to the external port, N_e. In other embodiments, the one or more external ports, Ne, may be merely conceptual and used to quantify one or more field intensities at one or more locations. The external port, N_e, may be assumed infinitesimal in area and/or volume and located at a particular radius-vector r¾.

[0097] Regardless of the number of external ports, N_e, the total number of ports N will correspond to the number of lumped antenna ports, N_a, and the number of external ports, N_e. In some embodiments, a common port (e.g., a waveguide or TL) associated with the antenna system may also be considered. In any such embodiments, the total size of the system matrices will be generally of size N, which does not grow exponentially with the degrees of freedom or number of variable impedance elements.

[0098] The S-Matrix element S_1N represents the complex magnitude of field (e.g., electric field) at a particular location in space, given by the radius vector r¾, normalized to the field magnitude at the input port. The absolute value \S_1N \, or the more algebraically convenient quantity |5_1W|², quantifies the quality of field concentration at that point. Maximizing this quantity (or minimizing in the case of forming nulls) represents a generalized beamforming algorithm.

[0099] In some embodiments, the location r¾, is in the far-field of the rest of the system, and the algorithm yields directive beams in the far-field. In other embodiments, the point r¾ is in the radiative near-field of the rest of the system, and the algorithm yields field focusing to that point. In still other embodiments, the point r¾ is within the reactive near-field of at least one part of the rest of the system, and the algorithm maximizes electric field intensity and electric energy density at that point.

[00100] To find all local optima and the global optimum we can use the equation q_n≡ _Λ/] which characterizes the individual port impedances z_n. The equation above, S = t/^†(l - ζΥ_άζ)(1 + ζΥ_άζ)^~1υ, is a rational (and meromorphic) analytical function of {q_n}.

[00101] To make this function bounded, and find its maxima that are attainable in a passive system, the function may be restricted to the multidimensional segment satisfying Re(z_n) > Ο, η = Ι, .,. , Ν. Equivalently, this condition is -π/2≤ argz_n < π/2, and consequently -π/4 < arg qr_n < π/4.

[00102] To reduce this problem to real values, each q_n variable can be expressed through real variables, q_n = p_n + ίξ_η. In this manner, the real valued function |5_1W |² is now a function of 2N real variables ρ_η, ξ_η, which is a rational function comprising a ratio of two 2N-variate polynomials.

[00103] In some embodiments, the resistance of each lumped element can be neglected by assuming Re(z_n) = 0, z_n = ix_n, with the real reactance values x_n. In such embodiments, the system as a whole is still assumed passive and lossy with the losses occurring on the paths between the ports and incorporated into the Z- Matrix (or Y-Matrix). This approximation satisfies the passivity constraints and also reduces the number of variables to N because z z→ i , and x is purely real.

[00104] The function |5_1W |² is necessarily bounded for a passive system, and therefore it has a finite global maximum as a function of real-valued variables Ρη, ξ_η. Moreover, it has a finite number of local extrema. These extrema can be found by solving a set of 2N multivariate polynomial equations given by the standard zero gradient condition at the extremum: ^{a | LJV L} = Q ^{A | IAF L} ₌ Q_{; N =} 1 N.

[00105] In the simplified approach above, there are N unknowns χ_η = _Λ/½ and N extremum conditions, so ^{a| ljv l} = 0, n = 1, ... , N.

dXn

[00106] Once these extrema are found, the extremal values of the function are evaluated numerically, and the global maximum is determined by choosing the largest local maximum. A similar approach can be performed to identify one or more minimums to attain a target radiation pattern with a null at one or more specific radius vectors r¾.

[00107] Numerical and symbolic-manipulation algorithms exist that take advantage of the polynomial nature of the resulting equations. For example, Wolfram Mathematica™ function Maximize supports symbolic solving of the global optimization problem for multivariate polynomial equations, unconstrained or with multivariate polynomial constraints. This function is based on a Groebner- basis calculation algorithm, which reduces the multidimensional polynomial system to a triangular system, which is then reduced to a single scalar polynomial equation by back-substitution. Sim ilar functionality exists in other software packages, including MATLAB™ with Symbolic Math Toolbox™, Maple™ and so on.

[00108] As previously discussed, once values are determined for each of the z_n for the variable or tunable lumped impedance elements associated with the sub- wavelength antenna elements, each of the impedance elements can be tuned. In some embodiments, the tuning is static and the impedance values are set at the manufacturing stage. In other embodiments, a physical stimulus (e.g. , mechanical, electric, electromagnetic, and/or a combination thereof) may be used to dynamically tune impedance elements to dynamically modify the radiation pattern of the antenna system during operation.

[00109] Depending on the manufacturing techniques employed (e.g. , 3D printing) the calculated values of optimum impedance values may translate trivially into the choices made for the selectable impedance elements. In contrast, for the dynamically adjustable, variable, or tunable impedance elements, there is generally a non-trivial relationship between the complex impedance of the elements and the stimuli that control them. In some embodiments, the relationship between the complex impedance of the impedance elements and the control inputs may be based on a magnitude of an applied signal. Appreciating that the magnitude of the stimulus may be binary in some embodiments (i.e. , on or off), the relationship may be modeled as z_n = f_n(s_n) , where s_n, is the real-valued magnitude of the stimulus. The function f_n(s_n) can be fitted with a polynomial order S, and substituted into |5_1W |². The functions f_n can be all the same when identical dynamically tunable elements are used, in which case there will be N extremum conditions for N real variables s_n, each of which is still a rational function.

[00110] In the lowest-order approximation, the fitting polynomial can be linear (S=1 ), in which case the complexity of the extremum problem is still ^{a |}^^1Jvl = 0,

"Xn n = Ι, .,. , Ν. The quality of a polynomial approximation depends greatly on the practically available range of the stimulus, or the range chosen for other practical considerations. Because the s_n variables are restricted to a finite interval, the optimization problem can be solved with the corresponding constraints. When the optimization problem is solved by exhaustive enumeration of the extrema, these constrains are applied trivially and the local extrema not satisfying the constraints are excluded from the enumeration.

[00111] A wide range of adaptive beamforming applications are contemplated and made possible using the systems and methods described herein. For example, in some embodiments, beamforming may include a multipath propagation channel involving one or more reflective, refractive, or generally scattering object. In many embodiments, the relevant properties of the multipath propagation channel are incorporated into the Z-Matrix. Numerical simulations that lead to a calculation of the Z-Matrix may include a model of such a channel. A model of the multipath propagation channel can be simulated using any of a wide variety of simulation software packages, including, for example, ANSYS HFSS, COMSOL RF, CST MWS, etc.

[00112] In some embodiments, a particular linear field polarization can be achieved by considering the output port to be a port susceptible to only one linear polarization. For instance, a lumped (electrically small, single-mode) port is susceptible to a linear polarization with the electric field directed across the gap of the port.

[00113] In some embodiments, a target radiation pattern may be identified that includes a combination of two linear polarizations, including without limitation a circular polarization, that can be achieved by considering two co-located output ports, each of which is susceptible to only one linear polarization. In such an embodiment, the system matrices may be slightly increased by the addition of more external ports, N_e, but the addition of a few external ports increases the complexity by a relatively small constant value and will not change the general course of the algorithms and methods described herein.

[00114] In some embodiments, multiple beams can be formed simultaneously (the process known as multi-beam forming) by considering M output ports located in different directions with respect to the rest of the system. The size of the system matrices may then correspond to N=Na+M+1 , which does not change the general course of the algorithm and does not exponentially increase the complexity. [00115] As previously discussed, approximate nulls of the field can be formed, either in the far-field or near-field, by considering a minimization problem for the rational function of the equations above. Similarly, a required level of sidelobe suppression for a target radiation pattern can be attained by maximizing the function F = |5_1W |² - a|5_{i w+1} | , where the N^th port measures the field intensity in one direction, the (N+1 )^th port measures field intensity in a specified sidelobe direction, and a is a selectable weight coefficient reflecting the degree to which sidelobe suppression should be achieved.

[00116] It is appreciated that the equation above can be readily generalized to include any number of sidelobes in any number of directions. Thus, it is appreciated that instead of optimizing the impedance values themselves, a function relating the impedance control inputs to the impedance values of the variable (i.e. , tunable) impedance elements may be substituted into the equations to allow for the direct optimization of the impedance control inputs.

[00117] The presently described systems and methods may be used to form beams or pattern radiation generation or scattering to optimize wireless power transfer between a transmitting antenna system and one or more receivers. Similarly, multiple transmitting antennas may be used to power a single receiver. The number of receivers to transmitters used for wireless power transfer may be N:M where N and M are integers that are not necessarily equal. The system may utilize the N transmitters and the M receivers and determine how to efficiently transmit power from the N transmitters to the M receivers in an efficient manner.

[00118] In one embodiment, the target electromagnetic radiation pattern is selected to optimize a power transfer from a first group of sub-wavelength antenna elements to an external lumped port that is collocated with a second group of sub-wavelength antenna elements. The first group of sub-wavelength antenna elements may be facing the second group of sub-wavelength antenna elements and/or may be mobile with respect to the second group of sub- wavelength antenna elements. In other embodiments, the second group of sub- wavelength antenna elements may be mobile with respect to the first group of sub-wavelength antenna elements.

[00119] Optimizing the power transfer may include maintaining the power transferred to a selected external lumped port below a threshold level or above a threshold level. For example, certain efficiencies might be attained above and/or below certain threshold levels. In cold situations, maintaining a transfer above at threshold level may prevent freezing and/or maintain various components of a charging or power system above a predetermined temperature. Optimizing the power transfer may include minimizing the power transferred to one lumped external port while maintaining the power transferred to another lumped external port at or above a threshold level.

[00120] Optimizing the power transfer may include minimizing the power transferred to one lumped external port while maintaining the power transferred to another lumped external port at or above a threshold level. The second lumped external port may be a virtual port corresponding to a location of a human or a location of a device. Again, optimizing the power transfer may include maintaining the power transfer above or below a threshold level. Part of the optimization may be based on heat generation associated with the power transfer above a threshold level or below a threshold level. In some embodiments, a receiver may communicate with a transmitter to ensure threshold levels are not exceeded. For example, it may be necessary to maintain a power transfer above a predetermined level to ensure that a liquid crystal based technology remains sufficiently warm to operate as intended.

[00121] In some embodiments, maintaining the power transfer below a threshold level may be beneficial if, for example, there may be a region of space within which we don't want to overexpose objects, equipment, or living things to power densities greater than a predetermined threshold. A wireless power transfer system may have a dynamic exclusion zone that can specifically limit a power density for given regions of space. Such exclusion zones may be dynamic in that they may change as the antenna (transmitter and/or receiver) moves and/or as other objects, equipment, and/or persons move within the environment. Identifying the zones that should be excluded may be a different problem solved in any number of ways as compared to the wireless power transfer system's ability to create such zones of exclusion.

[00122] Examples of components and devices that may be associated with a wireless power receiver include, but are not limited to, battery charging stations, cells within a battery, a rectifying circuit, personal electronic devices, cell phones, laptops, tablets, transformer circuits, frequency converter circuits, multiplier circuits, components of motor/electric/hybrid/fuel-cell vehicles, remotely operated vehicles, medical implants, and/or a medical device temporarily or permanently residing within a patient (e.g., within a lumen of a human).

[00123] FIG. 1 is a flow chart of one embodiment of a method 100 for radiation patterning by optimizing impedance values associated with an S-Matrix that includes at least one lumped port external to an antenna system. The method illustrated may be computer-implemented via software, hardware, firmware, and/or a processor or microprocessor. In other embodiments, the method may be implemented using an application specific integrated circuit, a field-programmable gate array, other hardware circuitry, integrated circuits, software, firmware, and/or a combination thereof. As illustrated, an S-Matrix may be determined that includes field amplitudes for each of a plurality of lumped ports, N, associated with an antenna device, at 1 10.

[00124] The N lumped ports may include a plurality of lumped antenna ports, N_a, wherein each lumped antenna port corresponds to an impedance value of a lumped impedance element in communication with at least one sub-wavelength antenna element of an antenna device, wherein the impedance value of each of the lumped impedance elements is variable based on one or more impedance control inputs, and at least one lumped external port, N_e, located physically external to the antenna device. In various embodiments, the S-Matrix may be expressible in terms of an impedance matrix, Z-Matrix, with impedance values, z_n, of each of the plurality of lumped ports, N.

[00125] Once the S-Matrix has been determined, a target radiation pattern of the antenna device may be defined in terms of target field amplitudes in the S-Matrix for the at least one lumped external port, N_e, at 120. An optimized port impedance vector, {z_n}, of impedance values for each of the lumped antenna ports, N_a, may then be determined, at 130, that results in an S-Matrix element for the at least one lumped external port, N_e, that approximates the target field amplitude for an operating frequency or operating frequency range.

[00126] FIG. 2 illustrates an antenna system comprising an array of sub- wavelength antenna elements 200, according to one simplified embodiment. The sub-wavelength antenna elements 200 may be associated with a plurality of variable or tunable impedance elements.

[00127] The array of sub-wavelength antenna elements 200 may form a two- dimensional array as illustrated, or may form a three-dimensional array. The lumped impedance elements may form any of a rectangular two-dimensional array, a square two-dimensional array, a triangular two-dimensional array, and a hexagonal two-dimensional array.

[00128] The lumped impedance elements may also be configured as a concentric-circular array with elements aligned along concentric circles. In other embodiments, the lumped impedance elements form a concentric-circular array with elements aligned along radial lines from a common center.

[00129] FIG. 3A illustrates a conceptual model of an antenna system 300 showing a section of an array of sub-wavelength antenna elements 301 with associated variable lumped impedance elements, z_n, 303 according to a simplified embodiment. As previously described, the sub-wavelength antenna elements 301 may have inter-element spacings that are substantially less than a free-space wavelength corresponding to an operating frequency or frequency range of the antenna system 300. For example, the inter-element spacings may be less than one-half or one-quarter of the free-space operating wavelength.

[00130] As shown, each of the sub-wavelength antenna elements 301 is associated with at least one lumped impedance element 303. A common TL 305 may be coupled to the sub-wavelength antenna elements via the lumped impedance elements and may be modeled as another lumped impedance element or may be incorporated based on the effects of the TL 305 or other common waveguide on each of the lumped impedance elements 303. Each lumped impedance element 303 may have a variable impedance value that is set during manufacture or that can be dynamically tuned via one or more control inputs. The 1 : 1 ratio of lumped impedance elements 303 and sub-wavelength antenna elements 301 is merely exemplary and other ratios are possible.

[00131] In some embodiments, the sub-wavelength antenna elements may be divided into two or more groups that are separated from one another by no more than half of an operating wavelength. Each group of sub-wavelength antenna elements may be spatially separated from each other group of sub-wavelength antenna elements by at least a distance exceeding that of half of an operating wavelength.

[00132] The separation of each group from each other may be greater than a Fraunhofer (far-field) distance associated with an aperture diameter of a largest of the at least two groups. In other embodiments, the separation from each group may be less than a Fraunhofer distance. In other embodiments, the separation of each group may be shorter than a diameter of a largest of the at least two groups or alternatively the separation distance may be associated with the free-space operation wavelength (e.g. , longer, the same as, or shorter). In many embodiments, the individual elements and/or groups of elements may be in the reactive near-field of one another. The groups of sub-wavelength elements may be part of a receiver antenna element physically coupled to a receiver device.

[00133] The array of antenna elements 300 need not be planar as illustrated in FIG. 3A, though it may be. In some embodiments, two groups of sub-wavelength antenna elements are coplanar with one another and at least one other group is non-co-planar with the first two, co-planar groups.

[00134] FIG. 3B illustrates a close-up view 350 of a model of a single sub- wavelength antenna element 360 with an associated lumped impedance element, Zn, 365, and an impedance control input 370 that can be used to control or vary the impedance of the lumped impedance element, z_n, 365, according to one simplified embodiment.

[00135] Sub-wavelength antenna element 360 may be arranged in an array and may be configured for submersion in a fluid, such as fresh water, salt water, brackish water, or a particular gaseous environment.

[00136] FIG. 4A illustrates an array of sub-wavelength antenna elements 450 and associated variable lumped impedance elements with variable impedances Zn, modeled as lumped ports, N_a, in an S-Matrix with a single external port, N_e, 475 that is located physically external to the antenna system 450, according to one simplified embodiment.

[00137] In various embodiments, the modeled lumped external port, N_e, 475 may be associated with a variable impedance element, as illustrated. In some embodiments, the lumped external port, N_e, 475 is modeled as an external port with an infinitesimal volume located at a particular radius-vector relative to the antenna device. The lumped external port, N_e, 475 may be in the far-field of the antenna device, the radiative near-field of the antenna device, or the reactive near-field of the antenna device.

[00138] In some embodiments, the lumped external port, N_e, 475 may comprise a virtual port, an external region of space assumed to be a void, a region of space assumed to be filled with a dielectric material, and/or a location in space assumed to be filled with a conductive, radiative, reactive, and/or reflective material. In at least some embodiments, the lumped external port, N_e, 475 comprises or corresponds to the location of a receiving antenna or portion thereof.

[00139] FIG. 4B illustrates a radiation pattern 480 formed to maximize a field amplitude of an S-Matrix element associated with an external port, N_e, 475 located physically external to the antenna system by adjusting the impedance values, z_n, associated with each of the lumped ports, N_a, defined by the sub-wavelength antenna elements and associated lumped impedance elements in the antenna system 450, according to one embodiment.

[00140] FIG. 4C illustrates a radiation pattern 480 formed to maximize a field amplitude of S-Matrix elements associated with two external ports, N_e, 475 located physically external to the antenna system and by minimizing the field amplitude of three other external ports, N_e, 476 according to one embodiment.

[00141] FIG. 5A illustrates an antenna system 550 comprising an array of sub-wavelength antenna elements and associated variable impedance lumped elements with two intended targets 590 and 595 for radiation patterning.

[00142] FIG. 5B illustrates an embodiment showing the modeling of the antenna system in an S-Matrix of field amplitudes of a plurality of ports, N, including lumped antenna ports, N_a, of the sub-wavelength antenna elements and associated variable impedance elements 550 and two lumped external ports, N_e, 575.

[00143] As previously described, multiple beams can be formed simultaneously or in switch-mode by considering M output ports (e.g., the two different external ports, Ne, 575) located in different directions and potentially very distant from one another. The size of the system matrices that must be optimized may then correspond to N=Na+M+1 , but again, this does not change the general course of the algorithm nor does this increase the complexity exponentially. [00144] As previously discussed, approximate nulls of the field can be formed, either in the far-field or near-field, by considering a minimization problem for the rational functions described in detail above. To attain a specific target radiation patter, a required level of sidelobe suppression can be attained by maximizing the function F = |5_1W |² - a|5_{i w+1} | , where the N^th port measures the field intensity in one direction, the (N+1 ) port measures field intensity in a specified sidelobe direction, where a is a selectable weight coefficient reflecting the degree to which sidelobe suppression should be achieved.

[00145] FIG. 5C graphically illustrates the results of adjusting one or more variable impedance control inputs to modify one or more impedance values of one or more of the variable lumped impedance elements associated with the sub- wavelength antenna elements of the antenna system 550 to attain a desired radiation pattern 580 based on the two lumped external ports, N_e, 575, and the associated targets 590 and 595.

[00146] FIG. 6 illustrates an edge-fed metamaterial surface antenna 625 for wireless power transfer transmitting a convergent beam 600 to an electrically small antenna 650, according to one embodiment. Gaussian beamforming can generally be understood as a generalization of the holographic beamforming techniques described above. Accordingly, many of the same features, variations, approaches, equations, and/or calculations may be applicable to Gaussian beamforming.

[00147] The illustrated edge-fed metamaterial surface antenna 625 may, as an example, be four wavelength wide. The width of the beam near the electrically small antenna 650 may be approximately two wavelengths wide. The holographic Gaussian beamforming techniques described herein allow for beams with a minimum waist size that is substantially smaller than the transmitting aperture diameter. Such convergent beams may only be possible within the Fraunhofer distance of the transmitter, where the Fraunhofer distance, DFr is defined as: DFr = 2 D²/A, where D represents the aperture diameter of the transmitter and λ represents the wavelength. The region within the Fraunhofer distance of the transmitter may be referred to as the radiative near-field (RaNF).

[00148] FIG. 7 illustrates a center-fed metamaterial surface antenna 725 for wireless power transfer transmitting a convergent beam 700 to an electrically small antenna 750, according to another embodiment. The center-fed metamaterial surface antenna 725 illustrates a narrow-waist beam transmitted from a transmitter for wireless power transfer. For both center- and edge-fed metamaterial surface antennas (e.g., 625 and 725), the relationship between minimum waist size, transmitter size, and transmission distance may be described as: Dmax = DTX DR_x / λ, where Dmax is the maximum transfer distance at which a receiver with an aperture diameter, DRX, that corresponds to the minimum waist diameter can be used with a receiver with an aperture diameter, DTX, can be used for a given wavelength, λ. Thus for distances up to Dmax, power transmission can be attained with 50-90% efficiency.

[00149] As an example, for λ = 3mm (100 GHz) and a receiver with an aperture diameter of 30 cm, the maximum transfer distance, Dmax, would be 100 multiplied by the transmitter aperture diameter, DTX. Thus, a transmitter with an aperture diameter of one meter could efficiently transmit up to 100 meters.

[00150] FIG. 8A illustrates a metamaterial surface antenna wireless power transmission system with utilizing two center-fed metamaterial surface antennas (one as a transmitter 825 and one as a receiver 875) with a transfer efficiency approximating 100% at one-half of the Fraunhofer distance. At one half of the Fraunhofer distance, the beam waist is approximately equal to the transmitter 825 diameter and the receiver 875 diameter of approximately the same size can be used to intercept nearly 100% of the transmitted power. As the transfer distance between the transmitter 825 and the receiver 875 increases to the Fraunhofer distance, the minimum achievable beam 800 waist approaches the far-field estimate which is roughly D²/A. Thus, for a fixed size receiver, efficiency decreases with the inverse square of the distance as is understood with far-field transmission.

[00151] FIG. 8B illustrates a zoomed in view of the transmitter 825 of the metamaterial surface antenna wireless power transmission system 800. The illustrated embodiment is approximately eight free-space wavelengths, λ, in diameter. The illustrated embodiment is simulated with 32 unit cells, with one tunable lumped element per cell, making the array pitch λ/4. As the Fraunhofer distance may be simply defined as 2*D²/A, one half of the Fraunhofer distance can be expressed as D²/A. As the diameter of the transmitter 825 is 8^*λ, the transfer distance amounts to (8*λ)²/λ, which is equal to 64λ. Described in terms of antenna aperture, the transfer distance equals eight antenna diameters.

[00152] Generally, the laws of diffraction limit the minimum possible beam waist to about half-wavelength. To achieve Gaussian beams with the tightest possible waists, the spatial Fourier spectrum of the radiated mode must contain transverse wavenumbers close to maximum free space number, and the spectrum must be accurately sampled.

[00153] In FIGS. 9A-9C, the number of radiating elements of the metamaterial surface transmitter is shown with two, four, and eight radiating elements. As illustrated in FIG. 9A, a transmitter 925 with only two elements creates a relatively inefficient metamaterial surface antenna that is only able to attain a resolution similar to that of a conventional phases array. The waist size of the transmitted beam 900 at the receiver 950 is relatively wide.

[00154] FIG. 9B illustrates a more efficient metamaterial surface antenna 926 with four sub-wavelength antenna elements per wavelength attaining an improved resolution 901 for a more efficient wireless power transfer to the receiver 950.

[00155] FIG. 9C illustrates an even more efficient metamaterial surface antenna 927 with eight sub-wavelength antenna elements per wavelength attaining even greater resolution 902 for improved wireless power transfer to the receiver 950.

[00156] FIG. 10A illustrates two relatively low-density metamaterial surface antennas 1025 and 1075. A beam 1000 transmitted from the transmitter 1025 to the receiver 1075 of the wireless power transmission system is efficiently collected. In the illustrated embodiment, both the transmitter 1025 and the receiver 1075 are center-fed and the transfer occurs within the Fraunhofer distance.

[00157] FIG. 10B illustrates a graph 1090 of power transmittance for a range of frequencies using the relatively low-density metamaterial surface antenna wireless power transmission system, according to one embodiment.

[00158] FIG. 1 1 A illustrates two relatively high-density metamaterial surface antennas 1 125 and 1 175 being used as transmitter 1 125 and receiver 1 175 in a wireless power transmission system. The wireless power transfer via the illustrated beam 1 100 occurs within the Fraunhofer distance. As shown in the graph 1 190 of FIG. 1 1 B, an improved power transmittance is possible for a range of frequencies using relatively high-density metamaterial surface antennas, according to various embodiment.

[00159] FIG. 12 illustrates a steerable metamaterial surface antenna wireless power transmission system, according to one embodiment. As illustrated, the beam 1200 can be transmitted at an angle relative to the normal of the transmitter 1225 and the receiver 1275. The ability to dynamically modify the angle at which the beam 1200 is transmitted to the receiver 1275, without requiring the angle of either the transmitter 1225 or the receiver 1275 to move, allows for efficient wireless power transfer to moving objects and devices.

[00160] FIG. 13A illustrates a graph of a metamaterial surface antenna wireless power transmission system with one metamaterial surface antenna acting as a transmitter 1325 and two metamaterial surface antennas acting as receivers 1375 and 1376, according to one embodiment. In some embodiments, the transmitter may transmit 50% of the power via split beams 1300 to each receiver 1375 and 1376. In alternative embodiments, the distribution of power may vary based on need, capabilities of both receivers, and/or various economic reasons.

[00161] FIG. 13B illustrates a frequency-normalized graph of power transmittance for the dual receiver wireless power transmission system of FIG. 13A, according to one embodiment. As illustrated, the power received by each receiver 1375 and 1376 adds up to nearly 100%.

[00162] FIG. 14 illustrates an examples of wireless power transfer system 1400 used to power an unmanned aerial vehicle 1490. The wireless power transfer system 1400 may include a metamaterial surface antenna transmitter 1425 that can transmit a relatively narrow beam to a metamaterial surface antenna receiver 1475. The beam may provide sufficient power to instantaneously power a motor of the unmanned aerial vehicle 1490. Alternatively, the beam may provide power that is used to charge batteries or a capacitor of the unmanned aerial vehicle 1490. The transmitter may automatically track the location of the unmanned aerial vehicle 1490 to ensure that the beam direction is correct. The location of the unmanned aerial vehicle 1490 may be provided to the wireless power transfer system 1400 through a third device configured to track location and/or based on real-time feedback from the unmanned aerial vehicle 1490. Zones of exclusion may be utilized by the wireless power transfer system to increase side-lobe suppression and/or otherwise not oversaturate certain zones in the environment. For example, it may be desirous to limit the power density that contacts the bird and/or tree 1430.

[00163] FIG. 15 is a flow chart of a wireless power transfer method 1500 utilizing at least one metamaterial surface antenna, according to various embodiments. The method illustrated may be computer-implemented via software, hardware, firmware, and/or a processor or microprocessor. In other embodiments, the method may be implemented using an application specific integrated circuit, a field-programmable gate array, other hardware circuitry, integrated circuits, software, firmware, and/or a combination thereof. As illustrated, an S-Matrix may be determined that includes field amplitudes for each of a plurality of lumped ports, N, associated with a wireless power transfer device, at 1505.

[00164] The N lumped ports may include a plurality of lumped antenna ports, N_a, wherein each lumped antenna port corresponds to an impedance value of a lumped impedance element in communication with at least one sub-wavelength antenna element of an antenna device, wherein the impedance value of each of the lumped impedance elements is variable based on one or more impedance control inputs, and at least one lumped external port, N_e, located physically external to the antenna device. If multiple receivers are used in the wireless power transfer system then multiple spatially separated lumped external ports, N_e, may be used. In various embodiments, the S-Matrix may be expressible in terms of an impedance matrix, Z-Matrix, with impedance values, z_n, of each of the plurality of lumped ports, N.

[00165] Once the S-Matrix has been determined, a target radiation pattern of the antenna device may be defined in terms of target field amplitudes in the S-Matrix for the at least one lumped external port, N_e, at 1510. For multiple receivers, the target radiation pattern may include one or more bifurcations of the transmitted beam (as illustrated in FIG. 13A. An optimized port impedance vector, {z_n}, of impedance values for each of the lumped antenna ports, N_a, may then be determined, at 1515, that results in an S-Matrix element for the at least one lumped external port, N_e, that approximates the target field amplitude(s) for an operating frequency or operating frequency range. [00166] FIG. 16 is a flow chart of a method 1600 of manufacturing a static wireless power transmitter, according to one embodiment. A first step for manufacturing a static wireless power transmitter may include a determ ination of a scattering matrix (S-Matrix) of field amplitudes for each of a plurality of lumped ports associated with a wireless power transmitter, at 1605. A target electromagnetic radiation pattern of the wireless power transmitter may be defined in terms of target field amplitudes in the S Matrix for the at least one lumped external port, at 1610.

[00167] An optimized port impedance vector of impedance values that results in an S-Matrix element that approximates the target field amplitude for an operating frequency may be determined, at 1615. Using the above information, any of a wide variety of manufacturing techniques may be utilized to form a plurality of sub- wavelength antenna elements configured to scatter received electromagnetic fields, at 1620.

[00168] Similarly, any of a wide variety of manufacturing techniques may be used to form a plurality of impedance elements in communication with the plurality of sub-wavelength antenna elements with impedance values corresponding to the optimized impedance vector, at 1625.

[00169] FIG. 17A illustrates a view of a conceptual model 1750 of a single sub- wavelength antenna element 1760 with an associated variable impedance element 1765, according to one simplified embodiment. FIG. 17A also shows an impedance control input 1770 that can be used to control or vary the impedance of the lumped impedance element 1765, according to various embodiments.

[00170] As described herein, the sub-wavelength antenna element 1760 may be arranged in an array and may be configured for submersion in a fluid, such as fresh water, salt water, brackish water, or a gaseous environment.

[00171] FIG. 17B illustrates a view of a conceptual model 1751 of a single sub- wavelength antenna element 1761 with an associated static or fixed impedance element 1766, according to one simplified embodiment. As illustrated, the Fixed, Zn, exemplifies a fixed impedance element 1766 that can be considered a lumped impedance value or part of a lumped impedance value with other impedance values associated with the system and/or the sub-wavelength antenna element 1761 . [00172] As described herein, the sub-wavelength antenna element 1761 may be arranged in an array and may be configured for submersion in a fluid, such as fresh water, salt water, brackish water, or a gaseous environment. In some embodiments, a combination of fixed 1750 and variable 1751 sub-wavelength antenna elements may be used in combination.

[00173] FIG. 18 is a flow chart of a method 1800 of manufacturing a variable or tunable wireless power transmitter, according to one embodiment. A wireless power transmitter might include an array or other formation of a plurality of sub- wavelength antenna elements configured to scatter received electromagnetic fields, at 1805. A plurality of variable impedance elements may be placed in communication with the plurality of sub-wavelength antenna elements, at 1810. A plurality of variable impedance control inputs may be connected, at 1815, to the plurality of variable impedance elements.

[00174] The variable impedance control inputs may be coupled to or otherwise controlled by a processor, at 1820. A scattering matrix (S-Matrix) of electromagnetic field amplitudes for each of a plurality of lumped ports may be determined, at 1825. A target electromagnetic radiation pattern of the wireless power transmitter may be defined in terms of target electromagnetic field amplitudes in the S-Matrix for the external port, at 1830.

[00175] An optimized port impedance vector that results in an S-Matrix element for the lumped external port that approximates the target electromagnetic field amplitude may be determined, at 1835. Finally, using the above information, a system could adjust at least one of the plurality of variable impedance control inputs to modify at least one of the impedance values of at least one of the plurality of variable impedance elements, at 1840.

[00176] Many existing computing devices and infrastructures may be used in combination with the presently described systems and methods. Some of the infrastructure that can be used with embodiments disclosed herein is already available, such as general-purpose computers, computer programming tools and techniques, digital storage media, and communication links. A computing device or controller may include a processor, such as a microprocessor, a microcontroller, logic circuitry, or the like. [00177] A processor may include a special purpose processing device, such as application-specific integrated circuits (ASIC), programmable array logic (PAL), programmable logic array (PLA), programmable logic device (PLD), field programmable gate array (FPGA), or other customizable and/or programmable device. The computing device may also include a machine-readable storage device, such as non-volatile memory, static RAM, dynamic RAM, ROM, CD-ROM, disk, tape, magnetic, optical, flash memory, or other machine-readable storage medium. Various aspects of certain embodiments may be implemented using hardware, software, firmware, or a combination thereof.

[00178] The components of the disclosed embodiments, as generally described and illustrated in the figures herein, could be arranged and designed in a wide variety of different configurations. Furthermore, the features, structures, and operations associated with one embodiment may be applicable to or combined with the features, structures, or operations described in conjunction with another embodiment. In many instances, well-known structures, materials, or operations are not shown or described in detail to avoid obscuring aspects of this disclosure.

[00179] The embodiments of the systems and methods provided within this disclosure are not intended to limit the scope of the disclosure, but are merely representative of possible embodiments. In addition, the steps of a method do not necessarily need to be executed in any specific order, or even sequentially, nor do the steps need to be executed only once. As described above, descriptions and variations described in terms of transmitters are equally applicable to receivers, and vice versa.

[00180] This disclosure has been made with reference to various exemplary embodiments, including the best mode. However, those skilled in the art will recognize that changes and modifications may be made to the exemplary embodiments without departing from the scope of the present disclosure. While the principles of this disclosure have been shown in various embodiments, many modifications of structure, arrangements, proportions, elements, materials, and components may be adapted for a specific environment and/or operating requirements without departing from the principles and scope of this disclosure. These and other changes or modifications are intended to be included within the scope of the present disclosure. [00181] This disclosure is to be regarded in an illustrative rather than a restrictive sense, and all such modifications are intended to be included within the scope thereof. Likewise, benefits, other advantages, and solutions to problems have been described above with regard to various embodiments. However, benefits, advantages, solutions to problems, and any element(s) that may cause any benefit, advantage, or solution to occur or become more pronounced are not to be construed as a critical, required, or essential feature or element. The scope of the present invention should, therefore, be determined by the claims that follow.

[00182] Various aspects of the subject matter described herein are set out in the following numbered clauses:

1 . An antenna system, comprising:

a plurality of sub-wavelength antenna elements;

a plurality of lumped impedance elements in communication with the plurality of sub-wavelength antenna elements;

a plurality of variable impedance control inputs configured to allow for the selection of an impedance value for each of the lumped impedance elements; a computer-readable medium providing instructions accessible to a processor to cause the processor to perform operations for radiation patterning, comprising:

determining a scattering matrix (S-Matrix) of electromagnetic field amplitudes for each of a plurality of lumped ports, wherein the lumped ports include:

a plurality of lumped antenna ports with impedance values corresponding to the impedance values of each of the plurality of lumped impedance elements; and

at least one lumped external port located physically external to the antenna system,

wherein the S-Matrix is expressible in terms of an impedance matrix, Z-Matrix, with impedance values of each of the plurality of lumped ports; identifying a target electromagnetic radiation pattern of the antenna system defined in terms of target electromagnetic field amplitudes in the S-Matrix for the at least one lumped external port;

determining an optimized port impedance vector {z_n} of impedance values for each of the lumped antenna ports that results in an S-Matrix element for the at least one lumped external port that approximates the target electromagnetic field amplitude for an operating frequency; and

adjusting at least one of the plurality of variable impedance control inputs to modify at least one of the impedance values of at least one of the plurality of variable lumped impedance elements based on the determined optimized {z_n} of the impedance values for the lumped antenna ports.

2. The system of clause 1 , wherein the plurality of sub-wavelength antenna elements is configured to scatter received electromagnetic fields

3. The system of clause 2, wherein the electromagnetic fields received by the plurality of sub-wavelength antenna elements are generated by a source external to the antenna system.

4. The system of clause 2, wherein the electromagnetic fields received by the plurality of sub-wavelength antenna elements are generated by at least one location on a transmission line.

5. The system of clause 4, wherein the plurality of sub-wavelength antenna elements receiving electromagnetic fields from the at least one location on a transmission line are evanescently coupled to that location.

6. The system of clause 2, wherein the electromagnetic fields received by the plurality of sub-wavelength antenna elements are generated by a slot in a transmission line.

7. The system of clause 2, wherein the electromagnetic fields received by the plurality of sub-wavelength antenna elements are generated by an inductive element or a capacitive element coupled to a transmission line.

8. The system of clause 2, wherein the electromagnetic fields received by the plurality of sub-wavelength antenna elements are generated by at least one transmission line.

9. The system of clause 2, wherein the electromagnetic fields received by the plurality of sub-wavelength antenna elements are generated by a transmission line selected from the group comprising a rectangular waveguide, a parallel-plate waveguide, a microstrip line, and a coaxial line.

10. The system of clause 2, wherein the electromagnetic fields received by the plurality of sub-wavelength antenna elements are generated by at least one of a short dipole antenna, a resonant dipole antenna, a half-wavelength dipole antenna, a loop antenna, a resonant loop antenna, a spiral antenna, a patch antenna, a horn antenna, and a dish antenna.

1 1 . The system of clause 10, wherein at least one element of the plurality of sub-wavelength elements is within the reactive near-field of the antenna generating electromagnetic fields.

12. The system of clause 1 1 , wherein the reactive near-field of the antenna generating electromagnetic fields being received by the plurality of sub- wavelength antenna elements is predominantly electric or predominantly magnetic.

13. The system of clause 1 , wherein each of the sub-wavelength antenna elements comprises an antenna element with a maximum dimension that is less than half of a wavelength of the smallest frequency in an operating frequency range.

14. The system of clause 1 , wherein a separation between each of the sub-wavelength antenna elements does not exceed half of a wavelength of a frequency within an operating frequency range.

15. The system of clause 1 , wherein all of the sub-wavelength antenna elements are within the near-field of each other.

16. The system of clause 1 , wherein all of the sub-wavelength antenna elements are within the reactive near-field of each other.

17. The system of clause 1 , wherein the plurality of sub-wavelength antenna elements is divided into at least two groups,

wherein sub-wavelength antenna elements within each group are separated from one another by no more than half of an operating wavelength, and wherein each group of sub-wavelength antenna elements is spatially separated from each other group of sub-wavelength antenna elements by at least a distance exceeding that of half of an operating wavelength. 18. The system of clause 17, wherein the separation of each group from each other group is greater than a Fraunhofer (far-field) distance associated with an aperture diameter of a largest of the at least two groups.

19. The system of clause 17, wherein the separation of each group from each other group is less than a Fraunhofer (far-field) distance associated with an aperture diameter of a largest of the at least two groups.

20. The system of clause 17, wherein the separation of each group from each other group is shorter than a diameter of a largest of the at least two groups.

21 . The system of clause 17, wherein the separation of each group from each other group is shorter than a free-space operational wavelength.

22. The system of clause 17, wherein the separation of each group from each other group is shorter than a free-space operational wavelength, and the elements within each group are within the reactive near-field of each other.

23. The system of clause 17, wherein at least one of the groups of sub-wavelength antenna elements is part of a receiver antenna element physically coupled to a receiver device.

24. The system of clause 17, wherein the at least two groups of sub- wavelength antenna elements comprise a first group of sub-wavelength antenna elements that is not co-planar with a second group of sub-wavelength antenna elements.

25. The system of clause 24, wherein the target electromagnetic radiation pattern is selected to optimize a power transfer from the first group of sub-wavelength antenna elements to the second group of sub-wavelength antenna elements.

26. The system of clause 24, wherein the target electromagnetic radiation pattern is selected to optimize a power transfer from the first group of sub-wavelength antenna elements to an external lumped port that is collocated with the second group of sub-wavelength antenna elements.

27. The system of clause 25, wherein the first group of sub-wavelength antenna elements is facing the second group of sub-wavelength antenna elements.

28. The system of clause 25, wherein the first group of sub-wavelength antenna elements is mobile with respect to the second group of sub-wavelength antenna elements, or the second group second group of sub-wavelength antenna elements is mobile with respect to the first group of sub-wavelength antenna elements.

29. The system of clause 25, wherein optimizing the power transfer comprises maximizing the power transfer efficiency.

30. The system of clause 25, wherein optimizing the power transfer comprises maintaining the power transferred to a selected external lumped port below a threshold level.

31 . The system of clause 25, wherein optimizing the power transfer comprises minimizing the power transferred to one lumped external port while maintaining the power transferred to another lumped external port at or above a threshold level.

32. The system of clause 25, wherein optimizing the power transfer comprises maximizing the power transferred to one lumped external port while maintaining the power transferred to another lumped external port at or below a threshold level.

33. The system of clause 32, wherein the second lumped external port is a virtual port corresponding to a location of a human or a device.

34. The system of clause 25, wherein optimizing the power transfer comprises maintaining the power transfer below a threshold level or above a threshold level.

35. The system of clause 25, wherein optimizing the power transfer comprises maintaining a heat generation associated with the power transfer above a threshold level or below a threshold level.

36. The system of clause 1 , wherein the antenna system is a wireless power transmitter and the at least one lumped external port corresponds to a wireless power receiver.

37. The system of clause 36, wherein the at least one lumped external port comprises a receiving antenna associated with the wireless power receiver.

38. The system of clause 36, wherein the wireless power receiver comprises a battery charging station.

39. The system of clause 36, wherein the at least one lumped external port comprises a cell within a battery. 40. The system of clause 36, wherein the at least one lumped external port is connected to an input port of a rectifying circuit.

41 . The system of clause 36, wherein the at least one lumped external port comprises a wireless power receiver associated with a personal electronic device, a cell phone, a laptop, or a tablet.

42. The system of clause 36, wherein the at least one lumped external port is connected to an input port of a transformer circuit, a frequency converter circuit, or a voltage multiplier circuit.

43. The system of clause 36, wherein the at least one lumped external port comprises a wireless power receiver associated with a vehicle.

44. The system of clause 43, wherein the vehicle comprises a remotely operated vehicle.

45. The system of clause 36, wherein the at least one lumped external port comprises a wireless power receiver associated with a medical implant, a medical device residing temporarily inside a patient, or a medical device passing through a lumen of a patient.

46. The system of clause 1 , wherein the operating frequency is: in a range of 30-300 kHz; in a range of 0.3-3 MHz; in a range of 3-30 MHz; within the ISM band centered at 13.56 MHz; 6.78 MHz; or within the ISM band centered at 40.68 MHz.

47. The system of clause 1 , wherein at least some of the sub- wavelength antenna elements comprise resonating elements.

48. The system of clause 47, wherein the resonating elements have at least one electric dipole resonance or at least one magnetic dipole resonance at a frequency within or proximate the operational frequency band.

49. The system of clause 47, wherein resonance frequencies of the resonating elements are controlled by the impedance values of the lumped impedance elements.

50. The system of clause 47, the resonating elements have resonances with quality factors exceeding 10, 100, or 1000.

51 . The system of clause 47, wherein the resonating elements include superconductors. 52. The system of clause 1 , wherein at least two of the sub-wavelength antenna elements comprise a metamaterial.

53. The system of clause 1 , further comprising a common transmission line (TL) coupled to the lumped impedance elements.

54. The system of clause 53, wherein the common TL is treated as another of the plurality of lumped ports in addition to the plurality of lumped antenna ports and the at least one lumped external port.

55. The system of clause 1 , wherein the at least one lumped external port comprises an external port with an infinitesimal volume located at a particular radius-vector relative to the antenna system.

56. The system of clause 1 , wherein the at least one lumped external port is in the far-field of the antenna system.

57. The system of clause 1 , wherein the at least one lumped external port is in the near-field of the antenna system.

58. The system of clause 57, wherein the near-field is a radiative near- field.

59. The system of clause 57, wherein the near-field is a reactive near- field.

60. The system of clause 1 , wherein the at least one lumped external port comprises a virtual external port.

61 . The system of clause 60, wherein the at least one lumped external port comprises a field probe.

62. The system of clause 61 , wherein the field probe corresponds to a field amplitude measured by a non-perturbing measurement.

63. The system of clause 60, wherein the at least one lumped external port comprises a numerical field probe.

64. The system of clause 63, wherein the numerical field probe corresponds to an electromagnetic field amplitude calculated using a numerical simulation.

65. The system of clause 60, wherein the virtual external port comprises a region of space assumed to be a void.

66. The system of clause 60, wherein the virtual external port comprises a region of space assumed to be filled with a dielectric material. 67. The system of clause 60, wherein the virtual external port comprises an electrically conductive portion of an object located physically external to the antenna system.

68. The system of clause 1 , wherein each lumped impedance element is associated with a unique impedance control input, such that the impedance value of each lumped impedance element is independently variable.

69. The system of clause 1 , wherein the impedance control input associated with at least one of the lumped impedance elements comprises a direct current (DC) voltage input, wherein the impedance value of the at least one lumped impedance element is based on the magnitude of the voltage supplied via the DC voltage input.

70. The system of clause 1 , wherein the impedance control input associated with at least one of the lumped impedance elements can be varied to adjust the impedance value of the at least one lumped impedance element, wherein the impedance control input comprises one of: an electrical current input, a radiofrequency electromagnetic wave input, an optical radiation input, a thermal radiation input, a terahertz radiation input, an acoustic wave input, a phonon wave input, a thermal conduction input, a mechanical pressure input and a mechanical contact input.

71 . The system of clause 1 , wherein each of the lumped impedance elements is modeled as an encapsulated two-port structure connected to the terminals of a lumped port.

72. The system of clause 1 , wherein the impedance value of at least one of the lumped impedance elements is variable based on one or more

electromagnetic impedance control inputs, one or more electrical impedance control inputs, or one or more mechanical impedance control inputs .

73. The system of clause 1 , wherein at least one of the lumped impedance elements comprises one or more of a resistor, a capacitor, an inductor, a varactor, a diode, a transistor, a liquid-crystal-filled tunable capacitor, a liquid-crystal-filled tunable resonator, a split ring resonator, an ELC resonator, a spiral resonator, a complementary split ring resonator, a complementary ELC resonator, a MEMS capacitor, a BST capacitor, a tunable ferroelectric capacitor, a tunable MEMS inductor, a diode, a PIN diode, an adjustable resistor, a HEMT transistor, a volatile memory element, a volatile memory element that includes a transistor and a capacitor, a volatile memory element that includes a flip-flop circuit, a non-volatile memory element, non-volatile memory element that includes a floating-gate transistor, a non-volatile ferroelectric RAM element, a non-volatile magnetoresistive RAM element.

74. The system of clause 73, wherein at least some of the lumped impedance elements are addressable using a row and column addressing scheme.

75. The system of clause 1 , wherein at least one of the lumped impedance elements is connected to and collocated with a tunable microelectronic circuit.

76. The system of clause 1 , wherein at least some of the lumped impedance elements are connected to and collocated with tunable microelectronic circuits addressable using a row and column addressing scheme.

77. The system of clause 1 , wherein at least one of the lumped impedance elements is connected to and collocated with a tunable microelectronic circuit located at the intersection of a word line and a bit line, and connected to both the word line and the bit line.

78. The system of clause 1 , wherein the lumped impedance elements form a two-dimensional array.

79. The system of clause 1 , wherein the lumped impedance elements form one of a rectangular two-dimensional array, a square two-dimensional array, a triangular two-dimensional array, a hexagonal two-dimensional array, a concentric-circular array with elements aligned along concentric circles, or a concentric-circular array with elements aligned along radial lines from a common center.

80. The system of clause 1 , wherein each of the sub-wavelength antenna elements has a spacing from its nearest neighbor that is substantially less than a free-space wavelength corresponding to the operating frequency.

81 . The system of clause 1 , wherein at least one of the plurality of sub- wavelength elements are submerged in a fluid.

82. The system of clause 81 , wherein the fluid is water, fresh water, salt water, or brackish water. 83. The system of clause 81 , wherein each of the sub-wavelength antenna elements has a spacing from its nearest neighbor that is substantially less than one wavelength in the fluid at the operating frequency.

84. The system of clause 80, wherein the operating frequency is: a microwave frequency; between 300 MHz and 300 GHz; a decimeter wave frequency; within the 2.45 GHz ISM band; a centimeter wave frequency; within the 5.8 GHz ISM band; within the 24.125 GHz ISM band; within the 61 .25 GHz ISM band; within the 94-96 GHz atmospheric transparency window; or a millimeter wave frequency.

85. The system of clause 80, wherein all of the inter-element spacings are less than one-half or one-quarter of the free-space wavelength.

86. The transmitter of clause 1 , wherein the target field amplitude in the S-Matrix for the at least one lumped external port comprises a null or a maximum in the field amplitudes of the target electromagnetic radiation pattern.

87. The system of clause 1 , wherein the at least one lumped external port comprises a plurality of lumped external ports all located external to the antenna system, and wherein the target electromagnetic field amplitudes in the S- Matrix of each of the plurality of lumped external ports correspond to a target electromagnetic radiation pattern of the antenna system for at least the operating frequency.

88. The system of clause 87, wherein each of the sub-wavelength antenna elements comprises an antenna element with a maximum dimension that is less than half of a wavelength of the smallest frequency in an operating frequency range.

89. The system of clause 87, wherein at least some of the sub- wavelength antenna elements comprise resonating metamaterial elements.

90. The system of clause 89, wherein the resonating elements have at least one electric dipole resonance or magnetic dipole resonance at a frequency within or proximate the operational frequency band.

91 . The system of clause 89, wherein resonance frequencies of the resonating elements are controlled by the impedance values of the lumped impedance elements. 92. The system of clause 87, wherein each of the plurality of lumped antenna ports are coupled to a common waveguide.

93. The system of clause 87, wherein the at least one lumped external port comprises a virtual external port.

94. The system of clause 93, wherein the virtual external port comprises a region of space assumed to be a void.

95. The system of clause 93, wherein the virtual external port comprises a region of space assumed to be filled with a dielectric material.

96. The system of clause 93, wherein the virtual external port comprises an electrically conductive portion of an object located physically external to the antenna system.

97. The system of clause 87, wherein the at least one lumped external port comprises a receiving antenna associated with an external device.

98. The system of clause 1 , wherein at least one of the plurality of lumped antenna ports is strongly mutually coupled to at least one other lumped antenna port.

99. The system of clause 1 , wherein each of the plurality of lumped antenna ports is strongly mutually coupled to at least one other lumped antenna port.

100. The system of clause 99, wherein all of the at least one lumped external ports are only weakly or negligibly coupled to any of the plurality of lumped antenna ports.

101 . The system of clause 1 , wherein determining the optimized {z_n} comprises calculating the optimized {z_n} using a mathematical optimization technique.

102. The system of clause 1 , wherein determining the optimized {z_n} comprises identifying an optimized {z_n} using a global optimization method.

103. The system of clause 102, wherein the global optimization method comprises at least one of the following: a stochastic optimization method, a genetic algorithm, a Monte-Carlo method, a Simulated Annealing algorithm, a Particle Swarm algorithm, a Pattern Search method, a Multistart algorithm and a Global Search algorithm. 104. The system of clause 1 , wherein determining the optimized {z_n} comprises a collective local optimization based on one or more initial guesses.

105. The system of clause 101 , wherein a local optimization method comprises a gradient-assisted optimization method.

106. The system of clause 105, wherein a gradient is calculated analytically using an equation relating an S-parameter to the Z-Matrix and the impedance vector {z_n}.

107. The system of clause 106, wherein the gradient-assisted

optimization method is also assisted by a Hessian matrix calculation.

108. The system of clause 107, wherein the Hessian matrix is calculated analytically using the equation relating the S-parameter to the Z-Matrix and the impedance vector {z_n}.

109. The system of clause 102, wherein the global optimization method comprises exhaustively determining all local extrema by solving a multivariate algebraic equation and selecting a global extremum from the determined local extrema.

1 10. The system of clause 102, wherein the global optimization method comprises determining a plurality of local extrema and selecting a global extremum from the determined local extrema.

1 1 1 . The system of clause 1 10, wherein the multivariate algebraic equation is a multivariate polynomial equation.

1 12. The system of clause 1 , wherein determining the optimized {z_n} comprises calculating the optimized {z_n} based on an optimization of complex impedance variables.

1 13. The system of clause 1 , wherein determining the optimized {z_n} comprises calculating the optimized {z_n} based on an optimization of the square roots of complex impedance values.

1 14. The system of clause 1 , wherein determining the optimized {z_n} comprises calculating the optimized {z_n} vector based on an optimization of reactances associated with the impedance values.

1 15. The system of clause 1 14, wherein the optimization is constrained to allow: only positive or inductive values of reactances; or only negative or capacitive values of reactances. 1 16. The system of clause 1 , wherein determining the optimized {z_n} comprises calculating the optimized {z_n} based on an optimization of resistivities of the impedance values.

1 17. The system of clause 1 16, wherein the optimization is constrained to allow only positive or passive values of resistivities.

1 18. The system of clause 1 , wherein determining the optimized {z_n} of impedance values for each of the lumped antenna ports comprises calculating the optimized {z_n} based on an optimization of the impedance control inputs

associated with the lumped impedance elements of each of the sub-wavelength antenna elements.

1 19. The system of clause 1 , wherein determining the optimized {z_n} comprises calculating the optimized {z_n} by optimizing the values of impedance control inputs that are related to the impedance values for each of the lumped antenna ports through at least one nonlinear function.

120. The system of clause 1 19, wherein the at least one nonlinear function relating the impedance values for each of the lumped antenna ports and the associated impedance control inputs, is fitted with a polynomial function.

121 . The system of clause 1 , wherein a physical location of the at least one lumped external port is associated with a propagation channel that includes an electromagnetically reflective surface.

122. The system of clause 1 , wherein at least one of the lumped external ports is associated with a multipath propagation channel that includes a refractive object.

123. The system of clause 121 , wherein the at least one lumped external port associated with the multipath propagation pattern is in the near-field.

124. The system of clause 1 , wherein identifying the target

electromagnetic radiation pattern of the antenna system in terms of the target electromagnetic field amplitude in the S-Matrix for the at least one lumped external port comprises:

identifying the target electromagnetic radiation pattern of the antenna system in terms of a target field amplitude for a single linear field polarization. 125. The system of clause 1 , wherein the at least one lumped external port comprises a plurality of lumped external ports all located external to the antenna system, and

wherein identifying the target electromagnetic radiation pattern of the antenna system in terms of the target field amplitude in the S-Matrix for the plurality of lumped external ports comprises:

identifying the target electromagnetic radiation pattern of the antenna system in terms of a target field amplitude for at least two linear polarizations.

126. The system of clause 1 , wherein a physical location of the at least one lumped external port and the target field amplitude in the S-Matrix from the at least one lumped external port are selected to decrease far-field sidelobes of the antenna system.

127. The system of clause 1 , wherein a physical location of the at least one lumped external port and the target electromagnetic field amplitude in the S- Matrix from the at least one lumped external port are selected to decrease a power level of at least one sidelobe of the antenna system.

128. The system of clause 1 , wherein a physical location of the at least one lumped external port and the target electromagnetic field amplitude in the S- Matrix from the at least one lumped external port are selected to change a direction of a strongest sidelobe of the antenna system.

129. The system of clause 1 , wherein a physical location of the at least one lumped external port and the target electromagnetic field amplitude in the S- Matrix from the at least one lumped external port are selected to increase uniformity of a radiation profile of the antenna system in the near-field.

130. The system of clause 1 , wherein a physical location of the at least one lumped external port and the target electromagnetic field amplitude in the S- Matrix from the at least one lumped external port are selected to minimize a peak value of field amplitudes in the near-field.

131 . The system of clause 1 , wherein a minimax approximation algorithm is used to minimize a peak value of field amplitudes in the near-field.

132. The system of clause 1 , wherein determining the optimized {z_n} of impedance values for each of the lumped antenna ports comprises determining an optimized set of control values for the plurality of impedance control inputs that results in a field amplitude for the at least one lumped external port in the S-Matrix that approximates the target field amplitude for an operating frequency.

133. A method for antenna radiation patterning, comprising:

determining a scattering matrix (S-Matrix) of field amplitudes for each of a plurality of lumped ports associated with an antenna device, including

a plurality of lumped antenna ports wherein each lumped antenna port corresponds to an impedance value of a lumped impedance element in communication with at least one sub-wavelength antenna element of an antenna device, and

at least one lumped external port located physically external to the antenna device,

wherein the S-Matrix is expressible in terms of an impedance matrix, Z-Matrix, with impedance values of each of the plurality of lumped ports;

identifying a target electromagnetic radiation pattern of the antenna device defined in terms of target field amplitudes in the S-Matrix for the at least one lumped external port; and

determining an optimized port impedance vector, {zn}, of impedance values for each of the lumped antenna ports that results in an S-Matrix element for the at least one lumped external port that approximates the target field amplitude for an operating frequency.

134. The method of clause 133, wherein the plurality of sub-wavelength antenna elements is configured to scatter received electromagnetic fields

135. The method of clause 134, further comprising generating the electromagnetic fields received by the plurality of sub-wavelength antenna elements via a source external to the wireless power transmitter.

136. The method of clause 134, further comprising generating the electromagnetic fields received by the plurality of sub-wavelength antenna elements via at least one location on a transmission line.

137. The method of clause 136, wherein the plurality of sub-wavelength antenna elements receiving electromagnetic fields from the at least one location on a transmission line are evanescently coupled to that location. 138. The method of clause 134, further comprising generating the electromagnetic fields received by the plurality of sub-wavelength antenna elements via a slot in a transmission line.

139. The method of clause 134, further comprising generating the electromagnetic fields received by the plurality of sub-wavelength antenna elements via at least one transmission line.

140. The method of clause 134, further comprising generating the electromagnetic fields received by the plurality of sub-wavelength antenna elements via a transmission line selected from the group comprising a rectangular waveguide, a parallel-plate waveguide, a microstrip line, and a coaxial line.

141 . The method of clause 134, further comprising generating the electromagnetic fields received by the plurality of sub-wavelength antenna elements are generated by at least one of a short dipole antenna, a resonant dipole antenna, a loop antenna, a resonant loop antenna, a spiral antenna, a half- wavelength dipole antenna, a patch antenna, a horn antenna, and a dish antenna.

142. The method of clause 141 , wherein at least one element of the plurality of sub-wavelength elements is within the reactive near-field of the antenna generating electromagnetic fields.

143. The method of clause 142, wherein the reactive near-field of the antenna generating electromagnetic fields being received by the plurality of sub- wavelength antenna elements is predominantly electric or predominantly magnetic.

144. The method of clause 133, wherein the impedance value of each of the lumped impedance elements is variable based on one or more impedance control inputs.

145. The method of clause 133, wherein each lumped impedance element is associated with a unique dielectric loading, such that the impedance value of each lumped impedance element is independently selectable.

146. The method of clause 133, wherein the antenna device is a wireless power transmitter and the at least one lumped external port corresponds to a wireless power receiver.

147. The method of clause 146, wherein the wireless power receiver is part of a battery charging station. 148. The method of clause 146, wherein the at least one lumped external port comprises a cell within a battery.

149. The method of clause 146, wherein the at least one lumped external port is connected to an input port of a rectifying circuit.

150. The method of clause 146, wherein the at least one lumped external port comprises a wireless power receiver associated with a personal electronic device, a cell phone, a laptop, or a tablet.

151 . The method of clause 146, wherein the at least one lumped external port is connected to an input port of a transformer circuit, a frequency converter circuit, or a voltage multiplier circuit.

152. The method of clause 146, wherein the at least one lumped external port comprises a wireless power receiver associated with a vehicle.

153. The method of clause 152, wherein the vehicle comprises a remotely operated vehicle.

154. The method of clause 146, wherein the at least one lumped external port comprises a wireless power receiver associated with a medical implant, a medical device residing temporarily inside a patient, or a medical device passing through a lumen of a patient.

155. The method of clause 133, wherein the operating frequency is: in a range of 30-300 kHz; in a range of 0.3-3 MHz; in a range of 3-30 MHz; within the

ISM band centered at 13.56 MHz; 6.78 MHz; or within the ISM band centered at 40.68 MHz. 0.3-3 MHz.

156. The method of clause 133, wherein at least some of the sub- wavelength antenna elements comprise resonating elements.

157. The method of clause 156, wherein the resonating elements have at least one electric dipole resonance or at least one magnetic dipole resonance at a frequency within or proximate the operational frequency band.

158. The method of clause 156, wherein resonance frequencies of the resonating elements are controlled by the impedance values of the lumped impedance elements.

159. The method of clause 156, the resonating elements have

resonances with quality factors exceeding 10, 100, or 1000. 160. The method of clause 156, wherein the resonating elements include superconductors.

161 . The method of clause 145, wherein the dielectric material comprises at least one material printed using a 3D printer and the dielectric value is selected based on a filling fraction of the at least one 3D-printed material.

162. The method of clause 145, wherein the dielectric material comprises at least one material printed using a 3D printer and the dielectric value is selected based on a dielectric constant of the at least one 3D-printed material.

163. The method of clause 145, wherein the dielectric material comprises a combination of at least two dielectric materials and the impedance value is based at least in part on the ratio of the two dielectric materials.

164. The method of clause 163, wherein the at least two dielectric materials are printed using a multi-material 3D printer and the dielectric value is selected based at least in part on a ratio of the at least two 3D-printed materials.

165. The method of clause 133, wherein each lumped impedance element is associated with a unique dielectric loading, such that the impedance value of each lumped impedance element is independently selectable.

166. The method of clause 133, wherein each of the sub-wavelength antenna elements comprises an antenna element with a maximum dimension that is less than half of a wavelength of the smallest frequency in an operating frequency range.

167. The method of clause 133, wherein a separation between each of the sub-wavelength antenna elements does not exceed half of a wavelength of a frequency within an operating frequency range.

168. The method of clause 133, wherein the plurality of sub-wavelength antenna elements is divided into at least two groups,

wherein sub-wavelength antenna elements within each group are separated from one another by no more than half of an operating wavelength, and wherein each group of sub-wavelength antenna elements is spatially separated from each other group of sub-wavelength antenna elements by at least a distance exceeding that of half of an operating wavelength. 169. The method of clause 168, wherein the separation of each group from each other group is greater than a Fraunhofer (far-field) distance associated with an aperture diameter of a largest of the at least two groups

170. The method of clause 168, wherein the separation of each group from each other group is less than a Fraunhofer (far-field) distance associated with an aperture diameter of a largest of the at least two groups

171 . The method of clause 168, wherein the separation of each group from each other group is shorter than a diameter of a largest of the at least two groups.

172. The method of clause 168, wherein the separation of each group from each other group is shorter than a free-space operational wavelength.

173. The method of clause 168, wherein at least one of the groups of sub-wavelength antenna elements is part of a receiver antenna element physically coupled to a receiver device.

174. The method of clause 168, wherein the at least two groups of sub- wavelength antenna elements comprise a first group of sub-wavelength antenna elements that is not co-planar with a second group of sub-wavelength antenna elements.

175. The method of clause 174, wherein the target electromagnetic radiation pattern is selected to optimize a power transfer from the first group of sub-wavelength antenna elements to the second group of sub-wavelength antenna elements.

176. The method of clause 175, wherein the target electromagnetic radiation pattern is selected to optimize a power transfer from the first group of sub-wavelength antenna elements to an external lumped port that is collocated with the second group of sub-wavelength antenna elements.

177. The method of clause 176, wherein the first group of sub-wavelength antenna elements is facing the second group of sub-wavelength antenna elements.

178. The method of clause 175, wherein the first group of sub-wavelength antenna elements is mobile with respect to the second group of sub-wavelength antenna elements, or the second group of sub-wavelength antenna elements is mobile with respect to the first group of sub-wavelength antenna elements. 179. The method of clause 175, wherein optimizing the power transfer comprises maximizing the power transfer efficiency.

180. The method of clause 175, wherein optimizing the power transfer comprises maintaining the power transferred to a selected external lumped port below a threshold level.

181 . The method of clause 175, wherein optimizing the power transfer comprises minimizing the power transferred to one lumped external port while maintaining the power transferred to another lumped external port at or above a threshold level.

182. The method of clause 175, wherein optimizing the power transfer comprises maximizing the power transferred to one lumped external port while maintaining the power transferred to another lumped external port at or below a threshold level.

183. The method of clause 182, wherein the second lumped external port is a virtual port corresponding to a location of a human or a device.

184. The method of clause 174, wherein optimizing the power transfer comprises maximizing the efficiency of the power transfer.

185. The method of clause 174, wherein optimizing the power transfer comprises maintaining the power transfer below a threshold level or above a threshold level.

186. The method of clause 174, wherein optimizing the power transfer comprises maintaining a heat generation associated with the power transfer above a threshold level or below a threshold level.

187. The method of clause 133, wherein at least some of the sub- wavelength antenna elements comprise resonating elements.

188. The method of clause 187, wherein the resonating elements have at least one electric dipole resonance or at least one magnetic dipole resonance at a frequency within or proximate the operational frequency band.

189. The method of clause 156, wherein resonance frequencies of the resonating elements are controlled by the impedance values of the lumped impedance elements.

190. The method of clause 133, wherein at least two of the sub- wavelength antenna elements comprise a metamaterial. 191 . The method of clause 133, wherein each of the plurality of lumped antenna ports are coupled to a common transmission line (TL).

192. The method of clause 191 , wherein the common TL is treated as another of the plurality of lumped ports in addition to the plurality of lumped antenna ports and the at least one lumped external port.

193. The method of clause 133, wherein the at least one lumped external port comprises an external port with an infinitesimal volume located at a particular radius-vector relative to the antenna device.

194. The method of clause 133, wherein the at least one lumped external port is in the far-field of the antenna device.

195. The method of clause 133, wherein the at least one lumped external port is in the near-field of the antenna device.

196. The method of clause 195, wherein the near-field is a radiative near- field.

197. The method of clause 195, wherein the near-field is a reactive near- field.

198. The method of clause 133, wherein the at least one lumped external port comprises a virtual external port.

199. The method of clause 198, wherein the at least one lumped external port comprises a field probe.

200. The method of clause 199, wherein the field probe corresponds to a field amplitude measured by a non-perturbing measurement.

201 . The method of clause 198, wherein the at least one lumped external port comprises a numerical field probe.

202. The method of clause 201 , wherein the numerical field probe corresponds to a field amplitude calculated using a numerical simulation.

203. The method of clause 198, wherein the virtual external port comprises a region of space assumed to be a void.

204. The method of clause 198, wherein the virtual external port comprises a region of space assumed to be filled with a dielectric material.

205. The method of clause 198, wherein the virtual external port comprises an electrically conductive portion of an object located physically external to the antenna device. 206. The method of clause 133, wherein the at least one lumped external port comprises a receiving antenna associated with an external device.

207. The method of clause 133, further comprising manufacturing the antenna device with a plurality of sub-wavelength antenna elements and lumped impedance elements corresponding to the determined optimized impedance matrix, wherein the impedance value of each of the manufactured lumped impedance elements is invariably assigned during a manufacturing process.

208. The method of clause 133, wherein each of the lumped impedance elements is tunable, such that an impedance value of each of the tunable, lumped impedance elements is variable based on a plurality of impedance control inputs, and wherein the method further comprises:

adjusting impedance values of at least some of the tunable, lumped impedance elements based on the determined optimized impedance matrix.

209. The method of clause 208, wherein each tunable, lumped

impedance element is associated with a unique impedance control input, such that the impedance value of each tunable, lumped impedance element is independently variable.

210. The method of clause 209, wherein the impedance control input associated with at least one of the tunable, lumped impedance elements comprises a direct current (DC) voltage input, wherein the impedance value of the at least one tunable, lumped impedance element is based on the magnitude of the voltage supplied via the DC voltage input.

21 1 . The method of clause 209, wherein the impedance control input associated with at least one of the lumped impedance elements can be varied to adjust the impedance value of the at least one lumped impedance element, wherein the impedance control input comprises one of: an electrical current input, a radiofrequency electromagnetic wave input, an optical radiation input, a thermal radiation input, a terahertz radiation input, an acoustic wave input, a phonon wave input, a thermal conduction input, a mechanical pressure input and a mechanical contact input.

212. The method of clause 133, wherein each of the lumped impedance elements is modeled as an encapsulated two-port structure connected to the terminals of a lumped port. 213. The method of clause 133, wherein the impedance value of at least one of the lumped impedance elements is variable based on one or more electromagnetic impedance control inputs, electrical impedance control inputs, or mechanical impedance control inputs.

214. The method of clause 133, wherein at least one of the lumped impedance elements comprises one or more of a resistor, a capacitor, an inductor, a varactor, a diode, a transistor, a liquid-crystal-filled tunable capacitor, a liquid-crystal-filled tunable resonator, a split ring resonator, an ELC resonator, a spiral resonator, a complementary split ring resonator, a complementary ELC resonator, a MEMS capacitor, a BST capacitor, a tunable ferroelectric capacitor, a tunable MEMS inductor, a diode, a PIN diode, an adjustable resistor, a HEMT transistor, a volatile memory element, a volatile memory element that includes a transistor and a capacitor, a volatile memory element that includes a flip-flop circuit, a non-volatile memory element, non-volatile memory element that includes a floating-gate transistor, a non-volatile ferroelectric RAM element, a non-volatile magnetoresistive RAM element.

215. The method of clause 133, wherein at least some of the lumped impedance elements comprise elements addressable using a row and column addressing scheme.

216. The method of clause 133, wherein at least one of the lumped impedance elements is connected to and collocated with a tunable microelectronic circuit.

217. The method of clause 133, wherein at least some of the lumped impedance elements are connected to and collocated with tunable microelectronic circuits addressable using a row and column addressing scheme.

218. The method of clause 133, wherein at least one of the lumped impedance elements is connected to and collocated with a tunable microelectronic circuit located at the intersection of a word line and a bit line, and connected to both the word line and the bit line.

219. The method of clause 133, wherein the lumped impedance elements form a two-dimensional array. 220. The method of clause 133, wherein the lumped impedance elements form one of a rectangular two-dimensional array, a square two-dimensional array, a triangular two-dimensional array, and a hexagonal two-dimensional array, a concentric-circular array with elements aligned along concentric circles, or a concentric-circular array with elements aligned along radial lines from a common center.

221 . The method of clause 133, wherein each of the sub-wavelength antenna elements has a spacing from its nearest neighbor that is substantially less than a free-space wavelength corresponding to the operating frequency.

222. The method of clause 133, wherein at least one of the plurality of sub-wavelength elements are submerged in a fluid.

223. The method of clause 222, wherein the fluid is water, fresh water, salt water, or brackish water.

224. The method of clause 222, wherein each of the sub-wavelength antenna elements has a spacing from its nearest neighbor that is substantially less than one wavelength in the fluid at the operating frequency.

225. The method of clause 133, wherein each of the sub-wavelength antenna elements has a spacing from its nearest neighbor that is substantially less than a free-space wavelength corresponding to the operating frequency.

226. The method of clause 225, wherein the operating frequency: is a microwave frequency; between 300 MHz and 300 GHz; a decimeter wave frequency; within the 2.45 GHz ISM band; a centimeter wave frequency; within the 5.8 GHz ISM band; within the 24.125 GHz ISM band; within the 61 .25 GHz ISM band; within the 94-96 GHz atmospheric transparency window; or a millimeter wave frequency.

227. The method of clause 225, wherein the inter-element spacings are less than one-half or one-quarter of the free-space wavelength.

228. The method of clause 133, wherein the target field magnitude in the S-Matrix for the at least one lumped external port comprises a null or a maximum in the field magnitudes of the target electromagnetic radiation pattern.

229. The method of clause 133, wherein the at least one lumped external port comprises a plurality of lumped external ports all located external to the antenna device, and wherein the target field amplitudes in the S-Matrix of each of the plurality of lumped external ports correspond to a target radiation pattern of the antenna device for at least the operating frequency.

230. The method of clause 229, wherein each of the sub-wavelength antenna elements comprises an antenna element with a maximum dimension that is less than half of a wavelength of the smallest frequency in an operating frequency range.

231 . The method of clause 229, wherein at least some of the sub- wavelength antenna elements comprise resonating metamaterial elements.

232. The method of clause 231 , wherein the resonating metamaterial elements have at least one electric dipole resonance at a frequency within or proximate the operational frequency band.

233. The method of clause 231 , wherein the resonating metamaterial elements have at least one magnetic dipole resonance at a frequency within or proximate the operational frequency band.

234. The method of clause 231 , wherein resonance frequencies of the resonating metamaterial elements are controlled by the impedance values of the lumped impedance elements.

235. The method of clause 229, wherein each of the plurality of lumped antenna ports are coupled to a common waveguide.

236. The method of clause 229, wherein the at least one lumped external port comprises a virtual external port.

237. The method of clause 236, wherein the virtual external port comprises a region of space assumed to be a void.

238. The method of clause 236, wherein the virtual external port comprises a region of space assumed to be filled with a dielectric material.

239. The method of clause 236, wherein the virtual external port comprises an electrically conductive portion of an object located physically external to the antenna device.

240. The method of clause 229, wherein the at least one lumped external port comprises a receiving antenna associated with an external device.

241 . The method of clause 133, wherein at least one of the plurality of lumped antenna ports is strongly mutually coupled to at least one other lumped antenna port. 242. The method of clause 133, wherein each of the plurality of lumped antenna ports is strongly mutually coupled to at least one other lumped antenna port.

243. The method of clause 133, wherein determining the optimized {z_n} comprises calculating the optimized {z_n} using a mathematical optimization technique.

244. The method of clause 133, wherein determining the optimized {z_n} comprises identifying an optimized {z_n} using a global optimization method.

245. The method of clause 244, wherein the global optimization method comprises at least one of the following: a stochastic optimization method, a genetic algorithm, a Monte-Carlo method, a Simulated Annealing algorithm, a Particle Swarm algorithm, a Pattern Search method, a Multistart algorithm, and a Global Search algorithm.

246. The method of clause 244, wherein the global optimization method is assisted by at least one local optimization method applied to at least one solution candidate or intermediate solution.

247. The method of clause 133, wherein determining the optimized Z-Matrix comprises a collective local optimization based on one or more initial guesses.

248. The method of clause 243, wherein a local optimization method comprises a gradient-assisted optimization method.

249. The method of clause 248, wherein a gradient is calculated analytically using an equation relating an S-parameter to the Z-Matrix and the impedance vector {z_n}.

250. The method of clause 249, wherein the gradient-assisted optimization method is also assisted by a Hessian matrix calculation.

251 . The method of clause 250, wherein the Hessian matrix is calculated analytically using the equation relating the S-parameter to the Z-Matrix and the impedance vector {z_n}.

252. The method of clause 244, wherein the global optimization method comprises exhaustively determining all local extrema. 253. The method of clause 252, wherein exhaustively determining all local extrema is performed by solving a multivariate polynomial equation and selecting a global extrema from the determined local extrema.

254. The method of clause 244, wherein the global optimization method comprises exhaustively determining all local extrema by solving a multivariate algebraic equation and selecting a global extremum from the determined local extrema.

255. The method of clause 244, wherein the global optimization method comprises determining a plurality of local extrema and selecting a global extremum from the determined local extrema.

256. The method of clause 255, wherein the multivariate algebraic equation is a multivariate polynomial equation.

257. The method of clause 133, wherein determining the optimized {z_n} comprises calculating the optimized {z_n} based on an optimization of complex impedance variables.

258. The method of clause 133, wherein determining the optimized {z_n} comprises calculating the optimized {z_n} based on an optimization of the square roots of complex impedance values.

259. The method of clause 133, wherein determining the optimized {z_n} comprises calculating the optimized {z_n} based on an optimization of reactances associated with the impedance values.

260. The method of clause 259, wherein the optimization is constrained to allow: only positive or inductive values of reactances; or only negative or capacitive values of reactances.

261 . The method of clause 133, wherein determining the optimized {z_n} comprises calculating the optimized {z_n} based on an optimization of resistivities of the impedance values.

262. The method of clause 261 , wherein the optimization is constrained to allow only positive or passive values of resistivities.

263. The method of clause 133, wherein determining the optimized {z_n} of impedance values for each of the lumped antenna ports comprises calculating the impedance control inputs that are related to the impedance values through a non-linear mapping. 264. The method of clause 133, wherein the S-Matrix comprises N values and the Z-Matrix comprises M values, where N and M are both integers and are different.

265. The method of clause 133, wherein the S-Matrix comprises N values and the Z-Matrix comprises M values, where N and M are both integers and equal to one another, such that there is a 1 : 1 mapping of S-Matrix values and Z-Matrix values.

266. The method of clause 133, 264, 265, or wherein a mapping from Z- Matrix values to S-Matrix values comprises a non-linear mapping.

267. The method of clause 266, wherein the mapping is expressible as one of a multivariate polynomial mapping and a single-variate polynomial, either of which has an order of P, where P is an integer between 1 and 5.

268. The method of clause 144, wherein determining the optimized {z_n} of impedance values for each of the lumped antenna ports comprises calculating the optimized {z_n} based on an optimization of the impedance control inputs

269. The method of clause 144, wherein determining the optimized {z_n} comprises calculating the optimized {z_n} by optimizing the values of impedance control inputs that are related to the impedance values for each of the lumped antenna ports through at least one nonlinear function.

270. The method of clause 269, wherein the at least one nonlinear function relating the impedance values for each of the lumped antenna ports and the associated impedance control inputs, is fitted with a polynomial function.

271 . The method of clause 133, wherein a physical location of the at least one lumped external port is associated with a propagation channel that includes an electromagnetically reflective surface.

272. The method of clause 133, wherein at least one of the lumped external ports is associated with a multipath propagation channel that includes a refractive object.

273. The method of clause 272, wherein the at least one lumped external port associated with the multipath propagation pattern is in the near-field. 274. The method of clause 272, wherein the at least one lumped external port associated with the multipath propagation pattern is in the reactive near-field, where the near-field comprises a radiative near-field.

275. The method of clause 133, wherein identifying the target

electromagnetic radiation pattern of the antenna device in terms of the target field amplitude in the S-Matrix for the at least one lumped external port comprises: identifying the target electromagnetic radiation pattern of the antenna device in terms of a target field amplitude for a single linear field polarization.

276. The method of clause 133, wherein the at least one lumped external port comprises a plurality of lumped external ports all located external to the antenna device, and

wherein identifying the target electromagnetic radiation pattern of the antenna device in terms of the target field amplitude in the S-Matrix for the plurality of lumped external ports comprises:

identifying the target electromagnetic radiation pattern of the antenna device in terms of a target field amplitude for at least two linear polarizations.

277. The method of clause 133, wherein a physical location of the at least one lumped external port and the target field amplitude in the S-Matrix from the at least one lumped external port are selected to decrease far-field sidelobes of the antenna device.

278. The method of clause 133, wherein a physical location of the at least one lumped external port and the target field amplitude in the S-Matrix from the at least one lumped external port are selected to decrease a power level of at least one sidelobe of the antenna device.

279. The method of clause 133, wherein a physical location of the at least one lumped external port and the target field amplitude in the S-Matrix from the at least one lumped external port are selected to change a direction of a strongest sidelobe of the antenna device.

280. The method of clause 133, wherein a physical location of the at least one lumped external port and the target field amplitude in the S-Matrix from the at least one lumped external port are selected to increase uniformity of a radiation profile of the antenna device in the near-field. 281 . The method of clause 133, wherein a physical location of the at least one lumped external port and the target field amplitude in the S-Matrix from the at least one lumped external port are selected to minimize a peak value of field amplitudes in the near-field.

282. The method of clause 133, wherein determining the optimized vector

{zn} of impedance values for each of the lumped antenna ports comprises determining an optimized set of control values for the plurality of impedance control inputs that results in a field amplitude for the at least one lumped external port in the S-Matrix that approximates the target field amplitude for an operating frequency.

283. A wireless power transfer system, comprising:

a plurality of sub-wavelength antenna elements configured to scatter received electromagnetic fields, wherein the plurality of sub-wavelength antenna elements is divided into two non-overlapping subsets, including a transmitting subset and a receiving subset;

a plurality of variable impedance control inputs configured to allow for the selection of an impedance value for at least some of the lumped impedance elements;

at least one computer-readable medium providing instructions accessible to at least one processor to cause the at least one processor to perform operations for wireless power transfer, comprising:

determining a scattering matrix (S-Matrix) of electromagnetic field amplitudes for each of a plurality of lumped ports wherein the lumped ports include:

a plurality of lumped antenna ports with impedance values

corresponding to the impedance values of the plurality of lumped

impedance elements associated with the transmitting subset of sub- wavelength antenna elements; and

a plurality of lumped external ports with impedance values

corresponding to the impedance values of the plurality of lumped impedance elements associated with the receiving subset of sub- wavelength antenna elements,

wherein the S-Matrix is expressible in terms of an impedance matrix, Z-Matrix, with impedance values of each of the plurality of lumped ports; identifying a target electromagnetic radiation pattern for wireless power transfer defined in terms of target electromagnetic field amplitudes in the S-Matrix for at least one of the plurality of lumped external ports.

284. The method of clause 283, further comprising determining an optimized port impedance vector {zn} of impedance values for at least one of the lumped antenna ports or one of the lumped external ports that results in an S- Matrix element for at least one lumped external port that approximates the target electromagnetic field amplitude for an operating frequency.

285. The method of clause 283, further comprising adjusting at least one of the plurality of variable impedance control inputs to modify at least one of the impedance values of at least one of the plurality of variable lumped impedance elements based on the determined optimized {zn} of the impedance values.

286. A wireless power transmission system, comprising:

a plurality of sub-wavelength antenna elements configured to scatter received electromagnetic fields;

a plurality of lumped antenna ports with impedance values corresponding to the impedance values of each of the plurality of lumped impedance elements; at least one lumped external port corresponding to a first wireless power receiver located physically external to the wireless power transmitter, and

at least one lumped external port corresponding to a second wireless power receiver located physically external to the wireless power transmitter,

identifying a target electromagnetic radiation pattern of the wireless power transmitter defined in terms of target electromagnetic field

amplitudes in the S-Matrix for the at least one lumped external port in each of the first wireless power receiver and the second wireless power receiver; and

determining an optimized port impedance vector {z_n} of impedance values for each of the lumped antenna ports that results in an S-Matrix element for the at least one lumped external port that approximates the target electromagnetic field amplitude for an operating frequency.

287. The wireless power transmission system 286, wherein the

instructions accessible to a processor are further configured to cause the processor to adjust at least one of the plurality of variable impedance control inputs to modify at least one of the impedance values of at least one of the plurality of variable lumped impedance elements based on the determined optimized {z_n} of the impedance values for the lumped antenna ports.

288. A method of manufacturing an antenna system, comprising:

determining a scattering matrix (S-Matrix) of field amplitudes for each of a plurality of lumped ports associated with an antenna, including

a plurality of lumped antenna ports wherein each lumped antenna port corresponds to an impedance value of a lumped impedance element in communication with at least one sub-wavelength antenna element of the antenna device, and

at least one lumped external port located physically external to the antenna device, wherein the S-Matrix is expressible in terms of an impedance matrix, Z Matrix, with impedance values of each of the plurality of lumped ports; identifying a target electromagnetic radiation pattern of the antenna device defined in terms of target field amplitudes in the S Matrix for the at least one lumped external port;

determining an optimized port impedance vector, {z_n}, of impedance values for each of the lumped antenna ports that results in an S-Matrix element for the at least one lumped external port that approximates the target field amplitude for an operating frequency;

forming a plurality of sub-wavelength antenna elements; and

forming a plurality of impedance elements in communication with the plurality of sub-wavelength antenna elements with impedance values

corresponding to the optimized impedance vector {z_n}.

289. A method of manufacturing an antenna system, comprising:

arranging a plurality of sub-wavelength antenna elements;

communicatively coupling a plurality of variable impedance elements in communication with the plurality of sub-wavelength antenna elements;

connecting a plurality of variable impedance control inputs to the plurality of variable impedance elements, such that the plurality of variable impedance control inputs allows for a selection of an impedance value for each of the lumped impedance elements;

communicatively coupling the variable impedance control inputs to a processor, such that the processor can control the selection of an impedance value for each of the lumped impedance elements via the variable impedance control inputs; and

providing a computer readable medium with instructions accessible to the processor to cause the processor to perform operations for wireless transferring power, the operations comprising:

determining a scattering matrix (S-Matrix) of electromagnetic field amplitudes for each of a plurality of lumped ports, wherein the lumped ports include: a plurality of lumped antenna ports with impedance values corresponding to the impedance values of each of the plurality of variable impedance elements; and

identifying a target electromagnetic radiation pattern of the antenna device defined in terms of target electromagnetic field amplitudes in the S-Matrix for the at least one lumped external port;

determining an optimized port impedance vector {z_n} of impedance values for each of the lumped antenna ports that results in an S-Matrix element for the at least one lumped external port that approximates the target electromagnetic field amplitude for an operating frequency; and adjusting at least one of the plurality of variable impedance control inputs to modify at least one of the impedance values of at least one of the plurality of variable impedance elements based on the determined optimized {z_n} of the impedance values for the lumped antenna ports.

290. The method of clause 288 or 289, wherein the sub-wavelength antenna elements are configured to scatter received electromagnetic fields.

291 . The method of clause 290, wherein sub-wavelength antenna elements are configured to scatter electromagnetic fields generated by a source external to the antenna device.

292. The method of clause 290, wherein the sub-wavelength antenna elements are configured to receive electromagnetic fields from the at least one evanescently coupled location on a transmission line.

293. The method of clause 290, wherein the sub-wavelength antenna elements are configured to scatter electromagnetic fields received from a slot in a transmission line.

294. The method of clause 290, further comprising forming a

transmission line coupled to the plurality of sub-wavelength elements, wherein the sub-wavelength antenna elements are configured to transmit electromagnetic fields received from the transmission line.

295. The method of clause 294, wherein the formed transmission line coupled to the plurality of sub-wavelength elements comprises a rectangular waveguide, a parallel-plate waveguide, a microstrip line, or a coaxial line.

296. The method of clause 290, further comprising electromagnetically coupling an antenna to the formed plurality of sub-wavelength elements.

297. The method of clause 296 wherein the electromagnetically coupled antenna comprises a short dipole antenna, a resonant dipole antenna, half- wavelength dipole antenna, a loop antenna, a resonant loop antenna, a spiral antenna, a patch antenna, a horn antenna, or a dish antenna.

298. The method of clause 296 wherein the electromagnetically coupled antenna is coupled reactively to the at least one element of the plurality of sub- wavelength elements.

299. The method of clause 298 wherein the reactive coupling between the electromagnetically coupled antenna and the at least one element of the plurality of sub-wavelength elements is predominantly magneto-inductive.

300. The method of clause 288 or 289, wherein each impedance element is associated with a unique dielectric loading, such that the impedance value of each impedance element is independently selected to correspond to the optimized impedance vector {z_n}.

301 . The method of clause 300, wherein a dielectric material for each impedance element is printed using a 3D printer and the dielectric value is selected by choosing a filling fraction of the at least one 3D-printed material having a known dielectric constant.

302. The method of clause 300, wherein a dielectric material for each impedance element is printed using a 3D printer with a single dielectric constant.

303. The method of clause 300, wherein the dielectric material comprises a combination of at least two dielectric materials and the impedance value is based at least in part on the ratio of the two dielectric materials.

304. The method of clause 303, wherein the at least two dielectric materials are printed using a multi-material 3D printer and the dielectric value is selected based at least in part on a ratio of the at least two 3D-printed materials. 305. The method of clause 300, wherein each lumped impedance element is associated with a unique dielectric loading, such that the impedance value of each lumped impedance element is independently selectable.

306. The method of clause 300, wherein forming the plurality of sub- wavelength antenna elements comprising printing at least a portion of each of the sub-wavelength antenna elements using a 3D printer.

307. The method of clause 300, wherein forming each of the impedance elements comprising printing at least a portion of each of the impedance elements using a 3D printer.

308. The method of clause 300, wherein forming an array of sub- wavelength antenna elements comprising forming an array of sub-wavelength antenna elements.

309. The method of clause 290, wherein the electromagnetic fields received by the plurality of sub-wavelength antenna elements are generated by a source external to the antenna device.

310. The method of clause 290, wherein the electromagnetic fields received by the plurality of sub-wavelength antenna elements are generated by at least one location on a transmission line.

31 1 . The method of clause 310, wherein the plurality of sub-wavelength antenna elements receiving electromagnetic fields from the at least one location on a transmission line are evanescently coupled to that location.

312. The method of clause 290, wherein the electromagnetic fields received by the plurality of sub-wavelength antenna elements are generated by a slot in a transmission line.

313. The method of clause 290, wherein the electromagnetic fields received by the plurality of sub-wavelength antenna elements are generated by an inductive element or a capacitive element coupled to a transmission line.

314. The method of clause 290, wherein the electromagnetic fields received by the plurality of sub-wavelength antenna elements are generated by at least one transmission line.

315. The method of clause 290, wherein the electromagnetic fields received by the plurality of sub-wavelength antenna elements are generated by a transmission line selected from the group comprising a rectangular waveguide, a parallel-plate waveguide, a microstrip line, and a coaxial line.

316. The method of clause 290, wherein the electromagnetic fields received by the plurality of sub-wavelength antenna elements are generated by at least one of a short dipole antenna, a resonant dipole antenna, a half-wavelength dipole antenna, a loop antenna, a resonant loop antenna, a spiral antenna, a patch antenna, a horn antenna, and a dish antenna.

317. The method of clause 288 or 289, wherein each of the sub- wavelength antenna elements comprises an antenna element with a maximum dimension that is less than half of a wavelength of the smallest frequency in an operating frequency range.

318. The method of clause 288 or 289, wherein a separation between each of the sub-wavelength antenna elements does not exceed half of a wavelength of a frequency within an operating frequency range.

319. The method of clause 288 or 289, wherein all of the sub-wavelength antenna elements are within the near-field of each other.

320. The method of clause 288 or 289, wherein all of the sub-wavelength antenna elements are within the reactive near-field of each other.

321 . The method of clause 288 or 289, wherein the plurality of sub- wavelength antenna elements is divided into at least two groups,

wherein sub-wavelength antenna elements within each group are separated from one another by no more than half of an operating wavelength, and wherein each group of sub-wavelength antenna elements is spatially separated from each other group of sub-wavelength antenna elements by at least a distance exceeding that of half of an operating wavelength.

322. The method of clause 321 , wherein the separation of each group from each other group is greater than a Fraunhofer (far-field) distance associated with an aperture diameter of a largest of the at least two groups.

323. The method of clause 321 , wherein the separation of each group from each other group is less than a Fraunhofer (far-field) distance associated with an aperture diameter of a largest of the at least two groups. 324. The method of clause 321 , wherein the separation of each group from each other group is shorter than a diameter of a largest of the at least two groups.

325. The method of clause 321 , wherein the separation of each group from each other group is shorter than a free-space operational wavelength.

326. The method of clause 321 , wherein the separation of each group from each other group is shorter than a free-space operational wavelength, and the elements within each group are within the reactive near-field of each other.

327. The method of clause 321 , wherein at least one of the groups of sub-wavelength antenna elements is part of a receiver antenna element physically coupled to a receiver device.

328. The method of clause 321 , wherein the at least two groups of sub- wavelength antenna elements comprise a first group of sub-wavelength antenna elements that is not co-planar with a second group of sub-wavelength antenna elements.

329. The method of clause 328, further comprising optimizing a power transfer from the first group of sub-wavelength antenna elements to the second group of sub-wavelength antenna elements.

330. The method of clause 328, further comprising optimizing a power transfer from the first group of sub-wavelength antenna elements to an external lumped port that is collocated with the second group of sub-wavelength antenna elements.

331 . The method of clause 329, wherein the first group of sub-wavelength antenna elements is facing the second group of sub-wavelength antenna elements.

332. The method of clause 329, wherein the first group of sub-wavelength antenna elements is mobile with respect to the second group of sub-wavelength antenna elements, or the second group of sub-wavelength antenna elements is mobile with respect to the first group of sub-wavelength antenna elements.

333. The method of clause 329, wherein optimizing the power transfer comprises maximizing the power transfer efficiency. 334. The method of clause 329, wherein optimizing the power transfer comprises maintaining the power transferred to a selected external lumped port below a threshold level.

335. The method of clause 329, wherein optimizing the power transfer comprises minimizing the power transferred to one lumped external port while maintaining the power transferred to another lumped external port at or above a threshold level.

336. The method of clause 329, wherein optimizing the power transfer comprises maximizing the power transferred to one lumped external port while maintaining the power transferred to another lumped external port at or below a threshold level.

337. The method of clause 336, wherein the second lumped external port is a virtual port corresponding to a location of a human or a device.

338. The method of clause 329, wherein optimizing the power transfer comprises maintaining the power transfer below a threshold level or above a threshold level.

339. The method of clause 329, wherein optimizing the power transfer comprises maintaining a heat generation associated with the power transfer above a threshold level or below a threshold level.

340. The method of clause 288 or 289, wherein the antenna device is a wireless power transmitter and the at least one lumped external port corresponds to a wireless power receiver.

341 . The method of clause 340, wherein the wireless power receiver comprises a battery charging station.

342. The method of clause 340, wherein the at least one lumped external port comprises a cell within a battery.

343. The method of clause 340, wherein the at least one lumped external port is connected to an input port of a rectifying circuit.

344. The method of clause 340, wherein the at least one lumped external port comprises a wireless power receiver associated with a personal electronic device, a cell phone, a laptop, or a tablet. 345. The method of clause 340, wherein the at least one lumped external port is connected to an input port of a transformer circuit, a frequency converter circuit, or a voltage multiplier circuit.

346. The method of clause 340, wherein the at least one lumped external port comprises a wireless power receiver associated with a vehicle.

347. The method of clause 346, wherein the vehicle comprises a remotely operated vehicle.

348. The method of clause 340, wherein the at least one lumped external port comprises a wireless power receiver associated with a medical implant, a medical device residing temporarily inside a patient, or a medical device passing through a lumen of a patient.

349. The method of clause 288 or 289, wherein the operating frequency is: in a range of 30-300 kHz; in a range of 0.3-3 MHz; in a range of 3-30 MHz; within the ISM band centered at 13.56 MHz; 6.78 MHz; or within the ISM band centered at 40.68 MHz.

350. The method of clause 288 or 289, wherein at least some of the sub- wavelength antenna elements comprise resonating elements.

351 . The method of clause 350, wherein the resonating elements have at least one electric dipole resonance or at least one magnetic dipole resonance at a frequency within or close to the operational frequency band.

352. The method of clause 350, wherein the resonance frequencies of the resonating elements are controlled by the impedance values of the lumped impedance elements.

353. The method of clause 350, the resonating elements have

resonances with quality factors exceeding 10, 100, or 1000.

354. The method of clause 350, wherein the resonating elements having resonances include superconductors.

355. The method of clause 288 or 289, wherein at least two of the sub- wavelength antenna elements comprise a metamaterial.

356. The method of clause 288 or 289, further comprising coupling a common transmission line (TL) to the lumped impedance elements. 357. The method of clause 356, wherein the common TL is treated as another of the plurality of lumped ports in addition to the plurality of lumped antenna ports and the at least one lumped external port.

358. The method of clause 288 or 289, wherein the at least one lumped external port comprises an external port with an infinitesimal volume located at a particular radius-vector relative to the antenna device.

359. The method of clause 288 or 289, wherein the at least one lumped external port is in the far-field of the antenna device.

360. The method of clause 288 or 289, wherein the at least one lumped external port is in the near-field of the antenna device.

361 . The method of clause 360, wherein the near-field is a radiative near- field.

362. The method of clause 360, wherein the near-field is a reactive near- field.

363. The method of clause 288 or 289, wherein the at least one lumped external port comprises a virtual external port.

364. The method of clause 363, wherein the at least one lumped external port comprises a field probe.

365. The method of clause 364, wherein the field probe corresponds to a field amplitude measured by a non-perturbing measurement.

366. The method of clause 363, wherein the at least one lumped external port comprises a numerical field probe.

367. The method of clause 366, wherein the numerical field probe corresponds to an electromagnetic field amplitude calculated using a numerical simulation.

368. The method of clause 363, wherein the virtual external port comprises a region of space assumed to be a void.

369. The method of clause 363, wherein the virtual external port comprises a region of space assumed to be filled with a dielectric material.

370. The method of clause 363, wherein the virtual external port comprises an electrically conductive portion of an object located physically external to the wireless power transmitter. 371 . The method of clause 288 or 289, further comprising submerging the plurality of sub-wavelength elements in a fluid.

372. The method of clause 371 , wherein the fluid is water, fresh water, salt water, or brackish water

373. The method of clause 371 , wherein each of the sub-wavelength antenna elements has a spacing from its nearest neighbor that is substantially less than one wavelength in the fluid at the operating frequency.

374. The method of clause 288 or 289, wherein each of the sub- wavelength antenna elements has a spacing from its nearest neighbor that is substantially less than a free-space wavelength corresponding to the operating frequency.

375. The method of clause 374, wherein the operating frequency is: a microwave frequency; between 300 MHz and 300 GHz; a decimeter wave frequency; within the 2.45 GHz ISM band; a centimeter wave frequency; within the 5.8 GHz ISM band; within the 24.125 GHz ISM band; within the 61 .25 GHz ISM band; within the 94-96 GHz atmospheric transparency window; or a millimeter wave frequency.

376. The method of clause 374, wherein all of the inter-element spacings are less than one-half or one-quarter of the free-space wavelength.

377. The method of clause 288 or 289, wherein the target field amplitude in the S-Matrix for the at least one lumped external port comprises a null or a maximum in the field amplitudes of the target electromagnetic radiation pattern.

378. The method of clause 288 or 289, wherein the at least one lumped external port comprises a plurality of lumped external ports all located external to the antenna device, and wherein the target electromagnetic field amplitudes in the S-Matrix of each of the plurality of lumped external ports correspond to a target electromagnetic radiation pattern of the antenna device for at least the operating frequency.

379. The method of clause 378, wherein each of the sub-wavelength antenna elements comprises an antenna element with a maximum dimension that is less than half of a wavelength of the smallest frequency in an operating frequency range. 380. The method of clause 378, wherein at least some of the sub- wavelength antenna elements comprise resonating metamaterial elements.

381 . The method of clause 378, wherein each of the plurality of lumped antenna ports are coupled to a common waveguide.

382. The method of clause 378, wherein the at least one lumped external port comprises a virtual external port.

383. The method of clause 382, wherein the virtual external port comprises a region of space assumed to be a void.

384. The method of clause 382, wherein the virtual external port comprises a region of space assumed to be filled with a dielectric material.

385. The method of clause 93, wherein the virtual external port comprises an electrically conductive portion of an object located physically external to the antenna device.

386. The method of clause 382, wherein the antenna device is a wireless power transmitter and the at least one lumped external port comprises a receiving antenna associated with a wireless power receiver.

387. The method of clause 288 or 289, wherein at least one of the plurality of lumped antenna ports is strongly mutually coupled to at least one other lumped antenna port.

388. The method of clause 288 or 289, wherein each of the plurality of lumped antenna ports is strongly mutually coupled to at least one other lumped antenna port.

389. The method of clause 388, wherein all of the at least one lumped external ports are only weakly or negligibly coupled to any of the plurality of lumped antenna ports.

390. The method of clause 288 or 289, wherein determining the optimized {z_n} comprises calculating the optimized {z_n} using a mathematical optimization technique.

391 . The method of clause 288 or 289, wherein determining the optimized {z_n} comprises identifying an optimized {z_n} using a global optimization method.

392. The method of clause 391 , wherein the global optimization method comprises at least one of the following: a stochastic optimization method, a genetic algorithm, a Monte-Carlo method, a Simulated Annealing algorithm, a Particle Swarm algorithm, a Pattern Search method, a Multistart algorithm and a Global Search algorithm.

393. The method of clause 288 or 289, wherein determining the optimized {z_n} comprises a collective local optimization based on one or more initial guesses.

394. The method of clause 390, wherein a local optimization method comprises a gradient-assisted optimization method.

395. The method of clause 394, wherein a gradient is calculated analytically using an equation relating an S-parameter to the Z-Matrix and the impedance vector {z_n}.

396. The method of clause 395, wherein the gradient-assisted optimization method is also assisted by a Hessian matrix calculation.

397. The method of clause 396, wherein the Hessian matrix is calculated analytically using the equation relating the S-parameter to the Z-Matrix and the impedance vector {z_n}.

398. The method of clause 391 , wherein the global optimization method comprises exhaustively determining all local extrema by solving a multivariate algebraic equation and selecting a global extremum from the determined local extrema.

399. The method of clause 391 , wherein the global optimization method comprises determining a plurality of local extrema and selecting a global extremum from the determined local extrema.

400. The method of clause 398, wherein the multivariate algebraic equation is a multivariate polynomial equation.

401 . The method of clause 288 or 289, wherein determining the optimized {z_n} comprises calculating the optimized {z_n} based on an optimization of complex impedance variables.

402. The method of clause 288 or 289, wherein determining the optimized {z_n} comprises calculating the optimized {z_n} based on an optimization of the square roots of complex impedance values. 403. The method of clause 288 or 289, wherein determining the optimized {z_n} comprises calculating the optimized {z_n} vector based on an optimization of reactances associated with the impedance values.

404. The method of clause 403, wherein the optimization is constrained to allow: only positive or inductive values of reactances; or only negative or capacitive values of reactances.

405. The method of clause 288 or 289, wherein determining the optimized {z_n} comprises calculating the optimized {z_n} based on an optimization of resistivities of the impedance values.

406. The method of clause 405, wherein the optimization is constrained to allow only positive or passive values of resistivities.

407. The method of clause 288 or 289, wherein determining the optimized {z_n} of impedance values for each of the lumped antenna ports comprises calculating the optimized {z_n} based on an optimization of the impedance control inputs associated with the lumped impedance elements of each of the sub-wavelength antenna elements.

408. The method of clause 288 or 289, wherein determining the optimized {z_n} comprises calculating the optimized {z_n} by optimizing the values of impedance control inputs that are related to the impedance values for each of the lumped antenna ports through at least one nonlinear function.

409. The method of clause 408, wherein the at least one nonlinear function relating the impedance values for each of the lumped antenna ports and the associated impedance control inputs, is fitted with a polynomial function.

410. The method of clause 288 or 289, wherein a physical location of the at least one lumped external port is associated with a propagation channel that includes an electromagnetically reflective surface.

41 1 . The method of clause 288 or 289, wherein at least one of the lumped external ports is associated with a multipath propagation channel that includes a refractive object.

412. The method of clause 41 1 , wherein the at least one lumped external port associated with the multipath propagation pattern is in the near-field.

413. The method of clause 288 or 289, wherein identifying the target electromagnetic radiation pattern of the antenna device in terms of the target electromagnetic field amplitude in the S-Matrix for the at least one lumped external port comprises:

identifying the target electromagnetic radiation pattern of the antenna device in terms of a target field amplitude for a single linear field polarization.

414. The method of clause 288 or 289, wherein the at least one lumped external port comprises a plurality of lumped external ports all located external to the antenna device, and

415. The method of clause 288 or 289, wherein a physical location of the at least one lumped external port and the target field amplitude in the S-Matrix from the at least one lumped external port are selected to decrease far-field sidelobes of the antenna device.

416. The method of clause 288 or 289, wherein a physical location of the at least one lumped external port and the target electromagnetic field amplitude in the S-Matrix from the at least one lumped external port are selected to decrease a power level of at least one sidelobe of the antenna device.

417. The method of clause 288 or 289, wherein a physical location of the at least one lumped external port and the target electromagnetic field amplitude in the S-Matrix from the at least one lumped external port are selected to change a direction of a strongest sidelobe of the antenna device.

418. The method of clause 288 or 289, wherein a physical location of the at least one lumped external port and the target electromagnetic field amplitude in the S-Matrix from the at least one lumped external port are selected to increase uniformity of a radiation profile of the antenna device in the near-field.

419. The method of clause 288 or 289, wherein a physical location of the at least one lumped external port and the target electromagnetic field amplitude in the S-Matrix from the at least one lumped external port are selected to minimize a peak value of field amplitudes in the near-field. 420. The method of clause 288 or 289, wherein a minimax approximation algorithm is used to minimize a peak value of field amplitudes in the near-field.

421 . The method of clause 288 or 289, wherein determining the optimized {z_n} of impedance values for each of the lumped antenna ports comprises determining an optimized set of control values for the plurality of impedance control inputs that results in a field amplitude for the at least one lumped external port in the S-Matrix that approximates the target field amplitude for an operating frequency.

422. The method of clause 289, wherein each impedance element is associated with a unique impedance control input, such that the impedance value of each impedance element is independently variable.

423. The method of clause 289, wherein the impedance control input associated with at least one of the impedance elements comprises a direct current (DC) voltage input, wherein the impedance value of the at least one impedance element is based on the magnitude of the voltage supplied via the DC voltage input.

424. The method of clause 289, wherein at least one of the impedance control inputs comprises one of: an electrical current input, a radiofrequency electromagnetic wave input, an optical radiation input, a thermal radiation input, a terahertz radiation input, an acoustic wave input, a phonon wave input, a thermal conduction input, a mechanical pressure input and a mechanical contact input.

425. The method of clause 289, wherein each of the impedance elements is modeled as an encapsulated two-port structure connected to the terminals of a lumped port.

426. The method of clause 289, wherein the impedance value of at least one of the impedance elements is variable based on one or more electromagnetic impedance control inputs, one or more electrical impedance control inputs, or one or more mechanical impedance control inputs.

427. The method of clause 289, wherein at least one of the impedance elements comprises one or more of a resistor, a capacitor, an inductor, a varactor, a diode, a transistor, a liquid-crystal-filled tunable capacitor, a liquid-crystal-filled tunable resonator, a split ring resonator, an ELC resonator, a spiral resonator, a complementary split ring resonator, a complementary ELC resonator, a MEMS capacitor, a BST capacitor, a tunable ferroelectric capacitor, a tunable MEMS inductor, a diode, a PIN diode, an adjustable resistor, a HEMT transistor, a volatile memory element, a volatile memory element that includes a transistor and a capacitor, a volatile memory element that includes a flip-flop circuit, a non-volatile memory element, non-volatile memory element that includes a floating-gate transistor, a non-volatile ferroelectric RAM element, a non-volatile

magnetoresistive RAM element.

428. The method of clause 427, wherein at least some of the impedance elements are addressable using a row and column addressing scheme.

429. The method of clause 289, wherein at least one of the impedance elements is connected to and collocated with a tunable microelectronic circuit.

430. The method of clause 289, wherein at least some of the impedance elements are connected to and collocated with tunable microelectronic circuits addressable using a row and column addressing scheme.

431 . The method of clause 289, wherein at least one of the impedance elements is connected to and collocated with a tunable microelectronic circuit located at the intersection of a word line and a bit line, and connected to both the word line and the bit line.

432. The method of clause 289, wherein the impedance elements form a two-dimensional array.

433. The method of clause 289, wherein the impedance elements form one of a rectangular two-dimensional array, a square two-dimensional array, a triangular two-dimensional array, a hexagonal two-dimensional array, a

concentric-circular array with elements aligned along concentric circles, or a concentric-circular array with elements aligned along radial lines from a common center.

434. The method of clause 289, wherein each of the sub-wavelength antenna elements has a spacing from its nearest neighbor that is substantially less than a free-space wavelength corresponding to the operating frequency.

Claims

What is claimed is:

1 . An antenna system, comprising:

a plurality of sub-wavelength antenna elements;

identifying a target electromagnetic radiation pattern of the antenna system defined in terms of target electromagnetic field amplitudes in the S-Matrix for the at least one lumped external port;

2. The system of claim 1 , wherein the plurality of sub-wavelength antenna elements is configured to scatter received electromagnetic fields

3. The system of claim 2, wherein the electromagnetic fields received by the plurality of sub-wavelength antenna elements are generated by a source external to the antenna system.

4. The system of claim 1 , wherein each of the sub-wavelength antenna elements comprises an antenna element with a maximum dimension that is less than half of a wavelength of the smallest frequency in an operating frequency range.

5. The system of claim 1 , wherein the antenna system is a wireless power transmitter and the at least one lumped external port corresponds to a wireless power receiver.

6. The system of claim 1 , wherein the operating frequency is: in a range of 30-300 kHz; in a range of 0.3-3 MHz; in a range of 3-30 MHz; within the ISM band centered at 13.56 MHz; 6.78 MHz; or within the ISM band centered at 40.68 MHz.

7. The system of claim 1 , wherein at least some of the sub-wavelength antenna elements comprise resonating elements.

8. The system of claim 7, wherein resonance frequencies of the resonating elements are controlled by the impedance values of the lumped impedance elements.

9. The system of claim 1 , wherein each lumped impedance element is associated with a unique impedance control input, such that the impedance value of each lumped impedance element is independently variable.

10. The system of claim 1 , wherein the impedance control input associated with at least one of the lumped impedance elements comprises a direct current (DC) voltage input, wherein the impedance value of the at least one lumped impedance element is based on the magnitude of the voltage supplied via the DC voltage input.

1 1 . The system of claim 1 , wherein at least one of the lumped impedance elements comprises one or more of a resistor, a capacitor, an inductor, a varactor, a diode, a transistor, a liquid-crystal-filled tunable capacitor, a liquid-crystal-filled tunable resonator, a split ring resonator, an ELC resonator, a spiral resonator, a complementary split ring resonator, a complementary ELC resonator, a MEMS capacitor, a BST capacitor, a tunable ferroelectric capacitor, a tunable MEMS inductor, a diode, a PIN diode, an adjustable resistor, a HEMT transistor, a volatile memory element, a volatile memory element that includes a transistor and a capacitor, a volatile memory element that includes a flip-flop circuit, a non-volatile memory element, non-volatile memory element that includes a floating-gate transistor, a non-volatile ferroelectric RAM element, a non-volatile magnetoresistive RAM element.

12. The transmitter of claim 1 , wherein the target field amplitude in the

S-Matrix for the at least one lumped external port comprises a null or a maximum in the field amplitudes of the target electromagnetic radiation pattern.

13. The system of claim 1 , wherein determining the optimized {z_n} comprises calculating the optimized {z_n} using a mathematical optimization technique.

14. The system of claim 1 , wherein determining the optimized {z_n} comprises calculating the optimized {z_n} based on an optimization of complex impedance variables.

15. The system of claim 1 , wherein determining the optimized {z_n} of impedance values for each of the lumped antenna ports comprises calculating the optimized {z_n} based on an optimization of the impedance control inputs associated with the lumped impedance elements of each of the sub-wavelength antenna elements.

16. The system of claim 1 , wherein determining the optimized {z_n} of impedance values for each of the lumped antenna ports comprises determining an optimized set of control values for the plurality of impedance control inputs that results in a field amplitude for the at least one lumped external port in the S-Matrix that approximates the target field amplitude for an operating frequency.

17. A method for antenna radiation patterning, comprising:

wherein the S-Matrix is expressible in terms of an impedance matrix,

Z-Matrix, with impedance values of each of the plurality of lumped ports;

determining an optimized port impedance vector, {z_n}, of impedance values for each of the lumped antenna ports that results in an S-Matrix element for the at least one lumped external port that approximates the target field amplitude for an operating frequency.

18. The method of claim 17, wherein the plurality of sub-wavelength antenna elements is configured to scatter received electromagnetic fields

19. The method of claim 18, further comprising generating the electromagnetic fields received by the plurality of sub-wavelength antenna elements via a source external to the antenna device.

20. The method of claim 17, wherein the impedance value of each of the lumped impedance elements is variable based on one or more impedance control inputs.

21 . The method of claim 17, wherein the antenna device is a wireless power transmitter and the at least one lumped external port corresponds to a wireless power receiver.

22. The method of claim 17, wherein the operating frequency is: in a range of 30-300 kHz; in a range of 0.3-3 MHz; in a range of 3-30 MHz; within the

23. The method of claim 17, wherein each of the sub-wavelength antenna elements comprises an antenna element with a maximum dimension that is less than half of a wavelength of the smallest frequency in an operating frequency range.

24. The method of claim 17, wherein at least some of the sub- wavelength antenna elements comprise resonating elements.

25. The method of claim 24, wherein resonance frequencies of the resonating elements are controlled by the impedance values of the lumped impedance elements.

26. The method of claim 17, wherein each of the lumped impedance elements is tunable, such that an impedance value of each of the tunable, lumped impedance elements is variable based on a plurality of impedance control inputs, and wherein the method further comprises:

27. The method of claim 26, wherein each tunable, lumped impedance element is associated with a unique impedance control input, such that the impedance value of each tunable, lumped impedance element is independently variable.

28. The method of claim 27, wherein the impedance control input associated with at least one of the tunable, lumped impedance elements comprises a direct current (DC) voltage input, wherein the impedance value of the at least one tunable, lumped impedance element is based on the magnitude of the voltage supplied via the DC voltage input.

29. The method of claim 17, wherein at least one of the lumped impedance elements comprises one or more of a resistor, a capacitor, an inductor, a varactor, a diode, a transistor, a liquid-crystal-filled tunable capacitor, a liquid-crystal-filled tunable resonator, a split ring resonator, an ELC resonator, a spiral resonator, a complementary split ring resonator, a complementary ELC resonator, a MEMS capacitor, a BST capacitor, a tunable ferroelectric capacitor, a tunable MEMS inductor, a diode, a PIN diode, an adjustable resistor, a HEMT transistor, a volatile memory element, a volatile memory element that includes a transistor and a capacitor, a volatile memory element that includes a flip-flop circuit, a non-volatile memory element, non-volatile memory element that includes a floating-gate transistor, a non-volatile ferroelectric RAM element, a non-volatile magnetoresistive RAM element.

30. The method of claim 17, wherein the target field magnitude in the S-

Matrix for the at least one lumped external port comprises a null or a maximum in the field magnitudes of the target electromagnetic radiation pattern.

31 . The method of claim 17, wherein determining the optimized {z_n} comprises calculating the optimized {z_n} using a mathematical optimization technique.

32. The method of claim 17, wherein determining the optimized {z_n} comprises calculating the optimized {z_n} based on an optimization of complex impedance variables.

33. The method of claim 20, wherein determining the optimized {z_n} of impedance values for each of the lumped antenna ports comprises calculating the optimized {z_n} based on an optimization of the impedance control inputs associated with the lumped impedance elements of each of the sub-wavelength antenna elements.

34. The method of claim 17, wherein determining the optimized vector